Duality Results in the Homogenization of Two-dimensional High-contrast Conductivities

The paper deals with some extensions of the Keller-Dykhne duality relations arising in the classical homogenization of two-dimensional uniformly bounded conductivities, to the case of high-contrast conductivities. Only assuming a L 1-bound on the conductivity we prove that the conductivity and its dual converge respectively, in a suitable sense, to the homogenized conductivity and its dual. In the periodic case a similar duality result is obtained under a less restrictive assumption.


Introduction
The homogenization of elliptic partial differential equations has had an important development for nearly forty years.During the seventies, the G-convergence of Spagnolo [24], and the H-convergence of Murat, Tartar [25], [23], as well as the study of periodic structures by Bensoussan, Lions, Papanicolaou [4] (see also [15]), laid the foundations of the homogenization theory in conduction problems with uniformly bounded (both from below and above) conductivities.
The boundedness assumption implies some compactness which preserves the nature of the homogenized problem.This is no more the case for high-contrast conductivities.Indeed, Khruslov was one of the first to derive vector-valued homogenized problems in the case of low conductivities [17], as well as nonlocal homogenized ones in the case of high conductivities [12] (see also [18] and [19] for various types of homogenized problems and complete references).In the case of high conductivities, the appearance of nonlocal effects is strongly linked to the dimension greater than two.So, the model example of nonlocal homogenization [12] in conduction is obtained from a three-dimensional homogeneous medium reinforced by highly conducting thin fibers which create a capacitary effect (see also [3], [6] and [10] for extensions and alternative methods).
Recently, Casado-Díaz and the first author proved in [5], [8], [9], that dimension two, contrary to dimension three or greater, induces an extra compactness which prevents from the nonlocal effects.In particular, an extension of the H-convergence is obtained in [8] for conductivities which are only bounded in L 1 but not in L ∞ .
The present paper deals with the duality relations arising in the two-dimensional homogenization.These relations were first noted by Keller [16] who obtained an interchange equality relating the effective properties of a two-phase composite when the conductivities are swapped.Following the pioneer work of Keller, Dykhne [11] (see also [21] and [13] for a more general approach) proved that, for any periodic, coercive and bounded matrix-valued function A, the homogenized matrix associated with the dual conductivity A T / det A (where A T denotes the transposed of A) is equal to A T * / det A * , where A * is the constant homogenized matrix associated with A. We refer to Chapters 3, 4 of [22] for a general presentation of the duality transformations.
Our contribution is the extension of the Dykhne duality relation to high-contrast two-dimensional conductivities.More precisely, consider an equicoercive sequence A n of (not necessarily symmetric) conductivity matrices, which is not uniformly bounded contrary to the classical case.Under the main assumption that det A n det A s n |A s n | weakly- * converges in the sense of the Radon measures to a bounded function, (1.1) (where A s n denotes the symmetrized of A n ), we prove (see Theorem 2.2) that the sequence A T n / det A n "H-converges" to A T * / det A * , when A n "H-converges" to A * , for suitable extensions of the Hconvergence (see Definition 2.1).As a consequence, we obtain (see Corollary 2.4) a compactness result for the opposite case of a uniformly bounded but not equicoercive sequence of conductivity matrices.We also prove a refinement (see Theorem 2.7) in the periodic case, i.e.A n (x) := A n ( x εn ) where A n is Y -periodic and ε n > 0 tends to 0, under the less restrictive assumption than (1.1) 2 The paper is organized as follows.In Section 2, we define some appropriate notions of Hconvergence and we state the main duality results for high-contrast conductivities, both in the nonperiodic and periodic framework.Section 3 is devoted to the proof of the homogenization results.

