El Hajji in [4] which generalizes - in the case of the linearized elasticity system - the notion ofH0-convergence introduced by M

ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ejde.math.unt.edu (login: ftp)

HOMOGENIZATION OF LINEARIZED ELASTICITY SYSTEMS WITH TRACTION CONDITION IN PERFORATED DOMAINS

MOHAMED EL HAJJI

Abstract. In this paper, we study the asymptotic behavior of the linearized elasticity system with nonhomogeneous traction condition in perforated do- mains. To do that, we use theH_e⁰-convergence introduced by M. El Hajji in [4] which generalizes - in the case of the linearized elasticity system - the notion ofH⁰-convergence introduced by M. Briane, A. Damlamian and P. Donato in [1]. We give then some examples to illustrate this result.

1. Introduction

The notion of H_e⁰-convergence was introduced by M. El Hajji in [4] for the study of the asymptotic behavior of the linearized elasticity system with homoge- neous traction condition in perforated domains. It translates the notion of H⁰- convergence introduced by M. Briane, A. Damlamian and P. Donato in [1] for the study of the diffusion system problem with homogeneous Neumann condition in perforated domains which generalizes in the case of perforated domains the H- convergence introduced by F. Murat and L. Tartar in [13], and the G-convergence for the symmetric operator introduced by S. Spagnolo in [14].

This paper is devoted to giving an application of theH_e⁰- convergence to study the asymptotic behavior of the linearized elasticity system with nonhomogeneous traction condition in perforated domains by using the convergence of a distribution defined from data on the boundaries of the holes. This result is the analogue for the linearized elasticity of Theorem 1 given by P. Donato and M. El Hajji in [3] as an application of theH⁰-convergence to the study of the nonhomogeneous Neumann problem.

In Section 2, we recall the definition ofH_e⁰-convergence introducing a definition of e-admissible set similar to that given by M. Briane, A. Damlamian and P. Donato in [1] for the H⁰-convergence and by F. Murat and L. Tartar in [13] for the H- convergence. In Section 3, we introduce the linearized elasticity problem and we give the main result. We establish then the proof making use of some preliminary results. In Section 4, we give some applications of this result - first for the case of periodic perforated domains by holes of size rε = ε where we use the results given by F. Lene in [11] and D. Cioranescu and P. Donato in [2]. Then we apply the results of section 3 and those of C. Georgelin in [8] and S. Kaizu in [10] when

1991Mathematics Subject Classification. 73C02, 35B27, 37B27, 35J55, 35J67.

Key words and phrases. Homogenization, Elasticity,H_e⁰-convergence, Double periodicity.

c1999 Southwest Texas State University and University of North Texas.

Submitted April 6, 1999. Published October 11, 1999.

1

rε≪ε. Finally, we apply section 3 to the case of a perforated domain with double periodicity (introduced by T. Levy in [12]) using the results given by M. El Hajji in [9] and P. Donato and M. El Hajji in [3].

2. Recall ofH_e⁰-convergence

In this section, we recall the definition ofH_e⁰-convergence introduced in [4]. First let us introduce the following notations.

Let Ω be a bounded open subset ofR^N,εthe general term of a positive sequence, andcdifferent positive constants independent ofε. We introduce the following sets:

Ms={symmetric linear operatorsl:R^N →R^N2}, L(Ms) ={linear operatorsp:Ms→ Ms}, Ls(Ms) ={symmetric operatorsp∈ L(Ms)}, Me(α, β; Ω) =

A∈L^∞(Ω,Ls(Ms)), A(x)ξ·ξ≥α|ξ|²,

A⁻¹(x)ξ·ξ≥β|ξ|², ∀ξ∈ Ls(Ms), x a.e. ∈Ω .

In what follows, we use the Einstein summation convention, that is, we sum over repeated indices. We denote bye(·) the symmetric tensor of elasticity defined by

e(u) = (eij(u))ij where eij =1 2

∂ui

∂xj +∂uj

∂xi

.

We denote by Sε a compact subset of Ω. We denote the perforated domain by Ωε= Ω\Sε. We denote byχε the characteristic function of Ωεand we set

Vε=

v∈[H¹(Ωε)]^N, v|∂Ω= 0 , (1) which equipped with theH¹-norm forms a Hilbert space.

