Marc Briane
email: mbriane@insa-rennes.fr

Laurent Pater
email: laurent.pater@ens-cachan.org

Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension three

Keywords: homogenization, high-contrast conductivity, magneto-transport, strong field, two-phase composites

Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension

three.

Introduction

The mathematical theory of homogenization for second-order elliptic partial differential equations has been widely studied since the pioneer works of Spagnolo on G-convergence [START_REF] Spagnolo | Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche[END_REF], of Murat, Tartar on H-convergence [START_REF] Murat | Séminaire d'Analyse Fonctionnelle et Numérique[END_REF][START_REF] Murat | Topics in the Mathematical Modelling of Composite Materials[END_REF], and of Bensoussan, Lions, Papanicolaou on periodic structures [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF], in the framework of uniformly bounded (both from below and above) sequences of conductivity matrixvalued functions. It is also known since the end of the seventies [START_REF] Fenchenko | Asymptotic of solution of differential equations with strongly oscillating matrix of coefficients which does not satisfy the condition of uniform boundedness[END_REF][START_REF] Ya | Composite Media and Homogenization Theory[END_REF] (see also the extensions [START_REF] Bellieud | Homogenization of elliptic problems in a fiber reinforced structure[END_REF][START_REF] Briane | Fibered microstructures for some nonlocal Dirichlet forms[END_REF][START_REF] Briane | Homogenization of high-conductivity periodic problems: Application to a general distribution of one-directional fibers[END_REF][START_REF] Ya | Homogenization of Partial Differential Equations[END_REF]) that the homogenization of the sequence of conductivity problems, in a bounded open set

Ω of R 3 , div (σ n ∇u n ) = f in Ω u n = 0 on ∂Ω, (1.1)
with a uniform boundedness from below but not from above for σ n , may induce nonlocal effects. However, the situation is radically different in dimension two since the nature of problem (1.1) is shown [START_REF] Briane | Nonlocal effects in two-dimensional conductivity[END_REF][START_REF] Briane | Asymptotic behaviour of equicoercive diffusion energies in dimension two[END_REF] to be preserved in the homogenization process provided that the sequence σ n is uniformly bounded from below. H-convergence theory includes the case of non-symmetric conductivities in connection with the Hall effect [START_REF] Hall | On a new action of the magnet on electric currents[END_REF] in electrodynamics (see, e.g., [START_REF] Landau | Électrodynamique des Milieux Continus[END_REF][START_REF] Omar | Elementary Solid State Physics: Principles and Applications[END_REF]). Indeed, in the presence of a constant magnetic field the conductivity matrix is modified and becomes non-symmetric. Here, we consider an idealized model of an isotropic conductivity σ(h) depending on a parameter h which characterizes the antisymmetric part of the conductivity in the following way:

• in dimension two, σ(h) = αI 2 + βhJ, J := 0 -1 1 0 , (

where α, β are scalar an h ∈ R,

• in dimension three,

σ(h) = αI 3 + βE (h), E (h) := 0 -h 3 h 2 h 3 0 -h 1 -h 2 h 1 0 , (1.3)
where α, β are scalar and h ∈ R 3 .

Since the seminal work of Bergman [START_REF] Bergman | Self-duality and the low field Hall effect in 2D and 3D metal-insulator composites. Percolation Structures and Processes[END_REF] the influence of a low magnetic field in composites has been studied for two-dimensional composites [START_REF] Milton | Classical hall effect in two-dimensional composites: A characterization of the set of realizable effective conductivity tensors[END_REF][START_REF] Bergman | Macroscopic conductivity tensor of a threedimensional composite with a one-or two-dimensional microstructure[END_REF][START_REF] Briane | Homogenization of the two-dimensional hall effect[END_REF], and for columnar composites [START_REF] Bergman | Duality transformation in a three dimensional conducting medium with two dimensional heterogeneity and an in-plane magnetic field[END_REF][START_REF] Bergman | Magnetotransport in conducting composite films with a disordered columnar microstructure and an in-plane magnetic field[END_REF][START_REF] Bergman | Exact between magnetoresistivity tensor components of conducting composites with a columnar microstructure[END_REF][START_REF] Grabovsky | An application of the general theory of exact relations to fiber-reinforced conducting composites with hall effect[END_REF][START_REF] Grabovsky | Exact relations for effective conductivity of fiber-reinforced conducting composites with the hall effect via a general theory[END_REF].

The case of a strong field, namely when the symmetric part and the antisymmetric part of the conductivity are of the same order, has been also investigated [START_REF] Bergman | Strong-field magnetotransport of conducting composites with a columnar microstructure[END_REF][START_REF] Bergman | Recent advances in strong field magnetotransport in a composite medium[END_REF]. Moreover, dimension three may induce anomalous homogenized Hall effects [START_REF] Briane | Homogenization of the three-dimensional hall effect and change of sign of the hall coefficient[END_REF][START_REF] Briane | Giant hall effect in composites[END_REF][START_REF] Briane | An antisymmetric effective hall matrix[END_REF] which do not appear in dimension two [START_REF] Briane | Homogenization of the two-dimensional hall effect[END_REF].

In the context of high-contrast problems the situation is more delicate when the conductivities are not symmetric. An extension in dimension two of H-convergence for non-symmetric and nonuniformly bounded conductivities was obtained in [START_REF] Briane | Two-dimensional div-curl results. application to the lack of nonlocal effects in homogenization[END_REF] thanks to an appropriate div-curl lemma. More recently, the Keller, Dykhne [START_REF] Keller | A theorem on the conductivity of a composite medium[END_REF][START_REF] Dykhne | Conductivity of a two-dimensional two-phase system[END_REF] two-dimensional duality principle which claims that the mapping

A → A T det A (1.4)
is stable under homogenization, was extended to high-contrast conductivities in [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF]. However, the homogenization of both high-contrast and non-symmetric conductivities has not been precisely studied in the context of the strong field magneto-transport especially in dimension three. In this paper we establish an effective perturbation law for a mixture of two high-contrast isotropic phases in the presence of a magnetic field. The two-dimensional case is performed in a general way for non-periodic and periodic microstructures. It is then compared to the case of a three-dimensional fiber-reinforced microstructure.

In dimension two, following the modelization (1.2), consider a sequence σ n (h) of isotropic twophase matrix-valued conductivities perturbed by a fixed constant h ∈ R, and defined by

σ n (h) := (1 -χ n ) α 1 I 2 + β 1 h J + χ n α 2,n I 2 + β 2,n h J , (1.5)
where χ n is the characteristic function of phase 2, with volume fraction θ n → 0, α 1 > 0, β 1 are the constants of the low conducting phase 1, and α 2,n → ∞, β 2,n are real sequences of the highly conducting phase 2 where β 2,n is possibly unbounded. The coefficients α 1 and β 1 , respectively α 2,n and β 2,n also have the same order of magnitude according to the strong field assumption.

Assuming that the sequence θ -1 n χ n converges weakly- * in the sense of the Radon measures to a bounded function, and that θ n α 2,n , θ n β 2,n converge respectively to constants α 2 > 0, β 2 , we prove (see Theorem 2.2) that the perturbed conductivity σ n (h) converges in an appropriate sense of Hconvergence (see Definition 1.1) to the homogenized matrix-valued function

σ * (h) = σ 0 * α 1 , α 2 + α -1 2 β 2 2 h 2 + β 1 h J, (1.6)
for some matrix-valued function σ 0 * which depends uniquely on the microstructure χ n in the absence of a magnetic field, and is defined for a subsequence of n. The proof of the result is based on a Dykhne transformation of the type

A n → (p n A n + q n J) -1 + r n J -1 , (1.7)
In the periodic case, i.e. when σ n (h)(•) = Σ n (•/ε n ) with Σ n Y -periodic and ε n → 0, we use an alternative approach based on an extension of Theorem 4.1 of [START_REF] Briane | Asymptotic behaviour of equicoercive diffusion energies in dimension two[END_REF] to ε n Y -periodic but non-symmetric conductivities (see Theorem 3.1). So, it turns out that the homogenized conductivity σ * (h) is the limit as n → ∞ of the constant H-limit (σ n ) * associated with the periodic homogenization (see, e.g., [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF]) of the oscillating sequence Σ n (•/ε) as ε → 0 and for a fixed n. Finally, the Dykhne transformation performed by Milton [START_REF] Milton | Classical hall effect in two-dimensional composites: A characterization of the set of realizable effective conductivity tensors[END_REF] (see also [START_REF] Milton | The Theory of Composites[END_REF], Chapter 4) applied to the local periodic conductivity Σ n and its effective conductivity (σ n ) * , allows us to recover the perturbed homogenized formula (1.6). An example of a periodic cross-like thin structure provides an explicit computation of σ * (h) (see Proposition 3.2).

