mathematical problems in elasticity

MATHEMATICAL BLEMS

IN ELASTICITY

HOMOGENIZATlOt ' O.A. Oleinik A.S. Shamaev

/ G.A. Yosifian

MATHEMATICAL PROBLEMS IN ELASTICITY AND HOMOGENIZATION

STUDIES IN MATHEMATICS AND ITS APPLICATIONS

VOLUME 26

Editors: J.L. LIONS, Paris

G . PAPANICOLAOU, New York H. FUJITA, Tokyo

H.B. KELLER, Pasadena

NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO

MATHEMATICAL PROBLEMS IN ELASTICITY AND HOMOGENIZATION

O.A. OLEINIK Moscow University, Korpus 'K'

Moscow, Russia

and

A.S. SHAMAEV G.A. YOSIFIAN

Institute for Problems and Mechanics Moscow, Russia

NORTH-HOLLAND AMSTERDAM LONDON NEW YORK -TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. SARA BURGERHARTSTRAAT 25

P.O. BOX 21 1,1000 AE AMSTERDAM, THE NETHERLANDS

Library of Congress Cataloging-In-Publication Data

Oleinik. 0. A. Mathematical problens In elasticity and homogenlzatlon / O.A.

Oleinlk. A.S. Shamaev. G.A. Yoslflrn p . cn. -- (Studles in nathenatlcs and its applications ; v.

26 ) Includes blbllographical references. ISBN 0-444-88441-6 talk. paper) 1. Elasticlty. 2. Homogenlzatlon (Dlfferential equations)

I. Shamaev. A. S. 11. Yosiflan. G. A. 111. Title. IV. Series. PA93 1 .033 1992 6311.382--dc20 92- 15390

CIP

ISBN: 0 444 88441 6

0 1992 O.A. Oleinik, A.S. Shamaev and G.A. Yosifian. All rights reserved

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CONTENTS

PREFACE

CHAPTER I: SOME MATHEMATICAL PROBLEMS OF THE THEORY

OF ELASTICITY

$1. Some Functional Spaces and Their Properties.

Auxiliary Propositions

$2. Korn's Inequalities

2.1. The First Korn Inequality

2.2. The Second Korn lnequality in Lipschitz Domains

2.3. The Korn Inequalities for Periodic Functions

2.4. The Korn Inequality in Star-Shaped Domains

53. Boundary Value Problems o f Linear Elasticity

3.1. Some Properties of the Coefficients o f the

Elasticity System

3.2. The Main Boundary Value Problems for the System

of Elasticity

3.3. The First Boundary Value Problem

(The Dirichlet Problem)

3.4. The Second Boundary Value Problem

(The Neumann Problem)

3.5. The Mixed Boundary Value Problem

$4. Perforated Domains with a Periodic Structure.

Extension Theorems

4.1. Some Classes o f Perforated Domains

4.2. Extension Theorems for Vector Valued Functions

in Perforated Domains

vi Contents

4.3. The Korn Inequalities in Perforated Domains 5 1

55. Estimates for Solutions of Boundary Value Problems

of Elasticity in Perforated Domains 55

5.1. The Mixed Boundary Value Problem 55

5.2. Estimates for Solutions of the Neumann Problem

in a Perforated Domain 56

56. Periodic Solutions of Boundary Value Problems

for the System of Elasticity

6.1. Solutions Periodic in All Variables

6.2. Solutions of the Elasticity System Periodic in

Some of the Variables

6.3. Elasticity Problems with Periodic Boundary

Conditions in a Perforated Layer

57. Saint-Venant's Principle for Periodic Solutions

of the Elasticity System 67

7.1. Generalized Momenta and Their Properties 67

7.2. Saint-Venant's Principle for Homogeneous Boundary

Value Problems 7 1

7.3. Saint-Venant's Principle for Non-Homogeneous

Boundary Value Problems 73

58. Estimates and Existence Theorems for Solutions

of the Elasticity System in Unbounded Domains

8.1. Theorems of Phragmen-Lindelof's Type

8.2. Existence of Solutions in Unbounded Domains

8.3. Solutions Stabilizing to a Constant Vector at

Infinity

59. Strong G-Convergence of Elasticity Operators 98

9.1. Necessary and Sufficient Conditions for the Strong

G- Convergence 98

9.2. Estimates for the rate of Convergence of Solutions of

the Dirichlet Problem for Strongly G-Convergent Operators 111

Contents

CHAPTER II:

HOMOGENIZATION OF THE SYSTEM OF LINEAR ELASTICITY.

COMPOSITES AND PERFORATED MATERIALS 119

51. The Mixed Problem in a Perforated Domain with the

Dirichlet Boundary Conditions on the Outer Part of

the Boundary and the Neumann Conditions on the Surface

o f the Cavities 119

1.1. Setting of the Problem. Homogenized Equations 119

1.2. The Main Estimates and Their Applications 123

52. The Boundary Value Problem with Neumann Conditions

in a Perforated Domain 134

2.1. Homogenization o f the Neumann Problem in a Domain 52 for a Second Order Elliptic Equation with Rapidly

Oscillating Periodic Coefficients 134

2.2. Homogenization of the Neumann Problem

for the System o f Elasticity in a Perforated Domain.

Formulation of the Main Results 140

2.3. Some Auxiliary Propositions 142

2.4. Proof o f the Estimate for the Difference between

a Solution o f the Neumann Problem in a Perforated

Domain and a Solution o f the Homogenized Problem 149

2.5. Estimates for Energy Integrals and Stress Tensors 157

2.6. Some Generalizations 158

53. Asymptotic Expansions for Solutions o f Boundary

Value Problems o f Elasticity in a Perforated Layer 163

3.1. Setting of the Problem 163

3.2. Formal Construction o f the Asymptotic Expansion 164

3.3. Justification o f the Asymptotic Expansion.

Estimates for the Remainder 171

54. Asymptotic Expansions for Solutions of the Dirichlet

Problem for the Elasticity System in a Perforated Domain 178

4.1. Setting o f the Problem. Auxiliary Results 178

4.2. Justification o f the Asymptotic Expansion

55. Asymptotic Expansions for Solutions of the Dirichlet

Problem for the Biharrnonic Equation. Some Generalizations

for the Case o f Perforated Domains with a Non-Periodic

Structure

5.1. Setting o f the Problem. Auxiliary Propositions

5.2. Justification o f the Asymptotic Expansion for Solutions

o f the Dirichlet Problem for the Biharmonic Equation

5.3. Perforated Domains with a Non-Periodic Structure

56. Homogenization of the System of Elasticity with

Almost-Periodic Coefficients

6.1. Spaces of Almost-Periodic Functions

6.2. System of Elasticity with Almost-Periodic

CoefFicients. Almost-Solutions

6.3. Strong G-Convergence o f Elasticity Operators with

Rapidly Oscillating Almost-Periodic CoefFicients

57. Homogenization of Stratified Structures

7.1. Formulas for the Coefficients o f the Homogenized

Equations. Estimates of Solutions

7.2. Necessary and Sufficient Conditions for Strong

G-Convergence o f operetors Describing

Stratified Media

58. Estimates for the Rate of G-Convergence o f

Higher-Order Elliptic Operators

8.1. G-Convergence o f Higher-Order Elliptic Operators

(the n-dimensional case)

8.2. G-Convergence o f Ordinary Differential Operators

Contents

185

Contents

CHAPTER Ill:

SPECTRAL PROBLEMS

$1. Some Theorems from Functional Analysis.

Spectral Problems for Abstract Operators 263

1.1. Approximation of Eigenvalues and Eigenvectors of

Self-Adjoint Operators 263

1.2. Estimates for the Difference between Eigenvalues and

Eigenvectors o f Two Operators Defined in Different Spaces 266

$2. Homogenization of Eigenvalues and Eigenfunctions o f

Boundary Value Problems for Strongly Non-Homogeneous

Elastic Bodies 275

2.1. The Dirichlet Problem for a Strongly G-Convergent

Sequence o f Operators 275

2.2. The Neumann Problem for Elasticity Operators with

Rapidly Oscillating Periodic Coefficients in a

Perforated Domain 279

2.3. The Mixed Boundary Value Problem for the System o f

Elasticity in a Perforated Domain 286

2.4. Free Vibrations o f Strongly Non-Homogeneous

Stratified Bodies 290

$3. On the Behaviour o f Eigenvalues and Eigenfunctions

o f the Dirichlet Problem for Second Order Elliptic

Equations in Perforated Domains 294

3.1. Setting of the Problem. Formal Constructions 294

3.2. Weighted Sobolev Spaces. Weak Solutions o f a Second

Order Equation with a Non-Negative Characteristic Form 296

3.3. Homogenization o f a Second Order Elliptic Equation

Degenerate on the Boundary 308

3.4. Homogenization of Eigenvalues and Eigenfunctions

of the Dirichlet Problem in a Perforated Domain 313

$4. Third Boundary Value Problem for Second Order

Elliptic Equations in Domains with Rapidly Oscillating

Contents

Boundary

4.1. Estimates for Solutions

4.2. Estimates for Eigenvalues and Eigenfunctions

95. Free Vibrations of Bodies with Concentrated Masses

5.1. Setting of the Problem

5.2. The case -oo < m < 2, n > 3 5.3. The case m > 2, n 2 3 5.4. The case m = 2, n > 3

96. On the Behaviour of Eigenvalues o f the Dirichlet

Problem in Domains with Cavities Whose Concentration

is Small

97. Homogenization of Eigenvalues o f Ordinary Differential

Operators

98. Asymptotic Expansion o f Eigenvalues and Eigenfunctions

o f the Sturm-Liouville Problem for Equations with Rapidly

Oscillating Coefficients 356

§9. On the Behaviour of the Eigenvalues and Eigenfunctions

o f a G-Convergent Sequence o f Non-Self-Adjoint Operators 367

REFERENCES 383

PREFACE

Homogenization o f partial differential operators is a new branch of the the-

ory of differential equations and mathematical physics. It first appeared about

two decades ago. The theory of homogenization had been developed much

earlier for ordinary differential operators mainly in connection with problems

o f non-linear mechanics.

In the field o f partial differential equations the development of the homoge-

nization theory was greatly stimulated by various problems arising in mechanics,

physics, and modern technology, requiring asymptotic analysis based on the

homogenization o f differential operators. The main part o f this book deals with

homogenization problems in elasticity as well as some mathematical problems

related t o composite and perforated elastic materials.

The study of processes in strongly non-homogeneous media brings forth a

large number o f purely mathematical problems which are very important for

applications.

The theory o f homogenization o f differential operators and its applications

form the subject o f a vast literature. However, for the most part the material

presented in this book cannot be found in other monographs on homogeniza-

tion. The main purpose o f this book is t o study the homogenization problems

arising in linear elastostatics. For the convenience o f the reader we collect in

Chapter I most o f the necessary material concerning the mathematical theory

o f linear stationary elasticity and some well-known results o f functional anal-

ysis, in particular, existence and uniqueness theorems for the main boundary

value problems o f elasticity, Korn's inequalities and their generalizations, a

priori estimates for solutions, properties o f solutions in unbounded domains

and Saint-Venant's principle, boundary value problems in so-called perforated

domains. These results are widely used throughout the book and some o f them

are new.

xii Preface

In Chapter II we study the homogenization of boundary value problems

for the system of linear elasticity with rapidly oscillating periodic coefFicients

and in particular homogenization of boundary value problems in perforated

domains. We give formulas for the coefficients o f the homogenized system

and prove estimates for the difference between the displacement vector, stress

tensor and energy integral of a strongly non-homogeneous elastic body and

the corresponding characteristics o f the body described by the homogenized

system. For some elastic bodies with a periodic micro-structure characterized

by a small parameter e we obtain a complete asymptotic expansion in E for

the displacement vector.

A detailed consideration is given in Chapter II t o stratified structures which

may be non-periodic. Some general questions o f G-convergence o f elliptic

operators are also discussed.

The theory o f free vibrations o f strongly non-homogeneous elastic bodies is

the main subject o f Chapter Ill. These problems are not adequately represented

in the existing monographs.

In the first part of Chapter Ill we prove some general theorems on the

spectra o f a family o f abstract operators depending on a parameter and defined

in different spaces which also depend on that parameter. On the basis of these

theorems we study the asymptotic behaviour of eigenvalues and eigenfunctions

o f the boundary value problems considered in Chapter II and describing non-

homogeneous elastic bodies. This method is also applied t o some other similar

problems. We prove estimates for the difference between eigenvalues and

eigenfunctions o f the problem with a parameter and those o f the homogenized

problem.

Apart from the homogenization problems of Chapter II, the general method

suggested in §I, Chapter Ill, is also used for the investigation of eigenvalues and

eigenfunctions o f differential operators in domains with an oscillating boundary

and of elliptic operators degenerate on a part of the boundary o f a perforated

domain. This method is also applied in this book to study free vibrations of

systems with concentrated masses.

The theorems of 51, Chapter Ill, about spectral properties o f singularly

perturbed abstract operators depending on a parameter can be used for the in-

... Preface xl11

vestigation o f many other eigenvalue problems for self-adjoint operators. Some

abstract results for non-selfadjoint operators and their applications are given

in 58, Chapter Ill. Although the methods suggested in this book deal with stationary problems,

some of them can be extended to non-stationary equations.

With the exception o f some well-known facts from functional analysis and

the theory o f partial differential equations, all results in this book are given

detailed mathematical proof.

This monograph is based on the research of the authors over the last ten

years.

We hope that the results and methods presented in this book will promote

further investigation o f mathematical models for processes in composite and

perforated media, heat-transfer, energy transfer by radiation, processes of dif-

fusion and filtration in porous media, and that they will stimulate research in

other problems o f mathematical physics, and the theory o f partial differential

equations.

Each chapter is provided with its own double numeration o f formulas and

propositions, the first number denotes a section o f the given chapter. In

references t o other chapters we always indicate the number o f the chapter

where the formula or proposition referred to occurs. When enumerating the

propositions we do not distinguish between theorems, lemmas, etc.

The authors express their profound gratitude t o W. Jager, J.-L. Lions,

G. Papanicolaou, and I. Sneddon, for their remarks, advice and many useful

suggestions in relation t o this work.

This Page Intentionally Left Blank

CHAPTER l SOME MATHEMATICAL PROBLEMS OF THE THEORY OF

ELASTICITY

This chapter mostly contains the results concerning the system of linear

elasticity, which are widely used throughout the book. Here we introduce

functional spaces necessary t o define weak solutions o f the main boundary

value problems o f elasticity as well as solutions of some special boundary value

problems which are needed in Chapter II to obtain homogenized equations and

in Chapter Ill t o study the spectral properties of elasticity operators describing

processes in strongly non-homogeneous media.

Some results o f this chapter are very important for the mathematical theory

o f elasticity. Among these are Korn's inequalities in bounded and perforated

domains, strict mathematical proof o f the Saint-Venant Principle, asymptotic

behaviour a t infinity o f solutions of the elasticity problems, etc. On the basis of

the well-known Hilbert space methods we give here a thorough consideration

to the questions o f existence and uniqueness of solutions for boundary value

problems of elasticity in bounded and unbounded domains, and we obtain es-

timates for these solutions.

$1. Some Functional Spaces and Their Properties.

Auxiliary Propositions

In this section we define the principal functional spaces and formulate some

theorems from Functional Anlysis t o be used below. The proof of these the-

orems can be found in various monographs and manuals (see e.g. [40], [106],

[107], [1171, [1081).

Points o f the Euclidean space Rn are denoted by x = (xl, ..., x,), y =

(yl, ..., Y,), = (tl, ..., tn) etc.; A stands for the closure in IR" of the set A.

Let R be a domain o f Rn, i.e. R is a connected open set in Rn. If not

2 I. Some mathematical problems of the theory of elasticity

indicated otherwise we assume R t o be bounded.

For the main functional spaces we use the following notations:

C,"(R) is the space of infinitely differentiable functions with a compact

support belonging t o R; ck(f=l) consists o f functions defined in f=l and possessing all partial deriva-

tives up t o the order [k] which are continuous in 0 and satisfy the Holder

condition with exponent k - [k], provided that k - [k] > 0; [k] stands for the

maximum integer not larger than k. LP(R) ( 1 5 p 5 m) is the space o f measurable functions defined in R and

such that the corresponding norms

I l f l l~m(n) = ess SUP If I i f p = m n

are finite. For p = 2 we get the Hilbert space L2(R) with a scalar product

(u , v)o = 1 u(x)v(x)dx ; n

Hm(R) (for integer m > 0) is the completion o f C m ( n ) with respect t o the

norm

(1.1)

dlalu where Dau = , a is a multi-index, a = ( a l , ..., an), la1 = a l + ax:' ... ax;, ... + a,, aj are non-negative integers.

H,"(R) is the completion o f C,"(R) with respect t o the norm (1.1).

By dR we denote the boundary o f the domain R. Throughout the book we shall mostly deal with domains whose boundary

is sufficiently smooth, in particular with Lipschitz domains and domains with

the boundary of class C' which are defined as follows.

Denote by CR,L the cylinder

$1. Some functional spaces and their properties 3

where L, R are positive constants, $ = ( y l , ..., yn-1). We call R a Lipschitz domain if for any point x0 E d R one can introduce

orthogonal coordinates y = C ( x - xO), where C is a constant ( n x n ) matrix,

such that in coordinates y the intersection of d R with cR,L is given by the

equation yn = cp($), where p($) satisfies the Lipschitz condition in {$ : 161 < R ) with the Lipschitz constant not larger than L and

The numbers R a n d L are assumed t o be the same for any point xO E d R and

depend only on R. We say that the boundary d R of R belongs to the class CT if the functions

cp($) defined above belong to CT(I$I < R ) , 0 < r .

Let 7 be a subset o f dR. Suppose that R is a Lipschitz domain and 7 has

a positive Lebesgue measure on dR. For a set y of this type one can introduce

the following spaces o f functions vanishing on 7 , and spaces of trace functions:

H m ( R , y ) (for integer m > 0) is the completion with respect t o the norm

(1.1) of the subspace o f C m ( f i ) formed by all functions vanishing in a neigh-

bourhood of y ; obviously H m ( R , d R ) = H r ( R ) ;

~ " + + ( y ) is the factor space Hm+'(R)/Hm+'(R, y ) . We say that a function u E Hm+'(R) coincides on y with a function

cp E Hm+'(R) together with its derivatives up t o the order m, if u - cp E

Hm+'(fl, 4. As usual the norm in ~ " + + ( y ) is

= inf {llv + v I I ~ m + l ~ n ~ , v E ~ ' ~ ' ( 0 , y ) ) . V

Under the above assumptions on y the space ~ ~ + f r ( y ) is non-trivial, since

Hm+'(R) does not coincide with Hm+'(R, y ) . This fact is due t o

Lemma 1.1 (The Friedrichs Inequality).

Let R be a bounded Lipschitz domain and let y be a subset of its boundary

4 I. Some mathematical problems o f the theory of elasticity

80. Suppose that y has a positive Lebesgue measure on d o . Then for any

cp E H 1 ( R , y ) the inequality

d p acp holds wi th a constant C independent of cp; V p - (- , ..., -).

6x1 ax, If ./ = aR, then (1.2) holds for any bounded domain R and any cp E HA(R).

The proof of this lemma as well as some more general results o f this type

can be found in [117], [62].

Since constant functions belong t o H1(R) and inequality (1.2) obviously

does not hold for cp = const., we conclude that H1(R,7) # H1(R) . It follows

that we also have Hmtl(R,y) # Hm+l(a) . By H-'(R) is denoted the space dual t o H1(R , d R ) H,'(R). Some properties of functions defined in Lipschitz domains are given in the

next theorem. Results o f this kind in a much more general situation are proved

in 1481, [117], [67].

Theorem 1.2.

Let R be a bounded Lipschitz domain. Then

1. The imbedding o f H1(R) in L2(R) is compact.

2. If 0 C R0 and R0 is a domain of R n , then each v E H 1 ( R ) can be extended

t o R0 as a function 6 E H1(RO) such that

where C is a constant depending on R only.

3. Each function w E H1(R) possesses a trace on an (see [67], [117]) be-

longing t o L2(aR) and such that

where C1 is a constant depending on R only.

51. Some functional spaces and their properties 5

4. Functions w E H1(R) such that w dx = 0 satisfy the PoincarC inequality / n

with a constant C2 depending only on R.

5. H1(R) consists o f all functions which belong t o L2(R) together with their

first derivatives.

We assume that the domains considered henceforth at least have a Lipschitz

boundary unless pointed otherwise.

In order t o study homogenization problems for differential equations we

shall also need the following spaces o f periodic functions.

Let Zn be the set of all vectors z = ( z l , ..., 2,) with integer components.

By s,(G) we denote the shift o f the set G by the vector z , i.e. s,(G) = z + G. For the given G the set of all x such that E - ~ X E G is denoted by EG.

We say that an unbounded domain w has a 1-periodic structure, if w is

invariant with respect to all the shifts s,, z E Zn. Note that w is also assumed

t o be an open connected set of Rn. The spaces o f periodic functions are defined as follows:

&(G) is the space of infinitely differentiable functions in ij which are

1-periodic in x l , ..., x,;

w ~ ( w ) is the completion o f &(G) with respect t o the norm in H1(wnQ) ,

Q = { x : 0 < xj < 1, j = 1 , ..., n); e r ( w ) is the space o f infinitely differentiable functions in w that are 1-

periodic in xl , ..., x,, and vanish in a neighbourhood o f dw; 0

W (w) is the completion o f 6 r ( w ) with respect t o the norm in H1(w n Q ) .

A function cp(x) is said t o be 1-periodic in x and belonging t o H1(w n Q ) ,

if cp is an element o f W;(W) .

Let w be an unbounded domain with a 1-periodic structure. Set


Denote by H' (w(a, b)) the completion with respect t o the norm in H1 (&(a, b))

of the space o f infinitely differentiable functions in w(a, b) which are 1-periodic

In XI, ..., xn-l.

Elements o f H1(w(a, b)) can be referred t o as functions in H1 (;(a, b)).

1-periodic in xl, ..., x,-~.

Consider a set y on ~ w ( u , b) such that y is invariant under the shift by any

vector z = (2,O) E Zn.

W e w r i t e u = v o n y f o r u , v E H1(w(a,b)), i f u - V E ~ ' ( & ( a , b ) , ~ n

a&(a, b)) . Note that Hm(R), H,"(R) are Hilbert spaces with the scalar product

and W;(W), H1(w(a,b)) are also Hilbert spaces with the scalar product o f

H1(w n Q). HI (;(a, b)) respectively.

Many problems considered in this book involve vector-valued and matrix-

valued functions, whose components belong t o one o f the spaces defined above.

For such cases we shall adopt the following conventions.

For column vectors u = (ul, ..., un)*, v = (vl, ..., vn)* by (u, v) we denote

the sum uiv;, and as usual lul = (u,u)lI2. Here and in what follows summation

over repeated Latin indices from 1 t o n is assumed; the sign * denotes the

transpose of a matrix, however in the case of column vectors this sign is

sometimes omitted unless that leads t o a misunderstanding.

For matrices A and B with elements ai, and bij respectively we set

(A, B) = aijbij , IAI = (A, A ) ' / ~ . (1.8)

If vectors u, v or matrices A, B have elements belonging to a Hilbert space

'Id with a scalar product ( a , we shall often use the following notation:

and write u ,v E 7-t; A, B E 'Id instead of u,v E 7-tn; A, B E 'Idn2. The proof o f uniqueness and existence theorems for solutions o f various

boundary value problems considered below is based on the following well-

$1. Some functional spaces and their properties

known

Theorem 1.3 (Lax, Milgram).

Let H be a Hilbert space and let a(u, v ) be a bilinear form on H x H such

that

Then for any continuous linear functional 1 on H (i.e. 1 E H*) there is a unique

element u E H such that

E(v) = a ( u , v ) for any v E H

(see [134]).

The Sobolev imbedding theorem (see [117]) yields

Lemma 1.4. n n

Let R c Rn be a bounded Lipschitz domain and 1 - - + - 2 0 . Then for 2 s

any u E H 1 ( R ) the inequality

holds with a constant C independent o f u.

Denote by p ( x , A ) the distance in Rn of the point x E Rn from the set

A c lRn.

Lemma 1.5.

Let R be a bounded domain with a smooth boundary and Bs = { x E

R , p (x ,aR) < 61, 6 > 0. Then there exists b0 > 0 such that for every

6 E (0,6,,) and every v E H 1 ( R ) we have


where c is a constant independent of 6 and v.

Proof. Due to the smoothness o f dR there is a sufFiciently small 60 > 0 and

a family o f smooth surfaces S,, T E [O,dO], such that S, is the boundary of

a domain R, C 0 , R, 3 R,, if T' > T , R0 = 52, C,T 5 p(x,dR) 5 c27 if

x E ST , T E [ O , bO], CI, c2 = const, R\R, > B,. By virtue o f the imbedding theorem (see Theorem 1.2) we have

J Iv12dS < c3 Ilv/lZ,i(nT1 < cs llvllZ,l(n) 3 T E 10,601 7

S,

where cg is a constant independent o f T . Integrating this inequality with respect

to T from 0 t o 6, we get

2 I I v I I L ~ ( B ~ ) 5 ~ 4 6 II~llLl(n, . This inequality implies (1.12). Lemma 1.5 is proved.

Let Cl be a bounded domain with a Lipschitz boundary. Denote by 2(IRn x

R) the set o f all functions f ((, x ) which are bounded and measurable in (t, x ) E

Rn x R, 1-periodic in < and Lipschitz continuous with respect t o x uniformly

in ( E Rn i.e.

I f ( ( , 2) - f (t, xO)I 5 Cf 12 - xOI (1.13)

for any x,xO E 0, ( E I?, where c, is a constant independent o f x, xO, (.

Lemma 1.6.

Let g(C,x) E i (Rn x R) , / g(( ,z)d< = 0 for any z E a. Then the inequality

8

holds for every u,v E H1(R) , where c is a constant independent o f e E (O,l) ,

U , v. Moreover, if F ( ( , x ) E i (Rn x R) , then for any 1C, E L1(R) we have

§1. Some functional spaces and their properties 9

where P ( x ) = / F ( ( , x ) d ( , Q =]0, I[.= {C : 0 < < 1, j = 1, ..., n). Q

Proof. Denote by I' the set o f all z E Zn such that s ( z + Q ) C R. Set

$I1 = U e ( z + Q ) , G = R\nl. Let us consider the functions m ( x ) , C(x) , z E P

~ ( x ) which are constant on every ~ ( z + Q ) and are given by the formulas

~ ( x ) = E - ~ J u (x )dx for x E E ( Z + Q ) . ++Q)

Then we have

Let xO, x E ~ ( z + Q ) . Since g( ( , x ) satisfies the Lipschitz condition in x

and its mean value in ( vanishes for any fixed x , it follows that

Obviously, the estimate (1.17) holds for almost all x E R1.

The PoincarC inequality (1.5) in ~ ( z + Q ) yields

110 - C I I L Z ( ~ ~ ) I Cle IIVVIIL~(~~) ,

1 1 ~ - ~ 1 1 ~ 2 ( n ~ ) 5 Cie IIVUIIL~(~~, . By the definition o f ~ ( x ) we get

I. Some mathematical problems o f the theory o f elasticity

The set G belongs t o the Cza-neighbourhood of dR (C2 = const), and there-

fore according t o Lemma 1.5 we have

The last integral in (1.16) is equal t o zero. It follows from (1.16) by virtue

of (1.18), (1.19), (1.20) and the Holder inequality that

where C5 is a constant independent o f E . These inequalities imply (1.14).

Let us prove (1.15). For any $ E C1(fi) the convergence (1.15) is obviously

a direct consequence o f the inequality (1.14) for u = $, v = 1, g(E,x) =

F ( t , x ) - fi(x).

Approximating a given 4 E L1(R) by functions in C1(Q) and taking into

account the fact that F((,x) is bounded, we easily obtain (1.15) for any func-

tion $ E L1(R). Lemma 1.6 is proved.

Corollary 1.7.

Let w be an unbounded domain with a 1-periodic structure and let { $ c ) , (9,)

be two sequences o f functions in LZ(R n EW) such that

$1. Some functional spaces and their properties 11

l l ~ e - ~II~2(nncw) + 0

where $, cp E L 2 ( R ) . Then for any f ( < , x ) E L(R" x R ) we have

nncw n where

F ( x ) = m e 4 8 n w ) (f (., 1.)) E / f (C, x ) d t - (1.23)

Qnw Proof. I t is easy t o see that

The last two integrals tend to zero as E + 0 due t o (1.21). Setting F ( < , x ) =

f ( < , z ) x w ( ( ) in Lemma 1.6, where x u ( ( ) is the characteristic function of the

domain w , we get

This convergence and (1.24) imply (1.22) since

Lemma 1.8.

Let a ( ( ) be a bounded function which is piecewise smooth and 1-periodic in

<. Let / a(<)d( = 0. Then there exist bounded piecewise smooth functions

0 a a 4 0 a , (< ) , i = 1, ..., n, which are 1-periodic in < and such that a ( < ) = - at;

h f . Let us use the induction with respect to the number o f independent

variables. For n = 1 the assertion of Lemma 1.8 is evident since one can

take a l ( t l ) = a ( t ) d t . Assume that the lemma holds in the case of n - 1 I 0


independent variables. Let = (i , tn), ( E IRn-', and let a(<) satisfy the

conditions of the lemma. Set

1

The functions b j ( t ) , j = 1 , ..., n, are 1-periodic in ( and

a b j ( t ) a ( ( ) = - + ~ ( i ) . (1.25)

atj

Obviously, / c(()d( = 0 , where Q = {i : 0 < t j < 1 , j = I, ..., n - 1).

s Since c ( i ) depends on n - 1 variables it follows from the above assumption

n-1 ac. that c ( [ ) = -2.. Therefore, taking into account (1.25) we obtain the

j=1 a& needed representation for a ( ( ) . Lemma 1.8 is proved.

52. Korn 's inequalities

52. Korn's lnequalities

lnequalities of Korn's type are essential for establishing the solvability of

the main boundary value problems of elasticity as well as for getting estimates

of their solutions.

In this section we denote by u, v the vector valued functions u = (ul, ..., u,),

v = (vl, ..., v,), and Vu, e(u) stand for matrices whose elements are

aui ( V U ) ; ~ = -

a x j

respectively.

We obviously have

In the theory of elasticity u = (ul, ..., u,) is the displacement vector and

e(u) is the strain tensor.

2.1. The First Korn Inequality

Theorem 2.1.

Let R be a bounded domain o f P. Then every vector valued function u E

H,'(R) satisfies the inequality

Proof. Since C,"(R) is dense in H,'(R), it is sufficient t o prove (2.2) for

functions in C,"(R).

By virtue o f the Green formula we get

14 I. Some mathematical problems o f the theory o f elasticity

for any u E C r ( R ) . Therefore (2.2) is valid for u since the second integral in

the right hand side o f the last equality is non-negative. Theorem 2.1 is proved.

Note that inequality (2.2) of Theorem 2.1 holds for any bounded domain

R even if its boundary dR is non-regular.

2.2. The Second K o r n Inequality in Lipschitz Domains

The inequality

for any u = (u l , ..., un) E H1(R) is called the Second Korn Inequality.' In

contrast t o the First Korn Inequality the proof o f (2.3) is rather complicated

and requires some additional conditions on R. Inequality (2.3) as well as some

more general inequalities o f this type under various assumptions on the domain

R are proved in numerous papers (see e.g. the references in [42]).

Using the method suggested in [42] we give here a simple proof for the

Second Korn Inequality in a domain with a L ipxhi tz boundary. This proof is

essentially based on the next two lemmas.

We assume R t o be a bounded Lipschitz domain o f IRn. By p(x) is denoted

the distance from the point x t o dR; we denote by A the Laplace operator.

Lemma 2.2.

Let v E C w ( R ) n L2(R) , p2Av E L2(R). Then pVv E L2(R) and the

estimate

holds with a constant c independent o f v.

Proof. The function p(x) satisfies the inequality p(x) - p(y) 5 lx - yl for any

x,y E 0. Indeed, denote by z, the point o f aR such that p(y) = ly - zyl.

'One may omit the proof of (2.3) at first reading. A more simple proof of the Second Korn Inequality for star-shaped domains is given in $2.4.

$2. Korn's inequalities 15

Then p ( ~ ) - ~ ( y ) I )x-z,J-Jy-z,J 5 )x-z,- y+z,\. Thus p(x) is Lipschitz

continuous in R and therefore p(x) possesses bounded weak derivatives o f first

order in 0.

Taking into account these properties o f p(x) and using the Green formula

in R(&) = 52 n { x : p(x) > 61, we obtain

I t follows that

where c2 is a constant independent of 6. Making 6 tend t o zero in this inequality

and taking into account the fact that p(x) > 6 in we get

for any domain G such that G c 0 , where the constant c3 does not depend

on G. Therefore (2.4) is satisfied and pVv E LZ(R) . Lemma 2.2 is pr0ved.n

Lemma 2.3. d2w

Let w E C W ( R ) n L2(R) , p - E L2(R) . Then w E H 1 ( R ) and az;dxj

where the constant C does not depend on w.

Proof. I t is easy t o see that for any scalar function f E C1[O, b] we have


Using the mean value theorem let us choose T such that

This inequality together with (2.6) yields

where Cl is a constant independent o f f. Let us cover R by the domains R i , i = 0 , 1 , ..., N , such that Ro =

{ x : p ( x , d R ) > 61, 6 = const > 0 , and Ri = { X : $ i ( ~ ' ) < xki < Gi(x1) + bi, xi = ( x l ,..., xki-l, xki+l , ..., I,,) , x' E R : ) , i = 1 , ..., N , 1 5 ki 5 n, (possibly after an orthogonal transformation o f the variables x ) , where the

functions $i are Lipschitz continuous and d R n d R i = { x : xki = $i (x i ) , x' E 52:). By virtue of Lemma 2.2 we find that

where R t f 2 is the 612-neighbourhood o f Ro, the constant C2 depends only on

6.

Suppose that the domain Ri is defined by the conditions: $(XI) < xk < aw

$ ( x i ) + bi, x' E 0:. Setting b = bi, f = -, t = xk in (2.7) and considering a x j

aw - as a function o f xk, we get from (2.7) a x j

§2. Korn 's inequalities 17

Since +(xl) satisfies the Lipschitz condition, it is easy t o see that I+(xl) +a - xkl 5 Cp(x), where the constant C depends only on the Lipschitz constant

for +(xl). Therefore integrating (2.9) over R: and making a tend to zero we

find

provided that 6 is chosen sufficiently small. Summing up these inequalities

with respect to i from 1 t o N and using (2.8) we obtain

It follows that estimate (2.5) is valid since p(x) 2 6 > 0 in 0;. Lemma 2.3 is

proved.

Theorem 2.4 (The Second Korn Inequality).

Let R be a bounded Lipschitz domain. Then each vector valued function

u E H1(n) satisfies the inequality (2.3) with a constant C depending only on

0.

Proof. Obviously we can restrict ourselves t o the case of u E Cm(f i ) . By a 2 ~ ; a a

the definition of the matrix e(u) we have - = 2 - eij(u) - - e j j (u ) 8x3 ax j ax;

(there is no summation over i, j).

Consider the following equations


Set Fj = 0 outside R , i, j = 1, ..., n. Let v; E H,'(Ro) be a solution o f the

equation (2.10) in a smooth domain Ro such that c Ro. According t o the

well-known a priori estimate we have

This inequality can be easily obtained by virtue o f the Friedrichs inequality and

the integral identity for solutions o f the Dirichlet problem for equation (2.10).

Set v = ( v l , ..., vn)*, w = u - V . Then

~ ( e i j ( w ) ) = 0 in fl , e;,(w) E C m ( R ) , i, j = 1, ..., n . Due t o (2.11) we get

where the constant C3 does not depend on u. Therefore using (2.4) we find

that

I t is easy t o see that

Therefore (2.13) yields the inequality

Combining this inequality with estimate (2.5) of Lemma 2.3 we establish

$2. Korn's inequalities

Since w = u - v the above estimate implies

Therefore owing t o (2.11) we find that (2.3) is satisfied. Theorem 2.4 is

proved.

In applications it is often important to have another version o f the Second

Korn Inequality, namely the inequality

which holds for v belonging t o a subspace V of H1(R). Subspaces V of that

kind will often be dealt with below.

Denote by R the linear space of rigid displacements o f Rn, i.e. the set

of all vector valued functions q = (ql, ...,vn) such that 7 = a + A s , where

a = (al, ..., a,) is a vector with constant real components, A is a skew-

symmetric (n x n)-matrix with real constant elements. Here 7, a, x are

column vectors.

It is easy t o see that R is a linear space o f dimension n ( n - 1)/2 + n.

Theorem 2.5.

Let R be a bounded Lipschitz domain and let V be a closed subspace of vector

valued functions in H1(R), such that V n R = {0), where R is the space o f

rigid displacements. Then every v E V satisfies the inequality (2.14).

Proof. Suppose that the assertion o f Theorem 2.5 does not hold. Then there

is a sequence o f vectors urn E V such that

Since the imbedding H1(R) c L2(R) is compact (see Theorem 1.2), it follows that there is a subsequence mj + oo such that for some v E L2(R) we have vrnl + v in L2(R). According to Theorem 2.4 the Second Korn

Inequality (2.3) is valid in R, and therefore

I. Some mathematical problems of the theory of elasticity

This estimate and (2.15) show that urn) -+ v in H1(R) as mj + co. Since V is a closed subspace of H1(R) , by virtue of (2.15) we conclude that

The last equality implies that -

avi avh - + - = O , i , h = l , ..., n . ax,, axi

Les us show that any v satisfying (2.16) belongs t o R. Consider the mollifiers for v :

where v = 0 outside R , p( ( ) E C r ( R n ) . p ( ( ) 2 0 , p(()d( = 1. ~ ( 6 ) = 0 m n

forl(1 > 1. One can easily verify (see e.g. (1171, [311) that v' E CW(G) and

vc -t v in H1(G) as E -+ 0 for any subdomain G such that G C 52.

It follows from (2.16) that for sufficiently small E

Since the vc are smooth in G these equations imply

a 2 ~ ; - a 2 ~ ; , a2vi - - a 2 q -- in G . dxkaxh dxiaxh axiaxk dxhdxk

Therefore v f = a t x j + bf, where at j , bf are constants such that atj = -af i .

Due to the convergence o f vc t o v in H1(G) as E + 0 we have v E 72.

Thus v E V n R, I I v I I ~ I ( ~ ) = 1, which is in contradiction with the condition

V fl R = (0). Theorem 2.5 is proved.

Corollary 2.6.

In Theorem 2.5 one can take as V one of the spaces

V = {V E H1 ( R ) : ( v , V ) ~ I (n) = 0 Vq E R} ,

$2. Kern 's inequalities 21

We shall now give some other examples of spaces V whose elements satisfy

the inequality (2.14). Spaces of this type are often used below t o establish the

existence of solutions o f boundary value problems for the elasticity system and

to obtain estimates for these solutions.

Theorem 2.7.

Let R be a bounded domain with a Lipschitz boundary. Suppose that the set

y c dR can be represented in the form x, = c p ( i ) , where 3 = (XI, ..., 1,-1) varies over an open subset of Rn-', ~ ( 3 ) is a Lipschitz continuous function.

Then each vector valued function v E H1(R,y) satisfies the inequality (2.14).

Proof. If we show that H1(R, y) n R = {0), then in order t o obtain (2.14)

we can use Theorem 2.5 with V = H1(R, y).

Let r] E H1(R, y) n R. Therefore 77 = 0 on y. Every rigid displacement

has the form r] = b + Ax , where A is a skew-symmetric matrix with constant

elements, and b is a constant vector. Since the system A x + b = 0 is linear,

it is obvious that the ( n - 1)-dimensional surface y = { x : x, = cp(3)) must belong to a hyperplane, provided that A # 0. Therefore the dimension

of the space formed by all solutions o f system A x + b = 0 is not less than

n - 1, and consequently this system can have a t most one linearly independent

equation. Thus any two equations of the system are linearly dependent, and

therefore since all elements on the main diagonal o f A vanish, the coefficients

by xl, ..., x, vanish, too. Hence r] = 0. Theorem 2.7 is proved.

2.3. The Korn Inequalities for Periodic Functions

Here we establish the Korn inequalities similar t o (2.14) for 1-periodic vec-

tor valued functions.

Theorem 2.8.

Let w be an unbounded domain with a 1-periodic structure and let w n Q be

a domain with a Lipschitz boundary. Then for any v E W;(W) such that


the inequality

holds with a constant C independent of v.

Proof. Denote by V the linear space consisting of all restrictions t o w n Q of

vector valued functions in W;(W) satisfying the conditions (2.17). It is easy t o

see that V is a closed subspace o f H1(w fl Q) and that any rigid displacement

1-periodic in x is a constant vector. Therefore if v E V n R then by virtue of

(2.17) we have v = 0. Now Theorem 2.5 for R = w n Q yields the inequality

(2.18). Theorem 2.8 is proved.

The Second Korn inquality of type (2.14) for functions 1-periodic in 2 =

( x l , ..., x , - ~ ) is the result of

Theorem 2.9.

Let w be an unbounded domain with a 1-periodic structure and let the do-

mains w(a ,b ) , Lj(a,b) (0 < a < b < m) be defined by (1.6). Suppose

that ;(a, b) has a Lipschitz boundary. Then for any vector valued function

v E H' (,(a, b) ) such that / v dx = 0 the following inequality holds

&(ah )

where c is a constant independent of v.

The proof of Theorem 2.9 is almost exactly a repetition o f that o f Theorem

2.8. It should only be noted that a rigid displacement 1-periodic in i is also a

constant vector.

$2. Korn's inequalities

2.4. The Icorn Inequality in Star-Shaped Domains

In many applications it is important t o know the nature o f the dependence

of the constants in Korn's inequalities on the geometric properties of the do-

main. This dependence can be characterized on the basis o f the elementary

proof of the Korn inequality in a star-shaped domain, which is given in this

section.

Korn's inequalities in unbounded domains and some more general inequali-

ties of that type for the norms in LP(R) and in weighted spaces were considered

in [42], [43], [68], [46].

A domain R is said t o be star-shaped with respect to a ball G belonging

t o R, if the segment connecting any two points x E G, y E 51 lies in R.

Theorem 2.10.

Suppose that R is a bounded domain o f diameter R and R is star-shaped with

respect to the ball QR1 = { x : 1x1 < R 1 ) . Then for any u = (u l ,..., u,) E

H 1 ( R ) we have the inequality

where C1, Cz are constants depending only on n.

Proof. Obviously it is sufficient to prove (2.20) for smooth vector valued

functions u ( x ) . Let R1 = 1. By Cj we denote here constants which can

depend only on n . Let v = ( v l , ..., v,) be a solution o f the system

a a At); = 2 ( 2 - eik(u) - - ekk(u)) in

k = l axk ax;

with the boundary conditions

Multiplying (2.21) by v, and integrating by parts in R the resulting equality,

we find that


J IVv12dx 5 C3 I l e (u ) l k (n ) . (2.23) n

Set w = u - v. For any smooth V = (Vl, ..., V,) the following identities are

valid

Therefore due t o (2.24), (2.21) we have

It follows from (2.23) that

Therefore by virtue of (2.26) and Lemma 2.2 we get

where p = p ( x ) is the distance from x E R t o dR. It follows from (2.28) and

(2.24) that

Let us apply the following inequality

where C is a constant independent of a and f. The proof of (2.30) follows

immediately from (2.6).

Let us apply (2.30) t o the function f = dw;/dxj and the segment AP

belonging t o the segment OP, where P is any point on aR, 0 is the origin.

Considering P as the origin, we obtain

52. Korn's inequalities

Let us choose the point A such that A E QR,. IA l = A E [f , l ] ,

where dw is the area element on the unit sphere. Such a choice o f A is possible

due t o the mean value theorem. Obviously (2.31) implies

Let us integrate (2.33) over the unit sphere. Since the domain R is star-

shaped with respect t o QRl with R1 = 1 it follows that IP - XI < p(x)R. Therefore (2.32), (2.33) yield


Estimate (2.20) with R1 = 1 follows from (2.23), (2.29), (2.34), since

w = u - v. The inequality (2.20) with any R1 > 0 can be obtained from

(2.20) with R1 = 1, if one passes t o the variables y = x/R1.

Remark 2.11.

The coefficient by the second term in the right-hand side of (2.20) is asymptot-

ically exact and cannot be improved in the following sense. Let u = Ax + B,

where A is a skew-symmetrical matrix with constant elements, B is a constant

vector. Then (2.20) holds (in the form of an equality) with the coefficient

C2(R/R1)", provided that R has the volume o f order Rn.

Remark 2.12.

The inequality o f type (2.20) holds for any bounded smooth domain 0 (and

even for a Lipschitz domain), since such a domain is a union of a finite number

of star-shaped domains.

Remark 2.13.

Using a slightly more detailed analysis in the proof of Theorem 2.10 we can find

a more exact coefFicient by the first integral in the right-hand side o f (2.20).

Namely, under the assumptions of Theorem 2.10 the following inequalities of

Korn's type are valid

In order t o prove (2.35) we should use the inequality

32. Korn's inequalities

where C is a constant independent o f a and f . This inequality can be easily

obtained from the Hardy inequality (see e.g. (421, [44]). For the proof of (2.36)

the inequality (2.37) should be replaced by the following one

where C is a constant independent o f a and f (see [152]).

Estimate (2.35) cannot be improved in the following sense. Consider a vec-

tor valued function u = $(Ax + B ) , where A is a constant skew-symmetrical

matrix, B is a constant vector, $ E C m ( R n ) , G(z) = 0 in QR,, $(x) = 1

outside of QzR1 = {x : 1x1 < 2R1} , QzR, C 0 . Then (2.36) (in the form of

an equality) holds for u(x) with the coefficient C1(R/R1)" by the first integral

in the right-hand side, provided that R has the volume of order R".

Theorem 2.14.

Suppose that R satisfies the conditions o f Theorem 2.10 and u E H1(R) . Then

where y is the distance of QR, from 6'0.

Proof. Let cp E C F ( R ) , cp = 1 in QR,, 0 5 9 5 1 in R. Then according t o

Theorem 2.1 we have


It follows that

\(Vu\(12(pRl) 5 2 \le(u)l/t(o) + C3Y2 11~112qn) . (2.39)

Estimates (2.20), (2.39) imply (2.38). Theorem 2.14 is proved.

Theorems 2.10, 2.14 can be applied to study homogenization problems in domains having the form of lattices, carcasses, frames, etc.

53. Boundary value problems o f l inear elas ticity

53. Boundary Value Problems of Linear Elasticity

3.1. Some Properties of the Coeficients of the Elasticity System

In a domain R c Rn consider the differential operator o f linear elasticity

Here u = (ul, ..., u,) is a column vector with components ul, ..., u,, Ahk(x)

are (n x n)-matrices whose elements af/(x) are bounded measurable functions

such that

where {qih) is an arbitrary symmetric matrix with real elements, x E R , tcl, ~2 = const > 0.

We say that a family of matrices Ahk, h , k = 1, ..., n, belongs to class

E(rcl, n2), if their elements a? ' are bounded measurable functions satisfying

conditions (3.2), (3.3). In this case we also say that the corresponding elasticity

operator t belongs t o class E(K~, nz).

The operator L defined by (3.1) can also be written in coordinate form as

follows

a au j ( u ) - ( a x ) -) , i = 1, ... , n .

axh axk

In the classical theory of linear elasticity for a homogeneous isotropic body

the coefficients o f operators (3.4) are given by the formulas

where X > 0, p > 0 are the Lam6 constants, bij is the Kronecker symbol:

6,j = 0 for i # j, 6ij = 1 for i = j . In this case we have

for any symmetric matrix {qih). Moreover, the family of the matrices Ahk,

h , k = 1, ..., n , belongs t o the class E(2p,2p +nX). Indeed, i t is obvious that

nl = 2p, and the estimate KZ 5 2p + nX follows from (3.5), since


Thus the elasticity operator corresponding t o a homogeneous isotropic body

has the form

where aZul

Ul,hk = axhaxk

In order t o study the boundary value problems for the system o f elasticity

we briefly describe some simple properties of the elasticity coefficients. These

properties are easily obtained from the relations (3.2), (3.3) and will be fre-

quently used below.

With each family o f matrices Ahk(x) o f class E(nl, n2) for any fixed x we

associate a linear transformation M of the space of (n x n)-matrices, which

maps a matrix ( with elements tjk into the matrix M( with the elements

Then according t o (1.8) we have

Denote by €* the transpose o f the matrix €.

Lemma 3.1.

Let Ahk, h , k = 1, ..., n, be a family of matrices o f class E(nl, K ~ ) . Then for

any ( n x n)-matrices € = {&), 71 = {v ;~) with real elements the following

conditions are satisfied

Proof. By virtue of the first inequality in (3.2) we obtain that

$3. Boundary value problems of linear elas ticity

Due t o (3.3) and (3.6) the bilinear form (M( ,n) can be considered as a

scalar product in the space of symmetric (n x n)-matrices. Therefore by (3.2),

(3.3) and the Cauchy inequality we get

1 (Mt,n) = 4 (M(t + t * ) , n + v*) 5 It + t * I In + s* l .

It follows from (3.2) and (3.3) for 7 = ( t + t * ) that

KI It + <*I2 I (M(t + E*),E + E*) = 4 ( M t , t ) .

Lemma 3.1 is proved.

Lemma 3.2.

Each operator (3.1) o f class E(nl, n2) ( K ~ , tc2 > 0) is elliptic, i.e.

det lla!:thtkll # 0 for 161 # 0 < = ([I, ..., tn) .

Proof. Consider the following quadratic form

for a fixed t # 0. If J(q) = 0 it follows from (3.9) that

Multiplying each of these equations by Jivh and summing with respect t o i, h

from 1 t o n we obtain Itirli12 + 1t121912 = 0. Therefore 77 = 0. Thus J(7) > 0

for 9 # 0. Lemma 3.2 is proved.


3.2. The Main Boundary Value Problems for the System of Elasticity

Let L: be an elasticity operator o f type (3.1) belonging t o class E ( n l , n z ) ,

n l , n2 > 0 , and let R be a bounded domain o f Rn occupied by an elastic body.

The displacement vector is denoted by u = ( u l , ..., tin)*.

The following boundary value problems are most frequently considered in

the theory of linear elasticity.

The first boundary value problem (the Dirichlet problem)

involves finding the displacement vector u at the interior points of the elastic

body for the given displacements u = @ at the boundary and the external

forces f = ( f i , ..., f,) applied t o the body.

The second boundary value problem (the Neumann problem)

i.e. a t the points of the boundary the stresses u ( u ) = cp are given. Here

v = (4, ..., vn) is the unit outward normal t o dR.

The third boundary value problem (the mixed problem)

It is assumed here that the boundary d R of R is a union o f two sets and S such that r n S = 0.

In order t o prove existence and uniqueness of solutions of these problems, it

is necessary t o impose certain restrictions on dR, r, S , which will be specified

below.

$3. Boundary value problems o f linear elasticity 33

In $6 we shall also consider some other boundary value problems for the

system of elasticity, in particular problems with the conditions o f periodicity in

some of the independent variables.

Let u = (u1 , ..., u,) be the displacement vector and let e(u) be the

corresponding strain tensor, i.e. e(u) is a matrix with elements ei j (u) = 1 dui duj

= - (- + -). 2 axj axi Set

Then taking into account (3.7), (3.8) for = V u , [* = ( V u ) ' , we find

3.3. The First Boundary Value Problem (The Dirichlet Problem)

Let R be a bounded domain o f Rn (not necessarily with a Lipschitz bound-

ary), fj E L 2 ( R ) , j = 0,1, ..., vz , cp E H 1 ( R ) .

We say that u ( x ) is a weak solution o f the problem

a? L ( u ) = f " + - in 0 , u=cp on 8 0 , ax, (3.14)

if u - 9 E HA(R) and the integral identity

holds for any v 6 H,'(R).

Theorem 3.3. There exists a weak solution u ( x ) of problem (3.14), which is unique and

satisfies the estimate

where the constant %(a) depends only on nl, nz in (3.3) and the constant in

the Friedrichs inequality (1.2) for 7 = 30.

