Applicable Analysis Homogenization of Boundary-Value ...fian-pages.lebedev.ru/andrey/apan99.pdf · should be independently verified with primary sources. ... Russian Academy of Sciencies

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by:On: 4 January 2010Access details: Access Details: Free AccessPublisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Applicable AnalysisPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713454076

Homogenization of Boundary-Value Problem in a Locally PeriodicPerforated DomainGregory A. Chechkin a; Andrey L. Piatnitski b

a 1Department of Differential equations Faculty of Mechanics and Mathematics, Moscow StateUniversity, Russia, Moscow b Lebedev Phisycal Institute, Russian Academy of Sciencies, Russia,Moscow

To cite this Article Chechkin, Gregory A. and Piatnitski, Andrey L.(1999) 'Homogenization of Boundary-Value Problem ina Locally Periodic Perforated Domain', Applicable Analysis, 71: 1, 215 — 235To link to this Article: DOI: 10.1080/00036819908840714URL: http://dx.doi.org/10.1080/00036819908840714

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t713454076

http://dx.doi.org/10.1080/00036819908840714

http://www.informaworld.com/terms-and-conditions-of-access.pdf

Applicable Analyris. Val. 71(1-4). pp. 215-235 Reprints available directly f m the Publisher Photocopying permitted by license only

O 1999 OPA (Overseas Publishers Association) N.V. Published by license under

the Gordon and Breach Science Publishers imprint.

Printed in Malaysia.

Homogenization of Boundary-Value Problem in a Locally Periodic Perforated Domain Communicated by B. Vainberg

Gregory ~ . ~ h e c h k i n b , Andrey L. piatnitskid

Department of Differential equations Faculty of Mechanics and Mathematics

Moscow State University Moscow 119899, Russia

Lebedev Phisycal Institute Russian Academy of Sciencies

Leninski pr., 53 Moscow 117924, Russia

Abstract

We consider a model homogenization problem for the Poisson equation in a locally periodic perforated domain with the smooth exterior boundary, the Fourier boundary condition being posed on the boundary of the holes. In the paper we construct the leading terms of formal asymptotic expansion. Then, we justify the asymptotics obtained and estimate the residual.

AMS: 35B27, 35B40

KEY WORDS: Homogenization, locally periodic perforation.

(Received for Publication June 1998)

Downloaded At: 18:03 4 January 2010

G.A. CHECHKIN AND A.L. PIATNITSKI

In memory of Professor Landis Evgeni Mikhailovich

Introduction.

Recent years many mathematical works were devoted to the asymptotic analysis of problems in perforated domains. Various homogenization results have been achieved for periodic, almost periodic and random structures. We mention here the general frameworks 1151, 1161, 1211, [24], [27], [28], where the detail bibliography can be found.

In the paper we consider perforated media with locally periodic microstructure in the presence of a small dissipation at the boundary of the cavities. Corresponding mathematical description involves the Fourier boundary condition with a small parameter cQ, characterizing the dissipation. The effective characteristics of the media depend essentially on the value of a. Earlier, similar problems for purely periodic structures were investigated in the works 161, [lo], [ l l ] , [12], where the general convergence results were obtained for various values of a; namely, the case -1 < a < 1 was considered in [lo], the case a 5 -1 in [ll] and the case a > 1 in [12]. The Stokes and Navier-Stokes systems in perforated domains were studied in [13]. Also there is an interesting work 1141, devoted to the problem in domains with "small" cavities. It should be noted that the case of Neumann homogeneous boundary conditions were primely studied in 191, [22], [30].

When studying a locally periodic perforation, we encounter an additional difficulty: the fact that the geometry of the cavities is not fixed. One can apply the compensated compactness method [23] or the two-scale convergence method [l] to obtain the limit problem, but these methods do not allow to estimate the residual. Previously, locally periodic perforated structures have been studied in [20], [8], where, by means of the two-scale convergence method, the homogenized problem has been constructed and the weak convergence of solutions has been proved. In [26] another approach was used for study the problems in perforated domains with an arbitrary density of cavities. In the present paper we use the asymptotic expansion technique [2], [3] that requires the regularity of data but gives the estimates of the rate of convergence.

