Homogenization for a Fredholm alternative in

periodically perforated domains

Alain Damlamian (Université Paris-Est Créteil)Joint work with D. Cioranescu, and G. Griso

ISFMA SYMPOSIUM:

HOMOGENIZATION AND MULTISCALE ANALYSISShanghai Jiaotong University

3-7 October 2011

Plan

0. The Fredholm alternative

1. Presentation of perforated domains

2. The pure Neumann problem in perforated domains

3. A brief summary of the unfolding method in perforated domains

4. A framework for the homogenization of the pure Neumann problem in perforateddomains

5. The Homogenization result : unfolded limit, homogenized problem, corrector result

Main reference: D. Cioranescu, A. Damlamian, P. Donato, G. Griso, R. Zaki: Theperiodic unfolding method in domains with holes (accepted in SIMA)

The Neumann problem with a zero order term (no Fredholm Alternative) was pre-sented by Doina Cioranescu in her lecture Monday. See also the appendix of [2](G. Allaireand F. Murat, Homogenization of the homogeneous Neumann problem with nonisolatedholes, Appendix with A.K. Nandakumar, Asymptotic Analysis 7 (1993), 81–95.)

The Fredholm alternative

The Fredholm alternative is associated with a pure Neumann problem (i.e., withoutzero order term) on an open domain O such as :

−div(A∇u) = f in O;∂u

∂nA= g on ∂O,

where f is in L2(O) , g in L2(∂O) and A(x) = (aij(x))1≤i,j≤n is a bounded and uniformlyelliptic matrix field.

Its variational formulation is to find u in H1(O) such that

�

O

A∇u∇v dx =

�

O

fv dx+

�

∂O

gv dσ, ∀v ∈ H1(O). (0.1)

Note that if g �≡ 0, the last integral makes sense only under some minimal regularitycondition on ∂O (e.g. Lipschitz). If g ≡ 0, such requirement is not necessary.

It is well-known that provided O is bounded connected and with Lipschitz boundary,this problem has a unique solution up to a constant if and only if the following Fredholmcondition is satisfied: �

O

f(x) dx+

�

∂O

g dσ = 0. (0.2)

The usual proof uses a duality argument, which is specific to the linear case (Lax-Milgram’s theorem does not apply here for lack of coerciveness).

Another approach to obtain existence and uniqueness of the solution for (0.1) is to useLax-Milgram’s theorem applied in a space V (O) which is a Hilbert space for the norm�w�V (O)

.= �∇w�L2(O). The “natural” candidate is

V (O).= {w ∈ H1(O),MO(w) = 0}, (0.3)

where MO(w) denotes the average1

|O|

�

O

w dx, (which makes sense only if O is of finite

measure). Then, the fact that V (O) is a Hilbert space is equivalent to the fact that thePoincaré-Wirtinger inequality holds for H1(O):

∃C ∈ R+ such that v ∈ H1(O) ⇒ �v −MOv dx�L2(O) ≤ C�∇v�L2(O). (0.4)

Clearly, this inequality requires that O be connected. It is always satisfied for O boundedwith Lipschitz boundary, but holds in much more general cases. A large class of domainswhich satisfy the Poincaré-Wirtinger inequality is that of John domains, and many ofthem do not have Lipschitz boundaries (John domains were introduced by Fritz Johnin one of his important papers on elasticty, in particular in connection with the Korninequality, see [8]).

The Lax-Milgram theorem applies to the problem in the following form:

Find u ∈ V (O) such that

�

O

A∇u∇v dx =

�

O

fv dx+

�

∂O

gv dσ, ∀v ∈ V (O), (0.5)

In this formulation, there is no Fredholm condition. However, in full generality, theproblem corresponds to a different formulation in the sense of distributions, namely

−div (A∇u) = f −

1

|O|(

�

O

f dx+

�

∂O

g dσ) in O

A∇u · ν = g on ∂O.(0.6)

The Fredholm condition reappears here as a Fredholm correction�

O

f dx+

�

∂O

g dσ

which should vanish in order for the right-hand side of (0.6) to be simply f .

Under these conditions, problem (0.1) has many solutions w, all differing by a constantfrom the unique solution u of (0.5).

What can be said of this problem when O is the periodically perforated domain Ω∗εde-

fined in the next section? As can be expected, connectedness and the Poincaré-Wirtingerinequality will play a central role.

1 A brief presentation of perforated domains

In this section, we begin by giving the setup and notations for periodically perforateddomains.

Let b = (b1, . . . , bn) be a basis in Rn. Set

G =�ξ ∈ Rn | ξ =

n�

i=1

kibi , (k1 , . . . , kn) ∈ ZZn�. (1.1)

By Y , we denote the reference cell, which in the simplest case is the open parallelotopgenerated by the basis b,

�y ∈ Rn | y =

n�

i=1

yibi , (y1 , . . . , yn) ∈ (0 , 1 )n�. (1.2)

For the general case of Y having the paving property, we refer to [4].

Let S be a closed strict subset of Y and denote by Y ∗ the part occupied by the materiali.e. Y ∗ = Y \S. The sets S and Y ∗ are the reference hole and perforated cell, respectively(see Figure 1).

Figure 1. The Y ∗ = Y \ S

We introduce the following notations :

(Rn)∗ = Rn \�

ξ∈G

(ξ + S), (Rn)∗ε= ε(Rn)∗ = Rn \

�

ξ∈G

ε(ξ + S).= Rn \ Sε. (1.3)

By this definition, (Rn)∗εis Rn εG-periodically perforated by εS.

Let now Ω be an open bounded connected subset of Rn with Lipschitz boundary, andconsider the sets

�Ωε = interior� �

ξ∈Ξε

ε�ξ+ Y

��, Λε = Ω \ �Ωε, where Ξε =

�ξ ∈ G, ε(ξ+ Y ) ⊂ Ω

�. (1.4)

Figure 2. The sets �Ωε (in grey) and Λε (in green)

The set �Ωε is the interior of the largest union of ε(ξ + Y ) cells (ξ ∈ G), such thatε(ξ+Y ) are included in Ω, while Λε is the subset of Ω containing the parts from ε

�ξ+Y

�

cells intersecting the boundary ∂Ω (see Figure 2).

As one can easily see from the pictures below, for a connected Ω, the set �Ωε is notnecessarily connected. But, if ∂Ω is Lipschitz, eventually, for small enough ε, �Ωε isconnected.