Proof. It follows from (3.15) that w = u - cp must satisfy the integral identity

I. Some mathematical problems of the theory o f elasticity

for any v E H,1(R). Note that due t o the Friedrichs inequality (1.2), the First

Korn inequality (2.2) and estimates (3.13) the quadratic form

satisfies the conditions o f Theorem 1.3, if we take as H the space o f a l l vector

valued functions with components in H,'(fl).

Obviously the right-hand side of (3.17) defines a continuous linear func-

tional on v E H t ( R ) . Therefore by Theorem 1.3 there is a unique element

w E H i ( R ) satisfying the integral identity (3.17). Setting u = w + cp we

obtain the solution o f the problem (3.14).

Let us prove the estimate (3.16). Set w = u - cp, v = u - cp in (3.17).

Then by virtue of the Friedrichs inequality (1.2), the First Korn inequality (2.2)

and estimate (3.13) we find

where the constant C3 depends only on KI, ~2 and the constant in (1.2). Since

I llull - llcpll 1 5 I I u - (pll, the estimate (3.18) implies (3.16). Theorem 3.3 is

proved.

53. Boundary value problems of linear elas ticity 35

The details, concerning the smoothness o f the solutions obtained in Theo-

rem 3.3, are given a thorough consideration in the article [17] which contains in

particular the proof o f the fact that the smoothness o f d R , the data functions

f", cp and the coefficients of L guarantee the smoothness of the weak solution

u ( x ) of problem (3.14).

Denote by H - ' ( 0 ) the space o f continuous linear functionals on the space

o f vector valued functions with components in H,'(R). As usual the norm in

H - ' ( 0 ) is defined by the formula

It follows from the proof of Theorem 3.3 that

defines a continuous linear functional on H,'(R), namely

fc.1 = J [ ( f O , v ) - (f', -31 dx n Xi

for any v E H,'(R). We obviously have

n

l l f l l ~ - l (n ) I C lI fmll~2(n) , C = const . m=O

On the other hand, for any f E H- ' (R) there exist functions f m E L 2 ( R ) , m = 0, ..., n, such that

in the sense o f the integral identity (3.19), and

Indeed, by the Riesz theorem (see [107]), every continuous linear functional

f ( v ) on H i ( R ) can be represented as a scalar product in H,'(R), i.e. there is

a unique element u E Ht(R) such that


Setting v = u in (3.22) and taking into consideration the definition o f the

norm in H-'(a), we find that

Setting f0 = u, f i = - e , by virtue o f (3.22), (3.23) we obtain the repre-

sentation (3.20) and the estimate (3.21).

Remark 3.4.

In the special case when ip = 0 in (3.14), we can consider the problem

for any f E H- ' (R), since f can be represented in the form (3.20). Then by

Theorem 3.3, due to (3.21) we have

where the constant C depends only on 61, rc2, and the constant in the

Friedrichs inequality (1.2) for y = do.

3.4. The Second Boundary Value Problem (The Neumann Problem)

In this section we assume R t o be a bounded domain with a Lipschitz

boundary. Let S1 be a subset o f dR with a positive ( n - 1)-dimensional

Lebesgue measure on dR. Set

We say that u ( x ) is a weak solution of the problem

where fj E L2(R), j = 0, ..., n, ip E L2(S1), if the integral identity

$3. Boundary value problems o f l inear elas t ici ty

holds for any v E H 1 ( R ) . Note that if dR, fj, cp, Ahk are not smooth, the boundary conditions in

(3.27) are satisfied only in a weak sense, namely in the sense of the integral

identity (3.28). The integral over S1 in the right-hand side of (3.28) exists

due t o the estimate

I I v I I ~ 2 ( ~ ~ ) 5 C ( R ) IIvIIH1(n) (3.29)

for any v E H 1 ( R ) , which follows from Proposition 3 o f Theorem 1.2.

Theorem 3.5.

Suppose that

for any rigid displacement q E R. Then there exists a weak solution u(x) of

problem (3.27). This solution is unique (to within an additive rigid displace-

ment) and satisfies the inequality

Here the constant ~ ~ ( $ 2 ) depends only on n l , n2, the constants in (3.29) and

in (2.14) when V is a closed subspace o f H 1 ( R ) orthogonal t o R with respect

t o the scalar product in L 2 ( R ) or H1(R) .

Proof. Let H = V in Theorem 1.3, where V is either of the spaces defined in

Corollary 2.6. Since inequality (3.29) is valid for the elements o f V, it is easy

t o see that the right-hand side o f the integral identity (3.28) is a continuous

linear functional on v E H . By the same argument that has been used in the

proof of Theorem 3.3, due t o the Second Korn inequality and the estimate

(3.13), we find that the bilinear form in the left-hand side o f (3.28) satisfies


the conditions of Theorem 1.3. Thus there is a unique element u E H such

that the integral identity (3.28) holds for all v E H . For v E R the left-hand

side o f (3.28) is equal to zero due t o the fact that L(v ) = 0 in R, o ( v ) = 0

on dR; the right-hand side o f (3.28) is also equal t o zero for v E R, since

we have assumed that conditions (3.30) are satisfied. Therefore the integral

identity (3.28) holds for all v E H1(R) , which means that u ( x ) is a solution

of problem (3.27).

Estimate (3.31) can be obtained from (3.28) for v = u , the Second Korn

inequality and (3.13), (3.29). Theorem 3.5 is proved.

Remark 3.6. In Theorem 3.5 we can choose a solution u(x) orthogonal in L2(R) or H1(R)

to the space of rigid displacements R. For such u ( x ) we have the following

estimate

where the constant C2(R) depends on the same parameters as the constant

Cl (R) in (3.31). This fact is due t o the Second Korn inequality (2.14) (see

Theorem 2.5).

Remark 3.7. Similarly t o the case of the Dirichlet problem one can prove the smoothness of

weak solutions o f the Neumann problem, provided that the coefficients a f / ( s ) ,

the boundary of R, and the data cp, f", i = 0, ..., n, in (3.27) are smooth (see

[I71 1.


In a bounded domain R C Rn we consider the following boundary value

problem for the operator C of class E(n l , n2) , n l , n2 > 0:

53. Boundary value problems o f linear elas ticity

where f j E L Z ( R ) , j = 0,1, ..., n, cp E LZ(S1) , E H 1 f 2 ( y ) , v = ( v l , ..., v,) is the unit outward normal to dR.

Before giving a definition of a solution of the mixed problem we impose

the following restrictions on a R , y , S1 , S Z .

1. d R = 7 U $ U S2 and y, S1, S2 are mutually disjoint subsets of dR.

2. R is a domain with a Lipschitz boundary d o , y contains a subset satisfying

the conditions of Theorem 2.7.

Note that all further results are also valid under weaker assumptions on d R and y which guarantee the inequalities (1.2), (2.14).

We define a weak solution of problem (3.33) as a vector valued function

u E H 1 ( R ) satisfying the integral identity

for any v E H 1 ( R , y ) , and such that u = iP on y (i.e. u - E H 1 ( R , y ) ) . Note that by the definition o f ~ l / ~ ( y ) we can consider @ as an element of

H 1 ( R ) .

Theorem 3.8.

There exists a weak solution u ( x ) of problem (3.33). This solution is unique

and satisfies the estimate


where the constant C ( 0 ) depends only on 6 1 , K ~ , the constant in (3.29)

and the constants in the Korn inequality (2.14) for vector valued functions in

H 1 ( R , -y) (see Theorem 2.7).

Proof. From (3.34) we conclude that w = u - @ must satisfy the integral identity

for any v E H 1 ( R , y ) . Due t o Proposition 3 of Theorem 1.2 the inequality

(3.29) holds for a l l v E H 1 ( R , -y), and according t o Theorem 2.7 the inequality

(2.14) is also valid for such v.

Inequalities (2.14) and (3.13) show that the bilinear form in the left-hand

side o f (3.36) satisfies all assumptions of Theorem 1.3 with H = H 1 ( R , -y).

By virtue of (3.29) the right-hand side of (3.36) defines a continuous linear

functional on H'(O,y ) . It follows from Theorem 1.3 that there is a unique

element w E H1(O,-y) satisfying the integral identity (3.36). Obviously u =

w+@ is the solution of problem (3.33). Let us prove estimate (3.35). Setting

v = w in (3.36) by virtue of (2.14) and (3.13), we have

Therefore taking into account (3.29) for v = w, we find that

Therefore

$3. Boundary value problems of linear elasticity 41

since w = u - 9. Note that in the proof of the last estimate we can replace 9 by any & such

that 9 - 6 E H1(R, y), and this would not affect the constant C3 which does

not depend on 9.

Thus by the definition of the norm in H ' / ' ( ~ ) we obtain (3.35) from (3.37). Theorem 3.8 is proved. •


$4. Perforated Domains with a Periodic Structure. Extension Theorems

4.1. Some Classes of Perforated Domains

Let w be an unbounded domain of Rn with a 1-periodic structure, i.e. w

is invariant under the shifts by any z = (zl, ..., z,) E ZZn.

Here we also use the notation:

Q = { x : O < x j < l , j = 1 , ..., n ) ,

p(A, B) is the distance in Rn between the sets A and B, E is a small positive

parameter.

In what follows we shall mainly deal with domains w satisfying

Condition B (see Fig. 1):

B1 w - is a smooth unbounded domain of Rn with a 1-periodic structure.

B2 The cell o f periodicity w n Q is a domain with a Lipschitz boundary.

B3 Theset Q\G and the intersection o f Q\w with the 6-neighbourhood (6 < i) of dQ consist of a finite number of Lipschitz domains separated from each

other and from the edges of the cube Q by a positive distance.

Fig..

$4. Perforated domains with a periodic structure 43

We shall consider two types of bounded perforated domains Re with a pe-

riodic structure characterized by a small parameter e.

A domain Re of t ype I has the form (see Figs. 1, 2, 3):

where R is a bounded smooth domain o f Rn, w is a domain with a 1-periodic

structure satisfying the Condition B; Re is assumed to have a Lipschitz bound-

ary.

R

Fig.

Fig..

The boundary of a domain Re of type I can be represented as dRc = I',US,,

where r, = d R n ew, Se = (dRe) n R. A domain Re of t ype II has the form (see Figs. 4, 5a, 5b):


where R is a bounded smooth domain.

TE is the subset of Zn consisting of all z such that

E is a small parameter.

Fig..

0;

Fig. 5a.

Q1

Fig. 5b.

$4. Perforated domains with a periodic structure 45

We assume that R1, R;, RE (the sets of interior points o f nl, n;, a') are

bounded Lipschitz domains.

The boundary 8Rc of a domain RE of type II is the union o f df l and the

surface SE c R of the cavities, S, = ( d V ) n R.

4.2. Extension Theorems for Vector Valued Functions in Perforated Domains

In order t o estimate the solutions o f the above boundary value problems

for the system of elasticity in perforated domains RE we shall construct exten-

sions t o R of vector valued functions defined in RE and prove some inequalities

(uniform in E ) for these extensions.

Lemma 4.1.

Let G c 2) c Rn and let each of the sets G , V , V\G be a non-empty bounded

Lipschitz domain (see Fig. 6). Suppose that y = ( 8 G ) n V is non-empty. Then

for vector valued functions in H~(v\G) there is a linear extension operator

P : H'(D\G) + H 1 ( V ) such that

where the constants cl , ..., c4 do not depend on w E H~(D\G) .

Fig..


Proof. Let us first show that each w E H~('D\G) can be extended as a

function 6 E H 1 ( V ) satisfying the inequality

with a constant c independent o f w.

Indeed, consider the ball B c Rn containing a neighbourhood o f the

set V . According t o Proposition 2 of Theorem 1.2 the function w can be

extended from 'D\G to the entire ball B as a function w1 E H 1 ( B ) . Taking

the restriction of w1 on V we get a function 6 which satisfies the inequality

(4.9).

Denote by W the weak solution o f the following boundary value problem

for the system of elasticity

where C is an arbitrary operator o f class E(n1, K Z ) with constant coefficients.

Note that the last boundary condition in (4.10) should be omitted if d G n d V =

0. By Theorem 3.8 W exists and satisfies the inequality

Therefore due t o (4.9) we obtain

Set

w ( x ) for x E V\G , P ( w ) =

W ( x ) for x E G .

I t is easy to see that P ( w ) is a vector valued function in H 1 ( V ) . By virtue of

(4.10) we have Pv = 17 for any q E R. Taking into account (4.11) and the

Korn inequality (2.3) in DIG (see Theorem 2.4) we conclude that estimates

(4.5), (4.6) hold with constants cl , cz depending only on G and 'D.

Let us prove the estimate (4.8) for Pw. Suppose that (4.8) does not hold.

Then there is a sequence o f vector valued functions vN E H~(v\G) such that

54. Perforated domains w i th a periodic structure 47

I I P v N I I ~ ~ ( v ) 5 ci l l vNII~ l (v \~) , (4.13)

but

Ile(PvN)II~2(v) 2 N l le(vN)II~2(v\~) , (4.14)

Without loss o f generality we can assume that (vN,r])dx = 0 for any rigid I V\G

displacement q, since P(v t r ] ) = Pv + r] due t o (4.10), (4.12), and for

any bounded domain wo and any v E H1(wo) we have le(v + q)I2dx = J wo 1 l e ( ~ ) ( ~ d x . By (4.15) and the Second Korn inequality (2.14) in D\G (see

wo Corollary 2.6) we get

Thus vN -+ 0 as N -+ m in H'(D\G), and therefore I I P v ~ I I ~ I ( ~ ) --, 0 as

N -+ CCJ due t o (4.13). On theother hand, (4.14) implies that I l e ( P ~ ~ ) 1 1 ~ 2 ( ~ ) 2 1. This contradiction establishes the inequality (4.8).

To prove (4.7) we choose a constant vector C such that 1 P ( w + C)dx = 0. Because of the Poincarh inequality (1.5) in D\G it

V\G follows from (4.5) that

Therefore (4.7) is valid since V C = 0, PC = C . Lemma 4.1 is proved.

Theorem 4.2 (Extension o f functions in perforated domains of type 11). Let Re be a perforated domain of type II. Then for vector valued functions in

H1(Rc) there is a linear extension operator P, : H1(Rc) -+ H 1 ( R ) such that


for any v E H1(Rc) , where the constants q , ..., c4 do not depend on E , v .

P,,,f. Let v ( x ) E H1(Rc). Set V ( J ) = v ( E [ ) and fix z E T,, where Tc is

the index set in the definition o f a perforated domain Rc o f type I I (see (4.3)).

Consider the function V ( [ ) in the Lipschitz domain w n ( z + Q ) . By Lemma

4.1 one can extend V ( J ) as a vector valued function PIV E H1(z + Q ) such

that

Extending V ( [ ) in this way for every z E T, we get a vector valued function

PIV which satisfies the inequalities (4.21) for any z E Tc with constants

16 , ..., IC3 independent o f z .

If the distance between Q\G and dQ is positive (i.e. Q\G lies in the interior

of cube Q ) , then the function ( P ~ v ) ( ~ ) is the extension whose existence is x

asserted by Theorem 4.2, and therefore we can take (Pcv) (s ) = (P,v)( - ) . E

where V ( J ) = v(E[) .

However, if Q\L;) has a non-empty intersection with dQ (as in Fig. I), the function P I V ( J ) may not belong t o HI(&-'R), since its traces on the

$4. Perforated domains wi th a periodic structure 49

adjacent faces o f the cubes z + Q, z E T,, do not necessarily coincide. In a

neighbourhood o f such faces we shall change PIV as follows.

For 1 = 0,1 set &Q = U { t E dQ, tk = I}. k = l

Due t o Condition 63 on w the intersection of the 6-neighbourhood of d Q

with Q\W consists o f a finite number o f Lipschitz domains separated from

each other and from the edges o f Q by a positive distance larger than some

61 E (0,1/4). For 1 = 0 and 1 = 1 denote by those o f the domains

just mentioned whose closure has a non-empty intersection with d,Q (see Fig.

7). Therefore each ~f lies in the 6-neighbourhood of dQ and is adjacent to a

face o f Q lying on the hyperplane tk = I for some I c .

1 -.p: d

.__I

Fig. 7. -

Let the domain R1 and the set T, E Zn be the same as in the definition

of a perforated domain Re of type II (see Figs. 4, 5a, 5b). Denote by T,' the

set of z E T, such that (T,! + z) n d ( ~ - l O ~ ) # 0 for some j = 1, ..., ml. The extension PIV(t) constructed above is such that PIV E H1(g) for

any open g C e-'R which has no intersection with any of the domains + z,

z E T,, yj + z , z E T,'. Let us change PIV in these domains so as t o obtain

a function in HI(&-'a).

Fig.. The domains G1, ..., GN are shaded pale.


Denote by GI, ..., GN all mutually non-intersecting domains having the

form either $ + z , z E T, or yi + z , z E T,' (see Fig. 8). Obviously

p(G,, Gt) > 61 for s # t. The number N tends t o infinity as E t 0, however,

GI, ..., GN are the shifts o f a finite number o f bounded Lipschitz domains.

Consider the extension PIV(e). We have constructed the sets GI, ..., GN in

such a way that the set dG1 U ... U dGN contains all those parts o f the faces

of the cubes z + Q, z E T,, where the traces o f PIV(J) may differ. Set

Go = G1 U ... U GN. Then one clearly has PIV E H'(E-'R\Go). Denote by

G~ the 61/2-neighbourhood o f Gj. By virtue of Lemma 4.1 let us extend PIV to each of the sets Gj as a

vector valued function P2V satisfying the following inequalities

IIVCP~VIIL~(G,) I M3 IIVCP~VIIL~(G,\G,~ ,

Ilec(P~V)11~2(c,) 5 M4 I~~c(P~V)I ILZ(C, \G~) 7

depend on V, j .

and such that P2q = q if q E R, where the constants Ml, ..., M4 do not

Set U(J) = (PIV)(J) for J E (c-'R)\G0, U(J) = (P2V)(J) for J E GO. 2

Applying the estimates (4.21). (4.22) we finally conclude that u(-) can be &

taken as the extension (Pev)(x) satisfying the conditions (4.21). Theorem 4.2

is proved.

Theorem 4.3 (Extension of vector valued functions in perforated domains of

type 1). Let Re be a perforated domain o f type I and let Ro be a bounded domain such

that fi C 00, p(dRO,R) > 1. Then for every sufficiently small E there exists

a linear extension operator PC : H1(RE,rE) -+ H,'(Ro) such that

$4. Perforated domains with a periodic structure

for any u E H1(Rc, re), where the constant C1, C2, C3 do not depend on E , u.

Moreover, (P,u)l, = 0 for any open g such that g C Ro\R, if E is suffi-

ciently small.

Proof.' Denote by Tc the set of all z E Zn such that ~ ( z + Q n w ) n 52 # 0. Let @ be the interior o f U ~ ( z + Q n w), and let fil be the interior of

zETe

U E ( Z + Q) . For each u E H1(RC,r,) we introduce the following vector PET*

valued function

u ( x ) , X E R C ,

0 , X E ~ ; \ R ,

0 , x E Ro\fil . It is easy t o see that U ( x ) E ~ ' ( f i f ) . According t o Theorem 4.2 one can

extend U ( x ) t o the domain Ro. Denote this extension by P ~ U , and set

P,u = Feu. Obviously the conditions (4.23)-(4.25) are satisfied. The last

statement of the theorem holds since Pcu = 0 in Ro\fil. Theorem 4.3 is

proved. •

4.3. The Kern Inequalities in Perforated Domains

In this section we prove the Korn inequalities (with constants independent

of E ) for perforated domains Re of types I and II. These results are widely used

in Chapter I I for the homogenization of various elasticity problems.

Theorem 4.4 (Korn's inequalities in perforated domains of type 11).

Let Rc be a perforated domain o f type II. Then for any vector valued function

u E H1(Rc) the inequality

he proof is based on the extension of a function u from H1(Qc, r,) by u = 0 outside il and the subsequent application of Theorem 4.2 in a new perforated domain which is different from that of Theorem 4.2 but is also of type 11.


holds with a constant C independent o f u, E .

Moreover, if one of the following conditions is satisfied

( u , ~ ) H l ( n . ) = O , V q E R , or

(u,71)~2(nr) = 0 , Vq E R , then

IIuIIHl(ne) 5 Cl ~ ~ ~ ( u ) I I L ~ ( S Z ~ ) 3

where the constant Cl does not depend on u , E .

Proof. The estimate (4.26) immediately follows from the Korn inequality (2.3)

in R (see Theorem 2.4) and the extension Theorem 4.2. Indeed, let P, be the

extension operator constructed in Theorem 4.2. Then

Suppose now that u ( x ) satisfies (4.27). Then

for any rigid displacement q E R . Let P,u E H 1 ( R ) be the extension of u

constructed in Theorem 4.2. Denote by qo the orthogonal projection of P,u

on R with respect to the scalar product in H 1 ( R ) . Then

Due to the Corollary 2.6 we have

since I(e(Pcu - q0)llL2(n) = I ~ ~ ( P , u ) ~ ( ~ z ( ~ ) . By virtue o f (4.30) and Theorem

4.2 the last inequality yields

54. Perforated domains with a periodic structure

L Cq lle(Pcu)Ili2(n) 5 C5 lle(u)112L2(n*) . Suppose that (4.28) is satisfied. Then

I I u I I ~ ~ ( ~ c ) 5 IIu - r111iz(n*) , V7) 'rl 7 2 . (4.32)

Choosing 7 = q0 such that (4.27) holds for u - qo, we obtain by (4.29) for

u - qo, that

Therefore,

IIuIIZ2(n*) L C6 Ile(u)II22(n*)

by virtue o f (4.32). This inequality together with (4.26) implies (4.29) for vec-

tor valued functions u(x) satisfying (4.28). Theorem 4.4 is proved.

Let us now prove the Korn inequality in a perforated domain 0' o f type

I for vector valued functions in H1(Rc) vanishing on re. Note that Theorem

2.7 provides an inequality of this kind with a constant which may depend on E ,

however, in what follows we need the inequality with a constant independent

of E.

Theorem 4.5.

Let W be a perforated domain of type I. Then for any vector valued function

v E H'(Rc, r c ) the inequality

is valid, where C is a constant independent of E and v.

Proof. Let v E H1(Rc, I',) and denote by P,v E Hi (no ) the extension o f v t o

the domain Ro constructed in Theorem 4.3. Due t o Theorem 2.1 the vector

valued function Pcv satisfies the Korn inequality of type (2.2) in no. Therefore

by (4.25) we have

I. Some mathematical problems of the theory o f elasticity

IIvIIHl(n*) 5 IIPcvII~l(no) 5 cl Ile(Pcv)IIL2(no) 5

5 C2 l le(v)Il~2(n~) ,

where the constants C1, Cz do not depend on E , v. Theorem 4.5 is proved.0

Directly from Theorem 4.2 and Proposition 3 of Theorem 1.2 we obtain

Lemma 4.6. Let Re be a perforated domain of type II. Then

for any v E H1(Rc) , where C is a constant independent o f E , v.

$5. Estimates for solutions of boundary value problems of elasticity 55

$5. Estimates for Solutions o f Boundary Value Problems o f Elasticity in

Perforated Domains

In $3 existence and uniqueness o f solutions for the main boundary value

problems of linear elasticity were established together with the estimates of

these solutions through the norms o f the given functions. If the domain occu-

pied by the elastic body or the coefficients o f the system depend on a parameter

E , the constants in these estimates may depend on E . In this section we show

that for perforated domains Rc defined in $4 the constants in estimates of type

(3.31), (3.35) can be chosen independent o f E , provided that the coefficient

matrices o f the elasticity system belong to the class E ( n l , n 2 ) with n l , K Z

independent o f E .


Let R" be a perforated domain of type I (see (4.1)), dRc = Sc U rC, where

S, is the surface o f the cavities, S, = R n dRc , I?, = d R n d R c .

Consider the following boundary value problem

where fj E L 2 ( R c ) , j = 0, ..., n, E H1(R' ) , L is an elasticity operator of

type (3.1) belonging t o the class E ( n l , n 2 ) .

In the general situation this problem was considered in $3 (see Theorem

3.8). The next theorem represents a more precise version o f Theorem 3.8 for

perforated domains RE.

Theorem 5.1.

Let RE be a perforated domain o f type I and let the coefficient matrices o f the

operator L belong t o the class E ( n 1 , n2 ) with constants n l , n2 > 0 indepen-

dent o f E . Then there exists a weak solution u ( x ) of problem (5.1), which is

unique and satisfies the inequality

I. Some mathematical problems o f the theory of elasticity

where C is a constant independent o f E .

m. Existence and uniqueness o f the solution o f problem (5.1) follow im-

mediately from Theorem 3.8 with S1 = 0, S2 = Sc, y = r e . As stated in

Theorem 3.8, the constant C in (5.2) depends only on tcl, K Z , and the con-

stant in the Korn inequality (4.33) for vector valued functions in H1(R', re). According t o Theorem 4.5 the last constant can be chosen independent of E ,

and therefore (5.2) holds with a constant C which is also independent o f E .

Theorem 5.1 is proved.

Remark 5.2.

Every vector valued function f 0 E L2(Rc) defines a continuous linear func-

tional I(v) on H1(RC,I',) by the formula l(v) = ( f O , ~ ) ~ z ( ~ . ) . Denote by

1 1 fOII* the norm o f this functional in the dual space ( ~ ' ( f l ' , re))*. Then

Obviously 1 1 fO) l* 5 1 1 f O 1 l L ~ ( n e ) . It follows from the proof of Theorem 3.3 that

we can replace the estimate (5.2) by

5.2. Estimates for Solutions of the Neumann Problem in a Perforated

Domain

In a perforated domain Re of type II consider the second boundary value

problem o f elasticity

af i L (u ) = p + - in Rc ,

dxi (5.5)

o(u ) = (P + V; fi o n aR , a ( u ) = u; f' on Sc ,

§5. Estimates for solutions of boundary value problems of elasticity 57

where

In contrast t o Theorem 3.5 the next theorem establishes estimates uniform

in E. for the solutions of problem (5.5).

Theorem 5.3.

Let RE be a perforated domain o f type 11, and

for any rigid displacement 7 E R. Suppose that the coefficient matrices of

the operator L belong to the class E(lcl , K Z ) with ~ 1 , ~2 > 0 independent of

E. Then problem (5.5) has a unique solution u (x ) such that

( u , ~ ) ~ l ( n . ) = O , V v E R ,

and

where C is a constant independent o f e.

Proof. Existence and uniqueness o f a solution of problem (5.5) follow from

Theorem 3.5 and Remark 3.6. We also have the estimate (5.9) for u (x ) with

a constant C depending only on ~ 1 , nz and the constant in the Second Korn

inequality (4.29), which does not depend on E . Therefore (5.9) holds with a

constant independent of E . Theorem 5.3 is proved.

In order to study the spectral properties o f the Neumann problem of type

(5.5) (see Ch. Ill) we shall need the following auxiliary boundary value problem

in the domain R" of type II:


where fj E L 2 ( R c ) , j = 0, ..., n, cp E L 2 ( a R ) , the matrices Ahk(x ) belong t o

the c lan E( / c l , / c 2 ) , p (x) is a bounded measurable function in Rc such that

We say that u ( x ) is a weak solution of problem (5.10) if u ( x ) E H'(Rc) and the integral identity

holds for any w E H ' ( R c ) . Denote by a (u , w) the bilinear form in the left-hand side o f (5.12). This

form satisfies all conditions o f Theorem 1.3 for H = H1(R' ) with constants c l , c2 independent o f E. This fact is due to the Korn inequality (4.26). Therefore

existence, uniqueness and estimates of solutions o f problem (5.10) are proved

on the basis o f (5.12) in the same way as Theorems 3.5, 3.8. We have thus

established

Theorem 5.4.

Let Rc be a perforated domain o f type II, and let the family of matrices A h k ( x ) , h , k = 1, ..., n, belong t o the class E ( n l , K ~ ) . Suppose that conditions (5.11)

are satisfied and the constants Q, c , , n l , nz do not depend on E. Then problem

(5.10) has a unique solution u ( x ) , and this solution satisfies the estimate

where C is a constant independent o f E.

$6. Periodic solutions o f boundary value problems 59

$6. Periodic Solutions o f Boundary Value Problems for the System of

Elasticitv

To study homogenization problems for the system of elasticity we need

existence theorems for some special boundary value problems.

6.1. Solutions Periodic in All Variables

Let w be an unbounded domain with a 1-periodic structure, which satisfies

Condition B of $4, Ch. I.


w is 1-periodic in x , I w d x = O , Qnw I

where the vector valued functions F j ( x ) are I-periodic in x , F j E L2(w n Q ) , j = 0, ..., n, the family of matrices A ~ ~ ( x ) belongs t o the class E ( K , , rcz) and

their elements a f i ( x ) are 1-periodic in x.


w E w ~ ( w ) such that w d z = 0 , and the integral identity I Qh

dv = / [ ( F m , -) - (PO, v)] d.

ax, Qnw

holds for any v E W;(W) .

Theorem 6.1.

Let F O d x = 0 . Then problem (6.1) has a unique solution, and this solu- J

Q n w tion satisfies the estimate


where the constant C depends only on nl, K * , w.

The proof o f this theorem rests upon Theorem 1.3 and is quite similar t o

the proof o f Theorem 3.5. In this case one should take as H the space o f vector

valued functions v E W;(W) such that v d x = 0; the Korn inequality is J Qnw

furnished by Theorem 2.8.

In what follows we shall often use the fact that solutions o f problem (6.1)

are piece-wise smooth, provided that the coefficients of the system (6.1) and

the functions F j , j = 0, ..., n, are piece-wise smooth and may loose their

smoothness only on surfaces which do not intersect dw. Let us consider these

questions more closely.

We assume that there are mutually non-intersecting open sets Go, ..., G ,

with a 1-periodic structure and such that G j c w , j = 0,1 , ..., m; G j n d w = 0 , j = 1, ..., m; Go = w\(G1 U ... U G,); GI , ..., G , have a smooth boundary.

We say that a function cp which is 1-periodic in x belongs t o class 6 (cp is

called piece-wise smooth in w and smooth in a neighbourhood o f dw) if cp has

bounded derivatives of any order in G j , j = 0,1 , ..., m.

Theorem 6.2.

Let w ( x ) E W;(W) be a weak solution o f problem (6.1), and suppose that the

elements of A h k ( x ) , F j ( x ) belong t o class 6. Then w also belongs t o 6, i.e.

w is piece-wise smooth in w and smooth in a neighbourhood of dw.

Proof. The smoothness of w in a neighbourhood o f dw follows from the gen-

eral results on the smoothness of solutions of the elasticity system near the

boundary (see [17]).

Let x0 E d G j , xO 6 dw, and consider the set G j n { x : lx - xO1 < 6 ) =

q;(xo). It is shown in [17], Section 13, Part I, that for sufFiciently small 6 the

function w has bounded derivatives of any order in qj6(x0). The smoothness

of w at the interior points of w , which do not belong t o d G j , is also proved

in (17). Therefore w E d.

$6. Periodic solutions o f boundary value problems

6.2. Solutions of the Elasticity System Periodic in Some of the

Variables

Let the coefficient matrices Ahk(x) of the differential operator C belong

to the class E ( K , , K ~ ) , and suppose that their elements are 1-periodic in ? =

( 5 1 , ..., xn-1). In this section w is an unbounded domain with a 1-periodic structure, which

satisfies the Condition B of $4 (see Fig. I), the domains w(a , b) and ;(a, b)

are defined by (1.6).

Set

Let gt be a non-empty open set belonging t o f i t and invariant with respect

t o the shifts by any vector z = ( z l , ..., 0 ) E Zn. Set

Fig..


w is 1-periodic in P , w d x = O , O(5.b)

where $.. $I,, F are vector valved functions 1-periodic in 2 . Fj E L2 (;(a, b ) ) ,

j = 0 ,..., n ; $, E L2(g , ) ,$bE L2(gb), 0 I a < b < m, un = -1 on g,, vn = 1 on gb. The domain &(a, b) is assumed t o have a Lipschitz boundary.



w E H 1 ( u ( a , b ) ) such that for any v E ~ l ( u ( a , b ) ) the following integral

identity is valid:

Theorem 6.3.

Let

Then there exists a weak solution w o f problem (6.6), which is unique, and w

satisfies the estimate

where C is a constant depending only on w, a , b, nl, n2.

This theorem is proved in a similar way t o Theorem 3.5. In this case we

take as H the subspace o f B1 ( u ( a , b ) ) formed by all vector valued functions

v such that / vds = 0. Then the Second Korn inequality follows from

Theorem 2.9. To estimate the right-hand side of (6.7) we should use the

inequality

§6. Periodic solutions of boundary value problems 63

which holds due to Proposition 3 of Theorem 1.3 and the Korn inequality

(2.19).

Let us also establish the existence and uniqueness of the solution of the

following mixed boundary value problem:

where a(?), $a(? ) . F j ( x ) . j = 0 , ..., n, are 1-periodic in 2, Fj E L ~ ( s ( ~ , b) ) ,

+b E L2( ib) , E ~ ' / ~ ( d . ) . A vector valued function w is called a weak solution of problem (6.11), i f

w E ~ l ( w ( a , 6 ) ) . w = Q on d., and the integral identity

is satisfied for any v E H' (@(a, 6)) n H1(B(a , b) , d.).

Theorem 6.4.

There exists a weak solution w ( x ) o f problem (6.11), which is unique and

satisfies the inequality

where C is a constant depending only on w , K , , ~ 2 , a, b.


Proof. By virtue o f Theorem 2.7 the Korn inequality (2.19) holds for any

v E ~l (w(a, b)) n ~ ' ( 3 ( a , b), i.) ( i .e u = 0 on 9.). Moreover, it follows

from Proposition 3 of Theorem 1.2 and the Korn inequality, that

Taking into account the inequalities (2.9), (6.14) and following the proof

of Theorem 3.8, we establish the existence o f the solution of problem (6.11)

and the validity o f the estimate (6.13).

6.3. Elasticity Problems with Periodic Boundary Conditions in a

Perforated Layer

In this section Re denotes the perforated layer

where w is an unbounded domain with a 1-periodic structure, w satisfies the

Condition B of 54, d = const 2 1 is a parameter, C-' is a positive integer.

Set

In Re consider the following boundary value problem:

The coefficient matrices of operator L are assumed t o be o f class E(n1, nz),

their elements are functions 1-periodic in 2, fj, iP1, iP2 are also 1-periodic in

$6. Periodic solutions of boundary value problems

We define a weak solution o f problem (6.15) as a vector valued function

u E ~ ' ( f l ' ) such that u = (P1 on ro, u = (P2 on rd and u satisfies the integral

identity

for any v E H1(nC), v = o on ro u rd (i.e. v E H1(nc) n ~ l ( f i ~ , F O u Fd)).

Theorem 6.5.

There exists a weak solution u(x ) o f problem (6.15) which is unique. Moreover,

u(x) satisfies the inequality


This theorem can be proved similarly t o Theorems 3.8 and 6.4 by virtue of

the following

Lemma 6.6.

Every vector valued function v E H1(ne) vanishing on r o U r d satisfies the

inequalities

where the constants C1 and C2 do not depend on E , d, v.

Proof. This lemma is established by the same argument as Theorems 4.2, 4.3

and is also based on the construction o f suitable extensions o f vector valued

functions defined in RE.

Let v E H'(s~'), v = 0 on ro U rd. We extend v t o ~w as follows


Set

By analogy with the proof o f Theorem 4.2 we can extend 6 to the entire layer

B as a function p v E H'(B) such that I?v = 0 for rc, = -1, z, = d + 1,

and

It is shown below that

Inequalities (6.19)-(6.21) imply (6.17), (6.18).

To complete the proof o f Lemma 6.6 let us outline the method t o obtain

(6.20), (6.21).

Obviously (6.20) is a kind o f Friedrichs' inequality, which holds since

for any w E C ~ B ) such that w(i, -1) = 0.

The estimate (6.21) is similar to the First Korn Inequality. It can be proved

in the same way as (2.2) in Theorem 2.1. To this end we approximate P v by a

sequence o f smooth vector valued functions wm, which are 1-periodic in i and

vanish in a neighbourhood of the hyperplanes x, = -1, x, = d + 1. Then,

similarly t o the proof of Theorem 2.1, we integrate by parts over B taking into

account the 1-periodicity of wm in i.

§ 7. Saint - Venant 's principle for periodic solutions

57. Saint-Venant's Principle for Periodic Solutions o f the Elasticity

System

Initially formulated in 1851 Saint-Venant's Principle has ever since been

widely used t o study various theoretical as well as practical problems in me-

chanics. The mathematical expression o f Saint-Venant's Principle, its applica-

bility and formal justification were and still are the subject o f intensive research

(see e.g. [94], [133], [126], [37], [153]).

Roughly speaking St. Venant's Principle asserts that if the forces statically

equivalent t o zero are applied to a part V of the body R, then the energy

contained in a subdomain V' of R is small, provided that the distance between

V' and V is sufficiently large.

Fig. 10.

In the case of an elastic cylinder St. Venant's Principle implies that if the

applied forces are nonvanishing only on an end-face of the cylinder and the

mean values of these forces and of their moments are equal t o zero, then

the solution of the corresponding boundary value problem has the form of a

boundary layer near the end-face.

In this book the asymptotic properties of solutions o f the elasticity system,

which are closely related to Saint-Venant's Principle, will be used t o construct

boundary layers for the asymptotic expansions of solutions o f the elasticity

system with rapidly oscillating periodic coefficients.

7.1. Generalized Momenta and Their Properties

In this section w is an unbounded domain with a 1-periodic structure sat-

isfying the Condition B of $4.

We introduce the following notation


S ( a , b) = (dw) fl { x : a < x , < b} ,

The coefficient matrices A h k ( x ) of operator C are assumed t o belong t o

class E ( n l , n2 ) , n l , nz = const > 0 , and their elements a;hjk(x) are functions

1-periodic in P = ( z l , ..., x,-1).

A vector valued function u ( x ) is called a 1-periodic in i solution of the

system


if u E ~ l ( w ( t ~ , t ~ ) ) and for any u E ~ ' ( w ( t , , t ~ ) ) such that v = 0 on

r,, U rt, the following integral identity holds:

au av A" - , -)dx = ! ( a x , a x ,

oi(t1,tz)

A vector valued function u ( x ) is called a weak 1-periodic in 2 solution of

system (7.2) in w ( 0 , m ) with the boundary conditions (7.3) on S ( O , m ) , if

u ( x ) is a weak 1-periodic in P solution of (7.2) with the boundary conditions

(7.3) for every t l , t2 such that 0 5 tl < t z < m.

It is assumed that fl E ~ ~ ( L j ( t ~ , t ~ ) ) , j = 0 , .., n, 0 5 t l < t 2 < m , and

the vector valued functions f j are 1-periodic in P. Note, that if t 2 = rn then

the functions fj may not belong t o L2 ( L j ( t 1 , m)) .

5 7. Saint- Venant 's principle for periodic solutions 69

For a weak 1-periodic in 2 solution u(x ) of system (7.2) in w(tl , t2) with

the boundary conditions (7.3) on S ( t l , t 2 ) we introduce the vectors P(t , u ) , which are called generalized momenta, setting

Existence of P(t , u ) follows from

Lemma 7.1.

Suppose that the vector valued function f" is such that

and u(x ) is a weak 1-periodic in f solution o f system (7.2) in w(t l , t z ) with

the boundary conditions (7.3) on S ( t 1 , t ~ ) . Then the generalized momenta

P(t , u ) satisfy the following conditions

au P( t , U ) = lim s-' Ank - dx =

s - + + ~ J axk &(t , t+s)

P(tl', u ) - P(tl, u) = / f 0 dx = J f" d i + / f n d i , (7.8)

3(t1,t11) r,, rill

where tl < t' < t" < t2.

Proof. If the coefficients of system (7.2), the functions f j , j = 0, ..., n, and

u(x ) are sufficiently smooth, the relations (7.7) are obvious, and integration

by parts directly results in (7.8). Consider now a weak solution u(x) . Let e l , ..., en be the standard basis o f Rn. Take v = t9(xn)er in the integral

identity (7.4), where 29(xn) is a continuous scalar function such that 6 ( t ) = 1,


29(xn) = 0 for t l < xn < t - hl and for t + h2 < xn < t 2 , d ( x n ) is linear

on each of the segments [t - h l , t ] , [ t , t + h2] , h l , h2 being sufficiently small

positive constants. Then due t o (7.4) we have

= ( f O , v ) d x - h;' J ( f " , e r ) d x + h;' J ( f n , e r ) d x , 3 ( t - h 1 ,t+hz) &(t-hl , t ) G ( t , t t h z )

It follows due to (7.6) that the first and the second integrals in the left-hand

side of this equality have finite limits as hl + $0 or h2 + +O respectively.

Making hl tend to zero in (7.9) and then making h2 tend to zero, we obtain

(7.7).

Let us prove (7.8). Set v = dl(xn)er in the integral identity (7.4), where 29'

is a continuous function such that d l ( t l ) = dl(t l l) = 0, 29 = 1 on (t l+h, t"-h),

d(x,) is linear on [t', t' + h] and on [t" - h , t ] , h > 0 is sufficiently small. I t

thus follows from (7.4) that

This relation together with (7.6) yields (7.8). Lemma 7.1 is proved.

If the functions f j , j = 0 , ..., n, and u as well as the elements o f matrices

Ahk are sufficiently smooth, it is easy t o see that

In the rest of Chapter I it is assumed that for systems o f type (7.2) condi-

tions (7.6) are always satisfied for every t E ( t l , t 2 ) .

$7. Saint-Venant 's principle for periodic solutions

7.2. Saint- Venant 's Principle for Homogeneous Boundary Value

Problems

O f primary importance in Continuum Mechanics is Saint-Venant's Principle

for bodies of cylindrical type with the conditions u(u) = 0 on the lateral part

of the boundary. The details concerning this case can be found in [94].

In applications t o the theory of homogenization it is necessary t o have

estimates which express Saint-Venant's Principle for various boundary value

problems with periodic boundary conditions.

Theorem 7.2 (Saint-Venant's Principle).

Let s, h be integers such that s > h > 0, and let u(x) be a weak 1-periodic

in 2 solution of the system


Let P ( s + 1,u) = 0. Then

where A is a positive constant independent o f u, s h; A depends only on 2 - hk bui duj h ( 0 , l ) and the coefficients of (7.10); I&(u)l - a . . - - .

" dxh dxk

Proof. S e t g = 3 ( s - h , s + l + h ) , g l =3 (s -h , s ) ,g2 = L ( s + l , s + l + h ) .

Let {urn) be a sequence of vector valued functions in ~ ' ( u ( 0 , m)) 1-periodic

in 2 and such that urn -+ u in H1(g) as m m. We define the scalar

function a(.,) setting @(xn) = exp [A(x. - (s - h))] for x, t [s - h, s],

@(xn) = exp(Ah) for x, E [s, s + 11, @(xn) = exp[A(s + 1 + h - x,)] for

x, E [s + 1,s + 1 + h], where A is a positive constant t o be chosen later.

Taking v = ( a - l)um in the integral identity for u(x) in g, we obtain


t=O Let us fix t and choose a constant vector C which satisfies the condition

Then by virtue of the PoincarC inequality (2.3) in R = w:, the Second

Korn inequality (2.19) and (3.13) we get

where Mo is a constant independent o f t and rn.

Taking into consideration (7.14) and the fact that P ( x n , u ) = 0 for x, E

(S - h , s ) , 1F1 = A@ for xn E ( s - h, s). exp(At) 5 l ( x n ) < exp [A( t + I ) ] 5 n

for x E w:, we obtain

112 112

- < G M ~ A ~ ~ ( ~ + ~ ) (I q u ) I 2 d x ) ( m X ) 5

< &M0AeA JE(u)12 l dx + om , - (7.15)

w : where C2 is a constant independent o f s , t , h and E , -+ 0 as m + m. We

deduce from (7.15) that

$7. Saint-Venant's principle for periodic solutions

A similar inequality holds for g2, and can be proved in the same way as (7.16).

Making m tend to infinity we find from (7.16) and (7.13) that

J jE(u)12(@ - 1)dx 5 CM0AeA IE(u)12 @ dx . 9 g1 ug2

Estimate (7.12) follows from this inequality if we choose the constant A

such as to satisfy the condition CMoAeA = 1.


Another version of Saint-Venant's Principle is given by

Theorem 7.3.

Let w(x) be a weak 1-periodic in ? solution o f the system

L ( w ) = 0 in w(0, k + N) , where k > 0, N > 0 are integers, and

Let P(t ,w) = 0 for t E (0, k + N). Then

J IE(w)12dx 5 e-AN J IE(w)12dx , W,k) b(O,k+N)

where A is the constant from Theorem 7.2.

7.3. Saint- Venant 's Principle for Non-Homogeneous Boundary Value

Problems

Consider ( n - 1)-dimensional open sets g j C rj, j = 0,1,2, ... , such that

g j # Q,g, = g o + ( O ,..., O , j ) , g j + r = g j for all z = (21, ..., 2,-,,0) E En.

Existence of such g, is guaranteed by the Condition B of 54 on the domain w.

Set


Let us first prove some auxiliary results.

Lemma 7.4.

Let cp E L2(&), $ E L2(4N) and

for some integer N > 0. Then there exists a weak 1-periodic in d solution of

the problem

where v = (4 , ..., v,) is the unit outward normal t o aw(0, N ) . Moreover,

U(x) satisfies the inequality

where C is a constant independent o f N, and

Qm = (mes Go)-' (&(b, fOdx - J $ 2 )

B N

m = 1 , ..., N-1, $ o = p , $ J ~ = - - $ . (7.21)

Proof. Existence of the solution U(x) of problem (7.19) follows directly from

Theorem 6.3, since (6.8) holds with a = 0, b = N due t o (7.18).

Let us prove (7.20). Setting v = U in the integral identity (6.7), we obtain

§ 7. Saint- Venant 's principle for periodic solutions 75

Denote by Vm, m = 1, ..., N , weak 1-periodic in i solutions of the following

boundary value problems

where do, ..., d N are vector valued functions defined by (7.21), (ul , ..., u,)

is the unit outward normal t o d w ( m - 1 , m ) . Let us check the solvability

conditions of type (6.8) for problems (7.23).

For m = 1 using (7.18), (7.21), we find

J d o d i - J $ l d i = / p d i - J f O d x + J d d i =

Bo 8 1 Bo G(1,N) BN

= / p d i + J $ d i - / l p d l - / d d i + J f O d x =

Po BN Po BN 4 0 , l )

= J f O d x .

G ( 0 , l )

For m = N it follows from (7.21) that


I f m = 2, ..., N - 1, relations (7.21) yield

= J f O d x . G(m-1,m)

Thus the solvability conditions hold for problems (7.23), and therefore

according to Theorem 6.3 the solutions Vm, m = 1, ..., N , exist and satisfy

the inequalities

where C is a constant independent o f m, N . I t follows from the integral identity for Vm that

+ J ($,, u)d? - / ($m-1, u)d? . (7.25) Brn gm-1

Summing up these equalities with respect to m from 1 to N , we find

5 7. Saint-Venant 's principle for periodic solutions

Comparing this relation with (7.22) we conclude that

This inequality together with (7.24) yields (7.20). Lemma 7.4 is proved.

By the same argument we establish

Lemma 7.5. Let U(x) be a weak 1-periodic in P solution of the problem

Then U(x) satisfies the following inequality

where C is a constant independent of N , and

$ J ~ = (mes ijo)-' - (m,N) l J

m = 0 , 1 , ..., N - 1 , $ J ~ = - I I , .

78 I. Some mathematical problems of the theory of elasticj ty

M. Consider a vector valued function w such that w E H' (w(0, N ) ) ,

w = on Yo, w = 0 in w(1/2, N ) . It follows from the integral identity for U that

= / I(?, a(u - " ) ) - (lo, U - w ) ] dx t axi

3 ( O , N )

Denote by Vm 1-periodic in i weak solutions o f problems (7.23) with $m

given by the formulas (7.28). The solvability o f these problems is established

similarly to the solvability o f the corresponding problems in the proof of Lemma

7.4. The functions Vm satisfy the inequalities (7.24), where 40,$1, ..., $N are

defined by (7.28). The integral identity for Vm implies

+ J m u - - J (+m-l, u - w)d? . Bm Sm-1

Summing up these equalities with respect t o m from 1 to N , and taking

into consideration the fact that U - w = 0 on go , we obtain

From this relation and (7.29) we conclude that

§ 7. Saint- Venant 's principle for periodic solutions

This inequality and (7.24) imply (7.27). Lemma 7.5 is proved.

Lemma 7.6.

Let u E H' (w(0, N)), u = 0 on ro. Then

where Mo is a constant independent of N and u.

Proof. Consider a vector valued function w which is a weak 1-periodic in 2 solution of the problem

L(w) = u in w(0, N) ,

u(w) = -U o n gp, ,

u(w)=O o n dw(O,N)\(roUgN), w = O on F o .

1 By virtue o f Lemma 7.5 w satisfies the inequality

J


Setting v = u in the integral identity for w we obtain

This inequality and (7.32) yield (7.30). Lemma 7.6 is proved.

For some applications it is important t o have an extension of Theorem 7.2

to a more general situation, namely, to the case o f non-zero boundary condi-

tions, external forces and generalized momenta. Saint-Venant's Principle for

solutions of a non-homogeneous system of elasticity is expressed by

Theorem 7.7 (Generalized Saint-Venant's Principle).

Let u ( x ) be a weak 1-periodic in 2 solution of the system

a f i C ( u ) = p + - in w ( t l , t 2 )

ax;


where t 2 > t l + 2, t l , t 2 are positive integers, and for any t E ( t l , t 2 ) let

conditions (7.6) be satisfied.

Then for any integer s, h > 0 such that s - h > t l , s + 1 + h < t 2 the

inequality

5 7. Saint- Venant 's principle for periodic solutions 81

holds for u(x ) . Here C is a constant independent o f s , h; A is the constant

from Theorem 7.2.

Proof. Consider a vector valued function U ( x ) which is a 1-periodic in 3

solution of the problem

o(U) = vif' on dw(s - h,s + h + l)\(gs-h U gs+h+l) , J where c p , $ are constant vectors chosen in such a way that

P(s - h, U ) = P(s - h, U ) ,

P ( s + h + l , u ) = P ( s + h + l , U ) .

We have

P ( s - h , U ) = - o(U)di.= - J y d i + J f n d 3 . J r a - h i s - h r r - h

Now we can find 1C, and cp from (7.37):

Let us show that the solvability conditions for problem (7.36) with the

above chosen $, cp are satisfied. Indeed by virtue o f (7.8) we obtain


- r d i + J f " d i = J f O d x .

r a + h + l r a - h &(a-h,s+h+l)

Therefore according to Lemma 7.4 a solution of problem (7.36) exists and


/ Ie(U)12dx 5 G(s-h,a+h+l)

where

Since

it follows from (7.37) that

$m = (mes i0)-' f O d x - P ( s - h ,u ) - G(8-h+m,s+h+l)

Therefore

§ 7. Saint- Venant 's principle for periodic solutions 83

I t is easy to see that u - U is a weak 1-periodic in 2 solution of system

(7.10) with the boundary conditions (7.11). Moreover, P ( u - U,s - h ) = 0.

Then by Theorem 7.2 we have for u - U :

J I B ( U - u)12dx 5 e - ~ ~ J J E ( U - u)12dx . G(s,s+l) G(s-h,s+l+h)

This inequality and (3.13) imply

Estimate (7.35) follows from this one and (7.39), (7.40). Theorem 7.7 is

proved.


$8. Estimates and Existence Theorems for Solutions o f the Elasticity

System in Unbounded Domains

In this section we use the notation of $7.

8.1. Theorems of Phragmen-Lindelof 's Type3

The classical Phragmen-Lindelof's theorem for the Laplace equation has

been the subject o f various generalizations for elliptic equations and systems

(see the review [49]).

The next theorem is closely related to the generalized Saint-Venant Prin-

ciple (see Theorem 7.7) and can be considered as a theorem of Phragmen-

Lindelof's type.

Theorem 8.1.