In the section 1 we introduce necessary notation, construct the family of domains, depending on a small positive parameter E and pose the problem to be studied.

The sections 2 and 5 deal with a formal interior asymptotic expansion of the solution for a = 1 and a > 1, respectively.

The technical results obtained in the section 3, allow to justify the asymptotic expansion and to estimate the discrepancy.

Theorem 1 proved in the section 4, states that for a = 1 two terms of the interior asymptotic expansion provide the precision of order f i in H1-norm.

Theorem 2 proved in section 6, states that for a > 1 two terms of the interior asymptotic expansion provide the precision of order max (6, &"-I) in H1 -norm.

Theorem 3 from the last section states that in the case a < 1 the uniform estimate of solution is of order max (&, &I-") in H1-norm.


PERIODIC PERFORATED DOMAIN

1 Statement of the problem. First we define a perforated domain. Let R c IRd, d 2 2, be a smooth bounded domain. Denote

1 1 J " = { ~ E z ~ : d i s t ( ~ j , d R ) > ~ & } , O = { ( : - - < ( j < - , j = l ,..., d}.

2 2

Given an 1-periodic in ( smooth function F(x , () such that F(X, ()I > const > 0, [€an -

F(x,O) = -1, V C F # 0 as ( E o\{O}, weset

and introduce the perforated domain as follows:

We also use the following notation fl; = R\ (J ( ~ ( 0 + j)). Afterwards, we will . i€JC

often interprete 1-periodic in 6 functions as functions defined on d-dimensional torus Td - {( : E E IRd/Zd}.

According to the above construction the boundary dRC consists of dR and the boundary of the cavities S, c R, SE = (dRE) n R.

We investigate the asymptotic behavior of solution u,(x) as E + 0 of the following boundary-value problem in the domain Rc :

where n, is the internal normal to the boundary of "holes", q(x,() is a sufficiently smooth 1-periodic in ( function.

Definition 1 Function uc E H1(Rc, dR) is a solution of problem ( I ) , if the following integral identity

holds true for any function v E H1(RE, do ) .

Here we use the standard notation H1(Rc, dR) for the closure of the set of Cm(r ) - functions vanishing in a neighborhood of dR, by the H1(R" norm.

In what follows we show that a = 1 is a critical value for problem (1); the dissipation dominates if a < 1 and is neglectable if a > 1.


G.A. CHECHKIN AND A.L. PIATNITSKI 218

Part I

The case ac = 1.

2 The formal homogenization procedure.

In this section we construct the leading "locally periodic" terms of the formal asymptotic expansion and, then, find the limit problem. To this end we represent the solution uc(x) to problem (1) in the form of asymptotic series

Substituting expression (3) in equation (1) and taking into account an evident relation

we obtain after simple transformations the formal equality

to be satisfied on S,. Note that the normal vector n, depends on x and 5 in Stc. Considering, as usually,

x and [ = as independent variables, we represent n, in Stc in the following form:

x nc(., -1 E = %Oj,== + N x , C)It=,, (7)

where 6 is a normal to S ( X ) = (6 I F(x, () = 01,

n: = n' + O(E).



Figure 1: Cell of periodicity

Acul (x , t ) = 0 in w, au1(x7 0

= - (Vx(u0(x)), fi) on S(x), (8)

to be solved in the space of 1-periodic in E functions; here x is a parameter, w := (5 E Td I F (x , E ) > 0). This is the standard "cell" problem appearing in case of Neumann conditions on the boundary of holes. The solvability condition

for problem (8) is clearly satisfied, and its solution forms the first "internal7' corrector in (3).

At the next step we collect all the terms of order EO in (5) and of order E' in (6). This yields


220 G.A. CHECHKIN AND A.L. PIATNITSKI

The 1-periodic in 5 solution of the latter problem is the second term of the internal asymptotic expansion of u,(x).