Notations
• Ω denotes a bounded open subset of R 2 ; • I denotes the unit matrix in R 2×2 , and J the rotation matrix of angle 90 • ; • for any matrix A in R 2×2 , A T denotes the transposed of the matrix A, A s denotes its symmetric part in such a way that A = A s + aJ, where a ∈ R; • for any matrices A, B ∈ R 2×2 (even non-symmetric), A ≤ B means that A s ≤ B s , i.e., for any ξ ∈ R 2 , Aξ • ξ ≤ Bξ • ξ; • | • | denotes both the euclidian norm in R d and the subordinate norm in R 2×2 , i.e., for any A ∈ R 2×2 , |A| := sup {|Ax| : |x| = 1}, which agrees with the spectral radius of A if A is symmetric; • for any α, β > 0 , M (α, β; Ω) denotes the set of the matrix-valued functions • for Y := (0, 1) 2 and for V := L p , W 1,p , V # (Y ) denotes the Y -periodic functions which belong to V loc (R 2 ); • for any locally compact subset X of R 2 , M(X) denotes the space of the Radon measures defined on X; • c denotes a constant which may vary form a line to another one.
2 Statement of the results

The general case
We consider a sequence of two-dimensional conduction problems in which the conductivity matrixvalued is either not uniformly bounded from above or (exclusively) not equicoercive.As a consequence, either the associated flux is not bounded in L 2 or the associated potential is not bounded in H 1 .To take into account these two degenerate cases we extend the definition of the classical Murat-Tartar H-convergence (see [23]) by the following way: Definition 2.1.Let α n and β n be two sequences of positive numbers such that α n ≤ β n , and let A n be a sequence of matrix-valued functions in M (α n , β n ; Ω) (see (1.3)).
• The sequence A n is said to H(M(Ω) 2 )-converge to the matrix-valued function A * ∈ M (α, β; Ω), with 0 < α ≤ β, if for any distribution f in H −1 (Ω), the solution u n of the problem satisfies the convergences where u is the solution of the problem We denote this convergence by • The sequence A n is said to H(L 2 (Ω) 2 )-converge to the matrix-valued function A * ∈ M (α, β; Ω), with 0 < α ≤ β, if for any function f in L 2 (Ω), the solution u n of (2.1) satisfies the convergences where u is the solution of (2.3).We denote this convergence by A n The main result of the paper is the following: Let Ω be a bounded open set of R 2 such that |∂Ω| = 0. Let α > 0, let β n , n ∈ N, be a sequence of real numbers such that β n ≥ α, and let A n be a sequence of matrix-valued functions (not necessarily symmetric) in M (α, Then, there exists a subsequence of n, still denoted by n, and a matrix-valued function ii) In addition to the assumptions of i), assume that there exists a constant C 0 > 0 such that, for Then, we have Remark 2.3.The part i) is a two-dimensional extension of the H-convergence for unbounded sequences of equicoercive matrix-valued functions.It was first proved in [8] under the following assumption: there exists a constant γ > 0 and ā (2.9) Assumption (2.9) is more restrictive than (2.5) since which converges to a bounded function in the weak- * sense of the measures on Ω, hence convergence (2.5).The proof of (2.6) is quite similar to the one in [8] up to a few extra computations (see [20] for details).
On the contrary, the part ii) of Theorem 2.2 is a new result which extends the duality result obtained by Dykhne [11] for periodic and uniformly bounded conductivities to non-periodic and nonuniformly bounded ones.Condition (2.7) is a technical assumption we need in the non-symmetric case.Indeed, (2.7) clearly holds with Part ii) will be proved in Section 3.
Theorem 2.2 implies the following H-convergence result for uniformly bounded sequences of matrixvalued functions which are not equicoercive: Corollary 2.4.Let Ω be a bounded open set of R 2 such that |∂Ω| = 0. Let β > 0 and let α n be a sequence of real numbers such that 0 < α n ≤ β.Let B n be a sequence of matrix-valued functions in M (α n , β; Ω).Assume that there exist a function a in L ∞ (Ω) such that and a constant C 0 > 0 such that, for any n ∈ N, Then, there exists a subsequence of n, still denoted by n, and a matrix-valued function (2.12) Proof.The sequence A n defined by Then, by the part i) of Theorem 2.2, the sequence A n (up to a subsequence) ; Ω .Therefore, by the part ii) of Theorem 2.2, B n H(L 2 (Ω) 2 )converges to the matrix-valued function The matrix-valued function B * clearly belongs to the set M (α, β; Ω), with α := 2 a L ∞ (Ω) −1 , which concludes the proof.

