Homogenization of a Transmission Problem in Solid Mechanics

Journal of Mathematical Analysis and Applications 233, 659᎐680 Ž1999. Article ID jmaa.1999.6327, available online at http:rrwww.idealibrary.com on

Homogenization of a Transmission Problem in Solid Mechanics L. Baffico and C. Conca Departamento de Ingenierıa Uni¨ ersidad de Chile, Casilla 170r3ᎏCorreo ´ Matematica, ´ 3, Santiago, Chile Submitted by Firdaus E. Udwadia Received June 5, 1997

In this paper we study a simplified model of the behavior of a 3-D solid made from two elastic homogeneous materials separated by a rapidly oscillating interface. We study the asymptotic beha¨ ior of the solution of such model using homogenization tools and a compactness result. We obtain the homogenized equation, and by studying its coefficients, we find some properties of the limiting material. 䊚 1999 Academic Press Key Words: boundary homogenization; transmission problems

1. INTRODUCTION In this paper we study a simplified mathematical model that describes the behavior of a 3-D solid body made from two elastic materials which are separated by a periodically oscillating interface with period ⑀ ) 0 and a constant amplitude. This work is related to the study of transmission problem posed in a bounded domain with a rapidly oscillating interface Žsee w3x.. The model involves the classical system of linear elasticity in a bounded domain ⍀ ; ⺢ 3, with an interior boundary ⌫ ⑀ that represents the interface between both elastic materials. We consider homogeneous conditions on the external boundary of ⍀ and use continuity boundary conditions on ⌫ ⑀. In order to simplify the elasticity system, we consider a particular kind of elastic homogeneous material. The solution ª u ⑀ of such a system represents the amount of deformation induced on the solid by the action of an external force. Our aim is to study the asymptotic behavior Žas ⑀ tends to zero. of the solution ª u ⑀. We use classical homogenization tools Žsee, e.g., the books by A. Bensoussan, J.-L. w8x. and a F. Murat’s Lions, G. Papanicolaou w2x and E. Sanchez-Palencia ´ 659 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

660

BAFFICO AND CONCA

w6x compactness result to prove that the sequence ª u ⑀ converges to ª u 0 as ⑀ ª0 tends to zero, where u is the solution of an elliptic boundary value system Žthe homogenized equation. that we are to explicitly find. Our study generalizes previous results obtained by R. Brizzi in the case of an elliptic scalar equation Žsee w3x.. From a physical point of view, this limiting analysis can be seen as a mixing process of materials in the region near the rapidly oscillating interface. The homogenized equation corresponds to a generalized elasticity system whose coefficients, arranged in a fourth-order tensor, can be seen as the elasticity coefficients of the limiting material. Studying the homogenized operator, we can conclude that the new material behaves, in the region near the interface, like a nonhomogeneous anisotropic elastic material. In the regions where no mixing process occurs, the new material still behaves like the original materials, i.e., with the same elastic coefficients. The homogenized problem can also be deduced using the two scales asymptotic de¨ elopment method Žsee w2x. which consists in proposing an asymptotic ansatz of the solution ª u ⑀ using functions that depend upon two variables, the microscopic and the macroscopic scales. With this expansion and by means of a formal calculus, the homogenized problem can be found. An important difference between the problem treated in this paper and other problems in homogenization theory is that the microscopic scale has one spatial dimension less than the macroscopic one. Let us remark that the 2-D case is a straightforward consequence of the 3-D case studied in this paper. However, in the 2-D case it is possible to explicitly obtain the coefficients in the homogenized problem and to check technical hypotheses Žsee w1x.. In Section 2, we present the geometry of the domain with its rapidly oscillating interface and we formulate our model. From its variational formulation, we obtain a first result concerning the convergence of ª u ⑀ 4⑀ ) 0 ⑀ and  ␴ 4⑀ ) 0 , the stress tensor. In Section 3, we formulate the homogenized problem and we present our main convergence result. Section 4 is devoted to proving this result. Finally, in Appendix A, we prove a result concerning the homogenized coefficientsᎏwhich is necessary for the existence and uniqueness of the homogenized problemᎏand in Appendix B we briefly expose the 2-D case.

2. PRESENTATION OF THE MODEL 2.1. The Geometry of the Problem In this section, we describe the domain ⍀ ; ⺢ 3 with its rapidly oscillating interface. To this end, let Y sx0, T1w=x0, T2 w; ⺢ 2 , Ti ) 0 Ž i s 1, 2.,

TRANSMISSION PROBLEM HOMOGENIZATION

661

and let h: Y ª ⺢ be a smooth function such that Ži. h < ⭸ Y s h1 , where h1 s max hŽ y .< y g Y 4 and h1 ) 0. Žii. There exists y 0 g Y such that hŽ y 0 . s 0 and ⵜy hŽ y 0 . s 0. Let z 0 ) h1 ) 0 be given. We define ⍀ 11 s  Ž y, z . g ⺢ 2 = ⺢ < h Ž y . - z - z 0 , y g Y 4 , ⍀12 s  Ž y, z . g ⺢ 2 = ⺢ < yz 0 - z - h Ž y . , y g Y 4 . Let ⌫ 1 be the interface between these sets, i.e., ⌫ 1 s  Ž y, z . g ⺢ 2 = ⺢ < h Ž y . s z, y g Y 4 , Using these elements, the so-called reference cell ⍀ is built up as follows Žsee Fig. 1, left.: ⍀ s ⍀11 j ⌫ 1 j ⍀ 12 Ž s Y = yz 0 , z 0 w . . If we intersect ⍀ with the hyperplane w Z s z x, where 0 - z - h1 , we obtain Y and its following decomposition ŽFig. 1, right. Y s Y *Ž z . j ␥ Ž z . j O Ž z .

FIG. 1. The reference cell ⍀ Žleft. and the decomposition of Y Žright..

662

BAFFICO AND CONCA

where O Ž z . Žresp., Y *Ž z ., ␥ Ž z .. is defined as  y g Y < hŽ y . - Žresp., ) , s. z 4 . Note that Y *Ž z . s Y y O Ž z . . Let ⑀ be a positive parameter. Extending h by Y-periodicity 1 we can introduce x ⍀ 1⑀ s Ž x, z . g ⺢ 2 = ⺢ < h - z - z0 , x g Y ⑀ x ⍀ ⑀2 s Ž x, z . g ⺢ 2 = ⺢ < y z 0 - z - h , xgY ⑀ and the rapidly oscillating interface is therefore defined by x ⌫ ⑀ s Ž x, z . g ⺢ 2 = ⺢ < h s z, x g Y Ž see Fig. 2, left . . ⑀ Finally, as in Fig. 2 right, we set ⍀ 1 s Y = x h1 , z 0 w, ⍀ m s Y = x0, h1w and ⍀ 2 s Y = x yz 0 , 0w. We used ⍀ m to denote the region near the interface, because it is in this region where we obtain mixed material in the limit as ⑀ goes to zero.