Definition 1 (e-admissible set). The set Sε is said to be admissible (in Ω) for the linearized elasticity if

every function inL^∞(Ω) weak? ofχε is positive almost everywhere in Ω, (2) and for each ε there is an extension operatorPε from Vε to [H₀¹(Ω)]^N and there exists a real positiveC such that

i) Pε∈ L Vε,[H₀¹(Ω)]^N , ii) (Pεv)|Ωε=v, ∀v∈Vε,

iii) ke(Pεv)k_[(L2(Ω)]^N2 ≤Cke(v)k_[(L2(Ωε)]^N2, ∀v∈Vε.

(3) Remark 1.1) As an example of an e-admissible set, one can consider the case of a periodic function on a perforated domain by holes of size ε or rε (see F. Lene [11] and C. Georgelin [8]). One can consider also a perforated domain with double periodicity introduced by T. Levy in [12] (see also M. El Hajji [9]).

2). Observe that ifSεis admissible in the sense of definition 1, then we have a Korn inequality in Ωεindependent ofε, i.e.,

k∇vk_[L2(Ωε)]^N2 ≤C(Ω)ke(v)k_[L2(Ω)]^N2, ∀v∈Vε. Indeed, from the Korn inequality in Ω and (3 iii) one has

k∇vk_[L2(Ωε)]^N2 ≤k∇(Pεv)k_[L2(Ω)]^N2

≤c(Ω)ke(Pεv)k_[L2(Ω)]^N2

≤C(Ω)ke(v)k_[L2(Ω)]^N2, ∀v∈Vε.

To give the definition ofH_e⁰-convergence, we introduce the adjoint operatorP_ε^? ofPεdefined from [H⁻¹(Ω)]^N toV_ε⁰ by

hPε^?f, viV_ε⁰,V_ε=hf, Pεvi_[H⁻¹(Ω)]^N,[H¹₀(Ω)]^N

= XN i=1

hfi,(Pεv)iiH⁻¹(Ω),H₀¹(Ω), ∀v∈V_ε⁰.

Definition 2. LetA^ε∈Me(α, β; Ω) andSε be e-admissible in Ω. One says that the pair (A^ε, Sε)H⁰-converges toA⁰ (in the sense of the linearized elasticity) and we denote this (A^ε, Sε)*^He0 A⁰ if for each functionf in [H⁻¹(Ω)]^N, the solutionu^ε of −div (A^εe(u^ε)) =P_ε^?f in Ωε,

(A^ε(x)e(u^ε))·n= 0 on∂Sε, u^ε= 0 on∂Ω,

(4) satisfies

Pεu^ε* u weakly in [H₀¹(Ω)]^N, A^εe(u]^ε)* A⁰e(u) weakly in L²(Ω)N²

,

(5) whereuis the solution of the problem

−div A⁰e(u)

=f in Ω,

u= 0 on∂Ω, (6)

andev is the extension by zero to Ω of the functionv defined in Ωε.

Remark 2. 1) If Sε is empty, the H_e⁰-convergence reduces to the notion of H- convergence in elasticity introduced by G.A. Francfort and F. Murat in [7].

2). The system (4) is equivalent to the system

− ∂

∂yjσ^ε_ij(u^ε) = (P_ε^?f)i in Ωε

σ^ε_ij(u^ε)·nj= 0 on∂Sε, u^ε= 0 on∂Ω,

where σ_ij^ε(u^ε) =A^ε_ijkhekh(u^ε),A^ε= (A^ε_ijkh), and whose variational formulation is written as: Findu^ε∈Vεsuch that

Z

Ωε

σ_ij^ε(u^ε)eij(v)dx=hP_ε^?f, viV_ε⁰,V_ε. We can rewrite this problem in the form: Findu^ε∈Vεsuch that

Z

Ωε

A^εe(u^ε)e(v)dx=hPε^?f, viV_ε⁰,V_ε.

Some examples will be given in section 4, when we apply the main result of this paper.

3. The main result

In this section, we establish a property of theH_e⁰-convergence, and apply it to the study of the asymptotic behavior of the linearized elasticity system with nonho- mogeneous traction condition. This result is analogous to the linearized elasticity of Theorem 1 in [9] given as an application of the H⁰-convergence to the study of

the nonhomogeneous Neumann problem. We then give some examples to illustrate this result.