To make a comparison with dimension three we restrict ourselves to the ε n Y -periodic fiberreinforced structure introduced by Fenchenko, Khruslov [START_REF] Fenchenko | Asymptotic of solution of differential equations with strongly oscillating matrix of coefficients which does not satisfy the condition of uniform boundedness[END_REF] to derive a nonlocal effect in homogenization. However, in the present context the fiber radius r n is chosen to be super-critical, i.e. r n → 0 and ε 2 n | ln r n | → 0, in order to avoid such an effect. Similarly to (1.5) and following the modelization (1.3), the perturbed conductivity is defined for h ∈ R 3 , by

σ n (h) := (1 -χ n ) α 1 I 3 + β 1 E (h) + χ n α 2,n I 3 + β 2,n E (h) , (1.8)
where χ n is the characteristic function of the fibers which are parallel to the direction e 3 . The form of (1.8) ensures the rotational invariance of σ n (h) for those orthogonal transformations which leave h invariant. Under the same assumptions on the conductivity coefficients as in the two-dimensional case, with θ n = π r 2 n , but using a quite different approach, the homogenized conductivity is given by (see Theorem 4.1)

σ * (h) = α 1 I 3 + α 3 2 + α 2 β 2 2 |h| 2 α 2 2 + β 2 2 h 2 3 e 3 ⊗ e 3 + β 1 E (h). (1.9)
The difference between formulas (1.6) and (1.9) provides a new example of gap between dimension two and dimension three in the high-contrast homogenization framework. As former examples of dimensional gap, we refer to the works [START_REF] Briane | Homogenization of the two-dimensional hall effect[END_REF][START_REF] Briane | Homogenization of the three-dimensional hall effect and change of sign of the hall coefficient[END_REF] about the 2d positivity property, versus the 3d nonpositivity, of the effective Hall coefficient, and to the works [START_REF] Briane | Asymptotic behaviour of equicoercive diffusion energies in dimension two[END_REF][START_REF] Fenchenko | Asymptotic of solution of differential equations with strongly oscillating matrix of coefficients which does not satisfy the condition of uniform boundedness[END_REF] concerning the 2d lack, versus the 3d appearance, of nonlocal effects in the homogenization process.

The paper is organized as follows. Section 2 and 3 deal with dimension two. In Section 2 we study the two-dimensional general (non-periodic) case thanks to an appropriate div-curl lemma. In Section 3 an alternative approach is performed in the periodic framework. Finally, Section 4 is devoted to the three-dimensional case with the fiber-reinforced structure.

Notations

• Ω denotes a bounded open subset of R d ;

• I d denotes the unit matrix in R d×d , and J := 0 -1 1 0 ;

• for any matrix A in R d×d , A T denotes the transposed of the matrix A, A s denotes its symmetric part;

• for h ∈ R 3 , E (h) denotes the antisymmetric matrix in R 3×3 defined by E (h) x := h × x, for x ∈ R 3 ; • for any A, B ∈ R d×d , A ≤ B means that for any ξ ∈ R d , Aξ • ξ ≤ Bξ • ξ;
we will use the fact that for any invertible matrix

A ∈ R d×d , A ≥ αI d ⇒ A -1 ≤ α -1 I d ;
• | • | denotes both the euclidean norm in R d and the subordinate norm in R d×d ;

• for any locally compact subset X of R d , M(X) denotes the space of the Radon measures defined on X;

• for any α, β > 0, M(α, β; Ω) is the set of the invertible matrix-valued functions

A : Ω → R d×d such that ∀ ξ ∈ R d , A(x)ξ • ξ ≥ α|ξ| 2 and A -1 (x)ξ • ξ ≥ β -1 |ξ| 2 a.e.

in Ω;

(1.10)

• C denotes a constant which may vary from a line to another one.

In the sequel, we will use the following extension of H-convergence and introduced in [16]:

Definition 1.1 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 (1.10)).

The sequence A n is said to H(M(Ω) 2 )-converge to the matrix-valued function A * if for any distribution f in H -1 (Ω), the solution u n of the problem

div (A n ∇u n ) = f in Ω u n = 0 on ∂Ω,
satisfies the convergences

u n -⇀ u in H 1 0 (Ω) A n ∇u n -⇀ A * ∇u weakly- * in M(Ω) 2 ,
where u is the solution of the problem

div (A * ∇u) = f in Ω u = 0 on ∂Ω.
We now give a notation for H(M(Ω) 2 )-limits of high-contrast two-phase composites. We consider the characteristic function χ n of the highly conducting phase, and denote ω n := {χ n = 1}. Notation 1.1 A sequence of isotropic two-phase conductivities in the absence of a magnetic field is denoted by

σ 0 n (α 1,n , α 2,n ) := (1 -χ n )α 1,n I 2 + χ n α 2,n I 2 , (1.11
)

with lim n→∞ α 1,n = α 1 > 0 and lim n→∞ |ω n | α 2,n = α 2 > 0, (1.12)
and its H(M(Ω) 2 )-limit is denoted by σ 0 * (α 1 , α 2 ).

2 A two-dimensional non-periodic medium

A div-curl approach

We extend the classical div-curl lemma.

Lemma 2.1 Let Ω be a bounded open subset of R 2 . Let α > 0, let ā ∈ L ∞ (Ω)
and let A n be a sequence of matrix-valued functions in L ∞ (Ω) 2×2 (not necessarily symmetric) satisfying

A n ≥ αI 2 a.e. in Ω and det A n det A s n |A s n | ⇀ ā ∈ L ∞ (Ω) weakly- * in M(Ω). (2.1)
Let ξ n be a sequence in L 2 (Ω) 2 and v n a sequence in H 1 (Ω) satisfying the following assumptions: (i) ξ n and v n satisfy the estimate

ˆΩ A -1 n ξ n • ξ n dx + ||v n || H 1 (Ω) ≤ C; (2.2) (ii) ξ n satisfies the classical condition div ξ n is compact in H -1 (Ω). (2.3)
Then, there exist ξ in L 2 (Ω) 2 and v in H 1 (Ω) such that the following convergences hold true up to a subsequence ξ n ⇀ ξ weakly- * in M(Ω) 2 and ∇v n ⇀ ∇v weakly in L 2 (Ω) 2 .

(2.4)

Moreover, we have the following convergence in the distribution sense

ξ n • ∇v n ⇀ ξ • ∇v weakly in D ′ (Ω).
Proof of Lemma 2.1. The proof consists in considering the "good-divergence" sequence ξ n as a sum of a compact sequence of gradients ∇u n and a sequence of divergence-free functions J∇z n . We then use Lemma 3.1 of [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF] to obtain the strong convergence of z n in L 2 loc (Ω). Finally, replacing ξ n by ∇u n + J∇z n , we conclude owing to integration by parts.

First step: Proof of convergences (2.4).

An easy computation gives

A -1 n s -1 = det A n det A s n A s n . (2.5)
The sequence ξ n is bounded in L 1 (Ω) 2 since the Cauchy-Schwarz inequality combined with the weak- * convergence of (2.1), (2.2) and (2.5) yields

ˆΩ |ξ n | dx 2 ≤ ˆΩ A -1 n s -1 dx ˆΩ A -1 n s ξ n •ξ n dx = ˆΩ det A n det A s n |A s n | dx ˆΩ A -1 n ξ n •ξ n dx ≤ C.
Therefore, ξ n converges up to a subsequence to some ξ ∈ M(Ω) 2 in the weak- * sense of the measures. Let us prove that the vector-valued measure ξ is actually in L 2 (Ω) 2 . Again by the Cauchy-Schwarz inequality combined with (2.1), (2.2) and (2.5) we have, for any

Φ ∈ C 0 (Ω) 2 , ˆΩ ξ(dx) • Φ = lim n→∞ ˆΩ ξ n • Φ dx ≤ lim sup n→∞ ˆΩ det A n det A s n |A s n | |Φ| 2 dx 1 2 ˆΩ A -1 n ξ n • ξ n dx 1 2 ≤ C ˆΩ ā|Φ| 2 dx 1 2
, which implies that ξ is absolutely continuous with respect to the Lebesgue measure. Since ā ∈ L ∞ (Ω), we also get that

ˆΩ ξ • Φ dx ≤ ||Φ|| L 2 (Ω) 2 hence ξ ∈ L 2 (Ω) 2 .
Therefore, the first convergence of (2.4) holds true with its limit in L 2 (Ω) 2 . The second one is immediate.

Second step: Introduction of a stream function. By (2.3), the sequence u n in H 1 0 (Ω) defined by u n := ∆ -1 (div ξ n ) strongly converges in H 1 0 (Ω):

u n -→ u in H 1 0 (Ω). (2.6)
Let ω be a regular simply connected open set such that ω ⊂⊂ Ω. Since by definition ξ n -∇u n is a divergence-free function in L 2 (Ω) 2 , there exists (see, e.g., [START_REF] Girault | Finite Element Approximation of the Navier-Stokes Equations[END_REF]) a unique stream function z n ∈ H 1 (ω) with zero ω-average such that ξ n = ∇u n + J∇z n a.e. in ω.

