homogenization, multiple scales problems in biomathematics, mechanics, physics and numerics, cimpa...

Editors: Assyr Abdulle, Jacek Banasiak, Alain Damlamian and Mamadou Sango
GAKUTO International Series Math. Sci. Appl., Vol.** (2009) Multiple scales problems in Biomathematics, Mechanics, Physics and Numerics, pp. i-v
GAKKOTOSHO
Preface
This volume contains a collection of lectures presented at the 2007 CIMPA-UNESCO- South Africa School “Multiple scales problems in Biomathematics, Mechanics, Physics and Numerics” held at the African Institute of Mathematical Sciences in Muizenberg, South Africa on 6th-18th of August 2007. The School primarily focused on the presentation of the state of the art in homogenization theory, multiscale methods for homogenization problems, asymptotic methods as well as modeling issues in a variety of applications in physics and biology involving multiple scales.
This School was organized under the auspices of the CIMPA (Centre International de Mathematiques Pures et Appliquees), a non-profit international organization based in Nice, France, whose purpose, as a Category 2 Institute of UNESCO, is to promote international cooperation in higher education and research in mathematics for the benefit of developing countries (see http://www.cimpa-icpam.org).
The School consisted of 8 courses delivered by 10 invited lecturers coming from Cameroon, France, Scotland and South Africa and it attracted 43 participants from countries as diverse as Burkina Faso, Cameroon, the Democratic Republic of Congo, Morocco, Nigeria, Scotland, Slovakia, South Africa, Tunisia and Zimbabwe. The lectures targeted postgraduate students and young researchers and thus contained a blend of educational material with survey of cutting edge research.
In parallel with the School, two mini-workshops were organized: one on functional ana- lytic methods in applied sciences and the other on numerical methods for problems arising from multiple scales models. The talks at the mini-workshops were presented by regular participants of the School and invited speakers.
This volume is based on the courses given by the invited lecturers. It also contains three invited lectures delivered during the workshop. The first part of the book gives a thorough survey of recent developments in homogenization theory including the periodic unfolding method and the sigma-convergence theory. This provides an introduction to these topics as well as an account of its recent developments. New results in the homogenization of linear and nonlinear elliptic eigenvalue problems in domains with fine grained boundaries are also presented. On the computational side, new numerical methods, the so-called heterogeneous multiscale method, is discussed for homogenization problems. Several numerical examples and a detailed convergence theory of various numerical methods based on finite elements are presented. Here again, the lecture allows for a rather complete presentation of the developments of this new method, successful in several applications over the past few years.
ii CIMPA School Cape Town 2007
The second part of the book discusses the asymptotic analysis of singularly perturbed problems, applications of the asymptotic analysis to biology as well as numerical methods for singularly perturbed problems. Here the reader can see how modeling based on the recognition of multiple time scales in a complex model allows to construct a systematic way of aggregating variables leading to a significant decrease in the dimension of the models. This, in turn, greatly facilitates its robust analysis without compromising accuracy of the results. The lectures of this second part cover aggregation methods in discrete time population models and describe a modified classical Chapman-Enskog asymptotic procedure which can be used for aggregation of variables in continuous time population models and kinetic models. They also provide a survey of numerical methods designed to treat singularly perturbed problems.
A. Damlamian was the main organizer of the School from the CIMPA side, which funded participation of African students from outside South Africa, including two scholarships specif- ically targeted for young women mathematicians. The local organizing committee consisted of M. Sango and J. Banasiak. However, the School would not have been possible without the significant help and contributions of many other people and institutions. Special thanks must go to Prof F. Hahne and the staff of AIMS for providing an excellent infrastructure and support throughout the School. Thus, the organizers were able to focus purely on academic matters - a rare feat as far as organization of conferences is concerned. AIMS also provided financial support for some of the participants. The organizers are also extremely grateful for financial support received from the National Research Foundation of South Africa, which funded all South African participants as well as supported two lecturers of the School. Or- ganizers received generous support from the French Embassy in South Africa, the Hanno Rund Fund of the School of Mathematical Sciences of the University of KwaZulu-Natal and the Commission for Development and Exchange of the International Mathematical Union. Last but not least, the School would not have been successful without the enthusiasm of the students who duly attended and actively participated in all lectures and activities for the full two weeks.
December 2008
The web page of the school is : http://maths.za.net/index.php?cf=5.
Multiple scales problems in Biomathematics, Mechanics, Physics and Numerics iii
List of participants
School Lecturers
A. Abdulle (UK) P. Auger (France) J. Banasiak (SA) A. Damlamian (France) P. Donato (France) F. Ebobisse Bille (SA) G. Nguetseng (Cameroon) E. Perrier (France) B.D. Reddy (SA) M. Sango(SA)
Invited Workshop Lecturers
School Participants
J. Absalom (Zimbabwe) V. Aizebeokhai (Nigeria) A. Al Ahouel (Tunisia) A. Barka (Morocco) J. Busa (Slovakia) A. Chama (SA) A. Chirigo (Zimbabwe) A. Goswami (SA) S. Faleye (SA) W. Lamb (UK) M. A. Luruli (SA) J. Malka Koubemba (DRC) A. Masekela (SA) B. Matadi Maba (SA) K. Matlawa (SA)
V. Melicher (Slovakia) F. Minani (SA) J. Mtimunye (SA) B. Nana Nbendjo (Cameroon) L. Nkague Nkamba (Cameroon) K. Okosun (Nigeria) S. C. Oukoumie Noutchie (SA) J. Y. Semegni (SA) L. Signing (Cameroon) J. M. Tchoukouegno Ngnotchouye (SA) A. Traore (Ivory Coast) A. Udomene (Nigeria) J. Urombo (Zimbabwe) T.T. Yusuf (Nigeria) J. d.D. Zabsonre Jean de Dieu (Burkina Faso)
iv CIMPA School Cape Town 2007
Multiple scales problems in Biomathematics, Mechanics, Physics and Numerics v
CONTENTS
Part I. Homogenization, elasticity and multiscale methods
1) The Periodic Unfolding Method in Homogenization (D. Cioranescu, Alain Damla- mian and G. Griso) page 1
2) The periodic unfolding method in perforated domains and applications to Robin problems (D. Cioranescu, P. Donato and R. Zaki) page 37
3) Homogenization of linear and nonlinear spectral problems for higher-order elliptic problems in varying domains (M. Sango) page 69
4) Σ-Convergence of Parabolic Differential Operators (G. Nguetseng) page 95
5) The Finite Element Heterogeneous Multiscale Method: a computational strategy for multiscale PDEs (A. Abdulle) page 135
6) Mathematical Aspects of Elastoplasticity (F. Ebobisse Bille and B. D. Reddy) page 185
Part II. Asymptotic analysis, numerical methods and applications
1) Asymptotic analysis of singularly perturbed dynamical systems of kinetic type (J. Banasiak) page 221
2) Aggregation methods of time discrete models: review and application to host- parasitoid interactions (P. Auger, C. Lett and T. Nguyen-Huu) page 257
3) Numerical schemes that preserve properties of the solutions of the Burgers equation for small viscosity (R. Anguelov, J.K. Djoko and J.M.-S. Lubuma) page 279
4) Implicit-Explicit (IMEX) Schemes and Relaxation Systems (M.K. Banda) page 303
5) Numerical Methods for Multi-Parameter Singular Perturbation Problems (K.C. Patidar) page 329
PART I
Homogenization, elasticity
Multiple scales problems in Biomathematics, Mechanics, Physics and Numerics, pp. 1–35
The Periodic Unfolding Method in Homogenization
D. Cioranescu, Alain Damlamian & G. Griso
Abstract: We give a detailed presentation of the Periodic Unfolding Method and how it applies to periodic homogenization problems. All the proofs are included as well as some examples.
1. Introduction.
The notion of two-scale convergence was introduced in 1989 by G. Nguetseng [27] and further developed by G. Allaire [1] with applications to periodic homogenization. It was generalized to some multi-scale problems by A. I. Ene and J. Saint Jean Paulin [17], G. Allaire and M. Briane in [2], J. L. Lions, D. Lukkassen L. Persson and P. Wall [25],
In 1990, T. Arbogast, J. Douglas and U. Hornung [5] defined a “dilation” operation to study homogenization for a periodic medium with double porosity. This technique was used again by A. Bourgeat, S. Luckhaus and A. Mikelic [7], G. Allaire and C. Conca [3], G. Allaire, C. Conca and M. Vanninathan [4], M. Lenczner [20-21], M. Lenczner et all [22-24], D. Lukkassen, G. Nguetseng and P. Wall [26], by J. Casado Daz [8] , J. Casado Daz et al. [9-11].
It turns out that the dilation technique reduces two-scale convergence to weak convergence in an appropriate space. Combining this approach with ideas from Finite Element approximations, we give here a very general and quite simple method, the “periodic unfolding method”, to study classical or multi-scale periodic homogenization. It a fixed-domain method (the dimension of the fixed domain depends on the number of scales) that applies as well to problems with holes and truss-like structures or in linearized elasticity. We pre-
2 D. Cioranescu, Alain Damlamian & G. Griso
sented this method in [12]. A preliminary version of the proofs can be found in the survey of A. Damlamian [15]. Here the complete proofs of the results announced in [12] are given as well as more recent developments.