Let the vector valued functions f j , j = 0, ..., n, satisfy the inequalities

where cl, al are positive constants; and let u ( x ) be a weak 1- periodic in 2

solution of the system

such that

P(O, U ) = - / f " d x + J f n d 2 ,

4 0 , ~ ) f a

3Theorems of Phragmen-Lindelof's type describe the behaviour of solutions of elliptic boundary value problems in unbounded domains. There are many results of this kind. Of particular interest here are theorems which give sufficient conditions for the decay at infinity of solutions belonging to classes of functions whose growth at infinity is not too rapid.

$8. Estimates and existence theorems 85

where c is a constant independent o f s , 60 = const, 0 < 60 5 A, A is

the constant from Theorem 7.2. Then there exist constants c2 , c3, a2 , a3

independent o f s and a constant vector w, such that

Proof. By virtue o f the formulas (7.8), (8.3) we have

- - - f O d x + J r d ? .

G(8,oo) P.

Therefore, taking into account inequalities (8.1), we get

Setting h = [s /2] in (7.35) and using (8.4), (8.7), (8.1) we establish the

inequality (8.5).

Let us prove estimate (8.6). For every s = 0,1 ,2 , ... set

w, = ( m e s ~ ( 0 , I ) ) - ' / u(x )dx . (8.8)

G ( s , s + l )

In the domain w(s , s + 2 ) consider a weak 1-periodic in i solution o f the

problem

where X , is the characteristic function of the set g. It follows from (8.8) and

the integral identity for the solution of problem (8.9) that


By virtue of Theorem 6.3 we have \ \ E ( V ) \ \ ~ ~ . 5 C, where C is (w(s,9+2))

a constant independent o f s. Therefore due t o the inequalities (8.5) proved

above we find

Jw, - w , + ~ J 5 cexp(-aos) , a0 = const > 0 .

Therefore, there is a vector w, = lim w,. Moreover, a - w

where the constants K1, do not depend on s , t. Making t tend to infinity

in this inequality we obtain

In order t o prove the estimate (8.6) we apply the Korn inequality (2.19) in

L ( s , s + 1). We have

where is a constant independent of s . Now we obtain estimate (8.6) from

this one and (8.5). Theorem 8.1 is proved.

$8. Estimates and existence theorems 87

Remark 8.2.

Suppose that under the assumptions of Theorem 8.1 we have f" = 0, i =

1 , ..., n. I f f0 and the coefFicients of system (8.2) are sufficiently smooth for

large x , it follows from the a prion' estimates for solutions o f elliptic systems

(see [I], [17]) that for large s we have

Moreover, Theorem 8.1 and the imbedding theorem (see [117]) imply for

m > n/2 - 2 the inequality

max lu - wmI 5 c [ ~ X P ( - Q S ) + l l f O l l H m (G(s - l , s+2) )

J(s,s+l) I

holds with constants C , a3 independent of s.

8.2. Existence of Solutions in Unbounded Domains

In this section we consider existence o f solutions for the following boundary

value problem

a? L ( u ) = f 0 + - in w(0, co) ,

8x1 u = @ on T o , o(u)= v i f i on S ( O , c o ) , I (8.11)

u is 1-periodic in 2 .

Solutions of similar problems are used in Chapter II for the construction of

boundary layers in the homogenization theory.

It is assumed in (8.11) that @ E ~ 1 / 2 ( i ' ~ ) is 1-periodic in 2, fj are 1-

periodic in 2 and belong t o ~ ~ ( h ( t , , t , ) ) for any t l . t 2 such that 0 5 t1 < t 2 < c o , j = 0 , 1 ,..., 72.

We say that u ( x ) is a weak solutiaon of problem (8.11) i f u = @ on rO, u ( x ) belongs t o 8 l ( u ( t , , t 2 ) ) for any t l , t 2 such that 0 5 t1 < t2 < m, and

u ( x ) satisfies the integral identity (7.4).


Estimates o f Saint-Venant's type (see Theorems 7.2, 7.3, 7.4) make it

possible t o prove existence and uniqueness of solutions for problem (8.11) in

classes o f functions growing at infinity.

Theorem 8.3.

Suppose that

where M,6 = const, A is the constant from Theorem 7.2, 0 < 6 5 A. Then

for any constant vector q = ( q l , ..., q,) there is a unique weak solution u ( x ) of

problem (8.11) such that P ( 0 , u) = q and the following estimate is satisfied

r

where C is a constant independent o f k ; 61 is an arbitrary constant from the

interval ( 0 , 6 ) .

Proof. Denote by v N a weak 1-periodic in 2 solution o f the problem (7.26)

with

It is easy t o see that

Indeed, due t o (7.8) we have

Therefore taking into account (8.14) and the formula

58. Estimates and existence theorems

p ( N , v N ) = / (1di.+ / f ' d i , BN PN

we find that

Since 11, is given by (8.14), therefore the functions $, in (7.28) are

11, , .=(rne~i j~) - ' ( -~- &(o,m) / f O d i + / f n d i ) . (8.11)

Po I t thus follows from (7.27) and Lemma 7.5 that

where C is a constant independent o f N .

The function vk+N+l - v ~ + ~ satisfies all the conditions o f Theorem 7.3,

since from (8.15) we have

P(0, Vk+N+' - V k + N ) = 0 .

Therefore from (7.17) we get

Taking into account (8.17) we conclude from the above inequality that

90 I. Some mathematical problems of the theory of elastjcity

Let us estimate the last sum in the right-hand side o f (8.18). W e have

where 61 is any constant from the interval (0,6), C depends on b1 and does

not depend on N , k. To obtain the last inequality we also used the conditions (8.12) and the fact that mesG(0, m ) 5 c l m .

Thus (8.12), (8.18), (8.19) yield

Therefore

$8. Estimates and existence theorems

where Mz is a constant independent of k and N I t follows from (8.20) that

Therefore

where is any constant from (0,6), the constants M, do not depend on k, s, t. Inequality (8.21) implies that

Note that vkt" vktstt = 0 on rO. Therefore applying Theorem 2.7 to

vkts - vkt'+t we deduce that


where the constant C1 depends on k but does not depend on s , t.

It follows from (8.22), (8.23) that for any k the sequence vS converges

in B1(w(O, k)) as s + m to a vector valued function u. Making s tend t o

infinity in the integral identity for v h e see that u is a solution of problem

(8.11).

Making t tend t o infinity in (8.21) for s = 0 and using (8.17) for N = k, we obtain

2

+ \ l @ ( l ~ l l l ( i . o ) , 1 where M8 is a constant independent of k. This proves inequality (8.13).

To complete the proof o f Theorem 8.3 we need t o show that P(0, u ) = q.

According t o (7.8) we have for s 5 m:

P(s , vm) -P(O,vm)= / f o d x + / f n d i - 1 f n d i . ~ ( 0 , s ) f , f o

Integrating both sides of this equality from 0 t o t we get

Passing here t o the limit as m 4 m we obtain the above equality with vm

replaced by u. Passing t o the limit as t + +O in the equality for u we find that

58. Estimates and existence theorems 93

P(0, u ) = q. The uniqueness of u ( x ) follows from Theorem 7.3. Theorem 8.3

is proved.

8.3. Solutions Stabilizing to a Constant Vector at Infinity

Existence of solutions for problem (8.11) and their estimates in the case of

external forces, which rapidly decay at infinity, are established by

Theorem 8.4.

Suppose that inequalities (8.1) are satisfied. Then there exists a unique solu-

tion o f problem (8.11), such that

Moreover there is a constant vector C , such that

where M I , M2, a0 are positive constants independent of s.

Proof. I t is obvious that conditions (8.1) imply (8.12). Set

q = J f n d i - J f O d z (8.26) Po G(O,w)

in Theorem 8.3. Let u ( x ) be the solution of problem (8.11) whose existence

is asserted by Theorem 8.3 with P(0, u ) = q.

Our aim is t o show that u ( x ) satisfies inequalities (8.24), (8.25).

We first check that estimates (8.4) hold for u ( x ) . Indeed, for f j , j =

0 , ..., n, inequalities (8.12) are valid with 6 = A. Therefore inequalities (8.13)

hold for u ( x ) with 61 = 3A/4 < 6. I t follows from (8.13) that estimates

(8.4) hold with 60 = A/4. Thus we can use Theorem 8.1, which implies the estimates (8.5), (8.6).

By virtue o f the Korn inequality o f type (2.3) in G(s , s + I ) , we have


Therefore (8.5), (8.6) yield (8.25).

Let us prove estimate (8.24). Consider the vector valued functions vN

constructed in the proof o f Theorem 8.3. These functions satisfy inequalities

(8.17). Taking as q in (8.17) the vector given by (8.26), and passing to the

limit as N + m we get estimate (8.24). Here we also used estimates (8.1)

and the convergence of vN to u in H' ( ~ ( 0 , k ) ) as N -+ m for any fixed k .

This convergence was established in the proof of Theorem 8.3. Theorem 8.4

is proved.

where e l , ..., en is the standard basis of Rn. According to Theorem 8.3 v' can be chosen so as t o satisfy the inequalities

For the vector C, in (8.25) we can obtain an explicit formula which ex-

presses C, in terms of fj, j = 0 , ..., n , and the boundary values o f u ( x ) on

ro . To this end we shall need some auxiliary functions v', r = 1, ..., n , whose

existence is guaranteed by Theorem 8.3.

By v', r = 1 , ..., n , we denote weak solutions o f the following boundary

value problems

By Lemma 7.6 we also have

L ( v r ) = 0 in w(0, m) , \

V' = 0 on I'o , a ( v P ) = 0 on S ( 0 , m ) ,

P(0 , v r ) = -er ,

v' is 1-periodic in ? , J

>

58. Estimates and existence theorems 95

Theorem 8.5.

Suppose that all conditions o f Theorem 8.4 are satisfied. Then the constant

vector C , = ( c k , ..., c&) in (8.25) is given by the formulas

dvr CL = / [ ( f O . v ' ) - (fi, g ) ] d ~ xi +

~ ( O , O O )

where V' are the solutions of problems (8.27) satisfying the inequalities (8.28),

(8.29).

Note that if v' and the coefficients of the operator L: are sufficiently

smooth, then the integral

is defined in an obvious way. Let us give a meaning t o this integral when v r ,

r = 1, ..., n , are weak solutions of problem (8.27).

It is easy to see that

for smooth v' and any scalar function qjs E C 1 ( u ( O , m ) ) such that qj6 is

1-periodic in 5, $6 = 1 in w(0,6) , $6 = 0 in w ( 2 S , m ) , 6 = const > 0. I t

follows from the integral identity for v' that the integral on the right-hand

side o f (8.31) does not depend on $6 and 6. That is why we can consider this

integral as / (u(vT) ,u)di : in the case of weak solutions v'.

P o

Proof of Theorem 8.5. Fix an integer s > 1 and consider a scalar function

cp(x,) E C O ( R 1 ) such that cp(x,) = 1 for x, E (O,s), cp(x,) is linear for

x, E [s,s + 11, cp(x,) = 0 for x, E [s + 1, m ) . Set v = cpv' in the integral

identity (7.4) for u with t l = 0, t2 = s + 1. Taking into account (8.31) and

the integral identity for v' we find


dyvT du dyvT J [(fO, uvr ) - (f', dx = - J (A" - , -)dx = G(O,s+l)

dxk dxh G(s,s+l)

It is easy t o see that the second and the fourth integrals in the right-hand

side of the last equality are bounded by

and therefore these integrals decay exponentially as s + co due t o (8.25),

(8.29), (8.28).

Consider the third integral in the right-hand side of (8.32). Using the definition of P(t ,vT) and the fact that P( t , vT) = P(O,vT) = -eT, we obtain

Therefore, if we make s tend t o infinity, the formula (8.32) reduces to (8.30).


$8. Estimates and existence theorems 9 7

Remark 8.6. Theorem 8.5 implies that under the assumptions of Theorem 8.4 conditions of

decay for a solution of problem (8.11) read

where are the boundary values of u(x) on ro; r = 1, ..., n.


$9. Strong G-Convergence o f Elasticity Operators

Homogenization of differential operators considered in the next chapter is

closely associated with the notion o f strong G-convergence. The theory o f G- convergence and strong G-convergence was developed by many authors (see

[22]-[24] and the review [148]). The initial works on the subject date back t o

the 60's and belong to S. Spagnolo, ([118], (1191).

9.1. Necessary and Suficient Conditions for the Strong G-Convergence

Consider a sequence of the elasticity operators

where a E ( 0 , l ) is a small parameter; A';j(x), i, j = 1, ..., n , is a family of

matrices of class E(rc1, r c 2 ) ; rcl, n2 are positive constants independent of a ; R is a bounded Lipschitz domain o f IRn.

We also consider another elasticity operator

of class E ( i l , i2), where 21, i2 are positive constants which may differ from

K 1 , K 2 .

A sequence o f operators {L, ) is called strongly G-convergent t o operator

2 as a + 0 ( L , a k ) , if for any f E H-' (0) the sequence uc E H i ( R )

of solutions of the problems

converges to u0 E Hi(R) weakly in H,'(R) as a -+ 0, where u0 is the solution

of the problem

moreover,

. . auc .. . . auo 7i(x) A:J - t 9 i (2 ) E AZ3 - weakly in L2(R)

axj axj

59. Strong G-convergence of elasticity operators

as E -+ 0, i = 1,2, ..., n (see [148]).

Remark 9.1.

In the above definition o f the strong G-convergence it is sufficient to require

that

uc -+ u0 and y: -+ ji as E -+ 0

for any f belonging t o a subspace V c H - ' ( 0 ) dense in H - ' ( 0 ) . Indeed, let

us show that in such a case uc -+ u0 and yf -+ ji for any f E H - ' ( 0 ) .

Consider a sequence f m E V, such that f m -+ f in the norm of H - ' ( 0 )

as m -+ m. Denote by u',, 6, solutions o f the following problems

. - a v , Let us introduce matrices r c ( v ) and r ( v ) whose columns are A:3 -, 2 = ax

J

- . . av . 1, ..., n , and AZ3 -, z = 1, ..., n , respectively. Then for any vector valued axj function v E HA(S2) and any matrix valued function w E L2(S2) we have

It is easy t o see that the right-hand sides of these equalities converge t o zero

as E -+ 0, since by Theorem 3.3 and Remark 3.4 (see (3.25))

with a constant C independent of e , m, and


as E + 0 for a fixed m due t o the definition o f strong G-convergence with

f = f r n E V . The matrices r C ( u c ) , f'(uO) with columns yf, +', i = 1 , ..., n , are some-

times called weak gradients.

Of great importance for the theory of strong G-convergence is the following

Condition N (see [148]).

We say that a sequence o f the elasticity operators {LC) satisfies the Con- dition N, if there exist matrices * j ( x ) , i , j = 1, ..., n , and matrices N,"(x) E

H 1 ( R ) , s = 1, ..., n , such that for E + 0 we have

N1. N,"+O weaklyin H 1 ( R ) , s = 1 , ..., n ;

- . . a ~ , j Ai3 f A! - + A: + Ai j (x ) weakly in L 2 ( R ) ,

8x1

N3. a - (A: - Aij) -+ 0 in the norm of H - ' ( 0 ) , axi

Note that in the Condition N, the family o f matrices k j ( x ) i, j = 1, ..., n ,

is not assumed t o define the coefficients o f an elasticity system, i.e. relations

of type (3.2), (3.3) are not imposed on k j ( x ) . Obviously it only follows from

the Condition N that the elements of the matrices A i j ( x ) belong t o L 2 ( R ) . However, as it is shown below (see Theorem 9.1), the Condition N actually

implies relations (3.2), (3.3) for Aij, and therefore their elements are bounded

measurable functions.

Theorem 9.1.

Suppose that the Condition N holds for the sequence of operators {LC) of

class E ( n l , n 2 ) and n l , n2 are positive constants independent of E . Then for

any cp E C r ( R ) we have

59. Strong G-convergence of elasticity operators 101

where the matrix A* is the transpose o f A; E is the unit matrix with elements

6ij, bpk is the Kronecker symbol.

Moreover, the family of matrices A ~ P , q , p = 1, ..., n , belongs to the class

E(IE~, I E ~ ) and therefore defines a system of linear elasticity.

Proof. Let us first establish formula (9.4). Denote by J,4p the integral in the

right-hand side of (9.4). Then

where J;, ..., Ji successively stand for the integrals on the left-hand side of

the last equality. Let us estimate these integrals.

Taking into consideration the fact that a weakly convergent sequence in a

Hilbert space is bounded and that the imbedding H1(R) c L2(R) is a compact

one, we deduce from the Condition N1 that


N,S -+ 0 strongly in L2(R) , aNi - -+ 0 weakly in L ~ ( R ) , dx j

as E -+ 0, s, j = 1 ,..., n , where C = const and does not depend on E . It is

easy t o see that

Therefore Jf -+ 0 as E -t 0 by virtue of (9.7) and the Condition N3.

Using the Holder inequality and the fact that the elements of matrices A: are

bounded uniformly in E , we conclude that

Thus J i -+ 0 as E -+ 0 due t o (9.6), (9.7). From (9.6) we get Jg -+ 0 as

E -+ 0, and the Condition N2 implies that J,' converges t o the left-hand side

of (9.4) as E + 0. Thus formula (9.4) is proved.

Now let us show that the family of matrices Apq, p, q = 1, ..., n , belongs t o

the class E(rcl, K ~ ) , i.e. that their elements iif,P(x) satisfy the relations (3.2),

(3.3).

The equality 6fX = i i i follows directly from the Condition N2 and relations

(3.2) for the elements of Azq.

In order to prove that iifX = iijh(; let us note that these relations are equiv-

alent to A P ~ = (29~) ' . The last equality follows from (9.4) and the equality

A: = (A:')' which holds due t o (3.2) for the elements of matrices A:.

Now let us prove the inequalities (3.3) for kj(x). First we obtain the

lower bound. Let {qih) be a symmetric (n x n)-matrix with constant elements.

Denote by gk the column vector whose components are qlk, ...,q,k, and by

gh* the line (gh l , ...,ghn). By virtue of (9.4) we have

$9. Strong G-convergence of elasticity operators

for any cp E Cr(R), cp > 0. Set

It is easy t o see that Ck(c, x) is a column vector with components

where N,Pis are the elements of matrices N:. Denote by J, the integral in (9.9) after the limit sign. Then

JE = 1 cp ai:h(x) C;,(r, 2) hi(€, x)dx n

According to Lemma 3.1 we have

I t is easy t o see that

where 11f: = NA, qsq. Therefore

Let us multiply this equality by cp(x) 2 0 and integrate it over R. Then

due t o (9.11) and the relation

we get


where p, -, 0 as E -+ 0 owing t o (9.6), (9.7).

Since the second and the third integrals in the right-hand side of the last

inequality are non-negative, it follows from (9.10) that

Passing here t o the limit as e --t 0, by virtue of (9.9) we obtain the inequality

Since ,(x) is an arbitrary non-negative function in C,""(R) the last inequal-

ity yields the lower bound in (3.3).

Let us establish the upper bound in (3.3) for the elements of matrices A P ~ .

Fix a symmetric ( n x n)-matrix r] = {v ih) with constant elements. We have

just shown (see (9.4), (9.5)) that for any cp E C,"(fl), cp 2 0, the following

relations are valid

Therefore

I t follows that

$9. Strong G-convergence o f elasticity operators

for a subsequence E' --t 0, since according to Lemma 3.1 we have

Due to the conditions (A?)* = A:j we get

Therefore

Since 779* AZP f 5 n2 q ih q i h by virtue of the Condition N2 we obtain the

inequality T 5 n 2 q i h q i h cpdx, which implies the upper bound in (3.3). J n


Theorem 9.2.

Suppose that Condition N is satisfied for the sequence o f elasticity operators

{LC) of class E ( n l , n Z ) , and K I , n2 are positive constants independent of E .

Then {LC) is strongly G-convergent t o an elasticity operator 2 as E + 0, and

the coefficient matrices $ j ( x ) of belong t o the class E ( n l , n 2 ) .

Proof. We have already established in Theorem 9.1 that the matrices k j ( x )

define a system of elasticity and belong t o the class E ( K , , K ~ ) with n l , n2 the

same as for operators L,. Let us prove the strong G-convergence of L, t o c as e + 0.

By virtue of (3.21) and the representation (3.20) for the elements of

H - ' ( a ) the Condition N3 can be rewritten in the form


FjS + 0 strongly in L2(R) as E + 0, j = 0, ..., n , s = 1, ..., n. Here we have

also used the relations

(see the proof of Theorem 9.1).

Consider the vector valued function cpuc, where cp is an arbitrary scalar

function in C,"(R), and uc is a weak solution of the problem


d,uC = / [F;cpuc - F : ~ -1 ax; dx .

n

By the definition of a weak solution o f problem (9.15) we have

Subtracting (9.16) from (9.17) we get

Theorem 3.3 implies

$9. Strong G-convergence of elasticity operators

. auc where yj = Azk - and C1, Cz are constants independent o f E .

ax k Due to the weak compactness of a ball in a separable Hilbert space and the

compactness o f the imbedding H1(R) c L2(R) , the inequalities (9.19) imply

that there exist vector valued functions U E H,'(R), ?j E L2(R) such that

uc' --t U weakly in H i ( R ) and strongly in L2(R) , I (9.20) Yj jj weakly in L2(R) , j = 1, ..., n ,

for a subsequence E' -t 0.

Note that by virtue o f (9.6), (9.7), (9.13), (9.19) the first integral in

the left-hand side of (9.18) and the integral in the right-hand side of (9.18)

converge to zero as E -t 0. Therefore we deduce from (9.18) that

where p,, -t 0 as E' -+ 0. Since uc' - U -+ 0 strongly in L2(R) as E -+ 0, we

can pass t o the limit in (9.21) as E' -+ 0. Then taking into account (9.14),

(9.20) and the Condition N2 we see that the first integral in the left-hand side

of (9.21) is infinitely small as E' -t 0, and the second integral converges to

Therefore

since cp is an arbitrary function in C,"(R). Let us show that U ( x ) is a weak solution of the problem

By the definition of a weak solution o f problem (9.15) we have


for any matrix M ( x ) E H,'(R). Passing to the limit in this integral identity as E' -+ 0, by virtue o f (9.20),

(9.22) we obtain

Therefore U(x) is indeed a weak solution o f problem (9.23).

The above considerations show that from any sequence (u", $,, ..., y,",) we

can always extract a subsequence such that u"' -+ U weakly in H t ( R ) and

y:,, -+ 7' weakly in L2(R) as E" + 0. Therefore the sequence o f operators

{L,) is strongly G-convergent to 2 as e -+ 0. Theorem 9.2 is proved.

Theorem 9.3 (On the uniqueness of the strong G-limit).

Let L % 2 and L, 2 as E -+ 0, where {L,) is a sequence o f the

elasticity operators o f class E ( K ~ , K ~ ) , K ~ , K~ are positive constants indepen-

dent of E , 2, 2 are elasticity operators with bounded measurable coefficients.

Then the coefficients o f operators 2 and i? coincide almost everywhere in R.

Proof. Let 6 be any vector-valued function wi th components in C,"(R). Set

f = 2 6 and consider a sequence us E H,'(R) of the solutions o f the following

problems

By virtue o f the strong G-convergence o f C, t o J! and we have

uc + weakly in H,'(R) , . . auS * . . 8.;

A:J dz, + A" - weakly in L 2 ( R ) , i = 1 ,..., n , ax . . a u c -. 86

A' a, + A - weakly in L2(R) as e + O , i = 1 ,..., n , axj .... \'3

where At3, A are respectively the coefficient matrices o f the operators 2, i . .. . a.ri : i j 86

Therefore A'j - = A -. almost everywhere in R for any f i E C,"(O). It ax j axj

$9. Strong G-convergence o f elasticity operators 109

:ij

follows that A'j = A almost everywhere in R. Theorem 9.3 is proved.

Theorem 9.4.

Let {C,) be a sequence of elasticity operators belonging t o class E ( K * , K ~ )

with positive constants K , , tc2 independent o f E , and let i be an elasticity

operator. Then C, E as E + 0 , if and only if the Condition N is satisfied

for the coefficient matrices of the operators LC and E . Proof. SufFiciency o f the Condition N for the strong G-convergence o f L, t o

2 is established in Theorem 9.2. Let us prove the necessity. Suppose that G LC ==+ k as E + 0. Consider a sequence of matrices B!, j = 1 , ..., n , such

that Bj are weak solutions of the problems

It is easy t o see that IIBjllH;(n, 5 C with C = const independent of E ,

since the elements of matrices A:(x) are bounded uniformly in E and we can

apply Theorem 3.3. Due to the weak compactness o f a ball in a separable

Hilbert space there is a subsequence E' + 0 such that

B$ + B{ weakly in H 1 ( R ) as E ' + O , j = 1 ,..., n .

Let us define the matrices fi:, as weak solutions of the following boundary

value problems

Set N:, = -B::+ Mil. Since LC is strongly G-convergent t o L?, it follows that

M:,-+B{ weak ly in H t ( R ) , j = l ) ...) n ) 1 dM2: * . d B : (9.25)

A$ - + A:/ - weakly in L2(R) , i) j = 1 , ..., n . 8x1 8 x 1

Therefore the Condition N1 is satisfied for the sequence E' + 0, i.e. N;' + 0

weakly in H 1 ( R ) . a

Since the elements o f the matrices E A: - N! + A?, i, j = 1, ..., n , 8x1

are bounded in the norm of L2(R) uniformly in E , it follows that there is a

subsequence E" + 0 o f the sequence E' + 0 such that


a A:,, - N:, + A:, + A:(X? weakly in L2(R) , (9.26)

8x1

where A: are matrices with elements in L2(R).

Let us consider the Condition N3 for the sequence E" + 0 and the matrices

N!,, :

a - . . . . a a , , a . . - (A:?, - A:J) = - (A;:, -) ax; ax, + - A;?, -

8x1 ax, a . . .. . a . . a * . a & . .

- - A'3 = L B ; - -A:' = - (A" - -A') . (9.27) ax; * ax; 6's; 8x1

The integral identity for problem (9.24) yields

where M is any (n x n)-matrix with elements in HJ(R). Passing t o the limit

in this equality as E" + 0 and using (9.26), we obtain

a - . a ~ i . . Therefore, - ( A - - A ) = 0 and by (9.27) we find that ax, axl a - (e;, - A?) = 0. Thus the Condition N with the matrices A: is satisfied ax: fo r the subsequence E" -+ 0.

It follows from Theorem 9.2 and the uniqueness of the strong G-limit

(Theorem 9.3) that A? = h'j almost everywhere in R.

Let us show that the Condition N holds for the entire sequence E + 0.

Define matrices Nj as weak solutions of the problems

a a ~ , j a - ( A -) = - ( A - A ) , N: E H$) . ax, ax1 ax;

(9.28)

It follows from (9.27) that these relations hold for E = a", ~2 = N:.

Therefore, from any sequence N: defined by (9.28) we can extract a subse-

quence which satisfies the Conditions N1-N3 with matrices A'j(x). Hence the

whole sequence N j satisfies the Condition N. Theorem 9.4 is proved.

$9. Strong G-convergence o f elasticity operators 11 1

Corollarv 9.5.

Let { L C ) be a sequence o f operators of class E ( n l , K ~ ) with n l , nz > 0 inde-

pendent of E , and let L, I? as E + 0. Then the coefFicient matrices of

the operator I? also belong to the class E ( K ~ , ~ 2 ) .

9.2. Estimates for the Rate of Convergence of Solutions of the

Dirichlet Problem for Strongly G-Convergent Operators

It was shown in the previous section that the Condition N guarantees only

weak convergence in HA(R) of solutions u' of problems (9.2) t o a solution of

problem (9.3). However, if one imposes some additional restrictions on the

convergence of the functions in the Condition N, i t becomes possible to obtain

estimates for the difference u0 - uc - v, in the norm of H 1 ( R ) , where v, is

the so-called corrector.

We assume here that the boundary of the domain R and the coefficients

of the G-limit operator are smooth.

To characterize the degree of deviation of the coefficients of L, from those

of I? we introduce the following functional spaces.

Denote by H - m l W ( R ) , (m 2 0 is an integer) the space whose elements

are distributions o f the form

where f , E L W ( R ) . The norm in H-"vW(R) is defined as

where the infimum is taken over all representations of f in the form (9.29).

Lemma 9.6.

Let g = V " g, E H-mlW ( R ) , g, E L m ( R ) , u E H m ( 0 ) . Then one can I4Sm

define an element ug E H - m ( 0 ) by the formula


Moreover

Proof. Let us show that (9.30) correctly defines a continuous linear functional

on H r ( R ) . Indeed, let g = Dag; be another representation of the lalSm

element g E H-"tm(R), g; E L M ( R ) . Then for any 1C, E C F ( R ) the following

identity holds in the sense of distributions

= (-l)Ial 1 g,~1C,dx . (9.32) la l lm n

Since gk, g, have bounded norms in L M ( R ) , the last equality is valid for all II, such that DalC, E L 1 ( n ) , la1 5 m, and in particular for 1C, = up. The inequal-

ity (9.31) follows from (9.30) and the definition o f the norms in H-"vM(R)

and H-"(R). Lemma 9.6 is proved.

We say that a sequence o f the elasticity operators {L,) of class E ( n l , K ~ )

( K ~ , K Z = const > 0) satisfies the Condition N', if there exist matrices a i j ( z ) ,

i, j = 1, ..., n , N,d(x) E H1(R) n L M ( R ) , s = 1 , ... , n , such that

A , .

N'2. aNj Aal = A: -

C - + A: + Aij(x) in the norm o f ax r

N'3. a - (a: - a i j ) + 0 in the norm of H - l v M ( R ) ax;

as E + 0.

It is easy t o check that Condition N' implies Condition N. Therefore the

matrices a i j define an elasticity operator k which also belongs t o the class

E ( K I , KZ).

Let us introduce the following parameters to characterize the rate o f con-

vergence in Conditions N'l-N'3:


,8, = max - kjllH-l,m(n) , a,j=l, ..., n

yc = max j=l, ..., n

Theorem 9.7.

Suppose that the operators t,, E satisfy the Condition N', and the coefficients

i'hjk(x) of the operator k are smooth functions. Then the solutions of problems

(9.2), (9.3) with f E H 1 ( R ) satisfy the inequalities

where the constants K 1 , K2 do not depend on E , v' is the solution of the

Dirichlet Problem

auO Proof. Set 6 = uO+N,d-. Applying the operator LC t o uc--6+vc we obtain

a x , the following equalities wh~ch are understood in the sense o f distributions


According to (9.39) F;, F,E E H - ' ( R ) and

where ye, P, are defined by (9.35), (9.34), the constant c is independent of E .

It is easy t o see that F,' also belongs t o H- ' (R) and

where cl is a constant independent o f E , a, is defined by (9.33).

Since uc - 6 + v' E H,'(R), it follows from (9.39)-(9.42) and Remark 3.4

that

where c2 is a constant independent of E . Since k is an elliptic operator with

smooth coefficients, the well-known a pm'ori estimates for solutions o f elliptic

boundary value problems (see [ I ] , [17]) yield

IIu011~m+2(n) 5 cm I l f ~ I H ~ ( R ) , m = 0,1,2, ... . (9.44)

These inequalities and (9.43) imply (9.36), (9.37). Theorem 9.7 is proved.

Thus i t is evident that in order t o estimate the difference between uc and

uO it suffices t o construct matrices N,J satisfying the Conditions N'l-N'3 and

then estimate a,, PC, r c , IIvcIIHl(C2)* 11vC11~2(~).

Let us give the simplest example in which the Condition N' is satisfied.

$9. Strong G-convergence of elasticity operators 115

Example 9.8. Let A y ( x ) -+ a ' j ( x ) in the norm of L m ( R ) as E 4 0, i, j = 1, ..., n. Set

N,"(x) 0 in 0 , s = 1 , ..., n. Then the Conditions N'l-N'3 are satisfied with

a, = 0, P,,Y, 5 SUP I I A ~ - A i j l l L r n c n ) . Therefore i,j=l, ..., n

C = const.

In fact, according to Theorem 9.7 we should have placed 1 1 f l l H l c n ) in-

stead o f 1 1 f l lLz(n) in the right-hand side of the last inequality. Neverthe-

less in this situation, as one can see from the proof of estimate (9.41), we

have ((F,'((H-i(n) 5 C sup ((A: - A i j l l L m ( n ) I I u O I I ~ Z ( ~ ) . Therefore estimate i , j

(9.45) is valid.

Now we consider a less trivial example, when the Condition N' is satisfied

(see also Chapter 11, $8).

Let the coefficient matrices A?($) of the operators L, have the form A';~(x) = . . x

At3( - ) , i, j = 1, ..., n , and let the elements o f the matrices Ai j ( ( ) be E

smooth functions 1-periodic in (. Operators of this kind in a much more gen-

eral situation will be studied in Chapter II, where another approach is suggested

in relation t o such problems. x

Let us define the matrices N,"(x) setting N,b(x) = E N ' ( - ) , where N s ( ( ) E

are 1-periodic in ( solutions of the system

As it was shown in $6.1, this system possesses a solution in the class o f

smooth functions 1-periodic in 5. Let us define the coefficient matrices Aij for the operator k , which is the

strong G-limit of the sequence {LC) as E -+ 0. Set


where (f) = J f ( ( )d(, Q = {( : 0 < t j < 1, J' = 1 , ..., n}.

Q Let us show that the matrices A?, i i j , N," satisfy the Condition N1. The Condition N'1 holds since N S ( t ) are smooth. Moreover a, < CE,

C = const.

Equations (9.46) show that the Condition N13 is also satisfied. It is easy

to see that y, = 0.

Consider now the Condition N12. Obviously & ( x ) - k j ( x ) r B i j ( 4 ) , E

and B"( ( ) are matrices whose elements are smooth functions 1-periodic in

. Moreover 1 ~ ' ~ ( ( ) d ( = 0, by virtue of (9.47). According t o Lemma 1.8 Q

. . x d B t J ( - ) = E - q i j ( e , x ) , where the elements o f the matrices F ; ' ~ ( E , x ) are

E ax, smooth functions uniformly bounded in E, x . It follows that

.. x 11 B1~(--)IIH < C E . Hence PC 5 C E , C = const.

In order t o obtain an effective estimate for uC-u0 we must have an estimate

for IIvCIIHlcfl). Let cp,(x) be a truncating function such that

It is easy t o see that v" is a solution of the problem

and

a ayc auO ~ N P duo aZu0 - = ~ - N ~ - + c p , - - + ~ c p ~ N p - a x j dxj dx,

. (9.50) dEj dxp d t P d x

dNP Since the elements of the matrices NP, - , p, j = 1, ..., n , are bounded

functions we have atj

where cl is a constant independent of E , Kc is the set of all x E R such

that cp,(x) # 0. It is obvious that Ii', lies in the 2~-neighbourhood of 8 0 .

Therefore by Lemma 1.5 we have IIVUOII~?(~.) c ~ E ~ I ~ ~ ~ U O I I ~ ( ~ ) . Thus


Il*cll~l(n) I c3&'J2 IIuOIIp(n) , ~3 = const .

Applying Theorem 3.3 to the solution of problem (9.49) we get

IIvCIIHl(n) 5 c ~ E " ~ IlfllLP(CI) .

Therefore we can deduce from Theorem 9.7 that

CHAPTER ll

HOMOGENIZATION OF THE SYSTEM OF LINEAR

ELASTICITY.

COMPOSITES AND PERFORATED MATERIALS

This chapter deals with homogenization problems in the mechanics of

strongly non-homogeneous media. Most of the results are obtained for the

system of linear elastostatics with rapidly oscillating periodic coefficients in

domains which may contain small cavities distributed periodically with period

E. In mechanics, domains of this type are referred to as perforated. The

main problem consists in constructing an effective medium, i.e. in defining the

so-called homogenized system with slowly varying coefficients and finding its

solutions which approximate the solutions of the given system describing a

strongly non-homogeneous medium.

In Chapter II we give estimates for the closeness between the displace-

ment vector, the strain and stress tensors, and the energy o f a strongly non-

homogeneous elastic body and the corresponding properties o f the body char-

acterized by the homogenized system under various boundary conditions.

Homogenization problems for partial differential equations were studied by

many authors, (see e.g. [5], [3], [110], [148], [82], [83] and the bibliography

given there as well as at the end of the present book).

$1. The Mixed Problem in a Perforated Domain with the Dirichlet

Boundary Conditions on the Outer Part o f the Boundary

and the Neumann Conditions on the Surface of the Cavities

1.1. Setting of the Problem. Homogenized Equations

Let R' = R n EW be a perforated domain o f type I, defined in $4, Ch. I.

In RE we consider the following boundary value problem

11. Homogenization of the system of linear elasticity

where Ahk(J) are ( n X n)-matrices o f class E ( K ~ , K Z ) , K I , ~2 = const > 0

whose elements aihjk(() are functions 1-periodic in J . It is also assumed that

a: / ( f ) are piece-wise smooth in w and the surfaces across which they or their

derivatives may loose continuity do not intersect dw i.e. the functions a:!

belong t o the class 6' defined in $6.1, Ch. I.

Existence and uniqueness of solutions o f problem (1.1) for f" E L 2 ( n E ) ,

QC E H 1 ( R E ) are guaranteed by Theorem 5.1, Ch. I.

Our aim is t o study the behaviour of a solution u' of problem (1.1) as E + 0

and t o estimate the closeness of uE t o uO, which is a solution o f a boundary

value problem in the domain R for the homogenized system of elasticity with

constant coefficients. Using the approximate solutions thus obtained we shall

calculate effective characteristics such as energy, stress tensor, frequencies of

free vibrations, etc., of a perforated strongly non-homogeneous elastic body,

whose elastic properties can be described in terms o f problem (1.1).

The homogenized system corresponding t o problem (1.1) has the form

where the coefficient matrices ( P , ~ = 1 , ..., n ) are given by the formula

and matrices N * ( J ) are solutions o f the following boundary value problems for

the system of elasticity

4 N 9 ) = -ukAkq on dw , 1 (1.4) N q ( J ) is 1-periodic in J , / N q ( W = 0 ,

Qnw

Q = { J , O < ( j < l , j = l ,..., n ) .

51. Mixed problem in a perforated domain 121

Existence of the matrices Nq follows directly from Theorem 6.1, Ch. I.

According to Theorem 6.2, Ch. I, the elements o f the matrices Nq are

piecewise smooth functions in w belonging t o the class 6. System (1.2) can also be derived by the method of multi-scale asymptotic

expansions which is thoroughly described in numerous sources (see e.g. [3], [5],

[110]). We shall not reproduce here this well-known procedure since for the

system of linear elasticity it is essentially the same as for second order elliptic

equations (see e.g. [5]).

Theorem 1.1.

The homogenized system (1.2) is a system of linear elasticity, i.e. the elements

of the matrices Ak' satisfy the conditions

for any symmetric matrix 7 = { y i h ) , where it1, k2 are positive constants. In

other words the operator k belongs to the class E ( k l , k 2 ) .

Proof. In the special case of w = Rn, i.e. R' = R , the relations (1.5), (1.6) 2

can be obtained from Theorem 9.2, Ch. I, since the matrices N,9(x) ENq(-) E

and ak' satisfy the Condition N which can be easily verified on account of (1.3),

(1.4). In the general case when w may not coincide with Rn, i.e. RE may be

a perforated domain in the proper sense, Theorem 9.2, Ch. I is not applicable,

and we shall use another method to prove the relations (1.5), (1.6).

Let C be a column vector with components el , ..., en. Denote by C* the line

( e l , . . . ,en) . By A' we denote the transpose of the matrix A. Thus A( = y is

a column vector with components yj = a&, j = 1, ..., n , and y* = ( * A is a

line with components yj = ciaij , j = 1 , ..., r ~ .

It is easy t o see that the second equality in'(1.5) follows directly from (1.3)

and the properties of the elements of the matrices APq(t), since

122 11. Homogenization of the system of linear elasticity

where NA, are the elements of the matrices Nq.

Let us establish the first equality in (1.5), which is equivalent to ( A ~ ' J ) * =

A ~ P .

It follows from the integral identity for solutions o f problem (1.4) that for

any matrix M ( J ) E W ; ( U ) we have

dM dNq d M - J 6 ~ ~ j ( t ) - d t = J .

Q n w % Q n w

Making use of the relations ( A k j ( t ) ) * = A j k ( t ) , (AB)* = B*A* for

matrices A, B , we obtain from (1.8) that

dN9* dM* dM* - J = A j k ( 0 , d t = J ~ ~ ~ ( o ~ d t . (1.9)

Qnw Q n w

Setting M = NP* in (1.9) and taking into account (1.3) and the relation

(Apj)* = Ajp we find

I t follows that the coefFicient matrices of the homogenized system can be

written in the form

51. Mixed problem in a perforated domain 123

Replacing p by q and q by p in this formula and taking the transpose of

the equality obtained, we see that h ' q = ( A ~ P ) ' .

In order t o prove the inequalities (1.6) let us note that iiP/qihvjk = qh*AhkVk

where sk is a column with components qlkr ..., qnk, and qh* = (v lh , ...? qnh).

For any symmetric matrix 71 with constant elements q;h we obtain due to (1.10)

that

Let w = (Nq + tqE)qq be a vector valued function with components

wl, ..., w,. It then follows from (1.11) that

dw* dw 8gq,pl)jq = (mes Q n w)-I / - - dt =

Qnw X j

= (mes Q n w)-' - d( . Qnw

Suppose that for a symmetric matrix q we have iiY;qipqJq = 0. It then

follows from (1.12) and the estimate (3.13), Ch. I, that I l e ( ~ ) 1 I ~ 2 [ ~ ~ ~ ) = 0.

Therefore w is a rigid displacement (see the proof o f Theorem 2.5, Ch. I). On

the other hand w ( t ) = ( N Q + (,E)qq. Therefore due t o the periodicity of

Nq(6) the vector valued function Nqqq must be constant, and the matrix q

must be a skew-symmetrical one. It follows that q = 0. Thus i i~~qi ,qjq > 0

for 7 # 0, which proves the lower bound in (1.6). The upper bound in (1.6)

holds because o f the formula (1.7) for 2:;. Theorem 1.1 is proved.

1.2. The Main Estimates and Their Applications

Let us take as an approximation t o the solution of problem (1.1) the fol-

lowing vector-valued function

124 II. Homogenization of the system of linear elasticity

where NP(E) are the matrices defined by (1.4) and uO(x) is the solution of the

problem

Theorem 1.2.

Suppose that uc(x) is a weak solution of problem (1.1) in W, f' E L2(Rc) ,

iPc E H'(Rc) , f0 E H1(R), iPO E H3(R) and uO(x) is a weak solution of the

homogenized problem (1.14). Then

where C is a constant independent o f E , the norm 11 . 11, is defined by (5.3),

Ch. I.

Proof. Applying the operator LC t o uc - ii we obtain the following equalities

which hold in the sense o f distributions

Since the matrices N 8 satisfy the equations (1.4), it follows that

Lc(uC - ii) =

§1. Mixed problem in a perforated domain 125

Define the matrices NPq(<) ( p , q = 1, ..., n ) as weak solutions o f the


NPq is 1-periodic in ( , / N p q ( ( ) d < = 0 . Qnw 1

The existence of NPq(<) follows from Theorem 6.1, Ch. I and the equalities

(1.3). Thus we deduce from (1.16), (1.17) that

Therefore

a L C ( U ' - ~ ) = fE - f0 +EFO + + -Fk, (1.18)

dxk where

Let us consider now the boundary conditions on Sc for uc - 12. We have


By virtue of the boundary conditions on a w for Nq and NPq it follows that

On the outer part o f the boundary o f Re we have

Let us show that

where c is a constant independent o f E . To this end it suffices t o find a vector

valued function \kc E H 1 ( O c ) such that V!, + EN' E H1(R",I ' , ) ,

We define Q e ( x ) as follows. Let cp , be a scalar function in Cw(Q) such

that cpc(x) = 1 if p ( z , a R ) 5 E . cp.(x) = o if p ( x , 80) 2 2&, o 5 cp,(x) 5 1,

IVcpl < c2&-l.

Set

It is easy t o see that Q , E H 1 ( R c ) and

81. M ixed problem in a perforated domain

a Q c acp, duo dN"uO - = - & - N S - - E c p c - - - a x j d x j ax. dxj dx ,

Therefore taking into account the properties of cp, and the fact that the matri-

ces N s ( [ ) and dNB(E)/dEj have bounded elements, we obtain the inequality

~ I P C I I H ~ ( O ~ ) 5 ~3 ( I I ~ ~ I I H ' ( K . ) + I I U ~ I I H ~ ( K C , ) . (1.24)

By virtue o f Lemma 1.5, Ch. I, we get

I I u O I I H ~ ( K . ) I c4&lI2 11~011~2(n) 3

where c4 is a constant independent of E . This inequality together with (1.24)

yields (1.23). Therefore estimate (1.22) is valid.

On the basis o f (1.18), (1.20), (1.21), (1.22) we conclude that u' - ii is

a weak solution o f the following mixed boundary value problem studied in 55,

Ch. I:

Here 4, satisfies the inequality (1.22) and

where the constant cg does not depend on E , since the elements of the matrices

Ahk, NP, Npq are piecewise smooth functions (see Theorem 6.2, Ch. I). I t

follows by virtue of Theorem 5.1, Ch. I, and Remark 5.2 that

This inequality implies (1.15) since due to the a priori estimates for solutions

of elliptic systems (see [I]) we have:



We now prove some important results which follow from Theorem 1.2.

Formula (1.13) for an approximate solution of problem (1.1) allows us t o

estimate some effective characteristics o f strongly non-homogeneous bodies,

in particular the stress tensor and the energy.

Let R' be a subdomain o f R with a smooth boundary. Set

The integrals Ec(ue), Eo(uO) represent the energy contained in R' n R' and R' respectively.

Theorem 1.3 (On the Convergence o f the Energy).

Suppose that all conditions o f Theorem 1.2 are satisfied. Then

where c is a constant independent of E .

Proof. It follows from Theorem 1.2 that

I I~f(x)l l~2(n*) 5

5 s + ~ l @ ~ l l ~ ~ n ( s n ) ) + + IIfO - f ' l l * + 11,' - @'l lx~t~~r.~] ,

where q, is a constant independent o f E . Therefore

$1. Mixed problem in a perforated domain 129

where

dN" Since the elements of the matrices Aij, - are bounded (see Theorem

X i 6.2, Ch. I) we get

This inequality together with (1.29) yields

Therefore

Let us introduce the matrices

a a Hat(<) r - (N" + t S ~ ) ~ ' j ( < ) - (N t + <tE) -

X i at j - (mes Q n w ) k t .


In the rest o f the proof it is assumed that the matrices A i j ( ( ) , N s ( ( ) ,

dN"/d ( i are defined in Rn and are equal t o zero in Q\w. Thus we obtain

from (1.10)

Note that due t o our assumptions we can replace the domain o f integration

0' n R' in (1.30) by 0'. Therefore after a suitable transformation, (1.30)

becomes

From this equality and (1.32) we conclude that

Note that due t o Theorem 6.2, Ch. I, the elements o f the matrices Hst are

bounded functions.

Denote by Jc the set of all vectors z E Zn such that E ( Q + Z ) c 0' and

by J: denote the set of all z E Zn such that E ( Q + z) n dR' # 0. Then

Ec ( u c ) - (mes Q n W ) E0(u0) =

auO* duo = x, - H a t ( - ) - dx +

dx , e dxt ' E J * ~ ( ~ + g ) n n ~

It is clear that the first sum in the right-hand side o f (1.36) can be repre-

sented in the form

$1. Mixed problem in a perforated domain 131

where G, is an open set lying in the 6-neighbourhood of dR1 and 6 is o f order

E. Therefore using Lemma 1.5, Ch. I, we deduce that

Consider now K 2 , i.e. the second sum in the right-hand side of (1.36).

Denote by R1' the set formed by the cubes ~ ( z f Q), when z takes values in

J,. Set

auO J - d z f o r z ~ a ( z + Q ) . 7 t ( x ) = - - mes EQ

c(z+Q) t

The vectors y t (x ) are constant on each o f the sets ~ ( z + Q). We have


mes EQ duo 2 duo 2 ' zEJ. ' ( m e s ~ ~ ) ~ ~ ~ S E Q J lzl dx = J lzl dz . ++Q) 0''

Taking into account the Paincark inequality in E ( Z + Q ) we obtain

Let us estimate the sum IG. Due t o (1.33) the last integral in (1.38) vanishes

since rt are constant on each o f the sets E ( Z + Q) for z E J,. It follows that

11. Homogenization o f the system o f linear elasticity

Therefore taking into account (1.39), (1.40) we obtain

The inequality

0 IIU IIH~(*) 5 C ~ ( I I P I I ~ ( ~ ) + I I ~ O I I H ~ , ~ ~ ~ ~ ) )

and (1.31), (1.36), (1.37), (1.41) imply estimate (1.27). Theorem 1.3 is

proved.

Let us consider now the convergence of the stress tensors, i.e. o f matrices

whose columns are

where u' is the solution o f problem (1.1).

The stress tensor corresponding t o the homogenized problem (1.14) has

the form

In the homogenization theory the matrices with columns u,P, u,P are also referred

to as weak gradients or flows.

In the next theorem it is assumed that

and that

Theorem 1.4.

Suppose that the conditions o f Theorem 1.3 are satisfied. Then

$1. Mixed problem in a perforated domain

x auO 110,' - (mes Q n w)u,P - G"(-) -11 a ax, L2(n) '

where c is a constant independent o f E , and the matrices GP'J(J) are defined

by the formulas

aN8 GP"J) = AP8(<) + Api - - a< i

( m e s ~ n w ) A ~ ~ for J E Q ~ W , (1.45)

GP" (() = -(mes Q n w),&'~ for J E Q\w .

Moreover, if f c = f O , @' = @O then

u,"(x) --+ (mesQ nw)~,P(x) weakly in ~ ~ ( 0 ) as E + 0

Proof. Let us make use of the relations (1.28), (1.29) which hold due t o

Theorem 1.2. Then according t o (1.42) we have

- dNa duo - (apa + A~ -) - + (x) . ati axs

Therefore taking into account (1.43) we get

u,P(x) - (mes Q n w)ag(x) =

This equality and (1.29) imply (1.44).

The weak convergence of u,P(x) t o (mes Q ~ w ) u , P ( x ) for fE = f O , = Q0

follows from (1.44) and the fact that GP"(J)d( = 0 which implies the weak J Q

auO convergence Gpa(-) - -t 0 in L2(R) as E + 0 by virtue o f Lemma 1.6, Ch. ax, I. Theorem 1.4 is proved. .

134 11. Homogenization o f the system o f linear elasticity

$2. The Boundary Value Problem with Neumann Conditions

in a Perforated Domain

Results similar t o those of $1 for the mixed problem can also be proved for

the Neumann problem in a perforated domain Rc of type II (see $4, Ch. I).

However, in the last case some difficulties arise in obtaining estimates for the

boundary values o f the conormal derivative of a rapidly oscillating corrector x duo

which has the form EN'( - ) - outside a neighbourhood of do. Therefore E ax,

in order t o clarify the main ideas used in the proof o f an analogue to Theorem

1.2 we shall first consider the Neumann problem in a domain R independent

of E for a single second order elliptic equation with rapidly oscillating periodic

coefficients. It should be noted that the absence of cavities makes the proof

much simpler.

2.1. Homogenization of the Neumann Problem in a Domain R for a Second Order Elliptic Equation with Rapidly Oscillating

Periodic Coefficients

In a bounded smooth domain R consider the following Neumann problem

where (y, ..., un) is the unit outward normal t o dR. It is assumed that ai j ( [ )

are smooth functions in R n , 1-periodic in [, and such that

where 61, K~ = const > 0. The functions f and cp are sufficiently smooth and

satisfy the condition of solvability of problem (2.1)

J f d x = J c p d ~ . (2.2) n an

We define the functions NP(t ) , p = 1 , ..., n , as solutions of the problems

$2. Boundary value problem with Neumann conditions

Set

Q = { x : O < x 3 < l , j = l ,..., n ) .