LetA^ε= (a^ε_ijkh)∈Me(α, β; Ω) and letSε be e-admissible in Ω such that Sεhas boundary∂Sε of classC¹. (7) We consider the linearized elasticity system

−div (A^εe(u^ε)) = 0 in Ωε, (A^ε(x)e(u^ε))·n=g^ε on∂Sε,

u^ε= 0 on∂Ω,

(8) where

g^ε∈[H^−1/2(∂Sε)]^N. (9) It is well known that (8) has a unique solution. Our aim is to study the asymptotic behavior of the solution u^ε as ε approaches zero. To do that, we introduce a vectorial distributionν_g^ε defined in Ω by

hνg^ε, ϕi_[H⁻¹(Ω)]^N,[H¹₀(Ω)]^N =hg^ε, ϕi_[H^−1/2(∂S_ε)]^N,[H^1/2(∂S_ε)]^N, ∀ϕ∈[H₀¹(Ω)]^N. (10) It is easy to check that this defines ν_g^ε as an element of [H⁻¹(Ω)]^N, and if ν_g^ε ∈ [L²(Ω)]^N, we deduce from the Riesz Theorem that ν_g^ε is a measure. The following theorem shows that the convergence ofu^εcan be deduced from the H_e⁰- convergence of (A^ε, Sε) and the convergence of ν_g^εin [H⁻¹(Ω)]^N.

Theorem 1. Let {u^ε} be the sequence of the solutions of (8). Suppose that (7) is satisfied and that

i) (A^ε, Sε)*^H0e A⁰,

ii) there existsν∈[H⁻¹(Ω)]^N such that ν_g^ε→ν strongly in[H⁻¹(Ω)]^N. (11) Then

i) Pεu^ε* u weakly in[H₀¹(Ω)]^N,

ii) A^εe(u]^ε)* A⁰e(u) weakly in L²(Ω)N²

,

(12) whereuis the solution of the problem

−div A⁰e(u)

=ν inΩ,

u= 0 on ∂Ω. (13)

Proof. Observe first, by using (3 ii) and (10), that

hg^ε, vi_[H^−1/2(∂S_ε)]^N,[H^1/2(∂S_ε)]^N =hνg^ε, Pεvi_[H⁻¹(Ω)]^N,[H₀¹(Ω)]^N, ∀v∈Vε. Hence, problem (8) is equivalent to the problem

−div (A^εe(u^ε)) =P_ε^∗ν_g^ε in Ωε, (A^ε(x)e(u^ε))·n= 0 on∂Sε,

u^ε= 0 on∂Ω,

since both of the two systems have the variational formulation: Find u^ε∈Vεsuch

that Z

Ωε

A^εe(v^ε)e(v)dx=hν_g^ε, PεviV_ε⁰,V_ε, ∀v∈Vε. (14)

Let us show that there existcindependent ofεsuch that

kPεu^εk_[H₀¹(Ω)]^N ≤c . (15) By takingu^ε as a test function in the variational formulation of (14) one obtains

Z

Ωε

A^εe(u^ε)e(v)dx=hνg^ε, Pεu^εiV_ε⁰,V_ε. From (3 iii) and the fact thatA^ε∈Me(α, β,Ω) one deduces that

ke(Pεu^ε)k²_[L2(Ω)]^N2 ≤C Z

Ωε

e(u^ε)e(u^ε)dx

≤C α

Z

Ωε

A^εe(u^ε)e(u^ε)dx

≤cν_g^ε

[H⁻¹(Ω)]^Nke(Pεv^ε)k_[L2(Ω)]^N2. Hence (15) gives (11 ii). One may deduce (up to a subsequence) that

Pεu^ε* u^? weakly in [H₀¹(Ω)]^N. (16) Consider now the solutionv^εof the problem

−div (A^εe(v^ε)) =P_ε^∗ν in Ωε, (A^ε(x)e(v^ε))·n= 0 on∂Sε,

v^ε= 0 on∂Ω.

(17) From (11 i), one deduces that

i) Pεv^ε* v weakly in [H₀¹(Ω)]^N,

ii) A^εe(v]^ε)* A⁰e(v) weakly in L²(Ω)N²

,

(18) wherev is the solution to (13).

On the other hand,w^ε=u^ε−v^εis the solution to

−div (A^εe(w^ε)) =P_ε^∗ ν_g^ε−ν

in Ωε, (A^ε(x)e(w^ε))·n= 0 on∂Sε,

w^ε= 0 on∂Ω.

(19) By choosingw^ε as a test function in the variational formulation of (19) and (3) and the fact thatA^ε∈Me(α, β,Ω), one has

k(Pεw^ε)k²_[L2(Ω)]^N2 ≤Cke(w^ε)k²_[L2(Ωε)]^N2

≤C α

Z

Ωε

A^εe(w^ε)e(w^ε)dx

=chνg^ε−ν, Pεw^εi_[H⁻¹(Ω)]^N,[H₀¹(Ω)]^N. SincePεw^εis bounded in [H₀¹(Ω)]^N, one deduces from (12 ii) that

hν_g^ε−ν, Pεw^εi_[H⁻¹(Ω)]^N,[H₀¹(Ω)]^N →0, which implies that

Pεw^ε→0 strongly in [H₀¹(Ω)]^N. (20) This, with (18) proves that in (16) one hasu^?=u.