(2.7)

Third step: Convergence of the stream function z n .

Since ∇u n is bounded in L 2 (Ω) 2 by the second step, ξ n is bounded in L 1 (Ω) 2 by the first step and z n has a zero ω-average, the Sobolev embedding of W 1,1 (ω) into L 2 (ω) combined with the Poincaré-Wirtinger inequality in ω implies that z n is bounded in L 2 (ω) and thus converges, up to a subsequence still denoted by n, to a function z in L 2 (ω). Moreover, let us define

S n := J -1 (A -1 n ) s J -1 .
The Cauchy-Schwarz inequality gives

ˆω S -1 n ∇z n • ∇z n dx = ˆω J -1 (A -1 n ) s J∇z n • ∇z n dx = ˆω(A -1 n ) s J∇z n • J∇z n dx = ˆω(A -1 n ) s ξ n -∇u n • ξ n -∇u n ] dx ≤ 2 ˆω(A -1 n ) s ξ n • ξ n dx + 2 ˆω(A -1 n ) s ∇u n • ∇u n dx = 2 ˆω A -1 n ξ n • ξ n dx + 2 ˆω A -1 n ∇u n • ∇u n dx.
The first term is bounded by (2.2) and the last term by the inequality A -1 n ≤ α -1 I 2 and the convergence (2.6). Therefore, the sequences v n := z n and, by (2.14), S n satisfy all the assumptions of Lemma 3.1 of [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF] since, by (2.5),

S n = det A n det A s n J -1 A s n J.
Then, we obtain the convergence z n -→ z strongly in L 2 loc (ω).

(2.8)

Moreover, the convergence (2.6) gives

ξ = ∇u + J∇z in D ′ (ω).
(2.9)

Fourth step: Integration by parts and conclusion.

We have, as J∇v n is a divergence-free function,

ξ n • ∇v n = (∇u n + J∇z n ) • ∇v n = ∇u n • ∇v n -div (z n J∇v n ) . (2.10)
The strong convergence of ∇u n in (2.6), the second weak convergence of (2.4) justified in the first step and (2.8) give

∇u n • ∇v n -div (z n J∇v n ) -⇀ ∇u • ∇v -div (zJ∇v) in D ′ (ω). (2.11)
We conclude, by combining this convergence with (2.10), (2.9) and integrating by parts, to the convergence

ξ n • ∇v n -⇀ ∇u • ∇v -div (zJ∇v) = (∇u + J∇z) • ∇v = ξ • ∇v weakly in D ′ (ω).
for an arbitrary open subset ω of Ω.

For the reader's convenience, we first recall in Theorem 2.1 below the main result of [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF] concerning the Keller duality for high contrast conductivities. Then, Proposition 2.1 is an extension of this result to a more general transformation.

Theorem 2.1 ( [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF]) Let Ω be a bounded open subset 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 matrixvalued functions (not necessarily symmetric) in M(α, β n ; Ω). Assume that there exists a function

ā ∈ L ∞ (Ω) such that det A n det A s n |A s n | ⇀ ā weakly- * in M(Ω).
(2.12)

Then, there exist a subsequence of n, still denoted by n, and a matrix-valued function A * in M(α, β; Ω), with β = 2||ā|| L ∞ (Ω) , such that

A n H(M(Ω) 2 ) -⇀ A * and A T n det A n H(M(Ω) 2 ) -⇀ A T * det A * . (2.13) Proposition 2.1
Let Ω be a bounded open subset of R 2 such that |∂Ω| = 0. Let p n , q n and r n , n ∈ N be sequences of real numbers converging respectively to p > 0, q and 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 in M(α, β n ; Ω) (not necessarily symmetric) satisfying

r n A n is bounded in L ∞ (Ω) 2×2 and det A n det A s n |A s n | ⇀ ā ∈ L ∞ (Ω) weakly- * in M(Ω), (2.14)
and that B n = (p n A n + q n J) -1 + r n J -1 is a sequence of symmetric matrices.

(2.15)

Then, there exist a subsequence of n, still denoted by n, and a matrix-valued function

A * in M(α, β; Ω), with β = 2||ā|| L ∞ (Ω) , such that A n H(M(Ω) 2 ) -⇀ A * and (p n A n + q n J) -1 + r n J -1 H(M(Ω) 2 ) -⇀ pA * + qJ. (2.16)
Remark 2.1 Proposition 2.1 completes Theorem 2.1 performed with the transformation

A -→ A T det A = J -1 A -1 J, (2.17)
to other Dykhne transformations of type (see [START_REF] Milton | The Theory of Composites[END_REF], Section 4.1):

A -→ (pA + qJ) -1 + rJ -1 = pA + qJ (1 -rq)I 2 + rpJA -1 (2.18) Remark 2.2
The convergence of r n to r = 0 is not necessary but sufficient for our purpose. If r = 0, the different convergences are conserved but lead us to the expression

pA * + qJ = B * (1 -qr)I 2 + p rJA * . (2.19)
Proof of Proposition 2.1. The proof is divided into two steps. In the first step, we use Lemma 2.1 to show the H(M(Ω) 2 )-convergence of A n := p n A n + q n J to pA * + qJ. In the second step, we build a matrix Q n which will be used as a corrector for B n and then use again Lemma 2.1.

First step:

A * = pA * + qJ.
First of all, thanks to Theorem 2.2 [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF], we already know that, up to a subsequence still denoted by n, A n H(M(Ω) 2 )-converges to A * . We consider a corrector P n associated with A n in the sense of Murat-Tartar (see, e.g., [START_REF] Murat | Topics in the Mathematical Modelling of Composite Materials[END_REF]), such that, for λ ∈ R 2 , P n λ = ∇w λ n is defined by

div(A n ∇w λ n ) = div(A * ∇(λ • x)) in Ω w λ n = λ • x on ∂Ω (2.20)
Again with Theorem 2.2 of [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF] and Definition 1.1, we know that P n λ converges weakly in L 2 (Ω) 2 to λ and

A n P n λ converges weakly- * in M(Ω) to A * λ. Since, for any λ, µ ∈ R 2 , α||∇w µ n || 2 L 2 (Ω) 2 ≤ ˆΩ A n ∇w µ n • ∇w µ n dx = ˆΩ A * µ • ∇w µ n dx ≤ 2||ā|| L ∞ (Ω) |µ| |Ω| 1/2 ||∇w µ n || L 2 (Ω) 2 and ˆΩ A -1 n A n ∇w λ n • A n ∇w λ n dx = ˆΩ A n ∇w λ n • ∇w λ n dx,
the sequences ξ n := A n ∇w λ n and v n := w µ n satisfy (2.2) and (2.3). This combined with (2.14) implies that we can apply Lemma 2.1 to obtain

∀ λ, µ ∈ R, A n P n λ • P n µ -⇀ A * λ • µ in D ′ (Ω).
(2.21)

We denote A n := p n A n + q n J and consider δ n such that δ n J := A n -A s n . Then, the matrix

A n satisfies A n ξ • ξ = p n A n ξ • ξ ≥ p n α|ξ| 2 . (2.22) Moreover, det A n = p 2 n det A s n + (p n δ n + q n ) 2 ≤ p 2 n (det A s n + 2δ 2 n ) + 2q 2 n ≤ 2p 2 n det A n + 2q 2 n ≤ C det A n ,
the last inequality being a consequence of A n ≥ αI 2 . This inequality gives, by (2.14),

det A n det A s n | A s n | = det A n p 2 n det A s n p n |A s n | ≤ C det A n det A s n |A s n | ≤ C. (2.23)
Then by (2.22), (2.23) and [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF] again, up to a subsequence still denoted by n, A n H(M(Ω) 2 )converges to A * and we have, by the classical div-curl lemma of [START_REF] Murat | Topics in the Mathematical Modelling of Composite Materials[END_REF] for

JP n λ • P n µ and (2.21), ∀ λ, µ ∈ R, (p n A n + q n J)P n λ • P n µ = p n A n P n λ • P n µ + q n JP n λ • P n µ D ′ (Ω) -⇀ pA * λ • µ + qJλ • µ,
that can be rewritten A * = pA * + qJ.