The perforated domain Ω∗εis obtained by removing from Ω the set of holes Sε

Ω∗ε= (Rn)∗

ε∩ Ω. (1.5)

The following notations will be used (see Figure 3):

�Ω∗ε= �Ωε ∩ Ω∗ε, Λ∗ε = Λε ∩ Ω∗ε. (1.6)

Figure 3. The sets Ω∗ε, �Ω∗

ε(in dark blue) and Λ∗

ε(in light green)

For a connected Ω, the sets Ω∗εare not necessarily connected, even when �Ωε is con-

nected. Furthermore, and even if ∂Ω is Lipschitz, it is not necessarily so of ∂Ω∗ε(there

can be cusps).

Here is such an example

But one should not think this is a fluke. It can happen even with an analytic boundary∂Ω:

or a straight boundary:

However, we can see that the “core” of Ω∗εis connected provided the perforated unit

cell Y ∗ is itself connected and is connected with its neighboring cells (see Hypothesis H2below).

Because of this difficulty, we have to adapt and modify the approximate problem.

2 The Neumann problem in perforated domains with

Fredholm alternative

We would like to consider the homogenization of the Neumann problem whose varia-tional formulation is

Find uε in V (Ω∗

ε) such that

�

Ω∗ε

Aε∇uε ∇v dx =

�

Ω∗ε

fεv dx+

�

∂Ω∗ε

gεv dσ, ∀v ∈ V (Ω∗

ε),

(2.1)where fε is given in L2(Ω∗ε) and gε in L

2(∂Ω∗ε).

As we have seen, it is not true in general that Ω∗εis connected, which precludes that

it satisfies the Poincaré-Wirtinger inequality. The “natural” solution would be to assumethat the data fε and gε vanish near the boundary. This would in essence set the problemin the “core” connected component of Ω∗

ε.

However, the boundary of Ω∗εnear the boundary of Ω can be so irregular (including

cusps) that there can exist functions with gradient in L2(Ω∗ε) which are not integrable in

Ω∗ε. Consequently, the space V (Ω∗

ε) does not make sense (nor the corresponding space for

the “core” of Ω∗ε). These difficulties are specific to the case of the Fredholm alternative

(they do not appear when there is a zero oder term which make the problem coercive, in[2] and [4]) and the main contribution of this paper is to give a way to overcome them.

To do so, we turn to the unfolding method.

3 A brief summary of the unfolding method in per-

forated domains

For z ∈ Rn, [z]Y denotes the unique (up to a set of measure zero) integer combination�n

j=1 kjbj of the periods such that z − [z]Y belongs to Y (see Figure 4).

Figure 4. Definition of [z]Y and {z}Y

Set now{z}Y = z − [z]Y ∈ Y a.e. for z ∈ Rn.

The unfolding operator T ∗ε

for functions defined on the perforated domain Ω∗εwas

introduced in [4] as follows:

Definition 3.1. For any function φ Lebesgue-measurable on �Ω∗ε, the unfolding operator

T ∗ε

is defined by

T∗

ε(φ)(x, y) =

φ

�ε

�x

ε

�

Y

+ εy�

a.e. for (x, y) ∈ �Ωε × Y ∗,

0 a.e. for (x, y) ∈ Λε × Y∗.

(3.1)

For φ Lebesgue-measurable on Ω∗ε, we denote T ∗

ε(φ|�Ω∗ε

) simply by T ∗ε(φ).

This operator maps functions defined on the oscillating domain �Ω∗ε, to functions defined

on the fixed domain Ω×Y ∗. Its main properties are recalled in the two propositions below(see [4] for their proofs).

Proposition 3.2. For p ∈ [1,+∞[, the operator T ∗εis linear and continuous from L

p(Ω∗ε)

to Lp(Ω× Y ∗) . For every φ in L1(Ω∗

ε) and w in Lp(Ω∗

ε),

(i)1

|Y |

�

Ω×Y ∗T

∗

ε(φ)(x, y) dx dy =

�

�Ω∗εφ(x) dx,

(ii) �T ∗ε(w)�Lp(Ω×Y ∗) ≤ | Y |

1/p�w�Lp(Ω∗ε).

Proposition 3.3. Let p belong to [1,+∞[.

(i) For w ∈ Lp(Ω),

T∗

ε(w) → w strongly in Lp(Ω× Y ∗).

(ii) Let wε be in Lp(Ω∗

ε) such that ||wε||Lp(Ω∗ε) ≤ C. If

T∗

ε(wε) � �w weakly in Lp(Ω× Y ∗),

then its extension by 0 into the holes �wε satisfies

�wε �|Y ∗|

|Y |M

Y ∗( �w) weakly in Lp(Ω)

(iii) For wε in Lp(Ω∗

ε) such that �wε�Lp(Ω∗ε) is bounded, the following are equivalent:

a) There is w ∈ Lp(Ω) such that

�wε �|Y ∗|

|Y |w weakly in L

p(Ω).

b) All the weak limit points W in Lp(Ω×Y ∗) of the sequence {T ∗

ε(wε)} have the same

average over Y∗(this average MY ∗(W ) being just w).

The next results, which will be used in the sequel, are consequences of these proposi-tions.

1

Corollary 3.4. Let p be in [1,∞] and p� be its conjugate.

(i) Both sequences {1Ω∗ε} and {1�Ω∗ε} converge weakly-∗ in L∞(Ω) to the constant

|Y ∗|

|Y |and

consequently

limε→0

|Ω∗ε| = lim

ε→0|�Ω∗

ε| =

|Y ∗|

|Y ||Ω|. (3.2)

(ii) Let wε be in Lp(Ω∗

ε) such that

�wε �|Y ∗|

|Y |w weakly in L

p(Ω).

Then,

limε→0

MΩ∗ε(wε) = limε→0

M�Ω∗ε(wε 1�Ω∗ε) = MΩ(w). (3.3)

In particular, for Ψ in Lp(Ω),

MΩ∗ε(Ψ 1Ω∗ε) → MΩ(Ψ). (3.4)

To state corrector results, we will make use of the averaging operator U∗ε, the adjoint

of T ∗ε. Its definition and properties are recalled below.

Definition 3.5. For p in [1,+∞], the averaging operator U∗ε: Lp(Ω × Y ∗) �→ Lp(Ω∗

ε) is

defined as

U∗

ε(Φ)(x) =

1

|Y |

�

Y

Φ�ε

�x

ε

�

Y

+ εz,�x

ε

�

Y

�dz a.e. for x ∈ �Ω∗

ε,

0 a.e. for x ∈ Λ∗ε.

