Γ -convergence and H -convergence of linear elliptic operators

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1 Available online at J. Math. Pures Appl ) Γ -convergence and H -convergence of linear elliptic operators Nadia Ansini a, Gianni Dal Maso b, Caterina Ida Zeppieri c, a Dipartimento di Matematica G. Castelnuovo, Sapienza Università di Roma, Piazzale Aldo Moro 2, Roma, Italy b SISSA, Via Bonomea 265, Trieste, Italy c Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, Bonn, Germany Received 10 March 2012 Available online 12 September 2012 Abstract We consider a sequence of linear Dirichlet problems as follows { divσε u ε ) = f in, u ε H 1 0 ), with σ ε ) uniformly elliptic and possibly non-symmetric. Using purely variational arguments we give an alternative proof of the compactness of H -convergence, originally proved by Murat and Tartar Elsevier Masson SAS. All rights reserved. Résumé On considère une suite de problèmes de Dirichlet linéaires définis par { divσε u ε ) = f dans, u ε H 1 0 ), où σ ε ) est non-symétrique et uniformément elliptique. En utilisant une approche purement variationnelle on donne une démonstration alternative de la compacité de la H -convergence de Murat et Tartar Elsevier Masson SAS. All rights reserved. MSC: 35J15; 35J20; 49J45 Keywords: Linear elliptic operators; Γ -convergence; H -convergence 1. Introduction The notion of H -convergence was introduced by Murat and Tartar in [9,10] to study a wide class of homogenization problems for possibly non-symmetric elliptic equations. Let σ ε L ; R n n ) be a sequence of matrices satisfying uniform ellipticity and boundedness conditions on a bounded open set R n. We say that σ ε H -converges to * Corresponding author. Current address: Institut für Numerische und Angewandte Mathematik, Universität Münster, Einsteinstr. 62, Münster, Germany. addresses: ansini@mat.uniroma1.it N. Ansini), dalmaso@sissa.it G. Dal Maso), zeppieri@uni-bonn.de C.I. Zeppieri) /$ see front matter 2012 Elsevier Masson SAS. All rights reserved.

2 322 N. Ansini et al. / J. Math. Pures Appl ) amatrixσ 0 L ; R n n ) satisfying the same ellipticity and boundedness conditions if for every f H 1 ) the sequence u ε of the solutions to the problems { divσε u ε ) = f in, u ε H 1 0 ), 1.1) satisfy u ε u 0 weakly in H 1 0 ) and σ ε u ε σ 0 u 0 weakly in L 2 ; R n), where u 0 is the solution to { divσ0 u 0 ) = f in, u 0 H 1 0 ). The notion of Γ -convergence was introduced by De Giorgi and Franzoni in [4,5] to study the asymptotic behavior of the solutions of a wide class of minimization problems depending on a parameter ε>0, which varies in a sequence converging to 0. Let X, d) be a metric space and let F ε : X R be a sequence of functionals, we say that F ε Γd)-converges to a functional F 0 : X R if for all x X we have i) liminf inequality) for every sequence x ε ii) limsup inequality) there exists a sequence x ε d x in X F 0 x) lim inf F εx ε ); d x in X such that F 0 x) lim sup F ε x ε ). It has been proved that when σ ε is symmetric, Eq. 1.1) has a variational structure since it can be seen as the Euler Lagrange equation associated with F ε u) = 1 σ ε x) u udx fudx, 2 or, equivalently, as the solution to the minimization problem min { F ε u): u H 1 0 )}. 1.2) Therefore in this case 1.2) provides a variational principle for the Dirichlet problem 1.1) and the convergence of the solutions of 1.1) can be equivalently studied by means of the Γ -convergence, with respect to the weak topology of H0 1), of the associated functionals F ε or in terms of the G-convergence of the uniformly elliptic, symmetric matrices σ ε ) see De Giorgi and Spagnolo [6]). In this paper we consider the equivalence between H -convergence and Γ -convergence in the possibly nonsymmetric case. To every elliptic matrix σ L ; R n n ) we associate a suitable quadratic integral functional F : L 2 ; R n ) H0 1 ) [0, + ) see 2.12)) and we consider the Γ -convergence with respect to the distance d defined by d α, ϕ), β, ψ) ) = α β H 1 ;R n ) + divα β) H 1 ) + ϕ ψ L 2 ). We prove Theorem 3.2) that the H -convergence of σ ε to σ 0 is equivalent to the Γd)-convergence of the functionals F ε corresponding to σ ε to the functional F 0 corresponding to σ 0.In[2] this result was proved using compactness properties of H -convergence [9,10], while in the present paper the equivalence is obtained as a consequence of a general compactness theorem for integral functionals with respect to Γd)-convergence [1]. Moreover, as a consequence of the results proved in [1], we also give an independent proof Theorem 3.1) of the compactness of H -convergence based only on Γ -convergence arguments.