Second step: B

* = A * . Let θ ∈ C 1 c (Ω) and P n a corrector associated with A n , such that, for λ ∈ R 2 , P n λ = ∇ w λ n is defined by div( A n ∇ w λ n ) = div A * ∇(θλ • x) in Ω w λ n = 0 on ∂Ω. (2.24)
By Definition 1.1, we have

w λ n -⇀ θλ • x weakly in H 1 0 (Ω), A n ∇ w λ n -⇀ A * ∇(θλ • x) weakly- * in M(Ω) 2 . (2.25) Let us now consider B n = A -1 n + r n J -1 .
B n is symmetric and so is its inverse :

B -1 n = A -1 n + r n J = ( A -1 n + r n J) s = ( A -1 n ) s .
We then have, by a little computation (like in (2.5)) and (2.23),

det B n det B s n |B s n | = |B n | = A -1 n s -1 = det A n det A s n | A s n | ≤ C. (2.26)
For any ξ ∈ R 2 , the sequence

ν n := (I + r n J A n ) -1 ξ satisfies, by (2.14), |ξ| ≤ 1 + ||r n A n || L ∞ (Ω) 2×2 |ν n | ≤ 1 + p n ||r n A n || L ∞ (Ω) 2×2 + q n r n |ν n | ≤ (1 + C)|ν n |, hence B n ξ •ξ = A n ν n •(I +r n J A n )ν n = A n ν n •ν n = p n A n ν n •ν n ≥ p n α|ν n | 2 ≥ α p n (1 + C) 2 |ξ| 2 ≥ C|ξ| 2 (2.27)
with C > 0. Therefore, with (2.27) and (2.26), again by Theorem 2.2 of [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF], up to a subsequence still denoted by n, B n H(M(Ω) 2 )-converges to B * .

Let

ψ ∈ C 1 c (Ω) and R n be a corrector associated to B n , such that, for µ ∈ R 2 , R n µ = ∇v µ n is defined by div (B n ∇v µ n ) = div B * ∇(ψµ • x) in Ω v µ n = 0 on ∂Ω.
(2.28)

By Definition 1.1, we have the convergences

v µ n -⇀ ψµ • x weakly in H 1 0 (Ω), B n ∇v µ n -⇀ B * ∇(ψµ • x) weakly- * in M(Ω) 2 . (2.29)
Let us define the matrix Q n := (I + r n J A n ) P n . We have

B n Q n = ( A -1 n + r n J) -1 (I + r n J A n ) P n = ( A -1 n + r n J) -1 ( A -1 n + r n J) A n P n = A n P n . (2.30)
We are going to pass to the limit in D ′ (Ω) the equality given by (2.30) and the symmetry of B n :

A n P n λ • R n µ = B n Q n λ • R n µ = Q n λ • B n R n µ. ( 2
ˆΩ A n -1 ξ n • ξ n dx + ||v n || H 1 0 (Ω) = ˆΩ A n P n λ • P n λ dx + ||v µ n || H 1 0 (Ω) dx ≤ C by (2.
A n P n λ • R n µ -⇀ A * ∇(θλ • x) • ∇(ψµ • x) in D ′ (Ω). (2.32)
On the other hand, we have the equality

Q n λ • B n R n µ = B n R n µ • P n λ + B n R n µ • r n J A n P n . ( 2
ˆΩ B n -1 ξ n • ξ n dx + ||v n || H 1 0 (Ω) = ˆΩ B n R n µ • R n µ dx + || w λ n || H 1 0 (Ω) dx ≤ C by (2.
B n R n µ • P n λ -⇀ B * ∇(ψµ • x) • ∇(θλ • x) in D ′ (Ω). (2.34)
The convergence of the right part of (2.33) is more delicate. The demonstration is the same as for Lemma 2.1. Let ω be a simply connected open subset of Ω such as ω ⊂⊂ Ω. The function

A n P n λ -A * ∇(θλ • x)
is divergence-free and we can introduce a function z λ n such as

A n P n λ = A * ∇(θλ • x) + J∇z λ n , (2.35)
z λ n -→ 0 strongly in L 2 loc (ω). (2.36)
The equality

B n R n µ • r n J A n P n λ = r n B n R n µ • J A * ∇(θλ • x) -r n B n R n µ • ∇z λ n = r n B n R n µ • J A * ∇(θλ • x) -r n div(z λ n B n R n µ) + r n z λ n div B * ∇(θλ • x)
leads us, by (2.29), (2.36) and the convergence to 0 of r n , like in the demonstration of Lemma 2.1, to

B n R n µ • r n J A n P n -⇀ 0 in D ′ (ω). ( 2
A * ∇(θλ • x) • ∇(ψµ • x) = B * ∇(ψµ • x) • ∇(θλ • x) in D ′ (ω).
We conclude, by taking θ = 1 and ψ = 1 on ω and taking into account that B * is symmetric and ω, λ, µ are arbitrary, that:

B * = A * = pA * + qJ.

An application to isotropic two-phase media

In this section, we study the homogenization of a two-phase isotropic medium with high contrast and non-necessarily symmetric conductivities. The study of the symmetric case in Proposition 2.2 permits to obtain Theorem 2.2 by applying the transformation of Proposition 2.1. We use Notation 1.1.

χ n θ n H( --⇀ a ∈ L ∞ (Ω) weakly- * in M(Ω). (2.38)
We assume that there exists α 1 , α 2 > 0 and two positive sequences

α 1,n , α 2,n ≥ a 0 > 0 verifying lim n→∞ α 1,n = α 1 and lim n→∞ θ n α 2,n = α 2 , (2.39)
and that the conductivity takes the form

σ 0 n (α 1,n , α 2,n ) = (1 -χ n )α 1,n I 2 + χ n α 2,n I 2 .
Then, there exists a subsequence of n, still denoted by n, and a locally Lipschitz function

σ 0 * : (0, ∞) 2 -→ M(a 0 , 2||a|| ∞ ; Ω) such that ∀ (α 1 , α 2 ) ∈ (0, ∞) 2 , σ 0 n (α 1,n , α 2,n ) H(M(Ω) 2 ) -⇀ σ 0 * (α 1 , α 2 ). (2.40)
Proof of Proposition 2.2. The proof is divided into two parts. We first prove the theorem for α 1,n = α 1 , α 2,n = θ -1 n α 2 , and then treat the general case. [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF] implies that for any α ∈ (0, ∞) 2 , there exists a subsequence of n such that σ 0 n (α) H(M(Ω) 2 )-converges in the sense of Definition 1.1 to some matrix-valued function in M(a 0 , 2||a|| ∞ ; Ω).

First step: The case α 1,n = α 1 , α 2,n = θ -1 n α 2 . In this step we denote σ 0 n (α) := σ 0 n (α 1 , θ -1 n α 2 ), for α = (α 1 , α 2 ) ∈ (0, ∞) 2 . Theorem 2.2 of
By a diagonal extraction, there exists a subsequence of n, still denoted by n, such that

∀ α ∈ Q 2 ∩ (0, ∞) 2 , σ 0 n (α) H(M(Ω) 2 )
-⇀ σ 0 * (α).

(2.41)

We are going to show that this convergence is true any pair α ∈ (0, ∞) 2 . We have, by (2.38), for any

α ∈ Q 2 ∩ (0, ∞) 2 , |σ 0 n (α)| = (1 -χ n )α 1 + χ n α 2 θ n H( --⇀ α 1 + α 2 a ∈ L ∞ (Ω) weakly- * in M(Ω) (2.42)
and, since θ n ∈ (0, 1),

∀ ξ ∈ R 2 , σ 0 n (α)ξ • ξ = α 1 (1 -χ n )|ξ| 2 + χ n α 2 θ n |ξ| 2 ≥ min(α 1 , α 2 )
|σ 0 * (α)λ| ≤ 2|λ| (α 1 + α 2 ||a|| ∞ ) . (2.44) For any α ∈ Q 2 ∩ (0, ∞) 2 and λ ∈ R 2 , consider the corrector w α,λ n associated with σ 0 n (α) defined by    div σ 0 n (α)∇w α,λ n = div σ 0 * (α)λ in Ω, w α,λ n = λ • x on ∂Ω, (2.45)
which depends linearly on λ.