½ ½

½

5

ž /

ž /

ž /

5

5

2.2. Setting the Model Up Let us fill the regions ⍀ 1⑀ ; ⍀ and ⍀ ⑀2 ; ⍀ with two different elastic materials. Let ª u ⑀i : ⍀ ⑀i ª ⺢ 3, i s 1, 2, be the functions that represent the

FIG. 2. The rapidly oscillating interface Žleft. and its homogenized version Žright.. 1 A function f : ⺢ 2 ª ⺢ is Y-periodic if f Ž y q Ý2is 1 k i Ti e i . s f Ž y . for all y g Y and for all k i g ⺪ Ž i s 1, 2., where e1 s Ž1, 0.T and e2 s Ž0, 1.T .

TRANSMISSION PROBLEM HOMOGENIZATION

663

small deformations inside both materials. By e x, z Žª u i⑀ ., i s 1, 2, we mean ª⑀ the linear strain tensor associated with u i , i.e., e x , z Žª u i⑀ . s

1

žⵜ 2

x, z

u i⑀ . q ⵜx , z Ž ª u i⑀ . Žª

T

/,

and ␴i ⑀ , i s 1, 2, stands for the stress tensor, i.e.,

␴i ⑀ s ␭ i divx , z ª u i⑀ I q 2 ␮ i e x , z Ž ª u i⑀ . , where ␭ i G 0 and ␮ i ) 0 are the Lame´ coefficients of the ith material, i s 1, 2. To simplify, we consider the case ␭ i s 0. This mathematical simplification implies that Poisson’s ratio 2 for each material is equal to zero. The assumption ␭ i s 0 is possible since Poisson’s ratio can vary between y1 and 1r2, although this physical parameter is in practice always strictly positive. More details on the theory of elasticity can be found in the book by L. D. Landau and E. M. Lifschitz w5x. Therefore, the stress tensor in our model is now given by

␴i ⑀ s 2 ␮ i e x , z Ž ª u ⑀i .

i s 1, 2.

The functions ª u ⑀i must satisfy the following system

¡ydiv

u Ž P. ~ ⑀

ª

x, z

Ž 2 ␮i e x , z Ž ªu ⑀i . . s fi

in ⍀ ⑀i

ª

ª⑀ i

s0

on ⌫i , i s 1, 2

ª⑀ u1

sª u 2⑀

on ⌫ ⑀

¢2 ␮ e

1 x, z

u1⑀ . ª n s 2 ␮2 e x , z Žª u 2⑀ . ª n Žª

on ⌫ ⑀

ª

where f i g L2 Ž ⍀ . 3 represents the density of external forces acting on the solid body, and ª n means the exterior normal vector to ⍀ 1⑀. The second equation in Ž P .⑀ is the homogeneous condition on the external boundaries of ⍀ 1⑀ and ⍀ ⑀2 , the third and fourth equations in Ž P .⑀ are the continuity boundary conditions on the interface ⌫ ⑀ for the deformations ª u ⑀i , and for ⑀ª the stresses ␴i n. ª Let ª u⑀ s ª u1⑀ ␹ ⍀ 1⑀ q ª u 2⑀ ␹ ⍀ ⑀2 , and f ⑀ , ␮ ⑀ defined in a similar way, where ␹ ⍀ ⑀i stands for the characteristic function of ⍀ ⑀i , i s 1, 2. Then, the variational formulation of Ž P .⑀ is

¡Find u g H Ž ⍀ . Ž PV . ~ ¢H 2 ␮ e Ž u . : e ª⑀

⑀

⑀

⍀

2

x, z

1 0

ª⑀

3

such that

x, z

ª

3 ¨ . dx dz s H f ⑀ ⭈ ª ¨ dx dz ᭙ ª ¨ g H01 Ž ⍀ . . Žª

⍀

Poisson’s ratio is the ratio of the transverse compression to the longitudinal extension.

664

BAFFICO AND CONCA

It is easy to prove that this problem has a unique solution ª u ⑀ g H01 Ž ⍀ . 3 ª⑀ 1 3 and the sequence  u 4⑀ ) 0 is bounded in H0 Ž ⍀ . Žsee w7x.. Therefore, the sequence ␴ ⑀ s 2 ␮ ⑀ e x, z Žª u ⑀ . is bounded in L 3,2 s Ž ⍀ ., the space of 3 = 3 symmetric second-order tensor with coefficients in L2 Ž ⍀ .. Then the following proposition holds. PROPOSITION 2.1. Ža. There exists ª u* g H01 Ž ⍀ . 3 and a subsequence of  u 4⑀ ) 0 , which we still denote by ⑀ , such that ª⑀

ª⑀

u ©ª u* in H01 Ž ⍀ . ᎏweakly, 3

3

and in L2 Ž ⍀ . ᎏstrongly.

Žb. There exists ␴ * g L 3,2 s Ž ⍀ . and a subsequence of  ␴ ⑀ 4⑀ ) 0 , which we still denote by ⑀ , such that

␴ ⑀ © ␴ * in L 3,2 s Ž ⍀ . ᎏweakly. 3. THE MAIN CONVERGENCE RESULT 3.1. The Homogenized Problem Let us consider the following periodic system for ␹ k l s ŽŽ ␹ k l .1 , Ž ␹ k l . 2 .T in Y,

Ž P . kl

½

ydivy Ž 2 ␮ e y Ž ␹ k l . . s divy Ž 2 ␮ Mk l . in Y

␹k l

Y-periodic

1 F k, l F 2

where Mk l , 1 F k, l F 2, is the 2 = 2 matrix defined by

w Mk l x i j s

1 2

Ž ␦i k ␦ jl q ␦il ␦ jk . .

Since ␮ Ž y, z . s ␮ 1 ␹OŽ z .Ž y . q ␮ 2 ␹ Y *Ž z .Ž y ., then Ž P . k l is parameterized by z gx0, h1w. For a fixed z, it is easy to show that problem Ž P . k l has a unique solution in Ž H噛1 Ž Y .r⺢. 2 , where H噛1 Ž Y .r⺢ is the space of Y-periodic 1 Ž 2. H loc ⺢ functions defined up to an additive constant Žproofs of existence and uniqueness for similar periodic problems can be found in C. Conca w4x and E. Sanchez-Palencia w8x.. We assume that the solution of Ž P . k l , as function of Ž y, z . g Y = x0, h1w Ž i s 1, 2., satisfies the following regularity hypothesis:

¡Ž a. ␹

kl

g L2loc Ž 0, h1 ; H噛1 Ž Y .