The periodic unfolding method is essentially based on two ingredients. The first one is the unfolding operator Tε defined in Section 2, where its properties are investigated. Let be a bounded open set and Y a reference cell in IRn. By definition, the operator Tε associates to any function v in Lp(), a function Tεv in Lp(ε×Y ), where ε is the smallest finite union of cells εY containing . An immediate (and very interesting) property of Tε is that it enables to transform any integral over in an integral over ε × Y . Indeed, one has (Proposition 2.6)∫

Tε(w)(x, y) dx dy, ∀w ∈ L1(). (1.1)
If {vε} is a bounded sequence in Lp(), weakly converging to v in Lp(), the sequence {Tε(vε)} weakly converges to v in Lp(× Y ) (Proposition 2.10). This allows to show that the two-scale convergence in the Lp()-sense of a sequence of functions {vε}, is equivalent to the weak convergence of the sequence of unfolded functions {Tε(vε)} in Lp( × Y ) (Proposition 2.12).
In Section 2 are also introduce an local average operatorMε
Y and an averaging operator
Uε, the later being in some sense, the inverse of the unfolding operator Tε. The second ingredient of the periodic unfolding method consists of separating the char-
acteristic scales by decomposing every function belonging to W 1,p() in two parts. In Section 3 this is achieved by using the local average. In Section 4, the original proof of this scale-splitting, inspired by the Finite Element Method, is given. The confrontation of the two method of Sections 3 and 4, is interesting in itself (Theorem 3.5 and Proposition 4.7). In both approaches, is written as = 1
ε + ε2 ε where 1
ε is a macroscopic part, designed not to capture the oscillations of order ε (if there are any ), while the microscopic part 2
ε is designed to do so. The main result states that from any bounded sequence {wε} in W 1,p(), weakly convergent to some w, one can always subtract a subsequence (still denoted {wε}) such that wε = w1
ε + εw2 ε with
(ii) Tε(wε) w weakly in Lp(;W 1,p(Y )),
(iii) Tε(w2 ε) w weakly in Lp(;W 1,p(Y )),
(iv) Tε(∇wε) ∇xw +∇yw weakly in Lp(× Y ),
(1.2)
where w belongs to Lp(;W 1,p per(Y )). Convergence (1.2)(iii) shows that if the proper scaling
is used, oscillatory behaviour can be turned into weak (or even strong) convergence, at the price of an increase in the dimension of the problem.
The Periodic Unfolding Method in Homogenization 3
In Section 5 we apply the periodic unfolding method to a classical periodic homogenization problem. We point out that in the framework of this method, the proof of homogenization result is elementary. It relies essentially on formula (1.1), on the properties of Tε, and on convergences (1.2).
Section 6 is devoted to a corrector result which holds without any additional regularity on the data (contrary to all previous proof, see [6], [13] or [28]). This result follows from the use of the averaging operator Uε. The idea of using averages to improve corrector results first appeared in G. Dal Maso and A. Defranceschi [14].
The periodic unfolding method is particularly well-suited for the case of multi-scale problems. This is shown in Section 7 by a simple reiteration argument.
2. Unfolding in Lp−spaces
2.1. The unfolding operator Tε. In IRn, let be an open set and Y a reference cell (ex. ]0, 1[n, or more generally a set having the paving property with respect to a basis (b1, · · · , bn) defining the periods).
By analogy with the notation in the one-dimensional case, for z ∈ IRn, [z]Y denotes the unique integer combination
∑n j=1 kjbj of the periods such that z− [z]Y belongs to Y , and
set
{z}Y = z − [z]Y ∈ Y a.e. for z ∈ IRn.
Then for each x ∈ IRn, one has
x = ε ([x ε
Definition of [z]Y and {z}Y
We use the following notations: ε = interior
{ x ∈ ,
} .
(2.2)
The set ε is the smallest finite union of εY cells contained in , while Λε is the subset of containing the parts from εY cells intersecting the boundary ∂ (See Figure below).
DEFINITION 2.1. For φ Lebesgue-measurable on , the unfolding operator Tε is defined
as follows:
Tε(φ)(x, y) =
0 a.e. for (x, y) ∈ Λε × Y.
Observe that the function Tε(φ) is Lebesgue-measurable on × Y and vanishes for x outside of the set ε.
The domains ε and Λε
The following property of Tε is a simple consequence of Definition 2.1 for v and w
Lebesgue-measurable, it will be used extensively :
Tε(vw) = Tε(v) Tε(w). (2.3)
REMARK 2.2. For f measurable on Y , in order to define the sequence fε given by fε(x) = f (x ε
) , it is customary to extend the function f by Y−periodicity to the whole of IRn. With
notation (2.1), it seems simpler to define fε by fε(x) = f ({x
ε
} Y
extend f .
PROPOSITION 2.3. For f measurable on Y , and for fε defined in the previous remark,
one has
0 a.e. for (x, y) ∈ Λε × Y. (2.4)
If f belongs to Lp(Y ), 1 ≤ p <∞, and if is bounded, then
Tε(fε|)→ f strongly in Lp(× Y ).
REMARK 2.4. For f in Lp(Y ), 1 < p < ∞, it is well-known that fε| converges weakly
in Lp() to the mean value of f on Y , and not strongly unless f is a constant. Therefore,
Proposition 2.3 shows that the strong convergence of the unfolding of a sequence does not
imply strong convergence of the sequence itself.
Like in classical periodic homogenization, two different scales appear in Definition 2.1: x, the “macroscopic” scale gives the position of a point in the domain , while
x
ε , the
“microscopic” one, gives the position of a point in the cell Y . The unfolding operator doubles the dimension of the space and put all the oscillations in the second variable, separating in this way, the two scales (see Figures below).
fε(x) = 1 4
Tε(fε) for the function fε(x) above and ε = 1 6
The next two results, essential in the study of the properties of the unfolding operator, are also straightforward from Definition 2.1.
PROPOSITION 2.5. For p ∈ [1,+∞[, the operator Tε is linear and continuous from Lp() to Lp(× Y ) . For every φ in L1() and w in Lp()
(i) 1 |Y |
Proof. Recalling Definition 2.2 of ε, one has
1 |Y |
∫ ×Y
∫ ε×Y
On each (εξ + εY ) × Y , by definition, Tε(φ)(x, y) = φ(εξ + εy) is constant in x. Hence, each integral in the sum on the right hand side successively equals∫
(εξ+εY )×Y Tε(φ)(x, y) dx dy = |εξ + εY |
∫ Y
ε
φ(x) dx which gives the result.
Property (iii) in Proposition 2.5 shows that any integral of a function on , is “almost equivalent” to the integral of its unfolded on × Y , the ”integration defect” arises only from the cells intersecting the boundary ∂ and is controlled by its integral over Λε.
The next proposition, which we call unfolding criterion for integrals (u.c.i.), is a very useful tool when treating homogenization problems.
PROPOSITION 2.6. (u.c.i.) If {φε} is a sequence in L1() satisfying∫ Λε
|φε| dx→ 0,
Based on this result, we introduce the following notation:
Notation. If {wε} is a sequence satisfying u.c.i., we write∫
wεdx Tε' 1 |Y |
∫ ×Y
Tε(wε) dxdy.
PROPOSITION 2.7. Let {uε} be a bounded sequence in Lp() with p ∈]1,+∞] and v ∈ Lp ′ () (1/p+ 1/p′ = 1), then∫

∫ ×Y
Tε(uε)Tε(v) dxdy. (2.5)
Suppose ∂ bounded. Let {uε} be a bounded sequence in Lp() and {vε} a bounded
sequence in Lq() with 1/p+ 1/q < 1, then∫
uεvεdx Tε' 1 |Y |
Proof. Observe that
Consequently, by the Lebesgue’s Dominated Convergence Theorem one gets ∫
Λε
Λε
|uεv| → 0. This proves (2.5). If ∂ is bounded,
then one immediately has 1Λε → 0, when ε→ 0 in Lr() for every r ∈ [1,∞), in particular
for 1 r = 1
p + 1 q , and this implies (2.6).
COROLLARY 2.8. Let p belong to ]1,+∞[, let {uε} be a sequence in Lp() such that
Tε(uε) u weakly in Lp(× Y ),
and {vε} be a sequence in Lp ′ () (1/p+ 1/p′ = 1) with
Tε(vε)→ v strongly in Lp ′ (× Y ) and
∫ Λε
∫ ×Y
u v dxdy.
Proof. The result follows from the fact that the sequence {uε vε} satisfies the u.c.i. by the Holder inequality.
We now investigate the convergence properties related to the unfolded operator when ε→ 0. For φ uniformly continuous on , with modulus mφ, it is easy to see that
sup x∈ε,y∈Y
|Tε(φ)(x, y)− φ(x)| ≤ mφ(ε).
So, as ε goes to zero, even though Tε(φ) is not continuous, it converges to φ uniformly on . By density, and making use of Proposition 2.5, further convergence properties can be expressed using the mean value of a function defined on × Y :
DEFINITION 2.9. For Φ ∈ Lp( × Y ), the mean value M Y
(Φ) : Lp( × Y ) → Lp() is
defined as follows:
Φ(x, y) dy. a.e. for x ∈ . (2.7)
Observe that an immediate consequence of this definition is the estimate
M Y
(Φ)Lp() ≤ |Y |− 1 p ΦLp(×Y ), for every Φ ∈ Lp(× Y ).
PROPOSITION 2.10. Let p belong to [1,+∞[.
(i) For w ∈ Lp(),
(ii) Let {wε} be a sequence in Lp() such that
wε → w strongly in Lp().
Then
Tε(wε)→ w strongly in Lp(× Y ).
