HOMOGENIZATION OF ELLIPTIC PROBLEMS IN PERFORATED DOMAINS

on his 70th birthday

HOMOGENIZATION OF ELLIPTIC PROBLEMS IN PERFORATED DOMAINS

WITH MIXED BOUNDARY CONDITIONS

DOINA CIOR€NESCU and A. OULD HAMMOUDA

We consider a class of second order elliptic problems on perforated domains with small holes of sizeεδ, distributed periodically with the period ε. A non homo- geneous Neumann condition is prescribed on the boundary of some holes; on the boundary of the other ones, it is a homogeneous Dirichlet condition that is con- sidered. We are interested here to give the limit behaviour of the problems when ε→0andδ=δ(ε)→0. To do so, we apply the periodic unfolding method intro- duced in [5], that allows us to consider general operators with highly oscillating (withε) coecients and rather general geometries.

AMS 2000 Subject Classication: Primary 49J45; Secondary 35B27, 74Q05.

Key words: homogenization of Neumann problems, perforated domain with small holes, periodic unfolding method.

1. INTRODUCTION

The aim of this work is to apply the periodic unfolding method introduced by Cioranescu, Damlamian and Griso [5] to the homogenization of a class of elliptic second-order equations with highly oscillating coecients, in perforated domains in Rⁿ(n >2), with mixed-type conditions on the boundary of holes.

The holes we consider here are ε-periodically distributed and their size r(ε)is such that r(ε)/ε→ 0. Throughout the paper, we call such holes small holes.

We consider the case where in each period there are two dierent kinds of holes:

some are of size of order ofεδ₁, and the other ones of size of order ofεδ₂ with δ1 =δ1(ε) and δ2 =δ2(ε),δ1 → 0 and δ2 → 0 asε→ 0. On the boundary of holes of sizeεδ1 a non homogeneous Neumann condition is prescribed while on the boundary of holes of size εδ₂ and on the exterior boundary of the domain, a homogeneous Dirichlet condition is imposed.

The asymptotic behaviour of the homogeneous Dirichlet problem for the Poisson equation in perforated domains with holes of sizeε^k, k >0, was studied

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 56, 389406

by Cioranescu and Murat [11]. They showed that the size ε^n/n−2 is critical in the following sense: the limit problem not only contains the Laplacian but also an additional zero order term, called strange term in [11], depending on the capacity of the set of holes in the limit. Conca and Donato [12] studied the non homogeneous Neumann problem for the Laplacian in the same geometrical framework. Now, the critical size is of order ε^n/n−1, and the contribution of the holes in the limit is reected by an additional right-hand side integral term.

The case of mixed-type boundary conditions on the holes has been stud- ied by Cardone, D'Apice and De Maio [2]. They consider the following setting:

the size of the holes is of order of ε and a homogeneous Neumann bound- ary condition is assumed on their boundaries, except on a at portion of size ε^n/n−2, where a homogeneous Dirichlet condition is prescribed. In the limit problem, as expected, the strange term appears again. A related problem was studied by Corbo Esposito, D'Apice and Gaudiello [13] where the holes are now of the critical size ε^n/n−1. A non homogeneous Neumann condition is imposed on the boundary of each hole, except a at portion of size ε^n/n−2, where a homogeneous Dirichlet condition is given. In the limit problem the two additional terms appear, the one giving the contribution of the Neumann condition as in [12], and the strange term corresponding to the critical size for Dirichlet conditions from [11]. In all these papers, standard variational homogenization methods are used. In particular, they need to introduce ex- tension operators (since the domains are changing with ε) and to construct test functions, specic for each situation.

In the present paper we take the advantage of the simplicity of the pe- riodic unfolding when applied to perforated domains, as it can be seen in Cioranescu, Donato and Zaki [8][10]. Indeed, the periodic unfolding, being a xed-domain method, no extension operator is needed. Moreover, it does not use any construction of special test functions and so, one can treat general second order operators with highly oscillating (inε) coecients, which was not the case in the papers quoted above.