As an approximation to the solution of problem (2.1) we take the function

x duO(x) 6 = u O ( x ) + & N d ( - ) - ,

& a x ,

where u0 is the solution of the homogenized Neumann problem

duo . . duo - = 6'3 - - v; = cp on dR . auA ax

In analogy with (1.16) simple calculations show that

Let us define the functions N i s ( J ) as solutions of the problems

Then

136

where


dNia d2u0 FO - akh - - k - k = 1, ..., n , ath dXSdx; ' (2.8)

dNi"uO Fl = -akh - - a i j ~ s d3u0

a& axsaxiaxk axsaxiaxj a

(2.9)

Consider the boundary conditions

a . . - 6 ) - (uC - 6 ) a'' v; = ~ Y A , d x j

- . . duo d N s duo d2u0 - y - a t J v i -+ - -) = ( a x j atj ax, axsaxj . . a N s duo . . d2u0 - -a'J - vi - - &a'3ViNs - . . auO

- + (iY3 - at3)ui - . d f j dxa dx,dxj d z j

Therefore

d x . . x d N S duo - - 6 ) = [;is - ass(-) - (-1 -1 vi - - ~ V A . E E atj dx ,

a2u0 - &at3viNS - . (2.10)

dx,dxj

We introduce the following notation

&la ' - aia( f ) - a''(() - a N s ( < ) , ati

i j = (E l , . . + , E j - l , f j + l , ..., En) E Rn-l

$ = { ( I : & = t , o < f i < l , l # j ) .

Lemma 2.1. Functions a i d ( t ) defined by (2.11) satisfy the relations

a i a - Q ( f ) = 0 , s = 1, ..., n ; at; J d S ( [ ) d i j = o , 4 E R' , s, j = 1, ..., n , s;

(there is no summation over j ) .

92. Boundary value problem with Neumann conditions 137

Proof. The equalities (2.12), (2.13) follow directly from (2.3) and the definition o f hhk. Let us prove (2.14). Denote by Q;,,, the set

Multiplying (2.13) by tj - t l and integrating over we get for each j

= / a i a ( 0 ( t 2 - t l )d& - / d S ( ( ) d ( . g2 Q:, t ,

Setting t l = t 2 - 1 and taking into account (2.12) we obtain (2.14) for t = t 2 .


Lemma 2.2.

Let a'"(( ) be functions in H 1 ( Q ) 1-periodic in ( and satisfying the conditions

(2.12)-(2.14). Then for any v E H 1 ( R ) the following inequalities are valid

where c is a constant independent of E , v.

Proof. Denote by I: the set o f a l l indices z E Zn such that ~ ( z $ Q) C R ,

P ( E ( Z + Q ) , d o ) 2 E . Set R1 = U ( E Z + Q ) . It is easy to see that ~ € 1 :

d x 0 = - (aik(--))vdx = aikvivdS -

ax; n\nl

J an

dv - J a i k u ; v d ~ - J aik - dx , IC = 1, ..., n

dzi an, R\RI

Therefore

11. Homogenization of the system o f linear elasticity

We clearly have

Let us estimate the first integral in the right-hand side of (2.16).

It is easy t o see that an1 consists o f (n - 1)-dimensional faces o f the cubes I

E ( Z + Q ) for some z E If. Denote by uj', ..., uj) the (n - 1)-dimensional faces

of the cubes E ( Z + Q) for z E If such that ujk is parallel to the hyperplane

xj = 0 and lies on an1, j = 1 ,..., n. We thus have

Denote by qj" the cube ~ ( z + &) whose surface contains the set u,9. It is

obvious that among the cubes q;, j = 1, ..., n; s = 1, ..., l j , there cannot be

more than 2n identical ones.

Due t o the condition (2.14) for any u; we have

since ui = 0 for i # j (there is no summation over j). Set

Thus the function ~ ( x ) is constant on each surface a,".

Taking into account (2.18) and the fact that not more than 2n cubes q,"

can have a non-empty intersection we obtain


Here we have also used the inequality

which can be proved i f we pass t o the variables [ = e-'x and apply Proposi-

tions 3 and 4 of Theorem 1.2, Ch. I, in the domain R = E-'qj".

The relations (2.16), (2.17), (2.19) imply (2.15). Lemma 2.2 is proved.

Therefore due t o (2.7)-(2.10) we have

where Fl, F t , F2 are bounded uniformly with respect t o E .

Setting w = uc - 6 + vc where 7' is a constant such that w dx = 0, we / n

obtain from (2.21)

auO Applying Lemma 2.2 t o v = - w and using the Poincare inequality (1.5),

ax, Ch. I, we find that

140 II. Homogenization of the system of l inear elasticity

Thus we have actually proved

Theorem 2.3.

Suppose that uc, u0 are solutions o f problems (2.1), (2.5) respectively, and f ,

cp are smooth functions satisfying the solvability conditions for problems (2.1),

(2.5). Then there is a constant qc such that

where c is a constant independent o f E.

In the same way as it was done in $1 we can obtain estimates for the close-

ness o f energy integrals and weak gradients related t o problems (2.1), (2.5). In

this section we omit the consideration of these questions. However, estimates

of this kind for the system of elasticity in a perforated domain are established

below.

2.2. Homogenization of the Neumann Problem for the

System of Elasticity in a Perforated Domain.

Formulation of the Main Results

In the rest o f this section Rc denotes a perforated domain of type II defined

in $4, Ch. I. The boundary dRc is a union o f d R and the surface of the cavities

S, c R. In Rc we consider the boundary value problem of Neumann type for the

system of linear elasticity:

It is assumed that the elements of the coefficient matrices Ahk satisfy

the same conditions as the coefficients of the system (1.1), fC E L 2 ( R c ) ,

@ E L 2 ( d R ) and satisfy the conditions of solvability for problem (2.22), i.e.

'$2. Boundary value problem with Neumann conditions

where R is the space of rigid displacements.

Existence and uniqueness ( to within a rigid displacement) of a solution of

this problem follow from Theorem 5.3, Ch. I.

Consider also the Neumann problem for the homogenized elasticity operator

(mes Q n w)6(u0) = Go on 8 0 , J duo

where 6(u0) vhahk - , the matrices ahk are defined by the formulas (1.3), axk

11,' E L2(dR) , f 0 E L2(R) , satisfy the solvability conditions

(mes Q n w ) - I I (Go, l ) d S = / ( f O , l ) d x Vq E R . an R

It is important t o note that the factor mes Q n w appears in the Neumann

conditions on dR. This factor is equal t o 1 if RE coincides with R (see $2.1,

formula (2.5)).

To characterize the closeness between functions f O , 11,' and fc, 11,' we

introduce the following notation.

For any vector valued functions f E L2(Rc) , 11, E L2(dR) the scalar prod-

ucts ( f , v ) ~ z ( ~ . ) , ($, ~ ) ~ 1 ( ~ ~ ) define continuous linear functionals on H1(R'), and therefore f and 1C, can be considered as elements of the dual space

H1(Rc)*. Let us denote the norms of the respective functionals as 1 1 f llH1.,

~ ~ ~ ~ ~ H ~ * ~ i.e.

Note that I l f I I H I * 5 I l f l l ~ z ( n * ) ~ l l 1 1 , l l ~ l - I c I I ~ I I L Z ~ ~ ~ ) . We seek an approximate solution of problem (2.22) in the form

x duo a ( x ) = uo + + E ~ ~ ( X ) N ~ ( - ) - .

E dx,

Here u0 is the solution o f problem (2.23), N Y t ) are solutions of problems

(1.4), cp(x) is a truncating function which satisfies the following conditions:


cp E C c ( R ) , IVql l CE-' , cp = 0 in R\R1 , (2.25)

cp(x) = 1 for x E R1 such that p(x,dRl) 2 CIE ,

where cl, c are constants independent of E ; R1 is defined by formula (4.3),

Ch. I.

In contrast t o the case, considered in 52.1, o f a single second order elliptic

equation in a non-perforated domain, here the truncating function cp enters

the expression for C (cf. (2.4)) since the solution u' is considered in the per- x

forated domain Rc but the matrices N S ( - ) are in general not defined in a &

neighbourhood o f dR.

The main result of this section is

Theorem 2.4.

Suppose that fc E L2(Re), f0 E H1(R) , @ E L2(dR) , 6' E H ~ / ~ ( ~ R ) . Then the solutions u', u0 of problems (2.22), (2.23) respectively satisfy the

following inequality

where c is a constant independent o f E ; 7" is a rigid displacement which may

depend on E .

The proof o f this theorem is given in Section 2.4 and is based on the lem-

mas established in the next section.

2.3. Some Auxiliary Propositions

Let us introduce the notation

d N S a''(<) = 2' - A"(<) - Ai j ( t ) - , i, s = 1, ..., n

atj


Lemma 2.5.

The matrices a i s ( ( ) satisfy the following conditions

J d s ( ( ) d i j = (mes 2: - mes Q n w ) i j S , 5;

(there is no summation over j).

Proof. Equalities (2.27), (2.28) follow directly from (1.3), (1.4).

Let us prove (2.29). Multiplying the system (1.4) by ( ( j - t l ) E , where E is the unit matrix, and integrating over Qj,,, n w, ( t l < t z ) , we obtain

Each integral in (2.30) over d(Qi,,, n w ) can be represented as a sum of

integrals over the sets

Q:,naw, S:,u$,, (j $ u s ; . r= l + I

Since

and the integrands are 1-periodic in (,, r # j , r = 1, ..., n, it follows from

(2.30) that

II. Homogenization of the system of linear elasticity

(there is no summation over j).

Setting t, = t2 - 1 in this equality and taking into account (1.3) we find

that

It follows from the definition o f a'" that

A . A .

= (mes S,3)A3' - (mes Q n w ) k s =

= (mes j;j - mes Q n w)Ajs .


Remark 2.6.

I f the domain Re is not perforated, i.e. w = Rn, Rc = R, then A . / a j S ( t ) d i j = 0, since mes S,3 = rnes Q n w = 1 (see Lemma 2.1).

9;

Lemma 2.7.

Let al, ..., a2, be (n - 1)-dimensional faces of the cube EQ = {x : 0 < x j < E, j = 1, ..., n). Then each u E H1(eQ) satisfies the inequality


where c is a constant independent of i, j, E .

M. Set o1 = { x l = 0) n EQ, o2 = { x 2 = O} n E Q , S1 = & - l o l , s - -1

2 - a a2. Consider the points 6' = (0, y2, y3, ..., y,), G2 = ( y 2 , 0 , y,, ..., y,) on the faces Sl, S2 of cube Q. The segment g ( t , y2, y3, ..., y,) = tjjl +( l -t)j j2

for t E [O,1] belongs t o Q . It is easy t o see that for any v E H 1 ( Q ) we have

Integrating this equality with respect t o y2, ..., y, from 0 to 1 we obtain the

estimates

&-(,-I) J v2 do - &-(n-l) / v 2 d o 4 CE-"E I v I 1VrvI d x 0 1 0 2

/ EQ

which imply (2.31). Estimate (2.31) for other faces can be proved in a similar

way.

In the next two lemmas we establish some inequalities, uniform in E , for

functions defined on the set d R 1 which is the boundary of the domain O 1 given

by formula (4.3), Ch. I. The domain R 1 depends on a and its boundary 6'01

consists of the (n - 1)-dimensional faces of the cubes ~ ( z + Q), z E T,.

Denote by a,!, ..., o) the faces o f the cubes a ( z + Q) for r E T, parallel t o

the hyperplanes x j = 0 , j = 1 , ..., n , and laying on d o 1 . Then

146 II. Homogenization o f the system o f l inear elasticity

The cube ~ ( z + Q ) , z E T,, on whose boundary lies the set ojS is denoted by

qjJ It is easy t o see that among gj, j = 1, ..., n , s = 1, ..., lj, the number of

the identical cubes is not greater than 2n.

Lemma 2.8.

Let u E H1(R). Then

where c is a constant independent of E.

Proof. According t o (2.32) dR1 consists of the sets ufi, and each ufi is an

(n - 1)-dimensional face o f the cube qg. The boundary dR is a smooth

surface, therefore each cube qj" possesses a face u,j such that u,,j is parallel

to the hyperplane xm(j,,) = 0 and u,,j is the orthogonal projection along the

axis of a surface SaVj C dfl which is given by the equation

and c le 5 Ix - yl 5 c2c for x E o,,j, y E Ss,j ,

where constants cl, cz, M do not depend on E , s, j.

Denote by QSj the set formed by the segments orthogonal to u,j and

connecting the points o f a,,. and SaVj. Then using a suitable diffeomorphism

mapping QSj t o EQ and taking into account Lemma 2.7 we find that

2 IIullr2cS,,) 5 ~ ( I I ~ I I ~ ( s , , , + I I ~ I I ~ ~ ( Q , , ) ) . Therefore by Lemma 2.7 we get

2 IIuIIL~(~;, 5 c l ( l l ~ l l t ( ~ , , ) + llullBl(Q.,,)) .

Summing these equalities with respect t o s, j we obtain estimate (2.33),

since due t o the smoothness o f dR there is an integer k independent of E

and such that each Q,, can have a non-empty intersection only with a finite

number o f QI,$ which is not greater than k. Lemma 2.8 is proved.

Lemma 2.9. Let the matrices yhk(x) E Lm(dfll) be such that


[ yhk(x)ds = 0 (there is no summation over h ) ,

where u r are the same as in (2.32), y = const. Then for any vector valued

functions uO E H3(SZ), w E H1(LR) the inequality

holds with a constant c independent o f E .

Proof. Consider a function r (x ) defined almost everywhere on dR1 by the

formula

r (x ) = (mes a;")-' duo

uhyhk - dS for x E 07 J ax, ~ i "

Obviously r (x ) is constant on each 0;". Therefore setting

&(x) = (mes oY)-'

i"

and taking into account (2.34) and the Poincarh inequality in a;" we obtain

'1 mes 0; = 5 C

(rnesq12 j=1 s=l


It follows from Lemma 2.8 that

J Ir12dS < ~ 2 ~ ~ 7 ~ IIuoIIZx3(n) anl

where c2 is a constant independent o f E .


Due t o (2.36) and Lemma 2.8 we have

Let us estimate the second integral in the right-hand side of (2.37). Define

the vector valued function ~ ( x ) on dR1 by the formula

r ) (x ) = (rnesd,)-' / w d x for x E 0;

s!"

Therefore

Here we have used the fact that among q; the number of identical cubes is

not greater than 2n; and we have applied the inequalities (2.20) for v = w.

By the definition of r we have


Therefore since q ( x ) is constant on each at, we find by virtue o f (2.39), (2.36)

and Lemma 2.9 that -

This inequality together with (2.37), (2.38) yields (2.35). Lemma 2.9 is

proved.

2.4. Proof of the Estimate for the Digerence between a Solution of

the Neumann Problem in a Perforated Domain and a Solution

of the Homogenized Problem

In this section we give proof o f Theorem 2.4.

Let us apply the operator C, t o the vector valued function uc - 6, where

ii is given by (2.24). Then

a auc a a auO r . (~€ - ii) = - (A" ,) - - (ahk - (,O + EPN' -)) = axh xk axh ax ,

- d due d duo 8 - duo duo - - (A" -) - - ( A h * -) + - (Ahk - - -) - axh axk axh axk axh axk ax k

- - d(cpNa) duo a [Ahk(& - a2u0 - + E ~ N ' -)] = dxh ax/; dx , dxkdxs

a duo auO ~ ( P N ~ ) auO = f'- f '+ - (A" - - - & a h 3 - axh dxk dxk d x j -) dxk -


Taking into account the equations (1.4) for N" we obtain

a duo - 6) = f' - f 0 + - ((1 - $,)(A" - -1 + dzh axk

+ [Ahk - Ah* - &Ah; + 8 9 A d N k duo -f - [Ahk - Ahk - E ~ h l -1 - - axh d x j dxk

d d2 uO - - ! & A h j N k ! f ] - E * A h k N s

B x ~ a x j dxk axh dxkdx,

d d2u0 - E(P - ( A ~ ~ N ~ ) - -

axh axkdx,

- d3u0 a duo

= I' - f0 + - [ (1 - $,)(A" - A") -1 + dxkdxhdxs dxh dxk

d N k d A S h ~ ' aZu0 a$, auO + [A" - ~ h * - & ~ h j - - dx; +- T I ax, a ~ ,

- E ~ A ~ ~ N ~ d3u0

dxkdxhdx, '

Let us define the matrices N h k ( J ) as weak solutions of the following bound-

ary value problem

$2. Boundary value problem with Neumann conditions 151

a dNhk

- - a a~~ - - ( A " N ~ ) - ~ h j - - Ahk + Ahk in w , (2.40) a t s a t j

a ( N h k ) = -vsAshNk on I ~ W , N h k ( t ) is 1-periodic in [ .

Then

a auO Lc(uc - i) = f' - f0 + - [(I - p) (Ahk - Ah*) -1 +

axh a s k

d dNhk d2u0 d p duo + pa - (ul 8--) - + - ahk - - a x j ( I dxkdxh dxh axk

d~ hk d2 U O - - & A NS - - axh d ~ ~ d ~ ,

- a p ~ h k ~ s d3u0 - -

d ~ k d ~ ~ d ~ ,

a a a~~~ a 2 ~ '

d p dNhk d2u0 - & ~ j l - -- ~ p ~ j l - a N h k d3u0

a x j dxkdxh axkdxhdx j +

W e thus have


duo a p h . duo \

~1 h - ( - 1 - c p ) ( a h k - ~ h k ) - - E - A 3N - , dxk axj axk

dNhk d2u0 F? = &pAjl - - , 3 X I axkdxh

d p dNhk a2u0 +-&-AjI- -- 8 9 hk d2u0 E - A N S - (2.42)

d x j atl dxhdxk axh d2kdx8 ' . dNhk @uO

= -&pA" - - d3u0

atl dxkdxhdx j dxkdxhdxs ' d p duo = -'p-, axh dxk

Consider the boundary conditions for u' - ii:

. . due .. a auO gC(ue - ii) = A'jv, - - AtJvi - (uo + € p N 8 -) =

dx j dxj 8x8

. . due . due .. duo = (1 - ~Y)~,A'" + +uiA'J - - uiA'J - - . . d p duo ,cViAS3 - N 8 - - axj a x j dxj axj axs

. . d N 8 duo - ,cviA'3p - - - d2u0 E ~ ~ A ~ ~ ~ N ~ - - -

d x j ax, dx,dxj

duE duo A . . duo = (1 - c p ) ~ i j v i - + (1 - (p)vi(,$'j - ~ i j ) - - (1 - p ) ~ 1 3 v i - - a x j d x j d x j

.. duo - cpV,A'3 - - a N j duo , c ~ , ( p ~ ~ l - - - .. d(p duo ,cViA13 - NS - -

a x j axl dxj axj ax, a2 uo due A . . duo

- &viA"pN8 - - - (1 - p ) ~ " v ; - - (1 -cp)A"vi - + axsaxj a x j d x j

.... duo + (1 - p)vi(AV - A") - - p (v,A'j + EV,A'I d x j

. d(p duo d2u0 - ,rviAt3 - N s - - EyiA'~cpNs --- d x j d z , dxsdx j '

By virtue o f the boundary conditions in (1.4), (2.40) we have

dN' cp(v i~G + E U , A ~ I -) = o ,

8x1

- E U ~ ~ A ~ ~ N " = E ~ u ~ A ~ ~ - a N J s anc . at1

Therefore taking into account (2.42) we find that


Set w = u" - 4 + qE, where q' is a rigid displacement suchlthat

( w , ~ ] ) ~ I ( ~ . ) = O for any E R .

Due to the boundary conditions for uO, ue and the fact that cp = 0 in O\Ol

it follows from (2.41), (2.43) that

Let us estimate the integrals in the right-hand side of this equality. Note,, 89 that owing to (2.25) the functions - and 1 - cp vanish in {x : x E ax

O1,p(x,aO1) 2 cis) and IsVcpl 5 c, where c, cl are constants independent

o f E. Therefore by Lemma 1.5, Ch. I, we obtain

where c2 is constant and does not depend on E .

I t follows from (2.42) that

where c3 is a constant independent o f E.

Taking into account (2.27), (2.28) and setting a = mes Q n w we get

154 11. Homogenization o f the system o f l inear elasticity

- duo duo

- / ( h a h k - , w ) d ~ + / (viAij - , w ) d ~ + a x k ax anl\s. anl ns,

It should be noted here that in the integral over (aR1)\Sc the normal v is

exterior t o dR1, whereas in the integral over dRl n S,, the normal v is exterior

to RE. The last two integrals on the right-hand side o f (2.47) can be estimated

by

similarly to (2.45).

Let us introduce the matrices phk([) setting

Then

It follows from (2.47), (2.48) that


lJll I IJ21 + C E " ~ ~ ~ U O I I ~ ~ ( ~ ) l l w l l H l ( n e ) , where

duo J2 = - / (phk($)uh - , W ) ~ S + Q\w J ( $ O , W ) ~ S +

a x k mesQ n w anl an

The integral identity for u0 yields that

Therefore by virtue o f Lemma 1.5, Ch. I,

/ ( ~ O , w ) d S = mesQ n u an

where

IJ3I 5 ~ 1 / 2 ( ~ l ~ 0 1 1 ~ z ( n ) Ilwllel(n*) + Ilf011r2cn) llwllxl(n*,) . (2.52)

W e thus obtain from (2.50), (2.51) that

duo Jz = / ((mes g\w),Ahk - v h p k - , w) d~ +

an1 dxh

+ J (Go - @,w)dS + J3 - an

Set

-yhk = (mes Q \ w ) A ~ ~ - phk

in Lemma 2.9. I t is easy to verify that conditions (2.34) are satisfied for -yhk. Indeed due to (2.48) and (2.29) we have

156 11. Homogenization o f the system of linear elasticity

J -yhk d~ = an-' (mes Q\w)A~* - / p h k d s - / p h k d s =

ohm o r \ s * q n S c

= an-'(mes Q\w)Ahk - / ah* d~ - (mes or n sC)Ahk =

a,"\&

- (mes or n sC)Ahk = en-' (mes Q\w - mes a-' (or\$) + t rnes Q n w - rnes €-'(or n sC))Ahk = 0 ,

since

m e s Q \ w t m e s Q ~ w = 1 ,

mes E-l (op\S,) + mes E-' (or n S,) = 1

We conclude from (2.52), (2.53) and Lemma 2.9 that

I t follows from (2.44), (2.45), (2.46), (2.49), (2.54) that

From the well-known results on the smoothness of solutions of elliptic

boundary value problems we have

since d R is a smooth surface and f0 E H 1 ( R ) , 4' E H ~ / ~ ( ~ R ) .

Therefore by virtue of Theorem 4.4, Ch. I , the inequalities (2.55), (2.56)

yield (2.26). Theorem 2.4 is proved.


2.5. Estimates for Energy Integrals and Stress Tensors

Slightly modifying the proof o f Theorems 1.3 and 1.4 on the convergence

of energy integrals and stress tensors one can establish similar theorems in the

case o f the Neumann problem. To this end we should use estimate (2.26)

instead of (1.15).

Theorem 2.1Q (On the convergence of energy integrals).

Suppose that all conditions of Theorem 2.4 are satisfied and E,(uc), Eo(uO)

are defined by (1.25), (1.26). Then

where C is a constant independent of E ; uE , uO are solutions o f problems

(2.22), (2.23) respectively.

The proof of this theorem in the main repeats that o f Theorem 1.3. How-

ever, slight modifications should be made. In particular we consider the solu-

tions uO and uc such that

J ( u ' , q ) d z = ~ ( u 0 , q ) d z = O , V ~ E R . (2.58)

nz nc

This choice of ue and u0 is possible since solutions of problems (2.22),

(2.23) are defined t o within a rigid displacement. In this case one can take

qc = 0 in (2.26), and use the estimates

which are well known from the theory o f elliptic equations (see 111).

Similarly t o Theorem 1.4 we establish

158 II. Homogenization o f the system o f linear elasticity

Theorem 2.11 (On the convergence of stress tensors).

Suppose that all the conditions o f Theorem 2.4 are satisfied and u E , uO are

orthogonal t o the space o f rigid displacements as in (2.58). Let the stresses

a,P(x), a:(x) be defined by the formulas (1.42), (1.43). Then

auO 110: - (mes Q n w ) o i - GPq(-) - 8 ax, 11L21fi) 5

where c i s a constant independent o f E , the matrices GPq are defined by (1.45).

Moreover

UP(.) 4 (mes Q n w)u:(x) weakly in L 2 ( R ) as E + 0

2.6. Some Generalizations

For the homogenization o f eigenvalues and eigenfunctions related t o the

Neumann problem (2.22) for the system of elasticity in a perforated domain

we shall need some results on homogenization of an auxiliary system.

Consider the Neumann problem

L c ( u e ) - pe(x)ue = f" in Re ,

and also the corresponding homogenized problem

L(uO) - pO(x)uO = f0 in ,

(mes Q n w)&(uO) = $0 on d a ,

where operators L,, E are the same as in problems (2.22), (2.23), the functions

PC E L w ( R c ) , PO E L m ( R ) are such that

52. Boundary value problem with Neumann conditions 159

and constants co, cl, c2, cg do not depend on E .

In Theorem 2.4 we established the closeness of solutions of problems (2.60),

(2.61) when p, = 0, po = 0. If we introduce a parameter characterizing the

closeness o f p, t o po it becomes possible to prove a similar theorem for the

problems (2.60), (2.61) under the conditions (2.62).

In particular it is o f interest to consider the case in which p,(x) = 2

p ( ; , x ) , p ( ( ,x ) is 1-periodic in ( and satisfies the Lipschitz condition with

respect to x E R uniformly in <, i.e. p((, x ) E J!,(R" x 0 ) in terms of Lemma

1.6, Ch. I. Let pO(x) -- (p ( . , z ) ) , where (p( . , x ) ) is defined by (1.23), Ch. I, and is

equal t o the mean value o f p([ ,x) with respect to t .

I t follows from Lemma 1.6, Ch. I, that for any vector valued functions

u, v E H'(RE) we have

Indeed, set g(<,x) = (p (< ,x ) - po(x ) ) xw( ( ) in Lemma 1.6, Ch. I, where

xw(<) is the characteristic function of the domain w with a 1-periodic structure.

It is easy t o see that g(<,x) E L(nn x !=I),

g((,x)d( = 0. Consider the extensions P,u, P,v of u , v t o the domain R 6 which were constructed in Theorem 4.2, Ch. I. Then

Note that the set fl\R1 belongs t o a 6-neighbourhood o f dR and 6 is of

order E . Therefore applying Lemma 1.6, Ch. I, t o estimate the first term in


the right-hand side o f this inequality, and Lemma 1.5, Ch. I, to estimate the

second term, we obtain

qc 5 cl& IIpeuII~l(n) IIPevIIH1(n)

This estimate together with (4.17), Ch. I, yields (2.63). x

Therefore the functions p(- , x) and po(x) are close in the sense o f the E

inequality (2.63).

In a more general situation we shall characterize the closeness of p, and po

by the norm

where the supremum is taken over all vector valued functions u, v in H 1 ( R c ) .

Relation (2.64) implies that for any u , v E H1(Rc) we have

It is easy t o see that estimate (2.63) implies

Lemma 2.12. x

Let pc(x) = p ( - , x ) , po ( p ( . , x ) ) , p ( t , x ) E ~ ( I R " x fi). Then E

where c is a constant independent of E .

Theorem 2.13.

Suppose that f' E LZ(Rc) , f 0 E H 1 ( R ) , $f E L2(aR) , go E H ~ I ~ ( ~ R ) ,

PO E C 1 ( Q ) and ue, u0 are the solutions of problems (2.60), (2.61) respectively. Then

x duo Ilue - u0 - a p N 8 ( - ) - 1 1 5

E dx , HIPc)


where the constant C does not depend on E , the function cp is defined by the

conditions (2.25) and is the same as in Theorem 2.4.

The proof of this theorem is almost identical to that o f Theorem 2.4. Here

we briefly outline its main steps referring t o the proof o f Theorem 2.4.

An approximate solution of problem (2.60) is sought in the form

x duo ii = uO + &(pNd(-) - ,

E ax,

where u0 is the solution o f problem (2.61), N" are the same as in Theorem

2.4.

Applying the operator C, - p,I to uc - ii we obtain

C,(uC - ii) - p,(u" - fi) =

where F;, q, e, e, are defined by the formulas (2.42) and

For u,(uE - ii) the formula (2.43) remains valid.

Setting w = uC -ii we obtain from the integral identity for problem (2.68),

(2.43) the following relation which replaces (2.44):

Owing to (2.65), (2.69) we have


Formulas (2.45)-(2.50) remain the same.

In order to obtain (2.53) one should use the integral identity in R\RI for

the solution u0 of problem (2.61).

Further changes in the proof of Theorem 2.4 are obvious.

$3. Asymptotic expansions for solutions o f boundary value problems 163

$3. Asymptotic Expansions for Solutions of Boundary Value Problems

o f Elasticitv in a Perforated Laver


Consider a domain RE of the form

where w is an unbounded domain with a 1-periodic structure satisfying the

Condition B of $4.1, Ch. I , E > 0 is a small parameter, and E-' is an integer

number.

Set

If w # Rn, then RE is a perforated layer.

In 0' we consider the following boundary value problem

Here A h k ( [ ) are (n x n ) matrices o f class E( tc l , t c 2 ) ( t c l , tc2 > 0 ) whose

elements a;/ ( ( ) are functions 1-periodic in (.

Existence, uniqueness and estimates for solutions o f problem (3.1) under

suitable assumptions on f, Q 1 , a2 are established in Section 6.3, Ch. I (The-

orem 6.5).

In this section it is assumed that f E C"(Rn) , iP3 E Cm(Rn-I) , f, iP j

are 1-periodic in 2, j = 1 ,2 .

164 II. Homogenization of the system o f linear elasticity

Our aim is t o find an asymptotic expansion for the solution u' in powers

of the small parameter E and t o obtain an estimate for the remainder.

In the case o f a single second order elliptic equation such an expansion was

constructed in [ l o l l . Here we reproduce the results obtained in [87].

For any integer k > 0 the solution ue of problem (3.1) can be represented

in the form

where PA(( ,&) are n x n matrices such that

and PA, are 1-periodic in (, PA1(() and PA,([,(, - f ) define boundary lay-

ers, the components of the vectors Y;(x,E) are polynomials in s whose co-

efficients can be expressed in terms of solutions o f boundary value problems

for the homogenized system of elasticity with constant coefficients in the layer

{ x : 0 < x, < d ) , the remainder pk(s ,x ) satisfies the inequality

with a constant Mk, independent o f E .

3.2. Formal Construction of the Asymptotic Expansion

We seek the solution o f problem (3.1) in the form

In contrast t o Chapter I, here for the sake o f convenience we use the

following notation

Dav = d'v

, a = ( w ,... , a ' ) , ( Q I ) = ~ , axa1 ... dx,,

a, takes the values 1, ..., n; N a ( ( ) are matrices whose elements are 1-periodic

in J; ve(x) = (v f , ..., v i ) is a vector valued function 1-periodic in 4. Substituting the series (3.2) in (3.1) and taking into account that

$3. Asymptotic expansions for solutions of boundary value problems 165

we obtain the formal equality

03 aN ( I ) dD0v, + E-I x &I C A ~ ~ ( I ) A - + I=O (a)=l 8Ij dxk

Here we used the following notation

a + A a l j ( o N"2 ... " , ( I ) + A""(0Na3 ,,, ( I )

for (a) > 2,

a a a a I (ak'(<) 6 T ; N ] ( I ) ) + BF; (Aka1 ( < ) N o ( ( ) ) t


for ( a ) = 1,

for (cr) = 0.

Substituting the series ( 3 . 2 ) in the boundary conditions (3.1) we obtain

the equalities

For x E dRc\(I'o U I'd), E = C ' X , we have

w

+ aka' ( ~ ) ~ a ~ . . . a ~ ( t ) ) D a v e ( x ) s C &I-' C Ba(C)Daul (x) 2 0 I=O (a)=l

where

B,(F) = v k ( ~ * j ( t ) aNa;i;(t) + A ~ ~ ~ ( ~ ) N , ~ . . . , ~ ( O ) ( 3 . 3 )

for ( a ) > 0 and


for ( a ) = 0.

Let us represent N,(t) in the form

where N:(t) are matrix valued functions 1-periodic in J, N i , Nz define bound-

ary layers near the hyperplanes x, = 0 and x, = d respectively.

Set N: = E , N,' = No2 = 0, where E is the unit ( n x n)-matrix. Denote

where N,P = 0, if the length o f the index a is negative. Set

The matrices N:(t) are defined as solutions of the recurrent sequence of

problems

N:(t) is 1-periodic in t , J N:([)dt = 0 , Qnw

(a) = 1,2, ... .

Existence and uniqueness o f N: can be easily established by induction due

t o Theorem 6.1, Ch. I. We define the matrices NA, N: successively with respect t o ( a ) = 1 ,2 , ...

as the solutions o f the problems


where h i , h i are (n x n)-matrices with constant elements chosen in such a

way that the inequalities

d 5 c;eXp(-K;(- - S)) ,

E

hold with constants C,", C,", K:, n i independent o f E .

Existence o f the solutions for problems (3.6), (3.7) and existence o f the

constant matrices h i , h i can be proved by induction on the basis o f Theorem

8.4, Ch. I. Note that because o f the boundary condition in (3.7) on the hyperplane

d tn = - , the matrices N,2 and h i depend on E . If d is a multiple o f E it follows E

from the 1-periodicity in J o f the matrices Ahk(J) that the dependence of

N:([) on E is determined by the relation

53. Asymptotic expansions for solutions o f boundary value problems 169

where N:([) are solutions of the corresponding sequence of problems of type

(3 .7 ) in w ( - m , 0 ) with the boundary conditions

Obviously the matrices N:, k i do not depend on E .

Having thus defined N,P, p = 0,1,2,, let US substitute v, in (3 .1 ) . We get

the formal equalities

I t is easy t o verify that here

and the boundary conditions on (dOE)\(ro U r d ) are satisfied due t o the

boundary conditions in (3 .5 ) , ( 3 .6 ) , ( 3 .7 ) for the matrices N:, NA, N:. Note that by virtue o f (3.4) the constant matrices h0,,,2 are defined by the

formulas

h:l,2 = (mes Q n w ) - l J ( ~ " l " ~ ( t ) + ~ " l j - ) d t 8% . Qnw

at j

Comparing these equalities with (1 .3) we conclude that hyj = aij, i, j =

1, ..., n, i.e. h:j are the coefficient matrices o f the homogenized elasticity sys-

tem.

Let us seek v, in the form of a series

Substituting v, given by (3 .13) in (3 .10) and taking into account (3.12) we obtain the following formal equalities


Therefore by virtue o f (3.10) we find that

Consider the first equality in (3.11). Due t o (3.13) it is obvious that

By virtue o f (3.9) we have

Therefore the first equality in (3.11) yields

In the same way we find that

Equating the terms o f the same order with respect t o & in (3.14), (3.15), (3.16) we get the following recurrent sequence of problems for V,(x):


Here

j

P - @ P , (PP=-C C h E V m K - l , p = 1 , 2 , 'Po - I=1 (")=I jt2

q0 = f , ~j = - C C h 0 , 2 ) " ~ , + ~ - ~ , (3.18)

1=3 (m)=l

j' = 1,2, ... . Existenceof Vj followsfrom Theorem 6.5, Ch. I, when w = IRn and the

coefficients of the elasticity system are independent of E .

3.3. Justzjication of the Asymptotic Expansion.

Estimates for the Remainder

In the previous section we constructed a formal asymptotic expansion for

the function ue which is the solution of problem (3.1). This asymptotic ex-

pansion has the form (3.2) where 1% = N: + NA + N:, N:, NA, N: are

solutions of problems (3.5), (3.6), (3.7) respectively, v, is given by (3.13), V, are solutions o f the problems (3.17).

Let us seek an approximate solution o f problem (3.1) in the form

where k

In the next theorem we give an estimate for the remainder term of the

asymptotic expansion for the solution u' of problem (3.1).

Theorem 3.1.

Let uc be the solution o f problem (3.1). Then for each integer k 3 0 we have


where Mk is a constant independent o f E , u ( ~ ) is defined by the formula (3.19).

Before giving the proof o f this theorem let us establish the necessary esti-

mates for the matrices N:, p = 0,1,2.

Lemma 3.2.

The solutions N:, p = 0 ,1 ,2 , of problems (3.5), (3.6), (3.7) satisfy the

inequalities

where ME, cj , yj are positive constants independent of E

Proof. Let us establish (3.22) for p = 0. By induction with respect t o (cr) =

0,1,2, ... we obtain from Theorem 6.1, Ch. I, that

Changing the variables x = E< and taking into account the 1-periodicity

of N i ( J ) and the fact that the domain fY contains not more than (d + + cells E ( Z + Q n w ) , z E Zn, we get (3.22) for p = 0.

Let us prove (3.22) for p = 1.

Summing estimates (3.8) with respect t o s = 1,2, ... we deduce that

where M, is a constant independent of E . Passing t o the variables x = E(

in this inequality we obtain (3.22) for p = 1 since N : ( t ) are 1-periodic in [ and the domain he contains exactly domains E(Z + d(0, d ) ) , z E Zn,

E Z = (2,O).


Estimate (3.22) for p = 2 is proved in the same way as for p = 1. However, in this case one should use the inequalities (3.9) instead of (3.8).

Estimates (3.23), (3.24) follow directly from (3.8), (3.9) and the definition of the norms in H ' / ~ ( F ~ ) , ~ l / ~ ( f ' ~ ) . Lemma 3.2 is proved. 0

Proof o f Theorem 3.1. Let us show that the vector valued function u(k) given

by (3.19) is the solution of the problem

a L , ( u ( ~ ) ) = f + E ~ + ' ~ ~ ( X , E ) + E ~ + ' - dm ( x , E ) in R' ,

ax , 1 dk)(?, d ) = i p 2 ( f ) + ~ ~ + l d ~ ( f , E ) o n I'd , 1

where

Mo, M I , Mz are constants independent of E .

Then estimate (3.21) would follow directly from Theorem 6.5, Ch. I.

Consider first the boundary conditions for u @ ) ( x ) . We have


where

and hz are assumed to be zeros if the length o f the index cr is negative or is

larger than k + 1. Due t o the conditions (3.18) we have

Taking into account the smoothness of V, in the layer {x : 0 < x, < d )

we conclude from (3.23), (3.28), (3.29) that ~ ( ~ ) ( ? , d ) = @2(? )+~k f1292 (? ,~ )

and the second inequality (3.27) is satisfied.

In the same way we prove that u ( ~ ) ( ? , o ) = Q1(?) + ~ ~ + ' t 9 ~ ( ? , & ) and that

the first inequality (3.27) is satisfied.

Let us now calculate u , ( u ( ~ ) ) on dRE\(ro u rd). Setting 5 = E-'x, due t o (3.3) we have


Here we used the equality Ba(() = 0 for E a-' ( ~ R ' \ ( I ' ~ u I 'd)) which holds

owing to the boundary conditions in (3.5), (3.6), (3.7).

Substituting dk) in (3.1) we obtain

Since N:, N i , N: are solutions of problems (3.5), (3.6), (3.7) respectively

we can replace the expression in the square brackets in (3.31) by

Therefore

Let us transform the expression (3.32) setting ft: = h: for ( a ) 5 Ic + 2, LO, = 0 for ( a ) 2 k + 3. W e have


Therefore it follows from (3.17), (3.18), (3.33) that

where

Estimate (3.26) holds due to the inequalities (3.22), Lemma 3.2 and the

smoothness of V,. It is obvious that satisfies (3.25). Theorem 3 .1 is proved.

Remark 3.3.

It follows from the estimate (3.21) and the equalities (3.19), (3.20) that

53. Asymptotic expansions for solutions of boundary value problems 177

where J ~ ~ ( X , E ) J I ~ ~ ( ~ ~ , 5 M with a constant M independent o f E . Therefore

where llq1(x,~)llL2(n.) I MI, and constant Ml does not depend on E, N, = 0

for negative (a). In particular we obtain for k = 0

where C is a constant independent o f E.

It also follows from (3.37) for k = 0 that

where C1 = const and does not depend on E .

It is important t o note that having taken into account the boundary layers

we obtain in the first approximation an estimate o f order E for the remainder

term, whereas without the boundary layers we can only get an estimate of

order as in Theorem 1.2 with a0 = aE, f0 = f" (see estimate (1.15)).


$4. Asymptotic Expansions for Solutions of the Dirichlet Problem

for the Elasticity System in a Perforated Domain

Here we consider asymptotic expansions in E for solutions o f the Dirichlet

problem for the elasticity system in a perforated domain RE with a periodic

structure. The displacement vector is assumed t o vanish on the surface o f the

cavities S,. Similar asymptotic expansions for solutions o f the Dirichlet problem for the

equation AuE = f in a perforated domain RE were obtained in [52], where

the estimates for the remainder term were proved in the case f E C r ( R ) .

In order t o justify the asymptotic expansion, when f ( x ) is sufFiciently smooth

and may be non-vanishing in a neighbourhood of d R , we construct boundary

layers which exponentially decay in x with the increase o f the distance from x

to dR.

4.1. Setting of the Problem. Auxiliary Results

Consider a perforated domain RE = R n E W , where w is an unbounded

domain o f Rn with a 1-periodic structure, i.e. w is invariant under the shifts

by the vectors z E Zn. It is assumed here that Q\w contains a surface of

class C 1 and R is a smooth bounded domain.

Note that in this section we do not impose any restrictions on the smooth-

ness o f w.

In R' we shall study the following Dirichlet problem

where Ah'(<) are (nx n)-matrices o f class E ( n l , n 2 ) and their elements a$(<)

are 1-periodic in (. The aim o f this section is t o justify the asymptotic expansion

u E ( x ) %' x & I t 2 x NN,(&, ()Do f ( x ) , 1=0 (a)=/

$4. Asymptotic expansions for solutions of the Dir ichlet problem 179

for solutions of problem (4.1). Here a, ID" are the same as in $3, No([) are

matrices of the form N, = NL + N:, where the elements of N t are functions x

defined in w and 1-periodic in (, the elementsaof N~(E, -) decay exponentially E

in Re with the increase of the distance from x t o dR, N:, N: do not depend

on f. Let us now prove some auxiliary propositions to be used below for the jus-

tification o f the asymptotic expansion (4.2).

Lemma 4.1.

For any vector valued function w E HA(R') the following inequalities are valid

where M is a constant independent of E.

W f . The inequality (4.4) follows directly from the First Korn inequality

(2.2), Ch. I, in R, applied t o the function 6 E HA(R) such that 6 = w in Re, 6 = 0 in R\Rc.

Let us prove (4.3). Obviously we can assume that w is defined in Rn and vanishes in Rn\Re. Denote by Tc the set o f all z E Zn such that

E(Z + Q ) n R # 0 and consider the function W([) = w(EJ). Taking into

account the properties o f dw and the fact that W = 0 on dw, we can apply

the Friedrichs inequality o f Lemma 1.1, Ch. I, to W(t) in ( z + Q) n w . We

thus get

(4.5) ( ~ + Q ) n w ( ~ + Q ) n w

Summing up these inequalities with respect to z E TE and passing t o the

variables x = E[ we obtain (4.3). Lemma 4.1 is proved.

Lemma 4.2.

Let U(x) E H1(Rc) be a weak solsution of the problem


where fj E L 2 ( R c ) , j = 0 ,..., n , iP E H 1 ( R c ) . Then

where the constants C, C1 do not depend on E .

Proof. It follows from the integral identity of type (3.5), Ch. I, for w = U -

that

Due t o (3.13), Ch. I, we have

This inequality combined with (4.3), (4.4), (4.8) implies

where the constants K2, do not depend on E .

Therefore estimates (4.7) are valid, since w = U-iP. Lemma 4.2 is proved.

The next theorem shows in particular that the solutions o f problem (4.6)

have the form o f a boundary layer in the vicinity o f d R , provided that = 0

$4. Asymptotic expansions for solutions o f the Dir ichlet problem 181

on (an.) n R and f ' ( x ) , i = 0, ..., n , rapidly decay in Re with the growth of

the distance from x to do .

Consider a scalar function ~ ( x ) E C 1 ( n ) such that T = 0 in a neighbour-

hood of 8 0 , T 2 0 in R, (Vr1 5 M = const. It is assumed that E is so

small that there is a subdomain R1 c R whose closure a' consists o f the cubes

EQ + E Z with z belonging to a set T, C Zn, and dR1 lies in the neighbourhood

of dR where T = 0.

Theorem 4.3.

Let U ( x ) be a weak solution of problem (4.6) with @ ( x ) = 0 on (aRE)\dR

(i.e. Q, E H1(RE, S,). Then

where K, 6 are positive constants independent of e.

Proof. Set v = (ep7 - l )U in the integral identity (3.15), Ch. I, for U ( x ) ,

where p = const > 0 is a parameter to be chosen later. We have

dU dU dU d r / (A" - --, -) exp pr dx = - / (Ahk - , U ) p - exp(pr)dx - ax, ax, n a n axk dzh

Since T = 0 outside R', we find by virtue o f (3.13), Ch. I, that


+ C4 / le(U)12dx . (4.10) R

Due t o the Korn inequality for vector valued functions w E H 1 ( Q n w , a w n

Q ) (i.e. w = 0 on (dw) n Q) we have

This inequality follows from Theorem 2.7, Ch. I, if we extend w as zero t o

Q\w and note that Q\w contains a surface o f class C1. Passing t o the variables t = E-'x in (4.11) we obtain for any w, = E(Q n

w ) + E Z c Oc n O', z E T e , the following inequalities

54. Asymptotic expansions for solutions of the Dirichlet problem 183

Setting p = a(2 a&)-', where a = const E (0 , l ) will be chosen later,

we get from (4.12)

for any w, C RE no'. Therefore

By (4.11) we obtain

I t follows that

a Therefore since p = -

2 d G ~ we find from (4.13) that

< C,(a + a') / l ~ U l ' e x p ( p ~ ) d x + CSI J I~ (u) I ' e x p ( p ~ ) d x . ncnnl ncnnt

1 Thus for all a E (0,min(l , =)) we have

J I v u ~ ' e xp (p r )dx 5 Cia J l e ( u ) / ' e x p ( p ~ ) d x . (4.14) ncnnl Rcnnl

I t follows from (4.10), (4.13), (4.14) that

184 II. Homogenization o f the system o f linear elasticity

t / le(U)12dx . (4.15) n Choosing u sufficiently small and independent o f E and taking into account

u that p = - we obtain from (4.15), (4.14), (4.13) the estimate (4.9).

2 G & Theorem 4.3 is proved.

For the justification o f the asymptotic expansion (4.1) we shall also use the

following result.

Consider the boundary value problem for the system of elasticity

w = 0 on aw , w is 1-periodic in J , where Ahk(E) are matrices o f class E(rcl, n2 ) , w = ( w l , ..., w,)', 3j E L2(Qf l w ) , 33 are 1-periodic in t , j = 0,1 , ..., n.

A weak solution of problem (4.16) is defined as a vector valued function

w E$ ( w ) = r/i/;(w) n H 1 ( Q flu, Q n 8u) which satisfies the integral identity

(6.2), Ch. I, for any v E$ ( w ) .

Theorem 4.4.

There exists a weak solution of problem (4.16) which is unique and satisfies

the estimate n

I l w l ! ~ l ( ~ n w ) 5 IIFjll~2(~m) j = O

54. Asymptotic expansions for solutions o f the Dirichlet problem 185

The proof of this theorem is based on Theorem 1.3, Ch. I, and is quite

similar t o that o f Theorems 6.1, 3.5, Ch. I.

4.2. Justification of the Asymptotic Expansion

Let us substitute the series

in the equations (4.1). Formal calculations similar t o those o f 53.2 yield

(4.18) I=O ( , ) = I

where

Let us seek N , ( [ ) in the form N, = N,O([) + NA([ ) , where N:( [ ) are 0

matrices whose elements are 1-periodic in [ functions belonging t o W (w), and x

the elements of N:( - ) decay exponentially with the increase o f the distance E

from x to dR. We introduce the notation

T , O E I , T t ~ 0 , 1


where I is the unit matrix.

Define the matrices N,O(J) as weak solutions of the problems

The matrices Nt(J) are defined as weak solutions o f the problems

Existence of N:, NA can be easily proved by induction with respect to 1 on the basis o f Theorems 4.3, 4.4, Ch. I.

Lemma 4.5. x

The matrices N:( - ) satisfy the following inequalities E

where C, are constants independent o f E .

x x x Proof. Relations (4.19), (4.20), (4.21) show that N , ( - ) = N : ( - ) + N ~ ( E , -)

E E E are solutions of the following boundary value problems:

Let us use induction with respect t o 1. For 1 = 0 i t follows from (4.23) due to (4.7) with = 0 that

§4. Asymptotic expansions for solutions o f the Dir ichlet problem 187

where Co is a constant independent o f E .

Let 1 = 1. By virtue of (4.7), (4.24) we get

These inequalities and (4.26) imply that for k <_ 1 we have

Suppose now that the inequalities (4.27) hold for k 5 1 - 1. Let us show

that they also hold for k = 1.


Therefore due to (4.27) for k 5 1 - 1 we obtain (4.27) for k = 1.

The elements o f the matrices N:(() are 1-periodic in J . Therefore esti-

mates (4.22) for j = 0 are obvious. For j = 1 estimates (4.22) follow from

(4.22) for j = 0 and the inqualities (4.27). Lemma 4.5 is proved.

Lemma 4.6. 2

The elements o f matrices N:(a, -) are o f boundary layer type, i.e. for any €

subdomain R0 such that no c R the following inequalities are valid

where C,, y are positive constants independent of E .

k f . Consider a domain R' such that 0' c R, Q0 c R' and the distance


between 52' and dR' is larger than K > 0, where K is a constant independent

of E , and 0' consists of the cubes E ( Q + z), z E T , for some subset T C Zn. The parameter E is assumed so small that R' with the above properties exists.

Let us construct a scalar function T ( X ) such that T E C 1 ( n ) , T r 1 in RO, K

T - 0 outside the --neighbourhood of RO, JVTJ 5 C K - ' , C = const. 2 x

Using the induction with respect t o s = 0 , 1 , 2 , ... , let us prove that N:( - ) E

for a = (a1, . . . ,a,) satisfy the inequalities

where C,, 6 are positive constants independent of E .

Let us first show that (4.29) holds for the matrix N,' which is a solution

of the problem

x Since N,'(-) = 0 on dRE\bQ, we can apply the estimate (4.9) of Theorem

E 4.3 t o N,'. We get

< K J IV.NiJ2dx. n *

This inequality together with (4.22) implies (4.29) for N,'. Fix a positive

integer s and suppose that (4.29) is valid for all NA with a = (a, , ..., a,),

1 < s - 1. Let us show that (4.29) holds for NA with 1 = s, where NA is a solution of the problem

x Taking into account the fact that N:( - ) = 0 on BRE\BR and using the

e estimate (4.9) o f Theorem 4.3 applied to N;,,,, , , , we obtain

$4. Asymptotic expansions for solutions o f the Dir ichlet problem 189

67 + c 4 E 2 J IN: a...a* 1' ~ x P ( - ) ~ x + E - ~ J IN:^...^^ I' ~ X P ( - ) ~ X . E E

n*nnf ncnni 6r I Estimating the first integral in the right-hand side o f this inequality by

(4.22), and applying the assumption o f induction to the other integrals, we

get (4.29) for NL

The estimates (4.28) follow from (4.29), since T r 1 in RO. Lemma 4.6 is

proved.

Theorem 4.7.

Let u E ( x ) be a weak solution o f problem (4.1) with f E CS+'(i=l). Set

u:(x) = 2 of+' Na(;)Da f ( x ) , 1=0 (a)=[ 1 . .

v:(x) = 2 ol+' N,o(:)D" f ( x ) , f=o (a)={ J

where N, = Nz + NA, N:, NA are weak solutions of problems (4.20), (4.21)

respectively. Then

l l " z ( ~ ) - uC(x) I I~ l (n*) 5 CO&~+' I l f llca+2(n) , (4.31)

where R0 is a subdomain of R such that no c R, the constants Co, C1 do

not depend on E ; Cl may depend on RO.

Proof. Let us apply the operator LC to u: - uc. Assuming N, = 0 in (4.17)

for (a) 2 s we obtain in the same way as (4.18) that


Note that, because of (4.22), the L 2 ( R e ) norms of the elements of the x d x

matrices N , ( - ) , - N, ( - ) are bounded by a constant independent of E . E atj E

Therefore applying Lemma 4.2 with @ = 0, fj = 0, j = 1 ,..., n , to u: - uc

we get the estimate (4.31).