(iii) For every relatively weakly compact sequence {wε} in Lp() the corresponding Tε(wε) is relatively weakly compact in Lp(× Y ). Furthermore, if
Tε(wε) w weakly in Lp(× Y ),
then
(iv) If Tε(wε) w weakly in Lp(× Y ), then
wLp(×Y ) ≤ lim inf ε→0
|Y | 1 p wεLp(). (2.8)
(v) Suppose p > 1 and let {wε} be a bounded sequence in Lp(). Then, the following
assertions are equivalent:
(a). Tε(wε) w weakly in Lp(× Y ) and lim sup ε→0
|Y | 1 p wεLp() ≤ wLp(×Y ),
(b). Tε(wε)→ w strongly in Lp(× Y ) and
∫ Λε
|wε|p → 0.
Proof. (i) The result is obvious for any w ∈ D(). If w ∈ Lp(), let φ ∈ D(). Then, by using (iv) from Proposition 2.5,
Tε(w)− wLp(×Y ) = Tε(w − φ) + ( Tε(φ)− φ
) + (φ− w)Lp(×Y )
≤ 2|Y | 1 p w − φLp() + Tε(φ)− φLp(×Y ),
hence,
lim sup ε→0
Tε(w)− wLp(×Y ) ≤ 2|Y | 1 p w − φLp(),
from which statement (i) follows by density.
(ii) From Proposition 2.5 (iv), one has the estimate
Tε(wε)− Tε(w)Lp(×Y ) ≤ | Y | 1 p wε − wLp(), ∀w ∈ Lp(),
hence (ii). (iii) For p ∈ (1,∞), by Proposition 2.5 (iv), boundedness is preserved by Tε. Suppose that Tε(wε) w weakly in Lp(× Y ) and let ψ ∈ Lp′(). From Proposition 2.7∫

∫ ×Y
Tε(wε)(x, y) Tε(ψ)(x, y) dx dy.
In view of (i), one can pass to the limit in the right-hand side to obtain
lim ε→0
w(x, y) dy } ψ(x) dx.
For p = 1, one uses the extra property satisfied by weakly convergent sequences in L1(), in the form of the De La Vallee-Poussin criterion (which is equivalent to relative weak compactness): there exists a continuous convex function Φ : IR+ 7→ IR+, such that
lim t→+∞
Φ(t) t
{∫ ×Y
( Φ |Tε(wε)|
} is bounded,
which completes the proof of weak compactness of Tε(wε) in L1( × Y ) in the case of with finite measure. For the case where the measure of is not finite, a similar argument shows that the equi-integrability at infinity of the sequence {wε} carries over to {Tε(wε)}.
If Tε(wε) w weakly in L1(× Y ), let ψ be in D(). For ε sufficiently small, one has∫
wε(x)ψ(x) dx = 1 |Y |
Tε(wε)(x, y) Tε(ψ)(x, y) dx dy.
In view of (i), one can pass to the limit in the right-hand side to obtain
lim ε→0
w(x, y) dy } ψ(x) dx.
(iv) Inequality (2.8) is a simple consequence of Proposition 2.5 (ii).
(v) From Proposition 2.5 (i), one has for any φ in L1(),
1 |Y |
∫ ×Y
1 |Y | Tε(wε)pLp(×Y ) +
∫ Λε
This identity implies the required equivalence.
Concerning the converse of (ii) in Proposition 2.10, Remark 2.4 shows that it is not true.
REMARK 2.11. A consequence of (iii) of Proposition 2.10, together with (iv) of Propo-
sition 2.5, is the following. Suppose the sequence {wε} converges weakly to w in Lp(). Then Tε(wε) is relatively weakly compact in Lp(× Y ), and each of its weak-limit points
w, satisfies M Y
(w) = w.
Now recall the following definition from G. Nguetseng [27 ] and G. Allaire [1 ]: Two-scale convergence. Let p ∈]1,∞[. A bounded sequence {wε} in Lp() two-scale converges to some w belonging to Lp( × Y ), whenever, for every smooth function on × Y , the following convergence holds:∫

w(x, y)(x, y) dxdy.
The following result reduces two-scale convergence to a mere weak Lp(×Y )-convergence of the unfolded.
PROPOSITION 2.12. Let {wε} be a bounded sequence in Lp() with p ∈]1,∞[. The
following assertions are equivalent :
i). {Tε(wε)} converges weakly to w in Lp(× Y ), ii). {wε} two-scale converges to w.
Proof. To prove this equivalence, it is enough to check that for every in a set of ad- missible test-functions for two-scale convergence (for instance, D(, Lq(Y ))), Tε[(x, x/ε)] converges strongly to in Lq(× Y ). This follows from the definition of Tε, indeed
Tε [ ( x, x
+ εy, y ) .
REMARK 2.13. Proposition 2.12 shows that the two-scale convergence of a sequence in
Lp(), p ∈]1,∞[, is equivalent to the weak−Lp( × Y ) convergence of the unfolded se-
quence. Notice that by definition, to check the two-scale convergence one has to use
special test functions. To check a weak convergence in the space Lp( × Y ), one makes
simply use of functions in the dual space Lp ′ (×Y ). Moreover, due to density properties,
it is sufficient to check this convergence only on smooth functions from D(× Y ).
2.2. The averaging operator Uε In this section, we consider the adjoint Uε of Tε which we call averaging operator. To do so, let v be in Lp(× Y ) and let u be in Lp
′ (). We have successively,
∫ ε×Y
This gives the formula for the averaging operator Uε.
DEFINITION 2.14. For p in [1,∞], the averaging operator Uε : Lp( × Y ) → Lp() is
defined as
) dz a.e. for x ∈ ε,
0 a.e. for x ∈ Λε.
Consequently, for ψ ∈ Lp() and Φ ∈ Lp′(× Y ), one has
∫
Φ(x, y) Tε(ψ)(x, y) dxdy. (2.9)
Note that if Φ is continuous on × Y , it is not the case for Uε(Φ) on . As consequence of the duality (Holder’s inequality), and of Proposition 2.5 (iv), we get
immediately
PROPOSITION 2.15. Let p belong to [1,∞]. The averaging operator is linear and contin-
uous from Lp(× Y ) to Lp(). Moreover, for 1 p + 1
p′ = 1,
The operator Uε maps Lp(×Y ) into the space Lp(). It allows to replace the function x 7→ Φ
( x, {x ε
} Y
) which is meaningless in general, by a function which always makes sense.
This implies that the largest set of test functions for two-scale convergence is actually the set Uε(Φ) with Φ in Lp
′ (× Y ).
It is immediate from its definition, that Uε is almost a left-inverse of Tε since
Uε ( Tε(φ)
0 a.e. for x ∈ Λε, (2.11)
for every φ in Lp(), while
Tε(Uε(Φ))(x, y) =
1 | Y |
0 a.e. for (x, y) ∈ Λε × Y, (2.12)
for every Φ in Lp(× Y ).
PROPOSITION 2.16. (Properties of Uε). Suppose that p is in [1,+∞[. (i) Let {Φε} be a bounded sequence in Lp(×Y ) such that Φε Φ weakly in Lp(×Y ). Then
Uε(Φε) MY (Φ) =
Φ( · , y) dy weakly in Lp().
(ii) Let {Φε} be a sequence such that Φε → Φ strongly in Lp(× Y ). Then
Tε(Uε(Φε))→ Φ strongly in Lp(× Y ).
(iii) Suppose that {wε} is a sequence in Lp(). Then, the following assertions are
equivalent:
(b) wε 1 ε − Uε(w)→ 0 strongly in Lp().
(iv) Suppose that {wε} is a sequence in Lp(). Then, the following assertions are
equivalent:
∫ Λε
(d) wε − Uε(w)→ 0 strongly in Lp().
Proof.(i) This follows from Proposition 2.10(ii) by duality for p > 1. It still holds for p = 1 in the same way as the proof of Proposition 2.10(ii). Indeed, if the De La Vallee-Poussin
criterion is satisfied by the sequence {Φε}, it is also satisfied by the sequence {Uε(Φε)}, since for F convex and continuous, Jensen’s inequality implies
F (Uε(Φε))(x) ≤ Uε(F (Φε))(x).
(ii) The proof follows the same lines as that of (i)-(ii) of Proposition 2.10. (iii) (a)=⇒(b) simply follows from the application of inequality (2.10) to the function Φ .= Tε(wε)− w, making use of (2.11).
(b)=⇒(a): by Proposition 2.10 (ii), Tε(wε − Uε(w)) → 0 strongly in Lp( × Y ), then from the result of (ii) above, Tε(wε)→ w strongly in Lp(× Y ). (iv) (c)=⇒(d) follows from (iii) and the second condition of (a).
(d)=⇒(c) follows from (iii) since Uε(w) vanishes on Λε by definition.
REMARK 2.17. In view of Proposition 2.16 (i), if Tε(wε)→ w weakly in Lp(×Y ), then
wε 1 ε − Uε(w) converges weakly to 0 in Lp().
The converse cannot make sense. Indeed, let (wε) be such that wε 1 ε − Uε(w) con-
verges weakly to 0 in Lp(). Choose any non-zero v with M Y
(v) = 0. Since Uε(v)
converges weakly to M Y
(v) = 0 by Proposition 2.16 (i), it follows that the weak limit
of wε 1 ε − Uε(w) is also the weak limit of wε 1
ε − Uε(w + v) making it impossible to
conclude that Tε(wε) converges weakly (would it be to w or to w + v?)
Comparing the situations for strong and weak convergences, if v is such that wε 1 ε −
Uε(w + v) and wε 1 ε − Uε(w) converge strongly to 0, then v = 0, while a weak conver-
gence will only imply that M Y
(v) = 0 .