2

. l Ž L2loc Ž x 0, h1 w= ⺢ 2 . .

2

Ž H 1 .~ Ž b . ⭸ Ž ␹ . g L2 0, h ; L2 Ž Y . l L2 Ž x 0, h = ⺢ 2 . Ž k l i . loc Ž 1 噛 . loc 1

¢

⭸z 1FiF2

665

TRANSMISSION PROBLEM HOMOGENIZATION

Now, we consider the following periodic scalar problem in Y

Ž P. k

½

ydivy Ž ␮ⵜy ␸ k . s divy Ž 2 ␮ e k .

␸k

in Y

Y-periodic

1FkF2

where e k is the kth vector of the canonical basis of ⺢ 2 . Ž P . k is also parameterized by z gx0, h1 w and, for a fixed z, this equation has a unique solution in H噛1 Ž Y .r⺢. We assume that the following regularity hypothesis holds for its solution:

¡Ž a. ␸ g L Ž0, h ; H Ž Y . . l L Ž x0, h w=⺢ . ⭸␸ Ž H 2 .~ ¢Ž b. ⭸ z g L Ž0, h ; L Ž Y . . l L Ž x0, h w=⺢ . . 2 loc

k

k

1 噛

1

2 loc

1

2 loc

2 噛

2

1

2 loc

2

1

Using ␹ k l and ␸ k we construct AŽ z ., a fourth-order tensor whose coefficients are defined by

a i jk l

¡2 ␮ w M Ž z . s~a ¢2 ␮ w M

x

1 m i jk l

kl i j

2

kl i j

h1 - z - z 0 0 - z - h1

x

1 F i , j, k , l F 3,

yz 0 - z - 0

where the coefficients in the region ⍀ m are defined by

¡1


HY2 ␮ ž w M

1 1

H␮ 2
H␮ 2
a imjk l s

~1

1

H␮ 2
H␮ 2
¢


x q e y Ž ␹k l .

kl i j

ž ž

2 ␦k i q

ž ž

2 ␦k j q

2 ␦li q

2 ␦l j q

HY2 ␮ dy

⭸␸ k

/

⭸ yi ⭸␸ l ⭸ yi

/

⭸␸ k ⭸␸ l ⭸ yj

dy

/

⭸ yj

/

dy

dy dy

ij

/ dy

if if if if

if

½ ½ ½ ½ ½ ½

1 F i, 1 F k,

jF2 lF2

1 F i F 2, 1 F k F 2,

js3 ls3

1 F i F 2, k s 3,

js3 1FlF2

i s 3, 1 F k F 2,

1FjF2 ls3

i s 3, k s 3,

1FjF2 1FlF2

isjs3 ksls3 otherwise

if

Ž< Y < denote the Lebesgue’s measure of the set Y .. This tensor satisfies

666

BAFFICO AND CONCA

PROPOSITION 3.1. The coefficients of A are such that: Ža. a i jk l Ž z . s a k l i j Ž z . s a i jl k Ž z ., ᭙1 F i, j, k, l F 3, ᭙ z gx yz 0 , z 0 w. Žb. There exists ␤ ) 0 such that for all ␰ , 3 = 3 symmetric second-order tensor,

Ž A Ž z . ␰ . : ␰ G ␤␰ : ␰ , ᭙ zgx yz 0 , z 0 w . Proof. See Appendix A. Let us now introduce the homogenized problem

¡Find u ~ ¢H Ž Ae

ª0

Ž PV . H

⍀

3

g H01 Ž ⍀ . such that

x, z

ª

3 u0 . . : e x , z Ž ª ¨ . dx dz s H f ⭈ ª ¨ dx dz ᭙ ª ¨ g H01 Ž ⍀ . , Žª

⍀

ª

ª

where f is the weak limit of f ⑀ ; see Ž3. below. Using Proposition 3.1, we conclude that Ž PV .H has a unique solution ª u 0 g H01 Ž ⍀ . 3 Žsee w7x.. Remark 3.1. The homogenized problem Ž PV .H can be deduced using the two-scale asymptotic de¨ elopment method Žsee w2, 4, or 8x for more details.. In this case the scales to be considered are the microscopic one, y s xr⑀ g ⺢ 2 , and the macroscopic scale, Ž x, z . g ⺢ 3. The ansatz to be used is ª⑀

u Ž x, z . s ª u 0 Ž x, z, xr⑀ . q ⑀ ª u1 Ž x, z, xr⑀ . q ⑀ 2 ª u 2 Ž x, z, xr⑀ . q ⭈⭈⭈ ,

where ª u k : ⍀ = Y ª ⺢ 3, k s 0, 1, 2, . . . , are functions such that ª u k Ž x, z, ⭈ . is Y-periodic, for all Ž x, z . in ⍀. A detailed use of this method can be found, e.g., in C. Conca w4x. Remark 3.2. The homogenized equation Ž PV .H is the variational formulation of a generalized elasticity system where the stress tensor and the linear strain tensor are related by 3

w ␴ x i j s Ý ai jk l Ž z . e x , z Ž ª u.

kl

,

1 F i , j F 3,

Ž 1.

k , ls1

or ␴ s Ae x, z Žª u.. The coefficients a i jk l of A can be seen as the elasticity coefficients of the homogenized material. From the definition of A, we observe that in the regions ⍀ 1 and ⍀ 2 the homogenized material still behaves as the original materials did, with Lame’s ´ coefficients ␮ 1 and ␮ 2 , respectively. Instead, in ⍀ m the coefficients depend upon z gx0, h1w. Therefore, the homogenized material in the region near the interface behaves like a nonhomogeneous anisotropic elastic material Žsee w7x..

TRANSMISSION PROBLEM HOMOGENIZATION

667

3.2. The Con¨ ergence Result The main convergence theorem is THEOREM 3.1. If Ž H 1., Ž H 2. and Ž H 3. hold, then the sequence of solutions ª u ⑀ 4⑀ ) 0 of Ž PV .⑀ con¨ erges to ª u 0 in H01 Ž ⍀ . 3 ᎏweakly, where ª u 0 is the unique solution of the homogenized problem Ž PV .H . Hypothesis Ž H 3. in Theorem 3.1 is a technical hypothesis that we present in Section 4.3.