To prove (4.32) i t suffices to observe that N , = N: + N: and N i satisfy

the inequalities (4.28). Theorem 4.7 is proved.

55. Some generalizations for the case o f perforated domains 191

$5. Asymptotic Expansions for Solutions of the Dirichlet Problem for

the Biharmonic Equation. Some Generalizations for

the Case o f Perforated Domains with a Non-Periodic Structure

5.1. Setting of the Problem. Auxiliary Propositions

The methods suggested in $4.1 and $4.2 can also be used t o justify asymp-

totic expansions for solutions o f the Dirichlet problem for higher order elliptic

equations. In this section we consider a special case which is particularly

important for mechanics, namely, the Dirichlet problem for the biharmonic

equation:

and obtain a complete asymptotic expansion for solutions of this problem.

Here RE is a perforated domain of type I with a periodic structure described

in 54.1, f ( x ) is a sufficiently smooth function in R; v is the outward normal.

We seek the asymptotic expansion for the solution o f problem (5.1) in the

form

u: = E'+' N,(E,[)V" f ( x ) , [ = E-'x , l=O ( + I

(5.2)

where D", a are the same as in $3.2.

We shall prove that solutions of (5.1) admit asymptotic expansions of type

(5.2) after establishing some preliminary results.

Lemma 5.1.

For any v E H i ( R E ) the following inequality is satisfied

a2v a2v where E2(u) = (- -) ' I2 , MI is a constant independent of E .

dx;dxj dx;dxj

Proof. Obviously it is sufficient to prove (5.3) for v E C,"(RE). Set v = 0 in

Rn\Rc and denote by T' the set o f all z E iZn such that ~ ( z + Q) n R # 8. Consider the function W ( [ ) = v(e[ ) . Since W = 0 in Rn\w, the Friedrichs

inequality for each o f the sets w, = z + Q yields

II. Homogenization o f the system o f l inear elasticity

Summing these inequalities with respect t o z E TE and passing t o the

variables x = E < , we obtain (5.3). Lemma 5.1 is proved.

Let O E H 2 ( R E ) , fj E L 2 ( f l E ) , j = 0,1,2, ..., n.

We say that U ( x ) is a weak solution o f the problem

if W = U - @ belongs to H i ( f l e ) and satisfies the integral identity

for any v E H,2(RE). Denote by H ~ ( W ) the completion with respect t o the norm IIvIIKlcsnw, of

the functions v ( t ) such that v E C 2 ( 3 ) , v = 0 in a neighbourhood o f dw and v ( J ) is 1-periodic in J . Here w is an unbounded domain with a 1-periodic

structure, the same as in s4.1.

We say that w is a weak solution o f the problem

where Fj E L 2 ( u n Q), F j ( J ) are 1-periodic in J , j = 0, ..., n , if w E H ; ( W )

and satisfies the integral identity

$5. Some generalizations for the case of perforated domains 193

for any v E H;(W).

The existence and uniqueness of solutions of problems (5.4), (5.6) follow

from Theorem 1.3, Ch. I .

Lemma 5.2.

A weak solution U(x) of problem (5.4) sdtisfies the following inequalities

where K1, I<2, I<3 are constants independent of E .

Proof. Set v = W = U - @ in the integral identity (5.5) for v = W . We get


It follows that

Since W = U - O, this inequality implies (5.8).

Due t o (5.3) we have

llVW11~2(n*) I M I E l lEz(W)II~~(nc) .

From these inequalities and (5.11) we obtain (5.9), (5.10), since W = U - cP.


Let 7 ( x ) be a function o f class C 2 ( n ) such that 7 = 0 in a neighbourhood

of aR, 7 > 0 in R. Consider a subdomain R' defined just before Theorem 4.3

and assume that 7 = 0 outside 0'.

Theorem 5.3. a@ Let U ( x ) be a weak solution of problem (5.4), cP = - = 0 on aRe\aR (i.e.

av @ E HZ(Rc , aRc\aR)). Then

55. Some generalizations for the case of perforated domains 195

where KO > 0 , 6 > 0 are constants independent o f E . (Note that I(o and S can depend on 52' and 1 1 ~ ( x ) 1 1 ~ ~ ( ~ ) . )

Proof. For any v(x) E HZ(Rc) the function U ( x ) satisfies the integral identity

Set v = (epT - 1)U, where p > 0 is a parameter t o be chosen later. W e have

Since T - 0 outside of R', we obtain by virtue of the Holder inequality and

(5.13) that


In the same way as in the proof o f Theorem 4.3 t o obtain (4.14), we find

that

where p = u / I ( a , I( is a constant independent of E , u E (0 , l ) is a constant

to be chosen later.

Since U ( x ) can be approximated in the norm of H2(R") by functions van-

ishing in a neighbourhood o f BRe\aR it follows that inequality similar t o (5.15)

holds for the first derivatives o f U ( x ) , i.e.

/ IVU12e"dx < K2a2 1 I E ~ ( U ) ~ ~ ~ ' ~ ~ X . (5.16) nennl ncnnl

Estimates (5.15), (5.16) yield

where K 2 , I(3 are constants independent o f E .

From (5.14), (5.16), (5.17) we obtain

55. Some generalizations for the case o f perforated domains 197

(5.18)

n *

where p = u I K E . If we choose a sufficiently small but independent of c, we

get from (5.18) the following inequality

where M I , M2, M3 are constants independent of c.

Estimate (5.12) follows from (5.19), (5.16), (5.17). Theorem 5.3 is proved.

5.2. Justification of the Asymptotic Expansion for Solutions of

the Dirichlet Problem for the Biharmonic Equation

Suppose that f E C"t4(fi) in (5.1). Let us seek an asymptotic expansion

for the solution o f (5.1) in the form (5.2) where N O ( € , [ ) = N:( [ ) + N:(e,E), x

N:(() are 1-periodic in (, N;(E, -) are functions o f boundary layer type in E

RE, which decay exponentially with the increase of the distance from x to dR. It is easy t o verify that

Therefore


From (5.2), (5.20) we obtain

where 6,, is the Kronecker symbol.

Let us define the functions Na(e,t) as weak solutions of the following


a A ~ N ,,,, = - ~ - A ~ N , , - ~ ~ , , , , A N 4 a 2 N ~ in & - ~ a e

ata1 O - at at,, 7

a1

$5. Some generalizations for the case o f perforated domains 199

On the basis of Theorem 1.3, Ch. I, we can easily prove by induction that

N o r ( [ ) exist.

Let us show that N , ( E , ~ ) = N: + NA, where N : ( [ ) are functions 1-

periodic in t and belonging to H ~ ( w ) ; N:(E, f ) are of boundary layer type in

Re. Set

Define the functions N:( t ) as solutions o f the following boundary value

problems

Obviously Theorem 1.3, Ch. I, guarantees the existence of N,O([). In the domain &-'RE define the functions N: as weak solutions o f the

Dirichlet problems


Obviously N , = N,O + N t .

Lemma 5.4. x x

The functions N:(-) , N ~ ( E , -) satisfy the inequalities E E

where the constants M, do not depend on E ; j = 0 , l .

This lemma is proved by induction in the same way as Lemma 4.5.

Lemma 5.5. x

The functions N ~ ( E , -) are of boundary layer type, i.e. for any subdomain Q0 e

such that !=lo c R the following inequalities hold

where C,, y are positive constants independent o f E , (C, and y may depend

on RO).

Proof. The estimate (5.26) is obtained in the same way as (4.28) in Lemma

4.5. Let us indicate the main steps of the proof.

Consider a subdomain R' c R which consists of the cubes ~ ( z + Q) for

some z E Zn, and let ~ ( x ) E C 2 ( O ) possess the same properties as in the

proof of Lemma 4.5. x .

The function N t 1 . , , , , ( ~ , -) IS a weak solution o f the problem E

55. Some generalizations for the case of perforated domains 201

aN: Since No = - = 0 on dRE\dR, we can apply Theorem 5.3 to U = N:.

av Due to (5.12) for U = N:l,,.,m we get

where IC2 is a constant independent of E.

From these inequalities and (5.25) we obtain by induction with respect t o

m = 0,1,2, ... that

where the constant I~,,,,,,, does not depend on E . Taking into account that

T = 1 on RO, we obtain the estimates (5.26). Lemma 5.5 is proved.

Theorem 5.6 (On the asymptotic expansion o f solutions of problem (5.1)).

Let u E ( x ) be a weak solution o f problem (5.1) and let f E C s f 4 ( f i ) ,

ti:(.) = 2 mlt4 x N,(E, f )Do f ( x ) , I=O (,)=I E

202 11. Homogenization of the system of l inear elasticity

where N , ( E , [ ) = N: + NA, N:, N i are weak solutions of problems (5.23),

(5.24). Then

where C1, C2 are constant independent o f E , R0 is a subdomain o f R such

that fiO c R, the constant C2 may depend on RO.

Proof. I t is easy t o see that by virtue o f (5.21), (5.22) u: - u' is a weak

solution of the problem

Due t o the estimates (5.8) we get

m=s+l as, ..., om=]

$5. Some generalizations for the case o f perforated domains 203

3+4 n

+ E ~ + ~ E ~ C C IINa5...~,ll~~(n*) I l f Ilcs+4(ii) . m=s+l as, ..., o,=l I

Since Na = Nz + N:, we obtain by virtue o f (5.25) that

This estimate and (5.9), (5.10) imply (5.27). Inequalities (5.28) follow from

(5.27), (5.26). Theorem 5.6 is proved. 0

5.3. Perfarated Domains with a Non-Periodic Structure

Analysing the proof o f Theorems 4.3, 5.3, we can easily see that estimates

similar t o (4.9), (5.12) can also be obtained in the case o f some non-periodic

structures. d*

Suppose that a subdomain R' c R is such that fi' c R and fi' = U B,', s=l

where B: are bounded domains of Rn such that B; n B; = 0 for i # j .

Suppose also that I':, s = 1 , ..., d,, are closed sets r: c @ and for each

v E C1(&) such that v = 0 in a neighbourhood o f r:, the Friedrichs inequality

holds with a constant C * independent o f c and s.

Let T ( X ) be a function in C2(fi) such that T = 0 in R\S2', T 2 0 in R, ) ) ~ ) ) ~ z ( ~ ) 5 M * , where M* is a constant independent of E, s.

Theorem 5.7.

Let U ( x ) be a weak solution of the boundary value problem


acp where iP E H Z ( R c ) , cP = - = 0 on R' n d o c , fj E L2(R') , j = 0 , ..., n.

a v Then for U the estimate (5.12) is valid with constants KO > 0, 6 > 0 depend-

ing only on C* and M*. acp Suppose that fj = 0, j = 0 , ..., n in R e n R 1 , @ = - = 0 on R 1 n d R e and

d v the domain R0 is such that no c R', p(dRO, d o ' ) 2 K > 0 with x independent

of E . Then the solution U ( x ) satisfies the inquality

where C > 0 is a constant depending only on C * , RO. The estimate o f type (5.12) in this case is proved by the same argument

as Theorem 5.3. The estimate (5.30) follows from (5.12) if we take ~ ( x ) such

that T ( X ) = 1 in RO, T ( X ) = 0 outside the ~/2-neighbourhood o f RO, the

C 2 ( n ) norm of ~ ( x ) is bounded by a constant independent o f E.

Consider now the system of elasticity.

Suppose that the sets rz, s = 1, ..., d,, are such that for each v E C 1 ( & ) , v = 0 in a neighbourhood o f l?: the following inequality is valid

where C; is a constant independent o f E . let ~ ( x ) E C 1 ( O ) , ~ ( x ) = 0 outside

R', 1 1 ~ 1 ) ~ 1 ( ~ ) < M;, where M: is a constant independent o f E .

Theorem 5.8.

Let U ( x ) be a weak solution o f the boundary value problem for the elasticity

system

where fj E L2(R') , j = 0 ,..., n , @ E H1(R ' ) , @ = 0 on R' n do' and

matrices A h k ( x , & ) belong t o the class E(nl, n 2 ) with n l , n2 > 0 independent

$5. Some generalizations for the case of perforated domains 205

of E . Then for U ( x ) the estimate (4.9) holds with constants I( > 0, S > 0

depending only on C: in (5.31), M:, K,, KZ.

Suppose that f j G 0, j = 0, ..., n , in RE n R', = 0 on 0' n 80' and

the domain R0 c R' is such that p(8R0,dR') 2 K > 0, where 6 is a constant

independent of E . Then the solution U ( x ) satisfies the inequality

where C is a constant depending only on C;, M;, K,, nz , RO. The estimate (4.9) in this case is proved in the same way as the corre-

sponding estimates in Theorem 4.3. The inquality (5.32) follows from (4.9),

if we take T ( X ) such that T = 1 on RO, T = 0 outside the ~/2-neighbourhood

of RO, I ( T ( I ~ ~ ( ~ ) is bounded uniformly in E .


$6. Homogenization of the System of Elasticity with

Almost-Periodic CoefFicients

In this section we consider homogenization o f solutions o f the Dirichlet

problem for the system of elasticity with rapidly oscillating almost-periodic co-

efficients.

6.1. Spaces of Almost-Periodic Functions

Denote by TriglRn the space o f real valued trigonometric polynomials.

Thus Trig Rn consists o f a l l functions which can be represented in the form

of finite sums

U(Y) = ctexp {i(y,E)) , C

y,[ E R n , (y,() = y;&, ct = = const . (6.1)

The completion of TrigRn in the norm sup Iu(y)l is called the Bohr R"

space of almost-periodic functions and is denoted by A P ( R n ) (see [50], [51]).

The space of all finite sums having the form (6.1) and such that Q = 0 is 0

denoted by Trig Rn. Let 1I, E LLc(EP). We say that M ($1 is the mean value of +, if

$(e-'z) + M ($1 weakly in L ~ ( G ) as e -+ 0

for any bounded domain G C Rn. I t is well known that for any function g E L:,,(Rn), which is T-periodic in

y, the mean value exists and is equal to

[O,TIn={y : O s y j S T , j = l , ..., n ) .

Thus each function belonging t o TrigRn possesses a finite mean value,

and therefore we can introduce in Tr igRn the scalar product defined by the

formula

$6. Homogenization o f the system o f elasticity 207

The completion of T r i g R with respect t o the norm corresponding t o the

scalar product (6.2) is denoted by B 2 ( R n ) and is called the Besicovitch space

of almost-periodic functions.

We keep the symbol M ( $9 ) for the scalar product of the elements II, and

g in B 2 ( R n ) . As before we say that a matrix (or vector) valued function belongs t o one

of the spaces Trig Rn, B 2 ( R n ) , A P ( R n ) , if its components belong t o the

corresponding space. In this case the mean value is a matrix (or vector) whose

components are the mean values o f the components o f the given function. We

shall also use the notation (1.8), (1.9), Ch. I, for matrix (or vector) valued

functions.

As usual e(u) denotes the symmetric matrix with elements e l j (u) = auj 1 (k + -), where u is a vector valued function u ( y ) = ( u l , ... ,u,).

2 a y , 8 ~ 1

Lemma 6.1.

Suppose that f , g E Trig Rn, and u = ( u l , ..., u,) E Trig IRn. Then

Moreover for any functions Flh E Trig Rn such that Flh = Fhl, 1 , h = 1 , ...,n ,

there is a vector valued function w E Trig Rn such that

Proof. Note that

M {ei("t)) = 0 for [ # 0 . Let

f = C f E e ' ( ~ ' P ) , = C g,e'("d E 7


Then by virtue of (6.6) we have

Let us prove inequality (6.4). Let u = (ul, ..., u,), uj = 4 e ' ( ~ ~ ) . Then E

due to (6.6) we find that

Eta

This implies (6.4).

Let us show now the existence of the solution of equations (6.5). Suppose

that

We seek w in the form w = C wCe"ylC). Then E

§6. Homogenization o f the system o f elasticity

Obviously for each # 0 the coefficients w: must satisfy the system

For each [ # 0 system (6.7) has a unique solution, since the corresponding

homogeneous system has only the trivial solution. Indeed, let # 0, c y = 0,

I , h = 1, ..., n. Then multiplying the equations (6.7) by and summing up

with respect t o I from 1 to n we obtain

Therefore w: = 0.

Let us replace by -( in (6.7) and write the complex conjugate equation.

One clearly has w: = wkt , since ckh = z'$. Lemma 6.1 is proved.

Consider the Hilbert space of (n x n)-matrices whose elements belong t o

B2 (F) and denote by W the closure in this space of the set

Elements o f W will be denoted by e, Z, etc.

The norm o f an element e E W is given by

M {eueu)'12 = M {(e, e))'I2 .

It should be noted that not every element e E W can be represented as

e = e(u) with u E B2(Rn). Nevertheless for every e E W there is a sequence

o f vector valued functions {u6) with components in T r i g R n and such that

M {(e - e(u6)I2) + 0 as 6 -, 0.

6.2. System of Elasticity with Almost-Periodic Coeflcients.

Almost-Solutions

Consider the system of linear elasticity

11. Homogenization o f the system of l inear elasticity

where A ~ ~ ( ~ ) are matrices of class E ( K ~ , c 2 ) , ~ 1 , K Z = const. > 0, whose

elements belong to A P ( R n ) , u = ( u l , ..., u,) , fj = ( f i j , ..., f n j ) are column

vectors, fjl = f i j E A P ( R n ) . In the general case o f almost-periodic coefficients in A P ( R n ) no proof

for the existence o f a solution u E B 2 ( B n ) of system (6.8) has yet been

found. However we can construct the so-called almost-solutions u6 o f (6.8)

with components in Trig Rn. This fact was established in [149].

Following [I491 we shall outline here a method for the construction o f such

almost-solutions.

Due t o the conditions (3.2), Ch. I, one can rewrite system (6.8) in the

form:

In the rest o f this paragraph we shall denote by qh the column ( q I h , ..., qnh)* of the matrix q with elements qih. Then system (6.9) becomes

where ek(u) = (elk(u), ..., e,,k(u))*. If the coefficients akk(y) and the functions f l j ( y ) are 1-periodic in y, then

the definition o f a weak 1-periodic solution u ( y ) of system (6.8) can be reduced

to the integral identity

for any v E w;(Rn), where f is a matrix with elements fib and

Let the coefficients a:/ be almost-periodic functions of class A P ( R n ) . Then in analogy with (6.10), (6.11) we consider the system

§6. Homogenization of the system of elasticity 21 1

and define a weak solution o f (6.12) as the element E E W, 2 = { E i j ) , which

satisfies the integral identity

for any e E W.

It follows from Lemma 3.1, Ch. I, that the bilinear form M ( ( M 2 , e ) ) is

continuous on W x W, i.e.

fot any 2,e E W , since for a ( y ) E A P ( R n ) , f E B 2 ( R n ) we have a f E

B 2 ( R n ) and Ilaf ( I B ~ R ~ ) 5 SUP lal l l f I I B ~ R ~ ) . R"

Moreover the condition (3.8), Ch. I, yields the inequality

for any e E W.

By virtue o f (6.14), (6.15) the bilinear form M { ( M 2 , e ) ) satisfies all

conditions of Theorem 1.3, Ch. I, with H = W. Therefore, the solvability of

problem (6.12) in W follows directly from Theorem 1.3, Ch. I.

Let us show that we can find vector valued functions = ( U f , ..., u:) E Trig Rn which approximate solutions o f the system (6.9) in the sense

o f distributions. To this end we need the following

Lemma 6.2.

Let f j , A h k E A P ( R n ) and let E E W be a weak solution o f system (6.12).

Then there exist sequences of vector valued functions U s E Trig Rn and ma-

trices gs E A P ( R n ) with columns g: = ( 9 4 , ..., g$) , gfj = g:,, j , 1 = 1, ..., n ,

such that

lim M {1g612) 4 0 , 6-0

lirn M (12 - e(u*)12) + 0 6-0

(6.17)

as 6 -+ 0, 6 > 0, and the integral identity

II. Homogenization o f the system of l inear elas t i c i ty

holds for any $(y) = (&, ..., 4,) E C,O"(Rn).

Proof. By the definition o f the space W we see that there is a sequence

Us E T r i g R n which satisfies the condition (6.17). Therefore due t o (6.13),

(6.14) we have

for any e E W, where y(6) -+ 0 as 6 + 0.

Set

Since the elements @ f h of matrices @6 belong t o AP(Rn), we can represent

Q6 in the form

where Q6, F6 , G6 are symmetric matrices with elements ath, G, Gfh, F6 E

Tr igRn , G6 E AP(Rn), and

lim M ( 1 ~ ~ 1 ~ ) = 0 . 6-0

(6.21)

Since

it follows from (6.19), (6.21) that

for any e E W, where+y1(6) -t 0 as 6 + 0.

According t o Lemma 6.1 there is a vector valued function w6 E Tr ig Rn such that


Multiplying each o f these equations by wf and summing wi th respect t o 1 from 1 t o n, we find by virtue o f (6.3) and Lemma 6.1 that

Therefore

It follows from (6.20), (6.17), (6.21) that M IF^^^) are bounded by a

constant independent o f 6. Therefore due t o (6.24), (6.25) we obtain

Obviously by virtue o f (6.20) we have

where g6 = e (w6) + G6; and the equations (6.27) hold in the sense of distri-

butions.

The convergence (6.16) is due to (6.26), (6.21), and the integral identity

(6.18) follows from (6.27) and the conditions (3.2), Ch. I, for ahk. Lemma 6.2

is proved.

The vector valued functions U6 are called almost-solutions o f system (6.9)

with almost-periodic coefficients.

Let us now establish some other properties o f the almost-solutions U s ,

which are essential for the study of G-convergence o f elasticity operators with

almost-periodic coefficients.

Lemma 6.3.

Suppose that fj, Ahk E AP(Rn), 2: is a weak solution o f system (6.9), 2: E W, and U6 (6 + 0 ) is a sequence o f almost-solutions o f system (6.9). Then for

any sequence E -+ 0 there exists a subsequence € 6 -+ 0 as 6 + 0 , such that

.c6 (u6 (6) + C 6 ) -+ 0 weakly in H1(O) , (6.28)


where cs is a constant vector,

rap(:) + M { A P ~ E ~ - f,} weakly in L 2 ( R ) , p = 1, ..., n , (6.29)

a x - r 6 h ( G ) + 0 in the norm of H-I (R) , (6.30) axh

as 6 -+ 0 , where

0 c Rn is a bounded Lipschitz domain.

Proof. Taking into account the inequality (6.4) o f Lemma 6.1, the fact that

U s E Trig Rn, and the convergence (6.17) we obtain

where I< is a constant independent of 6.

Denote by G 6 ~ " ( x ) the matrices whose elements are

a x G ~ : ( x ) - E - U f ( - ) E Trig Rn .

a x , E

Note that the matrices G61c are not necessarily symmetrical.

By the definition of mean value we have

Similarly

l im / (g6(Z)I2dx = ( m e s a ) M {1g61'} , E'O E

n

where g6 are the matrices from Lemma 6.2.

It is obvious that

56. Homogenization of the system of elasticity 215

M e ( u 6 ) ( 5 ) + M { M e ( u 6 ) } weakly in L2(R) as e + 0 . (6.35)

Moreover

G ~ , ' ( X ) + 0 weakly in L2(R) as e -+ 0 , (6.36)

dU6 x 0 since - (-) €Trig Rn and ei(: R, -i 0 weakly in L2(R) as E -+ 0 for J # 0.

~ Y I E Let V = {q1,v2, ...} be a countable dense set in the Hilbert space of a l l

matrices with elements in L2(R). For each 6 by virtue o f (6.33)-(6.36) we can find € 6 such that

f o r m m = 1 , 2 ,...; qm E V . It follows from (6.32), (6.39) that the norms llG6"611Lz(n) are bounded by

a constant independent o f 6, and inequalities (6.40) imply that for any qm E V

we have

Therefore

G6"6(x) + 0 weakly in L2(R) as 6 + 0

Set

where the constants c6 are chosen such that


Then due to the Poincari inequality we have

where c is a constant independent o f 6. Since the right-hand side o f (6.44)

is bounded in 6, it follows from (6.42). (6.44) that E ~ . V ~ ' ( ? ) + V weakly €6,

in H1(SZ) and strongly in L2(R) for a subsequence 6' + 0. Here we used the

weak compactness o f a ball in a Hilbert space and the compactness of the

imbedding H1(R) c L2(R). By virtue o f (6.43), (6.42) V = 0. Thus the convergence (6.28) is estab-

lished.

Since the elements a k of matrices are bounded, it follows from (6.17),

(6.37) that the norms J I M e ( u 6 ) (E) I l u ( n ) are bounded by a constant inde-

pendent o f 6, and

lim M { M e ( u 6 ) ) = M { M 2) . 6-0

Therefore we conclude from (6.41) that

M e ( u 6 ) ( f ) 3 M { M 2) weakly in LZ(SZ) as 6 --+ 0 . E6

x To complete the proof o f (6.29) it is sufficient t o observe that rs(-) are

€ 6 given by (6.31).

Let us prove (6.30). For any $ ( I ) = ($,, ..., $,) E C,"(R) due t o (6.31),

(6.18) we obtain

Therefore


This inequality together with (6.38), (6.16) implies (6.30). Lemma 6.3 is

proved.

6.3. Strong G-Convergence of Elasticity Operators with

Rapidly Oscillating Almost-Periodic Coeficients

In a bounded Lipschitz domain R consider the Dirichlet problem for the

system of elasticity

where f E H-'(R), matrices Ahk(y) belong t o the class E(nl, n2), n l , nz =

const > 0, and their elements akk(y) are almost-periodic functions of class

AP(Rn).

If matrices Ahk((y are 1-periodic in y, then according t o $1, Ch. II, the

homogenized elasticity system corresponding t o the strong G-l imit of the se-

quence {C,) has the following coefficients

where N: = (N:a, ..., N:,) is the s-th column of the matrix Nq, ejk(N:) = 1 dNZ* dN!a - (- + -), and the columns N: are 1-periodic solutions o f the system 2 a y j d ~ k

Setting A:q = (a;:, ..., a::), A:q = (A;:, ..., iK), we can rewrite (6.46),

(6.47) in vector form


Now let belong t o A P ( R n ) . I t was shown above that for fixed q, s

we can find weak solutions e"" E W (P" is a matrix with elements dg:) of the

system

which is similar t o (6.12) with

Set

and denote by ah¶ the matrices with the elements 6::

Theorem 6.4.

Suppose that APq(y) are matrices of class E(n1, n2) , 61, n2 = const > 0, and

their elements are almost-periodic functions belonging to A P ( R n ) . Then the sequence of operators

is strongly G-convergent t o the elasticity operator k whose coefficients are

given by (6.50).

Proof. Let us show that there is a sequence 6 -+ 0 and matrices N:, q =

1, ..., n . such that matrices A"(:), dhk satisfy the Condition N of $9, Ch.

I,'as 6 + 0, where dhk are matrices whose elements are defined by (6.50). By

virtue o f Theorem 9.2, Ch. I, this means that L,, 9 as 6 + 0. Due t o

the uniqueness of the strong G-limit (see Theorem 9.3, Ch. I) it follows that

L, S k as E --+ 0.

Fix q, s and consider the almost-solutions U& = ( U h S , ..., U:ns) of system

(6.48) constructed in Lemma 6.2. Set

$6. Homogenization of the system of elasticity 219

where c:, are constant vectors satisfying the condition (6.28) with Us = UsP,. Denote by N:(x) the matrices whose columns have the form (6.51). Let us

verify that the matrices N i , APQ - , h'4 satisfy the Condition N as 6 -+ 0. (2) Indeed, the Condition N 1 follows from (6.51) and (6.28). Consider the

Conditions N2, N3.

Due to (6.29)-(6.31) we have

weakly in LZ(R),

in the norm o f HW1(R), as 6 -t 0.

These relations show that Conditions N2 and N3 are satisfied, since due t o

(6.50) the expression in the right-hand side o f (6.52) is equal to ,@. Theorem

6.4 is proved.


57. Homogenization of Stratified Structures

7.1. Fonulas for the Coeficients of the Homogenized Equations.

Estimates of Solutions

Consider a sequence {LC) of differential operators o f the linear elasticity

system

belonging t o class E ( I c ~ , n2) with constant n l , n2 > 0 independent o f E , x (see

53, Ch. I). Here E is a small parameter, E E ( 0 , l ) ; the elements o f matrices

AF(t, y ) are bounded (uniformly in E) measurablefunctions of t E R 1 , y E Rn with bounded (uniformly in E) first derivatives in yl, ..., y,; p ( x ) is a scalar

function in C 2 ( o ) such that 0 5 p ( x ) 5 1 , ( V p l 2 const > 0; R is a bounded

smooth domain.

Let us also consider the following system of linear elasticity

whose coefficient matrices belong to E ( k l , 22) and k l , k2 are positive con-

stants which may be different from n1, n2; the elements of the matrices

a' j ( t , y ) are bounded measurable functions o f t E R1, y E Rn, possessing

bounded first derivatives in yl, ..., y,. In this section we consider the following Dirichlet problems

Problems of type (7.3) serve in particular t o describe stationary states of

elastic bodies having a strongly non-homogeneous stratified structure formed

by thin layers along level surfaces o f a function cp(x) (see [go]). Here we obtain estimates for the difference between the displacements uc

and u , the corresponding stress tensors and energies. We establish explicit

§7. Homogenization of stratified structures 221

dependence of the constants in these estimates on the coefficients of system

(7.3). We also obtain the necessary and sufficient conditions for the strong

G-convergence of the sequence {L,) t o the operator k as E + 0, and give

explicit formulas for the coefficients of k . The corresponding spectral problems are studied in $2, Ch. Ill.

Let the matrices N;(t, y), M$(t, y ) be defined by the formulas

where ( y 1 ( y ) , ..., y n ( y ) ) = (3, ..., *) = V y , B-I is the inverse matrix a y l a y n

of B. It will be proved in Lemma 7.5 that the matrix [cpl(y)cpk(y)A,kl(~, y)]-'

exists and that its elements are bounded functions (uniformly in c).

To characterize the closeness between solutions of problems (7.3), (7.4)

we introduce a parameter 6, setting

6, = max { I M G ( P ( X ) ~ X ) I , ~ N ; ( P ( x ) ~ x ) \ , X E R

l , i , j = 1 ,..., n

For a given matrix B with elements bra we set IBI = (bk'bk')1/2.

Theorem 7.1.

Let u", u be thesolutions of problems (7.3), (7.4) respectively, and u E H 2 ( R ) . Then the following estimates hold


. a u c . - . . a u ,

where yf A:] -, jt At3 -, 2 = 1, ..., n , the constants q, cl do not a x j a x j

depend on E

Proof. Define v c ( x ) as the solution of the problem

Then it is easy to calculate that

Therefore

The right-hand side of this equation is understood as an element of H - ' ( 0 ) . Let us show using the definition of S,, N,", MG that

$7. Homogenization of stratified structures 223

where IP;,(X,E)I 5 ~ 2 6 ~ ~ Icx,(x,~)I 5 c3&, Icx;,(x,~)l I ~46,) and the con-

stants c2, c3, c4 do not depend on E .

Indeed, we obviously have

Multiplying these equations by cpk and summing them up with respect t o aM;

k. we obtain (7.11) due t o the inequality 1-1 < c6,. ay1

Setting k = i in (7.14) we find by virtue of (7.5) that

This equality implies (7.12).

According to the formulas (7.6), (7.5) we have

A .

= A'" - A> v c p , c p j ~ y [ $ O ~ ~ ~ A , ~ ' ] - ' ( ~ ~ ~ - A r ) + ( Y ; , ( x , E ) =

Let us estimate the H-'(a)-norm of the right-hand side o f (7.10). For any

column vector $ = E C,"(R), due to (7.13), (7.11) we obtain

aMi; au a$ = - J ( - - , -)dx +

n at dx, ax;

du dlC, + J (ais(x ,&) - 7 -)dx = ax, axi

n

II. Homogenization of the sys tern of linear elasticity

Therefore, taking into account (7.12) and the definition of 6, we find that

where c5 is a constant independent of E .

Let us estimate the second term in the right-hand side of (7.10) in the

norm of H - ' ( 0 ) . Using the definition of 6, we get

I t thus follows from (7.10), (7.14), (7.15) that

5 7. Homogenization of stratified structures 225

where C, is a constant independent o f E and u.

Therefore by virtue o f Theorem 3.3, Ch. I, and Remark 3.4, Ch. I, we obtain

from (7.10), (7.16) the following inequality

where c8 is a constant independent o f E.

We now estimate the norm I l ~ , l l ~ l ( ~ ) . Set

where $, = 1 in the 6,-neighbourhood of dR, +, = 0 outside the 26,-

neighbourhood of dR, $, E CW(n), 0 5 $, 5 1, 6, IV$,I I const. It

follows from Theorem 3.1, Ch. I, that

Let us estimate IldcIIHl(n). We have

and therefore

where wl is the 26,-neighbourhood o f dR. By virtue of Lemma 1.3, Ch. I,

IlVuIIZzcw1) L cldc IIull&Zcn,. Hence

Estimates (7.17), (7.18) imply (7.7). Let us now prove (7.8). It follows from (7.7) that

where

IIqf Il~z(n) I ~ 1 4 6 , " ~ 1(~1(~2(n) .

Due to (7.13) we get

II. Homogenization o f the system o f linear elasticity

and thus the estimate (7.8) is valid. Theorem 7.1 is proved.

Corollary 7.2.

Suppose that the coefficients o f system (7.4) are smooth in fi and f E L 2 ( R ) , E H3I2(dR) . Then under the conditions o f Theorem 7.1 we have

where Q, cl are constants independent of E .

Estimates (7.20), (7.21) follow from (7.7), (7.8) due t o the inequality

which is known from the theory o f elliptic boundary value problems in smooth

domains (see [I]).

Now we shall obtain an effective estimate for the energy concentrated in a

part G of the stratified body R . Let G be a smooth subdomain o f R . We define the energies corresponding

to uc and u by the formulas

Theorem 7.3.

Let uc, u be the solutions o f problems (7.3), (7.4) respectively, u E H 2 ( 0 ) . Then

§ 7. Homogenization o f stratified structures

where c l (G) is a constant independent of E .

Proof. For the sake o f simplicity we prove this theorem assuming the elements

of the matrices A? t o be smooth functions. It is easy t o show using smooth

approximations for the coefFicients, that the result is valid if the coefFicients

are not smooth.


Taking into account (7.19), (7.11) we find

- - J [(dMi", Vk * au

G ax. , V V , ~ ax. ' az,) +

d2u , Nc -)dx -

G Ivv12 d x , dx,dxj


where lqZl I c36;l2 11~11&2(~). By virtue of (7.12) we have

du = - J (., L (a -ldX -

G axk 1 ~ ~ 1 2 ax, 3 ax j

pk au a au - J (a. - - - ( N C - ) )dx +

G 1 ~ ~ 1 2 ax, ' axk 3 axj

dML pk du du , NC -)ukdS . + aG a x j

Therefore it follows from (7.25) that

From (7.19) we obtain

§ 7. Homogenization o f stratified structures

where (p i I < ~6," ' ( ( u ( ( & = ( ~ ) . Since by the imbedding theorem we have IIVulJLa(ac) 5 c ~ ~ u I I ~ z ( ~ ) for

any u E H2(R) (see also Proposition 3 of Theorem 1.2, Ch. I), it follows from

(7.24)-(7.27) that the estimate (7.23) is valid. Theorem 7.3 is proved.

Corollary 7.4.

I f the coefficients of system (7.3) are smooth, it follows from (7.22). (7.23)

that

Note that the matrix [cpkcpl~,kl]- ' was used in (7.5) t o define N,', MG. Let us show that this matrix exists and its elements are bounded functions

(uniformly in E ) .

Lemma 7.5.

Let A'j(x), i, j = 1, ..., n, be matrices o f class E(nl, n2), where nl, n2 are

positive constants independent o f x. Let cp E C1(Q), (Vy( 2 const. > 0,

Vcp = (91, ..., cpn).

Then there exist two constants n3, n4, depending only on n l , n2 and cp,

such that for any E Rn

!%d. Set Tih = ( ~ i t h 4- 9 h E i in (3.3), Ch. I. Then

II. Homogenization o f the system o f linear elasticity

Set K ( x ) = cp,(x)cp,(x)APq(x). Then by (3.3), Ch. I, for any [ E Rn we get

where the constants cl, MI depend only on nl , K Z , (P. I t follows that I{-'

exists. Setting [ = K-lC we obtain

These inequalities imply (7.28). Lemma 7.5 is proved. •

7.2. Necessary and Suficient Conditions for Strong G-Convergence

of Operators Describing Stratified Media

In the case of stratified structures the general results on strong G-con-

vergence together with formulas (7.5) and Theorem 7.1 make it possible t o

formulate necessary and sufficient conditions for the strong G-convergence o f

the sequence {C,) to the operator 2 in terms o f convergence o f certain combi-

nations of the coefficients o f L,, and t o obtain for the coefficients o f E explicit

expressions involving only weak limits o f the above mentioned combinations of

the coefficients o f C,. We shall need some auxiliary results about compactness in functional spaces.

Denote by COIP the space of bounded measurable functions g(t , y), ( t , y ) E

[O, 11 x 0, equipped with the norm

t varies over a set of full measure.

By C1@ we denote the space o f functions g(t, y) such that g(t , y ) . 9 E a y j

Co8P, j = 1 ,..., n.

§ 7. Homogenization of stratified structures 231

Lemma 7.6.

Consider a family o f functions & ( t , Y ) whose norms in CotP are uniformly

bounded in E E ( 0 , l ) . Then there exists a subsequence E' -+ 0 and a function

@ E C 0 , P such that

& ( t , y ) -+ @ ( t , y ) weakly in ~ ~ ( 0 , l ) as E' -+ 0

for any y E a. Proof. Let V be a dense countable set in L 2 ( 0 , 1 ) . For a fixed v E V consider

the tamily o f functions o f y:

Due to the assumptions o f Lemma 7.6 this family is uniformly bounded

and equicontinuous with respect t o E .

Therefore by the Arzeli lemma there is a subsequence E' -+ 0 such that

f t t ( t , y ) v ( t ) d t -+ Q,(y) uniformly in y , (7.29) 0

where Q,(Y) is a function of y E n. Since V is a countable set, one can use

the diagonal process to construct a subsequence E' -t 0 such that (7.29) holds

for any v E V. Now let w be an arbitrary function in L 2 ( 0 , 1) and v j -+ w in L 2 ( 0 , 1 ) as

j -+ m, vj E V. Let us show that there exists Q,(y) such that

Qu,(y) -+ Q,(y) uniformly in a as j -+ oo .

Indeed, i t is easy t o see that

Choosing EO sufficiently small in order that for E' < EO we have


we get

IQv,(y) - * v k ( ~ ) I 5 c l l v j - vk11~2(0,1) + 612

for any j , k ; y E a. It follows that { Q , , ( y ) ) is a Cauchy sequence in c0(i?) and therefore there is a function Qw E C0(O) such that

Q v 1 ( y ) + Q W ( y ) uniformly in y E as j -t co .

Choosing a sufficiently large j in the inequality

we find that

uniformly in y E a. Obviously Q w ( y ) is a bounded linear functional on w E L 2 ( 0 , 1 ) for any

y E a. Therefore

where @ ( t , y ) E L 2 ( 0 , 1 ) for any y E a. Thus

for any w ( t ) E L 2 ( 0 , 1 ) . The function @ ( t , y ) satisfies the inequalities

-C (y' - y"(P 5 @ ( t , y') - @ ( t , y") 5 c l y l - ytl(* , (7.30)

5 7. Homogenization of stratified structures 233

owing t o the fact that iff, -+ f weakly in L2(0, 1) as E --+ 0 and m 5 fc < M ,

then m 5 f 5 M for almost all t E ( 0 , l ) . Therefore correcting, if necessary,

cP on the set o f measure zero we get iP E COl@ due t o (7.30). Lemma 7.6 is

proved.

Corollary 7.7.

Let {&(t , y ) ) , E E (0 , I ) , be a family of functions, whose norms in C'VP are

bounded uniformly in E . Then there exists a subsequence E' -+ 0 such that

weakly in L2(0, 1) for any y E 0, j = I , ..., n , where ~ E C1@.

Proof. It follows from lemma 7.6 that there is a subsequence E' -+ 0 such that

weakly in L2(0, 1) for all y E 0 where $,cpi E C0?O

Obviously for any g E C,"(R) we have

a'(t' ' ) in the sense of distributions. Since @, rl E Therefore p j ( t , y ) = - 8~ j

COIP the last equality holds in the classical sense for almost all t.

Lemma 7.8. Suppose that the functions &(t, y) are bounded in COIP uniformly with respect

to E E (0 ,1) , and that $,(t,y) --t 0 weakly in L2(0, 1) as E -+ 0 for every

y E 0. Then


in the norm of CO([O, 11 x a ) as E --t 0

a* . Moreover, if 2 , j = 1, ..., n , are also bounded in E , then 1 ayj 1 @,(cp(x), x ) -+ 0 weakly in H 1 ( R ) as E -+ 0

for any cp(x) E C 1 ( 0 ) .

Proof. The family {@,( t , y ) ) , E E (0 ,1 ) , is equicontinuous and uniformly

bounded in [O,1] x a . Therefore due t o the Arzela lemma there exists a function

$ ( t , y ) such that @,, -+ $ in the norm of CO([O, I ] x a ) for a subsequence

E' -+ 0. Since t

@.. = / A,(., y)dr -+ $ ( t , y ) for all t , y E [ O , l ] x 0, and $.(t, y) -+ 0 0

weakly in L2(0, 1) as E 4 0 for any fixed y E a , it follows that 11, = 0.

Let us prove that @,(cp(x), x ) -+ 0 weakly in H 1 ( R ) as E -+ 0 . Indeed, we

have already proved that @,(cp(x),x) -+ 0 in the norm o f L m ( R ) as E + 0 . a

Moreover the derivatives - @,(cp(x),z) are bounded uniformly in E . Thus axi due t o the compactness of a ball o f L 2 ( R ) there is a subsequence E' -+ 0 such

a that - @,t(cp(x), x ) -+ ~ ( x ) weakly in L Z ( R ) , and therefore x = 0. Lemma ax, 7.8 is proved.

We introduce the following notation for i, s = 1, ..., n:

$7. Homogenization of stratified structures

Let us now apply the general results, established in $9, Ch. I, on strong

G-convergence t o obtain the necessary and sufficient conditions for the strong

G-convergence o f operators describing stratified media, in terms of weak con-

vergence of the combinations (7.31) o f the coefficients o f system (7.1).

Theorem 7.9.

Suppose that the elements of the matrices A y ( t , y ) , i, j = 1, ..., n , have

norms in C'fP uniformly bounded in E. Then the sequence {L,) is strongly

G-convergent t o the operator as E --+ 0 if and only if the following conditions

are satisfied

weakly in L2(0, 1) as E -+ 0 for any y E 0.

Proof. Assume first that the conditions (7.32) are satisfied. Let us show that

in this case 6, + 0 as E + 0, where 6, is defined by formula (7.6). Indeed,

one can easily check that

A" -B: + ( B ; ) * ( B , O ) - ' B ; , = -bq ( (B')*(@)- l j j s . Therefore


Denote the integrands in the above formulas for N,'(t,y), M,",(t, y) by

n:(t, y) , rn;$(t, y) respectively. By virtue of (7.32) we have

n:( t ,y) ,rnk( t ,y) -+ 0 weakly in L2(0,1) (7.34)

as E -+ 0 for any y E fi. According t o Corollary 7.7 it follows that

weakly in L2(0, 1) as E -+ 0 for any y E a . as E -+ 0 for any y E a.

Lemma 7.8 and (7.34), (7.35) imply that the matrices N,'(t, y) , M;",(t, y ) , a a - N,E(t, y ) , - M;E,(t, y ) converge t o zero in the norm o f CO([O, 11 x a ) as 8~ j 8~ j E -+ 0. Therefore due t o (7.6) we have

Moreover, it follows from Lemma 7.8 that

N,'(cp(x), x) -+ 0 ML(cp(x), x ) -+ 0 (7.37)

weakly in H 1 ( R ) as E -+ 0 .

Taking into account (7.11), (7.36), (7.37) we find that

a - M i " , ( ~ ( x ) , x ) -+ 0 weakly in L2(R) as E + 0 , at

(7.38)

Pk a since - - Mi",(cp(x),x) -+ 0 weakly in L2(R) . IVPI2 a s k

Let us prove the strong G-convergence of L, t o c as E -+ 0.

Set f = L ( u ) E H V 1 ( R ) , iP = 0, u E C,"(R) in Theorem 7.1. Then

estimates (7.7), (7.8) are valid. By virtue o f (7.36), (7.37), (7.38) we have

u' -+ u weakly in H i ( R ) , -yf -+ +' weakly in L2(R)

57. Homogenization o f stratified structures 237

Now let us show that the set { E ( v ) , v E C F ( R ) ) is dense in H - ' ( 0 ) . Then

the convergence L, 3 E will follow from Remark 9.1, Ch. I. According

t o Remark 3.1, Ch. I, every g E H- ' (R) can be represented as g = k ( v ) ,

v E H,'(R), and for any f = L ( w ) E H - l ( R ) , w E (?,"(a), we have

This means that

Therefore choosing w E C,"(R) close t o v in H i ( R ) we get a functional

f = E ( w ) close t o g in H - ' ( a ) . Let us now prove that the conditions (7.32) are necessary for the strong

G-convergence of L, t o E. Suppose that L, 3 k as E + 0. Due to our assumptions about the coefficients of system (7.1) and Lemma

7.5, the elements o f matrices B l ( t , y ) , s = 0,1 , ..., n , B f j ( t , y ) , i, j = 1, ..., n ,

belong to C'vP and have norms in C1@ uniformly bounded in E. Therefore

by virtue o f Corollary 7.3 there exist matrices BO( t , y ) , B S ( t , y ) , B i j ( t , y ) ,

s , i, j = 1, ..., n , with elements in C 1 @ and such that for a sequence E' + 0

we have

weakly in L2(0, 1) for any y E a. Set

Define the matrices f i ~ ( t , y ) , ~ i E j ( t , y ) by the formulas (7.5) with k j ( r , y )

replaced by 2 j ( r , y ) and define 8c by (7.6) with N,E, MiEj replaced by N , E , M;. The same argument that we used a t the beginning o f the proof of this

theorem shows that


&I -t 0

N ~ ( ~ ( X ) , s) + 0 weakly in H 1 ( R ) , a - ~ : ( ~ ( 2 ) , 2 ) --t O weakly in L 2 ( R ) at

as E' t 0 .

Let ii E C r ( R ) . Denote by uc solutions of the following problems

Set

,k duC ' i k dii +y:=Ac - , ? ' = A -

dxk

Similar t o the proof of Theorem 7.1 we obtain the inequalities

- t dG u - ii - N -1 < lliill,plnl ,

a x , H1(Q) -

a Mfal aii < clJ;l2 I)iill~2(n) . II"t - 7; - at h / l a c o l -

Therefore by virtue o f (7.41) we have

u"' + ii weakly in H 1 ( R ) , y;, -+ 9' weakly in L 2 ( R ) (7.42)

as E' + 0.

Denote by u0 the solution of the problem

By the definition o f the strong G-convergence of L, t o i? and due to (7.42) 33 a3

we have u0 = 6, Ahk - = Ahk - almost everywhere in R. Since G is axk dxk

an arbitrary vector valued function from C r ( R ) , it follows that Ahk = ahk almost everywhere in R.

Thus we have shown that from any subsequence E" -t 0 we can extract

another subsequence E' + 0 such that relations (7.39) hold for B" B', - . . h . . . . A A . .

B'3 = B'J, s = 0 , ..., n , z , ~ = 1, ..., n , where B\ B'J are expressed in terms

of the coefficients of the G-limit operator by the formulas (7.31). Since

5 7. Homogenization o f stratified structures 239

{E") is an arbitrary subsequence, it follows that (7.32) is valid for E -+ 0.


In the proof of Theorem 7.9 we have actually established

Theorem 7.10.

Let the elements o f the matrices A y ( t , y) be such that their norms in C 1 @ are

uniformly bounded in E . Suppose that there exist matrices ~ " ( t , y), ~ ' j ( t , y ) ,

s = 0 , ..., n , i, j = 1, ..., n , such that (7.32) holds for the coefficients of

system (7.1). Then the sequence o f operators C, corresponding t o the co-

efficient matrices Ay(cp(x) , x) is strongly G-convergent to operator ,!? whose

coefficients a' j ( t , y ) have the form

a's = ( b ) * @ o ) - l B s - &S , z , s = l , ..., n . (7.43)

Let us consider some examples o f strongly G-convergent sequences {C,) which satisfy the conditions (7.32).

Theorem 7.11.

Suppose that the elements of the matrices A:]($) of class E ( K , , K ~ ) have the

form aY1(E-'zl), where a?,([) E A P ( R 1 ) are almost-periodic functions of

E E R1. Then the sequence C, strongly G-converges t o the operator k whose

matrices of coefficients are given by the formulas

where (Ai j ) is by definition the matrix with elements

Moreover estimates (7.7), (7.8) hold and 6, -+ 0 as E + 0 .

Proof. In the case under consideration we have A:j(t, y ) = Aij(&-It) , p ( x ) =

X I . Set

II. Homogenization of the system of l inear elasticity

( s ) = ( ~ l l (3))-' (A1] - A ' ~ ( s ) ) , 6..

'3 - A'3 . z i j ( s ) = ~ i l ( s ) ( ~ l l ( ~ ) ) - ~ (Alj - ~ l j ( s ) ) + A..(

The elements o f matrices I.;., Zij are almost-periodic functions, since for 1

any almost-periodic f , g, f 2 const > 0 , the functions fg and - are also

almost-periodic. f


Obviously ( Z i j ) = (5) = 0 and the elements o f N j , M,Fj are uniformly

bounded and equicontinuous. Therefore 6, -+ 0 in Theorem 7.1, since N j ,

M$ converge t o zero as E -+ 0 at any point x1 E ( 0 , l ) .

The strong G-convergence o f L, t o 2 follows from the conditions (7.32), t which hold due to the fact that f (-) -t ( f ) weakly in L2(0, 1) as E -+ 0 for E

any almost-periodic f. Theorem 7.11 is proved.

Let us consider some examples where the coefficients of c depend on 2.

We introduce a class A, consisting of functions f ( t , y) such that for some

~ f ( Y ) t 9 f ( t , Y ) we have

a f dcf ( y ) , gf, - The functions f ( t , y ) , -, c j ( y ) , - agf , 1 = 1, ..., n, are also 8 ~ 1 dYl - dYl

assumed t o be Holder continuous in y E R uniformly in t E [O,l], and such

that

where the constants co, a do not depend on t, o E (O, l] .

Set

5 7. Homogenization o f stratified structures

Obviously for any f E A, we have ( f ( a , y ) ) = c j ( y ) .

A few examples of functions that belong t o A, are listed below.

1. Functions f ( t , Y ) E C'lP that are 1-periodic in t belong t o A, with cr = 1 .

2. Consider a function f ( t ) of the form f ( t ) = M + cp(t) , where M = const.,

Ip( t ) l 5 C ( 1 + Itl)-N, N > 0 . We can easily check that f E dl, if N > 1;

f € & f o r a n y U E ( O , l ) , ifN= 1 ; f € A N , i f 0 < N < 1 .

3. The sum $1 + $2, where $1 E A,,, +2 E .Aaz, belongs t o A,, with

cr3 = min(cr1, a z ) , c r ~ , u z E ( 0 , 11.

Lemma 7.12.

Let f ( t , y ) E A, for some a E ( O , l ] , and let ( f ( . , y ) ) = 0 for all y E a. Then

where cl is a constant independent o f E , y , T

Moreover, for any y E 52 fixed, we have

weakly in L 2 ( 0 , 1 ) (as functions of 7 ) .

Proof. Let us prove (7.46) for a = 0 . Since ( f (., y ) ) = 0 , therefore

c j ( y ) = 0 in (7.44), and f ( s , y ) d s = g ( t , y ) . Setting s = E-'T we ob- i o

tain E-' jf f ( ~ - ' r , ~ ) d r = g ( t , y ) . Therefore setting T = ~ t by virtue of 0

(7.45) we get

Thus (7.46) is valid for a = 0.