REMARK 2.18. The condition (iii) (a) of Proposition 2.16 is used by some authors to
define the notion of “Strong two-scale convergence”. From the above considerations, con-
dition (c) of Proposition 2.16 (iv) is a better candidate for this definition.
2.3. The local average operator Mε
Y
Y : Lp() 7→ Lp(), is defined for any φ in
Lp(), p ∈ [1,+∞[, by
Y (φ) is indeed a local average, since
Mε
0 if x ∈ Λε.
REMARK 2.21. Note that Tε(Mε
Y (φ)) =Mε
Y (φ) on the set × Y for any φ in Lp().
PROPOSITION 2.22. (Properties of Mε
Y ). Suppose that p is in [1,+∞[.
(i) For any any φ in Lp(), one has
Mε
(ii) For φ ∈ Lp() and ψ ∈ Lp′(), one has
∫
Mε
Y (ψ) dx.
(iii) Let {wε} be a sequence such that wε → w strongly in Lp(). Then
Mε
Y (wε)→ w strongly in Lp().
The same result holds true with weak convergence in place of the strong one.
Proof. The proofs of (i) and (ii) are straightforward. The proof of (iii) is a simple consequence of (ii) of Proposition 2.10, and for the weak topology, of duality.
COROLLARY 2.23. Suppose that p is in [1,+∞[ . Let w be in Lp() and {wε} be a
sequence in Lp() satisfying
∫ Λε
wε → w strongly in Lp().
Proof. Since w does not depend on y, one has Uε(w) = Mε
Y (w) which, by Proposition
2.22 (iii), converges strongly to w. The conclusion follows from Proposition 2.16 (iv).
3. Unfolding and gradients
Now, we will examine the properties of unfolding in the case of W 1,p() spaces. Some results require no extra hypotheses, but many others are sensitive to the boundary conditions and the regularity of the boundary itself. In the next subsection we consider the former results, while the following subsections will deal with the latter.
Observe first that for w in W 1,p() one has
∇y(Tε(w)) = εTε(∇xw), ∀w ∈W 1,p() a.e. for (x, y) ∈ × Y. (3.1)
Proposition 2.5 (iv) implies that Tε maps W 1,p() into Lp(;W 1,p(Y )) .
PROPOSITION 3.1 (gradients in the direction of a period). Let k in [1, . . . , n] and
{wε} be a bounded sequence in Lp() with p ∈]1,+∞], satisfying
ε ∂wε ∂xk
≤ C. (3.2)
Then, there exist a subsequence (still denoted ε) and w ∈ Lp(× Y ) with ∂w
∂yk ∈ Lp(× Y ), such that
Tε(wε) w weakly in Lp(× Y ),
εTε (∂wε ∂xk
(3.3)
(weakly- ∗ for p =∞). Moreover, the limit function w is 1-periodic with respect to the yk coordinate.
Proof. Convergences (3.3) are a simple consequence of (3.1) and (3.2). It remains to prove the periodicity of w. For simplicity, we assume k = n and write y = (y′, yn), with y′
in Y ′ .= (0, 1)n−1 and yn ∈ (0, 1).
Let ψ ∈ D( × Y ′). By (3.3) Tε(wε) is bounded in Lp( × Y ′;W 1,p(0, 1)) so that Tε(wε)|{yn=s} is bounded in Lp( × Y ′) for every s ∈ [0, 1]. The result follows from the following computation with an obvious change of variable:∫
×Y ′
] ψ(x, y′) dx dy ′
= ∫
] dx dy′,
] dx dy′,
which goes to zero.
Applying the pevious result for all k = 1, ·, n at once, we get
COROLLARY 3.2. Let {wε} in W 1,p() with p ∈]1,+∞[, and assume that {wε} is a
bounded sequence in Lp() satisfying
ε∇wεLp() ≤ C.
Then, there exist a subsequence (still denoted ε) and w ∈ Lp(;W 1,p(Y )), such that
{ Tε(wε) w weakly in Lp(;W 1,p(Y )),
εTε(∇xwε) ∇yw weakly in Lp(× Y ).
Moreover, the limit function w is Y−periodic, i.e. belongs to Lp(;W 1,p per(Y )), where
W 1,p per(Y ) denotes the Banach space of Y−periodic functions inW 1,p
loc (IRn) with theW 1,p(Y ) norm.
COROLLARY 3.3. Let p be in ]1,+∞[ and {wε} be a sequence converging weakly in
W 1,p() to w. Then,
Tε(wε) w weakly in Lp(;W 1,p(Y )).
Furthermore, if wε converges strongly to w in Lp() (e.g. W 1,p() is compact in Lp()), the above convergence is strong.
Proof. By hypothesis, using (3.1) gives estimates
Tε(wε)Lp(×Y ) ≤ C,
∇y(Tε(wε))Lp(×Y ) ≤ εC,
so that there exist a subsequence (still denoted ε) and w in Lp(;W 1,p(Y )) such that
Tε(wε) w weakly in Lp(;W 1,p(Y )), (3.4)
with ∇yw = 0. Consequently, w does not depend on y, and Proposition 2.10 (iii) immediately gives that w = M
Y (w) = w. Moreover, convergence (3.4) holds for the entire
sequence ε. If wε converges strongly to w in Lp(), so does Tε(wε) by Proposition 2.10 (ii).
PROPOSITION 3.4. Suppose that p is in [1,+∞[. Let (wε) be a sequence which converges
strongly to some w in W 1,p(). Then,
(i) Tε(∇wε)→ ∇w strongly in Lp(× Y ),
(ii) 1 ε
where
Proof. Set
) ,
which has mean value zero in Y . Since ∇yZε = 1 ε ∇y ( Tε ( wε ))
= Tε ( ∇wε
∇yZε → ∇w strongly in Lp(× Y ),
which is the first assertion of the proposition. To prove (ii), recall the Poincare-Wirtinger inequality in Y
∀ψ ∈W 1,p(Y ), ψ −M
Y (ψ) Lp(Y )
Applying it to the function Zε − yc · ∇w, gives
Zε − yc · ∇wLp(×Y ) ≤ C∇yZε −∇wLp(×Y ), (3.6)
which concludes the proof.
THEOREM 3.5. Suppose that p is in ]1,+∞[. Let {wε} be a sequence which converges
weakly to some w in W 1,p() and strongly in Lp(). Up to a subsequence (still denoted
ε), there exists some w in Lp(;W 1,p per(Y )) such that
(i) Tε(∇wε) ∇w +∇yw weakly in Lp(× Y ),
(ii) 1 ε
(3.7)
(w) = 0.
Proof. Following the same lines as in the previous proof, introduce
Zε = 1 ε
) ,
which has mean value zero in Y . Since ∇yZε = Tε ( ∇wε
) , (ii) implies (i).
To prove (ii), note that the sequence {∇yZε} is bounded in Lp(×Y ). Hence, by (3.5),Zε− yc · ∇wLp(×Y ) is bounded, and there exists w in Lp(;W 1,p(Y )) such that, up to
a subsequence,
Since, by construction, M Y
(yc) vanishes, so does M Y
(w).
It remains to prove the Y−periodicity of w. This is obtained in the same way as in the proof of Proposition 3.1 using a test function ψ ∈ D(× Y ′). One has successively,∫
×Y ′
] ψ(x, y′) dx dy ′
= ∫
] dx dy′,
1 ε
] dx dy′.
− ∫
Similarly, noticing that (yc · ∇w)(y ′, 1)− (yc · ∇w)(y ′, 0) = ∂w
∂xn , we obtain∫
= ∫
∫ ×Y ′
w(x) ∂ψ
∂xn (x, y′) dx dy′.
This, with (3.8) and using convergence (3.7) (ii), shows that∫ ×Y ′
[ w(x, (y ′, 1))− w(x, (y ′, 0)
] ψ(x, y′) dx dy ′ = 0,
so that w is yn−periodic. The same holds in the directions of all the other periods.
Theorem 3.5 can be generalized to the case of W k,p()−spaces with k ≥ 1 and p ∈ ]1,+∞[ . To do so, introduce the notation Dr, r = (r1, . . . , rn) ∈ INn with |r| = r1 + . . .+ rn ≤ k:
Dr x =
Actually, the following result holds:
THEOREM 3.6. Let {wε} be a sequence converging weakly in W k,p() to w, k ≥ 1 and p ∈]1,+∞[. Then, there exist a subsequence (still denoted ε) and w in the space
Lp(;W k,p per (Y )) such that the following convergence holds:{
Tε(Dl xwε) Dl
Tε(Dl xwε) Dl
(3.9)
Furthermore, if wε converges strongly to w in W k−1,p() (e.g. W 1,p() is compact in
Lp()), the above convergences for |l| ≤ k − 1 are strong in Lp(;W k−l,p(Y )).
Proof. We briefly prove the result for k = 2. The same argument generalizes for k > 2. If |l| = 1, the first convergence in (3.9) follows directly from Proposition 2.11. Set
Wε = 1 ε2
Y
( ∇wε
)] The sequence {wε} is bounded in W 2,p(), hence proceding as in the proof of Proposition 2.22(iii), one obtains Wε
Lp(×Y )
) with |l| = 2.