4. PROOF OF THE CONVERGENCE RESULT The proof of Theorem 3.1 consists of several steps. First, using Proposition 2.1, we find the equation that satisfies ␴ *, the weak limit of  ␴ ⑀ 4⑀ ) 0 . The next steps are devoted to provingᎏusing classical homogenization techniques and a compactness resultᎏthat ␴ * and ª u*, the weak limit of ª u ⑀ 4⑀ ) 0 , are related by Ž1.. Toward that end, we identify each component of the limit tensor of  ␴ ⑀ 4⑀ ) 0 separately. Finally, using the equation for ␴ *, we conclude that ª u* is the solution of the homogenized problem Ž PV .H . 4.1. The Equation Satisfied by ␴ * We begin the homogenization process passing to the limit in equation ª Ž PV .⑀ . To this end, we first identify the weak limit of f ⑀ : Since

␹ ⍀ 1⑀ © ␪ and ␹ ⍀ ⑀2 © Ž 1 y ␪ . in L⬁ Ž ⍀ . ᎏweakly ), where

¡1

␪ Ž x, z . s

in ⍀ 1

~
¢0

Ž 2.


in ⍀ m in ⍀ 2 .

Then ª

ª

ª

f ⑀ © f 1 ␪ q f 2 Ž 1 y ␪ . in L2 Ž ⍀ . ᎏweakly. 3

Ž 3.

Using Proposition 2.1 and Ž3. we pass to the limit ⑀ ª 0 in Ž PV .⑀ and we obtain the following variational equation for ␴ *:

ª

ª

H⍀␴ *: e ª

ª

x, z

where f s f 1 ␪ q f 2 Ž1 y ␪ ..

3 ¨ . s H f ⭈ª ¨ ᭙ª ¨ g H01 Ž ⍀ . , Žª

⍀

Ž 4.

668

BAFFICO AND CONCA

4.2. Identification of w ␴ *x k l , 1 F k, l F 2, in ⍀ m We shall prove PROPOSITION 4.1. If Ž H 1. holds, then w ␴ ⑀ x k l © w ␴ *x k l in L2 Ž ⍀ m . ᎏ weakly Ž up to a subsequence., for 1 F k, l F 2, where

w ␴ *x k l s

1

2


Ý i , js1

½H

2␮

Y

ž

Mi j

kl

q e y Ž ␹i j .

kl

/ dy 5

ex, z Žª u* .

i j.

Proof. Let pk l : Y ª ⺢ 2 , 1 F k, l F 2, be the polynomial defined by p11 Ž y . s

y1 , 0

y 2r2 , y 1r2

p12 Ž y . s p 21 Ž y . s

0 p 22 Ž y . s y . 2

Note that e y Ž pk l . s Mk l , 1 F k, l F 2. Let w k l s pk l q ␹ k l and ␴ k l s 2 ␮ e y Ž w k l . Žremark that ␴ k l is a 2 = 2 matrix., and define the functions w k⑀ l Ž x, z . s ⑀ w k l

x

Ž x, z . g ⍀ m

ž⑀ / ,z

and

␴ k⑀l Ž x, z . s ␴ k l

x

Ž x, z . g ⍀ m

ž⑀ / ,z

Ždo not mistake the 2 = 2 matrix ␴ k⑀l for w ␴ ⑀ x k l , the Ž k, l .-element of tensor ␴ ⑀ .. As ␹ k l is the solution of Ž P . k l , we have divx ␴ k⑀l s 0 in ⍀ m .

Ž 5.

Since Ž H 1. is fulfilled, we can use classical arguments about the convergence of periodic functions, to conclude that for all ⍀⬘ ;; ⍀ m open sets w k⑀ l ª pk l

⭸ ⭸z ␴ k⑀l

Ž Ž wk⑀l . i . ª 0 ij

© ␴k l

ij

2

in L2 Ž ⍀⬘ . ᎏstrongly,

in L2 Ž ⍀⬘ . ᎏstrongly, in L2 Ž ⍀⬘ . ᎏweakly,

Ž 6.

Ž 1 F i F 2. ,

Ž 7.

Ž 1 F i , j F 2. ,

Ž 8.

669

TRANSMISSION PROBLEM HOMOGENIZATION

where ␴ k l is the average in Y of ␴ k l , i.e.,

␴k l s

1

HY2 ␮ Ž M


kl

q e y Ž ␹ k l . . dy.

Remark 4.1. From Ž5. and Ž8., we conclude that divx Ž ␴ k l . s 0 in ⍀ m .

Ž 9.

Now we can define the function ª w k⑀ l s Ž w k⑀ l , 0.T . Let ␾ g D Ž ⍀ m .. If we ª ª⑀ put ¨ s ␾ w k l in Ž PV .⑀ , after elementary calculations, we obtain

H⍀

␴ ⑀ ⵜx , z ␾ ⭈ ª w k⑀ l dx dz q

H⍀

m

2

q s

ÝH

2 ␮ ⑀ e x Ž u ⑀ . : e x Ž w k⑀ l . ␾ dx dz m

2 ␮⑀ e x , z Ž ª u⑀ .

js1 ⍀ m

H⍀

⭸ 3j

⭸z

Ž wk⑀ l . j ␾ dx dz

ª

f ⑀ ⭈ Ž␾ª w k⑀ l . dx dz,

Ž 10 .

m

where u ⑀ denotes the first two components of ª u ⑀ . On the other hand, if we ⑀ use ␾ u as a test function in Ž5., we have

H⍀

␴ k⑀l ⵜx ␾ ⭈ u ⑀ dx dz q

H⍀

m

2 ␮ ⑀ e x Ž w k⑀ l . : e x Ž u ⑀ . ␾ dx dz s 0.