The standard case of homogenization in perforated domains, i.e., with holes of size εwas studied via the periodic unfolding method in [8][10] where Robin or nonlinear boundary conditions were treated. To do so, a bound- ary unfolding operator was introduced and studied. For small holes, applica- tions of the same method to sieve type problems, can be found in Cioranescu, Damlamian, Griso and Onofrei [7] which also contains a complete list of the properties of the unfolding operator for xed domains, for perforated domains with holes of size εδ (δ is another small parameter), and of a boundary un- folding operator corresponding to these small holes. Let us mention que the unfolding operator for small holes appeared for the rst time (as a change of variables) in Casado-Díaz [3] and a boundary layer unfolding operator for

this case in Onofrei [16]. The situation from [12], of small holes of size ε^n/n−1 with non homogeneous Neumann conditions, was treated by unfolding in Ould Hammouda [15]. Our results here rely extensively on this last work.

The paper is organized as follows. In Section 2, following [5], [14] and [7], we recall the denitions and properties of the unfolding operator T_ε for xed domains and of the unfolding operator T_εδ for domains with small holes.

We also recall the properties of a boundary unfolding operator T_εδ^b that was introduced in [15]. Section 3 is devoted to the setting of the problem and to the proof of our main homogenization results, Theorems 3.2 and 3.3. We show that if δ₁ and δ₂ are chosen in order to get the critical size corresponding to Neumann, respectively, to Dirichlet small holes, the limit problem contains the two contributions of the holes, an additional right hand side term, and a strange term. Due to the oscillating character of the coecients in the original problem, in the homogenized equation, the partial dierential operator with constant coecients, is the standard homogenized one (see, for instance, Bensoussan, Lions and Papanicolaou [1]).

2. UNFOLDING OPERATORS 2.1. General notation

We start by introducing some general notation, in particular the denition of perforated domains with small holes, the geometric setting in this paper.

LetΩbe a bounded open set inRⁿsuch that|∂Ω|= 0, andY =

−¹₂,¹₂n

the periodicity (or reference) cell. Let now introduce the notation (2.1) Ωbε=interiorn [

ξ∈Ξ_ε

ε ξ+Yo

, whereΞε={ξ∈Zⁿ, ε(ξ+Y)⊂Ω}, and setΛε= Ω\Ωbε. The setΩbεis the largest nite union ofεY cells contained inΩwhileΛεis the subset ofΩcontaining the parts fromεY cells intersecting the boundary ∂Ω (see Figure 1).

Let B be an open set such that B ⊂⊂ Y. Introduce now the set Y_δ = Y \ δB¯ supposed to be connected;Y_δ correspond to the part occupied by the material in the cell Y.

The set B_ε,δ ofε-periodic holes of sizeεδ is dened as B_ε,δ= [

ξ∈Zⁿ

ε(ξ+δB).

The perforated domain Ω_ε,δ, with holes of sizeεδ is dened as

(2.2) Ω_ε,δ =n

x∈Ω

nx ε

o∈Y_δo .

Λε Ωε

Fig. 1. The domainsΩbεandΛε.

If{e₁, . . . , e_n} is the canonical basis of Rⁿ, for any z∈Rⁿwe denote by [z]_Y the unique integer combination Pn

j=1`_je_j such that z−[z]_Y belongs to Y. Set{z}_Y =z−[z]_Y ∈Y a.e. forz ∈Rⁿ (see for more details, [5], [6] and [14]). Then for each x∈Rⁿ, one has

x=εhx ε i

Y +nx ε

o

Y

a.e. forx∈Rⁿ (see Figure 2).

Fig. 2. The domainsΩbεandΛε.

2.2. The unfolding operator T_ε for xed domains

In this section we recall the general properties of the periodic unfolding operator introduced in [5], for more details see [7], and [14].

Denition 2.1. ForφLebesgue-measurable onΩ, the unfolding operator T_ε is dened as:

T_ε(φ)(x, y) = ( φ

εhx ε i

Y +εy

a.e. for (x, y)∈Ωb_ε×Y, 0 a.e. for (x, y)∈Λε×Y.

It is obvious from (2.1) that forv and wLebesgue-measurable we have (2.3) T_ε(vw) =T_ε(v)T_ε(w).

Proposition 2.2. Letφbe measurable on Y and extend it by periodicity to the whole of Rⁿ. Set

φ_ε(x) =fx ε

a.e. for x∈Rⁿ. Then

T_ε(φ_ε|_Ω)(x, y) =

( φ(y) a.e. for (x, y)∈Ωb_ε×Y, 0 a.e. for (x, y)∈Λε×Y.