This implies that the sequence {Wε} is bounded in Lp(;W 2,p(Y )). Therefore, there exist a subsequence (still denoted ε) and w ∈ Lp(;W 2,p(Y )) such that
Wε w weakly in Lp(;W 2,p(Y )), ∂Wε
∂yi =
(3.10)
xwε) Dl yw weakly in Lp(× Y ), |l| = 2. (3.11)
Now we apply Theorem 3.5 to each of the derivatives ∂wε ∂xi
, i ∈ {1, . . . , n}. There exist a
subsequence (still denoted ε) and wi ∈ Lp(;W 1,p per(Y )) such that M
Y (wi) ≡ 0 and
From (3.10) follows:
Set w = w − 1 2
n∑ i,j=1
∂xi∂xj . By construction, the function w belongs
to Lp(;W 2,p(Y )). Furthermore
Y (∇yw) = 0.
The last equality implies that w belongs to Lp(;W 2,p per(Y )). Finally from (3.12) one gets
Dl yw = Dl
which together with (3.11) proves the last convergence of (3.9).
COROLLARY 3.7. Let {wε} be a sequence converging weakly in W 2,p() to w, and
p ∈]1,+∞[. Then, there exist a subsequence (still denoted ε) and w in the space
Lp(;W 2,p per(Y )) such that
1 ε2
[ Tε(wε)−Mε
∂xi∂xj + w
weakly in Lp(;W 2,p(Y )), where w is such that M
Y (w) = 0.
4. Macro–micro decomposition: the scale-splitting operators Qε and Rε
In this section, we give a different method to prove Theorem 3.5. It was the original proof in [12], [15], and the contruction itself is useful later for corrector results. Since for these corrector results, a smooth boundary of the domain is necessary, we will assume such a regularity in this section (in the general situation, the contruction of this section can still be carried out locally).
The procedure is based on a splitting of functions φ in W 1,p() as
φ = Qε(φ) +Rε(φ),
where Qε(φ) is an approximation of φ having the same behavior as φ, while Rε(φ) is a remainder of order ε.
When considering the sequence {∇wε} where {wε} converges to w in W 1,p() we show that, while {∇wε} , {∇(Qε(wε))} and {Tε(∇Qε(wε))} have the same weak limit ∇w in Lp(), respectively in Lp(×Y ), the sequence {Tε(∇wε)} converges (up to a subsequence) in Lp(×Y ) to the limit ∇w+ r where r = ∇yw and is the weak limit of Tε
( ∇(Rε(wε))
) .
From now on, we suppose that is a bounded domain such that there exists a continuous extension operator P : W 1,p() 7→W 1,p(IRn) satisfying
P(φ)W 1,p(IRn) ≤ C φW 1,p(), ∀φ ∈W 1,p(),
where C is a constant depending only upon p and ∂.
The construction of Qε is based on the Q1−interpolate of some discrete approximation, as is customary in the Finite Element Method (FEM). The idea of using these type of interpolate was already present in G. Griso [19-20], for the study of truss-like structures. For the purpose of this paper, it is enough to take the average on εξ+ εY to construct the discrete approximations, but any other well-behaved average will do.
DEFINITION 4.1. For any φ in Lp(IRn), p ∈ [1,+∞[, the operator Qε : Lp(IRn) 7→ W 1,∞(IRn), is defined as follows
Qε(φ)(εξ) = Mε
Y (φ)(εξ) for ξ ∈ ε ZZ n,
and for any x ∈ IRn, we set
Qε(φ)(x) is the Q1 interpolate of the values of Qε(φ) at the vertices
of the cell ε [x ε
] Y
+ εY. (4.1)
For any φ in W 1,p(), the operator Qε : W 1,p() 7→W 1,∞() is defined by
Qε(φ) = Qε(P(φ))|, where Qε(P(φ)) is given by (4.1).
A straighforward computation gives the following estimates:
PROPOSITION 4.2 (properties of Qε on IRn ). For φ in Lp(IRn), 1 ≤ p ≤ ∞, there
exists a constant C depending only upon n and Y such that: Qε(φ)Lp(IRn) ≤ CφLp(IRn), ∇Qε(φ)Lp(IRn) ≤
C
C
ε1+n/p φLp(IRn).
For φ in Lp(IRn), 1 ≤ p <∞ we have the following convergences:{ Qε(φ) −→ φ strongly in Lp(IRn),
ε∇Qε(φ) −→ 0 strongly in (Lp(IRn))n.
Furthermore, for any ψ in Lp(Y )
Qε(φ)ψ ({ ·
ε
} Y
if ψ is in W 1,p per(Y ), then
Qε(φ)ψ ({ ·
ε
} Y
DEFINITION 4.3. The remainder Rε(φ) is given by
Rε(φ) = φ−Qε(φ) for any φ ∈W 1,p().
The following proposition is well-known from the Finite Elements Method:
PROPOSITION 4.4 (properties of Qε and Rε on W 1,p()). For any φ ∈W 1,p(), one
has
(ii). Rε(φ)Lp() ≤ εCφW 1,p(),
(iii). ∇Rε(φ)Lp() ≤ C∇φLp().
Moreover,
Lp()
≤ C
ε ∇φLp() for i, j ∈ [1, . . . , n], i 6= j. (4.4)
Up to the factor P, the constant C is the Poincare-Wirtinger constant for Y and depends
upon neither nor ε.
Proof. We start with φ in W 1,p(IRn). From Proposition 2.5 (i) and inequality (3.5 ), we get
φ−Mε
Y (φ)Lp(IRn×Y ) ≤ εC∇φLp(IRn). (4.5)
On the other hand, for any ψ ∈W 1,p(Y ∪ (Y + ei)), i ∈ {1, . . . , n}, we have
| M Y+ei
(ψ)−M Y
(ψ) |=| M Y
( ψ(·+ ei)− ψ(·)
≤ Cψ(·+ ei)− ψ(·)Lp(Y ) ≤ C∇ψLp(Y ∪(Y+ei)).
By a scaling argument and using Definition 4.1, this gives
|Qε(φ)(εξ)−Qε(φ)(εξ + εei)| ≤ εC∇φLp(ε(ξ+Y )∪ε(ξ+ei+Y )). (4.6)
for all ξ ∈ εZZn. Let x ∈ ε
( ξ + Y
) and set for every i = (i1, . . . , in) ∈ {0, 1}n,
x (ik) k =
If ξ ∈ εZZn, for every i ∈ {0, 1}n by definition we have
Qε ( φ ) (x) =
∑ i∈{0,1}n
Qε(φ) ( εξ + εi
) −Qε(φ)
n ,
and a same expression for the other derivatives. This last formula and (4.5)-(4.7) imply estimate (i) written in IRn.
Now, from (4.7), we get
φ(x)−Qε ( φ ) (x) =
n ,
and (ii) (again in IRn), follows by using estimate (4.5). Estimate (iii) is straightforward from the previous ones. In the spirit of Definition 4.3, if φ is in W 1,p(), estimates (i)-(iii) are simply obtained by taking the restrictions to of Qε(P(φ)) and Rε(P(φ)).
To finish the proof, it remains to show (4.4). To do so, it suffices to take the derivative
with respect to any xk with k 6= 1 in the formula of ∂Qε(φ) ∂x1
above and use estimate (4.6).
REMARK 4.5. Observe that by construction (see explicite formula (4.7)) , the function
Qε(φ) is separately piece-wise linear on each cell. Morover, the expression of ∂Qε(φ) ∂xk
shows that this function is independent of xk in each cell ε ( ξ+Y
) , for any k ∈ {1, . . . , n}.
PROPOSITION 4.6. Let {wε} be a sequence converging weakly in W 1,p() to w. Then,
the following convergences hold:
(ii). Qε(wε) w weakly in W 1,p(),
(iii). Tε(∇Qε(wε)) ∇w weakly in Lp(× Y ).
Proof. (i) and (ii). Statement (i) is a direct consequence of estimate (ii) in Proposition 4.4. It implies, together with estimate (i) of Proposition 4.4, convergence (ii).
(iii). Obviously,
From (4.4), ∂
ε for i, j ∈ [1, . . . , n], i 6= j.
Then, by Proposition 3.1, there exist a subsequence (still denoted ε) and wj ∈ Lp(× Y )
with ∂wj ∂yi ∈ Lp(× Y ), such that
Tε (∂Qε(wε)
εTε (∂2Qε(wε) ∂xi∂xj
weakly in Lp(× Y ),
where wj is yi−periodic with i 6= j. Moreover, from Remark 4.5, the function wj does not depend on yj , hence it is Y−periodic. But, see again Remark 4.5, wj is also piecewise linear with respect to any variable yi. Consequently, wj is independent of y. On the other hand, from (ii) above we have
∂Qε(wε) ∂xj
∂xj which shows that convergence (iii) holds for the
whole sequence ε.
PROPOSITION 4.7 (Theorem 3.5 revisited). Let {wε} be a sequence converging weakly
in W 1,p() to w. Then, up to a subsequence , and w ′
in the space Lp(;W 1,p per(Y )) such
that the following convergence holds: 1 ε Tε ( Rε(wε)
) w
′ weakly in Lp(× Y ).
Tε(∇wε) ∇w +∇yw ′
weakly in Lp(× Y ).
Actually, the connection with the w of Theorem 3.5 is given by:
w = w ′ −M
Y (w ′ ).
Proof. Due to the estimates of Proposition 4.4, up to a subsequence, there exists w ′
in Lp(;W 1,p
) w
Combining with convergence (iii) of Proposition 4.6, shows that
Tε ( ∇wε
) ∇w +∇yw
Y (w ′ ).
We end this section with a new characterization of the limit function w ′
in terms of ∇w and w given in Theorem 3.5 above.