Ž 11 .

m

If we subtract Ž11. from Ž10., we obtain

H⍀

␴ ⑀ ⵜx , z ␾ ⭈ ª w k⑀ l dx dz y m

2

q

ÝH

H⍀

␴ k⑀l ⵜx ␾ ⭈ u ⑀ dx dz m

2 ␮⑀ e x , z Ž ª u⑀ .

js1 ⍀ m

⭸ ⭸z

3j

ª

w k⑀ l . dx dz. Ž wk⑀ l . j ␾ dx dz s H f ⑀ ⭈ Ž ␾ ª ⍀m

Ž 12 . Using Proposition 2.1 and Ž3., Ž6., Ž7., Ž8. Žby considering ⍀⬘ ;; ⍀ m , an open set such that supp ␾ ; ⍀⬘., we can pass to the limit in Ž12., and obtain

H⍀

␴ *ⵜx , z ␾ ⭈ ª w k⑀ l dx dz y m

H⍀

␴ k lⵜx ␾ ⭈ u* dx dz s m

H⍀

ª

f* ⭈ ␾ª wUk l dx dz, m

670

BAFFICO AND CONCA

where ª wUk l s Ž pk l , 0.T . Integrating by parts the left-hand side of this equation, knowing that ␴ * satisfies Ž4. Žwith ª ¨ s ␾ª wUk l as a test function., and ␴ k l satisfies Ž9. Žwith ␾ u* as a test function., we have

H⍀

␴ *: e x , z Ž ª wUk l . ␾ dx dz s m

H⍀

␴ k l : e x Ž u* . ␾ dx dz. m

We also have ex, z Ž ª wUk l .

ij

s

½

w Mk l x i j

if 1 F i , j F 2

0

otherwise

Then, in the sense of D ⬘Ž ⍀ m ., we have

w ␴ * x k l s ␴ k l : e x Ž u* . s

2

1

Ý


i , js1

½H

2 ␮ w Mk l x i j q e y Ž ␹ k l .

ž

Y

ij

/ dy 5

e x Ž u* .

i j.

Since the following symmetric property holds Žsee Proposition 3.1.

HY2 ␮ ž w M

x q e y Ž ␹k l .

kl i j

ij

/ dy s H 2 ␮ ž Y

Mi j

kl

q e y Ž ␹i j .

kl

/ dy,

and w e x Ž u*.x i j s w e x, z Žª u*.x i j for 1 F i, j F 2, we finally conclude that for 1 F k, l F 2, 2

1

w ␴ *x k l s Ý < < i , js1 Y

½H

2␮

Y

ž

Mi j

kl

q e y Ž ␹i j .

kl

/ dy 5

ex, z Žª u* .

ij;

hence, Proposition 4.1 is proved. 4.3. Identification of w ␴ *x 3 k , 1 F k F 2, in ⍀ m Let ␸ k g H噛1 Ž Y .r⺢ be the solution of the periodic problem Ž P . k and let pk Ž y . s y k , 1 F k F 2ᎏnote that ⵜy pk s e k ᎏand set w k s 2 pk q ␸ k and ␰ k s ␮ⵜy w k . We define the functions w k⑀ s ⑀ w k

x

ž⑀ / ,z

Ž x, z . g ⍀ m

and

␰ k⑀ s ␰ k

x

ž⑀ / ,z

Ž x, z . g ⍀ m .

TRANSMISSION PROBLEM HOMOGENIZATION

671

Then we have divx ␰ k⑀ s 0 in ⍀ m ,

Ž 13 .

and for all ⍀⬘ ;; ⍀ m Žsince Ž H 2. holds., w k⑀ ª 2 pk

⭸ ⭸z

in L2 Ž ⍀⬘ . ᎏstrongly,

Ž wk⑀ . ª 0 in L2 Ž ⍀⬘ . ᎏstrongly, ␰ k⑀ ª ␰ k in L2 Ž ⍀⬘ . ᎏweakly, 2

Ž 14 . Ž 15 . Ž 16 .

where ␰ k is the average in Y of ␰ k , i.e.,

␰k s

1
HY␮ Ž2 e

k

q ⵜy ␸ k . dy.

Remark 4.2. From Ž13. and Ž16., we conclude that ␰ k satisfies ydivx ␰ k s 0.

Ž 17 .

We also need the following compactness lemma Žsee F. Murat w6x.: LEMMA 4.1. If the sequence  g ⑀ 4⑀ ) 0 belongs to a bounded set of Wy1 , p Ž ⍀ . for some p ) 2, and g ⑀ G 0, in the following sense: for all ␾ g D Ž ⍀ . , such that ␾ G 0, and ᭙ ⑀ ) 0, ² g ⑀ , ␾ : G 0. Then  g ⑀ 4⑀ ) 0 belongs to a compact set of Hy1 Ž ⍀ .. If ␰ k⑀ s ŽŽ ␰ k⑀ .1 , Ž ␰ k⑀ . 2 .T is assumed to fulfill

¡Ž a.

Ž H 3 .~

p Ž ⍀ m . and ½ Ž ␰ k⑀ . j 5 ⑀)0 locally bounded,  Ž ␰ k⑀ . j 4 ⑀)0 ; L loc

for some p ) 2 ⭸ Ž b. Ž Ž ␰ k⑀ . j . G 0 in the distribution sense, ⭸z

¢

then, using Lemma 4.1, we see that for all ⍀⬘ ;; ⍀ Ž j s 1, 2.,

⭸

⭸

Ž Ž ␰ k⑀ . j . ª ⭸ z Ž Ž ␰ k . j . in ⭸z

Hy1 Ž ⍀⬘ . ᎏstrongly.

Ž 18 .

672

BAFFICO AND CONCA

Let us now prove PROPOSITION 4.2. If Ž H 2. and Ž H 3. hold, then w ␴ ⑀ x 3 k ª w ␴ *x 3 k in L ⍀ . ᎏweakly, for 1 F k F 2, where 2Ž

w ␴ *x 3 k s

1

2


Ý js1

½H ž Y

␮ 2 ␦ jk q

⭸␸ j ⭸ yk

/ 5 dy

ex, z Žª u* .

3 j.

Proof. Let ␾ g D Ž ⍀ m . and consider ª w k⑀ s Ž0, w k⑀ .T . If we use the test ª ª⑀ Ž . function ¨ s ␾ w k in PV ⑀ we obtain, after algebraic developments,

H⍀

2

␮ ⑀ ⵜx u 3⑀ ⭈ ⵜx w k⑀ ␾ dx dz q

ÝH

js1 ⍀ m

m

2 ␮⑀

q

H⍀

s

m

H⍀

⭸ u 3⑀ ⭸ w k⑀ ⭸z

⭸z

␾ dx dz q

⭸ u ⑀j ⭸ w k⑀

␮⑀

⭸ z ⭸ xj

H⍀

␾ dx dz

␴ ⑀ ⵜx , z ␾ ⭈ ª w k⑀ dx dz m

ª

f⑀ ⭈ ␾ª w k⑀ dx dz,

Ž 19 .

m

where u ⑀j Ž j s 1, . . . , 3. denotes the jth component of ª u ⑀. ⑀ Ž . Now, if we multiply 13 by the test function ␾ u 3 , after carrying out an integration by parts, we have

H⍀ Ž ␰

⑀ k

⭈ ⵜx ␾ . u 3⑀ dx dz q

m

H⍀

Ž ␮ ⑀ ⵜx wk⑀ ⭈ ⵜx u 3⑀ . ␾ dx dz s 0.