3 N. Ansini et al. / J. Math. Pures Appl ) Notation and preliminaries In this section we introduce a few notation and we recall some preliminary results we employ in the sequel. For any A R n n we denote by A s and A a the symmetric and the anti-symmetric part of A, respectively; i.e., A s := A + AT 2, A a := A AT 2 where A T is the transpose matrix of A. We use bold capital letters to denote matrices in R 2n 2n. The scalar product of two vectors ξ and η is denoted by ξ η. Let be an open bounded subset of R n.for0<c 0 c 1 < +, Mc 0,c 1,) denotes the set of matrix-valued functions σ L ; R n n ) satisfying σx)ξ ξ c 0 ξ 2, or, equivalently, satisfying σx)ξ ξ c 0 ξ 2, Note that 2.1)or2.2)) implies that, σ 1 x)ξ ξ c 1 1 ξ 2, for every ξ R n, for a.e. x, 2.1) σ x)ξ ξ c 1 σx)ξ 2, for every ξ R n, for a.e. x. 2.2) 1 σx) c1 for a.e. x, and that necessarily c 0 c 1. To not overburden notation, in all that follows we always write σ in place of σx). Given σ Mc 0,c 1,) we consider the 2n 2n)-matrix-valued function Σ L ; R 2n 2n ) having the following block structure: σ s ) 1 σ s ) 1 σ a ) Σ := σ a σ s ) 1 σ s σ a σ s ) 1 σ a. 2.3) Notice that Σ is symmetric. Moreover, the assumption σ Mc 0,c 1,)easily implies that Σ is uniformly coercive see [2, Section 3.1.1] for the details); specifically, there exists a constant Cc 0,c 1 )>0, depending only on c 0 and c 1, such that Σw w Cc 0,c 1 ) w 2, 2.4) for every w R 2n, and a.e. in. If we consider the matrix-valued functions A,B,C L ; R n n ) defined as A = σ s) 1, B = σ s ) 1 σ a, C = σ s σ a σ s) 1 σ a, 2.5) the matrix Σ can be rewritten as ) A B Σ = B T. 2.6) C We notice that, for a.e. x, the matrix Σ belongs to the indefinite special orthogonal group SOn, n); i.e., ) 0 I ΣJΣ = J a.e. in, with J =, 2.7) I 0 where I R n n is the identity matrix see [2, Section 3.1.1]). Moreover, taking into account the symmetry of Σ,itis immediate to show that 2.7) is equivalent to the following system of identities for the block decomposition 2.6): AB T + BA = 0 AC + B 2 = I a.e. in. 2.8) CB + B T C = 0 Conversely, one can prove that, if M L ; R 2n 2n ) is symmetric and has the block decomposition ) A B M =, 2.9) C B T

4 324 N. Ansini et al. / J. Math. Pures Appl ) with A,B,C L ; R n n ), A and C symmetric, and det A 0, then the first two equations in 2.8) imply the third one, and 2.8) implies that M is equal to the matrix Σ defined in 2.3) with σ = A 1 A 1 B see [2, Proposition 3.1]). Throughout the paper the parameter ε varies in a strictly decreasing sequence of positive real numbers converging to zero. Let σ ε ) be a sequence in Mc 0,c 1,) and consider the sequence Σ ε ) L ; R 2n 2n ) defined by 2.3) with σ = σ ε.letq ε : L 2 ; R n ) L 2 ; R n ) [0, + ) be the quadratic forms associated with Σ ε ; i.e., a Q ε a, b) := Σ ε b Their gradients grad Q ε : L 2 ; R n ) L 2 ; R n ) L 2 ; R n ) L 2 ; R n ) are given by ) a b ) dx. 2.10) grad Q ε a, b) = A ε a + B ε b,b T ε a + C εb ), 2.11) where A ε,b ε, and C ε areasin2.5) with σ = σ ε. We also consider the quadratic forms F ε : L 2 ; R n ) H 1 0 ) [0, + ) defined by F ε α, ψ) := Q ε α, ψ). 