If φbelongs to L^p(Y), p∈[1,+∞[, and if Ω is bounded, then T_ε(φ_ε|_Ω)→φ d strongly in L^p(Ω×Y).

Let {v_ε} be a bounded sequence in L^p(Ω)such that v_ε →v strongly inL^p(Ω). Then

T_ε(vε)→v strongly inL^p(Ω×Y).

Remark 2.3. Observe that an equivalent way to dene φ_ε on Rⁿ, is to take simply

φ_ε(x) =φnx ε

o

Y

.

Proposition 2.4. Let p in [1,+∞[and v∈L^p(Ω). Then Z

Ω×Y

T_ε(v)(x, y) dxdy= Z

Ω

v(x) dx− Z

Λε

v(x) dx= Z

Ωbε

v(x) dx.

As a consequence of Proposition 2.4, (2.4)

Z

Ω

vdx− Z

Ω×Y

T_ε(v) dxdy

≤ Z

Λε

|v|dx,

and so, any integral of a function vonΩ, is almost equivalent to the integral of its unfolded on Ω×Y. The integration defect only comes from the cells intersecting the boundary∂Ωand is controlled by the right hand side integral

in (2.4). This remark led in [5] the introduction of the so-called unfolding criterion for integrals (u.c.i.):

Proposition 2.5. If{φ_ε}is a sequence inL¹(Ω)satisfyingR

Λε|φ_ε|dx→ 0,then

Z

Ω

φεdx− Z

Ω×Y

T_ε(φε) dxdy→0, property that is denoted

Z

Ω

φ_εdx '^Tε Z

Ω×Y

T_ε(φ_ε) dxdy.

This immediately justies the next result, essential when dealing with homogenization problems.

Proposition 2.6. Let {u_ε} be a bounded sequence in L^p(Ω) with p ∈ ]1,+∞]andv in L^p0(Ω) (1/p+ 1/p⁰ = 1). Then

(2.5)

Z

Ω

uεvdx '^Tε Z

Ω×Y

T_ε(uε)T_ε(v) dxdy.

Assume now that ∂Ωis bounded. Let{u_ε}be a bounded sequence inL^p(Ω)and {v_ε} a bounded sequence in L^q(Ω)with 1/p+ 1/q <1. Then

(2.6) Z

Ω

u_εv_εdx '^Tε Z

Ω×Y

T_ε(u_ε)T_ε(v_ε) dxdy.

The main results concerning the unfolding operatorT_ε is as follows Proposition 2.7. Let {w_ε} be a sequence in H¹(Ω) such that w_ε * w weakly in H¹(Ω). Then, up to a subsequence, there exists wb∈L² Ω;H_per¹ (Y) such that

(2.7) T_ε(∇w_ε)*∇_xw+∇_ywb weakly in L²(Ω×Y).

2.3. The unfolding operator T_ε,δ depending on two parameters ε and δ

In the next section we will consider domains perforated by small holes of size εδ, periodically distributed with period εY. This geometry of domains requires the introduction of a new unfolding operator T_ε,δ depending on both parameters ε and δ. As mentioned above, this operator was rst introduced in [3]. It was used for the study of reticulated structures in [4] and for sieve problems in [7].

Denition 2.8. For φ∈ L^p(Ω), p ∈ [1,∞[, the unfolding operator T_ε,δ : L^p(Ω)→L^p(Ω×Rⁿ) is dened as

T_ε,δ(φ)(x, z) =

( T_ε(φ)(x, δz) a.e. for(x, z)∈Ωb_ε×¹_δY,

0 otherwise.

Foru∈L²(Ω), from Denition 2.8 the estimates

(2.8)

(i) kT_ε,δ(u)k²_L2(Ω×

Rⁿ)≤ 1

δⁿkuk²_L2(Ω), (ii)

Z

Ω

udx−δⁿ Z

Ω×Rⁿ

T_ε,δ(u) dxdz

≤ Z

Λε

|u|dx, are straightforward.

The operator T_ε,δ was studied in details in [7]. To recall its properties (that will be widely used in the present paper), we need to introduce the notion of local average of a function.

Denition 2.9. The local averageM_Y^ε :L^p(Ω)7→L^p(Ω)is dened for any φ inL^p(Ω),1≤p <∞,as

M_Y^ε(φ)(x) = Z

Y

T_ε(φ)(x, y) dy.