REMARK 4.8. In the previous proposition, one can actually compute the average of w ′ .
It depends strongly on the choice of the cell Y and of the definition of Qε. In the case of
Y = (0, 1)n and the Definition 4.1, one can check the following :
M Y
(w ′ ) = −1
5. Periodic unfolding and the standard homogenization problem
Let α, β ∈ IR, such that 0 < α < β. Denote by M(α, β,O) the set of the n× n matrices A = (aij)1≤i,j≤n ∈ (L∞ (O))n×n such that for any λ ∈ IRn and a.e. on O,{
i. (A(x)λ, λ) ≥ α|λ|2,
ii. |A(x)λ| ≤ β|λ|.
be a sequence of non constant matrices such that
Aε ∈M(α, β,). (5.2)
For f given in H−1(), consider the Dirichlet problem{ −div (Aε∇uε) = f in
uε = 0 on ∂. (5.3)
By the Lax-Milgram theorem, there exists a unique uε ∈ H1 0 () satisfying∫

Aε∇uε∇v dx = f, vH−1(),H1 0 (), ∀v ∈ H1
0 (), (5.4)
which is the variational formulation of (5.3). Moreover, one has the apriori estimate
uεH1 0 () ≤
1 α fH−1(). (5.5)
Consequently, there exist u0 in H1 0 () and a subsequence, still denoted ε, such that
uε u0 weakly in H1 0 (), (5.6)
We are now interested in obtaining a limit problem, the so-called “homogenized” problem satisfied by u0. This is called standard homogenization and the answer, for some classes of Aε, can be found in many works , starting with the classical book A. Bensoussan, J.L. Lions and G. Papanicolaou [6] (see, for instance D. Cioranescu and P. Donato [13] and the references herein). We now recall it.
THEOREM 5.1 (standard periodic homogenization). Let A = (aij)1≤i,j≤n belong to
M(α, β, Y ), where aij = aij(y) are Y−periodic. Set
Aε(x) = ( aij
(x ε
a.e. on , (5.7)
Let uε be the solution of the corresponding problem (5.3) with f in H−1(). Then the
whole sequence {uε} converges to a limit u0 which is the unique solution of the homogenized
problem −div (A0∇u0) =
u0 = 0 on ∂,
(5.8)
where the constant matrix A0 = (a0 ij)1≤i,j≤n is elliptic and given by
a0 ij =MY
∂yk
) . (5.9)
In (5.9), the functions χj (j = 1, . . . , n), often referred to as correctors, are the solutions
of the cell systems −
(5.10)
As will be seen below, using the periodic unfolding, the proof of this theorem is elementary! Actually, with the same proof, a more general result can be obtained, with a sequence of matrices Aε.
THEOREM 5.2 (periodic unfolded homogenization). Let uε be the solution of prob-
lem (5.3) with f in H−1() and Aε satisfying (5.1)-(5.2). Suppose that there exists a
matrix B such that
Then there exists u0 ∈ H1 0 () and u ∈ L2(;H1
per(Y )) such that
0 (),
Tε(∇uε) ∇u0 +∇yu weakly in L2(× Y ),
(5.12)
and the pair (u0, u) is the unique solution of the problem
∀Ψ ∈ H1
1 |Y |
∫ ×Y
][ ∇Ψ(x) +∇yΦ(x, y)
(5.13)
Remark 5.3. Problem (5.13) is of standard variational form in the space
H = H1 0 ()× L2(; H1
per(Y )/IR).
Remark 5.4. Hypothesis (5.11) implies that B ∈M(α, β,× Y ).
Remark 5.5. If Aε is of the form (5.7), then B(x, y) = A(y). In the case where Aε(x) = A1(x)A2
(x ε
) , one has (5.11) with B(x, y) = A1(x)A2(y).
Remark 5.6. Let us point out that every matrix B ∈M(α, β,×Y ) can be approached by the sequence of matrices Aε in M(α, β,) with Aε defined as follows:
Aε =
Proof of Theorem 5.2. Convergences (5.12) follow from estimate (5.5), Proposition 2.10 and Theorem 3.5, respectively.
Let us choose v = Ψ, with Ψ ∈ D() as test function in (5.4). The integration formula (2.5) from Proposition 2.7, gives
1 |Y |
∫ ×Y
Tε' f,ΨH−1(),H1 0 (). (5.14)
We are allowed to pass to the limit in (5.14), due to (5.11) and (5.12), to get
1 |Y |
∫ ×Y
0 () (5.15)
which, by density, still holds for every Ψ ∈ H1 0 ().
Now, taking in (5.4), as test function vε(x) = εΨ(x)ψ (x ε
) , Ψ ∈ D(), ψ ∈ H1
1 |Y |
∫ ×Y
.
Since vε 0 in H1 0 (), we get at the limit
1 |Y |
∫ ×Y
] Ψ(x)∇yψ(y) dxdy = 0
which, due to the density of the tensor product D() × H1 per(Y ), is valid for all Φ in
L2(;H1 per(Y )).
Remark 5.7. As in the two-scale method, (5.13) gives u in terms of ∇u0 and yields the standard form of the homogenized equation, i.e., (5.8). In the simple case where A(x, y) = A(y) = (aij(y))1≤i,j≤n, it is easily seen that the limit matrix B is precisely A0
which was defined in Theorem 5.1 by (5.9)-(5.10). One also has
u = n∑ i=1
∂xi χi. (5.16)
PROPOSITION 5.8 (convergence of the energy). Under the hypotheses of Theorem
5.2, one has
lim ε→0
∫ ×Y
1 |Y |
∫ ×Y
f, uεH−1(),H1 0 ()
= f, u0H−1(),H1 0 () =
1 |Y |
∫ ×Y
which gives (5.17) as well as the convergence
lim sup ε→0
whence (5.18).
Remark 5.9. From the above proof, we also have the following convergence:
lim ε→0
Tε(∇uε)→ ∇u0 +∇yu strongly in L2(× Y ). (5.19)
Proof. We have successively
] .
Each term in the right hand side converges due to (5.12), Remark 5.9 and hypothesis (5.11) so that the limit is zero. Then convergence (5.19) follows from the ellipticity of Bε.
6. Some corrector results
Under additional regularity assumptions on the homogenized solution u0 and the corrector functions χj , the strong convergence for the gradient of u0 with a corrector is known (cf. [13], [14]). More precisely, suppose that ∇yχj ∈ (Lr(Y ))n, j = 1, . . . , n and ∇u0 ∈ Ls() with 1 ≤ r, s <∞ and such that 1/r + 1/s = 1/2. Then
∇uε −∇u0 −∇yu (·, · ε
)→ 0 strongly in L2().
Proposition 2.16 however gives a corrector result without any additional regularity as-
sumption on χj . In the fact, the proof of this corrector result (as given below), reduces to a few lines. We also include a new type of corrector.
THEOREM 6.1. Under the hypotheses of Theorem 5.2, one has
∇uε −∇u0 − Uε ( ∇yu
In the case B = A0 the function u0 + ε n∑ i=1
Qε (∂u0
has
Qε (∂u0
) −→ 0 strongly in H1(). (6.2)
Proof. From (5.18), (5.19) and Proposition 2.16 (iii), one immediately has
∇uε − Uε ( ∇u0
Uε ( ∇u0
) → ∇u0 strongly in L2(),
whence (6.1). From (4.3) in Proposition 4.2 the function u0 + ε n∑ i=1
Qε (∂u0
∇u0 + Uε ( ∇yu
∂xi
)] χ ({ ·
ε
} Y
) and one immediately has the strong convergence in L2() of the right hand side in the above equality. Thanks to (6.1) and the convergences in Proposition 4.2 one has (6.2).
7. Periodic unfolding and multiscales
In this section, we want to consider a “partition” of Y in two non-empty disjoint open subsets Y1 and Y2, i.e. such that Y1∩Y2 = /0 and Y = Y 1∪Y 2. We also introduce another unit periodicity cell Z and consider a matrix field Aεδ is defined by
Aεδ(x) =
A1
{x ε
} Y ∈ Y2,
where the two matrix fields A1 and A2 are defined on × Y and × Y × Z respectively. In this problem, there are two small scales, namely ε and εδ, associated respectively to
the cells Y and Z. Consider the solution uεδ ∈ H1 0 () of
∫
f w dx ∀w ∈ H1 0 ().
Suppose that A1 is in L∞(× Y ) and A2 in L∞(× Y ×Z). With standard ellipticity hypotheses it is easy to obtain some u0 such that, up to a subsequence,
uεδ u0 weakly in H1 0 ()).
Using the unfolding method for scale ε, as before we have
Qε ( uεδ ) u0 weakly in H1
0 (),
Tε(uεδ) u0 weakly in L2(; H1(Y )), 1 ε Tε ( Rε(uεδ)
) u weakly in L2(; H1(Y )),
Tε ( ∇uεδ
) ∇u0 +∇yu in L2(× Y ).
These convergences do not see the oscillations at the scale εδ. In order to capture them, one considers the restrictions to the set × 2 defined by
vεδ(x, y) .= 1 ε Tε ( Rε(uεδ)
) |2 .
Obviously,
vεδ u|2 weakly in L2(;H1(2)).
Now, we apply to vεδ, a similar unfolding operation for the variable y, thus adding a new variable z ∈ Z, denoted T yδ .
T yδ (vεδ)(x, y, z) = vεδ ( x, δ [y δ
] Z
+ δz )
for x ∈ , y ∈ 2 and z ∈ Z.