The periodic case
In this section we consider the case of highly oscillating sequences of conductivity matrices.Let Ω be a bounded open subset of R 2 , and let Y := (0, 1) 2 be the unit square of R 2 .Let A n be a sequence of , and let ε n be a sequence of positive numbers which tends to 0. We define the highly oscillating sequence associated with A n and ε n by For a fixed n ∈ N, let A * n be the constant matrix defined by where (2.17) Note that A * n is the H-limit of the oscillating sequence A n ( x ε ) as ε tends to 0 (see e.g. the periodic homogenization in [4]).Under the periodicity assumption (2.15) we can improve Theorem 2.2.To this end, we need a more general definition of H-convergence than the one of Definition 2.1: Definition 2.5.Let α n and β n be two sequences of positive numbers such that α n ≤ β n , and let A n be a sequence of matrix-valued functions in M (α n , β n ; Ω).
• The sequence A n is said to H s -converge to the matrix-valued function A * ∈ M (α, β; Ω), with 0 < α ≤ β, if for any function f in L 2 (Ω), the solution u n of problem (2.1) strongly converges in L 2 (Ω) to the solution u of problem (2.3).
We denote this convergence by A n Hs − A * .
• The sequence A n is said to H w -converge to the matrix-valued function A * ∈ M (α, β; Ω), with 0 < α ≤ β, if for any function f in L 2 (Ω), the solution u n of problem (2.1) weakly converges in L 2 (Ω) to the solution u of problem (2.3) and the flux A n ∇u n weakly converges to We denote this convergence by A n Hw − A * .
Remark 2.6.In the part i) of Definition 2.5 we have the strong convergence of the potential but not the convergence of the flux.This corresponds to the case of an equicoercive sequence of conductivity matrices without control from above.In the part ii) we have the weak convergence of both the potential and the flux.This corresponds to the case of a uniformly bounded sequence of conductivity matrices without control from below.
We have the following periodic homogenization result: Theorem 2.7.Let α > 0 and let β n be a sequence of real numbers such that β n ≥ α.Let A n be a sequence of Y -periodic matrix-valued functions (not necessarily symmetric) in M (α, β n ; R 2 ), and let A n be the highly oscillating sequence associated with A n by (2.15).i) Assume that the sequence A * n defined by (2.16) converges to A * in R 2×2 , and that the following limit holds Then, we have ii) In addition to the assumptions of i) assume that A n and A T n satisfy inequality (2.7), and that the solution u n of (2.1), with the matrix A T n / det A n , is bounded in L 2 (Ω) for any right-hand side f in L 2 (Ω).Then, we have which is clearly more restrictive than condition (2.18).The price to pay is that the sequence A n ∇u n is not necessarily bounded in L 1 (Ω) 2 .
In the part ii) of Theorem 2.7 we have to assume the L 2 (Ω)-boundedness of any solution of (2.1) with conductivity matrix A T n / det A n , since condition (2.18) does not imply it.To this end, it is sufficient to assume the existence of a constant C > 0 such that, for any n ∈ N,  [2] for the derivation of a similar estimate).The proof of (2.21) is based on the extension property established in [1] (see [20] for more details).
3 Proof of the results