It is classical that if{v_ε}is a bounded sequence inL^p(Ω)such thatv_ε→v strongly in L^p(Ω), then

(2.9) M_Y^ε(vε)→v strongly in L^p(Ω).

Now, we can list some of the properties of T_ε,δ from [7], needed in the next section.

Proposition 2.10. Supposen≥3and denote by2^∗ the Sobolev exponent

2n

n−2 associated with 2. Let ω be open and bounded in Rⁿ. Then k∇_z T_ε,δ(u)

k²

L²(Ω×¹

δY) ≤ ε²

δⁿ⁻²k∇uk²_L2(Ω), kT_ε,δ u−M_Y^ε(u)

k²_L2(Ω;L^2∗(Rⁿ))≤ Cε²

δⁿ⁻²k∇uk²_L2(Ω), kT_ε,δ u)k²_L2(Ω×ω) ≤ 2Cε²

δⁿ⁻²|ω|ⁿ²k∇uk²_L2(Ω)+ 2|ω| kuk²_L2(Ω), where C denotes the Sobolev-Poincaré-Wirtinger constant for H¹(Y).

An unfolding criterion for integrals also holds forT_ε,δ.

Proposition 2.11. If{w_ε}is a sequence inL¹(Ω)such thatR

Λε|w_ε|dx→ 0, then

(2.10) Z

Ω

w_εdx

Tε,δ

' δⁿ Z

Ω×Rⁿ

T_ε,δ(w_ε) dxdz.

If {u_ε} is bounded in L²(Ω)and {v_ε} is bounded in L^p(Ω)with p >2, then

(2.11) Z

Ω

u_εv_εdx

Tε,δ

' δⁿ Z

Ω×Rⁿ

T_ε,δ(u_ε)T_ε,δ(v_ε) dxdz.

2.4. The boundary unfolding operator T_ε,δ^b

We again use the notation from Section 2.1. From now on, we suppose that the setBhas a Lipschitz boundary. We dene a linear unfolding operator on the boundary of the holesB_ε,δ, specic to the case of domains with volume- distributed very small holes.

Denition 2.12. Let φ ∈ L^p(∂B_ε,δ), with p ∈ [1,+∞[. The boundary unfolding operator T_ε,δ^b is dened as

(2.12) T_ε,δ^b (φ)(x, z) =φ εhx

ε i

Y +εδz

a.e. for x∈Rⁿ, z ∈∂B.

For holes of size of order ofε(i.e., withδ= 1), such an operator, denoted T_ε^b, was introduced for the rst time in [8], its denition is exactly (2.12) with δ = 1. Most of the properties of T_ε,δ^b are almost transcriptions of the corresponding ones of T_ε^b and are obtained by a simple change of variables.

For more details we refer the reader to [15].

Let g belong to L²(∂B). Denote by M_∂B(g) its mean value on ∂B, namely,

M_∂B(g) = 1

|∂B|

Z

∂B

gds.

We now recall the following two propositions from [15], needed later on.

Proposition 2.13. Letφ∈L²(∂B_ε,δ). Then Z

∂Bε,δ

φ(x)ds= δⁿ⁻¹ ε

Z

Rⁿ×∂B

T_ε,δ^b (φ^ε,δ)(x, z) dxds.

Proposition 2.14. Letg∈L²(∂B) and set g_ε(x) =g1

δ nx

ε o

, for all x∈∂B_ε,δ. Then

(2.13)

Z

∂Bε,δ

gε(x)φds

≤Cδⁿ⁻¹

ε |M_∂B(g)|+εδ

k∇φk_(L2(Ω))^N.

Moreover, for all φ∈H¹(Ω), as ε→0, one has the convergence

(2.14) ε

δ^N−1 Z

∂B_ε,δ

gε(x)φ(x) ds→ |∂B|M_∂B(g) Z

Ω

φ(x) dx.