Let α ∈ Q 2 ∩ (0, ∞) 2 .
Let us show that the energies

ˆΩ σ 0 n (α)∇w α,λ n • ∇w α,λ n dx (2.46)
are bounded. We have, by (2.45), (2.44) and the Cauchy-Schwarz inequality

ˆΩ σ 0 n (α)∇w α,λ n • ∇w α,λ n dx = ˆΩ σ 0 * (α)λ • ∇w α,λ n -λ dx + ˆΩ σ 0 n (α)∇w α,λ n • λ dx = ˆΩ σ 0 * (α)λ • ∇w α,λ n dx -ˆΩ σ 0 * (α)λ • λ ≥0 dx + ˆΩ σ 0 n (α)∇w α,λ n • λ dx which leads us to ˆΩ σ 0 n (α)∇w α,λ n • ∇w α,λ n dx ≤ ˆΩ |σ 0 * (α)λ • ∇w α,λ n | dx + ˆΩ |σ 0 n (α)∇w α,λ n • λ| dx. (2.47)
On the one hand, the Cauchy-Schwarz inequality gives

ˆΩ |σ 0 n (α)∇w α,λ n • λ| dx 2 ≤ |λ| 2 ˆΩ |σ 0 n (α)| dx ˆΩ σ 0 n (α)∇w α,λ n • ∇w α,λ n dx that is ˆΩ |σ 0 n (α)∇w α,λ n • λ| dx 2 ≤ |λ| 2 |α| ˆΩ σ 0 n (α)∇w α,λ n • ∇w α,λ n dx. (2.48)
On the other hand, by (2.43) and the Cauchy-Schwarz inequality, we have

ˆΩ |σ 0 * (α)λ • ∇w α,λ n | dx ≤ 2|λ|(α 1 + α 2 ||a|| ∞ ) ˆΩ |∇w α,λ n | 2 dx ≤ 2|λ|(α 1 + α 2 ||a|| ∞ ) 1 α 1 + 1 α 2 ˆΩ σ 0 n (α)∇w α,λ n • ∇w α,λ n dx that is ˆΩ |σ 0 * (α)λ • ∇w α,λ n | dx ≤ C |λ| 2 |α| 1 α 1 + 1 α 2 ˆΩ σ 0 n (α)∇w α,λ n • ∇w α,
ˆΩ σ 0 n (α)∇w α,λ n • ∇w α,λ n dx ≤ C |λ| 2 |α| + |α| 2 (α 1 -1 + α 2 -1 ) =:M (α) (2.50)
where C does not depend on n nor α.

Let α ′ ∈ Q 2 ∩ (0, ∞) 2 .
The sequences ξ n := σ 0 n (α)∇w α,λ n and v n := w α ′ ,λ n satisfy the assumptions (2.2) and (2.3) of Lemma 2.1. By symmetry, we have the convergences

   σ 0 n (α)∇w α,λ n • ∇w α ′ ,λ n -⇀ σ 0 * (α)λ • λ weakly in D ′ (Ω), σ 0 n (α ′ )∇w α ′ ,λ n • ∇w α,λ n -⇀ σ 0 * (α ′ )λ • λ weakly in D ′ (Ω).
(2.51)

As the matrices are symmetric, we have

σ 0 n (α) -σ 0 n (α ′ ) ∇w α,λ n • ∇w α ′ ,λ n = σ 0 n (α)∇w α,λ n • ∇w α ′ ,λ n -σ 0 n (α ′ )∇w α ′ ,λ n • ∇w α,λ n , hence σ 0 n (α) -σ 0 n (α ′ ) ∇w α,λ n • ∇w α ′ ,λ n -⇀ σ 0 * (α) -σ 0 * (α ′ ) λ • λ weakly in D ′ (Ω). (2.52) Let λ ∈ R 2 .
We have, by the Cauchy-Schwarz inequality, with the Einstein convention

ˆΩ σ 0 n (α) -σ 0 n (α ′ ) ∇w α,λ n • ∇w α ′ ,λ n dx = ˆΩ\ωn |α 1 -α ′ 1 | ∇w α,λ n • ∇w α ′ ,λ n dx + ωn |α 2 -α ′ 2 | ∇w α,λ n • ∇w α ′ ,λ n dx ≤ |α 1 -α ′ 1 | ˆΩ\ωn |∇w α,λ n | 2 dx ˆΩ\ωn |∇w α ′ ,λ n | 2 dx + |α 2 -α ′ 2 | ωn |∇w α,λ n | 2 dx ωn |∇w α ′ ,λ n | 2 dx ≤ |α i -α ′ i | 1 α i ˆΩ σ 0 n (α)∇w α,λ n • ∇w α,λ n dx 1 α ′ i ˆΩ σ 0 n (α)∇w α ′ ,λ n • ∇w α ′ ,λ n dx.
This combined with (2.50) yields

ˆΩ σ 0 n (α) -σ 0 n (α ′ ) ∇w α,λ n • ∇w α ′ ,λ n ≤ C|λ| 2 |α i -α ′ i | |α i ||α ′ i | M (α) M (α ′ )
The sequence of (2.52) is thus bounded in L 1 (Ω) 2 which implies that (2.52) holds weakly- * in M(Ω). Hence, we get, for any ϕ ∈ C c (Ω), that

ˆΩ σ 0 * (α) -σ 0 * (α ′ ) λ • λ ϕ dx ≤ C |λ| 2 |α i -α ′ i | |α i ||α ′ i | M (α) M (α ′ ) ||ϕ|| ∞ . (2.53)
Then, the Riesz representation theorem implies that

σ 0 * (α) -σ 0 * (α ′ ) L 1 (Ω) 2×2 ≤ C |α i -α ′ i | |α i ||α ′ i | M (α) M (α ′ ).
Therefore, by the definition of M in (2.50), for any compact subset

K ⊂ (0, ∞) 2 , ∃ C > 0, ∀ α, α ′ ∈ Q 2 ∩ K, σ 0 * (α) -σ 0 * (α ′ ) L 1 (Ω) 2×2 ≤ C |α -α ′ |. (2.54)
This estimate permits to extend the definition (2.41) of σ 0 * on (0, ∞) 2 by [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF] implies that there exists a subsequence of n, denoted by n ′ , and a matrix-valued function σ * ∈ M(a 0 , 2||a|| ∞ ; Ω) such that

∀ α ∈ (0, ∞) 2 , σ 0 * (α) = lim α ′ →α α ′ ∈Q 2 ∩(0,∞) 2 σ 0 * (α ′ ) strongly in L 1 (Ω) 2×2 . (2.55) Let α ∈ (0, ∞) 2 . Theorem 2.2 of
σ n ′ (α) H(M(Ω) 2 ) -⇀ σ * .
(2.56)

Repeating the arguments leading to (2.54), for any positive sequence of rational pair (α q ) q∈N converging to α, we have

∃ C > 0, σ * -σ 0 * (α q ) L 1 (Ω) 2×2 ≤ C |α -α q |, (2.57)
hence, by (2.55), σ * = σ 0 * (α). Therefore by the uniqueness of the limit in (2.56), we obtain for the whole sequence satisfying (2.41)

∀ α ∈ (0, ∞) 2 , σ n (α) H(M(Ω) 2 ) -⇀ σ 0 * (α).
(2.58)

In particular, the function σ 0 * satisfies (2.54) and (2.55), i.e. σ 0 * is a locally Lipschitz function on (0, ∞) 2 .

Second step: The general case. We denote [START_REF] Briane | Duality results in the homogenization of two-dimensional highcontrast conductivities[END_REF] implies that there exists a subsequence of n, denoted by n ′ , such that

α n = (α 1,n , α 2,n ) and σ 0 n (α n ) = σ 0 n (α 1,n , α 2,n ). Theorem 2.2 of
σ 0 n ′ (α n ′ ) H(M(Ω) 2 )-converges to some tt σ * ∈ M(a 0 , 2||a|| ∞ ; Ω) in the sense of Definition 1.1.
As in the first step, for any α n ′ ∈ (0, ∞) 2 and λ ∈ R 2 , we can consider the corrector

w α n ′ ,λ n ′ associated with σ 0 n ′ (α n ′ ) defined by      div σ 0 n ′ (α n ′ )∇w α n ′ ,λ n ′ = div ( σ * λ) in Ω, w α n ′ ,λ n ′ = λ • x on ∂Ω, (2.59)
which depends linearly on λ. Proceeding as in the first step, we obtain like in (2.52), with α = (α 1 , α 2 ) the limit of α n according to (2.39),

σ 0 n ′ (α) -σ 0 n ′ (α n ′ ) ∇w α n ′ ,λ n ′ • ∇w α,λ n ′ -⇀ σ 0 * (α) -σ * λ • λ weakly in D ′ (Ω). (2.60)
Moreover, by the energy bound (2.50), which also holds for α n ′ , we have, for any ϕ ∈ D(Ω),

ˆΩ σ 0 n ′ (α) -σ 0 n ′ (α n ′ ) ∇w α n ′ ,λ n ′ • ∇w α,λ n ′ ϕ dx -→ n ′ →∞ 0.
This combined with (2.60), yields

ˆΩ σ 0 * (α) -σ * λ • λ ϕ dx = 0,
which implies that σ 0 * (α) = σ * . We conclude by a uniqueness argument.