Proposition 3.6. (Properties of U∗ε). Suppose that p is in [1,+∞[.

(i) Let {Φε} be a bounded sequence in Lp(Ω×Y ∗) such that Φε � Φ weakly in Lp(Ω×Y ∗).Then

�U∗ε(Φε) �

|Y ∗|

|Y |M

Y ∗(Φ) weakly in Lp(Ω).

(ii) Let {Φε} be a sequence such that Φε → Φ strongly in Lp(Ω× Y ∗). Then

T∗

ε(U∗

ε(Φε)) → Φ strongly in L

p(Ω× Y ∗).

(iii) Let wε be in Lp(Ω∗

ε). Then, the following assertions are equivalent:

(c) T ∗ε(wε) → �w strongly in Lp(Ω× Y ∗) and

�

Λ∗ε

|wε|p→ 0,

(d) �wε − U∗

ε( �w)�Lp(Ω∗ε) → 0.

In [4], the following geometrical hypothesis was introduced for the set Y ∗:

Hypothesis (Hp) The open set Y ∗ satisfies the Poincaré-Wirtinger inequality for theexponent p (p ∈ [1,+∞]) and for every vector bi, i ∈ {1, . . . , n}, of the basis of G, theinterior of Y ∗ ∪ (bi + Y ∗)) is connected.

As a consequence, the following compactness result was proved in [4].

Theorem 3.7. Under Hypothesis (Hp), suppose that wε in W1,p(�Ω∗

ε) satisfies

�wε�W 1,p(�Ω∗ε)≤ C.

Then, there exist w in W1,p(Ω) and �w in Lp(Ω;W 1,p

per(Y ∗)) with MY ∗( �w) ≡ 0, such that,

up to a subsequence,

(i)

�T

∗

ε(wε) → w strongly in L

p

loc(Ω;W 1,p(Y ∗)),

T∗

ε(wε) � w weakly in L

p(Ω;W 1,p(Y ∗)),

(ii) T ∗ε(∇wε) � ∇w +∇y �w weakly in Lp(Ω× Y ∗).

This compactness result is essential for homogenization problems. Since the unfoldingoperator T ∗

εtransforms functions defined on the oscillating domain Ω∗

εinto functions

defined on the fixed domain Ω × Y ∗, there is no need of any extension operator to thewhole of Ω. Therefore, regularity hypotheses on the boundary ∂S insuring the existenceof such extension operators, are not required (contrary to the “classical” methods, cf.references in [4]).

However, contrary to the standard case, the boundedness hypothesis of the previoustheorem is not readily satisfied in the case of the Fredholm alternative. Therefore a moreprecise result is needed.

3.1 A specific result for the Fredholm alternative case

To state this Proposition, we introduce the following notations (see Figure 4), whereρ(Y ) ≥ 2 diam(Y ) is defined in [4]:

�Ωε.=

�x ∈ Ω | dist(x, ∂Ω) > ερ(Y )

�, �Ω∗

ε= Ω∗

ε∩ �Ωε. (3.5)

Figure 4. The set �Ω∗ε(in dark blue)

It is easily seen that �Ωε ⊂ �Ωε, where �Ωε is defined by (1.4). Moreover, for ε small enough,the set �Ωε is connected due to the fact that Ω is connected with Lipschitz boundary.

Proposition 3.8. Suppose that hypothesis (Hp) holds. Then, the set�Ω∗εis included in a

single connected component of Ω∗ε.

Remark 3.9. In [4], it can be seen that if Y is a parallelotop, then ρ(Y ) = 2 diam(Y ).In general, ρ(Y ) can be explicitly computed and is related to the number of b-parallelotopsneeded to cover Y .

Proof of Proposition 3.8 . First note that �Ω∗εis included in the set �Ω∗∗

εintroduced in the

appendix of [4]. Furthermore, �Ω∗∗ε

is connected and is included in Ω∗ε.

If Y is a parallelotop, �Ω∗∗ε

coincides with a set similar to �Ω∗ε(only associated with the

basic cell 2Y ∗). To be precise, it is the interior of the union of cells ε(ξ + Y∗

) such thatε(ξ + 2Y ∗) is included in Ω∗

ε. Otherwise, it is explicitely constructed in the appendix of

[4].

Definition 3.10. Denote by C∗εthe connected component of Ω∗

εcontaining �Ω∗

ε.

In Figure 5, C∗εappears in blue.

Figure 5. Example of a disconnected set Ω∗ε; in blue is the set C∗

ε

Remark 3.11. All the connected components of Ω∗ε, others than C∗

ε, lie near the boundary

of Ω. Therefore, C∗εis the “core” connected component of Ω∗

ε.

We can now state the main result of this section.

Theorem 3.12. Let p be in ]1,+∞] and suppose that hypothesis (Hp) holds. Assumethat wε in W

1,ploc

(C∗ε) satisfies

�∇wε�Lp(C∗ε ) ≤ C and M�Ω∗ε(wε) = 0. (3.6)

Then, up to a subsequence (still denoted ε), there are two functions w in W1,p(Ω) and

�w in Lp(Ω;W 1,pper

(Y ∗)), such that

T∗

ε(wε1�Ωε) � w weakly in L

p(Ω× Y ∗) ∩ Lploc(Ω;W 1,p(Y ∗)),

T∗

ε

�(∇wε) 1C∗ε

�� ∇w +∇y �w weakly in Lp(Ω× Y ∗).

For p = +∞, the weak convergences above are replaced by weak-∗ convergences.

The proof uses two intermediary results, both in search of Poincaré-Wirtinger typeinequalities.

The first, based upon the equivalence of the uniform cone property and the Lipschitzboundary property, concerns the non perforated domains �Ωε.

Lemma 3.13 (Poincaré-Wirtinger inequality for �Ωε). Assume that Ω is a bounded domainwith Lipschitz boundary. Then there exist δ0 > 0 and a common Poincaré-Wirtingerconstant Cp for all the sets

�Ωε for ε ∈]0, δ0], i.e.

∀φ ∈ W1,p(�Ωε), ||φ−M�Ωε(φ)||Lp(�Ωε) ≤ Cp||∇φ||[Lp(�Ωε)],

where C is independent of ε.

The second result concerns perforated domains...

Proposition 3.14. Suppose that hypothesis (Hp) holds.

(a) The restriction w 1�Ω∗ε to�Ω∗εof every function w in W

1,ploc

(C∗ε) with ∇w in Lp(C∗

ε)

belongs to Lp(�Ω∗

ε) (and not only to Lp

loc(�Ω∗

ε)).