It is natural to represent the solution u l ( x , E ) of problem ( 8 ) in the form:

where 1-periodic vector-function M ( x , E ) = ( M I ( x , E ) , . . . , Md(x, I ) ) solves the problem

Now, (9) can be rewritten as follows

Buo(x) a 2 ~ i ( x - , E ) in @, -2 C - i,j=1 dxi atj axj

d2u0(x) a u 2 ( x 7 f l = - c ~ U O ( X ) aMi(x , El ,j-

dii ij=1 dzi ax j Mi(x,F)fij - C a.i axj i,j=l

(12)

~ U O ( X ) aMi(x ,E) - ~ ( x ~ E ) u o ( x ) - C - n'. - , x i a E j

-f: aU.0 n: on ~ ( x ) . \ i=l axi Writing down the compatibility condition in the last ~roblem, we get the following

equation:

~ U O ( X ) a2Mi(x , E ) d2u0(x) +2 C - ) d t = J ( z -Mi(., C)sj+

j i atj ax j i , j= l dxi dx j (13)

S

j,From (13) by the Stokes formula we derive the equation

d duo(x) + ID n wl f ( x ) = Q ( X ) U O ( X ) + z Ui(x)-, i=l


PERIODIC PERFORATED DOMAIN 22 1

to be the limit equation in f l . Here < . > means the integral over the set 0 n w, Q ( x ) = J q(x , t ) do, and Ui(x) = J ( v n : + n:) do.

S S Let us study in detail the functions U;(x) . Fortunatily, it is not necessary to

calculate U;(x) . Instead, taking into account the selfadjointness of the operators of the initial problems and the convergence of the corresponding belinear forms, we obtain that the G-limit operator is necessary selfadjoint. Hence, the limit equation (14) takes the form:

and, consequently,

Clearly ( d j i + y) is a smooth matrix, moreover, arguing like in [24] one can verify that this matrix is positively defined.

So, we find the homogenized problem:

The integral identity for problem (17) takes the form:

t ) ) ~ U O ( X ) a v ( x ) ' j = 1 ax; a x j

+ Q ( X ) U O ( X ) V ( X ) ) dx = (18)

for any function v c H 1 (0).

Remark 1 It should be noted that M;(x , 2) are not defined in the whole a. Applying the technique of the symmetric extension [19] allows to extend M ( x , E ) into the interior of the "holes" retaining the regularity of these functions. We keep the same notation for the extended functions.

The limit behavior of the solution of problem ( 1 ) is described by the following statement.



Theorem 1 Suppose that f (x) E c1(Itd) and that q(x, c) is smooth enough nonnegative function. Then, for any suficiently small E problem (1) has the unique solution and the following estimate

takes place, where uo and ul are solutions of problems (1 7) and (8) respectively, and Kl does not depend on E .

Remark 2 In fact, in the formulation of Theorem 1 the condition q(x,E) 2 0 can be replaced by the weaker condition Q(x) 2 0.

3 Preliminary lemmas. This section is devoted to various technical assertions, which will be used in the further analysis. Some of these assertions have been proved in [5 ] , [7] (see also [4]). We omit their proofs.

Lemma 1 Under the conditions of Theorem 1 the Friederichs type inequality

holds for any v E H1(S2', an), where C1 does not depend on E .

The next assertion is, in fact, a modified version of Lemma 5 from [7].

Lemma 2 If

then the following inequality

holds for any v(x) E H1(S2',aS2); the constant C3 does not depend on E .

Proof. By (20) the problem

has 1-periodic in E solution. Moreover, the solution is unique up to an additive constant. Let us multiply the equation (22) by the function v(x) E H1(S2',aR) and


PERIODIC PERFORATED DOMAIN 223

integrate it over the domain Re. Integrating by parts the left-hand side of the obtained formula gives

Here n/ is the unit normal to a(R\Z) . The lemma is proved. The following lemma allows to neglect the right-hand side of the equation ( 1 ) in

the thin layer Rr without deterioration of the estimate. Proof of this lemma is similar to the proof of Lemma 8 from [5].

Lemma 3 Suppose that ye is the solution of the problem

where he(x ) = f ( x ) for x E Rf and 0 otherwise. Then

I l ~ e I I ~ l ( i 7 ) 5 C4E.



The following assertion can be proved in the same way as Lemma 5 from [7].

Lemma 4 Suppose wC(x) E L,(R), and let IIe belong to {x E R I dist (x, 80) 5 Coe). Then the following inequality

holds for any v(x) E H1(RC,aR); the constant C5 does not depend on E .