3. HOMOGENIZATION RESULTS 3.1. Setting of the problem

As in Section 1, Ωis a bounded open set in Rⁿ such that |∂Ω|= 0. Let B and T be two open sets such that B ⊂⊂ Y,T ⊂⊂ Y and B∩T =∅. The part occupied by the material in the cell Y is now Yδ1δ2 = Y \ δ1T ∪δ2B supposed to be connected. Here, δ₁ and δ₂ are two small parameters going, to zero independently. The perforated domain Ω_ε,δ1δ2 where we will set the problem is obtained by removing fromΩthe set of holesB_εδ1 andT_εδ2, namely, (3.1) B_ε,δ1 = [

ξ∈Zⁿ

ε(ξ+δ1B), T_ε,δ2 = [

ξ∈Zⁿ

ε(ξ+δ2T).

An example of such a geometry is depicted in Figure 3 below.

. B

Y

δ₁B .

Ω

δ₂T T

Fig. 3. The perforated domainΩε,δ1δ2.

This means thatΩε,δ1δ2 has perforations of size of order ofεδ1 and of size of orderεδ₂ at the same time. Actually (see denition (2.2) for comparison), (3.2) Ωε,δ1δ2 = Ω\ Bε,δ₁ ∪Tε,δ₂

= n

x∈Ω

nx ε

o

∈Yδ1,δ2

o .

Assume that the matrix eldA^ε(x) = (a^ε_ij(x))1≤i,j≤n is such that there exist two real numbers α and β satisfying

(3.3) α|λ|² ≤(A^ε(x)λ, λ) and |A^ε(x)λ|²≤β(A^ε(x)λ, λ) for any λ∈Rⁿ and a.e. xinΩ.

Letg∈L²(∂B)and (recalling Remark 2.3), set (3.4) g_εδ1(x) =g1

δ1

nx ε

o for allx∈∂B_ε,δ1. For f ∈L²(Ω)consider the problem

(3.5)











−div(A^ε∇u_ε,δ

1δ2) =f inΩ_ε,δ1δ2, A^ε∇u_ε,δ

1δ2 ν_B=g_εδ1 on∂B_ε,δ1,

uε,δ₁δ2 = 0 on∂extΩε,δ1δ2 ∪∂Tε,δ₂,

where ∂extΩε,δ1δ2 is the exterior part of the boundary ∂Ωε,δ1δ2 and νB is the unit exterior normal to the set B. Observe that, by construction, ν_B also is the unit exterior normal to the setBε,δ₁ (see Figure 2).

We introduce the space

V^ε_δ1δ2 ={ϕ∈H¹(Ωε,δ1δ2)|ϕ= 0on ∂extΩε,δ1δ2∪∂Tε,δ₂}, and in the sequel will still denote by ϕinV_δ^ε

1δ2, its extension by zero in T_ε,δ2. Then the variational formulation of problem (3.5) is

(3.6)















Find u_ε,δ1δ2 ∈V^ε_δ

1δ2 satisfying Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2∇φdx= Z

Ωε,δ1δ2

f φdx+ Z

∂Bε,δ1

gεδ1φds

∀φ∈V^ε_δ1δ2.

The existence and uniqueness of the solution u_ε,δ1δ2 in the space V_ε,δ1δ2 are given by the Lax-Milgram theorem (due, in particular, to properties (3.3) of the matrix A^ε).

From now on, we assumen≥3. Suppose thatδ1 =δ1(ε)and δ2 =δ2(ε) satisfy

(3.7) k₁ = lim

ε→0

δⁿ⁻¹₁

ε , 0≤k₁ <∞ and k₂= lim

ε→0

δ

n 2−1 2

ε , 0≤k₂<∞.

We will study the asymptotic behaviour of problem (3.6) asε→0 under assumptions (3.7).

Remark 3.1. Observe thatk1 corresponds to the critical size of Neumann small holes from [12] while k₂ corresponds to that of Dirichlet small holes from [11].

3.2. Main results

LetKT be the functional space (see [7] for more details) (3.8) K_T ={ϕ∈L²_loc^∗(Rⁿ); ∇ϕ∈L²_loc(Rⁿ), ϕ= const.onT}.

We are now able to state the homogenization results concerning problem (3.6).

Theorem 3.2. (Unfolded limit problem). Suppose that (3.7) holds. Let A^ε satisfy (3.3) and suppose that, as ε → 0, there exists two matrix elds A and A0, such that

(3.9)

( T_ε(A^ε)(x, y)→A(x, y) a.e. inΩ×Y, T_ε,δ

2(A^ε)(x, z)→A0(x, z) a.e. inΩ×(Rⁿ\T).