Finally, one deduces from (20) and the fact thatA^ε∈Me(α, β,Ω) that kA^εe(w^^ε)k_[L2(Ω)]^N2 ≤cke(w^ε)k_[L2(Ωε)]^N2 ≤cke(Pεw^ε)k_[L2(Ω)]^N2 →0.

rεS Ωε

Figure 1. A periodic perforated domain

With the convergence (18 ii), it follows then that (12 ii) holds. ♦ Remark 3. As in the case of Theorem 1 of [9], from the linearity of the equation and the definition of theH_e⁰- convergence, the choice of an nonhomogeneous right-hand side of the equation (8) is not restrictive.

4. Some applications of the main result

The case of a periodic perforated domain. Let Y = [0, l₁[×..×[0, lN[ be the representative cell, S an open set of Y with smooth boundary ∂S such that S ⊂Y. Let rε be the general term of a positive sequence which converge to zero and satisfyingrε≤ε. One denote byτ(rεS) the set of all the translated ofrεS of the form (εkl+rεS), k ∈Z^N, kl= (k₁l₁, .., kNlN). It represents the holes inR^N.

One suppose that the holes τ(rεS) do not intersect the boundary ∂Ω. If Sε

design the holes contained in Ω, it follows that

Sεis a finite union of the holes, i.eSε=∪k∈Krε(kl+S).

Set Ωε= Ω\Sε, by this construction, Ωεis a periodic perforated domain by holes of sizerε(see Figure 1)

We propose to study the asymptotic behavior of the solutionv^ε of the system

−div (A^εe(v^ε)) = 0 in Ωε, (A^ε(x)e(v^ε))·n=h^ε on∂Sε,

v^ε= 0 on∂Ω,

(21) where

h^ε(x) =h(x

ε), h∈[L²(∂S)]^N Y-periodic. (22) We suppose that

εlim→0

ε^N

r^Nε⁻² = 0, (23)

and thatA^ε= (a^ε_ijkh) satisfies

a^ε_ijkh(x) =aijkh(x

ε), aijkh ∈Me(α, β;Y^∗). (24) In this case of a periodic perforated domain, the homogenization of system (21) has been studied by F. Lene in [11] for the caserε=ε, and C. Georgelin in [8] for the caserε≪ε. The results obtained allow us to deduce that

(A^ε, Sε)*^He0 A⁰, (25)

whereA⁰= (a⁰_ijkh) is defined by A⁰_ijkh= 1

|Y| Z

Y\SAijkhekh(χ^kh−P^kh)eij(χij−P^ij)dy, (26) where P^ij is the vector all of whose components are equal to zero except the i^th one, i.e., (P^ij)k=yjδki, and for allk, h= 1, .., N,χ^kh∈[H¹(Y \S)]^N Y-periodic, and is a solutin to

−div(Ae(χ^kh−P^kh)) = 0 inY \S, (Ae(χ^kh−P^kh))·n= 0 on∂S, ifrε=εand

A⁰_ijkh= 1

|Y| Z

Y aijkhekh(χ^kh−P^kh)eij(χij−P^ij)dy, (27) whereP^ij is the vector all of whose components are equal to zero except thei^thone which is equal toyj, i.e., (P^ij)k=yjδki, and for anyk, h= 1, .., N,χ^kh∈[H¹(Y)]^N Y-periodic is a solution to

−div(Ae(χ^kh−P^kh)) = 0 inY, ifrε≪ε.

On the other hand, from the results obtain by D. Cioranescu and P. Donato in [2] for the caserε=ε, and S. Kaizu in [10] for the caserε≪ε, we can deduce the following lemma.

Lemma 1 ([2],[10]). Let ν_h^ε be defined by (10). We suppose that (23) is satisfied and that the reference hole S is star-shaped if rε≪ε. Then

ε^N

rε^N−1ν_h^ε→ν in H⁻¹(Ω) strongly, (28) with

hν, viH⁻¹(Ω),H₀¹(Ω)=Ih

Z

Ωv dx ∀v∈H₀¹(Ω), (29) andIh= _|Y¹_|R

∂Sh ds.