2.12) For every λ,μ H 1 ), we consider the sequence of constrained functionals Fε λ,μ : L 2 ; R n ) H0 1) [0, + ] defined as follows: { Fε λ,μ Fε α, ψ) μ, ψ if div α = λ, α, ψ) := 2.13) + otherwise, where, denotes the dual paring between H 1 ) and H0 1). Given a symmetric matrix M L ; R 2n 2n ), we consider the quadratic functionals Q M : L 2 ; R n ) L 2 ; R n ) [0, + ) and F M : L 2 ; R n ) H0 1 ) [0, + ) defined by ) ) a a Q M a, b) := M dx and F b b M α, ψ) := Q M α, ψ). 2.14) Considering the block decomposition 2.9), the gradient of Q M is given by grad Q M a, b) = Aa + Bb,B T a + Cb ). 2.15) Finally, for every λ,μ H 1 ), we consider the constrained functional F λ,μ M : L2 ; R n ) H0 1 ) [0, + ] defined as follows F λ,μ M α, ψ) := { FM α, ψ) μ, ψ if div α = λ, + otherwise. Let w be the weak topology of L 2 ; R n ) H0 1) and let d be the distance in L2 ; R n ) H0 1 ) defined by d α, ϕ), β, ψ) ) := α β H 1 ;R n ) + divα β) H 1 ) + ϕ ψ L 2 ). The following result is proved in [1, Corollary 2.5]. Theorem 2.1. Let σ ε ) be a sequence in Mc 0,c 1,). There exist a subsequences of ε, not relabeled, and a symmetric matrix M L ; R 2n 2n ), such that the functionals F ε defined by 2.12) Γd)-converge to the functional F M defined in 2.14). Moreover, M is positive definite and satisfies the coercivity condition 2.4). The following result is a consequence of [1, Theorem 3.3] and of the stability of Γ -convergence under continuous perturbations. Theorem 2.2. Let σ ε ) be a sequence in Mc 0,c 1,)and let M L ; R 2n 2n ) be a symmetric, positive definite matrix satisfying 2.4). Assume that the functionals F ε defined by 2.12) Γd)-converge to the functional F M defined

5 N. Ansini et al. / J. Math. Pures Appl ) in 2.14). Then, for every λ,μ H 1 ), the functionals Fε λ,μ ) defined by 2.13) Γw)-converge to the functional F λ,μ defined by { F λ,μ FM α, ψ) μ, ψ if div α = λ, α, ψ) := + otherwise. For the reader s sake, here we briefly recall a fundamental tool we employ in what follows, the Cherkaev Gibiansky variational principle [3] see also Fannjiang and Papanicolaou [7] and Milton [8]), which will be presented in the notational setting which is suitable for our purposes. Loosely speaking, this variational principle amounts to associate to the two following Dirichlet problems { divσε u ε ) = f in, u ε H0 1), { div σ T ε v ε ) = g in, v ε H 1 0 ). 2.16) with f,g H 1 ), a quadratic functional whose Euler Lagrange equation is solved by a suitable combination of solutions to 2.16) and of their momenta. We set a ε := σ ε u ε and b ε := σ T ε v ε. 2.17) For every ε>0, λ,μ H 1 ) the unique minimizer α ε,ψ ε ) of Fε λ,μ satisfies the constraint div α ε = λ and the following system of Euler Lagrange equations: A ε α ε + B ε ψ ε ) βdx= 0, ) 2.18) B T ε α ε + C ε ψ ε ϕdx= μ, ϕ, for every β L 2 ; R n ) with div β = 0 and for every ϕ H 1 0 ). If u ε,v ε satisfy 2.16) then we can prove see [2, Section 3.2] for details) that the pair solves 2.18), with λ = f + g, μ = f g, and thus minimizes Fε. In the same way, it can be seen that the pair a ε + b ε,u ε v ε ) 2.19) minimizes Fε. a ε b ε,u ε + v ε ) 2.20) 3. The main result In this section we state and prove the main result of this paper: an alternative and purely variational proof of the sequential compactness of Mc 0,c 1,)with respect to H -convergence, originally proved by Murat and Tartar [9,10]. Theorem 3.1 Compactness of H -convergence). Let σ ε ) be a sequence in Mc 0,c 1,). Then there exist a subsequence not relabeled) and a matrix σ 0 Mc 0,c 1,) such that σ ε )H-converges to σ 0 and σ T ε )H-converges to σ T 0. Proof. By Theorem 2.1 there exist a subsequence of F ε, not relabeled, and a symmetric, positive definite matrix M L ; R 2n 2n ), with the block decomposition 2.9), such that F ε Γd)-converges to F M. In the rest of this proof we show that σ ε )H-converges to σ 0 and σ T ε )H-converges to σ T 0, where σ 0 := A 1 A 1 B. Let f,g H 1 ), letu ε,v ε be as in 2.16), and let a ε,b ε be as in 2.17). By standard variational estimates we have that u ε ) and v ε ) are bounded in H 1 0 ) while a ε) and b ε ) are bounded in L 2 ; R n ). Therefore, up to subsequences not relabeled),

6 326 N. Ansini et al. / J. Math. Pures Appl ) u ε u 0, v ε v 0 weakly in H 1 0 ) and a ε a 0, b ε b 0 weakly in L 2 ; R n), 3.1) for some u 0,v 0 H0 1) and a 0,b 0 L 2 ; R n ). Since a ε + b ε,u ε v ε ) are minimizers of Fε and these functionals Γ -converge to FM by Theorem 2.2, appealing to the fundamental property of Γ -convergence we find that lim Fε a ε + b ε,u ε v ε ) = F Similarly, since a ε b ε,u ε + v ε ) minimizes F ε M a 0 + b 0,u 0 v 0 ) = min FM. 3.2),wehavealso lim Fε a ε b ε,u ε + v ε ) = F M a 0 b 0,u 0 + v 0 ) = min FM. 3.3) Thanks to Theorem 2.2, 3.2), 3.3), and in view of [1, Proposition 2.6] we are now in a position to invoke the result about the convergence of momenta proved in [1, Corollary 4.6], hence we obtain grad Q ε a ε + b ε, u ε v ε ) grad Q M a + b, u v), 3.4) grad Q ε a ε b ε, u ε + v ε ) grad Q M a 0 b 0, u 0 + v 0 ) 3.5) weakly in L 2 ; R n ) L 2 ; R n ).By2.11) and 2.15), considering only the first component, we get A ε a ε + b ε ) + B ε u ε v ε ) Aa 0 + b 0 ) + B u 0 v 0 ), 3.6) A ε a ε b ε ) + B ε u ε + v ε ) Aa 0 b 0 ) + B u 0 + v 0 ) 3.7) weakly in L 2 ; R n ). Since by 2.5) A ε a ε +b ε )+B ε u ε v ε ) = u ε + v ε and A ε a ε b ε )+B ε u ε + v ε ) = u ε v ε, from 3.6) and 3.7) we deduce that u ε + v ε Aa 0 + b 0 ) + B u 0 v 0 ), 3.8) u ε v ε Aa 0 b 0 ) + B u 0 + v 0 ) 3.9) weakly in L 2 ; R n ). Hence, adding up 3.8) and 3.9) entails u ε Aa 0 + B u 0 in L 2 ; R n ), which gives u 0 = Aa 0 + B u 0 by 3.1). This implies with a 0 = σ 0 u 0, 3.10) σ 0 := A 1 A 1 B. 3.11) Since div a ε = f,by2.16) and 2.17) we get that div a 0 = f. Hence, 3.10) implies that u 0 is the solution to { divσ0 u 0 ) = f in, u 0 H 1 0 ). 3.12) So far we have proved that for every f H 1 ) the solutions u ε of 2.16) converge weakly in H0 1 ) to the solution u 0 of 3.12) and their momenta σ ε u ε converge weakly in L 2 ; R n ) to σ 0 u 0. Thus, to conclude the proof of the H -convergence of σ ε ) to σ 0 it remains to show that σ 0 belongs to Mc 0,c 1,). To this end, let u H0 1 ) and choose f := divσ 0 u); 3.13) in this way the solution u 0 of Eq. 3.12) coincides with u. Let ϕ Cc ).Usingϕu ε as a test function in the equation divσ ε u ε ) = f and then passing to the limit on ε we get ) fϕu 0 dx = lim fϕu ε dx = lim σ ε u ε u ε )ϕ dx + σ 0 u 0 u 0 ϕdx, 3.14)