We can now obtain a result for (perturbed) non-symmetric conductivities. Then, we will use a Dykhne transformation to recover the symmetric case following the Milton approach [START_REF] Milton | The Theory of Composites[END_REF] (pp. 61-65). This will allow us to apply Proposition 2.2. (2.61)

Consider the conductivity defined by

σ n (h) = (1 -χ n )σ 1 (h) + χ n θ n σ 2 (h) (2.62
)

where for j = 1, 2, σ j (h) = α j + hβ j J ∈ R 2×2 with α 1 , α 2 > 0 and (β 1 , β 2 ) = (0, 0).
Then, there exists a subsequence of n, still denoted by n, and a locally Lipschitz function

σ 0 * : (0, ∞) 2 -→ M min(α 1 , α 2 ), 2 |σ 1 | + |σ 2 | ||a|| ∞ ; Ω such that σ n (h) H(M(Ω) 2 ) -⇀ σ 0 * α 1 , α 2 + α -1 2 β 2 2 h 2 + hβ 1 J.
Proof of Theorem 2.2. We have

∀ ξ ∈ R 2 , σ n (h)ξ • ξ = (1 -χ n )α 1 |ξ| 2 + χ n θ n α 2 |ξ| 2 ≥ min(α 1 , α 2 )|ξ| 2 a.e. in Ω
and, by (2.61),

|σ n (h)| = (1 -χ n )|σ 1 (h)| + χ n θ n |σ 2 (h)|-⇀ |σ 1 (h)| + a|σ 2 (h)| ∈ L ∞ (Ω) weakly- * in M(Ω).
In order to make a Dykhne transformation like in p.62 of [START_REF] Milton | The Theory of Composites[END_REF], we consider two real coefficients a n and b n in such a way that

B n := a n σ n (h) + b n J a n I 2 + Jσ n (h) -1 = (p n σ n (h) + q n J) -1 + r n J -1
is symmetric. An easy computation shows that the previous equality holds when

p n := a 2 n a 2 n + b n , q n := a n b n a 2 n + b n and r n := 1 a n .
On the one hand, the estimates (3.39) and (3.40) with α 2,n = θ -1 n α 2 , β 2,n = θ -1 n β 2 , yield (note that they are independent of χ n )

p n ∼ n→∞ 1, q n -→ n→∞ -hβ 1 , r n -→ n→∞ 0 and ||r n σ n (h)|| ∞ ≤ C |σ 1 (h)| + |σ 2 (h)| .
(2.63)

On the other hand, as in Section 3.2, with Notation 1.1 and (3.34), we have

B n = σ 0 n α ′ 1,n (h), α ′ 2,n (h) , (2.64)
where

α ′ 1,n (h) = a n (α 1 + ihβ 1 ) + ib n a n + i(α 1 + ihβ 1 ) and α ′ 2,n (h) = a n (α 2 /θ n + ihβ 2 /θ n ) + ib n a n + i(α 2 /θ n + ihβ 2 /θ n ) . (2.65)
Hence, like in (3.41), we have

lim n→∞ α ′ 1,n (h) = α 1 and lim n→∞ θ n α ′ 2,n (h) = α 2 + α -1 2 β 2 2 h 2 .
(2.66)

We can first apply Proposition 2.2 with the conditions (2.64) and (2.66) to have the H(M(Ω) 2 )convergence of B n . Then, by virtue of Proposition 2.1, with (2.63) we get that

σ n (h) H(M(Ω) 2 ) -⇀ σ 0 * α 1 , α 2 + α -1 2 β 2 2 h 2 + hβ 1 J.

A two-dimensional periodic medium

In this section we consider a sequence Σ n of matrix valued functions (not necessarily symmetric) in L ∞ (R 2 ) 2×2 , which satisfies the following assumptions:

1 . Σ n is Y -periodic, where Y := (0, 1) 2 , i.e., ∀ n ∈ N, ∀ κ ∈ Z 2 , Σ n (. + κ) = Σ n (.) a.e. in R 2 , (3.1) 2 . Σ n is equi-coercive in R 2 , i.e., ∃ α > 0 such that ∀ n ∈ N, ∀ ξ ∈ R 2 , Σ n ξ • ξ ≥ α|ξ| 2 a.e. in R 2 . (3.2)
Let ε n be a sequence of positive numbers which tends to 0. From the sequences Σ n and ε n we define the highly oscillating sequence of matrix-valued functions σ n by

σ n (x) = Σ n x ε n , a.e. x ∈ R 2 . (3.3)
By virtue of (3.1) and (3.2), σ n is an equi-coercive sequence of ε n -periodic matrix-valued functions in L ∞ (R 2 ) 2×2 . For a fixed n ∈ N, let (σ n ) * be the constant matrix defined by

∀ λ, µ ∈ R 2 , (σ n ) * λ • µ = ˆY Σ n ∇W λ n • ∇W µ n dy, (3.4)
where, for any λ ∈ R 2 , W λ n ∈ H 1 ♯ (Y ), the set of Y -periodic functions belonging to H 1 loc (R 2 ), is the solution of the auxiliary problem

ˆY W λ n -λ • y dy = 0 and div Σ n ∇W λ n = 0 in D ′ (R 2 ) (3.5) or equivalently        ˆY Σ n ∇W λ n • ∇ϕ dy = 0, ∀ ϕ ∈ H 1 ♯ (Y ) ˆY W λ n (y) -λ • y dy = 0. (3.6) Set w λ n (x) := ε n W λ n x ε n , for x ∈ Ω, (3.7
(σ n ) * -→ σ * in R 2×2 . (3.9)
Consider, for f ∈ H -1 (Ω) ∩ W -1,q (Ω) with q > 2, the solution u n of the problem

P n -div(σ n ∇u n ) = f in Ω u n = 0 on ∂Ω. (3.10)
Then, u n converges uniformly to the solution u ∈ H 1 0 (Ω) of

P -div(σ * ∇u) = f in Ω u = 0 on ∂Ω. (3.11)
Moreover we have the corrector result, with the ε n Y -periodic sequence w n defined in (3.8):

∇u n - 2 i=1 ∂ i u ∇w i n -→ 0 in L 1 (Ω) 2 .
(3.12)

Remark 3.1 The first point of Theorem 3.1 is an extension to the non-symmetric case of the results of [START_REF] Briane | Asymptotic behaviour of equicoercive diffusion energies in dimension two[END_REF] and [START_REF] Briane | Uniform convergence of sequences of solutions of twodimensional linear elliptic equations with unbounded coefficients[END_REF]. The uniform convergence of u n is a straightforward consequence of Theorem 2.7 of [START_REF] Briane | Uniform convergence of sequences of solutions of twodimensional linear elliptic equations with unbounded coefficients[END_REF] taking into account that in the present case σ n ∈ L ∞ (Ω) 2×2 for a fixed n. The fact that f ∈ W -1,q (Ω) with q > 2 ensures the uniform convergence.

Proof of Theorem 3.1. Derivation of the limit problem P.

We only have to show that u is the solution of P in (3.11). We consider a corrector D w n : R 2 -→ R 2×2 associated with σ T n defined by

w n (x) := ε n W n x ε n = ε n W 1 n x ε n , ε n W 2 n x ε n where for i = 1, 2, W i n ∈ H 1 ♯ (Y )
is the solution of the auxiliary problem

ˆY W i n -e i • x dx = 0 and div Σ T n ∇ W i n = 0 in D ′ (R 2 ). (3.13)
Again, thanks to Theorem 2.7 of [START_REF] Briane | Uniform convergence of sequences of solutions of twodimensional linear elliptic equations with unbounded coefficients[END_REF], w n converges uniformly to the identity in Ω by the integral condition (3.13). Let ϕ ∈ D(Ω). We have, using the Einstein convention, by integrating by parts and by the Schwarz theorem (∂ 2 i,j ϕ = ∂ 2 j,i ϕ)

ˆΩ σ n ∇u n • ∇ (ϕ( w n )) dx = ˆΩ ∇u n • σ T n ∇ w i n (∂ i ϕ)( w n ) dx = ˆΩ σ T n ∇ w i n • ∇(u n ∂ i ϕ( w n )) dx =0 -ˆΩ σ T n ∇ w i n • ∇ w j n ∂ 2 i,j ϕ( w n ) u n dx = -ˆΩ σ n ∇ w i n • ∇ w i n ∂ 2 i,i ϕ( w n ) u n dx -ˆΩ σ T n ∇ w 2 n • ∇ w 1 n ∂ 2 2,1 ϕ( w n ) u n dx espace -ˆΩ σ T n ∇ w 1 n • ∇ w 2 n ∂ 2 1,2 ϕ( w n ) u n dx = -ˆΩ σ n ∇ w i n • ∇ w i n ∂ 2 i,i ϕ( w n ) u n dx -ˆΩ σ n ∇ w 1 n • ∇ w 2 n ∂ 2 1,2 ϕ( w n ) u n dx espace -ˆΩ σ T n ∇ w 1 n • ∇ w 2 n ∂ 2 1,2 ϕ( w n ) u n dx = -ˆΩ σ s n ∇ w i n • ∇ w i n ∂ 2 i,i ϕ( w n ) u n dx -2 ˆΩ σ s n ∇ w 1 n • ∇ w 2 n ∂ 2 1,2 ϕ( w n ) u n dx.
This leads us to the equality

f, ϕ( w n ) H -1 (Ω),H 1 0 (Ω) = ˆΩ σ n ∇u n • ∇ (ϕ( w n )) dx = -ˆΩ σ s n ∇ w i n • ∇ w j n ∂ 2 i,j ϕ( w n ) u n dx. (3.14)
To study the convergence of the last term of (3.14), we first show that σ s n ∇ w i n • ∇ w j n is bounded in L 1 (Ω). We have, by periodicity and the Cauchy-Schwarz inequality