Ž 20 .

m

We subtract Ž20. from Ž19.,

H⍀

␴ ⑀ ⵜx , z ␾ ⭈ ª w k⑀ dx dz q m

H⍀ Ž ␰

y

⑀ k

H⍀

H⍀

ª

m

⭈ ⵜx ␾ . u 3⑀ dx dz q

m

s

2 ␮⑀

f⑀ ⭈ ␾ª w k⑀ dx dz.

⭸ u 3⑀ ⭸ w k⑀ ⭸z

⭸z

2

ÝH

js1 ⍀ m

␮⑀

␾ dx dz ⭸ u ⑀j ⭸ w k⑀ ⭸ z ⭸ xj

␾ dx dz

Ž 21 .

m

Using Ž3., we can pass to the limit in the right-hand side of Ž21.. Using Proposition 2.1, Ž14., Ž15., Ž16., we do the same in the first, second, and third terms of the left-hand side of Ž21.. Let us rewrite the terms in the

673

TRANSMISSION PROBLEM HOMOGENIZATION

sum in the left-hand side of Ž21. as follows Ž j s 1, 2.

H⍀

⭸ u ⑀j ⭸ w k⑀

␮⑀

⭸ z ⭸ xj

m

⭸

Ž Ž ␰ k⑀ . j . ; ␾ u ⑀j ⭸z

¦

sy

␾ dx dz

;

H y1 Ž ⍀ ⬘., H 01 Ž ⍀ ⬘.

y

H⍀

Ž ␰ k⑀ . j u ⑀j m

⭸␾ ⭸z

dx dz,

where ⍀⬘ ;; ⍀ m is such that supp ␾ ; ⍀⬘. Using Proposition 2.1, Ž16. and Ž18., we can pass to the limit in the last equation and obtain

H⍀

␮

⑀

⭸ u ⑀j ⭸ w k⑀

␾ dx dz ª

⭸ z ⭸ xj

m

H⍀

Ž␰ k . j

⭸ uUj ⭸z

m

␾ dx dz,

1 F j F 2.

Then, passing to the limit in Ž21., we obtain

H⍀

␴ *ⵜx , z ␾ ⭈ ª wUk l dx dz y m

H⍀ ž ␰

k⭈

ⵜx ␾ uU3 dx dz q

/

m

s

H⍀

2

Ý H Ž␰ k . j

js1 ⍀ m

⭸ uUj ⭸z

ª

f* ⭈ ␾ª wUk dx dz,

␾ dx dz

Ž 22 .

m

where ª wUk s Ž0, 2 pk .T . Integrating by parts the first and second terms of this equation, using that ␴ * and ␰ k satisfy Ž4. and Ž17., respectively, we obtain y

H⍀

␴ *: e x , z Ž ª wUk . ␾ dx dz q m

q

H⍀

2

ÝH

js1 ⍀ m

Ž␰ k . j

⭸ uUj ⭸z

␾ dx dz

␰ k ⭈ ⵜx uU3 ␾ dx dz s 0

Ž 23 .

m

and in the distribution sense Žusing that ª wUk s Ž0, 2 pk .T and then, ªU ␴ *: e x, z Ž w k . s 2w ␴ *x 3 k ., 2

w ␴ *x 3 k s Ý js1

½

1

␮ < Y < HY

ž

2 ␦k j q

⭸␸ k ⭸ yj

/ 5 dy

ex, z Žª u* .

3 j.

Finally, since HY ␮ Ž2 ␦ k j q Ž ⭸␸ kr⭸ y j .. dy s HY ␮ Ž2 ␦ jk q Ž ⭸␸ jr⭸ y k .. dy Žsee Proposition 3.1., we conclude that 2

1

⭸␸ j

w ␴ *x 3 k s Ý ␮ 2 ␦ jk q dy < Y < HY ⭸ yk js1

½

ž

Proposition 4.2 is therefore proved.

/ 5

ex, z Žª u* .

3 j.

674

BAFFICO AND CONCA

Remark 4.3. By symmetry of the limit tensor ␴ *, we also have identified the coefficients w ␴ *x k 3 , k s 1, 2. 4.4. Identification of w ␴ *x 33 in ⍀ m Using Lemma 4.1 we shall prove the following PROPOSITION 4.3. The sequence w ␴ ⑀ x 33 4⑀ ) 0 is such that w ␴ ␧ x 33 © w ␴ *x 33 in L2 Ž ⍀ m . ᎏweakly, where

w ␴ * x 33 s

1
2 ␮ dy

½H 5 Y

e x , z Ž u* .

33 .

Proof. We use the method introduced in R. Brizzi w3x. We know that w ␴ ⑀ x33 s 2 ␮ ⑀ Ž ⭸ u 3⑀r⭸ z . and ␮ ⑀ s ␮ 1 ␹ ⍀ ⑀ l ⍀ q ␮ 2 ␹ ⍀ ⑀ l ⍀ ; then 1 m 2 m

w ␴ ⑀ x 33 s 2 ␮ 1 P1

ž

⭸ Ž u 3⑀ . 1 ⭸z

/

q 2 ␮ 2 P2

ž

⭸ Ž u 3⑀ . 2 ⭸z

/

,

where Ž u 3⑀ . i s u 3⑀ < ⍀ ⑀i Ž i s 1, 2., and the operator Pi Ž⭈. represents the extension by zero in ⍀ _ ⍀ ⑀i . Let ␺ i⑀ s Pi ŽŽ ⭸ Ž u 3⑀ . i .r⭸ z .; we see that it is a bounded sequence in 2Ž L ⍀ m .. Then there exists ␺ iU g L2 Ž ⍀ m . and a subsequence, such that

␺ i⑀ © ␺ iU

in L2 Ž ⍀ m . ᎏweakly.

Ž 24 .

It is possible to identify the functions ␺ iU : Let ␾ g D Ž ⍀ m .; then

¦

⭸

⭸z

Ž ␹ ⍀ 1⑀ l ⍀ m . ; ␾ u 3⑀

;

␺ 1⑀␾ dx dz y

sy

H⍀

H y 1 Ž ⍀ m . , H 01Ž ⍀ m .

m

H⍀

␹ ⍀ 1⑀ l ⍀ m u 3⑀ m

⭸␾ ⭸z

dx dz.