Let uε,δ₁δ2 be the solution of problem(3.6)with gεδ1 dened by(3.4). Then, up tp a subsequence, there exists a function u∈H₀¹(Ω)such that

(3.10) u^_ε,δ1δ2 →u weakly in L²(Ω).

Also, there exist ub∈ L²(Ω;H_per¹ (Y)) and U ∈L²(Ω;L²_loc(Rⁿ)), with U −k₂u in L²(Ω;KT), such that (u,u, Ub ) solves the equation

(3.11) Z

Y

A(x, y) ∇u(x) +∇_ybu(x, y)

∇φ(y) dy= 0

for a.e. x in Ωand all φ∈H_per¹ (Y). Next, (3.12)

Z

Rⁿ\T

A0(x, z)∇_zU(x, z)∇v(z) dz= 0, for a.e. x in Ωand all v∈K_T with v(T) = 0 and, nally,

Z

Ω×Y

A ∇_xu+∇_ybu

∇ψdxdy−k2

Z

Ω×∂T

A0∇_zU νTψds (3.13)

= Z

Ω

f ψdx+k₁|∂B|M_∂B(g) Z

Ω

ψ(x) dx

for all ψ∈H₀¹(Ω), where ν_T is the unit exterior normal to the setT.

The next theorem gives the classical (standard) form of the homogenized system (3.11)(3.13). To state it, we follow the procedure from [7], where more details can be found. Introduce rst the classical correctorsχbj,j= 1, . . . , n, for

the homogenization in xed domains (see for instance, [1]). They are dened by the cell problems

(3.14)















χb_j ∈L^∞(Ω;H_per¹ (Y)), Z

Y

(A(x, y)∇( ˆχ^j −y_j)∇φ= 0 a.e. x∈Ω,

∀φ∈H_per¹ (Y).

Let also χ be the solution of the cell problem corresponding to the small holes δ2T, namely,

(3.15)















χ∈L^∞(Ω;K_T), χ(x, T)≡1, Z

Rⁿ\T

tA₀(x, z)∇_zχ(x, z)∇_zΨ(z) dz= 0 a.e. forx∈Ω,

∀Ψ∈K_T withΨ(T) = 0, and set

(3.16) Θ(x) =

Z

∂T

tA₀(x, z)∇_zχ(x, z)ν_T dσ_z. We then have

Theorem 3.3. The limit function u ∈ H₀¹(Ω) given by Theorem 3.2 is the unique solution of the homogenized equation

(3.17)

−div(A^hom∇u) +k₁² Θu=f+|∂B|M_∂B(g), u= 0 on ∂Ω,

where A^hom is the classical homogenized matrix (3.18) A^hom_ij (x) =

Z

Y

a_ij(x, y)−

n

X

k=1

a_ik(x, y)∂χb_j

∂yk

(x, y) dy.

Remark 3.4. The contribution in the limit of the oscillations of the matrix A^εin the original problem(3.5), is reected by the appearance of the operator A^hom in the homogenized system (3.17). The contribution in the limit of the set of small Neumann holes Bε,δ₁, is the constant|∂B|M_∂B(g) =R

∂Bg ds.

The contribution of the set of small Dirichlet holes T_ε,δ2 is the zero order strange term k₁² Θu.

3.3. Proof of Theorem 3.2

In the proof of Theorem 3.2, we will use the following lemma from [7].

We adapted its statement to our situation.

Lemma 3.5. Letv in D(Rⁿ)∩KT (with KT dened by (3.8)). Set w_ε,δ2(x) =v(T)−v

1 δ₂

nx ε

o

Y

, x∈Rⁿ. Then

(3.19) w_εδ2 * v(T) weakly in H¹(Ω).