4 The basic estimate.

Proof of Theorem 1. We are going to estimate the H1-norm of the residual:

To this end we extend the functions M;(x, 5) in the layer Rf (see Remark 1 above) and substitute the expression

in the equation (1). Here we denote by x€(:) a smooth cut-off function 0 5 xC(:) 5 1, such that xC(:) = 0 if x E Rf and xE(T) = 1 if dist(x,af) 2 dist(Sc, Rf), moreover IVExe(.f)1 and lAtxe(()I are uniformly bounded. This yields



and

= -lOnwlf(x) in R,

we can transform (27) in the domain Rc\Rf as follows:



On the other hand, with the help of the Green formula one can transform the left-hand side of (31) as follows

+c J ( o z u l ( x , ~ l c = , , n c ) v ( x ) ds + E J ( v e w ( x , 51, n ~ x , 01 Ie=,v(x) ds+ s. L s,

'I, 5 ) ) c v ( x ) ds- (33)

aMi(x , 5 ) d2uo(x) --E / X c ( ~ ) ~ . U l ( x , [)Ic=. ~ ( 1 ) dx - -2 -1 v ( x ) dx-

' ns :13= atj a x i a x j t = j

nc

V ( X ) dx - J A , ~ ~ ( ~ ) v ( x ) dx- nc :,3=

a x j atj d z ; E=i n *



In view of the evident relation

a divt (- 8% (M,(x, t ) s ) ) dx; I t= I !=rdivx (A 8x.1 (M,(x, OF) 1=:) -

the Stokes formula gives

here we also used the fact that all the integrals containing the derivatives of x', are of order E. Now using (33) and the boundary condition in (17), we estimate the following expression


G.A. CHECHKIN AND A.L. PIATNITSKI

5 C6 E I l u ~ I I ~ l ( n * ) l l v I I ~ l ( ~ c ) -

The terms Il and I4 clearly satisfy the estimate

The identity I5 0 follows from the boundary condition of problem (8). Let us estimate the integral 16. Considering (16) it is easy to verify that

Applying the technique of the proof of Lemma 2, one can show that the latter relation implies the inequality

here we used the C1-smoothness of f ( x ) . By Lemma 3 one can assume that the function f ( x ) is equal to 0 in the layer Of. Then I 7 = 0. The term 13 can obviously be estimated as follows:

13 5 C~&IIVIIH~(W).



Finally, due to the properties of x C ( f ) one can apply Lemma 4 in order to estimate Is and 19. This gives

IIsI + 1191 5 clifiII~II~~(n~~ Substituting v = uo + &xEu1 - U, in (36) and taking into account all the estimates above, Lemma 1 and the evident relation Ileul(l - x ~ ) ( I ~ I ( ~ ~ ) 5 C12&, we obtain (19). The theorem is proved.

Part I1

The case a > 1.

5 The formal homogenization procedure.

This section deals with problem (1) in the case a > 1. Substituting the expression

( = 0 + a - l u l , - l ( x ) + u 0 1 ( x ) + E u 1 , (x, ) + (37)

in equation (1) and taking into account an evident relation (4), we obtain after simple transformations the following formal equality



Keeping in mind (7) and collecting all the terms with like powers of a in (38) and (39) , we arrive at the following auxiliary problems:

and problem ( 8 ) for uo,l ( x , E ) , to be solved in the space of 1-periodic in J functions. It follows from (40) that ul , - l does not depend on c. In fact, for our purposes it

suffices to put ul,-l r 0. Then u l , ~ z 0 solves (41) . At the next step we collect all the terms of order c0 in (38) and of order a' in (39) .

This yields

If we represent uot1(x , J ) = (V,uo(x) , M ( x , J ) ) , where 1-periodic vector-function M ( x , J ) = ( M l ( x , E ) , . . . , M d ( x , E ) ) solves problem ( l l ) , then (42) takes the form


PERIODIC PERFORATED DOMAIN 23 1

Writing down the compatibility condition in the last problem, we get the following equation:

In the same way as in Section 2 we find the homogenized problem:

t '- uo(x) = o on 80.

The integral identity for problem (45) reads

0

for any function v eH1 (o). . , he limit behavior of the solution of problem (1) is described by the following

statement.