Furthermore, there exists a constant Cε such that for every w in W1,ploc

(C∗ε) with ∇w in

Lp(C∗

ε)

�w −M�Ω∗ε(w)�

Lp(�Ω∗ε)≤ Cε�∇w�Lp(C∗ε ).

(b) For every function ϕ in W1,p(�Ω∗

ε), there exist two functions ϕ1 in W 1,p(�Ωε) and

ϕ2 ∈ W 1,p(�Ω∗

ε) and a constant Cp independent of ε, such that

(i) ϕ = ϕ1 + ϕ2, a.e. in �Ω∗ε,

(ii) �ϕ1�Lp(�Ωε) ≤ Cp�ϕ�Lp(�Ω∗ε)

, �∇ϕ1�Lp(�Ωε) ≤ Cp�∇ϕ�Lp(�Ω∗ε)

,

(iii) �ϕ2�Lp(�Ω∗ε)

≤ εCp�∇ϕ�Lp(�Ω∗ε), �∇ϕ

2�Lp(�Ω∗ε)

≤ Cp�∇ϕ�Lp(�Ω∗ε).

(3.7)

Proof. (a) Let w be in W 1,ploc

(C∗ε). By Hypothesis (Hp), it follows that |w|p is integrable

over every periodicity cell ε(ξ+Y ∗) included in C∗εhence over their union and in particular

over the smallest such union covering �Ω∗ε, which we temporarily denote by Z. We now

have �Ω∗ε⊂ Z ⊂ C∗

ε.

Now, w belongs to Lp(Z), and there exists a Poincaré-Wirtinger constant C �εfor

W1,p(Z) (not uniformly bounded!). Consequently,

�w −MZ(w)�Lp(Z) ≤ C�

ε�∇w�Lp(Z). (3.8)

Averaging over �Ω∗ε, and with Jensen’s inequality, this implies

|M�Ω∗ε(w)−MZ(w)| ≤ C

��

ε�∇w�Lp(Z),

and taking this back into (3.8) proves the claim since �Ω∗ε⊂ Z ⊂ C∗

ε.

(b) This corresponds to the decomposition w = Q∗ε(w) +R∗

ε(w), as introduced in Section

3.2.2 of [4], Propositions 3.6 – 3.8.

Combining the last two statements implies that the sets �Ω∗εsatisfy the Poincaré-

Wirtinger inequality with a constant independent of ε.

Theorem 3.15 (Poincaré-Wirtinger inequality for perforated domains). There is a con-stant Cp independent of ε such that for every function ϕ in W

1,p(�Ω∗ε),

�ϕ−M�Ω∗ε(ϕ)�

Lp(�Ω∗ε)≤ Cp�∇ϕ�Lp(�Ω∗ε)

. (3.9)

Proof. Using (b) of Proposition 3.14, we decompose ϕ = ϕ1 + ϕ2 and apply Lemma 3.13to ϕ1:

�ϕ1−M�Ωε(ϕ

1)�Lp(�Ωε) ≤ Cp�∇ϕ

1�Lp(�Ωε).

Taking the average over �Ω∗εgives

�M�Ω∗ε(ϕ1)−M�Ωε(ϕ

1)�Lp(�Ωε) ≤ Cp�∇ϕ

1�Lp(�Ωε),

hence,�ϕ

1−M�Ω∗ε

(ϕ1)�Lp(�Ωε) ≤ 2Cp�∇ϕ

1�Lp(�Ωε) ≤ C

�

p�∇ϕ�

Lp(�Ω∗ε).

Also by (3.7)(iii),�ϕ

2�Lp(�Ω∗ε)

+ |M�Ω∗ε(ϕ2)| ≤ εCp�∇ϕ�Lp(�Ω∗ε),

from which the result follows.

We now can give the proof of Theorem 3.12 recalled here:

Theorem (3.12). Let p be in ]1,+∞] and suppose that hypothesis (Hp) holds. Assumethat wε in W

1,ploc

(C∗ε) satisfies

�∇wε�Lp(C∗ε ) ≤ C, (3.10)

and M�Ω∗ε(wε) = 0. Then, up to a subsequence (still denoted ε), there are two functions

w in W1,p(Ω) and �w in Lp(Ω;W 1,p

per(Y ∗)), such that

T∗

ε(wε1�Ωε) � w weakly in L

p(Ω× Y ∗) ∩ Lploc(Ω;W 1,p(Y ∗)),

T∗

ε

�(∇wε) 1C∗ε

�� ∇w +∇y �w weakly in Lp(Ω× Y ∗).

For p = +∞, the weak convergences above are replaced by weak-∗ convergences.

Proof of Theorem 3.12. According to Proposition 3.14 (a) and (3.10), M�Ω∗ε(wε) exists sothat assuming M�Ω∗ε(wε) = 0 makes sense. By Theorem 3.15, �wε�W 1,p(�Ω∗ε) is bounded. Itfollows that, up to a subsequence, there exist W in Lp(Ω × Y ∗) and F in [Lp(Ω × Y ∗)]n

such thatT

∗

ε(wε1�Ωε) � W weakly in L

p(Ω× Y ∗),

T∗

ε

�(∇wε) 1C∗ε

�� F weakly in Lp(Ω× Y ∗).

(3.11)

But Theorem 3.7 applied in every relatively compact open subset ω of Ω implies that, upto a subsequence, there exist w in W 1,p

loc(Ω) and �w in Lp

loc(Ω;W 1,p

per(Y ∗)) with MY ∗( �w) ≡ 0,

withT

∗

ε(wε1�Ωε) � w weakly in L

p(ω;W 1,p(Y ∗)),

T∗

ε

�(∇wε) 1C∗ε

�� ∇w +∇y �w weakly in Lp(ω;Lp(Y ∗)),

(3.12)

for all ω. The above convergences imply w(x) = W (x, .) and ∇w = MY ∗(F ) in Lp(Ω). Itthen follows that w belongs to W 1,p(Ω) and �w to Lp(Ω;W 1,p

per(Y ∗)).

For p = +∞, the proof is similar, using the weak-∗ compactness criterion in L∞

spaces.

3.2 Unfolding for boundary integrals

We briefly recall the results of [4] concerning boundary integrals.

We suppose that p is in (1,+∞), that ∂S is Lipschitz and has a finite number ofconnected components. The boundary of the set of holes in Ω is ∂Sε ∩ Ω and we denoteby �∂Sε those that belong to the ε-cells which are included in �Ωε.