Consequently, we can apply Theorem 1 tou^ε=ε^Nv^ε/r^N_ε⁻¹andg^ε=ε^Nh^ε/r^N_ε⁻¹ to obtain the following theorem.

Theorem 2. Letv^εbe a solution of (21). Suppose that (22) and (23) are satisfied and that S is star-shaped if rε ≪ ε. Then there exists Pε an extension operator satisfying (3) such that

Pε( ε^N

r^Nε⁻¹v^ε)* v⁰ weakly in[H₀¹(Ω)]^N, A^εe( ε^N

r^Nε⁻¹

vf^ε)* A⁰e(v⁰) weakly in L²(Ω)N²

, wherev⁰ is the solution to

−div A⁰e(v⁰)

=ν inΩ,

v⁰= 0 on ∂Ω, (30)

withν defined by (29).

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,

P P P P P P i

3

:

Yε F

rε(Z+kz⁰)/ε rε(S+kz⁰)/ε

Figure 2. The reference cell

4.1. The case of a perforated domain with double periodicity. We consider the perforated domain Ωεdefined as Ωε= Ω\Sε, whereSεis a set with a double periodicity defined below. We adopt here the geometrical framework introduced in [5] and [6].

Assume thatY andZ are two fixed reference cells,

Y =]0, y^o₁[×...×]0, y^oN[, Z =]0, z₁^o[×...×]0, zN^o[. (31) We set

y⁰= (y₁^o, .., y_N^o), z⁰= (z₁^o, ..., z_N^o). (32) LetF⊂Y andS⊂Zbe two closed subsets with smooth boundaries and nonempty interiors.

Suppose thatrε andεare the general term of two positive sequences such that rε< εand

εlim→0

rε

ε = 0. (33)

We assume that for eachε >0 there exists a fineKε⊂Z^N, such that [

k∈K_ε

rε

ε(Z+kz⁰) =Y, (34)

and that

(∂F)∩( [

k∈K_ε

rε

ε(S+kz⁰)) =∅.

This means that for any ε the sets Y and Y \F are exactly covered by a finite number of translated cells of ^r_ε^εZ and ^r_ε^εS respectively. Denote

S_Y^ε = (Y \F)∩( [

k∈K_ε

rε

ε(S+kz⁰))

andYε=Y \S_Y^ε. From (34) it follows that there exist a finite setK⁰ε⊂Z^N such that

S_Y^ε = [

k∈K⁰_ε

rε

ε(S+kz⁰).

HenceS_Y^ε is a subset ofY\F of closed sets (“inclusions”) periodically distributed with periodicityrε/εand of the same size as the period (see Figure 2).

We also assume that for eachε >0, there exists a finite setHε⊂Z^N such that [

h∈Z^N

ε(S_Y^ε +hy⁰)∩Ω = [

h∈H_ε

ε(S_Y^ε +hy⁰) and we set

Sε = [

h∈H_ε

ε(S_Y^ε +hy⁰).

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A

A

A U

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εYε rε(S+hy^o) rε(Z+hy⁰)

εF

Figure 3. The perforated domain Ωε

Hence,∀ε >0, Ω and Ωεare exactly covered by a finite number of translated cells of εYε and εS_Y^ε respectively. Consequently, the structure of Ωε presents a double periodicity (ε and rε). The zones in which the inclusions are concentrated areε- periodic and of size ε. The inclusions in each zone arerε-periodic and of size rε

(see Figure 3).

Our aim is to apply Theorem 1 to this case of double periodicity with a matrix A^ε= (a^ε_ijkh) defined in (21) and satisfying

a^ε_ijkh(x) =aijkh(x ε, x

rε) i) aijkh Y ×Z−periodic

ii) aijkh ∈L^∞(Z, C⁰(Y)) oraijkh∈L^∞ Y, C⁰(Z) iii) aijkh =aijhk =ajikh

iv) ∃α >0 s.t. aijkh(y, z)ekheij ≥αeijeij, a.e. (y, z)∈Y ×Z

(35)

for any symmetric tensoreij, and h^ε is defined by

hε= (F^ε◦ Q^ε)h, hε∈H^−1/2(∂S^ε) (36) wherehisZ-periodic,h∈H^−1/2(∂S), and

hh,1iH^−1/2(∂S),H^1/2(∂S 6= 0. (37) The operatorQ^ε∈ L H^−1/2(∂S), H^−1/2(∂S_Y^ε)