For the sequence Ž ⭸r⭸ z .Ž ␹ ⍀ 1⑀ l ⍀ m .4⑀ ) 0 we use the Lemma 4.1 Žin R. Brizzi w3x it is shown that this sequence satisfies the hypothesis required in the compactness lemma., and for the right-hand terms we have Ž24., Proposition 2.1 and Ž2.. Then, passing to the limit

⭸


⭸z


¦ ž

sy

H⍀

/ ; ; ␾ uU3

H y1 Ž ⍀ m . ; H 01Ž ⍀ m .

␺ 1U␾ dx dz y m

H⍀



uU3

⭸␾ ⭸z

dx dz,

TRANSMISSION PROBLEM HOMOGENIZATION

675

and developing the duality product in the last equation, we obtain in the distribution sense the following identity:

␺ 1U

s

< O Ž z . < ⭸ uU3
⭸z

.

In the same way,

␺ 2U

s

< Y y O Ž z . < ⭸ uU3
⭸z

.

Finally, from Ž24. we know that w ␴ *x 33 s 2 ␮ 1 ␺ 1U q 2 ␮ 2 ␺ 2U ; then we conclude that

w ␴ * x 33 s s

1
2 Ž ␮1 < O Ž z . < q ␮ 2 < Y y O Ž z . <. 2 ␮ dy

½H 5

⭸ uU3

Y

⭸z

⭸ uU3 ⭸z

.

Hence, Proposition 4.3 is proved. 4.5. Identification of ␴ * in ⍀ 1 and ⍀ 2 Finally, to identify the limit ␴ * in ⍀ 1 and ⍀ 2 we prove the following. PROPOSITION 4.4.

For j s 1, 2, ␴ ⑀ < ⍀ j © ␴jU in Ls2 Ž ⍀ j . ᎏweakly, where

␴jU s 2 ␮ j e x , z Ž ª uUj . and ª uUj s ª u* < ⍀ j . Proof. From the a priori estimates ŽProposition 2.1., we know that u ©ª u* in H01 Ž ⍀ . 3 ᎏweakly and ␴ ⑀ © ␴ * in L 3,2 s Ž ⍀ . ᎏweakly, so, if we consider ª u ⑀⍀ j s ª u ⑀ < ⍀ j and ␴⍀⑀ j s ␴ ⑀ < ⍀ j Ž j s 1, 2., then

ª⑀

ª⑀ u⍀ j

©ª uUj

␴⍀⑀ j © ␴jU

3

in H 1 Ž ⍀ j . ᎏweakly in L 3,2 s Ž ⍀ j . ᎏweakly

Ž j s 1, 2 . Ž j s 1, 2 . ,

Ž 25 . Ž 26 .

where ␴jU s ␴ * < ⍀ j and ª uUj s ª u* < ⍀ j . ⑀ . We also know, from the definition of ␴ ⑀ , that ␴⍀⑀ j s 2 ␮ j e x, z Žª u⍀ j . Now, using Ž25. and Ž26., we conclude that

␴jU s 2 ␮ j e x , z Ž ª uUj . Proposition 4.4 is proved.

Ž j s 1, 2 . .

676

BAFFICO AND CONCA

4.6. Conclusion From Propositions Ž4.1., Ž4.2., Ž4.3., Ž4.4. and the definition of tensor A, we conclude that ␴ * and ª u* are related by Ž1.. Therefore, since ␴ * satisfies Ž4., ª u* is solution of Ž PV .H . Since all the steps shown above can be repeated for any weak accumulation point of ª u ⑀ 4⑀ ) 0 , we conclude that the solution of the homogenized problem Ž PV .H is the unique weak accumulation point of this sequence. Hence, the whole sequence converges weakly to this limit, and Theorem 3.1 is therefore proved.

APPENDIX A: PROOF OF PROPOSITION 3.1 To prove part Ža. of Proposition 3.1, we first study the coefficients of tensor A with indexes 1 F i, j, k, l F 2. The symmetry of these coefficients is evident when z gx h1 , z 0 w and z gx yz 0 , 0w. To prove the symmetry when z gx0, h1w Ži.e., in the region near the interface ., we use the bilinear form b: Ž H噛1 Ž Y .r⺢. 2 = Ž H噛1 Ž Y .r⺢. 2 ª ⺢, defined by bŽ ␹ , ␺ . s

1

2

2 ␮ e Ž ␹ . : e Ž ␺ . dy < Y < HY y

᭙␹ , ␺ g Ž H噛1 Ž Y . r⺢ . . Ž 27 .

y

Let w k l s pk l q ␹ k l , 1 F k, l F 2, and ␹ k l solution of Ž P . k l and pk l the polynomial defined in Section 4.2. Using properties of these functions we can easily see that b Ž w k l , wi j . s a imjk l .

Ž 28 .

Since bŽ⭈, ⭈ . is a symmetric bilinear form Žsee w4x., then a imjk l s a kml i j , for 1 F i, j, k, l F 2. On the other hand, as Mk l s Ml k , then by uniqueness of problem Ž P . k l , we have ␹ k l s ␹ l k Žup to an additive constant., and then a imjk l s a imjl k . We now study the coefficients a i jk l with j s l s 3 and 1 F i, k F 2. From the definition of tensor A, we have the symmetry of these coefficients when z gx yz 0 , 0w and z gx h1 , z 0 w. To prove the symmetry in the region near the interface, we now consider the bilinear form ˆ b: H噛1 Ž Y .r⺢ 1Ž . = H噛 Y r⺢ ª ⺢ defined by

ˆb Ž ␸ , ␰ . s

1 4
HY␮ⵜ ␸ ⭈ ⵜ ␰ . y

y

Let w k s 2 pk q ␸ k , where ␸ k is solution of Ž P . k and pk is the polynomial defined in Section 4.3. Doing elementary calculations we conclude that

ˆb Ž wk , wi . s ai3m k 3 ,

677

TRANSMISSION PROBLEM HOMOGENIZATION

m m and by symmetry of ˆ bŽ⭈, ⭈ ., we have a i3 k 3 s a k 3 i3 , 1 F i, j, F 2. By conm m struction of A, we also have a i3 k 3 s a i33 k , 1 F i, j, F 2. For the other nonzero terms Ži.e., a i33 l , a3 jk 3 and a3 j3 l ., the same method can be used. To prove part Žb., we first note that coerciveness when z gx h1 , z 0 w and z gx yz 0 , 0w is evident since ␮ i ) 0. When z gx0, h1 w, we must show that for all symmetric second-order tensors ␰ we have Ž A␰ .: ␰ G 0, and if Ž A␰ .: ␰ s 0, then ␰ s 0. Using the relationships between the coefficients of A and the bilinear forms b and ˆ b, we obtain

2

Ž A␰ . : ␰ s b

ž

2

wk l ␰ k l ,

Ý k , ls1

Ý k , ls1

2

qˆ b

ž

2

wk l ␰ k l q ˆ b

/ ž

2

Ý

wk ␰ k 3 ,

ks1

Ý wj ␰ j3 js1

/ ž q

2

Ý wl ␰ l3 , Ý wi ␰ i3 ls1

1
HY2 ␮ dy

is1

/

␰ 332 .