ˆΩ |σ s n ∇ w i n • ∇ w j n | dx = ˆΩ |Σ s n ∇ W i n • ∇ W j n | x ε n dx ≤ C ˆY |Σ s n ∇ W i n • ∇ W j n | dx ≤ C ˆY Σ s n ∇ W i n • ∇ W i n dx ˆY Σ s n ∇ W j n • ∇ W j n dx ≤ C (σ n ) * e i • e i (σ n ) * e j • e j
which is bounded by the hypothesis (3.9). Therefore,

σ s n ∇ w i n • ∇ w j n is bounded in L 1 (Ω). (3.15)
Due to the periodicity, we know that for i, j = 1, 2,

2σ s n ∇ w i n • ∇ w j n = σ T n ∇ w i n • ∇ w j n + σ T n ∇ w j n • ∇ w i n ⇀ (σ * ) T e i • e j + (σ * ) T e j • e i = 2 (σ * ) s e i • e j
weakly- * in M(Ω). Hence, we get that

σ s n ∇ w i n • ∇ w j n ⇀ (σ * ) s e i • e j weakly- * in M(Ω). (3.16)
Moreover, ∂ 2 i,j ϕ( w n ) u n converges uniformly to ∂ 2 i,j ϕ u. Thus, by passing to the limit in (3.14), we have, again with the Einstein convention

f, ϕ H -1 (Ω),H 1 0 (Ω) = -ˆΩ (σ * ) s e i • e j ∂ 2 i,j ϕ u dx = -ˆΩ σ * : ∇ 2 ϕ u dx.
Therefore, by integrating by parts and using ϕ = 0 on ∂Ω,

ˆΩ σ * ∇u • ∇ϕ dx = f, ϕ H -1 (Ω),H 1 0 (Ω) .
(3.17)

Proof of the corrector result

First of all, we show that the corrector function w n is bounded in H 1 (Ω) 2 . By the definition (3.8) of w n , the Y -periodicity of W e i n and the equi-coercivity of Σ n , we have, for i = 1, 2,

α ||∇w i n || 2 L 2 (Ω) 2 ≤ Cα ||∇W e i n || 2 L 2 (Y ) 2 ≤ C ˆY Σ n ∇W i n • ∇W i n dx = C (σ n ) * e i • e i (3.18)
which is bounded. This inequality combined with the uniform convergence of w n yields to the boundedness of w n in H 1 (Ω) 2 . Let us consider an approximation

u δ ∈ D(Ω) of u such that ||u -u δ || H 1 0 (Ω) ≤ δ. (3.19)
On the one hand, we have

ˆΩ σ n ∇u n • ∇ u n -u δ (w n ) dx = f, u n -u δ (w n ) H -1 (Ω),H 1 0 (Ω) .
Since w n converges uniformly to identity on Ω and is bounded in H 1 (Ω) (see (3.18)), with u δ ∈ D(Ω), u δ (w n ) converges weakly to u δ in H 1 0 (Ω). Hence, by the weak convergence of u n to u in H 1 0 (Ω) and (3.19), we can pass to the limit the previous inequality and obtain, for any δ > 0,

lim sup n→∞ ˆΩ σ n ∇u n • ∇ u n -u δ (w n ) dx = f, u -u δ H -1 (Ω),H 1 0 (Ω) ≤ Cδ. (3.20)
On the other hand, similarly to the proof of the first point (3.14), we are led to the equality

ˆΩ σ n ∇ u δ (w n ) • ∇ u n -u δ (w n ) dx = -ˆΩ σ s n ∇w i n • ∇w j n ∂ 2 i,j u δ (w n ) u n -u δ (w n ) dx. (3.21)
As in the first point, σ s n ∇w i n • ∇w j n is bounded in L 1 (Ω) (see (3.15)), u n converges uniformly to u and ∂ i,j u δ (w n ) converges uniformly to ∂ i,j u δ because u δ is a D(Ω) function. By passing to the limit in (3.21)

ˆΩ σ n ∇ u δ (w n ) • ∇ u n -u δ (w n ) dx -→ n→∞ -ˆΩ(σ * ) s e i • e j ∂ 2 i,j u δ u -u δ dx. (3.22)
Moreover, like in (3.17) we have

ˆΩ(σ * ) s e i • e j ∂ 2 i,j u δ u -u δ dx = ˆΩ σ * ∇u δ • ∇ u -u δ dx. (3.23)
By combining this equality with the convergence (3.22), we obtain the inequality

lim n→∞ ˆΩ σ n ∇ u δ (w n ) • ∇ u n -u δ (w n ) dx ≤ ˆΩ σ * ∇u δ • ∇ u -u δ (3.24) ≤ C|σ * | ||∇u δ || L 2 (Ω) 2 ||∇ u -u δ || L 2 (Ω) 2 ≤ Cδ. (3.25)
Thus, by adding (3.20) and (3.25), we have

lim sup n→∞ ˆΩ σ n ∇ u n -u δ (w n ) • ∇ u n -u δ (w n ) dx ≤ Cδ
which leads us, by equi-coercivity, to

lim sup n→∞ α ||∇(u n -u δ (w n ))|| 2 L 2 (Ω) 2 ≤ lim sup n→∞ ˆΩ σ n ∇ u n -u δ (w n ) • ∇ u n -u δ (w n ) dx ≤ Cδ. (3.26)
Thus, the Cauchy-Schwarz inequality, the boundedness of ∇w i n in L 2 (Ω) 2 (3.18) and the Einstein convention give, for any δ > 0,

||∇u n -∇w i n ∂ i u|| L 1 (Ω) 2 ≤ ||∇u n -∇w i n ∂ i u δ || L 1 (Ω) 2 + ||∇w i n ∂ i u δ -u || L 1 (Ω) 2 ≤ ||∇u n -∇w i n ∂ i u δ || L 1 (Ω) 2 + ||∇w i n || L 2 (Ω) 2 ||∂ i u δ -u || L 2 (Ω) ≤ ||∇u n -∇w i n ∂ i u δ || L 1 (Ω) 2 + Cδ ≤ ||∇u n -∇w i n ∂ i u δ (w n )|| L 1 (Ω) 2 + ||∇w i n ∂ i u δ -∂ i u δ (w n ) || L 1 (Ω) 2 + Cδ ≤ ||∇u n -∇w i n ∂ i u δ (w n )|| L 1 (Ω) 2 + ||∇w i n || L 2 (Ω) 2 ||∂ i u δ -∂ i u δ (w n )|| L 2 (Ω) + Cδ ≤ ||∇u n -∇w i n ∂ i u δ (w n )|| L 1 (Ω) 2 + C||∂ i u δ -∂ i u δ (w n )|| L 2 (Ω) + Cδ.
Since u δ ∈ D(Ω) and w n converges uniformly to the identity on Ω, the second term of the last inequality converges to 0. Hence, we get that

lim sup n→∞ ||∇u n -∇w i n ∂ i u|| L 1 (Ω) 2 ≤ lim sup n→∞ ||∇u n -∇w i n ∂ i u δ (w n )|| L 1 (Ω) 2 + Cδ. (3.27)
Finally, this inequality combined with (3.26) gives, for any δ > 0,

0 ≤ lim sup n→∞ ||∇u n -∇w i n ∂ i u|| L 1 (Ω) 2 ≤ C √ δ + Cδ,
which implies the corrector result (3.12).

Remark 3.2 If the solution u is a C 2 function, then the convergence (3.12) holds true in L 2 loc (Ω) since we may take u = u δ .

A two-phase result

Here, we recall a two-phase result due to G.W. Milton (see [START_REF] Milton | The Theory of Composites[END_REF] pp. 61-65) using the Dykhne transformation.

In order to apply the previous theorem, we reformulate Milton's calculus in such a way that every coefficient depends on n. We then consider, for a fixed n, the periodic homogenization of a conductivity σ n (h) to obtain (σ n ) * (h) through the link between the homogenization of the transformed conductivity and (σ n ) * (h) given by formula (4.16) in [START_REF] Milton | The Theory of Composites[END_REF]. Finally, we study the limit of (σ n ) * (h) through the asymptotic behavior of the coefficients of the transformation, and apply Theorem 3.1 in the example Section 3.3.