For a = 1 the estimate (7.46) is proved in the same way, since we can

differentiate (7.44) with respect t o yl, and d c j ( y ) / d y I = 0 .

The convergence (7.47) follows directly from (7.46). Indeed, due to (7.46)

we have


where 0 < a < b < 1 and X[a,b] is the characteristic function o f the interval

[a, 61. Approximating v E L2(0, 1 ) by linear combinations o f characteristic func-

tions and taking into account that f , d f / dy r are bounded, we get J %(: 0

, y)v(s)ds + 0 as E --t 0. Lemma 7.12 is proved.

For a given matrix B(t, y ) with elements B;j(t, y ) let (B( . , y ) ) be the ma-

trix with elements (B i j ( - , y)).

Theorem 7.13.

Let the elements of the matrices A:j have the form

and define for i, s = 1, ..., n , A" = {a;kj') the following matrices

Suppose that the elements o f BO(r, y ) , P ( r , y ) , BiS(r, y ) belong t o A, for some u E (0 , l . Then the sequence of operators t, corresponding t o

Q 1 ") the matrices x) is strongly G-convergent t o the operator k whose

coefficient matrices are

$7. Homogenization o f stratified structures 243

Moreover, the number 6, used in Theorem 7 . 1 satisfies the inequality 6, 5 CEO, where the constant c does not depend on E .

Proof. According to Lemma 7.12 we have

weakly in LZ(O, 1 ) as E -+ 0 for any y E a. Therefore due t o Theorem 7.9 one can take

B S ( t , y ) = (Bs ( . , y ) ) , &j( t , y ) = ( B i i ( . , y ) ) ,

Thus by virtue of (7 .43) the coefficients of the G-limit operator are given

by (7 .49) . Let us now show that 6, 5 e". It is easy t o see that

Therefore, from (7.48) we see that the matrices Njc(t, y ) , MiC,(t, y ) defined by

(7 .5 ) can be written in the form

7 7 Denoting the integrands in (7.50) by n j ( - , y ) , m i j ( - , y ) , respectively, we

E E see that the elements o f n j ( t , y ) , m i j ( t , y ) belong t o A, and ( n j ( . , y ) ) =

( m i j ( . , y ) ) = 0 . Thus by the definition of 6, and (7.46) we get 6, 5 E". Theorem 7.13 is

proved.


Corollary 7.14.

If cp(x) = x1 in Theorem 7.13, then the coefficients of the G-limit system are

given by the formulas

58. Estimates for the rate o f G-convergence

$8. Estimates for the Rate o f G-Convergence o f

Hieher Order E l l i ~ t i c O~era tors

8.1. G-Convergence of Higher Order El l ip t ic Operators

(the n-dimensional case)

In a smooth bounded domain R c Rn consider a differential operator of

the form

where aap(x) are bounded measurable functions in R , a,/? E Z;, (a( =

a1 + ... + a,, U(X) is a scalar function in H,"(R). We say that a differential operator L : H,"(R) + H-"(R) of the form

(8.1) belongs to the class E(Xo, XI, Xz), if its coefficients satisfy the following

conditions

for any u E CF(R) , where Ao, X I , A 2 are positive constants independent o f

U. It follows from the last inequality (see (1341, [9]) that for any t E Rn and

any x E R we have

where Q = const. > 0, ta = <p' ...<?. Thus every operator L of class E(X0, XI, Xz) is elliptic.

Now following [I481 we give the definition for the strong G-convergence of

a sequence of higher order elliptic operators.

We say that a sequence of operators {Lk) of class E(X0, XI, Xz) is strongly

G-convergent t o the operator k o f class ~ ( i ~ , X I , I , ) , if for any X > j, (j, =


const. > 0 ) and any f E H-"(R) the sequence o f solutions o f the Dirichlet

problems

converges in H,"(R) weakly as k t m to the solution u of the problem

and moreover, if the sequence of functions

converges in L 2 ( R ) weakly as k t oo to the functions

Here { a k p ( x ) ) and {Zl,p(x)) are the matrices of coefficients of operators

C k and respectively.

Note that the difference between the strong G-convergence and G-con-

vergence consists in the requirement of the weak convergence of the weak

gradients r , (uk , .Ck) t o r & , L ) in L 2 ( n ) as k -t oo.

It is shown in [I481 that the strong G-convergence o f C k to E as k t oo is equivalent to the following conditions, the so-called Condition N:

There exists a sequence of functions {N ,k (x ) ) such that

N 1 ~ , k E H m ( R ) , ~ , k + O weakly in H m ( R ) , l y l < m ;

N2 k a & ~ ' ~ p k + a t p -+ hop weakly i n L 2 ( R ) , I~l=m

N 3 Do($p - hop) + 0 in the no rm of H - " ( 0 ) , 1a1=m

$8. Estimates for the rate of G-convergence 247

A similar condition for the system o f linear elasticity was formulated in $9,

Ch. I.

If we impose some additional restrictions on the functions N,k we arrive at

a stronger condition (the so-called Condition N1) which not only implies the

weak convergence o f uk t o u in H,"(R) as k -+ co, but enables us to estimate

the difference between uk and u.

We say that a sequence of operators { C k ) E E(Xo,X1, X 2 ) with the rna-

trices o f coefficients {a$ (x ) ) , la[ , IPI 5 m , satisfies the Condition N' in 0,

if there exists an operator E E E ( ~ O , K ~ , i2) with the matrix of coefficients

{Zlap(x)) and a family of functions N,k E Hm(R) , lyl < m , such that

N'1 DaN,k E Lm(R) for la1 5 m , Iyl 5 m , DON; + O

in thenormof L M ( R ) , la1 < m , )yl 5 m ;

in the normof H-'*"(a), lal,IPI 5 m ;

in the norm of H-mlm(R) , 5 m as k + co

(for the definition o f H-"*"(R) see $9.2, Ch. I).

For the sake o f simplicity we assume that the coefficients 21,p are infinitely

smooth .

Let us introduce the parameters which characterize the rate of convergence

in the Conditions N'l , N12, N13. Set


,f?f) = max Ilidtp - i d a p l l ~ - ~ , m ( ~ ) , lol<m

IPlSm

a = max 1 oa(id:, - idar) I IP16m lol=m H-m$03(R)

Theorem 8.1.

Let the Condition N' hold for the operators C k , k . Then there is a real constant

ji such that for p > ji and s 5 m - 1 the solutions o f the Dirichlet problems

satisfy the inequalities

lluk - u l l ~ s ( n ) 5 IlvkllHs(Q) + + K[B!" I I ~ 1 1 ~ 1 c o ) + ( u k + BY + n) I I ~ I I L ~ ( Q ) ] , (8.11)

where I( is a constant independent of k , f , and vk is the solution o f the

Dirichlet problem

Proof. For any operator C E E ( X o , XI, Xz) there is a real constant jl depending

only on Xo, XI, Xz, and such that if p > f i , then the solution w of the Dirichlet

problem

58. Estimates for the rate of G-convergence


where c is a constant depending only on X o , X I , X z (see [9]). Let us choose

p > > 0 such that the solutions of the Dirichlet problems for operators

Lk + p , k + p satisfy (8.13) with a constant c the same for all I c . We shall use the following Leibnitz formulas (see [127]):

a a1 an Q! where ( ) = ( ) ( ) =

P , a! = al! ... an!, P I a means

P, P!(a-P)! that Pj < aj for each j = 1, ..., n.

Set

where u is the solution of problem (8.9) and N,k are the functions entering the

Condition N'. By virtue o f (8.14) we find

( L ( u ) , ) / a$ D@ U: Va v d x = l a l l m n IPlSm


Denote by JO the last integral. Then because o f (8.4) we have

where CI is a constant independent o f k.

Transposing the indices y and P in the integral next to the last one in

(8.16), we obtain

where J1, Jz , J3 stand for the respective integrals on the left-hand side of the

last equality.

From (8.4) we have

I t follows from (8.6) and Lemma 9.1, Ch. I, that


Let us estimate J I . Using (8.15) we find

Applying Lemma 9.1, Ch. I, and (8.7), (8.5) to estimate the first two integrals in the right-hand side of (8.21) we get

1 J ~ I 5 Cl [ 7 k l l u l l ~ m ( R ) IIvIIH"'(f2) + + /?!I) IIvIIHmm) ~ l u l l H z m * l ( R ) ] .

Thus

( ( . l k + P)u:,v) = ((i. + P ) U . V ) + e ( u , v )

for any v E H F ( f l ) , where

We obviously have u: - v k - u k E H F ( f l ) and


Hence, setting v = u i - vk - uk and using (8.23), (8.13), we obtain the

inequality

which implies (8.10), since for the solution u of problem (8.9) the following a

priori estimate is valid

(see [55]).

Due t o (8.4) we have for s 5 m - 1 the following inequality


One o f the simplest examples, when the Condition N' holds, is provided

by the sequence o f operators L k with coefficients akp(x) such that akp(x) +

Zl,p(z) in the norm of Lm(R) a s k + m.

In this case we can take N,k G 0. Obviously the Conditions N'l, N12, N'3

are satisfied and

where c is a constant independent of k,

$8. Estimates for the rate of G-convergence

According t o Theorem 8.1 we have

Actually one can prove a stronger inequality in the case under consideration,

namely:

To obtain (8.25) we note that in the proof of Theorem 8.1 the norm

1 1 f llHlcn, estimates I I u I I ~ ~ , + I ( ~ ) in (8.22). The norm I I u I I ~ z ~ + I ( ~ ) is needed

to estimate the first integral in the right-hand side o f (8.21). It is clear that

under our assumptions this integral can be estimated by

Let us now consider a less trivial example, when the Condition N' issatisfied.

Assume that the coefficients a k p ( x ) of operators Lk depend only on x l , i.e.

a!&(z) = akp(x l ). Let the coefficients Aap(z l ) of operator k be such that for all la1 5 m,

1/31 5 m , a = (m,O, ..., 0 ) we have

1 1 - atp Asp & Aaa -7, k -'^, +7, a:, a,, a,, a,, a:, a,, (8.26)

abuatp ~a,A,p a& - - -$ ZlaP - - weakly in ~ ' ( 0 , l ) as k -+ m , a:, a00

where 1 is such that R c { x : 0 < X I < I ) . Define the functions N j ( x l ) as solutions o f the equations

such that

11. Homogenization o f the system o f linear elasticity

It follows that the Condition N13 is satisfied and /If) t 0 as k t m,

-yk = 0 in (8.5), (8.7).

By virtue of (8.26) the right-hand side of (8.28) tends to zero weakly in d"

L2(0, I ) as k t oo. Therefore - N ; ( x ~ ) t 0 in the norm o f CO([O, 11) as dx:

k + m , s = O , l , ..., m - 1 . Owing t o (8.28), (8.4) one has

and a k t 0 as k t m . Thus the Condition N1 l is also valid.

Let us consider the Condition N12. We obtain due t o (8.28) that

- - -- ~*B.P G a p - ( a b 2 P aka

4,) . By virtue of (8.26) we have i ikp(xl) - i a p ( x l ) + 0 weakly in L2(0,1).

Therefore

21

@k,(z,) = / (i*,(s) - ~ , ~ ( s ) ) d s --+ 0 in the norm of CO([O, 11) 0

d Since hk0(x1) - ;,p(xi) = - @ k p ( x l ) , we can assume in (8.6) that

dx1

P?' = c max IaIsIPl<m

58. Estimates for the rate o f G-convergence

and p;) -+ 0 as k + oo, c = const.

Now in order to obtain an effective estimate for the closeness of uk(x) t o

u ( x ) it is sufficient t o estimate I I ~ ~ l l ~ r n ( n ) . We have

where cj are constants independent of k. Here we used the definition o f a k , the a prion' estimate (8.24), the in-

equality

IIuIIL*(an) 5 c I I u I I H # - ~ ( ~ ~ , lIuII~*+t(an) , > 0 t > 0

(see [9]), and the fact that Npk possess derivatives up t o the order m, which

are bounded uniformly in k. Define the parameter bk, by bk = rnax { a k , ,&)).

Then, according to Theorem 8.1, we obtain the inqualities

lluk - uIIHm-l(n) 5 c6:I2 I l f llHl(n) 7

8.2. G-Convergence of Ordinary Differential Operators

The results of the previous section are obviously valid for ordinary differ-

ential operators. However in the latter case we can obtain more accurate

estimates. Here we prove some theorems in this direction.


Let R = ( 0 , l ) and let .Lk, J? be ordinary differential operators of the form

Theorem 8.2.

Let u k , u be the solutions of the following Dirichlet problems

( L k + P ) U ~ = f , ( k + CL)U = f , u k , ~ E H,?(O, 1 ) , (8.34)

where C k , J? are ordinary differential operators (8.32), (8.33). Then

where the constant c does not depend on f, k,

Ak = max [ ( * - L ) d i l + ~ € [ O , l l mm a i m

+ max 11 [(':f:q hPq) - (aim';q - - a ) d (8.37') r E [ O , l ] 0 a k m

and N,k are the solutions of the equations

satisfying the boundary conditions

58. Estimates for the ra te o f G-convergence 257

Proof. It follows from the above result for higher order elliptic equations, whose

coefficients depend only on XI, that in order t o prove estimates (8.35), (8.36)

we have only t o estimate the functions vk which are solutions o f problems

(8.12), namely

For the functions N,k we have

Moreover, if p 5 2m - 1, it follows from Sobolev's lemma that

where c is a constant independent of u.

Therefore due t o (8.24) we get

Set

m dpu 'Pk = C Nk -

p=o dxp '


b!Pk)~, I ~ $ I 5 czAk Ilf ~ILz(o,I) . (8.43)

One can construct a continuous extension operator P mapping any pair o f

numerical sets {aj0)), {a!')), i = 0,1, ..., m - 1, into a smooth function cp(x)

defined on [O,1] and such that


Obviously Q ( X ) can be defined by the formula

where e p ) ( x ) , e l 1 ) ( z ) are smooth functions which satisfy the conditions

Therefore vk is the solution of the Dirichlet problem

L k ( v k ) + P V ~ = 0 on ( 0 , l ) , vk - cpk E H r ( 0 , l ) ,

where ~k are the functions defined by (8.44) with $1 = a!?, a!') = By vritue o f (8.43), (8.44) we get

Set wk = vk - ( p k . Then wk is the solution o f the Dirichlet problem

C k ( w k ) = C k ( Q k ) Wk E 1 ) .

Using the inequalities (8.13), (8.45), we find that

I I C k ( ~ l k ) l l ~ - m ( o , l ) 5 cs I I ~ l k l l ~ m ( 0 , l ) L G A ~ l l f I I L ~ ( o , I ) .

Hence l l w k l l ~ m 5 c7Ak 1 1 f l lLa (o , l ) and We finally obtain

1 1 v k l l ~ m I C S A ~ . I l f l l ~ 2 ( o , r ) . It is clear that ~k = ,Bf) = 0 ; , B f ' ) , c y k < c A k , c = const, and therefore

estimates (8.10), (8.11) imply (8.35). Theorem 8.2 is proved.

Remark 8.3.

Suppose that the coefficients a;, of the operators L k have the form a k , ( x ) =

a , , ( k x ) where a , , ( ( ) are 1-periodic bounded functions. It then follows from

(8.26) that the coefficients o f the G-limit operator 2 are given by the formulas


1 C

where p,p 9 rn - 1. (f) = / f ( t ) d< . We also have Ak 5 - . Moreover, the 0

k estimates (8.35) become

where c is a constant independent of k and f . d d - d d

If C k = - ( a k ( x ) -), L = - ( h ( x ) -), then the estimate (8.31) is . . d x d x d x d x

reduced to

where C is a constant independent o f k. Let us consider the latter case in more detail, so as t o obtain an explicit

expression for the constant C.


and

Therefore

M h ( t ) < - - max. *o .6[0,1]

2 6 0 11. Homogenization o f the system o f l inear elasticity

where so < a k ( x ) 5 M for any k = 1,2, ... and the function vk is such that

Due t o (8.24) there exists a constant R such that

The dependence of R on the coefficients of the operator 2 will be specified du 1

below. It is easy t o see that 11-11 < - 1 1 f l l L 2 ( o , l ) . Thus we have a x L2(0,1) - so

M R + 1 IIuk - ~ I l L 2 ( 0 , 1 ) 5 -

60 Ak I l f l l L 2 ( o , l ) + I I v k l l ~ 2 ( o , l ) 7

where

In order t o estimate the norm J J V ~ J J ~ Z ( ~ , ~ ) we apply the maximum principle,

which yields

Ivkl 5 max { l v k ( O ) l , l v k ( l ) l ) . =€[o , l l

Therefore

since obviously

Thus

Now let us estimate the constant R > 0 in terms of the ~oeff icients of the d u diL du

G-limit operator L. Squaring both sides o f the equation ii - = f - - - d x 2 d x d x

and integrating it over [ O , l ] , we obtain

$8. Estimates for the rate of G-convergence

I t follows that

dii where P > 0 is any constant such that max I,/ 5 P . Thus

z€[O,lI

2112 P2 112 and therefore we can take R = - (1 + -)

60 W e finally obtain the inequality

6:

where the constants M, R, 60 can be easily calculated for the given coefficients

o f the operators Ck, 2.

CHAPTER Ill

SPECTRAL PROBLEMS

$1. Some Theorems from Functional Analysis.

Spectral Problems for Abstract Operators

Here we formulate and prove some results in the spectral theory o f linear

operators, which are useful for applications considered below. Moreover in

$1 we prove theorems on the convergence of eigenvalues and eigenvectors of

a sequence of abstract self-adjoint operators depening on a parameter E and

defined on different Hilbert spaces which also depend on E . Such questions for

non-self-adjoint operators are considered in $9.

These theorems provide means for the investigation o f spectral problems

in the homogenization theory; they can also be applied t o study asymptotic

behaviour o f spectra o f some other singularly perturbed operators considered

in this chapter.

1.1. Approximation of Eigenvalues and Eigenvectors of

Self-Adjoint Operators

Following [I321 we give here a proof of an important lemma which has

wide applications for the approximation of eigenvalues and eigenvectors of

self-adjoint operators.

Let H be a separable Hilbert space with a real-valued scalar product

(u,v)H; and let A be a continuous linear operator d : H -+ H. By

lldll we denote the norm sup - IIAUIIH , where the supremum is taken over all llullH

ti E H , u # 0; and llullH as usual stands for (u,u)z2.

The space of all continuous linear operators A : H -+ H is denoted

by L ( H ) . It is well known that L ( H ) is a Banach space with the norm

I l d l l ~ c ~ ) = 11All.

264 111. Spectral problems

Lemma 1.1.

Let A : H -t H be a continuous linear compact self-adjoint operator in a

Hilbert space H . Suppose that there exist a real p > 0 and a vector u E H , such that llullH = 1 and

( IAu - pullH 5 a , a = const > o . (1.1)

Then there is an eigenvalue p, of operator A such that

Moreover, for any d > a there exists a vector 'll such that

and ii is a linear combination o f eigenvectors of operator A corresponding t o

eigenvalues of A from the segment [p - d, p + 4. Proof. Consider in H an orthonormal basis { ' P ~ ) , which consists of eigenvec-

tors o f A : Avk = p k ( ~ k , k = 1,2, ... . Such a basis exists according t o the

Hilbert-Schmidt theorem (see [40]). Then

The assumptions of Lemma 1.1 imply that

Let pi be the eigenvalue o f A such that Ip - pil = rnin Ip - pkl . We then k

have

and therefore Ip; - pI 5 a, since

§1. Some theorems f rom functional analysis

Let us prove the second statement of Lemma 1.1. Set

Then

where

Without loss o f generality we can assume that p # pj for any j . Therefore

ck = (pk - ~ ) - ' f f k . Set uo = x c,pl, where the sum is taken over all indices 1 such that

1

pi E [p - d , p + dl. We have

where the sum is taken over all k such that pk 6 [p - d , p + dl. Let us show

that ii = IIuolljjluo is the vector we seek.

Indeed, since Ilu - u o l l ~ = l l v l l ~ I ad- ' , l luol l~ I 1, l luol l~ 2 l l u l l ~ - l l v l l ~ , We have


266 III. Spectral problems

1.2. Estimates for the Difference between Eigenvalues and Eigen-

vectors of Two Operators Defined in Diflerent Spaces

In this section we prove some important theorems on the behaviour of

eigenvalues and eigenvectors o f a sequence o f abstract operators defined in

different Hilbert spaces under certain restrictions imposed on this sequence

(Conditions C1-C4). The Hilbert spaces can be chosen in such a way that

homogenization problems as well as many other singular perturbation problems

for differential operators can be associated with such sequences o f operators

satisfying Conditions C1-C4, which enable us t o study the corresponding spec-

tral properties.

Let 'FIE, X0 be separable Hilbert spaces with real valued scalar products

respectively, and let

be continuous linear operators, I m A C V C 7-10, where V is a subspace o f

7-10,

In the rest o f Chapter 111 we consider spaces 'Ido, 'H,, V and operators do, A, subject to the following Conditions C1-C4.

a. There exist linear continuous operators Re : 'Flo + 7-1, and a constant y > 0

such that

for any f0 E V. (If 'Flo = V = L2(R), 'FI, = L2(R" and R' is a perforated domain of

type I or II (see 54, Ch. I), then we can take as RE the restriction operator,

such that R e f = f In* for any f E L2(R). It is shown below that in this case

51. Some theorems from functional analysis

y = mes Q n w) .

C2. - Operators JZ, : 1-I, + 'He, Jb : 'Ho + 'Flo are positive, compact and

self-adjoint; their norms I l & l l , c c . H I ) are bounded by a constant independent of

&.

c.3. For any f E V

C4. - The family of operators {A,) is uniformly compact in the following sense.

From each sequence f' E 'H, such that sup 1 1 fcllnc < CX) , one can extract 6

a subsequence f" such that for some w0 E V

Remark 1.2.

Condition C1 implies that if the sequences f", gc and the elements f O , are

such that

1 1 f' - R, follw, + 0 , IJg' - R 6 g o ( l ~ , + 0 as & --+ 0 7

then

(f', gC)n. + 7 ( f 0 > ,s").HO (1.8)

Indeed, due t o (1.4), (1.7) we have

(f', gC)n. - ( R e f O, R,gO)n. = (f' - Ref0, gC)ne + + ( g c - REgO, REfO)nI 5 I l f ' - REfolln, llgelln. +

+ 119" - RcgOlln, IIR6f011n. + 0 & + 0 .


It is easy t o see that (1.4) implies the convergence ( R e f 0 , Reg0)%. -+ y ( f o , since ( u , v ) = 4-'(llu + v(I2 - I I u - v1I2).

Remark 1.3.

Condition C3 implies that i ff ' E ' F I E , f 0 E V and

[ I f ' - Refollnl -+ 0 as + 0 (1.9)

then

llAef' - RcAofOl l~ . -+ 0 as E -+ 0 , (1.10)

since IIA,f' - RcAofOllx, I IIAe(f' - R,f0)l17i, + IIAeR,f0 - R e d o f O I I ~ ~ and the norms of the operators are bounded by a constant independent &.

Consider the spectral problems for the operators A,, do:

d , u , k = p , k u ~ , k = 1 , 2 ,..., u ,k€ 'Hc ,

k p f 2 p: 2 1 pc"' , p; > 0 ,

( U ; , U ~ ) N , = 4 r n ,

where 61, is the Kronecker symbol: tilm = 0 for 1 # m, = 1 for 1 = m,

the eigenvalues p,k and ph, k = 1,2, ... , form decreasing sequences and each

eigenvalue is counted as many times as its multiplicity.

Our aim is t o estimate the difference between eigenvalues and eigenvectors

of problem (1.11) and those o f problem (1.12) for small E.

Theorem 1.4.

Let the spaces 'H,, 'HO, V and operators A,, A, R, satisfy Conditions C1-C4.

Then there is a sequence {@) such that P,k -+ 0 as E -+ 0, 0 < P,k < and

$1. Some theorems from functional analysis 269

where p,k, pk are the k- th eigenvalues o f problems (1.11), (1.12) respectively,

N(&,do) = { u E ?lo, &u = pku) is the eigenspace of operator do corre-

sponding t o the eigenvalue p;.

In order to prove this theorem let us first describe some properties o f op-

erators A, A.

Lemma 1.5.

Let u, E V and let {u,k), { p , k ) be sequences o f eigenvectors and eigenvalues

of problems (1.11) such that

for a fixed k. Then u, and p, are respectively an eigenvector and eigenvalue

o f d o , i.e. dOu, = p,u., U, # 0.

Proof. Setting f' = u,k, f 0 = u, in (1.9) and using (1.14), (1.10) we find

that



Due to the conditions (1.14) the first two terms in the right-hand side o f this

inequality converge to zero as e -+ 0, and the third term converges t o zero by

virtue o f (1.15). Thus

Hence, by (1.4) we deduce that Au, = p,u.. Due t o (1.14) I J u , ~ \ J ~ , -

llR,~,11~. -$ 0 as e -+ 0. Therefore according to (1.4) 71/2 IIu.llXo = 1,

which means that u, # 0. Lemma 1.5 is proved.

Lemma 1.6.

Suppose that Conditions C1-C4 are satisfied. Then

where pt, are the k- th eigenvalues o f problems (1.11), (1.12) respectively.

Proof. Let us first establish the inequalities

where Q, c ( j ) are constants independent o f E , q, does not depend on j .

The upper bound for p i follows from the fact that the norms of operators

A, are bounded uniformly in e.

Fix an integer j > 0. let > > ... > pi+' be eigenvalues of operator

and il;, ..., ilitl the corresponding eigenvectors such that ((iiL(17.1, = 1,

1 = 1, ..., j + 1. The fib exist since pk # 0, k = 1,2, ... , and each eigenspace

of do has a finite dimension.

Setting f = 6; for each k = 1, ..., j + 1 in Condition C3, we obtain from

(1.5) that

IIA,R,G~ - R,Aofi;lln, -+ 0 as E -+ 0

Therefore

JJA,R,E,~ -p,k~,~,k lJ~, -+ o ; t ~ E -4 0 .

Then by Lemma 1.1 for d = A,, H = 'HE, p = fit, u = II~~ckl l<fRe~k* there exists a sequence p,"(k"):

$1. Some theorems from functional analysis

where pr(klc) are eigenvalues of problem (1.11) such that

for all E smaller than some 6j. Therefore the inequality (1.16) is valid, since

pi 2 Pr(j9c) and p;(J+l") + hit1 as E + 0.

Taking into consideration the conditioan (1.16), the fact that Acu: = p:u:,

and using the diagonal process, we conclude from Condition C4 that there are

vectors u; E V and numbers p i such that

for a subsequence E' + 0, j = 1,2 , ... . It follows from (1.17), (1.18) that

According t o Lemma 1.5 u; is an eigenvector corresponding t o the eigenvalue . .

i.e. Aoui = p3,u:, u; # 0; 1 . 1

Setting f c = u i , f 0 = - u3,, g' = u,k, = ,US in (1.7) and using P: P*

(1.8), (1.19) we get

as E' + 0.

Let us show that the vectors U j = ( p i ) - 1 y 1 / 2 ~ i , j = 1,2 , ... , form an

orthonormal basis in 'Ho. Assume that this is not the case. Then there is a

vector U E V such that for some we have

Set f 0 = U in Condition C3. Then

III. Spectral problems

It follows that

where UC, = I I R c t U I I ~ ~ , R e , U , since

Let us apply Lemma 1.1 with

By Lemma 1.1 the relations (1.22) imply that there is a sequence o f eigenvalues

pet of the operators At which converges t o pgk as E' -t 0. Therefore due t o

(1.18) among the p i , j = 1 , 2 , ... , there is a ppk such that prk = pgk. Set

1 d = inf IPk - p i [

j

P:#P;

in Lemma 1.1 and suppose that the multiplicity o f p r k is equal t o 6, pFk =

... = p r k + ~ - l . Then the segment [pk - d , p $ + dl can contain only such

eigenvalues of the operator that coincide with p r k , and therefore by (1.18)

we see that for e' sufFiciently small the segment [pgk - d,pk + d] can contain mk+l-l only the eigenvalues p y k , ..., pc, of A,, corresponding to the eigenvectors

u F k , ..., uFk+I-l . By virtue o f Lemma 1.1 and (1.22), for E' sufficiently small 1-1 ,

there is a vector ii,, = x C : ~ U ; ~ + ~ , IIiiclllNzI = 1 , such that

Choosing a subsequence E" -+ 0 such that cf,, + cl as E" -t 0, we obtain

due t o (1.19), that

[ \ i ict j - R c ~ ~ i i * l l ~ z , , + 0 as E" -+ 0 ,

where

§ 1. Some theorems from functional analysis

Consequently by virtue o f (1.23) we have

Thus U is a linear combination o f the vectors urktj , j = 0,1 , ..., 1 - 1 , which

is in contradiction with (1.21).

Therefore the vectors U j , j = 1,2, ... , form an orthonormal basis in 'Flo.

Obviously we can assume that pi = and ui = U3 in (1.12), since Ui corresponds t o and p! 2 p2 2 ... . Lemma 1.6 is proved.

Proof o f Theorem 1.4. Fix k and consider the sequence ~,ku,k = dCu,k. Since

by Lemma 1.6 j ~ , k + p,k as E -+ 0 , it follows from the proof o f Lemma 1.6 (see

(1.19)) that there exists a sequence E' -+ 0 and a vector ut E N(&, do) c V , such that

Observing that the operators A, are self-adjoint we have

~,k(u,k, RCut)n. = (A,u,k, RCuf)x , = (u,k, d c ~ E u f ) n . . Therefore

0 = ~ ,k (u ; , R C ~ ~ ) H , - P: (u ,~ , R c u f ) ~ . + ~ : ( u , k , ~ c u f ) ~ . -

- ( U , ~ , A R ~ U : ) ~ * . Hence

(P," - P!)(u;, Rcu$)x. = (u!, dCRcu$ - ~ l o k ~ ~ u f ) ~ , . (1.25)

It follows from (1.24) and (1.8) that

Setting (u,k,, R . ,U: )~ , , = pk + actt where a,, --t 0 as E' --+ 0, we deduce

from (1.25), (1.24) that

111. Spectral problems

Therefore estimate (1.13) holds for the subsequence E' -+ 0 and @, = laell.

Let us prove that it also holds for E -t 0. For each fixed E E (0,l) denote by a, the infimum of P,k 1 0 such that the

estimate (1.13) is valid. It is easy t o see that 0 5 a, < and (1.13) is satis-

fied with P,k = a,. Let us show that a, -t 0 as E + 0. Suppose the contrary.

Then there is a subsequence E" -t 0 such that a,,, > c > 0. According t o

what has been proved above, there is a subsequence E' of the sequence E" such

that the estimate (1.13) holds with + 0. By the definition of a,) we have

a,, 5 P,k,,, which is inconsistent with the inequality a,~, > c > 0. Theorem 1.4

is proved.

Estimates for the difference between eigenvectors of problems (1.11), (1.12)

are established by

Theorem 1.7.

Let k 2 0, n 2 1 be integers such that pi > = ... = pt+m > pi+m+', i.e. the multiplicity o f the eigenvalue o f problem (1.12) is equal t o m,

p: = m. Then for any w E N ( ~ ; + ' , A ~ ) , I I W ~ ~ ~ , = 1, there is a linear

combination iic o f eigenvectors u,k+', ..., o f problem (1.11) such that

where Mk is a constant independent o f E .

Proof. Set H = 'He, A = A,, u = II~,wll;;:~,w, p = in Lemma 1.1,

and choose d > 0 so small that the segment - d,&' + dl contains

no eigenvalues o f d,, other than = ... = &". Since IIR,wII$. -t

7 llwllgo = 7 as E + 0, the existence o f .iie and the estimate (1.26) follow

directly from Lemma 1.1. Theorem 1.7 is proved.

$2. Homogenization of eigenvalues and eigenfunctions

$2. Homogenization of Eigenvalues and Eigenfunctions of Boundary

Value Problems for Strongly Non-Homogeneous Elastic Bodies

2.1. The Dirichlet Problem for a Strongly G-Convergent Sequence

of Operators

Let LC, k be the elasticity operators in a domain R, considered in $9, Ch.

I, and let LC be strongly G-convergent to as E + 0 ( L , 3 k ) .

Consider the following eigenvalue problems for operators L, and f?:

L,(ut) = -X,kp,(x)u~ in R , ut = 0 on dR ,

O < X f < X Z < . . . < X , k < ..., (2.1)

J P ~ ( x ) ( u : , u:)dx = 61, 7

n

where bl, is the Kronecker symbol, the eigenvalues of problems (2.1), (2.2)

form increasing sequences and each eigenvalue is repeated as many times as

its multiplicity. We impose the following restrictions on the scalar functions

P E ( x ) and P O ( X ) :

O < c o < p o ( ~ ) < ~ ~ ; o < C z I p C ( x ) I c 3 ;

p, + po weakly in LZ(R) as E -+ 0 , where c2, cg are constants independent of E .

Theorem 2.1.

If L, k as E + 0, and conditions (2.3) are satisfied, then

Moreover, suppose that the eigenvalue X o = Xit1 has multiplicity m, i.e.


and u ( x ) is the corresponding eigenfunction o f problem (2.2), Ilull.r,2(n) = 1.

Then there is a sequence o f functions {u,) such that ii, -+ u in L2(R) as E +

0, ii, is a linear combination of eigenfunctions of problem (2.1) corresponding

to the eigenvalues A;+', ..., A:+". To prove this theorem we shall reduce it t o Theorems 1.4 and 1.7 for

abstract operators, making a suitable choice o f spaces 'FI,, ?lo, V and operators

dc, do. Denote by 'If, the Hilbert space consisting of all vector valued functions u

with components in L2(R). The scalar product in 3-1, is defined by the formula

By 7Io = V we denote the space o f vector-valued functions with compo-

nents in L2(R) , where the scalar product is given by

Lemma 2.2.

Let the conditions (2.3) be satisfied. Then

provided that vc + vO, us -t u0 in the norm o f L2(R) as E 4 0.

Proof. It follows from (2.3) that

for any cp continuous in G. Taking into consideration the fact that p, are

bounded uniformly in E and that functions continuous in fi form a dense set

in L1(R) , we easily verify that (2.5) holds for any cp E L1(R) . Let v E -t vO,

uc + u0 in L2(R) as E -+ 0. Then

$2. Homogenization o f eigenvalues and eigenfunctions

Passing here t o the limit as E 4 0, we obtain (2.4). Indeed, the last two

integrals in the right-hand side of above equality converge t o zero as E --+ 0,

since uE, v" converge in the norm o f L2(R) and p, is bounded uniformly in

E . The difference of the first two integrals in the right-hand side converges t o

zero due to (2.5) for cp = (uO,vO). Lemma 2.2 is proved.

It follows from Lemma 2.2 that Condition C1 is satisfied if we take as RE the identical operator R,u = U . In this case y = 1.

Let us define operators A,, setting AEfE = uE, where uE is a solution of

the problem

It follows from Theorem 3.3, Ch. I, that the norms llA,ll are bounded by a

constant independent o f E . Operators A, are compact, owing to the compact

imbedding H1(R) c L2(R) and Theorem 3.1, Ch. I. The integral identity for

a solution o f problem (2.6) and the equality (Ahk)* = Akh yield

for any f ' ,gC E L2(R), where we = A,gE. Therefore A, is a positive self-adjoint operator in 3-1,. We take as & :

'Flo 4 No the operator which maps f 0 E 'Ho into the solution uO of the

problem


i.e. JZo f0 = uO. By the same argument that was used for the operators A, we

can show that is a positive compact and self-adjoint operator in ?to Thus

Condition C2 o f $1 is satisfied.

Consider now the Condition C3. Let f0 E V and define wc as a solution

of the problem

Since u0 = &fO is a solution o f problem (2.7), the G-convergence of LE t o

E as E -+ 0 implies that

wE - uO -+ O strongly in L 2 ( R ) . (2.8)

The function uE = A, f0 is a solution of problem (2.6) with = fO. Therefore

v E = uc - w C is a solution of the problem

It follows from the integral identity for vc that

The norms o f vc in H 1 ( R ) are bounded by a constant independent o f c , since

v E = uE - w E . Therefore vc' -t v0 weakly in H 1 ( 0 ) and strongly in L 2 ( R ) for

a subsequence E' -+ 0; v0 E H 1 ( R ) . It follows from Lemma 2.2 that the integral in the right-hand side of (2.10)

converges to zero as E' + 0, and therefore vc' -+ 0 strongly in H 1 ( R ) . Since

each sequence {vc) contains such a subsequence vE' -+ 0, it follows that

v' + 0 in H 1 ( R ) as& -+ 0. Thereforedue to (2.8) we have IJuO-uEllWc -+ 0.

This means that Condition C3 is satisfied.

Condition C4 is also valid owing to the compact imbedding H 1 ( R ) c L 2 ( R ) and the inequality Ildc f'l l H ~ ( n ) 5 c 1 1 f'll~*(~), where c is a constant

independent o f c . The last equality follows from Theorem 3.3, Ch. I.

It is easy t o see that in the case under consideration the eigenvalues of

problems (2.1), (1.11) and (2.2), (1.12) are related by the formulas

§2. Homogenization of eigenvalues and eigenfunctions 279

Thus we have shown that all conditions of Theorems 1.4, 1.7 are satisfied,

and therefore Theorem 2.1 follows directly from (2.11) and Theorems 1.4, 1.7,

since I(A,u - + 0 as E -+ 0 due to the Condition C3.

Note that Theorem 2.1 implies in particular the convergence o f the eigen-

values and eigenfunctions o f the elasticity operators with almost periodic coef-

ficients, considered in $6, Ch. II. In the case o f periodic coefficients it is possible

to give estimates for the difference between eigenvalues and eigenfunctions of

problem (2.1) and those of problem (2.2). Such estimates in a more general

situation o f perforated domains are obtained in Section 2.3.

2.2. The Neumann Problem for Elasticity Operators with Rapidly

Oscillating Periodic Coeficients in a Perforated Domain

In this section we study spectral properties of operators associated with

problems (2.22), (2.23), Ch. II. Here RE is a perforated domain of type 11, C, is an elasticity operator with rapidly oscillating periodic coefficients, L, is

given by (1.1), Ch. II, E is the corresponding homogenized operator whose

coefficients are defined by the formulas (1.3). Ch. II.

In order to simplify the derivation o f estimates for the difference of eigen-

values o f problems (2.22), (2.23), Ch. II, i t is convenient t o deal with suitably

"shifted" operators. To this end we consider the following eigenvalue problems

for operators of type (2.60), (2.61), Ch. II:


where 61, is the Kronecker symbol, the eigenvalues form increasing sequences

and each eigenvalue is counted as many times as its multiplicity.

It was shown in Section 2.2, Ch. II, that the operators C,, 2 are "close"

to each other in the sense that the solutions of problems (2.22), (2.23), Ch.

II, satisfy the inequalities (2.26), Ch. II. In contrast t o Section 2.1, here we

impose some additional restrictions on the scalar functions p, , po, namely

where c2, cg are constants independent o f E , the norm 1 1 1 . 1 1 1 is defined in

(2.64), Ch. II, and ( I l p o - p , ( l ( characterizes the closeness of the functions po

and p, .

Applying here the method suggested in Section 2.1 for G-convergent op-

erators, in order to compare the spectral properties o f problems (2.12) and

(2.13) we reduce these problems t o the form which allows us t o use Theorems

1.4, 1.7 for abstract operators in Hilbert spaces depending on a parameter.

The main result of the present section is

Theorem 2.3.

Let conditions (2.14) be satisfied. Then for the k - th eigenvalues o f problems

(2.12), (2.13) the estimate

52. Homogenization of eigenvdues and eigenfunctions 281

holds with a constant ck independent of E .

Moreover, if the multiplicity of the eigenvalue Xo = X f ; t l is equal to m, i.e.

and uo (x ) is the corresponding eigenfunction of problem (2.13), I I u ~ ~ ~ ~ z ~ ~ ) = 1,

then there is a sequence {u,) such that

where Mi is a constant independent of E and uo, uE is a linear combination of

eigenfunctions of problem (2.12) corresponding to A;+', ...,

Before giving a proof t o this theorem we establish some auxiliary results.

Let us introduce in L2(Rc) the following scalar product

and denote the obtained space by 3-1,. The space L2(R) equipped with the

scalar product

(uO, vO)n, = / po(x)(u0, vO)dx (2.18) R

is denoted by 3-10. Set V = 3-10 and take as RE in Condition C1 of $1 the

restriction operator

L ~ ( R ) 3 f -+ il,. E ~ ~ ( 0 ' ) . (2.19)

In order t o show that 7fo , 3-1,, V, R, satisfy Condition C1 we shall need

Lemma 2.4.

Let Re be a perforated domain of Type II, and let conditions (2.14) be satisfied.

Then for any uO, v0 E L2(R) we have


Proof. Set f ( ( , x ) = xw(E), II) = gC = pouO, cp = cpc = v0 in Corollary 1.7, Ch. I, where xu(<) is the characteristic function of the domain w. Then by

virtue o f (1.21), (1.22), Ch. I, we have

as E + 0. Formulas (4.2), (4.3), Ch. I, show that 0' = (I(l\Rl) U Rl n EW.

Therefore since the measure o f the set R\Rl is o f order E , the convergence

(2.21) implies that

as c -t 0. Taking into account estimate (2.65), Ch. II, and the fact that p , ,

PO are bounded and lllpc - polll -) 0 , we get for any u , v E H 1 ( R )

Obviously (2.23) is also valid for u = uO, v = v0 since we can approximate uO,

v0 by functions in H1(a). The convergence (2.20) follows from (2.22) and

(2.23) with u = uO, v = vO. Lemma 2.4 is proved.

Relations (2.19) and (2.20) show that for the above defined spaces 'He, 'Hop

V and operators Rc f = f In, Condition C1 is satisfied with 7 = mes Q fl w .

Let us introduce operators A, : 'Hc + IHc setting A, f' = u', where u'

is a solution of the problem

The existence of an upper bound for the norms 1]A,)1 independent of E

follows from Theorem 5.4, Ch. I, and the compactness of A, follows from the

compactness o f the imbedding H1(R") C L2(R'). Let us show that A, is a

positive self-adjoint operator in 'He.

Indeed, using the integral identity for solutions o f problem (2.24) and set-

ting wc = A,gc, we find


= J ( w C , pcf.)dx = (.%9', fe)n. nc

for any p , g c E 'He, since (Ahk)* = Akh. I t follows that operators A, : 'HE -+

'ME are positive and self-adjoint.

Denote by the operator mapping f0 E 'Ho in to the solution u0 of the

problem

Obviously is a positive compact self-adjoint operator in 'Ho.

Thus we have checked Condition C2 of $1. Consider now Condition C3.

Let us show that for any f0 E 'Ho we have

where .uE = A, fO , u0 = &fO. Let f" E H 1 ( R ) . Then

Since the norms IIAcll are bounded uniformly in E , i t follows that

where c is a constant independent of E . According to our choice o f operators

A, and & the functions we = d C f , w0 = do f are solutions of the problems


Estimate (2.67) of Theorem 2.13, Ch. II, implies that

By the definition of the norm I / . 1IH1. in Section 2.2, Ch. II, we have

Therefore from (2.28)-(2.31) we conclude that

Choosing J t o be such as t o make the first term in the right-hand side

of this inequality less than 613, and choosing €6 such that for e 5 56 the

second term be less than 613, we obtain the inequality (luE - u O ~ ~ ~ Z ( ~ . ) 5 C S for E 5 ~ 6 , c = const. Hence the convergence (2.26), which is equivalent t o

Therefore Condition C3 is also valid.

Let us establish Condition C4. Consider the extension operator PC of Theo-

rem 4.2, Ch. I, and suppose that 1 1 f ' l l . ~ . 5 C , where c is a constant independent

of E. Then due t o Theorem 5.4, Ch. I, we get

Therefore I I P C d c F l l H l c n ) 5 c3 and the constant c3 does not depend on E.

By virtue of the compact imbedding of H 1 ( R ) in L 2 ( R ) there is a vector

valued function w0 E H 1 ( R ) such that ((P,,&,f" - w O [ ~ ~ Z ( ~ ) -+ 0 for a

$2. Homogenization o f eigenvalues and eigenfunctions 285

subsequence E' --+ 0. This means that IIA,tf"' - R,,W~~~,~(~,I) + 0, and

therefore Condition C4 is satisfied.

Let us now consider the eigenvalue problem (1.11) and (1.12) for the op-

erators A,, do defined above. It is easy t o see that

According to the proof o f Lemma 1.6 we have

where the constants Q, c(k) do not depend on e. Taking into account (2.14)

we get for any f E H1(R)

where uc, u0 are solutions o f the problems (2.24), (2.25) with f c = f, f0 = f

respectively. Using the estimate (2.67) of Theorem 2.3, Ch. II, we obtain

where c2 is a constant independent of e. To derive (2.34) we also used the

estimate (2.31) for f = f. Thus Theorem 2.3 follows from Theorems 1.4, 1.7

and estimates (2.33), (2.34).

Corollary 2.5. x

Suppose that p.(x) = p ( ; , x), p(t7 x) is l-periodic in C E Fin and satisfies

the Lipschitz condition in x E uniformly in [. Then according to Lemma

2.12, Ch. II, we have lllp, - polll 5 E. In this case estimate (2.15) implies

where c i is a constant independent of E .


Remark 2.6.

Estimate (1.13) allows us t o obtain a more accurate expression for the constant

ck in (2.15). Indeed, according to (1.13) the constant ck in (2.15) can be

replaced by

where p: 4 0 as E -+ 0 and c is a constant independent of k, E . Note also

that in the proof of Lemma 1.6 it was established in particular that A,k 5 yk,

where -yk is a constant independent o f E .

2.3. The Mixed Boundary Value Problem for the System of Elasticity

in a Perforated Domain

Here we consider free vibrations o f elastic bodies with a periodic structure.

The boundary o f the body is free of external forces at the surface o f the cavities

and fixed at the outer part. The corresponding boundary value problem of

elasticity was studied in $1, Ch. II. It should be noted that in the case under

consideration Re is a perforated domain of type I, the elasticity operators LC are the same as in (1.1), Ch. II, and have rapidly oscillating coefficients, l? is

the corresponding homogenized operator whose coefficients are given by the

formulas (1.3), Ch. II. For these operators we consider the following eigenvalue

problems


where 6,, is the Kronecker symbol, the eigenvalues form increasing sequences

and each one is counted as many times as its multiplicity.

In Section 2.1 we already considered a particular case o f problems (2.37),

(2.38) when Rc = R, i.e. the domain RE is not a perforated one. Under the

assumptions (2.3) on p,, po we proved the convergence of the eigenvalues

of problem (2.37) t o the corresponding eigenvalues of problem (2.38). If

RE is a perforated domain and the elasticity coefficients are €-periodic, it

is possible t o obtain more accurate results compared with those of Section

2.1. In this case the key role is played by the closeness o f operators L, and

2, which is expressed in terms o f estimate (1.15), Ch. II, for solutions of

the corresponding boundary value problems. In similarity with the case of

the Neuman problem considered in Section 2.2, t o estimate the difference

of the respective eigenvalues o f problems (2.37), (2.38) we must have some

knowledge o f the closeness of the functions pE(x) and p,(x). It will be shown x

in particular that if p,(x) = p(-, x) and p([, z) E Lm(IRn x 0 ) is 1-periodic E

in E and satisfies the Lipschitz condition in x E 0 uniformly in ( then the

eigenvalues of problem (2.37) converge to those of problem (2.38) with

PO(X) = (mes Q n w)-' / P(E, x ) 4 1 (P(., x)) . Qnw

In this section the closeness of pc and po is characterized by the norm

Illplllo, which is defined by (2.64), Ch. II, where the supremum is taken over

all U, v E H 1 ( R c , re). For all u,v E H 1 ( R c , r C ) we obviously have


u. x

Suppose that p(E , x ) E L(R", f l ) (see (1.13), Ch. I), pc(x) = p ( - , x ) , pO(x) = E

( P ( . , 2 ) ) . Then

where c is a constant independent of a.

The proof of this lemma is based on an estimate o f type (2.63), Ch. II, for

vector valued functions u , v E H1(R',I' ,). This estimate can be obtained in

much the same way (with obvious simplifications) as the estimate (2.63), Ch.

II, for u , v E H 1 ( f l e ) .

We assume here that the functions p,, po in (2.37), (2.38) satisfy the

following conditions

where the constants c2, cg do not depend on E .

The closeness o f the eigenvalues and eigenfunctions o f problems (2.37), (2.38) is established by

Theorem 2.8.

Suppose that conditions (2.41) are satisfied. Then for the k - th eigenvalues A:, A; o f problems (2.37), (2.38) the estimate

holds with a constant ck independent of E .

Suppose that X o = A;" is an eigenvalue of problem (2.37) o f multiplicity

m, i.e.

xt, < xt;tl = ... = xt,+rn < xt,+rn+' (A: = 0) ,

and u o ( x ) is the corresponding eigenfunction such that 11uollL2(n) = 1. Then

there is a sequence ii, such that

$2. Homogenization of eigenvalues and eigenfunctions 289

where Mk is a constant independent o f E , U O ; 21, is a linear combination of

eigenfunctionsof problem (2.37) corresponding t o the eigenvalues A;+', ..., A:+".

The proof of this theorem can be easily reduced t o the verification of

Conditions C1-C4 and application of Theorems 1.4, 1.7, in the same way as in

the proof o f Theorem 2.3. In the case under consideration we take as 'He ('HO)

the space L2(R') ( L Z ( R ) ) with the scalar product (2.17) ((2.18)) respectively,

and set V = 'Ho. The operator Re is defined as the restriction t o Rc of vector

valued functions in L2(R). Operators A, : 'Hz -t ' H e , & : No -t No are

defined as follows

where u', u0 are solutions of the boundary value problems

The verification o f Condition C1 is based on

Lemma 2.9.

Suppose that R q s a perforated domain of type I and that the conditions (2.41)

are satisfied. Then for any uO, v0 E L2(R) the convergence (2.20) takes place.

This lemma is proved similarly t o Lemma 2.4; the convergence (2.22) fol-

lows from (2.21) since Rc = R n EW.

Conditions C1-C4 are checked similarly t o Section 2.2 with the following

modifications: problems (2.24), (2.25) should be replaced by (2.45), (2.46),

and instead o f Theorem 2.5, Ch. II, one should use Theorem 1.2, Ch. II; in the

proof o f C4 one should consider the extension Pcu constructed in Theorem

4.2, Ch. I, and use the compact imbedding


2 As in Corollary 2.5, if p, = p(- ,x) and po = (p ( . , x)), p(J , x) E L ( R ~ x

E R), then estimate (2.35) is also valid.

The inequality (1.13) allows us t o obtain a more accurate expression for the

constant ck in (2.42). Thus we can take as ck the constant defined by (2.36),

where X,k, XfS are the k- th eigenvalues of problems (2.37), (2.38) respectively.

2.4. Free Vibrations of Strongly Non-Homogeneous Stratijed Bodies

Consider the problems (2.1) and (2.2), where L,, k are elasticity operators,

studied in 57, Ch. II. It was shown in $7, Ch. II, that L, % 2 as E + 0,

provided that conditions (7.32), Ch. II, are satisfied. Therefore under the

conditions (7.32), the genreal Theorem 2.1 is valid.

In order to obtain estimates for the closeness of eigenvalues of problems

(2.1), (2.2) for stratified bodies, we shall assume that the coefficients of the

G-limit operator k are smooth in 0 and that in addition to the conditions

(2.3) on p,, po we have

where l l lpll lo is defined by (2.64), Ch. II, with Re = R, u , v E Ht(R).

Theorem 2.10.

Let LC, k be operators of the form (7.1), (7.2), Ch. II, and let C,, k sat-

isfy the conditions (7.32), Ch. II. Suppose also that the coefficients o f 2 are

smooth functions in 0. Then the eigenvalues o f problems (2.1), (2.2) satisfy

the inequality

where c k is a constant independent o f E , 6, is defined by (7.6), Ch. II.

Moreover, if Xo = A;+' is an eigenvalue of problem (2.2) o f multiplicity m,

1.e.