For a well-defined trace operator to exist from W 1,p(Y ∗) to W 1−1/p,p(∂S), we assumethat each component of ∂S has a Lipschitz boundary. Then, a well-defined trace operatorexists from W 1,p(�Ω∗

ε) to W 1−1/p,p(�∂Sε).

Definition 3.16. For any function ϕ Lebesgue-measurable on ∂�Ω∗ε∩ ∂Sε, the boundary

unfolding operator T bεis defined by

Tb

ε(φ)(x, y) =

φ

�ε

�x

ε

�

Y

+ εy�

a.e. for (x, y) ∈ �Ωε × ∂S,

0 a.e. for (x, y) ∈ Λε × ∂S.(3.13)

The operator T bεhas similar properties as the unfolding operators. In particular, the

integration formula, which reads�

�∂Sεϕ(x) dσ(x) =

1

ε|Y |

�

Ω×∂S

Tb

ε(ϕ)(x, y) dx dσ(y), (3.14)

transforms an integral on the rapidly oscillating set �∂Sε into an integral on a fixed setΩ× ∂S. The integration formula implies

�Tb

ε(ϕ)�Lp(Ω×∂S) = ε

1/p|Y |

1/p�ϕ�

Lp(�∂Sε). (3.15)

The following are Proposition 4.3 and 4.6 of [4].

Proposition 3.17. Suppose that v belongs to W1,p(�Ω∗

ε) and that g is in Lp

�(�∂Sε). Then

����

�∂Sεgv dσ(x)

��� ≤ C��T

b

ε(g)�

Lp� (Ω×∂S)�∇v�Lp(�Ω∗ε)

+1

ε

��M∂S(T bε (g))��Lp

� (Ω)�v�

Lp(�Ω∗ε)

�,

����

�∂Sεgv dσ(x)

��� ≤C

ε1/p�g�

Lp� (�∂Sε)

��v�Lp(Ω∗ε) + ε�∇v�Lp(Ω∗ε)

�.

Proposition 3.18. Let wε be in W1,p(Ω∗

ε). Suppose there exist w in W 1,p(Ω) and �w in

Lp(Ω;W 1,p

per(Y ∗)) such that,

T∗

ε(wε) � w weakly in L

p(Ω;W 1,p(Y ∗)),

T∗

ε(∇wε) � ∇w +∇y �w weakly in Lp(Ω× Y ∗),

(3.16)

with MY ∗( �w)(x) = 0 for a.e. x ∈ Ω. Suppose also that the following two convergences

hold for gε in Lp�(�∂Sε):

Tb

ε(gε) → g strongly in L

p�(Ω× ∂S),

1

εM∂S(T

b

ε(gε)) → G strongly in L

p�(Ω).

(3.17)

Then, �

�∂Sε

gε wε dσ(x) →|∂S|

|Y |

�

Ω

Gw dx+|∂S|

|Y |

�

Ω

M∂S(yMg) ·∇w dx

+1

|Y |

�

Ω×∂S

g �w dxdσ(y),(3.18)

where yM = y −MY ∗(y).

4 A framework for the Neumann problem in perfo-

rated domains with Fredholm alternative

Due to Propositions 3.8 and 3.14, a “natural” candidate for domain on which toconsider the approximate problem for (2.1) is the core set C∗

εintroduced in Definition

3.10. Recall that the boundary of C∗εcan be very irregular. Therefore one cannot make

sense of the condition requiring that the solution and test functions be of zero average onC∗εonly from their gradient. Some appropriate modifications are required together with

appropriate assumptions on the data fε and gε.

For the space, we will use

W (C∗ε).= {ϕ ∈ H1

loc(C∗

ε),∇ϕ ∈ L2(C∗

ε), and M�Ω∗ε(ϕ) = 0}. (4.1)

By (a) of Proposition 3.14, there exists a constant Cε (not necessarily bounded withrespect to ε) such that for all ϕ in W (C∗

ε),

�ϕ�L2(�Ω∗ε)

≤ Cε�∇ϕ�L2(C∗ε ), (4.2)

so that W (C∗ε) is a Hilbert space when endowed with the norm

�ϕ�W (C∗ε ).= �∇ϕ�L2(C∗ε ).

Concerning the data, we make the following assumptions:

�fε�L2(Ω∗ε) is bounded, fε vanishes outside�Ω∗εand satisfies M�Ω∗ε(fε) = 0

�T bε(gε)�L2(Ω×∂S),

1ε

��M∂S(T bε (gε))��L2(Ω)

are both bounded and

gε vanishes outside �Ω∗ε.(4.3)

Remark 4.1. The first condition on gε is equivalent to ε1/2 �gε�L2(�Sε) bounded. The second

condition is more abstract. It implies that

�

�Sεgε dσ is bounded (recall that the superficial

measure |�Sε| is of order 1/ε).

We now consider, the following problem:

Find uε in W (C∗

ε) such that for all ϕ ∈ W (C∗

ε)

�

C∗ε

Aε(x)∇uε(x)∇ϕ(x) dx =

�

�Ω∗εfε(x)ϕ(x) dx+

�

∂ �Sεgε(x)ϕ(x) dσ.

(4.4)

Theorem 4.2 (Existence and a priori estimates for this approximation). Suppose thatHypothesis (H2) is satisfied and (4.3) holds. Then, there is a unique function uε inW (C∗

ε), solution of (4.4).

It satisfies the following estimates with a constant C independent of ε:

�uε�W (C∗ε ) ≤ C(�fε�L2(Ω∗ε) + �Tb

ε(gε)�L2(∂S)). (4.5)

Proof. From Proposition 3.17, and inequality (4.2), it follows that the right hand side

of (4.4),

�

C∗ε

fε(x)ϕ(x) dx +

�

∂C∗ε

gε(x)ϕ(x) dσ, is bounded above by Cε�v�W (C∗ε ) (Cε is a

generic constant depending upon ε).

Existence and uniqueness of the solution uε now follow from the Lax-Milgram theorem.Unless Cε in (4.2) is bounded, this does not give a uniform bound. To obtain it, we returnto the variational inequality and make full use of Hypothesis (4.3), namely, the fact thatfε and gε vanish outside �Ω∗ε. Therefore,

α�∇uε�2L2(C∗ε )

≤ C(�uε�L2(�Ω∗ε)

+ �∇uε�L2(�Ω∗ε)).