Proof of Theorem 3.2. We start by establishing a priori estimates foru^∗_ε,δ in Ωε,δ1δ2. Considering uε,δ₁δ2 as a test function in (3.5), by (3.3) and (2.13) we obtain

αk∇u_ε,δ

1δ2k²

(L²(Ωε,δ1δ2))ⁿ ≤Ckfk_L2(Ω)ku_ε,δ

1δ2k_L2(Ωε,δ1δ2)

+Cδ₁ⁿ⁻¹

ε |M_∂B(g)|+εδ₁ k∇u_ε,δ

1δ2k_(L2(Ωε,δ1δ2))ⁿ, whence, by the Poincaré inequality and assumption (3.7),

(3.20) ku_ε,δ

1δ2k_H1(Ωε,δ1δ2)≤C,

so convergence (3.10) holds. Since ε→ 0 and δ₂ → 0, it follows from Propo- sition 2.7 (see for more details [15]) that, up to a subsequence, there exists a Y-periodicubinL²_loc(Ω;H_loc¹ (Y))such that

(3.21)







T_ε(uε,δ₁δ2)* u weakly in L²(Ω;H_loc¹ (Y)), T_ε(∇(u_ε,δ

1δ2))*∇u+∇_yub weakly in L²(Ω;H_loc¹ (Y)).

Moreover, by Proposition 2.10 (again up to a subsequence) there exist U in L²(Ω;L²_loc(Rⁿ)) and W in L²(Ω;L²_loc^∗(Rⁿ)) with ∇_zW in L²(Ω;L²_loc(Rⁿ)) such that

(3.22)





















 δ

n 2−1 2

ε T_ε,δ2(u_ε,δ1δ2)−M_Y^ε(u_ε,δ1δ2)11 δ2Y

* W weakly in L²(Ω;L²_loc^∗(Rⁿ)), δ

n 2−1 2

ε ∇_z T_ε,δ2(uε,δ₁δ2) 1¹

δ2Y *∇_zW weakly in L²(Ω;L²_loc(Rⁿ)), δ

n 2−1 2

ε T_ε,δ2(u_ε,δ1δ2)* U weakly in L²(Ω;L²_loc(Rⁿ)).

The argument used in the proof of Theorem 3.1 from [7] shows actually that U =W +k2u0 and ∇_zU =∇_zW, with W belonging to L²(Ω;KT). What is essential in this argument, is the convergence

δ

n 2−1 2

ε M_Y^ε(u_ε,δ1δ2)11

δ2Y →k₂u₀ strongly in L²(Ω;L²_loc(Rⁿ)), an easy consequence of (2.9).

Letψ∈ D(Ω)andφ∈C_per¹ (Y)vanishing in a neighborhood of the origin.

Forεand δ small enough, the functionΦ(·) =εψ(·)φ. ε

belongs toV_δ^ε

1δ2, so from (3.6) we have

ε Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2∇ψ φx ε

dx+

Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2 ψ∇φx ε

(3.23) dx

=ε Z

Ωε,δ1δ2

f ψ φx ε

dx.

Lettingε→0, all the terms go to zero, except the second one. Unfolding it by T_ε and recalling Proposition 2.4, yield

0 = Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2 ψ∇φx ε

dx

= Z

Ω×Y

T_ε(A^ε)T_ε(∇(u_ε,δ

1δ2))T_ε(ψ)∇φ(y) dxdy.

We can now let ε→ 0 in the last term, thanks to hypotheses (3.9) and convergences (3.21). We get immediately (3.11), since the limit is

ε→0lim Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2ψ∇φx ε

dx

= Z

Ω×Y

A(x, y)(∇u+∇_yu)∇φ(y)ψ(x) dxdyb = 0.

In order to obtain the other equations from the statement of Theorem 3.1, multiply (3.5) by wε,δ2ψ, where wε,δ2 was dened in Lemma 3.5, and ψ is in D(Ω). Since w_ε,δ = 0 on the set of holes T_εδ2, integrating by parts over Ω_ε,δ1δ2 yields

Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2∇w_εδ2 ψdx+ Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2 wεδ2 ∇ψ w_εδ2dx (3.24)

= Z

Ωε,δ1δ2

f w_εδ2 ψdx+ Z

∂Bε,δ1

g_ε,δ1w_εδ2ψds.

Unfolding withT_ε,δ2the rst integral above, the choice of the test function implies that u.c.i. is satised. So, by Proposition 2.11,

Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2∇w_εδ2 ψdx (3.25)

Tε,δ2

' δ₂ⁿ Z

Ω×Rⁿ

T_ε,δ2(A^ε)T_ε,δ2(∇u_ε,δ

1δ2)T_ε,δ2(∇w_ε,δ)T_ε,δ2(ψ) dxdy

= δ

n 2−1 2

ε Z

Ω×Rⁿ

T_ε,δ2(A^ε)δⁿ²T_ε,δ2(∇u_ε,δ

1δ2)(−∇_zv)T_ε,δ2(ψ), where we used the fact that T_ε,δ2(∇w_ε,δ) =−∇_zv (see Lemma 3.5).