/ Ž 29 .

From this equation, the positiveness of A is due to the positiveness of the bilinear forms Žsee w4 or 8x.. When Ž A␰ .: ␰ s 0, all the terms on the right-hand side of Ž29. are equal to zero, which implies that ␰ 33 s 0 and, by coerciveness of b and ˆ b, that ␰ k l s 0 Ž k, l s 1, 2. and ␰ k 3 s 0, k s 1, 2 Žsee, e.g., w4x.. Hence ␰ s 0. APPENDIX B: THE CASE ⍀ ; ⺢ 2 When ⍀ ; ⺢ 2 , it is possible to find explicit functions which are solutions of the periodic problems Ž P . k l , Ž P . k , and validate the hypotheses Ž H 1., Ž H 2., and Ž H 3. Žsee w1x.. In the 2-D case, the periodic problems Ž P . k l and Ž P . k become ordinary differential equations in x0, T w Žs Y . with periodic boundary conditions:

¡ d 2 ␮ d ␹ s d Ž 2 ␮ . in Y ž dy / dy and Ž P. ¢␹ Ž 0. s ␹ Ž T . ¡y d ␮ d␸ s d Ž 2 ␮ . in Y ~ dy ž dy / dy Ž P. ¢␸ Ž 0. s ␸ Ž T . . ~y dy

11

1

678

BAFFICO AND CONCA

It is easy to prove the following. For z gx0, h1w fixed, let ␸ and ␹ be defined by

PROPOSITION B.1.

¡

␸ Ž y. s

ž ~ž ¢ž

C

␮2 C

␮2 C

␮1

y2 y

if 0 - y - a Ž z .

/

y y

C

␮1 C

␮2

C

aq

y2 y

if a Ž z . - y - b Ž z .

/ ž / / ž / ␮1

C

Ž b y a. q

␮2

y2 y

if b Ž z . - y - 1

and

¡

␹ Ž y. s

ž ~ž ¢ž

C 2 ␮2 C 2 ␮2 C 2 ␮1

y1 y

if 0 - y - a Ž z .

/

y y

C 2 ␮1 C 2 ␮2

C

aq

y1 y

if a Ž z . - y - b Ž z .

/ ž / / ž / 2 ␮1

C

Ž b y a. q

2 ␮2

y1 y

if b Ž z . - y - 1

where O Ž z . sx aŽ z ., bŽ z .w and C s CŽ z. s 2

ž

1

1

dy < Y < HY ␮

y1

/

.

Then ␸ and ␹ are the unique solutions Ž up to an additi¨ e constant . in H噛1 Ž Y . of problems Ž P .1 and Ž P .11 , respecti¨ ely. Remark B.1. Studying the regularity of the solutions of the equations Ž P .1 and Ž P .11 one can validate hypotheses Ž H 1., Ž H 2., and Ž H 3. Žsee w1x.. In the 2-D case, the homogenized tensor has six nonzero coefficients,

¡2 ␮

a1111 s

~ 1 H2␮
¢2 ␮

h1 - z - z 0

1

2

ž

1q

⭸␹ ⭸y

/

dy

0 - z - h1 , z0 - z - 0

679

TRANSMISSION PROBLEM HOMOGENIZATION

¡2 ␮

a2222 s

h1 - z - z 0

1

~ 1 H 2 ␮ dy
¢2 ␮

2

0 - z - h1

¡␮

a1212 s a1221 s a2112 s a2121 s

~

z0 - z - 0 h1 - z - z 0

1

1 2
¢␮

HY␮

ž

2q

⭸␸ ⭸y

/

0 - z - h1

dy

z 0 - z - 0.

2

Remark B.2. If we denote by ␮* and ␮q the following positive constants,

␮* s

1
HY

␮ dy,

␮qs

ž

1
y1

1

dy

HY ␮

/

.

Žthese constants are the arithmetic and the harmonic media of ␮ in Y, respectively., then we note that a2222 s 2 ␮* and C Ž z . s 2 ␮q in ⍀ m . Since we know the explicit solutions of the periodic problems, we can easily show that the homogenized coefficients in the 2-D case are:

a1111

¡2 ␮ ¢2 ␮

s~2 ␮

1 q 2

h1 - z - z 0 0 - z - h1 , z0 - z - 0

a1212 s a1221 s a2112 s a2121

¡2 ␮ ¢2 ␮

s~2 ␮*

a2222

¡␮ s~␮ ¢␮

1 q 2

1

2

h1 - z - z 0 0 - z - h1 , z0 - z - 0

h1 - z - z 0 0 - z - h1 . z0 - z - 0

ACKNOWLEDGMENTS We gratefully acknowledge the French Government for its support. The second author is also partially supported by the Chilean Programme of Presidential Chairs in Sciences and by Conicyt under Fondecyt Grant No. 1970734.

REFERENCES 1. L. Baffico and C. Conca, Homogenization of a transmission problem in 2D elasticity, in ‘‘Computational Science for the 21st Century’’ ŽM.-O. Bristeau et al., Eds.., pp. 539᎐548, Wiley, New York, 1997.

680

BAFFICO AND CONCA

2. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, ‘‘Asymptotic Analysis for Periodic Structures,’’ North-Holland, Amsterdam, 1978. 3. R. Brizzi, Transmission problem and boundary homogenization, Re¨ . Mat. Apl. 15 Ž1994., 1᎐16. 4. C. Conca, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl. 64 Ž1985., 31᎐75. 5. L. D. Landau and E. M. Lifshitz, ‘‘Theory of Elasticity,’’ Collection Course of Theoretical Physics, Vol. 7, Addison᎐Wesley, London, 1959. 6. F. Murat, L’injection du cone ˆ positif de Hy1 dans Wy1 , q est compacte pour tout q - 2, J. Math. Pures Appl. 60 Ž1981., 309᎐322. 7. P. A. Raviart and J. M. Thomas, ‘‘Introduction ` a l’analyse numerique des ´ equations aux ´ derivees partielles,’’ Masson, Paris, 1983. ´ 8. E. Sanchez-Palencia, ‘‘Nonhomogeneous media and vibrating theory,’’ Lecture Notes in ´ Physics, Vol. 127, Springer-Verlag, Berlin, 1980.