Since M�Ω∗ε(uε) = 0, by Proposition 3.15, �uε�L2(�Ω∗ε) ≤ C2�∇uε�L2(�Ω∗ε), from which (4.5)follows.

Note that there is a Fredholm correction here which is

�

∂ �Sεgε(x) dσ.

2

5 Homogenization result 1: the unfolded limit

Theorem 5.1. Suppose that hypotheses (H2) and (4.3) hold. Assume furthermore that(3.17) is satisfied (for p = 2) by gε (which defines g and G), and that the extension by 0of fε to Ω, denoted �fε, satisfies

�fε �|Y ∗|

|Y |f weakly in L

2(Ω) (5.1)

(following the normalization of Proposition 3.3 (iii)).

Assume that for some matrix A,

T∗

ε

�A

ε�→ A a.e. in Ω× Y ∗.

Let uε be the solution of Problem (4.4). Then, there exist u in V (Ω) and �u in L2(Ω;H1per(Y ∗)),satisfying

(i) T ∗ε((uε) 1�Ω∗ε) � u weakly in L

2(Ω;H1(Y ∗)),

(ii) T ∗ε(∇uε1C∗ε ) � ∇u+∇y�u weakly in L

2(Ω× Y ∗),(5.2)

and the pair (u, �u) is the unique solution in of the problem

MΩ(u) = 0, MY ∗(�u) = 01

|Y |

�

Ω×Y ∗A(x, y)

�∇u(x) +∇y�u(x, y)

��∇Ψ(x) +∇yΦ(x, y)

�dxdy

=

�

Ω

�|Y ∗|

|Y |f(x) +

|∂S|

|Y |G(x)

�Ψ(x) dx

+|∂S|

|Y |

�

Ω

M∂S(y g(y)) ·∇Ψ(x) dx+1

|Y |

�

Ω×∂S

g(x, y)Φ(x, y) dx dσ(y),

∀Ψ ∈ V (Ω), ∀Φ ∈ L2(Ω; H1per

(Y ∗)),

(5.3)

Proof. Using estimates (4.5), convergences (5.2) follow from Theorem 3.12. This is up toa subsequence, but as usual, the uniqueness of the solution of the limit problem impliesthat the whole sequence converges.

At this point, the proof almost follows the standard way of using unfolding, using testfunctions of the form

vε(x).= Ψ(x) + ε ϕ(x)ψ

�x

ε

�−M�Ω∗ε

�ε ϕ(x)ψ

�x

ε

��

where Ψ is in C∞(Ω) with MΩ(Ψ) = 0, ϕ is in D(Ω) and ψ = ψ(y) is in H1per(Y∗).

vε(x).= Ψ(x) + ε ϕ(x)ψ

�x

ε

�−M�Ω∗ε

�ε ϕ(x)ψ

�x

ε

��

where Ψ is in C∞(Ω) with MΩ(Ψ) = 0, ϕ is in D(Ω) and ψ = ψ(y) is in H1per(Y∗). It

then follows by Proposition 3.3 (i) and (3.4) that

T∗

ε(vε) → Ψ strongly in L

2(Ω× Y ∗),

T∗

ε

�ϕψ

�·

ε

��→ Φ strongly in L2(Ω× Y ∗), with Φ(x, y) = ϕ(x)ψ(y).

(5.4)

Since∇vε(x) = ∇Ψ(x) + ϕ(x)(∇yψ)

�x

ε

�+ ε∇ϕ(x)ψ

�x

ε

�,

by Proposition 3.3 (i),

T∗

ε(∇vε) → ∇Ψ+∇yΦ strongly in L

2(Ω× Y ∗). (5.5)

This gives�

C∗ε

Aε∇uε ∇v

εdx =

1

|Y |

�

Ω×Y ∗T

∗

ε(Aε)T ∗

ε

�(∇uε)1C∗ε

�T

∗

ε(∇vε) dx dy

+

�

Λ∗ε∩C∗ε

Aε∇uε ∇v

εdx

→1

|Y |

�

Ω×Y ∗A(x, y)

�∇u0(x) +∇y�u0(x, y)

��∇Ψ(x) + φ(x)∇ψ(y)

�dx dy,

(5.6)

since ���

Λ∗ε∩C∗ε

Aε∇uε ∇v

εdx

�� ≤ C|Λ∗ε|1/2

�∇vε�L∞(Ω)�∇uε�L2(Ω∗ε) → 0

Concerning the right-hand side of (2.1), we make use of Proposition 3.18 for theboundary integral and of

limε→0

�

Ω∗ε

fε vε dx = limε→0

�

Ω

�fε vε dx =|Y ∗|

|Y |

�

Ω

f Ψ dx, ∀Ψ ∈ C∞(Ω). (5.7)

Equation (5.3) follows from the density of the functions of C∞(Ω) with zero averagein V (Ω) and that of the tensor product D(Ω)⊗H1

per(Y ∗) in L2(Ω;H1

per(Y ∗)).

Finally, by Corollary 3.4, MΩ(u) = 0, which implies the uniqueness for the solutionof (5.3).

6 Homogenization result 2: the homogenized prob-

lem

Classically, the homogenized limit problem can now be made explicit, making use ofthe so-called correctors χj, j = 0, . . . , n, which are solutions of the following cell-problems.

The cases j = 1, . . . , n are classical: the functions χj ∈ L∞(Ω;H1per(Y∗)), are, for a.e.

x in Ω, the solutions of the cell problems

−

n�

i,k=1

∂

∂yi

�aik(x, y)

�∂χj(x, y)∂yk

− δjk

��= 0 in Y ∗,

n�

i,k=1

aik(x, y)�∂χj(x, y)

∂yk− δjk

�ni = 0 on ∂S,

MY ∗(χj)(x, ·) = 0, χj(x, ·) Y -periodic.

(6.1)

We consider also the function χ0 in L2(Ω;H1per(Y∗)) which is the solution of

−

n�

i,k=1

∂

∂yi

�aik(x, y)

∂χ0(x, y)

∂yk

�= 0 in Y ∗,

n�

i,k=1

aik(x, y)∂χ0(x, y)

∂ykni = g(x, y) on ∂S,

MY ∗(χ0)(x, ·) = 0, χ0(x, ·) Y -periodic.

(6.2)

Remark 6.1. It is classical that problems (6.1) each have a unique solution since thecorresponding Fredholm condition is satisfied:

�

∂S

ai,j(x, y)ni dσ(y)−

�

∂S

n�

i=1

∂

∂yiai,j(x, y) dy = 0.