At this point, it is essential to remark that all the estimates and weak convergence
properties which were shown for the original unfolding Tε still hold for T yδ with
x being a mere parameter. For example, Proposition 4.6 and Theorem 3.5 adapted to this case imply :
T yδ ( ∇yvεδ
T yδ ( Tε ( ∇uεδ
)) ∇u0 +∇yu+∇zu weakly in L2(× 2 × Z).
Under these conditions, the limit functions u0, u and u are characterized in the following theorem:
Theorem 7.1. The functions
per(Y )/IR), u ∈ L2(× 2, H 1 per(Z)/IR)
are the uniquesolutions of the following variational problem:

}{ ∇Ψ +∇yΦ +∇zΘ
per(Y )/IR),∀Θ ∈ L2(× 2, H 1 per(Z)/IR)
The proof uses test functions of the form
Ψ(x) + εΨ1(x)Φ1
(x ε
) ,
where Ψ,Ψ1,Ψ2 are in D(), Φ1 in H1 per(Y ), Φ2 ∈ D(2) and Θ2 ∈ H1
per(Z). A more general approach to multiscale periodic homogenization in A. Damlamian and P. Donato [16] (where reiterated H0-convergence dealing with holes, is considered).
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Doina Cioranescu & George Griso LaboratoireJacques-Louis Lions Universite Pierre et Marie Curie (Paris VI) Boite courrier 187 4 Place Jussieu 75252 Paris Cedex 05 France Email: [email protected], [email protected]
Alain Damlamian Laboratoire d’Analyse et de Mathematiques Appliquees Universite Paris Est 94010 Creteil Cedex France Email: [email protected]
GAKUTO International Series
Math. Sci. Appl., Vol.** (2009) Multiple scales problems in Biomathematics,
Mechanics, Physics and Numerics, pp. 37–66
GAKKOTOSHO
and applications to Robin problems
D. Cioranescu, P. Donato and R. Zaki
Abstract: The periodic unfolding method was introduced in [C.R. Acad. Sci. Paris, Ser. I 335 (2002), 99-104] by D. Cioranescu, A. Damlamian and G. Griso for the study of classical periodic homogenization. The main tools are the unfolding operator and a macro-micro decomposition of functions which allows to separate the macroscopic and microscopic scales. In this paper, we extend this method to the homogenization in domains with holes, intro- ducing the unfolding operator for functions defined on periodically perforated domains as well as a boundary unfolding operator. As an application, we study the homogenization of some elliptic problems with a Robin condition on the boundary of the holes, proving convergence and corrector results.
1 Introduction.
The homogenization theory is a branch of the mathematical analysis which treats the asymptotic behavior of differential operators with rapidly oscillating coefficients. We have now different methods related to this theory:
• The multiple-scale method introduced by A. Bensoussan, J.-L. Lions and G. Papani- colaou in [2].
• The oscillating test functions method due to L. Tartar in [16].
• The two-scale convergence method introduced by G. Nguetseng in [15], and further developed by G. Allaire in [1].
38 D. Cioranescu, P. Donato and R. Zaki
Recently, the periodic unfolding method was introduced in [4] by D. Cioranescu, A. Damla- mian and G. Griso for the study of classical periodic homogenization in the case of fixed domains. This method is based on two ingredients: the unfolding operator and a macro- micro decomposition of functions which allows to separate the macroscopic and microscopic scales. The interest of the method comes from the fact that it only deals with functions and classical notions of convergence in Lp spaces. This renders the proof of homogenization results quite elementary. It also provides error estimates and corrector results (see [13] for the case of fixed domains). Here, we present the adaptation of the method to the homogenization in domains with holes introduced in [5] and [6]. We refer also to [7] for some complementary result and an application to a problem with nonlinear boundary conditions. We define in the upcoming section the unfolding operator for functions defined on periodically perforated domains. We also define in Section 5 a boundary unfolding operator, in order to treat problems with nonhomogeneous boundary conditions on the holes (Neumann or Robin type). The main feature is that, when treating such problems, we do not need any extension operator. Consequently, we can consider a larger class of geometrical situations than in [2], [5], and [9] for instance. In particular, for the homogenous Neumann problem, we can admit some fractal holes like the two dimensional snowflake (see [19]). For a general nonhomogeneous Robin condition, we only assume a Lipschitz boundary, in order to give a sense to traces in Sobolev spaces.
We also show in Section 4 a compactness result (Theorem 4.7) which states that any sequence {vε}, with vεH1(ε) ≤ C, defined on a space depending on ε, is mapped by the unfolding operator into a compact set in L2
loc( × Y ). This result is crucial for proving corrector results, as showed in Section 6.
The paper is organized as follows: In Section 2, we define the unfolding operator and prove some linked properties. In Section 3, we give the macro-micro decomposition of functions defined in perforated domains and in Section 4, we introduce the averaging operator and state a corrector result. The boundary unfolding operator, essential in this work, is introduced in Section 5, together with its main properties. Finally, Section 6 contains an application to the homogenization of an elliptic problem with Robin boundary condition.
2 The periodic unfolding operator in a perforated do-
main.
In this section, we introduce the periodic unfolding operator in the case of perforated domains. In the following we denote:
• an open bounded set in RN ,
• Y = N∏ i=1
[0, li[ the reference cell, with li > 0 for all 1 ≤ i ≤ N , or more generally a set
having the paving property with respect to a basis (b1, · · · , bN) defining the periods,
• T an open set included in Y such that ∂T does not contain the summits of Y . We can be, sometimes, transported to this situation by a simple change of period,
The periodic unfolding method in perforated domains 39
• Y = Y \ T a connected open set.
We define T ε =
Figure 1: The domain ε and the reference cell Y
We assume in the following that ε is a connected set. Unlike preceding papers treating perforated domains (see for example [5],[8],[9]) we can allow that the holes meet the boundary ∂. In the rest of this paper, we only take the regularity hypothesis
|∂| = 0. (1)
Remark 1 The hypothesis aforementioned is equivalent to the fact that the number of cells intersecting the boundary of is of order ε−N (we refer to [11, Lemma 21]).
Remark 2 An interesting example on the hypotheses aforementioned would be the lattice- type structures for which it is not possible, in some cases, to define extension operators. This situation happens if the holes intersect the exterior boundary ∂ (see [9],[10]).
In the sequel, we will use the following notation:
• for the extension by 0 outside ε (resp. ) for any function in Lp(ε) (resp. Lp()),
• χε for the characteristic function of ε,
• θ for the proportion of the material in the elementary cell, i.e. θ = |Y |
|Y | ,
• ρ(Y ) for the diameter of the cell Y ,
• T εint for the set of holes that do not intersect the boundary ∂.
By analogy to the 1D notation, for z ∈ RN , [z]Y denotes the unique integer combination j=N∑ j=1
kjbj , such that z− [z]Y belongs to Y . Set {z}Y = z− [z]Y (see Fig. 2). Then, for almost
every x ∈ RN , there exists a unique element in RN , denoted by [x ε
] Y , such that
Figure 2: The decomposition z = [z]Y + {z}Y
Definition 1 (Unfolding operator) Let ∈ Lp(ε), p ∈ [1,+∞]. We define the function Tε() ∈ Lp(RN × Y ) by setting
Tε()(x, y) = ( ε [x ε
] Y
Tε : ∈ Lp(ε) → Tε() ∈ Lp(RN × Y )
is called the unfolding operator.
Remark 3 Notice that the oscillations due to perforations are shifted into the second variable y which belongs to the fixed domain Y , while the first variable x belongs to RN . One see immediately the interest of the unfolding operator. Indeed, when trying to pass to the limit in a sequence defined on ε, one needs first, while using standard methods, to extend it to a fixed domain. With Tε, such extensions are no more necessary.
The main properties given in [4] for fixed domains can easily be adapted for the perforated ones without any major difficulty in the proofs. These properties are listed in the proposition below.
To do so, let us first define the following domain: ε = int(
ξ∈Λε
} .
The set ε is the smallest finite union of εY cells containing .
Figure 3: The domain ε
Proposition 4 The unfolding operator Tε has the following properties:
1. Tε is a linear operator.
2. Tε() ( x, {x ε
} Y
) = (x), ∀ ∈ Lp(ε) and x ∈ RN .
3. Tε(ψ) = Tε()Tε(ψ), ∀, ψ ∈ Lp(ε).
4. Let in Lp(Y ) or Lp(Y ) be a Y - periodic function. Set ε(x) = (x ε
) . Then,
5. One has the integration formula
∫
6. For every ∈ L2(ε), Tε() belongs to L2(RN × Y ). It also belongs to L2(ε × Y ).
7. For every ∈ L2(ε), one has
Tε()L2(RN×Y ) = √
|Y |L2(ε).
8. ∇yTε()(x, y) = εTε(∇x)(x, y) for every (x, y) ∈ RN × Y .
9. If ∈ H1(ε), then Tε() is in L2(RN ;H1(Y )).
10. One has the estimate
∇yTε()(L2(RN×Y ))N = ε √ |Y |∇x(L2(ε))N .
∫
+ εy ) dx dy,
since is null in the holes. The desired result is then straightforward.
N.B. In the rest of this paper, when a function ψ is defined on a domain containing ε, and for simplicity, we may use the notation Tε(ψ) instead of Tε(ψ|ε).
Proposition 5 Let ∈ L2(). Then,
1. Tε() → strongly in L2(RN × Y ),
2. χε θ weakly in L2(),
3. Let (ε) be in L2() such that
ε → strongly in L2().