is defined by hQ^εz, viH^−1/2(∂S_Y^ε), H^1/2(∂S_Y^ε)= X

k∈K⁰_ε

(rε

ε)^N−1hz, v◦σ⁻¹_ε iH^−1/2(∂S+kz⁰), H^1/2(∂S+kz⁰), (38) and the operatorF^ε∈ L(H^−1/2(∂S^ε_Y), H^−1/2(∂S^ε)) is defined by

hF^εu, φiH^−1/2(∂S^ε), H^1/2(∂S^ε)= X

h∈Hε

(ε)^N−1hu, φ◦τε⁻¹iH^−1/2(∂S_Y^ε+hy⁰), H^1/2(∂S_Y^ε+hy⁰), (39) whereσεandτεare the homotheties

σε: x−→ ε

rεx, τε: x−→ x

ε. (40)

From the result obtained in [9], we deduce that

(A^ε, Sε)*^He0 A⁰, (41)

whereA⁰= (a⁰_ijkh) is defined as follows: set

dijkh(y, z) = χF(y) +χY\F(y)χZ\S(z)

aijkh(y, z), (42) and forl, m= 1, .., N, letR^lm= R^lm_k

k=1,..,N be the vector defined by R^lm_k =zmδkl.

We denote byχ^lm=χ^lm(y, .) the unique function in [H¹(Z\S)]^N Z-periodic which is a solution to

− ∂

∂zj[dijkhe^z_kh(R^lm−χ^lm)] = 0 inZ\S dijkhe^z_kh(R^lm−χ^lm)·nj= 0 on∂S.

(43) We set

qijkh = 1

|Z| Z

Z\Sdijrse^z_rs(R^kh−χ^kh)dz.

Let P^lm = P_k^lm

k=1,..,N be the vector defined by P_k^lm = ymδkl, and let β^lm in [H¹(F)]^N Y-periodic, which is a solution to

− ∂

∂yj[qijkhe^y_kh(P^lm−β^lm)] = 0 inF, qijkhe^y_kh(P^lm−β^lm)·nj = 0 on∂F \∂Y .

(44) We define the homogenizated coefficients by

a⁰_ijkh= 1

|Y| Z

Y qijrse^y_rs(P^kh−β^kh)dy, (45) wherey= (yi)i=1,..,N andz= (zi)i=1,..,N.

Observe that the coefficients (a⁰_ijkh) are obtained by applying the homogenization process twice (see the classical methods of homogenization introduced by F. Murat and L. Tartar in [13] and S. Spagnolo in [14]). Indeed, first starting with the tensor (dijkh) and homogenizing with respect to Z, we obtain the tensor (qijkh). Then starting with (qijkh) and homogenizing with respect to Y, we obtain the tensor (a⁰_ijkh).

On the other hand, using the results obtain by P. Donato and M. El Hajji in [4], we obtain

Lemma 2. Let ν_ε^h be defined by (10), suppose that hh,1iH^−1/2(∂S),H^1/2(∂S) 6= 0, and that (33) is satisfied. Then

rεν_h^ε →ν strongly inH⁻¹(Ω), whereν is given by

hν, φiH⁻¹(Ω),H¹₀(Ω)=γθIh

Z

Ωφdx ∀φ∈H₀¹(Ω), (46) with

γ=

|YkZ|

|Y −F|− |S| ₋₁

, Ih=hh,1iH^−1/2(∂S),H^1/2(∂S), (47) andθ is defined by

θ= |F|

|Y|+|Y \F|

|Y|

|Z\S|

|Z| . Hence, we can apply Theorem 1 tou^ε=rεu^εand obtain

Theorem 3. Let v^ε be the solution of (21). Then there exists Pε an extension operator satisfying (3) such that

Pε(rεv^ε)* u⁰ weakly in [H₀¹(Ω)]^N, A^εe(r^εv^ε)* A⁰e(u⁰) weakly in L²(Ω)N²

, wherev⁰ is the solution of the problem

−div A⁰e(v⁰)

=ν inΩ,

v⁰= 0 on ∂Ω, (48)

whereA⁰= (a⁰_ijkh)is given by (43)-(45), andν defined by (46), (47).

Acknowledgments. It is a pleasure for the author to acknowledge his indebtedness to Patrizia Donato for her help and friendly suggestions throughout this work.

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Mohamed El Hajji

Universit´e de Rouen, UFR des Sciences UPRES-A 60 85 (Labo de Math.) 76821 Mont Saint Aignan, France

E-mail address: Mohamed.Elhajjiuniv-rouen.fr