Concerning problem (6.2), the corresponding Fredholm condition is simply�

∂S

g(x, y) dσ(y) = 0,

which is satisfied as a consequence of the two conditions of hypothesis (3.17).

Theorem 6.2. The homogenized strong formulation associated with Theorem 5.1, under

the hypotheses therein, is

−div (A0∇u) = f(x) +|∂S|

|Y ∗|

�G(x)−

�

Ω

G(x) dx�− div G in Ω,

A0∇u · n = G · n on ∂Ω,

MΩ(u) = 0,

(6.3)

where

G(x).=

|∂S|

|Y ∗|M∂S(yMg)(x)−MY ∗

�A(x, ·)∇yχ0(x, ·)

�in Ω,

which belongs to L2(Ω).

The homogenized matrix A0 = (a0

ij)1≤i,j≤n is the standard one: it is explicitely defined

by the following formula and is elliptic:

a0ij= MY ∗

�aij −

n�

k=1

aik∂χj

∂yk

�= MY ∗(aij)−MY ∗

� n�

k=1

aik∂χj

∂yk

�. (6.4)

Proof. Taking Ψ = 0 in problem (5.3) yields

�u(x, y) =n�

i=1

∂u

∂xi(x)χi(x, y) + χ0. (6.5)

Inserting into (5.3) with Φ = 0 gives

1

|Y |

�

Ω×Y ∗A(x, y)

�∇u(x) +

n�

i=1

∂u

∂xi(x)∇yχi(x, y) +∇yχ0

�∇Ψ(x)dxdy

=

�

Ω

�|Y ∗|

|Y |f(x) +

|∂S|

|Y |G(x)

�Ψ(x) dx+

|∂S|

|Y |

�

Ω

M∂S(y g(y)) ·∇Ψ(x) dx.(6.6)

The result follows by integrating over Y ∗, then interpreting the problem as for (0.6).

Remark 6.3. Here, the Fredholm correction is simply|∂S|

|Y |

�

Ω

G(x), since

�

Ω

f = 0

and

�

Ω

divG dx =

�

∂Ω

G · n dσ. It is indeed the limit of the Fredholm correction for the

approximate problems,

�

∂ �Sεgε(x) dσ.

Remark 6.4. The contribution of gε at the limit appears in particular as a non homoge-

neous Neumann condition on ∂Ω. In the “classical” case where

gε(x) = g({x/ε}Y ) 1�Sε with M∂S(g) = 0,

where g is in L2(∂S), one can see that g(x, y) ≡ g(y) while G ≡ 0. If furthermore A does

not depend upon x, then G ≡ 0.

7 Homogenization result 3: the corrector result

As is common with the unfolding method, under an additional hypothesis on fε onealso obtains the strong convergence of the energy:

Proposition 7.1. Suppose that hypotheses of Theorem 5.1 hold. Assume furthermore

that

T∗

ε(fε) is compact in L

2(Ω× Y ∗). (7.1)

Then

limε→0

�

C∗ε

Aε∇uε∇uε dx =

1

|Y |

�

Ω×Y ∗A�∇u+∇y�u

� �∇u+∇y�u

�dx dy (7.2)

Consequently,

T∗

ε(∇uε1C∗ε ) → ∇u+∇y�u strongly in L

2(Ω× Y ∗). (7.3)

Proof. Let F be a strong limit point of T ∗ε(fε) in L2(Ω× Y ∗) (which exits by assumption

(7.1)). According to Proposition 3.3 (iii), MY ∗(F ) = f .

Applying Proposition 3.2 (i), it follows that�

�Ω∗εfε uε dx =

1

|Y |

�

Ω×Y ∗T

∗

ε(fε)T

∗

ε(uε) dx dy.

Consequently,

limε→0

�

�Ω∗εfε uε dx =

|Y ∗|

|Y |

�

Ω

f u dx

for the considered subsequence associated to F . But this result holds independently ofF , hence holds for the whole sequence {ε}.

Furthermore, Proposition 3.18 implies that�

�∂Sεgε uε dσ(x) →

|∂S|

|Y |

�

Ω

Gudx+|∂S|

|Y |

�

Ω

M∂S(yMg) ·∇u dx

+1

|Y |

�

Ω×∂S

g �u dxdσ(y),(7.4)

The proof of (7.2) is now complete, since, on the one hand, by 4.4,�

C∗ε

Aε∇uε∇uε dx =

�

�Ω∗εfε uε dx+

�

�∂Sεgε uε dσ(x),

while on the other, by (5.3),

1

|Y |

�

Ω×Y ∗A�∇u+∇y�u

� �∇u+∇y�u

�dx dy =

�

Ω

�|Y ∗|

|Y |f(x) +

|∂S|

|Y |G(x)

�u(x) dx

+|∂S|

|Y |

�

Ω

M∂S(y g(y)) ·∇u(x) dx+1

|Y |

�

Ω×∂S

g(x, y) �u(x, y) dx dσ(y).(7.5)

The strong convergence (7.3) follows now by Lemma 5.8 of [4], taking into accountthat Aε 1C∗ε converges almost everywhere to A.

Corollary 7.2 (Corrector results). As ε → 0,

���∇uε −∇u−n�

i=1

U∗

ε

�∂u

∂xi

�U

∗

ε(∇yχi)

���L2(C∗ε )

→ 0. (7.6)

In the case where the matrix field A does not depend on x, the following corrector result

holds: ���uε − u− εn�

i=1

Qε

�∂u

∂xi

�χi

��·

ε

�

Y

����H1(�Ω∗ε)

→ 0. (7.7)

Proof. By construction, for i = 1, . . . , n, the function χi belongs to L∞(Ω;H1(Y ∗)). Dueto convergence (7.3) (note that Λε ∩ �Ω∗ε = ∅) , Proposition 3.6 (iii) gives

��∇uε1C∗ε − U∗

ε(∇u+∇y�u)

��L2(Ω∗ε)

→ 0. (7.8)

By Proposition 3.3 (i) and (6.5) this implies

���∇uε1C∗ε −∇u−n�

i=1

U∗

ε

�∂u

∂xi∇yχi

����L2(Ω∗ε)

→ 0, (7.9)

and gives convergence (7.6).

If A does not depend on x, this becomes

���∇uε1C∗ε −∇u−n�

i=1

U∗

ε

�∂u

∂xi

�∇yχi

��·

ε

�

Y

����L2(Ω∗ε)

→ 0, (7.10)

From (7.10), the proof of convergence (7.7) follows along the lines of that (5.31) in [4],taking also into account the fact that M�Ω∗ε(uε) = 0 and (3.9).