Proof of Theorem 2.2
Taking into account Remark 2.3 we focus on the part ii) of Theorem 2.2.Consider a sequence A n in M (α, β n ; Ω) which satisfies convergence (2.5) and H(M(Ω) 2 )-converges to A * in M (α, β; Ω), with 0 < α ≤ β, and set B n := J −1 A −1 n J. Let f ∈ L 2 (Ω) and let v n be the solution of the conduction problem (2.1) with conductivity matrix B n .The proof of the H(L 2 (Ω) 2 )-convergence (2.8) is divided into two steps.In the first step, we prove that the sequence v n strongly converges in L 2 loc (Ω) to some v ∈ H 1 0 (Ω), and that the flux B n ∇v n weakly converges to some ξ in L 2 (Ω).The second step is devoted to the determination of the limit ξ in order to establish convergence (2.8).
Moreover, by the Cauchy-Schwarz inequality combined with (2.14) we have .
Then, we deduce from the previous inequalities and (2.5) that The strong convergence of v n in L 2 loc (Ω) is a consequence of the following result which is proved in [8] (see the steps 3, 4 of the proof of Theorem 2.1 in [8], as well as the first step of Theorem 2.7 i), which uses similar arguments adapted to condition (2.18)): Lemma 3.1.Let S n be a sequence of symmetric matrix-valued functions in L ∞ (Ω) 2×2 such that there exist α > 0 and a ∈ L ∞ (Ω) satisfying Then, the sequence It remains to prove that v belongs to H 1 0 (Ω).Let Φ ∈ C 1 ( Ω) 2 .Using successively the Cauchy-Schwarz inequality and (3.2) we have . Therefore, passing to the limit in the previous inequality thanks to the weak convergence of v n , to equality (2.14) and to convergence (2.5), we get which implies that v belongs to H 1 0 (Ω).
Second step : Determination of the limit ξ of B n ∇v n .Let λ ∈ R 2 , θ ∈ C 1 c (Ω), and let w λ n be the solution of the problem By (2.6) and by virtue of Definition 2.1 we have the following convergences Now, we will pass to the limit in the product B n ∇v n • JA T n ∇w λ n by two different ways, which will give the desired limit ξ.
On the one hand, since B n = J −1 A −1 n J and J 2 = −I, we have Moreover, since J∇w λ n is divergence free, we have . Therefore, we obtain the first convergence On the other hand, consider a regular simply connected open subset ω of Ω.Since by definition (3.5) , there exists a stream function (see e.g.[14]) wλ n in H 1 (ω) uniquely defined by Since A T n ∇w λ n is bounded in L 1 (Ω) 2 by (3.6) and wλ n has a zero ω-average, the Sobolev imbedding of W 1,1 (ω) into L 2 (ω) combined with the Poincaré-Wirtinger inequality in ω implies that wλ n is bounded in L 2 (ω) and thus converges, up to a subsequence, to a function wλ in L 2 (ω).Moreover, by the Cauchy-Schwarz inequality and (3.5) we have, with in ω, hence wλ n strongly converges to wλ in L 2 loc (ω).Moreover, the second convergence of (3.6) and definition (3.8) imply that wλ has a zero ω-average and ∇ wλ = 0 in D (ω), hence wλ = 0 by the connectedness of ω.Therefore, by the uniqueness of the limit we get for the whole sequence wλ n −→ 0 strongly in L 2 loc (ω).
(3.9)By (3.8) we have Clearly, the sequence Moreover, the strong convergence (3.9) implies that Therefore, we obtain This combined with (3.7) yields (Ω) such that θ = 1 in ω in the former equality.Therefore, due to the arbitrariness of λ and ω we get the equality J∇v = A * Jξ a.e. in Ω, hence ξ = J −1 A −1 * J∇v = B * ∇v a.e. in Ω, which concludes the proof.

Proof of Theorem 2.7
Proof of the part i) of Theorem 2.7.The proof is similar to the one of the compactness result in [5].But there are extra difficulties since the conductivity matrices are not symmetric and the fluxes are not necessarily bounded in L 1 (Ω), due to the condition (2.18).We will give the main steps of the proof pointing out these difficulties.
Let u n be the solution of the conduction problem (2.1), where A n is the highly ocillating sequence (2.15).Let λ ∈ R 2 , and let V λ n be the unique solution of problem (2.17) with the matrix-valued function (A n ) T .Note that the matrix A * n defined by (2.16) and V λ n satisfy the relation Note that the second estimate of (3.10) and the α-coerciveness of A n imply that the sequence (