7 N. Ansini et al. / J. Math. Pures Appl ) where to compute the limit of the last term in 3.14) we appealed to the strong L 2 ) convergence of u ε to u 0.On the other hand, since by 3.12) from 3.14) we deduce that fϕu 0 dx = lim σ 0 u 0 u 0 )ϕ dx + σ ε u ε u ε )ϕ dx = σ 0 u 0 u 0 ϕdx σ 0 u 0 u 0 )ϕ dx, 3.15) for every ϕ Cc ). Hence, choosing ϕ 0, combining 3.15), the first condition in 2.1), and the equality u = u 0, we have σ 0 u u)ϕ dx c 0 lim inf u ε 2 ϕdx c 0 u 2 ϕdx, the second inequality following from u ε u 0 = u in L 2 ; R n ). Since this inequality holds true for every ϕ Cc ), ϕ 0, we get that σ 0 u u c 0 u 2 a.e. in, 3.16) for every u H0 1 ). Using the second condition in 2.2), we find σ 0 u u)ϕ dx c 1 1 lim inf σ ε u ε 2 ϕdx c 1 1 σ 0 u 2 ϕdx, since σ ε u ε σ 0 u 0 = σ 0 u in L 2 ; R n ). From the previous inequality we deduce σ 0 u u c 1 1 σ 0 u 2 a.e. in, 3.17) for every u H0 1 ). Finally, 2.2) follows from 3.16) and 3.17) by taking u to be affine in an open set ω. We now prove that σε T H -converges to σ0 T. Subtracting 3.9) from3.8) gives v ε Ab 0 B v 0 weakly in L 2 ; R n ); the latter combined with 3.1) imply that v 0 = Ab B v 0. We deduce then where b 0 = σ v 0, 3.18) σ := A 1 + A 1 B. 3.19) Since div b ε = g by 2.16) and 2.17), we get div b 0 = g, so that 3.18)impliesthatv 0 is the solution to { div σ v0 ) = g in, v 0 H 1 0 ). 3.20) As in the previous part of the proof, this implies that σ T ε H -converges to σ. We want to prove that σ = σ T 0. To this end, we argue as in the previous step. Let u, v H 1 0 ). We choose f := divσ 0 u) and g := div σ v) and we consider the corresponding solutions u ε and v ε of 2.16). Since u coincides with the solution u 0 of 3.12) and v coincides with the solution v 0 of 3.20), the H -convergence of σ ε entails u ε u 0 = u weakly in H 1 0 ) and σ ε u ε σ 0 u 0 = σ 0 u weakly in L 2 ; R n), while the H -convergence of σ T ε ) yields v ε v 0 = v weakly in H 1 0 ) and σ T ε v ε σ v 0 = σ v weakly in L 2 ; R n).

8 328 N. Ansini et al. / J. Math. Pures Appl ) Let ϕ Cc ); usingϕv ε as test function in the equation for u ε, we get fϕv ε )dx = σ ε u ε v ε )ϕ dx + σ ε u ε v ε ϕdx. Therefore, appealing to the strong L 2 ) convergence of v ε to v and using ϕv as a test function in 3.12), we obtain lim σ ε u ε v ε )ϕ dx = fϕv)dx σ 0 u v ϕdx = σ 0 u ϕv) dx σ 0 u v ϕdx= σ 0 u v)ϕ dx. 3.21) Moreover, arguing in a similar way, using now ϕu ε as test function in the equation for v ε, it is easy to show that ) lim σ T ε v ε u ε ϕdx= σ v u)ϕ dx. 3.22) Then 3.21) and 3.22) yield σ 0 u v)ϕ dx = σ u v)ϕ dx for every ϕ Cc ). Arguing as in the previous proof of 2.2) we deduce from this equality that for every ξ,η R n. This implies that σ = σ T 0 σ 0 ξ η = ση ξ a.e. in, a.e. in which concludes the proof of the theorem. Given σ 0 Mc 0,c 1,), the matrix Σ 0 and the functionals Q 0, F 0, and F λ,μ 0 are defined as in 2.6), 2.10), 2.12), and 2.13) with σ = σ 0. Theorem 3.2. Let σ ε ) be a sequence in Mc 0,c 1,) and let σ 0 Mc 0,c 1,). The following conditions are equivalent: a) σ ε H -converges to σ 0 ; b) σε T H -converges to σ 0 T ; c) F ε Γd)-converges to F 0 ; d) F λ,μ ε Γw)to F λ,μ 0 for every λ,μ H 1 ). Proof. The equivalence between a) and b) follows immediately from Theorem 3.1. The implication c) d) is given by Theorem 2.2. The implication d) a) is obtained in the proof of Theorem 3.1. It