52. Homogenization of eigenvalues and eigenfunctions 291

and uo(x) is the corresponding eigenfunction, IluollL~(n) = 1, then there is a

sequence { u , ) such that

where c is a constant independent of E , uo, and u, is a linear combina-

tion o f eigenfunctions o f problem (2.1), corresponding t o the eigenvalues

XL+l , ...,

This theorem is proved by the same argument as Theorems 2.3, 2.8. How-

ever instead o f the inequalities (2.67), (1.15), Ch. II, one should use the fol-

lowing estimate

c* [ I I ~ ' - f llH-'(n) + a:'2 1 1 f l l L 2 ( n ) ]

which holds for solutions o f the problems

LC(ue) = in Rc , uE E HJ (R) ,

k ( u O ) = f in R , u 0 € H ; ( R ) ,

where f', f E LZ(R). Let us prove the inequality (2.50).

Denote by iic a solution of the problem

Then estimate (7.7), Ch. II, with i9 = 0 holds for uc- t ic . The function uE-iic

is a solution of the problem

Therefore according to the inequality (3.25), Ch. I, we have

Hence the inequality (2.50) is valid.

Let us consider some examples of functions p c , po satisfying conditions

(2.47).


Let p,(x) = p ( W , x ) where cp(x) isdefined in Section 7.1, Ch. II, p(f , y ) E

belongs t o the class A,, a E (O,1) , po(x) = ( p ( . , x ) ) (see Section 7.2, Ch. 11). Our aim is t o show that

~ I ~ P ~ - P O I ~ ~ O - < C E ~ , c = c o n s t . (2.51)

0 bviousl y

p ( M ,2) - po(x) = E

- $0; a - - (Pi - 9 3 x 1 - - f A x ) ,

lVcp12 ax ; IVcpI2 where g:, fi denote the respective integrals in the right-hand side of the first

equality. Since (p(., y ) - po(y)) = 0, Lemma 7.2, Ch. II yields

where the constant c is independent of E . For any u , v E H ; ( f l ) we have

Due t o the inequality (2.52) and the fact that cp E C 2 ( n ) , ( V y I > c =

const > 0, it follows that the right-hand side of (2.53) can be estimated by C ~ E , ~ ~ u ~ ~ H 1 ( 0 ) ~ ) v ) ) ~ I ( ~ ) with c1 = const independent of E . Thus (2.51) is

established.

Therefore, if the coefficients o f the operators L,, ,? satisfy the conditions cp(x) of Theorem 7.13. Ch. Il, and p.(x) = p(- , x ) , p ( t , y ) E d o , then the

E estimates (2.48), (2.49) can be written in the form

a ~ i g f d 22.- f:] u v d x = 1 [ ( 1 - f - Vcp2

R

.

52. Homogenization of eigenvalues and eigenfunctions

c:, M = const 2 0.

In analogy with Remark 2.6, we can get a more precise expression for the

constants ck in (2.48) by using (1.13) and (2.32). In the case o f stratified

structures we can also replace ck in (2.48) by c', given by (2.36).


$3. On the Behaviour of Eigenvalues and Eigenfunctions of the

Dirichlet Problem for Second Order Elliptic Equations in

Perforated Domains

3.1. Setting of the Problem. Fornal Constructions

Here we consider free vibrations of a perforated membrane fixed at the

points o f its boundary.

In $4, Ch. II, we constructed complete asymptotic expansions for solutions

of the Dirichlet problem for the elasticity system in a perforated domain. Using

the same method we can construct asymptotic expansions for solutions of the

Dirichlet problem for a second order elliptic equation. In the latter case the

maximum principle and the well-known properties of the first eigenfunction

make it possible t o study the spectral properties of the corresponding operators.

Consider a family of second order elliptic operators

a x au x ~ ~ ( u ) = - (aij(-) -) - b(;)u ,

ax; E ax,

where E E ( 0 , l ) ; a i j ( t ) , b ( t ) are smooth functions o f 5 E Rn, 1-periodic in

[, and such that

It is assumed that Re is a perforated domain o f type I (see $4, Ch. I),

Re = R n EW and the domains R and w have smooth boundaries.

In this section we study the asymptotic behaviour (as E + 0) of the eigen-

values o f the following problem

$3. On the behaviour of eigenvalues and eigenfunctions 295

where P(( ) is a smooth function of < E Rn, l-periodic in [, p(() >_ GI =

const > 0; each eigenvalue is counted as many times as its multiplicity.

The question o f the behaviour o f A,k as E -+ 0 was considered before in

[130], [131], [54]. It is proved in these papers that A,k = E - ~ A ~ + At, where

A. > 0 is a constant independent o f k, E , and AS + At as E + 0, At is an

eigenvalue o f the Dirichlet problem in R for a second order elliptic operator

with constant coefficients. Here we not only prove the convergence o f At to

At, but also obtain the estimate IA,k - A;[ 5 C ~ E , ck = const, and study the

behaviour o f the eigenfunctions o f problem (3.3) as E -+ 0.

Let a ( [ ) be the eigenfunction corresponding t o the first eigenvalue A.

of the following boundary value problem in the unbounded domain w with a

l-periodic structure:

= 0 on aw , a ( [ ) is l-periodic in [ , I (3.4)

The boundary condition in (3.4) is understood in the sense that a([) belongs to the space $ ( w ) (see $1, Ch. I).

I t is well known (see [ l l ] , [64]) that a ( [ ) is a smooth function in w such

that a ( [ ) # 0 in w and IVF@I # 0 in a neighbourhood of dw.

Let us formally represent the k- th eigenfunction o f problem (3.3) in the

form

It is easy t o verify that v,k(x) must satisfy the relations

and


Thus we obtain an eigenvalue problem for a second order elliptic equation

degenerate on the subset S, of the boundary o f Cl".

Making a suitable choice of functional spaces for solutions o f the corre-

sponding degenerate boundary value problem, we shall reduce (3.6) t o an

eigenvalue problem for a positive compact self-adjoint operator in a Hilbert

space, and show that problem (3.6) has a discrete spectrum consisting of

eigenvalues

where each X i is repeated as many times as its multiplicity. If v,k is an eigen-

function o f problem (3.6) corresponding to X,k, then @(:)v3 belongs t o HA(Cle) and is an eigenfunction o f problem (3.3) corresponding to the eigenvalue

Applying the homogenization methods, developed in Chapter II and in $51,

2 , Ch. Ill, t o the degenerate operators

we obtain the estimates

where Xk is the k-th eigenvalue of the Dirichlet problem for a second order

elliptic equation with constant coefFicients. These coefficients are expressed

through the coefficients of operators (3.7) by means of the homogenization

procedure described in Chapter II.

3.2. Weighted Sobolev Spaces. Weak Solutions of a Second Order

Equation with a Non-Negative Characteristic Form

For our further consideration we shall need the following spaces o f periodic

functions:

§3. On the behaviour of eigenvalues a n d eigenfunctions

P1(w) is the completion o f e r ( w ) in the norm

PO(w) is the completion o f &?(w) in the norm

+(w) is the completion of e?(w) in the norm

Q = { ( : O < C < l , j = 1 , ..., n ) .

It is easy to see that if IIullPcw) = 0 and u is smooth, then u = 0 in

w. Indeed, suppose the contrary, i.e. that u + 0 at a point so. Then we

can assume that u > a. > 0 in a neighbourhood wo of xo, and therefore

/ Iv@12d( = 0. Multiplying the equation (3.4) by $(()@(<). where $([) E wo

C,OO(wo), $([) 2 0 in w and integrating over Q n w , we find that @ = 0 in wo,

which is impossible, since @ > 0 in Q n w.

Let us also consider the spaces $ (w), W;(W), introduced in $1, Ch. I.

Lemma 3.1.

The following imbeddings

(w) C w; (w) C P1(w) ,

Q1 (w) c Q(w) (3.10)

are continuous. Moreover, the imbedding (3.9) is compact, and for any v E

P1(w) we have @(()v(() E$' (w).

Proof. Let u E W;(W). Consider a function cp6 E &F(w) such that cp6(() = 1,

if ~ ( t , aw) > 26, cps(() = 0, if p ( t , dw) < 6, 0 I cps I 1, IVrcp61 < c6-l, c = const. Then


This implies (3.8), since 1 @ 1 5 c16 in the 26-neighbourhood o f dw, and there-

fore the right-hand side o f the above inequality tends to zero as 6 --+ 0.

Let us show now that for any u E 6 ' r ( w ) the following inequalities are

satisfied

Multiplying the equation (3.4) by cPu2, u E e?(w), we get

Therefore 112

Qnw

Hence (3.11) is valid. The inequality (3.12) follows from (3.11), since

53. On the behaviour o f eigenvalues and eigenfunctions 299

It is easy t o see that for any u E Q1(w) the inequalities (3.11), (3.12) are

also satisfied, and moreover e u E$ (w) . The continuity of the imbedding

(3.9) is obvious. Let us prove its compactness.

Consider a sequence {urn} of elements o f Q1(w) such that sup IlumllP1(,, m

< c < oo. It follows from (3.12) that Il@um(lH~(Qnw) < cl, where cl

is a constant independent of m. Due t o the compactness o f the imbed-

ding H 1 ( Q n w ) c L2(Q n w ) there is a subsequence m' -+ co such that

@urn' --+ w E H 1 ( Q n W ) in the norm o f L2(Q n w ) . It follows that @urn' is

a Cauchy sequence in L2(Q n w ) and therefore urn' is a Cauchy sequence in

c O ( w ) . Hence urn' + uO E QO(w). Lemma 3.1 is proved.

Lemma 3.2 (the Poincark inequality).

For any u E V 1 ( w ) such that

/ Q2ud[ = 0 , (3.13)

Qh

the inequality

I I U I I Z ~ ~ , , 5 c J l @ 1 2 lvcui2dt (3.14)

Qnw

holds with a constant c independent o f u .

Proof. Suppose the contrary. Then there is a sequence uN E c 1 ( w ) such that

1 N / I @ 1 2 l veuNl2dt 5 , , llu I I Q I ( ~ ) = 1 . (3.15)

Qnw

Due t o the compactness of the imbedding of v l ( w ) in QO(w) we can assume

that the sequence { u N ) is such that

/ Ie12 IuN - uNtt12dt -+ o as N -+ m . Qnw


Thus lluN -+ 0 and therefore there is a function u E P1(w) such

that lluN -ullol(,) -t 0 as N -t oo. Taking into account (3.15) we conclude

that lVEul = 0 almost everywhere in Q n w. Since iP vanishes only on Bw,

i t follows that u = const in Q n w and IIullplcW, = 1. This contradicts (3.13).


Let Rc be a perforated domain o f type I considered in $4, Ch. I. We in-

troduce the spaces V,'(R6), VO(Rc), V(Rc) as completions of C,""(Re) in the

respective norms:

Lemma 3.3. For any u E CF(Re) the following inequalities are satisfied

1 IV~@(:)(~ 1ul2dx 5 Q / + I V ~ U I ~ ) ~ ~ , (3.19) n e nr

I f u E V,'(Rc), then (9(f)u E HA(Rc). The imbedding V,'(Rc) c VO(R') is

compact and H1(Re,I',) c V,'(Rc).

Proof. The function a(;) satisfies the equation

x x and the boundary condition a ( - ) = 0 on d ~ w . Multiplying (3.21) by a ( - )u2

E E where u E CF(Re) , and integrating over Re, we find

'$3. On the behaviour of eigenvalues and eigenfunctions 30 1

a@ a@ = - J a . . - d@ du - 1 ~ 1 ~ d x - 2 a;j - @u - dx t " ax; axj n

J ax, a x j R e

Therefore

It follows that

x x Since I v , @ ( - ) ~ ~ = E-' I v ~ @ ( - ) ~ ~ , this inequality implies (3.19). Inequality

E E (3.20) follows from (3.22), since

For a fixed E , by virtue o f (3.20) the convergence u3 4 u in the norm x

of * ( R E ) as j -+ oo, uj E C?(Rc), implies that { @ ( - ) u i ) is a Cauchy 5

E

sequence in Hi(Ctc) and consequently @(-)uj --+ w in Hi(R') as j + m. E

The convergence o f uj t o u in V,'(RC) implies the convergence of @uj t o @u x

in L2(R') . Therefore u@ = w. This means that @(-)u E H,'(R') for any E

E v , ( n s ) .

The compactness o f the imbedding V,'(Re) c V O ( R ' ) for a fixed E can

be proved similarly to the compactness o f the imbedding Q1(w) c pO(w) in

Lemma 3.1. The imbedding H1(OC,r , ) c Vd(N) is established in the same

way as the imbedding W;(U) c p l ( w ) . Lemma 3.3 is proved.

Lemma 3.4.

Let the sequence u" E c ( R E ) be such that


Then there is a subsequence E' + 0 and a function u0 E H i ( R ) such that

l(uO - ~ " ' l l ~ ~ ( ~ ~ ' ) + 0 as E' + 0.

Proof. Note that the domain Re has the form R n EW, where w is a smooth

unbounded domain with a 1-periodic structure, w satisfies the Conditions

81-83 o f 54, Ch. I. Due t o the Conditions B1-B3 for any 6 < b0 (60 is

sufficiently small) there is a smooth unbounded domain w6 c w with a 1-

periodic structure, which also satisfies the Conditions B1-B3 and such that

0 < c16 < p(x,dw) < c26 for x E dws, c l , c2 are constants independent of

6. Set R; = R n E W ~ , rt = d R n d o ; , Sf = 80; n 0. It is easy t o see that

Ra c R' is also a perforated domain o f type 1, r,6 c I',. Since a(<) > 0 in w, it follows from (3.23) that

where cs is a constant independent of E ; 6 E (O,bo), u' E H1(R; , I ' t ) . For

a fixed 6 E (0,60) using Theorem 4.3, Ch. I, let us construct extensions

P:uc E ~ : ( f i ) of the functions uE t o a domain f i containing 0. According 1 t o Theorem 4.3, Ch. I, it follows from (3.24) that sup I I P : U ~ I I ~ ' ( ~ ) 5 cs

€

< m. Using the compactness o f the imbedding ~ i ( f i ) c ~ ~ ( f i ) let us choose

a subsequence E' + 0 such that

P:U" -+ u & ( x ) weakly in ~ , ' ( f i ) a n d strongly in ~ ~ ( f i ) .

(3.25)

By Theorem 4.3, Ch. I, we have u6 E HA(R). Thus

In complete analogy with the above considerations, for any E (0 ,6 ) we

can extract a subsequence E" + 0 of the sequence E' and find a function

UJ, E HA(R) such that

Let us show that us, = us. Indeed,

$3. On the behaviour of eigenvalues and eigenfunctions

The right-hand side of this inequality tends to zero as E" t 0. Setting f = 1,

$f = $ = (P' = (P = us, -us in Corollary 1.7, Ch. I, we see that the left-hand

side of (3.26) tends t o (mes ~ n w ~ ) ' / ~ llu6, Therefore us, -us = 0.

Thus we have shown that there is a function uo = us E HA(f l ) such that

for any 6 E (O,bO)


sup J lu'12 I v { Q ( ~ )12dx 5 c2 < m , n*

since the norms IIuCllvdcne) are bounded by a constant independent o f E . There-

fore, due to the fact that IV{iP([)I # 0 on dw, we have

sup J l u c 1 2 d r < c 3 < m , nc\n;

where c3 is a constant independent o f E ; 6 E (0,6,), i f 60 is sufficiently small.


Due to (3.28) and the boundary condition a ( ( ) = 0 on dw, for each o > 0

there is a 6 such that 1;' < 012 for all E'. Taking into account (3.27), let us

choose EO such that If' < 0 1 2 for all E' < EO. Hence the convergence t o zero

of the left-hand side o f (3.29) as E' -+ 0. Lemma 3.4 is proved.


As a consequence from Lemma 3.4 we get a Friedrichs type inequality for

functions in Vd(RE).

Lemma 3.5.

For any u E V,'(Rc) the following inequality of Friedrichs type

holds with a constant c independent o f E .

Proof. Suppose the contrary. Then there is a sequence E -+ 0 such that

U I d x < a c , / I@( ;)I2 IV. (3.31) n

where a, -+ 0 as E + 0. According t o Lemma 3.4 there is a subsequence

E' --+ 0 and a function uo E Hi(R) such that

Similarly t o the proof o f Lemma 3.4 consider the subdomain REg of RE for

a fixed 6 E (O,hO), and let P!uc be the extension o f ue from REg t o a domain x fi containing 0. Since @(-) 2 cg = const > 0 for x E @ and cs does not E

depend on E , it follows from Theorem 4.3, Ch. I, and inequalities (3.31) that

where c, cl are constants independent of E . On the basis o f (3.25) we can

assume that P ~ U " -+ uo(2) weakly in ~ , ' ( f i ) as E' -+ 0. Therefore, because

of (3.34) and (3.31) we have uo = const E H,'(R), which implies uo - 0. It

follows from (3.33) that I I u ~ ~ ~ ~ ~ ~ ( ~ ~ I ) -+ 0 as E' -+ 0. On the other hand, from

(3.31), (3.32) we have I I u ~ ~ ~ ~ ~ ~ ( ~ . I , -+ 1 as E' -+ 0. This contradiction shows

that inequality (3.30) is indeed valid. Lemma 3.5 is proved.


Let us also introduce the space V1(R") as the completion o f C"(ae) in

the norm (3.16).

Consider the following boundary value problem for a second order equa-

tion with a non-negative characteristic form, which is elliptic inside Re and is

degenerate on Sc c do':

where f j E VO(Re), j = 0, ..., n; 11, E V1(R"), operator M, is given by (3.7).

A weak solution of problem (3.35) is defined as a function u E V1(R')

such that u - 11, E V,'(Rc) and the following integral identity holds for any

w E G ( R E ) :

Theorem 3.6.

There is a unique weak solution u E V1(R') of problem (3.35). This solution


IIullvlcn*, 5 c 2 l l f i l lv~co, + Ildllvlcn*, 9 [ I (3.36) i=O

where c is a constant independent o f E , f', 11,.

The proof of this theorem is similar t o that of Theorem 3.8, Ch. I, and is

based on Theorem 1.3, Ch. I with H = V1(R") and on the Friedrichs inequality

(3.30).

In what follows we shall need a maximum principle for weak solutions of

problem (3.35).

Lemma 3.7 (The Maximum Principle).

Let u(x) be a weak solution of the problem


where II, E V 1 ( R c ) n C O ( @ ) , 5 M = const. Then lu(x)I 5 M almost

everywhere in Re.

Proof. Consider the domains fl; = R ~ E w & constructed in the proof o f Lemma

3.4. Denote by v6 a solution o f the problem

Since a ( ( ) vanishes only a t the points o f dw, this equation is elliptic in

a',, and therefore according t o the maximum principle

I t follows from the integral identity for solutions o f (3.38) that

a$ av = - / [m2aij - - + b ~ ' $ v ] dx ,

a x , a x j Q;

for any v E H:(R:), where w6 = v6 - II, E HG ( 0 : ) . Let us take v = w6 and extend it as zero t o Rc\R:. Then from (3.40) and

the Friedrichs inequality (3.30) we find that

where c is a constant independent of 6. Since w6 = 0 in RE\%, it fol-

low from (3.41) for 6 -r 0 that the sequence {wh) satisfies the condition

sup I I ~ ~ l l ~ ~ ~ ~ e ) < m. Due t o the compactness o f the imbedding V:(Rc) C 6

V O ( R e ) and the weak compactness of a ball in a Hilbert space, there is a

sequence 6 4 0 and a function wo E V,'(RE) such that

wg -i w0 weakly in V, ' (Rc) and strongly in V O ( R c ) . (3.42)

$3. On the behaviour o f eigenvalues and eigenfunctions 307

For a fixed v E Cp(Rc) the integral identity (3.40) holds for all sufficiently

small 6, since 0; C if < 6. Passing in (3.40) t o the l imit as 6 + 0 and

taking into account the uniqueness o f a solution o f problem (3.37), we find

that wo t $ = U, where u is a solution o f (3.37). It is easy t o see, by virtue

o f (3.42), that we have

and therefore ((u - + 0 for any open set G such that G c 0". It follows that lu(x)l 5 M , since 1v61 5 M. Lemma 3.7 is proved.

Let us consider the problem

A weak solution of this problem is defined as a function N E P1(w) satis-

fying the integral identity

for any $ E cl(w).

Let

in Theorem 1.3, Ch. I, and take the left-hand side o f (3.44) as the bilinear

form a(cp,+), and the right-hand side as l(y5). Then, using estimate (3.14) in

complete analogy t o Theorem 6.1, Ch. I, one easily establishes

Theorem 3.8. Suppose that


Then there is a unique (to within an additive constant) solution o f problem

(3.43). This solution satisfies the inequality

where q is a constant, c is a constant independent of N , Fi , i = 0,1, ..., n

3.3. Homogenization of a Second Order Elliptic Equation Degenerate on the Boundary

Let us now define the coefficients of the homogenized equation correspond-

ing t o the problem (3.35). Denote by N q ( < ) , q = 1, ..., n , solutions o f the

problems

Set

where

(3.49)

Let us show that iip,qpqq > c 1q(2, c = const > 0.

By virtue of (3.47) one can easily check that ii,, can be rewritten in the

forrn

Thus hPq = hqp. Due t o (3.50), setting w, = ( N , + t p ) q p , we obtain

53. On the behaviour of eigenvalues and eigenfunctions 309

If for some 77 # 0 we have apqqpq, = 0, then w, z ( N , + Ep)77, = const for

almost all ( E Q n w . Since N , ( ( ) are periodic, it follows that 77 = 0.

Set h = d a 2 ( ( ) b ( ( ) d t . ~ J n w

Thus we have defined the following second order elliptic operator with

constant coefficients

The next theorem establishes the closeness of a solution o f the boundary

value problem for Mc and a solution o f the boundary value problem for the

homogenized operator M.

Set

6 = d 0 2 ( ( ) p ( ( ) d ( ( 6 = d due to ( 3 . 4 ) ) / Qnw

Theorem 3.9.

Let u c , uO be solutions o f the boundary value problems

and f 0 E ~ ' ( a ) , f C E V O ( R c ) . Then

where c, cl are constants independent o f e.

Proof. Set

where v is a solution o f the problem


Therefore the function 4 belongs to Vd(RC).

Applying the operator M, to uc - 4 , in a similar way to (1.16), Ch. II, we

obtain the following equalities, which are understood in the sense of distribu-

tions:

a a auO Mc(uc - 4 ) = MC(uc) - - (ahkm2 - (UO + EN. -)) +

axh axk ax*

auO a auO + m2bu0 + &bm2Nj - = Mc(uc) - (m2ijhk -) + axj x h axk

a + - EC - E) - 8x1, 6x6 axk

8 aN, duo d2u0

auO + Q2bu0 + &ba2Nj - = M,(uc) - m2i$f(u0) + Q2(b - &)uO - ax j

duo dm2 8 - i h k - - + - [ ( a 2 & h k - @'ahk - Em2ahj

aNk duo dxk axh axh

a a2u0 auO - E - (ahk@'~. -) + & b ~ j @ ~ - = Me(uc) -

axh 8xk6x, ax j

duo dQ2 am2 duo - a2i$f(u0) + a2(b - i))uO - ihk - - + ahk - - - axk axh axh axk

a - - [m2a,,* + m

ax h

2 a~~ a + (m28hk - ahk - &@ ah' - - E - (ajh@ a2u0

axj axj - €@'ahkN,

puO auO + &bNjm2 - = axkaxhax, ax j

= pG2(f0 - fr) + (6 - p)@2 f O + m2(b - i))uO + dNk a(a2aihNk) + ( 0 2 i i h k - a2ahk - m2ahj - - at j ati +

$3. On the behaviour o f eigenvalues and eigenfunctions 311

Define the functions N,, ( t ) , B ( ( ) , R ( J ) as 1-periodic in J solutions of the

following boundary value problems

a a - ( a - B ) = ( b ) - ) a 2 in w , B E Q1(w) , at i at1

These problems are solvable, since the relations (3.48), (3.49), (3.52) allow

to apply Theorem 3.8.

We thus have

a a~ M.(ue - O) = pQ2(f0 - f') + a - (m2ai1 -) f O +

ax; at1

I t follows that u' - fi satisfies the equation

a M,(ue - 6 ) = a m 2 P + a - ( Q ~ F ~ ) + pm2(f0 - f') , (3.59)

ax , where


Due t o the periodicity o f R, B , Nhk, N8 we have

From this estimate and Theorem 3.6 with 11, - 0 we deduce that

where cl is a constant independent of e.

Since 6 has the form (3.57), it is easy t o see that the inequalities (3.55),

(3.56) will have been proved if we establish the estimates

where c2, cg are constants independent of E .

The proof o f the estimate (3.61) is based on the maximum principle for

solutions o f problem (3.58), established in Lemma 3.7. Let us show that 2

the functions N j ( - ) are continuous in @ and are bounded by a constant E

independent o f e.

It follows from Lemma 3.1 that @ ( O N , ( ( ) ~i (w). Since N, is a solution

of problem (3.47), we have

It follows that the function QN, = w E$ (w) satisfies the equation


since # 0 in w . Due t o the well-known results on the smoothness of solutions

of elliptic boundary value problems and our assumptions about the smoothness

of w and akj, the function w ( ( ) is smooth in 5. Moreover w = 0 on aw. Since

a(() = 0 on dw, but its gradient does not vanish in a neighbourhood o f dw,

the function N , r w/@ is continuous in w.

Thus by Lemma 3.7 we have Ivl 5 C ~ E IluOllclcn) and therefore the inequal-

ity (3.61) is satisfied.

Let us prove estimate (3.62). Let q ~ , ( x ) be a truncating function defined x auO

immediately after the formula (1.23), Ch. II. Set Q , = c p , ~ N , ( - ) - . Then E ax,

@, E V 1 ( R ' ) . It is easy t o see that

Therefore, taking into consideration the smoothness o f u0 and the periodicity

of N , , we obtain the inequality

where c5 is a constant independent of E . Using the integral identity for u - Qc

we get the estimate (3.62). Theorem 3.9 is proved.

3.4. Homogenization of Eigenvalues and Eigenfunctions of the

Dirichlet Problem in a Perforated Domain

Consider now the question o f the closeness of the eigenvalues and eigen-

functions of the following problems


~ ( v , k ) + X;,v,k = 0 in R , v,k E HA (0) ,

o < X ; < X : I . . . < X ; I ...,

I B V ~ V ~ ~ X = 61, , n

where

1 Theorem 3.10.

Let X,k and X b be the k- th eigenvalues of problems (3.63) and (3.64) respec-

tively. Then

where ck is a constant independent o f E .

Suppose that the multiplicity o f X o = A;+' is equal to m, i.e. X f , < X f ; t l =

... = x L + ~ < ~ f , + ~ + ' , XE = 0, and vo (x ) is an eigenfunction of problem (3.64)

corresponding t o Xo, IIvollLzcn, = 1. Then for every E E ( 0 , l ) there is a

function ve such that

where MI is a constant independent of E , vo; ve is a linear combination o f eigen-

functions o f problem (3.63) corresponding t o the eigenvalues A:+', ..., A:+".

Proof. Let us apply the abstract results obtained in Section 1.2. Denote by 'He the

space V O ( R e ) equipped with the scalar product

By 'Ho we denote the space L 2 ( R ) with the scalar product

( u , v ) ~ , , = / 6 uv dz . n


Set V = 3-10. We define Re as the restriction operator: Rcu = uc, uc = u on

Re, u E L2(R).

Let us check the conditions (1.4). According t o Lemma 1.6, Ch. I, and

inequalities (3.49), (3.52) we have

/ ~ u ~ l ~ ~ ( q ) @ ~ ( : ) d x -t / I ~ ~ / ~ d x = nc n

Thus Condition C1 of Section 1.2 is satisfied.

Let us introduce the operators A, : 3-1, -t 'H,, do : 3-10 -+ 3-10,

setting A, f' = uc, &fO = uO, where uc and u0 are solutions of problems

(3.53) and (3.54) respectively. Using the corresponding integral identities we

see that these operators are positive and self-adjoint. The compactness of A,

and d,, follows from the compactness of the imbeddings Vd(Rc) C VO(R')

and H,'(R) c L2(R) respectively. Due t o (3.36) the operators A, have norms

uniformly bounded in E . Therefore Condition C2 of Section 1.2 is also satisfied.

The validity of Condition C3 is guaranteed by the estimate (3.55) of The-

orem 3.9 and by the density o f C1(!?) in L2(R). Let us check the Condition C4. If sup 1 1 f ' l l N t < 00, then according to

c

(3.36) we have sup IIA. f' l lvlcn*, < 00 and therefore due t o Lemma 3.4 we c

can find a subsequence E' and a function w0 E H,'(R) c L2(R) such as t o

satisfy (1.6). Due t o the smoothness of eigenfunctions of problem (3.64) the estimate

(3.55) yields for any vk

Since the eigenvalues of problems (3.63), (3.64) and (1.11), (1.12) are

related by (2.11), the assertions o f Theorem 3.10 follow directly from Theorems

1.4, 1.7. Using the above results we can easily compare eigenvalues and eigenfunc-

tions of problems (3.3), (3.63), (3.64). Thus we have actually proved


Theorem 3.11.

Let At, A$, Xk be the k- th eigenvalues of problems (3.3), (3.63), (3.64) re-

spectively. Then

where A. is the first eigenvalue of problem (3.4), the constant ck does not

depend on E .

Suppose that the multiplicity o f the eigenvalue Xo = A;+' of problem (3.64)

is equal to m, i.e.

and vo(x) is an eigenfunction corresponding t o Xo. Then for each E there is a function Ue such that

where M,' is a constant independent of E , vo; U E is a linear combination of

eigenfunctions o f problem (3.3) corresponding t o the eigenvalues A:', ..., A:+"

$4. Third boundary value problem for second order elliptic equations 3 17

$4. Third Boundary Value Problem for Second Order Elliptic

Eauations in Domains with Ra~ id l v Oscillatinn Boundarv

4.1. Estimates for Solutions

Let R be a simply connected bounded domain in R2 whose boundary 8 0 is

smooth and is described by the natural parameter s , which takes values from

0 t o 1 and is equal t o the curve length counted from a fixed point on 8 0 .

In a neighbourhood of d R we introduce the coordinates ( s , t ) , where t is

the distance from a given point to d R along the normal t o d R containing this

point.

Consider the domain Rc c IR2 containing R and bounded by the curve

where E = l l m , m > 0 is integer, $([) is a smooth 1-periodic function of

6 E R1, $(<) 2 0. Thus for small e the domain RE has a rapidly oscillating

boundary. a Let L ( u ) = - a , .

au ( " ( 5 ) a,) be a second order elliptic operator whose

coefficients a i j ( x ) are smooth functions in R2 such that

a . , - a , . 13 - 3% 7 a i j ( ~ ) ~ i ~ j L KO 171' 7

KO = const > 0 , E R2 .

By a ( u ) on d R or on dRc we denote the conormal derivative a ( u ) = au

a;j - vj, where u = ( y , u 2 ) is the outward unit normal t o the boundary of ax;

the corresponding domain.

Consider the following boundary value problems


1

where I' = J (1 + (+'(s) l2)l l2ds, a ( x ) is a smooth function in R2, a ( x ) 2 0

a0 = const > 0.

Our aim is t o estimate the difference o f solutions o f problems ( 4 . 1 ) , (4 .2 )

in terms o f fO, f', and after that, following the general method developed in

Section 1.2, t o evaluate the closeness between eigenvalues and eigenfunctions

of operators corresponding t o problems (4.1) , ( 4 . 2 ) .

Set

where ds, is the element o f curve length on aRc . Weak solutions of problems (4 .1 ) , (4 .2) are defined as functions uc E

H 1 ( R c ) , u0 E H 1 ( R ) which satisfy the integral identities

for any v E H'(Rc) , w E H 1 ( R ) .

Theorem 4.1.

Let uc and uO be weak solutions o f problems ( 4 . 1 ) , ( 4 . 2 ) respectively. Then

the following estimate is valid

where c is a constant independent of E , f O , f'.

We first outline some auxiliary results to be used in the proof of Theorem

4.1.

Note that the existence, uniqueness and estimates (uniform in E ) for the

H1(R') norms o f solutions o f problem (4 .1 ) in terms o f 1 1 f ' l l L ~ ( n c , , can be

easily obtained from Theorem 1.3, Ch. I, and the following

54. Third boundary value problem for second order elliptic equations 319

Lemma 4.2.

There is a constant M independent o f E and such that

for any u E H 1 ( R L ) .

Proof. Since the diameter of RE is bounded by a constant independent of E ,

one can find a constant b such that b does not depend on E and 1 5 2-ebzl 5 2

for all x E RE. Set u = ( 2 - ebXl)v . Then

- / b(2 - ebz1)ebz1v2vldsE . an*

Therefore

+ / b(2 - ebz1)ebz1v2hdsE . an*

The estimate (4.4) follows from this inequality and the conditions imposed

on a i j ( x ) , a ( x ) , b. Denote by G6 the 6-neighbourhood o f d R and by the 6-neighbourhood

of the domain R, where 6 is sufficiently small.

In terms o f the coordinates ( s , t ) , introduced above in a neighbourhood of

80, one can write

S The parameter E is assumed t o be so small that 0 < E $ ( - ) < 612. Thus

E dRE c GbI2 .


Lemma 4.3.

For any v E H1(Rc) there is an extension Pcv E H1(R(6)) such that

IIpcvII~l(n(~,) 5 IIvIIH1(ne)

Moreover

IIvII~z(nqn) I c ~ E " ~ IIvII~lcnr) ,

where the constants GI, cl do not depend on E , v.

Proof. Fix v E H1(Re) . Let us consider v on G6 n Rc and extend i t t o the set

G6 as follows. First we pass from the coordinates s , t in G6 t o the coordinates s

S' = S , t' = t - E$(-) . In the variables s', t' the sets G6 n Rc, G6\RC have E

the form

(G6\Re)' = { ( s f , t ') : 0 < s' < 1 , 0 5 t' < 6 - ell(:)} .

Set

w(sl, t ') = v ( s ( s f , t '), t ( s f , t ' ) ) = v(s , t ) for t < E$( : ) , t' I O .

According t o Proposition 2 o f Theorem 1.2, Ch. I, the function w(sl , t') can

be extended from the set Go = {(s ' , t l ) : 0 5 s' I 1 , -26 5 t' 5 0 ) to the

set G = {(sl , t ' ) : 0 5 sf 5 1 , -26 5 t' 5 26) as a function Pw E ~ ' ( 6 ) such that Pw is 1-periodic in sf and I I P w I I ~ ~ ( ~ ) 5 c I I w I I ~ ~ ( ~ ~ ) , where c is

a constant independent o f w. Setting (Pcv) (s , t ) = v ( s , t ) for ( s , t ) E RE, S

(P.v)(s,t) = ~ w ( s , t - E $ ( ; ) ) for E$(:) < t < 6, we obtain the needed

extension.

Let us prove estimate (4.6). The set Rc\R lies in the 6-neighbourhood

of 8 0 , 6 is of order E . Therefore applying Lemma 1.5, Ch. I, in the domain

R(6)\0 we get



Lemma 4.4. 1

Let 7 ( 7 ) be a smooth 1-periodic function o f 7 E R1, such that / 7 (q )dq 0

= 0. Then for any u E H 2 ( R ) , v E H 1 ( R ) the following inequality is satisfied

where c is a constant independent of E , u , v.

The proof of this lemma can be obtained by the same method as the proof

o f Lemma 2.9, Ch. I; however, in the case under consideration we should take

as 0 1 the domain G = { ( s , t ) : 0 5 s 5 1 , -6 < t < 01, and instead o f the

sets 07 consider the sets a, = { ( s , t ) : t = 0 , ~ ( m - 1 ) 5 s 5 Em).

Proof o f Theorem 4.1. We can assume that the function u0 is extended t o

the domain R(&) in such a way that

The possibility of such extension is guaranteed by the smoothness o f dR. Let us write the integral identities for uc, uO:

and set v = uE - uO. Subtracting the second equality from the first one we

obtain


Passing from the coordinates x to s, t in the 6-neighbourhood of 80 and S S

setting w(s , t ) = a(s,t)uO(s, t )v(s , t ) ; g( - ) = (1 + l$t(-)12)112, we have E E

Applying Lemma 4.4 in the case of u = a(s,t)uO(s, t ) , v = v(s, t ) , ~ ( q ) =

g(q) - r, we get

It is easy to see that

4:) dw

w (s, E $ ( : ) ) - w(s, 0) = / ;il ( s , t )dt = 0

Therefore

L CI lluOllH~(nqn) I lvI I~l(n*\n) .

From the estimate (4.6) we have I l ~ ~ l l ~ l ( ~ c \ ~ ) 5 ~ ~ e ' ~ ~ llu0ll~2(na). Conse-

quently

1121 5 ~ 3 e ~ ~ ~ I I u O I I ~ ( n ) I Iv l I~l(n*\n) .

Therefore from (4.9), (4.10), (4.11) we get

$4. Third boundary value problem for second order elliptic equations 323

Let us estimate the remaining terms in the right-hand side of (4.8).By

vritue of (4.6) we find that

5 ~5 [I~uO~Iil(n*\n) IIvIIilcn*\n) + IIY - P I I L ~ ~ ~ , IlvII~lcn) +

+ IIPllLa(fic\f2) I I ~ I I L ~ ( ~ * \ ~ ) ] 5 [rl" IIuOIIi.(n) IIvIIi1(nC) + + IIP - ~ ' I I L . ( ~ ) I I V I I H ~ ( W ) + I I Y I I ~ ( ~ * \ ~ ) & ' / ~ I I V I I W ( ~ * ) ] +

(4.13)

Taking into account (4.12), (4.13) we deduce from (4.8) and Lemma 4.2 that

lluC - u0llLl(n*) < 5 c5 [a IIuOIIL~(n) + E IIYllbcn*\n, + Ilf' - f011Z2(n,] .

This inequality implies (4.3). Theorem 4.1 is proved. 0

4.2. Estimates for Eigenualues and Eigenfunctions

Consider the following spectral problems

L(ut) + Atut = 0 in fie , ut E H1(fie) ,

u(ut) + a(x)ut = O on dRe ,

Jufu:d~=6k{ , O < A ; < . . . ~ A ~ < ..., n= I


where the eigenvalues are enumerated in increasing order and according t o

multiplicity, as in §2.

To study the closeness o f A t t o A; we apply the general method described

in Section 1.2.

Set 'He = L2(n ' ) , 'Ho = L2(R) = V ;

Define the operator R, : L2(R) + L2(Rc) setting RE f = f ( x ) for x E R,

R, f = 0 for x E RE\R. It is obvious that Condition C1 holds with y = 1.

Let us introduce the operators A, : 'H, -+ 'H,, : 'Ho + 'Ho setting

AE fE = uE , do f 0 = uO, where uc, u0 are solutions of problems (4.1), (4.2)

respectively. It is easy to verify that A,, are positive compact and self-

adjoint operators and that due to Lemma 4.2 the norms IIAcll are bounded by

a constant independent of E .

Consider Condition C3. Let f 0 E L2(R) . Then ACRE f 0 = uc is the

solution of problem (4.1) with f' = f 0 in R, f" = 0 in RE\R. We clearly have

The first term in the right-hand side of this equality converges to zero as E + 0

due t o estimate (4.3), and the second term converges t o zero since the norms

lluEIIHl(n.) are bounded by a constant independent o f E and mesRE\R + 0

as E --+ 0.

Let us prove the validity o f Condition C4. Suppose that sup 1 1 f l l L ~ ( n c ) E

< co. Then sup I I u ' I I ~ I ( ~ ~ ) < co, uC = A, f " . E

Consider the extensions P,uc E H1(R(&)) of the functions uc, constructed

in Lemma 4.3. Due to the compactness of the imbedding H1(R(s) ) c L2(R(6)) there is a function U E H1(R(&)) and a subsequence E' + 0 such that

IIP,IU" - U I I L ~ ~ ~ ~ ~ , ) -t 0 as E' + 0

Then


This equality, together with (4.16) and Lemma 1.5, Ch. I, implies Condition

C4.

We have thus established that Conditions C1-C4 are satisfied and therefore

Theorems 1.4, 1.7 can be applied t o estimate the closeness of eigenvalues and

eigenfunctions of problems (4.1), (4.2) in exactly the same way as it was done

in 52 for the elasticity problems.

Theorem 4.5.

Let X,k, Xgk be the k- th eigenvalues of problems (4.14), (4.15) respectively.

Then

where ck is a constant independent of E .

Suppose that the multiplicity of the eigenvalue Ah+' = Xo is equal to rn,

i.e. X i < A?' = ... = A;+" < A;+"+', XE = 0, and uo is the eigenfunction of

problem (4.15) corresponding t o Xo, I I ~ ~ l l ~ 2 ~ ~ , = 1. Then there is a sequence

{ti,) such that

where MI is a constant independent o f E , UO; 21, is a linear combination o f eigen-

functions of problem (4.14) corresponding to the eigenvalues A:+', ..., A:+".

Remark 4.6.

The case G(7l) > 0, i.e. R C Re, has been considered merely for the sake of

simplicity. With the use of slightly more complex calculations, theorems on

the closeness of solutions and spectral properties of problems (4.14), (4.15)

can also be proved i f $(q) changes sign.

Remark 4.7.

Constructing suitable boundary layers we can also obtain estimates o f order E

for the difference of solutions of problems (4.1), (4.2).


Remark 4.8.

Methods used in this paragraph can also be applied in the case o f n in-

dependent variables, when the boundary dR in local coordinates has the

form {x : x, = $(?)I, and the perturbed boundary dRr has the form

{x : x, = $(2) + ~~(2)~(!)), where 2 = (zl, ..., x,-1) varies over a

bounded open set G c Rn-', g(2) E C,"(G), ~ ( 6 ) is a smooth function

1-periodic in q.

Remark 4.9.

A similar problem can be considered for the system o f linear elasticity.

The main results o f this paragraph were obtained by another method in [4]

(see also [110]).

55. Free vibrations of bodies with concentrated masses

$5. Free Vibrations o f Bodies with Concentrated Masses


We consider an eigenvalue problem for the Laplace operator with the Dirich-

let boundary condition and with a density function which is constant every-

where in a domain R c IR", n 2 3, except for a small neighbourhood o f one

o f its interior points, say 0. It is assumed that O is the origin o f Rn and R is a bounded smooth domain.

Here we study the following eigenvalue problem

where E > 0, form an increasing sequence and each eigenvalue is counted

as many times as its multiplicity; x ( ( ) is a bounded measurable function such

that x( ( ) > M = const > 0 for ( E G, x ( ( ) = 0 for ( $Z G, G is an open set

o f positive Lebesgue measure such that G c R, 0 E G. Our aim is to study

the asymptotic behaviour o f eigenvalues and eigenfunctions o f problem (5.1)

as E -+ 0 for n >_ 3 and various real values o f m.

There are three qualitatively different cases.

1. -00 < m < 2. For such values o f m the k-th eigenvalue o f problem (5.1)

converges t o the k-th eigenvalue o f the Dirichlet problem for the Laplace

equation in R.

2 . rn > 2. In this case X ~ E ~ - " , where X,k is the k - th eigenvalue of problem

(5.1), converges t o the k-th eigenvalue o f the following problem for the

Laplace operator in Rn


3. m = 2. The set o f the limiting points (as E -+ 0) o f the spectrum of

problem (5.1) is the union o f the spectrum o f the Dirichlet problem for the

Laplace operator in R and the spectrum of problem (5.2).

The behaviour o f the eigenvalues of problem (5.1) will be studied on the

basis of the general method suggested in $1. To this end we make a suit-

able choice o f spaces ?lo, %,, V and operators &, A,, R,, and check that

Conditions C1-C4 are satisfied.

Another approach t o the problem of free vibrations of bodies with concen-

trated masses is described in papers [log], [82], [72]-[74], [25]-[29], [156] (see

also [ I l l ] , [125]).

We shall need the following auxiliary propositions.

For any u E CF(lRn) ( n 2 3) the Hardy inequality

holds with a constant c independent o f u (see [42])

Lemma 5.1.

Let n >_ 3. Then for any u E HA(R)

Moreover the following inequality is satisfied:

/ lu12dx 5 CE' J lVzu12dx , (5.5) EG n

where c is a constant independent o f E , u; the sets G, R are the same as in

(5.1).

Proof. The estimate (5.5) follows directly from the Hardy inequality (5.3).

Let us establish the convergence (5.4).

For any 6 > 0 consider a function vs E C,OO(R) such that c1I2 11u -

v611H;(n) < 6, where c is the constant from inequality (5.5). Then apply-

ing estimate (5.5) to u - va, we obtain


This inequality implies (5.4) , since n > 2. Lemma 5 . 1 is proved.

Lemma 5 .2 .

Let ue(x) be a solution o f the problem

u ~ E H ~ ( R ) , a P € [ O , l ] , m>-oo, n 2 3 .

Then

where c is a constant independent o f a, P , m.

Proof. T h e integral identity combined with (5.5) and the Friedrichs inequality

yields


Hence the inequality (5.6). Lemma 5.2 is proved.

5.2. The case -oo < m < 2 , n 2 3

Denote by 3-1, and ?lo the space L2(R) equipped with the scalar product

and

respectively. We take Hi(R) as V. Set Ref0 = f0 for any f0 E 3-10 For

f0 E V we have by Lemma 5.1

This means that Condition C1 holds with 7 = 1.

Denote by A, : 'He -+ 3-1, the operator which maps a function f' E 3-1, into the solution ue of the Dirichlet problem

By & : 3-10 -r 3-10 we denote the operator mapping f0 E into the

solution u0 of the Dirichlet problem

One can easily verify that A, and & are positive compact self-adjoint

operators defined on 3-1, and 3-10 respectively. The inequality sup IIdclltciy., C

< m follows from (5.6) since for m < 2 by virtue o f (5.5) and the Friedrichs

inequality we have

$5. Free vibrations of bodies with concentrated masses

9 c, / IVuc12dx 5 c2 (1 + s-"x) 1 f.I2dx . a n J

Thus Condition C2 is established.

Let us show that Condition C3 is also satisfied. Set f 0 E XO. Then

where

Auc = - (1 + c-"~(:)) f O in R , uc E H: (R) ,

According to Lemma 5.2 with a = 0 we have

J 1v.(uc - uO) 12dx 5 C E ~ - ~ ~ J lf012dx . n CG

By virtue o f (5.5) and the Friedrichs inequality we find

5 c2 J lv(uC - u0)12dx . n

From this inequality and (5.11) we deduce that

IIdcRcfO - RclbfOIIk. 5 c3&2-2m 1 If012dx (5.12) CG

for any f 0 E XO, where c3 is a constant independent o f E and fO.

For f 0 E V = Hi(R) Lemma 5.1 implies that E-' 1 1 f O l l ~ z ( , ~ ) + 0 as

E + 0. Therefore convergence (1.5) follows from (5.12), since m < 2. This

shows the validity of the Condition C3.

Let us prove that the Condition C4 is also satisfied.

If sup 1 1 f . 1 1 . ~ ~ < cm, i t f ~ l l ows from (5.6) that sup I I u ' I I ~ ; ~ ~ , < cm, E C

where ue is the solution of problem (5.9). Therefore there exist a vector

w0 E H,'(R) = V and a subsequence E' + 0 such that


uc' 1 wO weakly in H,'(R) and strongly in L2 (R) . (5.13)

Thus due to the inequality (5.5) we have

5 J lur - ~ ~ l ~ d ~ + C ~ E ~ - " 1 IV(U' - w0)12dl. , n n

where uE = AcfE and c2 is a constant independent of E . From the above

inequality we obtain (1.6) by virtue o f the convergence (5.13) and the fact

that m < 2.

Thus the Conditions C1-C4 are valid and we can apply Theorems 1.4, 1.7.

The eigenvalue problem associated with the operator do has the form

Theorem 5.3. Let m < 2, n 2 3, and let Xi, X,k be the k - th eigenvalues o f problems (5.14),

(5.1) respectively. Then

where ck is a constant independent of E .

Suppose that the multiplicity o f the eigenvalue X o of problem (5.14) is

equal t o r , i.e. X o = A;+' = ... = A;+'. Then for any eigenfunction u0 of

problem (5.14) corresponding t o X 0 and such that I l ~ , , ( ) ~ z ( ~ ) = 1, there is a

linear combination iic of eigenfunctions of problem (5.1) corresponding t o the

eigenvalues A:+', ..., and such that

$5. Free vibrations of bodies with concentrated masses

where cl is a constant independent of E and uo.

Proof. It has been shown above that operators A,, do satisfy Conditions

C1-C4 and therefore Theorems 1.4, 1.7 are valid.

To obtain estimates (5.15), (5.16) from (1.13), (1.26) one has only to note

that p,k = (A:)-', p0 = (Xi)-1 and that each eigenfunction o f the operator

d,, is smooth. Therefore, by virtue o f (5.12) for f0 E N(&, d o ) we have

5.3. The case m > 2, n 2 3

Let us pass t o the variables ( = E-'x in problem (5.1), setting

Then problem (5.1) reduces to the following one

Let us study the behaviour of eigenvalues and eigenfunctions o f this problem

First we introduce an operator whose spectrum is formed by the limits of

eigenvalues of problems (5.18) as E + 0.

Denote by H the completion of C,"(Rn) with respect t o the norm

I I U I I ~ = J (1.1' I C I - ~ + IVCUI')~C . (5.19) Rn

By virtue o f the Hardy inequality (5.3) we have (IuIIH I co I I V [ U I I ~ Z ( ~ ~ )

for any u E H . Therefore the norms (5.19) and llVEu11~2(~n) are equivalent

in H .

Consider the following problem


We define a weak solution of problem (5.20) as a function uo E H which

satisfies the integral identity

By Theorem 1.3, Ch. I, this solution u0 exists and satisfies the inequality

The estimate (5.22) follows from (5.21) for v = u0 and the Hardy inequal-

ity.

Define the space 'Ho as L2(G) with the scalar product

In what follows we shall assume that all functions from 'Ho are defined

on Rn and vanish on Rn\G. Therefore we can consider each function from

L:,,(Rn) vanishing outside G as belonging t o No. Let us define the operator & : 3i0 -t 'Ho setting & f o = nG(()uO,

where K G is the characteristic function of the set G, u0 E H is the solution of

problem (5.20).

First we show that is a positive self-adjoint operator. Indeed, let do f 0 =

K ~ ( < ) u ' , &go = nG(<)vO, where u0 is the solution o f problem (5.20) and v0 is the solution o f problem (5.20) with f 0 = By the integral identity (5.21)

we have

These inequalities imply that is positive and self-adjoint. Let us prove

its compactness.

Suppose that sup (Ifd((no < co, fS = 0 outside G. Let ua be solutions 8

of the problems Acua = - x ( t ) f V n Rn, u" H . By definition we have

Af" = K ~ ( [ ) U ' . It follows from estimate (5.22) and the Hardy inequality

$5. Free vibrations of bodies with concentrated masses 335

that sup I l ~ ' l l ~ l ( ~ , ) < oo for any bounded measurable set GI containing G. 8

Therefore there exist a subsequence s' -+ 0 and an element u0 E L2(G) such

that u" + uO in the norm o f L2(G), and thus &f"' + m(t)uO in the norm

of 'KO ass1+ 0.

Consider the following Dirichlet problem

Lemma 5.4.

Let m 2 2. Then for any uC which is a solution of problem (5.23) the estimate

J I V ~ U ' I ~ ~ C 5 C 1 (.ern + X(O) 1f'l2dt (5.24) R* n

holds with a constant c independent o f e , a .

Proof. It follows from the integral identity for u-hat

Hence, choosing 6 small enough and taking into account the Hardy and

Friedrichs inequalities, we obtain (5.24). Lemma 5.4 is proved.

We define the space 'Kc as L2(Rc) with the scalar product

By Re : 'KO -+ 'KC we denote the operator extending f0 E L2(G) as zero t o

Rc\G. Set V = 'KO. Let us verify Condition CI. It is easy t o see that


moreover llRc f01In, --t 1 1 f O 1 l n o as E --t 0.