Theorem 2 Suppose that f (z ) E C1(IRd), and let q(x, 5) be a smooth nonnegative function. Then, for any suficiently small E problem (1) has the unique solution and the following estimate

takes place, where uo and uo,l are solutions of problems (45) and (8) respectively, and KZ does not depend on E .

The proof is similar to that of Theorem 1 and relies on the following two simple assertions:

Lemma 5 Under the conditions of Theorem 2 the inequality

holds for any v E H1(Re,dn).



Lemma 6 For any v E H1(Rc)

We omit their proof.

Part I11

The case a < 1.

6 The homogenization theorem.

In the case a < 1 the limit behavior of the solution of problem (1) is described by the following statement.

Theorem 3 Suppose that f (x) E c1(IRd) and that q(x, 5') is smooth enough strictly positive function. Then, for any suficiently small s problem (1) has the unique solution and the following estimate

takes place, K3 being independent of e.

Proof of Theorem 3. First let us note that Lemma 5 still holds under the conditions of Theorem 3. Writing down the integral identity for problem (I), by the Cauchy- Schwartz-Bunyakovskii inequality, we obtain the uniform boundedness of u,(x) in H1(Rc). Indeed,

Hence, I luc(J~l(n*) I (713. (50)

Let us recall the notation Q(x) = $ q(x, f ) do. Under the assumptions of the theorem S

the function Q(x) is uniformly positive and the estimate holds



the last inequality here can be proved in the same way as Lemma 2. On the other hand, from the integral identity we have

Combining the preceeding estimates and keeping in mind (50), we immediatly get (48). The theorem is proved.

Acknowledgment.

This work was partially supported by RFBR (RFFI), grant 98-01-00062. The final version of the paper was prepared during the stay of G.A.Chechkin in H@gskolen i Narvik (Norway) whose support is greatly appriciated.

References

[l] Allaire, G. 1992. Homogenization and Two-Scale Convergence. SIAM J. Math. Anal. 23: 1482-1518.

[2] Bakhvalov, N.S. 1974. Averaged characteristics of bodies with periodic structure. Doklady AN USSR. 218 (5) : 1046-1048.

[3] Bakhvalov, N.S. 1975. Averaging of partial differential equations with rapidly oscillating coefficients. Doklady AN USSR. 221 (3) : 516-519.

[4] Belyaev, A.G. 1990. On Singular Perturbations of Boundary - Value Problems (Russian). PhD Thesis. Moscow State University.

[5] Belyaev, A.G., and Piatnitski, A.L., and Chechkin, G.A. 1998. Asymptotic Be- havior of Solution for Boundary-Value Problem in a Perforated Domain with Oscillating Boundary. Siberian Math. Jour. 39 (4): 0000-0000.

[6] Brillard, A. 1983. Quelques questions de convergence pour des suites d'operateurs du calcul des variations (French). Thkse 3-kme cycle. UniversitC d'0rsay.



[7] Chechkin, G. A., and Friedman, A. , and Piatnitski, A.L. Ddcembre 1996. The Boundary Value Problem in Domains with Very Rapidly Oscillating Boundary. INRIA Rapport de Recherche No 3062. Sophia Antipolis: Unit6 de Recherche - Institut National de Recherche en Informatique et en Automatique.

[8] Chenais, D., and Mascarenhas, M.L., and Trabucho, L. 1997. On the Optimiza- tion of Non Periodic homogenized Microstructures. Moddlisation Mathe'matique et Analyse Nume'rique ( M A N ) 31 (5): 559-597.

[9] Cioranescu, D., and Saint Jean Paulin, J. 1979. Homogenization in open sets with holes. J. Math. Anal. Appl. 71: 590-607.

[lo] Cioranescu, D., and Saint Jean Paulin, J. 1992. Trussstructures, Fourier Condi- tions and Eigenvalue Problems. In: Boundary Variation, Ed. J.P.Zolezio, Lecture Notes in Control and Information Sciences, Berlin-New York: Springer Verlag. 178: 6 - 12.

[ll] Cioranescu, D., and Donato, P. 1997. On a Robin Problem in Perforated Do- mains. In: Homogenization and Applications to Material Sciences. Edited by D.Cioranescu, A.Damlamian, and P.Donato. GAKUTO International Series. Mathematical Sciences and Applications. Tokyo: Gakkotosho, 9: 123 - 136.