remains to prove that a) and b) imply c). By Theorem 2.1 we may assume that F ε Γd)-converges to F M where M L ; R 2n 2n ) is a positive definite, symmetric matrix satisfying the coercivity condition 2.4). To prove that M SOn, n) we consider the block decomposition 2.9). In Theorem 3.1 we proved that σ 0 = A 1 A 1 B and σ T 0 = σ = A 1 + A 1 B; hence, we immediately deduce that AB T + BA = 0 a.e. in. 3.23) It remains to prove the second condition in 2.8). Let us fix f,g H 1 ) and let u ε,v ε,a ε,b ε be as in 2.16) and 2.17). By 2.11), 2.15), 3.4), and 3.5) using only the second component we get B T ε a ε + b ε ) + C ε u ε v ε ) = σ ε u ε σ T ε v ε B T a 0 + b 0 ) + C u 0 v 0 ), 3.24) B T ε a ε b ε ) + C ε u ε + v ε ) = σ ε u ε + σ T ε v ε B T a 0 b 0 ) + C u 0 v 0 ) 3.25) weakly in L 2 ; R n ). Then, adding up 3.24) and 3.25) we get

9 N. Ansini et al. / J. Math. Pures Appl ) σ ε u ε B T a 0 + C u 0 weakly in L 2 ; R n ); on the other hand, since σ ε u ε = a ε a 0 weakly in L 2 ; R n ), we obtain a 0 = B T a 0 + C u 0. Since in the proof of Theorem 3.1 we already showed that a 0 = A 1 A 1 B) u 0, we finally obtain I B T ) A 1 A 1 B ) u 0 = C u 0 a.e. in. Therefore, suitably choosing f as in 3.13) and arguing as in the proof of Theorem 3.1 we can easily deduce that I B T ) A 1 A 1 B ) ξ = Cξ a.e. in, for every ξ R n, thus, by the arbitrariness of ξ R n, we get I B T ) A 1 A 1 B ) = C a.e. in. The latter combined with 3.23) leads to AC + B 2 = I a.e. in. 3.26) Eventually, by 3.23) and 3.26) we can apply [2, Proposition 3.1] and we deduce that M SOn, n) a.e. in and that M is equal to the matrix Σ defined in 2.3) with σ = A 1 A 1 B. Since we have also σ 0 = A 1 A 1 B,we conclude that M = Σ 0. Acknowledgements This material is based on work supported by the Italian Ministry of Education, University, and Research under the Project Variational Problems with Multiple Scales 2008 and by the European Research Council under Grant No Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture. C.Z. has been partially supported by the Deutsche Forschungsgemeinschaft through the Forschergruppe 797 and the Sonderforschungsbereich 611. References [1] N. Ansini, G. Dal Maso, C.I. Zeppieri, New results on Ɣ-limits of integral functionals, preprint, available at [2] N. Ansini, C.I. Zeppieri, Asymptotic analysis of non-symmetric linear operators via Γ -convergence, SIAM J. Math. Anal ) [3] A.V. Cherkaev, L.V. Gibiansky, Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli, J. Math. Phys. 1) ) [4] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell area, Rend. Mat ) [5] E. De Giorgi, T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. Fis. Natur ) [6] E. De Giorgi, S. Spagnolo, Sulla convergenza degli integrali dell energia per operatori ellittici del secondo ordine, Boll. Unione Mat. Ital ) [7] A. Fannjiang, G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math ) [8] G.W. Milton, On characterizing the set of possible effective tensors of composites: the variational method and the translation method, Comm. Pure Appl. Math. 43 1) 1990) [9] F. Murat, H -convergence, in: Séminaire d Analyse Fonctionelle et Numérique de l Université d Alger, [10] L. Tartar, Cours Peccot au Collège de France, Paris, 1977.