.11)
To prove the H s -convergence (2.19) it is enough to prove that where A * is the limit of A * n in R 2×2 , and u is the weak limit of u n in H 1 0 (Ω).To this end, we proceed in two steps.In the first step, we prove the convergence and in the second one, the convergence First step : Proof of (3.12).
Let ω be a regular simply connected subset of Ω, let v ∈ H 1 0 (Ω) be the solution of −∆v = f , and consider the stream function ũn ∈ W 1,1 (ω) defined by  (3.17) Following the procedure of [5], let us prove that ūn − ũn strongly converges to 0 on supp ϕ.By the Sobolev imbedding of W 1,1 in L 2 in each square ε n (k +Y ), k ∈ K n , (note that the following imbedding constant C is independent of the squares) combined with the Poincaré-Wirtinger inequality, and by the Cauchy-Schwarz inequality we have  In the symmetric case B n = B s n , the first author proved in [7] that, under the L 2 (Ω)-boundedness of any solution v n of − div (B n ∇v n ) = f ∈ L 2 (Ω), estimate (3.21) is a sufficient condition to obtain the H w -convergence of B n to B * .This compactness result can be easily extended (see [20] for details) to the non-symmetric case assuming that A n and A

First step :
Convergences of the sequences v n and B n ∇v n .Putting the function v n ∈ H 1 0 (Ω) as test function in the equation − div (B n ∇v n ) = f , we obtain by the Sobolev embedding of W 1,1 (Ω) into L 2 (Ω) combined with the Poincaré inequality and S n ≥ α I.Moreover, by (2.5) and (2.14) S n satisfies the weak convergence of (3.3), and by (3.2) v n satisfies (3.4).Lemma 3.1 thus implies that v n strongly converges to v in L 2 loc (Ω).
The last term is bounded by(3.6)  and by the inequality|A −1 n | ≤ α −1 .Therefore, the sequences v n := wλ n and S n = (B s n ) −1 of the first step satisfy the assumptions (3.3) and (3.4) of Lemma 3.1

(3. 18 )Y
Then, summing over k ∈ K n we get similarly to(3.15)Qn(ū n − ũn ) 2 dx ≤ c ε 2 n Y det A n det(A n ) s (A n ) s dy ω A n ∇u n • ∇u n + A −1 n ∇v • ∇v dx,(3.19)whichtends to 0 by(2.18).Therefore, we can replace ũn by ūn in(3.16).Now, consider the approximation of ∇ϕ by a function Φn constant in each square ε n (k + Y ) and such that |∇ϕ − Φn | ≤ c ε n .inequality and equality (2.14) with B n , imply that Y(V − V ) 2 dy ≤ c ) s −1 dy Y (B n ) s ∇V • ∇V dy = c Y det A n det(A n ) s (A n ) s dy Y B n ∇V • ∇V dy.This, combined with (2.18), yields the following estimate of the weighted Poincaré-Wirtinger inequality sup T n satisfy condition (2.7), or equivalently B n and B T n satisfy (2.11).Therefore, the H w -convergence (2.20) holds true sinceB n = A T n det A n and B * = A T * det A * ,which concludes the proof.
Then, the highly oscillating sequence A n defined by (2.15) satisfies the Poincaré inequality (2.21) (see e.g. .21) Example 2.9.Let E be a Y -periodic connected open set of R 2 , with a Lipschitz boundary, such that |Y ∩ E| > 0. Consider a Y -periodic symmetric matrix-valued function A n such that A n det A n ≥ I a.e. in E and A n det A n ≥ ε 2 n I a.e. in R 2 \ E, or equivalently A n ≤ I a.e. in E and A n ≤ ε −2 n I a.e. in R 2 \ E.
.2) Therefore, the sequences B n ∇v n • ∇v n and |∇v n | are bounded in L 1 (Ω), hence v n is bounded in L 2 (Ω) by (3.1).On the other hand, similarly to (2.13) inequality (2.7) implies that B T n B n ≤ C 0 B s n and |B n ∇v n