It is obvious from Denition 2.8 that

(3.26) T_ε,δ2(ψ)∇v →ψ∇v strongly in L²(Ω)×L²_loc(Rⁿ).

We are now able to let ε→0in (3.25) thanks to hypothesis (3.9) and conver- gences (3.21) and (3.26), to obtain

ε→0lim Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2∇w_ε,δψdx (3.27)

=−k₂ Z

Ω×(Rⁿ\T)

A₀(x, z)∇_zU(x, z)∇v(z)ψ(x) dxdz, which, by density, is true for any v∈K_T.

Unfolding the second integral in (3.24) byT_ε yields Z

Ω_ε,δ1δ2

A^ε∇u_ε,δ1δ2 wε,δ2 ∇ψdx '^Tε

'Tε

Z

Ω×Y

T_ε(A^ε)T_ε(∇u_ε,δ

1δ2)T_ε(wε,δ2)T_ε(∇ψ) dxdy, where we let ε→0, to get

ε→0lim Z

Ωε,δ1δ2

A^ε∇u_ε,δ

1δ2 w_ε,δ2 ∇ψdx (3.28)

=v(T) Z

Ω×Y

A ∇_xu+∇_ybu

∇ψdxdy,

where we used again hypotheses (3.9) and convergences (3.21), as well as Theo- rem 2.4 and convergence (3.19). Analogously,

(3.29) lim

Z

Ωε,δ1δ2

f w_εδ2 ψdx=v(T) Z

Ω

f ψdx.

For the fourth term we use Proposition 2.14, to get

ε→0lim Z

∂Bε,δ1

g_ε,δ1w_εδ2ψds=k1v(T)|∂B|M_∂B(g) Z

Ω

ψ(x) dx.

This, together with (3.27), (3.28) and (3.29), leads to the limit equation of (3.24):

v(T) Z

Ω×Y

A ∇u+∇_yub

∇ψdxdy−k₂ Z

Ω×(Rⁿ\T)

A₀∇_zU ∇v ψdxdy

=v(T) Z

Ω

f ψdx+k1v(T)|∂B|M_∂B(g) Z

Ω

ψ(x) dx,

for all ψ ∈H₀¹(Ω) and v ∈ K_T. Equation (3.12) is then obtained by taking v(T) = 0 while (3.13) follows by integrating by parts.

3.4. Proof of Theorem 3.3

The proof follows the reasoning from [7, Section 4.3]. We just emphasize the main points. The correctors dened by (3.14) enable us to express bu in equation (3.11) in terms of u as

bu(x, y) =−

n

X

j=1

∂u0

∂x_j(x)χbj(x, y).

Replacing this expression in (3.11), it is easily seen that the limit function u is solution of

Z

Ω

A^hom∇u₀∇ψdx−k₂ Z

Ω×∂B

A₀∇_zU ν_Bψdσ_z (3.30)

= Z

Ω

f ψ dx+k1|∂B|M_∂B(g) Z

Ω

ψdx,

withA^hom given by (3.18). Now, by integrating by parts in (3.15 ), one easily gets

Z

∂T

A₀∇_zU ν_T ds= Z

∂T

A₀∇_z(U−k₂u)ν_T ds=−k₂u Z

∂T

tA₀∇_zχ ν_T dσ

, which, replaced into (3.30) gives (3.17) with Θdened by (3.16).

It remains to show that the existence and uniqueness of the homogenized problem. To do so, it is sucient to notice that from (3.16) and using system (3.15) dening the corrector χ, one has

Θ(x) = Z

Rⁿ\B

A0(x, z)∇_zχ(x, z)∇_zχ(x, z) dz≥0.

We are thus enabled to apply Lax-Milgram theorem and so, to conclude the proof.

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Received 23 April 2008 Université Pierre et Marie Curie Paris 6 Laboratoire J.-L. Lions CNRS

Boîte courrier 187 4 place Jussieu 75005 Paris, France cioran@ann.jussieu.fr

and

Department of MathematicsENS P.O. Box 92 16050 Kouba, Algiers amar.ouldhamouda@ens-kouba.dz