Then,
Proof. 1. The first assertion is obvious for every ∈ D(). If ∈ L2(), let k ∈ D() such that k → in L2(). Then
Tε() − L2(RN×Y ) ≤ Tε() − Tε(k)L2(RN×Y ) + Tε(k) − kL2(RN×Y )
+k − L2(RN×Y ),
from which the result is straightforward.
∫
) .
On one hand, by using 1 and 7 of Proposition 2.5, we get as ε→ 0 ∫
RN×Y
lim ε→0
(Tε() − )2 dx dy = 0.
Therefore, assertion 3 holds true.
Proposition 6 Let ε be in L2(ε) for every ε, such that
Tε( ε) weakly in L2(RN × Y ).
Then,
(·, y)dy weakly in L2(RN).
Proof. Let ψ ∈ D(). Using 3 and 5 of Proposition 2.5, one has successively ∫
RN
This gives, using 1 of Proposition 2.6
∫
|Y |

Proposition 7 Let ε be in L2(ε) for every ε, with
ε L2(fε) ≤ C,
ε∇x ε(L2(fε))N ≤ C.
Then, there exists in L2(RN ;H1(Y )) such that, up to subsequences
1. Tε( ε) weakly in L2(RN ;H1(Y )),
2. εTε(∇x ε) ∇y weakly in L2(RN × Y ),
where y 7→ (., y) ∈ L2(RN ;H1
per(Y )).
∫
RN×Y
=
∫
=
∫
[ψ (x− εli −→ei , y) − ψ (x, y)] dx dy.
Passing to the limit, we obtain the result since ψ(x− εli −→ei , y)−ψ(x, y) → 0 when ε→ 0.
3 Macro-Micro decomposition.
Following [4], we decompose any function in the form
= Qε() + Rε(),
where Rε is designed in order to capture the oscillations. As in the case of fixed domains, we start by defining Qε() on the nodes εξk of the εY -lattice.
Here, it is no longer possible to take the average on the entire cell Y as in [4], but it will be taken on a small ball Bε centered on εξk and not touching the holes. This is possible using the fact that ∂T does not contain the summits of Y . However, Bε must be entirely contained in ε. To guarantee that, we are let to define Qε() on a subdomain of ε only. To do so, for every δ > 0, let us set
ε δ = {x ∈ ; d(x, ∂) > δ} and ε
δ = int(
δ
δ
Qε()(εξk) = 1
(εξk + εz)dz.
Observe that by definition, any ball Bε centered in a node of ε 2ερ(Y ) is entirely con-
tained in ε, since actually they all belong to ε ερ(Y ).
• We define Qε() on the whole ε 2ερ(Y ), by taking a Q1-interpolate, as in the finite ele-
ment method (FEM), of the discrete function Qε()(εξk).
• On ε 2ερ(Y ), Rε will be defined as the remainder: Rε() = −Qε().
Proposition 8 For belonging to H1(ε), one has the following properties:
1. Qε() H1(bε
2ερ(Y ) ) ≤ C
))N .
Proof. These results are straightforward from the definition of Qε. The proof, based on some FEM properties, is very similar to the corresponding one in the case of fixed domains (see [4]), with the simple replacement of Y by Y .
We can now state the main result of this section.
Theorem 9 Let ε be in H1(ε) for every ε, with εH1(ε) bounded. There exists in H1() and in L2(;H1
per(Y )) such that, up to subsequences
1. Qε( ε) weakly in H1
loc(),
loc(;H1(Y )),
4. Tε(∇x( ε)) ∇x+ ∇y weakly in L2
loc(;L2(Y )).
Remark 10 When comparing with the case of fixed domains, the main difference is that, since the decomposition was done on ε
2ερ(Y ), we have here local convergences only.
Proof of Theorem. Assertions 2, 3 and 4 can be proved by using the same arguments as in the corresponding proofs for the case of fixed domains. We consider here just the first assertion.
Let K be a compact set contained in . As d(K, ∂) > 0, there exists εK > 0 depending on K, such that
∀ε ≤ εK , K ⊂ ε 2ερ(Y ).
Hence,
Qε( ε) weakly in H1
loc().
What remains to be proved is that ∈ H1(). To do so, we make use of the Dominated Convergence theorem.
Let us consider the sequence (ε 1 N
)N . Observe that it is increasing. Indeed,
x ∈ ε 1 N
1 N+1
.
Moreover, for every N , there exists εN depending on ε 1 N
such that
⊂ ε 2ερ(Y ).
Let us define the sequence of functions (N)N for every N ∈ N as follows:
N = ||2 χε 1 N
.
Let us show that
One has successively
dx =
||2 dx,
∫
ε)2 L2(bε
2ερ(Y ) ) ≤ C,
whence (4). The next step is to prove that
the sequence (N)N simply converges towards ||2 . (5)
,
and x ∈ ε 1
x ∈ ε 1 N
χε 1 N
and this ends the proof of (5). Thanks to (3),(4) and (5), we can apply the Dominated Convergence theorem to deduce that
||2 ∈ L1() and lim N→∞
∫
4 The averaging operator Uε.
Definition 2 For ∈ L2(RN × Y ), we set
Uε()(x) = 1
) dz, for every x ∈ RN .
Remark 11 For V ∈ L1(RN × Y ), the function x 7→ V ( x, {x ε
} Y
) is generally not mea-
surable (for example, we refer to [5]-Chapter 9). Hence, it cannot be used as a test function. We replace it by the function Uε(V ).
The next result extends the corresponding one given in [4].
Proposition 12 One has the following properties:
1. The operator Uε is linear and continuous from L2(RN ×Y ) into L2(RN), and one has for every ∈ L2(RN × Y )
Uε()L2(RN ) ≤ L2(RN×Y ),
2. Uε is the left inverse of Tε on ε, which means that Uε Tε = Id on ε,
3. Tε (χεUε()) (x, y) = 1
|Y |
4. Uε is the formal adjoint of Tε.
Proof. 1. It is straightforward from Definition 4.1.
2. For every ∈ L2(ε), one has
Uε (Tε ()) (x) = 1
|Y |
] Y
3. Let ∈ L2 ( RN ) , one has
Tε (χεUε ()) (x, y) = Uε () ( ε [x ε
] Y
+ εy )
= 1
|Y |
and ψ ∈ L2 ( RN × Y
) , we have
|Y |
= 1
|Y |
= 1
|Y |
Uε() → strongly in L2(RN).
2. Let ∈ L2(RN × Y ). Then,
Tε (χεUε()) → strongly in L2(RN × Y ),
and
Proof. 1. If ∈ L2(RN), one has by definition
Uε()(x, y) = 1
But ( ε [ x ε
] Y
+ εz ) → (x) when ε → 0, and this explains the result.
2. It is a simple consequence of 1 in Proposition 2.6, and Proposition 2.7.
As in the case of fixed domains, one has
Theorem 14 Let ε be in L2(ε) for every ε, and let ∈ L2(RN × Y ). Then,
1. Tε( ε) → strongly in L2(RN × Y ) ⇐⇒ ε − Uε() → 0 strongly in L2(RN).
2. Tε( ε) → strongly in L2
loc(R N ;L2(Y )) ⇐⇒ ε−Uε() → 0 strongly in L2
loc(R N).
εUε ε)L2(RN×Y )
ε)L2(RN×Y )
ψ ≥ 0 and ψ = 1 on w.
Then, by using 1 of Proposition 2.6, one has
ε − UεL2(w) ≤ ψ ( ε − Uε
) L2(RN )
≤ C Tε (ψ) (Tε (ε) − Tε (χεUε ε)) L2(suppψ×Y )
≤ C ( Tε (ψ) (Tε (ε) − ) L2(suppψ×Y ) + Tε (ψ) (− Tε (χεUε
ε)) L2(suppψ×Y )
The converse implications are immediate.
This result is essential for proving corrector results when studying homogenization problems, as we show in Section 6. To apply it, the compactness result given by Theorem 4.7 below is crucial.
Let us first state the following proposition:
Proposition 15 For every ∈ H1(ε) one has
Rε()L2(ε) = −Qε()L2(ε) ≤ Cε∇(L2(ε))N .

Theorem 16 Let vε be in H1(ε) for every ε and v ∈ H1() such that
• vεH1(ε) is bounded,
Then,
loc(, L 2(Y )).
∫
(∫
∫
) .
loc().
|Tε (Qεv ε) − v|2 dx dy = 0.
∫
ω×Y
|Tε (vε −Qε (vε))|2 dx dy = Cvε −Qε (vε) 2 L2(ω∩ε)
≤ Cε2∇vε2 (L2(ε))N ,
2(Y )).

Remark 17 We can stress here one of the major properties of the unfolding operator. In- deed, it transforms any function defined on the perforated domain ε into a function Tε() defined on the fixed domain RN × Y . Theorem 4.7 actually states that any sequence {vε}, with vεH1(ε) ≤ C, is mapped into a compact set in L2
loc( × Y ).
5 The boundary unfolding operator.
We define here the unfolding operator on the boundary of the holes ∂T ε, which is specific to the case of perforated domains. To do that, we need to suppose that T has a Lipschitz boundary.
Definition 3 (Unfolding boundary operator) Suppose that T has a Lipschitz boundary, and let ∈ Lp(∂T ε), p ∈ [1,+∞]. We define the function T b
ε () ∈ Lp(RN × ∂T ) by setting
T b ε ()(x, y) =
( ε [x ε
ε () ∈ Lp(RN × ∂T )
is called

homogenization, multiple scales problems in biomathematics, mechanics, physics and numerics, cimpa...

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