We introduce operators A, : 'H, -t 'H,, setting A, f' = uc, where u" is

a solution o f the problem

We can easily check that A, is a positive self-adjoint and compact operator in

'He. It follows from the estimate (5.24) that sup lldcllt(ne, < m, since by E

the Hardy and Friedrichts inequalities we have

Consider now the Condition C3. Let f 0 E 'Flo. Then dofO = nc(<)uO,

where u0 is the solution o f problem (5.20); RE& f 0 = K G U O , d E R E f O = uC,

where us is the solution of the problem

ACuc = - (E" + X ( ( ) ) ~ G ( C ) ~ , uC E H;(Rc) . (5.26)

Therefore

IldcRcP - R c k f O l l f . = / (E" + x ) luc - u 0 ~ ~ ( € ) l ' d l . (5.27) n'

For uO - U' we have

Denote by w' a solution o f the problem

Since u0 is a harmonic function in Rn\G and u0 E H, it follows from the

results of [44], [45] that for sufficiently large we have


This inequality is based on the representation of uO(( ) in the form

u O ( [ ) = cn / fO(n) - s12-ndfl , cn = const . G

By the maximum principle we have

Then v" = u0 - u' - wc E H,'(Rc),

Applying Lemma 5.4 with uc = vE, CY = 0, KG = X , fC = foem and using

the Hardy and Friedrichs inequalities we get

From (5.32) taking into account (5.30), (5.31) we deduce

Due to (5.30) we have


am J I U O ~ ~ ~ E < %sm I I . f" l l$ (G, 7' ,.4-2n,.n-tdr . R*\G 1


for n = 3 ,

for n = 4 , (5.33) 1

for n > 4 ,

where MI, M2, M3 are constants independent o f E . Therefore

- R,A~OIIL 5 c [eZn-' + IlfOllb(nl 7 (5.34)

where

y3 = 1 , y4 = const E (O,1] , y, = 0 for n > 4 . (5.35)

Hence the validity of Condition C3.

Let us verify the uniform compactness of operators A, (Condition C4).

Suppose that sup (IfcllX, < M. It followsfrom (5.24) and the Hardy inequality C

(5.3) that sup I I U ' I I ~ ~ ( ~ ~ ) < M, where Q1 is any ball containing the set G c

and uc = A, f'. Due t o the compactness o f the imbedding H1(Q1) C LZ(QI) there exist a subsequence E' + 0 and a function G such that -+

0 as E' -+ 0. Setting wO(() = G(() for ( E G, wO(() = 0 for ( E Rn\G we

obtain that

By the Friedrichs inequality we get

55. Free vibrations of bodies with concentrated masses 339

The first term in the right-hand side o f this inequality converges to zero by

virtue o f (5 .36) , and it follows from ( 5 . 2 4 ) that the second term also converges

t o zero. This means that Condition C4 is satisfied.

Similarly t o Theorem 5.3, on the basis o f the estimate ( 5 . 3 4 ) and Theorems

1.4, 1.7 we can establish a theorem on the asymptotic behaviour of eigenvalues

and eigenfunctions of problem (5 .18) . The limit eigenvalue problem has the

form

A e U k = -A,kX(E)Uk in Rn , U k E H ,

J x ( O u k ( O u l ( O d E = , 1 (5 .37) G

o < A ; I A ; L . . . L A ; s ....

It follows from the estimate (5 .34) and Theorem 1.4 that eigenvalues o f

problems (5 .18) and (5 .37) satisfy the inequalities

where ck is a constant independent of E.

Theorem 1.7 implies that if U is an eigenfunction o f problem (5 .37) such

that 1 ~ ( t ) IU12d( = 1 and U corresponds t o the eigenvalue A. o f multiplicity G

r (A0 = A:+' = ... = A:+r), then there is a sequence Vc such that

and Vc is a linear combination of eigenfunctions o f the problem (5 .18) corre-

sponding t o the eigenvalues A:+', ..., A:+', the constant c, does not depend

on E and U .

Since the eigenvalues and eigenfunctions o f problems ( 5 . 1 8 ) and ( 5 . 1 ) for

m > 2 are related by (5 .17) , we have actually proved

Theorem 5.5.

For m > 2, n 2 3 the eigenvalues of problem ( 5 . 1 ) have the form


where ,8,k 5 C ~ ( E * - ~ + E ( ~ - Y ~ ) / ~ ) , Agk is the k - th eigenvalue of problem (5.37), y, is defined by (5.35).

Moreover, for any eigenfunction U o f problem (5.37) corresponding t o the

eigenvalue A. of multiplicity r (Ao = A:+' = ... = A:+') and such that

IIJTSUIIL2(c, = 1 there is a sequence of functions i i c ( x ) such that each i i c ( x )

is a linear combination of eigenfunctions of problem (5.1) corresponding to the

eigenvalues A:+', ..., A:+', and for V c ( [ ) = i i c ( ~ [ ) the estimate (5.39) is valid.

5.4. The case m = 2, n / 3

Consider the problem (5.1) f o r m = 2, n 2 3. The asymptotic behaviour of

eigenvalues of this problem as E -t 0 is determined by eigenvalues of problems

(5.37) and (5.14), namely by the eigenvalues of the following system

It is easy t o see that in fact we have an eigenvalue problem in the Hilbert

space ?lo = L2(G) r L2(R) whose elements are pairs o f functions ( ~ ( 0 , u ( x ) ) and the scalar product is given by the bilinear form

J X ( O U ( C ) V ( O ~ C + J U ( X ) V ( X ) ~ X . G n

Let us introduce the operator do : 'Ho + 'Ha associated with the prob-

lem (5.40) and mapping each element ( U ( [ ) , u ( x ) ) e ?lo into the element

( K ~ ( O V ( O , v ( x ) ) , where V ( ( ) , v ( x ) are solutions of the following problems

Here K G ( [ ) is the characteristic function o f the set G . It is easy t o verify that & is a positive compact self-adjoint operator in

?lo.

85. Free vibrations o f bodies w i th concentrated masses 34 1

.e the space 'Kc as L 2 ( R ) with the scalar product (5.7) for m = 2. As

V c rbb \ takethespace L 2 ( G ) x H i ( R ) . Let U E , lo, U = ( U ( [ ) , u ( x ) ) . We introduce the operator Rc : ?lo --+

?lc setting

x x R,U = u ( x ) + K G ( - ) E ' - " / ~ U ( - ) .

E E

Then

For any U E V we have u ( x ) E H:(fl). Therefore by Lemma 5.1 the first

integral in the right-hand side of the last equality converges t o I I u I I L ~ ( ~ , as

E + 0. Obviously the second integral converges t o

/ x(O lU(()12dt as E + 0, and the third integral converges t o zero. There- G fore I I R ~ U ~ ~ ~ ~ + IIUllwo for any u u V, which means that Condition C1 is

satisfied.

Define the operators A. : ?lc + 'He setting A, f' = uc, where u' is

the solution of the problem ( 5 . 9 ) with m = 2. It is easy t o see that A, are

compact positive self-adjoint operators. If sup I ( ff 117-1, < m, it follows from c

Lemma 5.2 with m = 2, that

sup 1 1 v 2 ~ C ( I ~ z ( n ) 5 c SUP llfc117-1. < 00 . (5 .41) c c

From ( 5 . 5 ) and the Friedrichs inequality we deduce that

Therefore due t o ( 5 . 4 1 ) we have


and thus the Condition C2 is also valid.

Let us consider the Condition C3.

For f0 E X O , f 0 = (BO(<) ,GO(x) ) We have & f O = ( K G ( c ) u ( E ) , ~ ( x ) ) I

where

A,, = -$O(x) in R , u E H,'(Q) ,

A,u(C) = -x(0Q0(C) u E H I (5.42)

x R , & ~ O = u ( x ) + n G ( E ) s 1 - n / 2 ~ ( - ) .

E E

On the other hand

w C E H i ( Q ) . j Denote by vc a solution o f the Dirichlet problem

Since U(E) is a harmonic function outside G, by analogy with (5.30) we have

and therefore

x I u ( ; ) ( < en-' l lQOl( t . (~ ) for x E .

It follows from the maximum principle that

where cl is a constant independent o f E . 2

The function W c ( x ) = u ( x ) + E ~ - ~ / ~ U ( ; ) - v' is a solution o f the prob-

lem

$5. Free vibrations of bodies with concentrated masses 343

Subtracting the equation (5.43) from (5.46) we obtain

x 5 Let us apply Lemma 5.2 with a = 0, x ( - ) = K G ( - ) . Then we have

E &

Taking into account (5.42), (5.43) we establish the following relations

To obtain the last inequality we made use o f the estimates (5.5) and (5.45).

Since the function U ( ( ) is harmonic outside G, by the same argument as in

Section 5.3 we conclude that


where a,(&) is defined by (5.33).

Thus from (5.48)-(5.50) we deduce

If f0 E V, then q0 E H i ( R ) , and by Lemma 5.1 the first term in the

right-hand side of (5.51) converges t o zero as E + 0. It thus follows from

(5.51) that Condition C3 is satisfied, i.e. relation (1.5) holds.

Note that if $O(x) is a smooth function, the inequality (5.51) implies that

where the constants cl, c2 depend on f0 but do not depend on E ; y, is the

same as in (5.35).

Let us establish now that Condition C4 is also valid. Suppose that

Due t o (5.53), the compact imbeddings H 1 ( R ) C L 2 ( R ) , H 2 ( R ) c H 1 ( R ) and the estimate I I v ~ I I ~ ~ ( ~ , 5 c 1 1 f L l l L z c n ) with a constant c independent o f E

(see [9]) there exist a subsequence E' -+ 0 and functions uO,vO E H,'(R) such that

ucr -t u0 weakly in H i ( R ) and strongly in L 2 ( R ) , I (5.54) vcr + v0 strongly in H,'(R) as e' + 0 .

Taking v E C,"(R) in the integral identity for wc we get


dw' dv - dx = E-' / x(:) / .v dx 5 n n

The first factor in the right-hand side of the above inequality is bounded

uniformly in E , and the second one tends to zero by virtue o f (5.4). Therefore

wE1 = uC1 - vC1 + 0 as c' -t 0 weakly in HA(R). It follows that v0 = uO.

The function W e ( ( ) = E"/~-'w'(E() is a solution o f the problem

and

IE"/~- ' f ' (c0l2d( = E-' / I f'12dx . G CG

Therefore sup IJWellH < w and there exist a subsequence E' -t 0 and a C

function W E H such that

W c 1 ( ( ) -t W ( ( ) weakly in H and strongly in L2(G1) (5.55)

for any bounded measurable set G1 c Rn. Obviously we can assume that the

subsequence E' + 0 in (5.54), (5.55) is the same one.

Denote by wO in the Condition C4 the pair ( K G ( ( ) w ( ( ) , u o ( x ) ) . Then.

taking into account that uc = v' + w' and applying Lemma 5.1 t o vE - uO,

we find


Passing in this inequality t o the limit with respect t o the subsequence E' -, 0

we see that, due t o (5.54) and (5.55), (1.6) holds. This means that Condition

C4 is satisfied and therefore we can use Theorems 1.4, 1.7 t o compare the

eigenvalues and eigenfunctions of problems (5.40), (5.1) for m = 2.

Theorem 5.6.

Let A,k and Ak be the k-th eigenvalues of problems (5.1) and (5.40) respec-

tively; m = 2. Then

where y, is defined by (5.35), ck is a constant independent of E .

Let A0 be an eigenvalue o f problem (5.40) of multiplicity r , As+' = ... =

As+' = AO. Let ii = (~(c), u(x)) be an eigenfunction o f (5.40) corresponding

to A0 such that llullRo = 1. Then for any E there is a linear combination iic

of the eigenfunctions of the problem (5.1) with m = 2, corresponding t o

A;+1, ...,A:+r, such that

where M, is a constant independent o f E , 0.

Note that in deriving the inequalities (5.56) from (1.13) we have taken

into account the inequality (5.52) and the smoothness o f eigenfunctions of

the Dirichlet problem for the Laplace equation.

55. Free vibrations of bodies with concentrated masses 347

Remark 5.7.

In the same way we can consider the cases n = 2, m E lR1 and n = 1,

m E R1.

Another approach t o the problems studied in this section was suggested in

[72]-[74], [82], [27]-[29], [log], [I1 11, [155].


$6. On the Behaviour of Eigenvalues of the Dirichlet Problem in

Domains with Cavities Whose Concentration is Small

Let R be a smooth bounded domain o f R3 and let Go = { x : 1x1 < 1)

be the unit ball. Set R, = R\ IJ (s3Go + 2 ~ 2 ) . Thus RE is a perforated zcz3

domain with ball-shaped cavities of radius s3 forming a 2s-periodic structure.

Consider the following boundary value problems

Let us estimate the norm Ilu, - ~ 1 1 ~ 2 ( ~ ~ ~ To this end, in accordance with

the method suggested in [lo], we define an auxiliary function w,(x), x E IR3

as follows

Theorem 6.1.

Solutions u, and u of problems (6.1) and (6.2) respectively satisfy the inequal-

ities

IIuC - uIIL~(R,) I CE Ilf llca(n) ,

where C,a = const > 0, C and a do not depend on s, f(x).

Proof. It is easy t o see that in EG~\E~GO the function w,(x) has the form

W, = (r-I - E-~)(E- ' - E-~ ) - ' , where r = r ( x ) is the distance from x t o the

origin.

Let us show that

$6. On the behaviour of eigenvalues of the Dirichlet problem 349

where Co > 0 is a constant independent of E. We have

where the constant C1 does not depend on E.

The estimate (6.3) follows from the last inequality, since the number of

the domains ( E G ~ \ E ~ G ~ ) + ~ E Z , z E iZ3, belonging t o 0, is of order E - ~ . Taking into account (6.1) and (6.2) we obtain the equalities

A(u, - UW,) = f - AUW, - ~ ( V U , VW,) - uAw, =

= f ( l - w,) - (Aw, - p)u + p(w, - 1)u - ~ ( V U , Vwe) =

which hold in the sense o f distributions.

Using (6.3) and the well-known Schauder estimate

llullc2+a(n) l C2 Ilf llcqn)

for solutions of problem (6.2) , we get

Let us estimate A4 in the norm of H-'(Re). We have

5 C sup IDaul I ~ W , - lIILz(n,) 5 Cs sup - ZER XER

la19 la152


By virtue o f the Schauder estimate it follows that

In order t o estimate the norm IIA211H-~cn,) we introduce an auxiliary func-

tion q, which is a solution o f the problem

where u is the outward unit normal to edGo. Obviously q,(x) is defined t o

within an additive constant. Choosing the suitable constant, we can assume

that q, = 0 on &dGo, since Go is a ball. Let us extend q, as zero to the

cube &QO = {x : -E < xj < E, j = 1, ..., n) and then to the whole R3 r2 E~

as a 2~-periodic function. Then q, = - - - in &(GO + 22), z E Z3, where 2 2

r = r ( x ) is the distance from x to the centre of the ball &(GO + 22) and

IVq,l < E. Moreover, q, satisfies the equation

where x,(x) = 1 for x E U &(GO + 2z), x,(x) = 0 for XE U &(Go + a€Z3 seZ3

2z), 6; is a distribution with support on the surface o f the ball &(Go + 22) and

such that

Indeed, setting T, = U &(Go + 22) we have for any cp E C,"(R3) Z E Z ~

where v is the unit outward normal t o aT,. aqc Since Aq, = 3 in T,, Aq, = 0 in R3\T,, - - - E, q, = 0 on dT,, we have a v l

§6. On the behaviour of eigenvalues of the Dirichlet problem 351

( ~ q . , p ) = 3 J ~ c - E J V ~ S = ( ~ X . , V ) - & c (J:,P). T. aT* t € Z 3

Therefore

On the other hand

where ye is a distribution with support in U e3(G0 + 2 z ) , i.e. ( y e , 4) = 0 , z€Z3

if 4 E C?(IR3) and vanishes on the set U E ~ ( G ~ + 2 ~ ) . The explicit form z€Z3

of yc is unnecessary for what follows.

Thus we have

Since the mean value of the function 3xe + p over the cube eQO is equal t o

zero, Lemma 1.8, Ch. I, provides the representation

where f,(() are bounded functions in R3. The inequality (Vq.1 < E shows that

a -Age = E -qf ,

ax,

where functions qf are bounded in IR3 uniformly with respect to e . Therefore

a E 2 V. = ( A w . - p)u = a (- hf (x ) ) u + 7.u + - ax, 1 - E ~

PU 9


where lhfl < C7 and C7 does not depend on E .

It follows that

I C E I I u I I H ; ( R ) . From (6.5)-(6.8) and the inequality

which holds for solutions o f the problem Au, = f , u, E HA(R,) , we obtain

the estimate

1Iuc - W~UIIH~(CI,) 5 C E I l f llc0(n) . This inequality together with (6.3) yield the estimates asserted in Theorem

6.1. The theorem is proved.

Now we can obtain estimates for the difference o f eigenvalues and eigen-

functions related t o problems (6.1), (6.2).

Set 'H, = L 2 ( R , ) , 'Ho = V = L2(R). As R, we take the operator restrict-

ing f € L2(R) t o the domain 0,. Define the operators A, and A0 by the

formulas: A, f = -u,, .&f = -u, where u,, u are solutions of the problems

(6.1), (6.2) respectively. Using Theorem 6.1 and the methods o f $2.2 we can

easily check that the Conditions C1-C4 are satisfied. Therefore we can apply

Theorems 1.4 and 1.7 to estimate the difference of the eigenvalues and the

eigenfunctions of the problems

$6. On the behaviour of eigenvalues of the Dirichlet problem 353

where A$, Xk form increasing sequences and each eigenvalue is counted as

many times as its multiplicity.

Theorem 6.2.

Let X,k, Xk be the k- th eigenvalues of the problems (6.9), (6.10) respectively.

Then

where c ( k ) is a constant independent o f E .

Moreover, if the multiplicity of the eigenvalue A'+' = Xo is equal t o m,

Xl+1 = ... = Xl+m , and uo is an eigenfunction of (6.10) corresponding t o Xo,

) ~ U ~ ) ) ~ Z ( ~ ) = 1 then there is a sequence {fie) such that

where Ml is a constant independent of E , uo; u, is a linear combination of

eigenfunctions of problem (6.9) corresponding to A;+', ...,

The method suggested in this section can also be applied to the case

n 2 3, as well as t o the general second order equations and the system

of elasticity, and for the case when Go is an arbitrary open set such that

Go C QO = {x : -1 < xj < 1, j = 1, ..., n) (see [154]).


57. Homogenization of Eigenvalues of Ordinary Differential Operators

We consider here a sequence of operators {Lk) such that Lk =$ J? as

k -t m, where Lk, E areordinary differential operators having the form (8.32),

(8.33), Ch. II, and satisfying the Condition N' of 58.1, Ch. II.

It is also assumed that

and that the problems

have unique solutions which satisfy the estimates

with constants Q, cl independent o f k and f ; the first eigenvalues of Lk, k are bounded from below by a positive constant independent o f k.

These assumptions imply that we can take p = 0 in Theorem 8.1, Ch. II.

Consider the eigenvalue problems for the operators Lk, E :

where A:, A' are enumerated in an increasing order and according t o their

multiplicity. It is also assumed that

$7. Homogenization o f eigenvalues of ordinary differential operators 355

where constants cz, c3 do not depend on k, the norm o f H-"1" is defined in

$9.2, Ch. I.

Theorem 7.1.

Let X i , A' be the I-th eigenvalues o f problems (7.2), (7.3) respectively. Then

where c, is a constant independent of k ; Ak are given by the formula (8.37),

Ch. II.

Let u be an eigenfunction of problem (7.3) such that llullL~ = 1 and u

corresponds t o the eigenvalue X 0 o f multiplicity r (A8+' = ... = A"+' = X 0 1. Then for any k there is a function iik such that

and iik is a linear combination o f eigenfunctions of problem (7.2) correspond-

ing t o the eigenvalues Xi+', ..., Xi+', c, is a constant independent o f k and u .

The proof o f this theorem is based on the abstract results obtained in

51.2, and is carried out in a similar manner t o the proof o f Theorem 2.3. In

the case under consideration N o , ('He = 'HFlllk), is the space L2(0 , 1 ) with 1 1

the scalar product 1 f g i d x . ( 1 fgpr d r ) . respectively, V = N O , Re is the J J 0 0

identical operator, &fO = uO, where u0 is a solution o f the Dirichlet problem

f?(uO) = / ; f O on ( 0 , l), u0 E H r ( 0 , l ) , Ak fk = uk, where uk is a solution of

the problem Lkuk = pk f k on ( 0 , I ) , uk E H r ( 0 , l ) . Due to the relations (7.1) for the coefficients o f C k and E , the operators

dk, satisfy the Conditions C1-C4 of $1, and therefore we can use Theo-

rems 1.4, 1.7 to estimate the closeness o f the corresponding eigenvalues and

eigenfunctions.


$8. Asymptotic Expansion of Eigenvalues and Eigenfunctions of the

Sturm-Liouville Problem for Equations with Rapdily

Oscillating Coefficients

In this section we shall construct complete asymptotic expansions for the

eigenvalues and the eigenfunctions of the following Sturm-Liouville problem

(see [143])

d x duk x x - dx ( a - E ) CEX + b ) + k ( ) p ( ) u = 0 , x E ( 0 , I] ,

I t is assumed that a ( < ) , b ( < ) , p ( < ) , a - ' ( f ) , a 1 ( J ) E K, where K is a set

of bounded continuous functions of [ E R1, and K satisfies the following

conditions:

1) K: is a ring containing all constants;

2) For any f E K there is a constant c f such that the function g ( x ) =

j f ( t ) d t + e l x belongs t o K. 0

Let us give some examples o f the sets which satisfy conditions I), 2).

I. The set o f all continuous T-periodic functions in R1

II. The set of all continuous functions which can be represented in the form

M + cp(x) , where M = const, Icp(x)l 5 C N ( ~ + for any integer

N .

Ill. The set formed by restrictions t o the line x; = p;t (i = 1, ..., n ) of smooth

functions, 2a-periodic in x l , ..., x, , where the constants p1, ..., p , are

such that

88. Asymptotic expansion of eigenvalues and eigenfunctions 357

C > 0, s > 0 are constants independent of m l , ..., mn; mj are arbitrary

integers, m: + ... + m: # 0 .

It is obvious that conditions 1) an&2) hold for classes I and II. Let us show

that class Ill also satisfies these conditions. To this end consider the Fourier

series

( m , x) = C mixi . i = l

The restriction of F t o the line x, = pit, i = 1 , ..., n , is F(p l t , ..., p,t), and

the primitive function corresponding to F has the form

The inequality (8.3) and the smoothness o f F guarantee the convergence of

the series (8.3). Therefore condition 2) is satisfied.

To construct asymptotic expansions for the eigenvalues and the eigenfunc-

tions of the problem (8.1) we shall need the following auxiliary propositions.

Lemma 8.1.

For each f E K:

x

Proof. According t o condition 2) the function h(x) = / f ( t )d t + 2ejx be- -2

longs to K and therefore is bounded in x E R1. This obviously implies the

convergence o f Lemma 8.1.

Lemma 8.2.

Let M ( ( ) E K: and (M) = 0. Then the equation


has a solution N ( ( ) which belongs to K and can be represented in the form

where

dN Moreover - E K.

dE Proof. Since (M) = 0, the primitive function L ( ( ) corresponding t o M(E) also belongs t o K. Due t o conditon 1) the set K is a ring, and therefore

L(J)a-I ( ( ) E K since a-I ( ( ) E K by assumption. The primitive o f a - ' ( ( ) L ( ( )

has the form P(( )+ ( L ( ( ) a - ' ( ( ) ) ( , where P E K. The primitive o f a - ' ( [ ) has

the form Q ( ( ) + (a-I( ( ) ) ( where Q E K. Choosing the constant C given by

(8.5) we see that the linear terms in the integrals entering (8.4) are mutually

reduced, and therefore N ( ( ) E K. Lemma 8.2 is proved.

Direct calculations show that the following lemma is valid.

Lemma 8.3.

The boundary value problem

d2u h - + Au = w ( x ) + Xwo(x) on [0, 11 , dx2

u ( O ) = a , u ( l ) = P ,

where A = ( ~ k ) ~ h , wO(x) = sin ~ k x , h > 0, A, a , P are constants, admits a

solution. if

X = 2xkh [ ( - l ) k + l p + a] - 2 sin xky w(y)dy . 1 The solution is given by the formula

$8. Asymptotic expansion o f eigenvalues and eigenfunctions 359

1 sin n f k - y) u ( x ) = a cos nkx + [w(Y) + X W O ( Y ) ] ~ Y +

0

+ C sin nkx , C = const

Now we formally construct asymptotic expansions for the eigenvalues and

the eigenfunctions o f the problem (8.1). Strict mathematical justification of

these expansions will be given later.

We seek the expansion of the eigenvalue X k ( & ) and of the corresponding

eigenfunction u$(x) o f problem (8.1) in the form

(the index k is omitted for the sake o f convenience). Here M > 2 is an integer,

N(',")((), v,(x) are the functions t o be determined, A,(&) are unknown real

numbers.

In what follows it is assumed that N('") are defined for all integers i, s,

and ~ ( ' 1 " ) = 0 for i < 0 or s < 0, or s > i. When the range o f summation

is not indicated, the sum is assumed to contain all terms with non-vanishing

functions ~ ( ' 9 " ) .

Let us substitute expressions (8.6), (8.7) for Xk(&), U $ in equation (8.1).

We get

d dN(OtO) dv, d d ~ w " + q ( ~ ( O N ' O * ) + a ( [ ) -1 dt -jE- + ( a ( [ ) -) d t v,) +


dN('y0) dv,

dlv, where F,O(x) is a sum of terms having the form stcp(() - , and 15 M + 2,

dx' t 2 0; y ( t ) is a bounded function.

Set N(OlO) r 1 , N('t0) 5 ~ ( ~ 1 ' ) = 0 and define N('gl)(() as a solution of

the problem

Existence of N('sl) follows from Lemma 8.2.

Define N ( ~ ? ~ ) ( ( ) as a solution of the problem

where h(2r2) is a constant given by the formula

$8. Asymptotic expansion of eigenvalues and eigenfunctions 361

Note, that due to Lemma 8.2 the right-hand side o f equation (8.8) belongs

to class K. We also define N ( ~ I O ) ( ( ) as a solution o f the problem

where h(210) is a constant such that

h(290) = X O ( P ( E ) ) + ( b ( € ) ) .

For the values of 1 larger than 2 we define functions N(' .") ( ( ) as solutions

of the problems

where

Let us introduce the following notation

Using induction over i, s we can successively find N('~") from (8.12). As a

basis of induction we take the functions N(Oy0), N ( ' - O ) , N ( ' ~ ' ) , N ( ~ ~ O ) , N ( ~ J ) ,

~ ( ~ 9 ~ ) defined above. It is easy to see that

ddv, C 6('+2.s) - + X i ( E ) ( p ) v6 (x) i=l s=o d x s


(8.14)

where F,'(x) has a form similar t o that of F f ( x ) .

Let us seek v,(x) in the form

Substituting this expression for v,(x) in (8.14) we obtain

y Ei ( = '-9' - -

dS X(i-p+2,8) 'Up

i=o p=o s=o dxS + ~ = o i - p p v p x ) +

Let us now define v p ( x ) , p = 0, ..., M - 2, as functions satisfying the

equations

and the boundary conditions

Let us single out those terms of (8.15), (8.16) which contain vi and rewrite

(8.15), (8.16) as follows.

For i = 0 we have

For i = 1,2, ..., M - 2 we obtain the following boundary value problems

$8. Asymptotic expansion of eigenvalues and eigenfunctions 363

i -1 i - p d% ( 0 ) v i ( ~ ) = - C C N('-P-")(o) A .

p=o s=o dxs '

Assuming X o , ..., X i - 1 , vo, ..., v i - l , N(OtO), ..., N(""" to be known, let us

choose A ; ( & ) such that the problem (8.18), (8.19) has a solution. Consider the k-th eigenvalue A. = X i of the problem (8.17). Then A0 = (p)- ' ( ( ~ k ) ' h ( ~ > ~ ) -

( 4 ) . Setting

in Lemma 8.3, we get r

i - 1 i - p - C - P ( ) m +

p=o a=O dx I


sin n k ( x - y ) 1 2 ic2 ~ ( i - p + ~ , s ) @ v P ( y ) - n k h ( 2 J )

0 p=o s=o d x s

Using the formulas (8.21), (8.22) and (8.18)-(8.20) we can easily construct

by induction the constants A , ( € ) and the functions v i ( z ) , ~ ( ~ ~ ~ 7 " ) provided that

Xo, ..., X i - 1 , 00, ..., 0, -1 , ..., N('+'V" are already known.

Thus we have constructed a formal asymptotic expansion for the eigenvalue

,Ik(€) of the Sturm-Liouville problem (8.1) and the corresponding eigenfunc-

tion u , k ( x ) .

Remark 8.4.

Formula (8.21) for X I is reduced to

1 1 1 ) 0 d v o ( 0 ) 3 d S v o ( y ) d y ,

- N ( ( ) - 2 / sin ~ k y C L ( 3 1 s ) -

0 [ d x 8 I where v o ( y ) = sin ~ k y . Since &(3*0) = j 1 ( 3 3 2 ) = 0 due t o (8.13), it follows that

hl = 0 for any k , provided that E-' is an integer and a ( ( ) is 1-periodic in (.

Note that the sequence of operators

58. Asymptotic expansion of eigenvalues and eigenfunctions 365

is strongly G-convergent t o the operator

d2 h(212) - + ( b ) .

d x 2

This fact was established in $8, Ch. II. It was also shown in 58, Ch. II, that

the eigenvalues o f the problems (8.1) and (8.17) satisfy the inequalities

[ (A: ) - ' - ( A n ) - ' [ 5 ~ ( A c + I I P c ( x ) - ( P ) I I H - ~ ~ ~ ) , (8.24)

C = const , A: = A n ( & ) , where

and by the definition of the norm in H-'9"

Due to the assumptions on the coefficients o f equation (8.1) the right-hand

side o f (8.24) converges t o zero as E + 0 . It follows from (8.24) that A: -+ A:

as E + 0 for any n.

On the other hand the formal asymptotic expansion constructed above

satisfies the following equalities

+ j ( ( h ) - h ( l ) ) d x I O E

A, = max ~ E [ 0 , 1 1

u i " ) ( o ) = E ~ - ~ $ ~ ( E ) , u L " ) ( l ) = E ~ - ~ $ I ( & ) , where

I I F E I I ~ 2 ( o , ~ ) I ~o , l $ o ( ~ ) I + I $ i ( & ) I I CI % , C i = const

h ( 2 1 2 ) ( t ) ) ldt [!(T-

and therefore according t o Lemma 1.1 there is an eigenvalue X i ( & ) of problem

(8.1) such that ~ A ' ( E ) - 5 f i M - l , c = const. Indeed, let $ , ( x ) be a

smooth function such that


Then 1$,(x)l I C 2 , where C2 does not depend on E by virtue of the maximum

principle. Therefore Lemma 1.1 can be applied t o the function uLM)(x) - E ~ $ , ( x ) and the operator A, defined on the space L2(0,1) with the scalar

1 x

product ( u , v ) = / p(-)uvdx and mapping f E L2(0, 1 ) into uc = A, f. E

0 where uc is a solution of the problem

It follows that, for sufficiently small E , 1 = k since X k ( & ) -+ X i , x ( ~ ) ( E ) -i

X,k as E -i 0 and X i has unit multiplicity, which implies that for sufFiciently

small E a neighbourhood of X k contains only one eigenvalue of operator LC

with homogeneous Dirichlet conditions.

The above considerations provide justification for the formal asymptotic

expansions (8.6). Thus we have proved

Theorem 8.5. The eigenvalues and the eigenfunctions o f the problem (8.1) satisfy the fol-

lowing inequalities:

where c l ( k ) , c2(k) are constants independent o f E .

$9. Eigenvalues and eigenfunctions o f a G-convergent Sequence 367

$9. On the Behaviour of the Eigenvalues and Eigenfunctions o f a

G-Convergent Sequence of Non-Self-Adjoint Operators

In $8, Ch. II, we introduced the notion of G-convergence o f operators

having the form

and belonging to the class E(Xo, XI, X z ) . In general the operators o f a G- converging sequence are not necessarily self-adjoint. The aim o f this para-

graph is to study the behaviour o f the eigenvalues and eigenfunctions of a

G-convergent sequence of non-self-adjoint elliptic operators and t o extend t o

this case the results of $2 on G-convergent sequences o f elasticity operators

which are self-adjoint.

We shall need some well-known (see [128]) results on the convergence of

the eigenvalues and eigenfunctions o f a sequence o f compact operators in a

Hilbert space. It is sufficient for our purposes here t o formulate and prove the

corresponding theorems in a less general form as compared with [128].

Let A E C ( H ) be a bounded linear operator in a separable Hilbert space

H with a complex valued scalar product.

By u ( A ) we denote the spectrum of the operator A , i.e. the set o f all points

p of the complex plane C 1 such that there is no bounded operator inverse t o

A - p I . Here I stands for the identity operator.

If pO E u ( A ) and there is an element x E H such that x # 0, ( A - p O I ) x =

0, then po is called an eigenvalue of A and x is an eigenvector corresponding

t o Po. If for some integer m > 1 we have ( A - poI)x # 0, ( A - poI)"x = 0 ,

then x is called a root vector corresponding t o po. By Ker A we denote the set { u E H , Au = 0 ) , Im A is the set consisting

o f such u E H that the equation A w = u admits a solution w E H . Let R ( p ) be a holomorphic function of p defined in a domain w c 6"

and taking values in the Banach space C ( H ) of bounded linear operators. Let

r be a closed curve limiting a subdomain w l , w l c w. Then the following

maximum principle holds for holomorphic functions with values in C ( H ) . The


norm R(p) for p E wl is not larger than the maximum of the norm o f R(p) on the curve r = awl.

We shall also use the following well-known results.

Theorem 9.1.

Let T E L(H) be a compact operator. Then

1) the conjugate operator T* is compact;

2) the set a ( T ) is discrete; if a is a limiting point of a ( T ) , then a = 0; the

set o(T)\{O) consists o f the eigenvalues o f the operator T ;

3) a(T*) is formed by the points complex conjugate t o those of u ( T ) ;

4) if p E a ( T ) , p # 0 and T has no root vectors corresponding t o p , then

the space H can be represented in the form

the dimension o f Ker(T - P I ) is finite and equal t o the dimension of

Ker(T' - P I ) ;

5) for each p # 0 the operator T - pI is of Fredholm type (see [40], p. 1071).

The sign $ denotes the direct sum of spaces, and 6 denotes the direct

sum of orthogonal spaces.

We say that the operator B E L ( H ) is o f Fredholm type, if the dimension

of KerB is finite and equal to the dimension of the orthogonal complement

of I m B in H.

Let {A,) be a sequence of compact operators in H , and let A E L(H)

be also a compact operator.

Definition 9.2.

The sequence {A,) is called compactly convergent to operator A as m -+ m,

if the following conditions are satisfied.

$9. Eigenval ues and eigenfunctions of a G-convergent Sequence 369

1. Amu + AU strongly in H for any u E H .

2. If {urn) is a sequence such that urn E H , IIumII < 1 , then the sequence

{Amurn) is a compact set in H .

Definition 9.3.

A sequence of operators {B,) (nor necessarily compact ones) is called properly

convergent t o operator B E C ( H ) as m -t oo, i f the following conditions are

satisfied.

1. B m u -+ B u strongly in H for any u E H .

2. If {urn) is a sequence such that urn E H , llurnll = 1 and {Bmum) is a

compact set in H , then {u,) is also a compact set.

Lemma 9.4.

Suppose that a sequence o f operators Am E L ( H ) is strongly convergent t o

operator A E L ( H ) , i.e. Amu + Au strongly in H for any u E H as m + KI.

Then

where C is a constant independent o f m.

The proof o f this lemma follows from the Banach-Steinhaus theorem (see

(1071, [134]), according t o which sup I I A r n l l ~ ( ~ ) < KI, if SUP llAm~ll m m

< oo for any u E H .

Lemma 9.5.

Suppose that {B,) is a sequence o f operators of Fredholm type properly con-

vergent to B E L ( H ) as m + oo, and B is an invertible operator. Then for

sufficiently large m operators B, are invertible and IIB,'~~.c(H) 5 C, where C

is a constant independent of m.

Proof. First we show that operators B;' exist for large m. Let us suppose

the contrary. Then there is a subsequence m' + oo such that B,I do not


admit bounded inverse operators. Since operators B, are o f Fredholm type

there is a sequence of vectors {x,,), lIx,,II = 1 , such that B,,x,t = 0. By

virtue o f the proper convergence o f B, to B, we can extract a subsequence

{x,rr) of {x,,) which strongly converges t o an element x E H such that

11x11 = 1. It follows that Bx = 0, since Bx = B,t(x - x,,) + ( B - B,I)X

and IIBrnllLcH, < C due t o Lemma 9.4. This fact is inconsistent with B being

an invertible operator.

Let us prove now the inequality IIBm1llccH, 5 C. Suppose the contrary.

Then we can choose a subsequence {xmj) such that llxrntll = 1 and B,,x,I +

0 strongly in H as m' + oo. Indeed, i f IIB;fII > C(ml) and C(ml) + m

as m' + oo, then there exists a sequence {y,,) such that IIy,~ll = 1

and (IB,3ymll( 2 C(ml). Set z , ~ = BG?~,~ . Then Brn,( l l~~, l (-~~rn,) = I j z ,~ I I -~y~ l + 0 as m' + oo and we can take as X,I the vectors llz,111-'z,1.

Due t o the proper convergence of B, to B there is a subsequence m" such

that X,II + x strongly in H as m" + co. Then B,,,X,I~ + Bx, since

Bx - B m ~ ~ m f r = B,rr(x - x,I,) + ( B - B,r,)x and therefore Bx = 0 which

contradicts the invertibility of B. Lemma 9.5 is proved.

Let w c 6'' be a subset o f the complex plane. Denote by N(w,A) the

span of all the eigenvectors o f A which correspond t o the eigenvalues o f A

belonging t o w. For example, if w = and po is an eigenvalue o f A, then

N(po, A) is the linear space of all eigenvectors o f A corresponding t o po.

Theorem 9.6. Suppose that A, + A compactly as m + oo, and A,, A are compact

operators of Fredholm type. Then the following assertions are valid.

1) If Po E u(A), PO # 0, then there is a sequence {P,), P , E ,-J(A,), such

that pm + po as m + oo.

2) If Pm E ,-J(Arn) and pm + p # 0, then p E u(A).

3) Ifurn E N(~rn,Arn) and p, + p # 0 , urn + u in H a s m + oo, then

" E N(P, A).

59. Eigenvalues and eigenfunctions of a G-convergent Sequence 371

Proof. Let us first establish 1). Suppose the contrary. Then there is a 6 > 0

and a sequence m' + oo such that the 6neighbourhood

o f po contains no spectral points o f operators A,, for sufficiently large m'.

Let = {p E C1, Ip - pol = 6) be a circle such that the ball Ip - pol 5 6

contains only one spectral point po of the operator A and does not contain

any spectral points of operators A,,.

By the same argument that was used in the proof o f Lemma 9.5, and taking

into account the compactness o f r6, we can easily show that for sufficiently

large m' there exist the inverse operators (A,, - pI)-', p E r6, and

where C is a constant independent o f m' and p E r6. Since, inside r6, there are no spectral points o f the operators A,,, the

above mentioned maximum principle for holomorphic operator valued functions

guarantees that the inequality (9.2) holds inside r6 and in particular

On the other hand, since po E u(A), there is an element uo such that

lluoll = 1 and (A - poI)uo = 0. Then (A,, - poI)uo + 0 strongly in H , which contradicts (9.3). Therefore for any 6 > 0 sufficiently small there is an

integer N such that for any m > N the inside o f r6 contains a spectral point

pm of operator A, and Ip, - pol I6 Therefore p, + po as m --+ m.

Now we prove assertions 2) and 3). Let p, + p and p # 0. Then there

are elements u, E H such that llurnll = 1, (A, - pmI)um = 0. Let us

pass to the limit in the last equality for a subsequence m' + m such that

urn) + u strongly in H. Such a subsequence exists due t o the proper conver-

gence of A, - p,I t o A - p I as m -+ m. Moreover A,Iu,, + Au, since

Amurn = A,(u, - u) + A,u and IIArnllc(~) 5 C by Lemma 9.4. It follows

that (A - pI)u = 0 and therefore p E u(A). Theorem 9.6 is proved.

In order t o estimate the difference between eigenvalues and eigenvectors of

operators A, and A we shall need the following


Lemma 9.7.

Let A, -t A compactly as m -+ oo and p , -+ pop where p,, po are eigen-

values of the compact operators A,, A respectively, pm # 0, po # 0. Then

the following assertions are valid.

1. Let P : H -+ H be an orthogonal projection on a finite dimensional

subspace V c H . Then B, = A, + P - p,I -, B = A + P - poI

properly as m -+ co; the operators B, are of Fredholm type.

2 . Suppose that operator A does not admit root vectors corresponding po

and {g,) is a sequence of vectors of H such that g , -+ 0 weakly in H ,

II(Ak - /imI)gmll -+ 0 as m -+ oo. Then g , -+ 0 strongly in I3 as

m -+ oo.

Proof. The convergence B,u -+ Bu as m -i oo follows from the definition

of the compact convergence of A, t o A. Obviously

where f l , ..., f 5 s an orthonormal basis of V = Im P. If the sequence {B,u,)

is compact and llurnll = 1, it follows from (9.4), by virtue of the uniform

compactness of A,, and the convergence p , -+ po # 0, that the sequence

{u,) is also compact in H. Operators B, are of Fredholm type, since B, =

(A , + P ) - p,I and operators A,, P are compact.

Let us prove the assertion 2. Denote by P E L(H) the operator o f orthog-

onal projection on the space Ker(A* - pol). Then according t o assertion 1 of

this lemma we have

properly as m -+ co, and operators B, are o f Fredholm type. Using assertion

4 o f Theorem 9.1 let us show that B is invertible. Indeed, if Bx = 0, then

x E Im(A - p o l ) n Ker(A - p o l ) = (0). I f x is orthogonal to Im B , then

x E Ker B* = ~ e r ( ( ~ * - p o l ) + P*) . Note that P is a self-adjoint operator.

Therefore (A* - poI)x + Px = 0 and thus

59. Eigenvalues and eigenfunctions of a G-convergent Sequence 373

Since I m B is a closed subspace of H, we have I m B = H. Thus, the

operator B is injective and by virtue of the Banach theorem (see [134]) B is

invertible.

It follows from Lemma 9.5 that for sufficiently large m there exist the

inverse operators B;' and J(BL-llJcH = IJBt;llJIqH) I C, where C is a

constant independent o f m.

Let us now prove the strong convergence g, -+ 0 as m -+ m. We have

as m + co, since by assumption //(A; - p,I)g,II -+ 0 and Pg, =

8

(gm, ?)fit where f l , ..., f a isanorthonormal basisin Ker(A*-pol) and i = l

therefore Pg, + 0 due t o the weak convergence t o zero o f g, as m -+ m.


Let M be a subspace o f H. For u E H set p(u, M) = inf Ilu - 1111. vEM

Theorem 9.8.

Suppose that A,, A are compact operators and the sequence {A,) is com-

pactly convergent t o A as m + m. Suppose also that p, -+ po, pm # 0,

PO # 0, p, E a(A,), po E c ( A ) , and the spectral point po corresponds

to eigenvectors only (not root vectors) o f operator A, urn E N(p,, A,),

llurnll = 1. Then the following estimates are valid


where the constants C1, C2 do not depend on m

Proof. Let PO E a(A), PO # 0, and let el , ..., eS be a basis of the eigenspace

N(p0,A) It is easy t o see that ((A, - pm~)e i , gm) = 0 for any sequence

9, E H such that gm E N(pm, A',), llgml( = 1. Fix any such sequence. We

have ((A - pm~)e ' , 9,) = ((A - ~ , ) e ' , 9,). It follows that

Let us show that we can choose the elements ei(") E N(po, A), so as t o have,

for sufficiently large m, the following inequality:

where a is a constant independent of m. Suppose that such a choice is

impossible. Then for some subsequence {g,~) we have

Due t o the compact weakness of a ball in a Hilbert space we can assume that

g , ~ -t g weakly in H as m' -+ m. Let us show that g E N(po, A*). We have

((A,. - p,11)~,1, 2) = 0 for any s E H. Hence (g,,, (A,, - pm,I)x) = 0.

Passing here to the limit as m' -r co we get (g, (A - poI)x) = 0 for any

x E H, and therefore g E N(pO, A*).

Let us show that g # 0. Suppose that g = 0. Then according t o the

assertion 2 of Lemma 9.7 the equality (A; - p,I)g, = O implies strong

convergence g , ~ -t 0 as m' --+ co, which contradicts the fact that I(g,(( = 1.

Therefore g # 0.

Due t o conditions (9.9) we have

Since A* has no root vectors corresponding t o pop the decompositions (9.1)

hold with T = A*. It follows from (9.10) that g E Im(A* - pol). We

have shown above that g E N(p0, A*) = Ker(A* - POI), and therefore g E

Im(A* - POI) n Ker(A* - POI). This inclusion contradicts the absence o f

root vectors of operator A' corresponding t o jio. Thus the inequality (9.8) is

proved.

§ 9. Eigenval ues and eigenfunctions of a G-convergent Sequence 375

The estimate (9.5) follows directly from (9.7) and (9.8).

Now we consider the closeness o f the eigenvectors urn of the operators A,

t o those of A.

Let

E N(pm, A m ) llumll = 1 , pm = inf IIum - u I I - U E N ( P O , A )

We first show that

where uO, are the eigenvectors o f A on which the infimum in the definition of

p, i s realized, a > 0 is a constant independent of m.


Suppose that the inequality (9.11) does not hold. Then one can find a sub-

sequence m' + co such that (A,, - pmlI) [11u$, - ~ , ~ l l - ~ ( u ; ~ - u,~)] + 0

strongly in H as m' + co. Due t o the proper convergence of the sequence

{A, - pmI) we have Ilu:, - u m l l l - l ( ~ ~ l - urn,) + u in H. Moreover it is

easy t o see that u E N(pO, A). Since 21% - Iluk -urn() u E N(po,A), we have

where ~ ( m ) + 0 as m = m' + co. This contradiction proves the validity of

estimate (9.11). Since 11uk11 5 Ilumll = 1, the relations (9.11), (9.12) yield

where C > 0 is a constant independent o f m. Theorem 9.8 is proved.

Let {L,) be a sequence of differential operators


(aap(E) are smooth 1-periodic in ( functions), which is strongly G-convergent

to the operator

as E + 0.

I t is easy t o verify that the results obtained in $8, Ch. II can be obviously

extended to the case o f operators with complex coefficients.

In this paragraph the operators LC are also assumed to satisfy the inequal-

ities

for any u E C r ( R ) , where C is a positive constant independent o f E, u; and

the operator 2 is assumed t o satisfy a similar inequality. It is easy t o see that

these conditions hold for the operators L,, 2 considered in $8, Ch. II, if one

adds the term pu t o LC and 2 with a sufFiciently large real constant p .

Inequalities of type (9.13) for C,, guarantee unique solvability o f the

problems

and the estimates

where the constants cl, cz do not depend on f, E .

We also assume that solutions o f the problem i ( u ) = f, u 6 Hr(0)

satisfy the following estimate

$9. Eigenvalues and eigenfunctions of a G-convergent Sequence 377

5 C l l f llHs(n) (9.14)

where C is a constant independent of f . Estimate (9.14) is always valid if the

coefficients of 2 and the domain R are sufFiciently smooth.

Let us define A,, A as operators from L2(R) t o L2(R) mapping a function

f E L2(Q) into the solutions o f the respective Dirichlet problems

where p,, ; ( x ) are bounded (uniformly in e) measurable functions.

Let us estimate the L2(R) norm of the difference u, -u = ( A , -A) f . De-

note by B,, B the operators mapping f E L2(R) into the respective solutions

of the problems

L ( u ) = f , U E H ~ ( R ) .

Then

(A, - A)f = B,p,f - B;f =

= BE ( ( P , - ~ ) f ) + (BC - B)b f . Let us estimate each term in the right-hand side o f the last equality. We have

I C2 l l ~ e - bIIH-l,m(n) I l f llH1(n) .

Theorem 8.1, Ch. II allows to obtain an estimate for (BE - B ) j f . In order

to apply Theorem 8.1, Ch. II, we have to check the Condition N' of Section

8.1, Ch. II. Let us define the functions N,k(x) as follows:

1 1 N , ~ ( x ) = - N,(kx) , k = -, 171 I m , km E

where N , ( [ ) E H1(Q) are 1-periodic in [ solutions of the equations


The solvability o f these equations can be proved by the standard method based

on the Lax-Milgram theorem.

Let us verify Condition N' in our case. We have

1 D: N: ( x ) = - - N 7 la1 = 7

where DaN,(() are smooth 1-periodic functions. Therefore

C and I(D6N,kllLm(n) 5 - for 161 5 m - 1. Thus Condition N1 l is valid with

k a k = ck-l.

We further have

weakly in L2(R) as k - m, where ( f ) = 1 f ( ( )d(, f (() E L1(Q) is 1-

Q periodic in (. Therefore 6kp(x) - G a p = f a ~ ( ~ l c = k z l where f(0 are 1-

periodic in ( and such that ( f a p ) = 0. By virtue o f Lemma 1.8, Ch. I, we

af'p , where fLp are bounded functions I-periodic in (. have f a p ( t ) = C - &i

Therefore

Since Ik-I f:p(kx)l 5 k-'C, by the definition o f the norm in H-'tm(R) we

obtain the inequality I16kp - 6aPIIH-~,m 5 Ck-l . It follows that &',&) 5 Ck-'. Thus we have proved the validity of Condition N'2. Condition N13

follows from the equations for IVY((). We obviously have 7 k = 0.

Let us estimate the norm llvkllo which enters the inequality (8.10), Ch. II. We have


To obtain the above inequalities we used the following facts: the a priori

estimate

for a solution w of the Dirichlet problem; the inequality

the trace estimates for functions in H 8 ( R ) (see [117]), the a pm'ori estimate

(9.14). Thus we have actually proved

Theorem 9.9.

For any f E H 1 ( R ) and the operators A,, A defined above the following

inequality is satisfied


Now we can consider the eigenvalue problems

where P,(x), @(x) are functions whose L W ( R ) norms are bounded by a con-

stant independent of E ,


as E --+ 0.

We say that an eigenvalue A. of problem (9.16) admits only eigenfunctions,

if the operator A has no root vectors corresponding to the eigenvalue p o = A,'.

Theorem 9.10.

Let A. be an eigenvalue of problem (9.16) which admits only eigenfunctions.

Then there is a sequence{X,,} o f eigenvalues o f the Dirichlet problem (9.15)

for the operators C,,, such that A,, + Xo as k + oo and the estimate

holds with a constant C independent o f a.

Eigenfunctions u,,(x) o f the Dirichlet problems (9.15) for operators L,, corresponding to the eigenvalues A,, satisfy the inequality

where C1 is a constant independent of E , M ( x ~ , E ) is the space o f all eigen-

functions of problem (9.16) corresponding t o Xo. If A,, -+ A. # 0 and A,, are

eigenvalues of problems (9.15) for C,,, then Xo is an eigenvalue o f problem

(9.16).

Proof. Let us first show that the strong G-convergence o f L, t o J? implies

the compact convergence of A, t o A . Indeed, let f E L2(R). Then there is

a function g E H1(R) such that 11 f - g(lLz(n) 5 a , where a is an arbitrarily

small real number. Obviously

(A, - A)f = (A, - A ) g + (A, - A ) ( f - 9) . Therefore

II(A= - A)f I l~z(n) I II(.A. - A)g l l~z(n) + Ca , C = const .

The first term in the right-hand side of the last inequality converges t o zero as

a --+ 0 due to Theorem 9.9, and a can be chosen arbitrarily small. Therefore

1I(A, - A ) f l lL~(n) -, 0 as E + 0 for any f E L2(R). For any sequence

f" E L2(R) such that 11 f'llvcn) = 1, the sequence {A, f } is a compact set


in L2(R) since IJSt,fCJJHrn(n) 5 C, with C = const independent of E. Hence

dE -+ A compactly as E -, 0, and estimates (9.17), (9.18) follow directly

from Theorems 9.9, 9.8.

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