8 Appendix: Proof of Lemma 3.13

The notation Ωδ in this section corresponds to �Ωε of the previous sections.

Let p in (1,+∞). From now on, Ω is a bounded open set with Lipschitz boundary inRn . It is well-known (see [3]) that this is equivalent to the fact that Ω has the uniformcone property which we recall below. In this definition, a cone is the convex envelope ofthe union of the origin and a closed ball which does not contain the origin.

Definition 8.1. The bounded open set Ω has the uniform cone property whenever thereexists a finite open cover {Uj}1≤j≤N of ∂Ω, and a corresponding family {Cj} of cones,each isometric to some fixed cone C, such that,

(i) for some strictly positive constant δ1,

�x ∈ Ω | dist(x, ∂Ω) < δ1

�⊂

N�

j=1

Uj,

(ii) for every j,

�

x∈Ω∩Uj

�x+ Cj

�⊂ Ω.

Let Ωδ denote the set

Ωδ =�x ∈ Ω | dist(x, ∂Ω) > δ

�.

Lemma 8.2. There exists δ0 > 0 such that all the sets Ωδ for δ ∈ (0, δ0], satisfy a uniformcone property (with a cone which depends only on δ0).

Proof. Since Ω has the uniform cone property, there exist δ1 and a finite open cover{Uj}1≤j≤N of ∂Ω satisfying (i) and (ii) of Definition 8.1. By Lebesgue’s lemma, there isa δ3 > 0 (the Lebesgue number of the open cover) such that for each x of ∂Ω, the ball ofcenter x and radius δ3 is included in at least one of the {Uj}1≤j≤N . Set also

δ0.=

1

3min

�δ1, δ3

�.

We use the notations of the figure below.

Consider the following open sets:

j ∈ {1, . . . , N}, U�

j=

�x ∈ Uj | dist(x, ∂Uj) > δ0

�.

By the choice of δ3, if follows that the family {U�j}1≤j≤N still covers ∂Ω, and actu-

ally covers the (23δ3)-neighborhood of ∂Ω. Consequently, for δ ∈ (0, δ0] the set�x ∈

Ωδ | dist(x, ∂Ωδ) < δ0�is included in

N�

j=1

U�

j. This is condition (i) of definition 8.1.

To prove condition (ii), set C�= 13 C, C

�j= 13Cj (which is isometric to C

�) and let j be

in {1, . . . , N}, x in Ωδ ∩ U�j. One can see that

Vj.= {z ∈ Rn, dist(z, x+ C �

j) < δ} ⊂

�

y∈B(x;δ)

�y + Cj

�.

(the Orange set is included in the Green set)

Furthermore, due to the uniform cone property the latter is included in Ω. Thus theset x+ C

�jis included in Ωδ. This proves the inclusion

�

x∈Ωδ∩U�j

�x+ C

�

j

�⊂ Ωδ,

and the proof of Lemma 8.2 is complete.

Corollary 8.3. From Lemma 8.2, for any ε ∈ (0, δ0] and each p ∈ (1,∞), there existsa continuous extention operator Pε from W

1,p(Ωε) into W 1,p(Ω), i.e. such that for allφ ∈ W 1,p(Ωε)

Pε(φ) ∈ W1,p(Ω), Pε(φ)|Ωε = φ.

The norm �Pε� is bounded uniformly with respect to ε.

Proof. This follows from the standard construction of the extension operator by local-ization using a partition of unity and by making use of an off-centered approximation ofidentity convolution kernel on each subdomain.

We can now give the proof of Lemma 3.13.

Proof of Lemma 3.13. Since Ω is bounded and connected with Lipschitz boundary, it hasa Poincaré-Wirtinger constant C0. Let φ be inW 1,p(Ωε). We apply the Poincaré-Wirtingerinequality to the extension Pε(φ) of φ to get

||Pε(φ)−MΩ�Pε(φ)

�||Lp(Ω) ≤ C0||∇Pε(φ)||[Lp(Ω)]n ≤ C0�Pε�||∇φ||[Lp(Ωε)]n .

Here MΩ�Pε(φ)

�is the mean value of Pε(φ) over Ω. Let C be a common upper bound

for 2C0�Pε�, ε ∈ (0, δ0]. Then,

||φ−MΩ

�Pε(φ)

�||Lp(Ωε) ≤ ||Pε(φ)−MΩ

�Pε(φ)

�||Lp(Ω) ≤

1

2C||∇φ||[Lp(Ωε)]n .

Taking the mean value over Ωε implies ||MΩ(φ)−MΩ�Pε(φ)

�||Lp(Ωε) ≤

12C||∇φ||[L

p(Ωε)]n ,which, when added to the previous inequality, ends the proof.

We end this section by a result which could be of interest by itself.

Lemma 8.4. Under the same hypotheses, assume that 1 ≤ p < ∞ and let {uε} be asequence of functions belonging to W

1,p(Ωε) satisfying

||uε||W 1,p(Ωε) ≤ C,

where C is a finite constant. Then (up to a subsequence) there is a function u ∈ W 1,p(Ω)with

�uε −→ u strongly in Lp(Ω).

Proof. We start by extending uε by using the extension operator Pε. So there exist asubsequence, still denoted ε, and a function u ∈ W 1,p(Ω) such that

Pε(uε) � u weakly in W1,p(Ω),

Pε(uε) −→ u strongly in Lp(Ω).

Since the measure |Ω \ Ωε| goes to zero,�

Ω

|u− �uε|p =�

Ωε

|u− Pε(uε)|p +

�

Ω\Ωε

|u|p

→ 0,

which concludes the proof.

References

[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1965.

[2] G. Allaire and F. Murat, Homogenization of the homogeneous Neumann problemwith nonisolated holes, including Appendix with A.K. Nandakumar, AsymptoticAnalysis 7 (1993), 81–95.

[3] D. Chenais, On the existence of a solution in a domain identification problem,Journal Math. Anal. and Appl., 52 (1975), 189-219.

[4] D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodicunfolding method in domains with holes, submitted.

[5] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogeniza-tion, C. R. Acad. Sci. Paris, Série 1, 335 (2002), 99–104.

[6] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method inhomogenization, SIAM J. of Math. Anal. Vol. 40, 4 (2008), 1585-1620.

[7] G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot.Anal., 40 (2004), 269–286.

[8] F. John, Rotation and strain, Comm. Pure Appl. Math. 14, pp. 391-413, 1961.

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