[12] Cioranescu, D., and Donato, P. 1988. HomogCnbation du problkme de Neumann non homogkne dans des ouverts perfor&.. Asymptotic Analysis 1: 115 - 138.

[13] Conca, C. 1985. On the application of the homogenization theory to a class of problems arising in fluid mechanics. J. Math. Pures Appl. 64: 31-75.

[14] Conca, C., and Donato, P. 1988. Non-homogeneous Neumann problems in domains with small holes. Mode'lisation Mathdmatique et Analyse Numdrique ( M A N ) 22 (4 ) : 561-607.

[15] Hornung, U. (ed.) 1997. Homogenization and Porous Media. IAM, Volume 6. Berlin, New York: Springer Verlag.

[16] Jikov, V.V., and Kozlov, S.M., and Oleinik, O.A. 1994. Homogenization of Dif- ferential Operators and Integral Functionals. Berlin-New York: Springer Verlag.

[17] Ladyzhenskaya, O.A. 1973. Boundary-Value Problems of Mathematical Physics. Moscow: Nauka.

[18] Landis, E.M., and Panasenko, G.P. 1977. A Theorem on the Asymptotics of Solutions of Elliptic Equations with Coefficients Periodic in All Variables Except One. Dokl. Akad. Nauk SSSR 235: 1253-1255. (English translation in Soviet Math. Dokl.)

[19] Lions, J.-L., and Magenes, E. 1968. Probldmes aux limites non homogdnes et applications. Volume I. Paris: Dunod.



[20] Mascarenhas, M.L., and PoliBevski, D. 1994. The Warping, the Torsion and the Neumann Problems in a Quasi - Periodically Perforated Domain. Mode'lisation Mathe'matique et Analyse Nume'rique ( M A N ) 28 (1): 37-57.

[21] Marchenko, V.A., and Khruslov, E.Ya. 1974. Boundary Value Problems in Do- mains with Fine-Grainy Boundary. Kiev: Naukova Dumka.

1221 Mortola, S., and Profeti, A. 1982. On the convergence of the minimum points of non equicoercive quadratic functionals. Comm. Partial Diferential Equations 7: 645-673.

[23] Murat, F., and Tartar, L. 1984. Calcul des variations et homoge'ne'isation. R 84012. Paris. Universite Pierre et Marie Curie, Centre National de la Recherche Scientifique, Laboratoire d'analyse numkrique.

[24] Oleinik, 0 . A., Shamaev, A. S. and Yosifian, G. A. 1992. Mathematical Problems in Elasticity and Homogenization. Amsterdam: North-Holland.

[25] Oleinik, O.A., and Iosif'yan, G.A. 1980. On the Behavior at Infinity of Solutions of Second Order Elliptic Equations in Domains with Noncompact Boundary. Math. Sb. 112 (154): 588-610. (English translation in Math. USSR Sb.).

[26] Oleinik, O.A., and Shaposhnikova, T.A. 1997. On the Homogenization of the Poisson Equation in Partially Perforated Domain with the Arbitrary Density of Cavities and Mixed Conditions on their Boundary. Rendiconti Lincei: Matemat- ica e Applicazioni, ser IX. 8 (3): 129-146.

[27] Sanchez-Palencia, E. 1987. Homogenization Techniques for Composite Media. Berlin-New York: Springer-Verlag.

[28] Sanchez-Palencia, E. 1980. Non Homogeneous Media and Vibration Theory. Berlin, New York: Springer-Verlag.

[29] Sobolev, S.L. 1991. Some Applications of Functional Analysis in Mathematical Physics. Third Edition. Translations of Mathematical Monographs Serie, Volume 90. Providence, Rhode Island: AMS Press.

[30] Tartar, L. 1977. Probldmes d'homoge'ne'isation duns les e'quations aux de'rive'es partielles, Cours Peccot Colldge de France. In: H-convergence Ed. F.Murat. SCminaire d'Analyse Fonctionnelle et NumCrique, 1977178, Universitk d'Alger.


Applicable Analysis Homogenization of Boundary-Value ...fian-pages.lebedev.ru/andrey/apan99.pdf · should be independently verified with primary sources. ... Russian Academy of Sciencies

Documents