In this section we consider a two-phase periodic isotropic medium. Let χ n be a sequence of characteristic functions of subsets of Y . We define for any α 1 > 0, β 1 ∈ R, any sequences α 2,n > 0, β 2,n ∈ R and any h ∈ R, a parametrized conductivity Σ n (h):

Σ n (h) = (1 -χ n )(α 1 I 2 + hβ 1 J) + χ n (α 2,n I 2 + hβ 2,n J) in Y.
(3.28)

We still denote by Σ n (h) the periodic extension to R 2 of Σ n (h) (which satisfies (3.1)). We assume that Σ n (h) satisfies (3.2), and define σ n (h) by (3.3) and (σ n ) * (h) by (3.4).

We have the following result based on an analysis of [START_REF] Milton | The Theory of Composites[END_REF] (pp. 61-65). Assume that the effective conductivity in the absence of a magnetic field

σ 0 n * (γ 1,n , γ 2,n ) is bounded when lim n→∞ γ 1,n = α 1 and lim n→∞ γ 2,n α 2,n = γ 2 > 0. (3.30)
Then, there exist two parametrized positive sequences

α ′ 1,n (h), α ′ 2,n (h) such that lim n→∞ α ′ 1,n (h) = α 1 and α ′ 2,n (h) ∼ n→∞ α 2 2 + h 2 β 2 2 α 2 2 α 2,n , (3.31)
and

(σ n ) * (h) = σ 0 n * α ′ 1,n (h), α ′ 2,n (h) + hβ 1 J + o n→∞ (1) (3.32)
where

σ 0 n * α ′ 1,n (h), α ′ 2,n (h) is bounded. Remark 3.3
In view of condition (3.29), the case where β 2,n tends to β 1 corresponds to perturb the symmetric conductivity

σ s n = (1 -χ n )α 1 I 2 + χ n α 2,n I 2 by σ s n + β 1 J + o n→∞ (1).
Then it is clear that

(σ n ) * (h) = σ s n * + β 1 J + o n→∞ (1)
.

Proof of Proposition 3.1. The proof is divided into two parts. After applying Milton's computation (pp. 61-64 of [START_REF] Milton | The Theory of Composites[END_REF]), we study the asymptotic behavior of the different coefficients. t First step: Applying Dykhne's transformation through Milton's computations.

In order to make the Dykhne's transformation following Milton [START_REF] Milton | The Theory of Composites[END_REF] (pp. 62-64), we consider two real coefficients a n and b n such that

σ ′ n := a n σ n (h) + b n J a n I 2 + Jσ n (h) -1 = a n σ n (h) + (a n ) -1 b n J a n I 2 + Jσ n (h) -1 (3.33)
is symmetric and, more precisely, according to Notation 1.1, reads as

σ ′ n = (1 -χ n )α ′ 1,n (h)I 2 + χ n α ′ 2,n (h)I 2 = σ 0 n α ′ 1,n (h), α ′ 2,n (h) . (3.34)
Then, using the complex representation

αI 2 + βJ ←→ α + βi (3.35)
suggested by Tartar [START_REF] Tartar | Private communication to G[END_REF], the constants a n , b n must satisfy

α ′ 1,n (h) = a n (α 1 + ihβ 1 ) + ib n a n + i(α 1 + ihβ 1 ) ∈ R and α ′ 2,n (h) = a n (α 2,n + ihβ 2,n ) + ib n a n + i(α 2,n + ihβ 2,n ) ∈ R, (3.36)
which implies that

b n = -a 2 n hβ 1 + a n ∆ 1 a n -hβ 1 = -a 2 n hβ 2,n + a n ∆ 2,n a n -hβ 2,n . (3.37)
Denoting

∆ 1 := α 2 1 + h 2 β 2 1 and ∆ 2,n := α 2 2,n + h 2 β 2 2
,n (thanks to (3.29), n is considered to be larger enough such that β 2,n -β 1 = 0 and a n is real), the equality (3.37) provides two non-zero solutions for a n :

a n = ∆ 2,n -∆ 1 + (∆ 2,n -∆ 1 ) 2 + 4h 2 (β 2,n -β 1 )(β 2,n ∆ 1 -β 1 ∆ 2,n ) 2h(β 2,n -β 1 ) , (3.38)
and

a - n = ∆ 2,n -∆ 1 -(∆ 2,n -∆ 1 ) 2 + 4h 2 (β 2,n -β 1 )(β 2,n ∆ 1 -β 1 ∆ 2,n ) 2h(β 2,n -β 1
) .

The value (3.38) is associated with a positive matrix σ ′ n , while a - n leads us to the negative matrix σ - n = -J(σ ′ n ) -1 J -1 to exclude (see [START_REF] Milton | Classical hall effect in two-dimensional composites: A characterization of the set of realizable effective conductivity tensors[END_REF] for more details). t Second step: asymptotic behavior of the coefficients and the homogenized matrix.

One the one hand, by the equality (3.38) combined with (3.29), we have

lim n→∞ a n h(β 2,n -β 1 ) α 2 2,n = α 2 2 + h 2 β 2 2 α 2 2
which clearly implies that

a n ∼ n→∞ α 2 2 + h 2 β 2 2 α 2 2 α 2 2,n h(β 2,n -β 1 ) and a n -hβ 2,n ∼ n→∞ α 2 2,n h(β 2,n -β 1 ) . (3.39)
On the other hand, (

and the first equality of (3.37) give

b n = -a n hβ 1 + ∆ 1 + o n→∞ (1). (3.40) From (3.29), (3.38), (3.39) and (3.40)
we deduce the following asymptotic behavior for the modified phases:

lim n→∞ α ′ 1,n (h) = α 1 and lim n→∞ α ′ 2,n (h) α 2,n = α 2 2 + h 2 β 2 2 α 2 2 . (3.41)
To consider σ ′ n * , we need to verify that σ ′ n is equi-coercive. We have, by denoting for any

ξ ∈ R 2 , ν n = a n I 2 + Jσ n (h) -1 ξ, ∀ ξ ∈ R 2 , σ ′ n ξ • ξ = a n σ n (h) + b n J ν n • a n I 2 + Jσ n (h) ν n = (a 2 n + b n )σ n (h)ν n • ν n and, because a -1 n σ n (h) is bounded in L ∞ (Ω) 2×2 by (3.39), ∀ ξ ∈ R 2 , |ξ| = a n ν n + Jσ n (h)ν n ≤ a n (1 + C)|ν n |.
The equi-coercivity of σ n (h) gives

∃ C > 0, ∀ ξ ∈ R 2 , σ ′ n ξ • ξ ≥ C (1 + C) 2 a 2 n + b n a 2 n |ξ| 2 (3.42)
that is, for n larger enough, by (3.39) and (3.40), σ ′ n is equi-coercive. We can now apply the Keller-Dykhne duality theorem (see, e.g., [START_REF] Keller | A theorem on the conductivity of a composite medium[END_REF][START_REF] Dykhne | Conductivity of a two-dimensional two-phase system[END_REF]) to equality (3.33)

to obtain (σ ′ n ) * = a n (σ n ) * + b n J a n I 2 + J(σ n ) * -1 . (3.43)
Moreover, by inverting this transformation, we have

(σ n ) * (h) = a n I 2 -(σ ′ n ) * J -1 a n (σ ′ n ) * -b n J . a α 1 I 2 + β 1 hJ α 2,n I 2 + β 2,n hJ 2t n 2t n ℓ 1 x 1 x 2
(σ n ) * (h) = (σ ′ n ) * - b n a n J + o n→∞ (1) = (σ ′ n ) * + hβ 1 J + o n→∞ (1), (3.44)
which concludes the proof taking into account (3.34).

To derive the limit of

σ 0 n * (α ′ 1,n (h), α ′ 2,n (h)
), we need more information on the geometry of the high conductive phase. To this end, we study the following example.

A cross-like thin structure

We consider a bounded open subset Ω of R 2 with a Lipschitz boundary, a real sequence ε n converging to 0, and f ∈ H -1 (Ω) ∩ W -1,q (Ω) with q > 2. We define, for any h ∈ R, α 1 , β 1 > 0 and positive sequences t n ∈ (0, 1/2], α 2,n , β 2,n , a parametrized matrix-valued function Σ n (h) from the unit rectangular cell period Y := (-ℓ 2 , ℓ 2 ) × (-1 2 , 1 2 ), with ℓ ≥ 1, to R 2×2 , by (cf. figure 3.1)

Σ n (h) := α 2,n I 2 + β 2,n hJ in ω n := {(x 1 , x 2 ) ∈ Y | |x 1 |, |x 2 | ≤ t n } α 1 I 2 + β 1 hJ in Y \ ω n (3.45)
Denoting again by Σ n (h) its periodic extension to R 2 , we finally consider the conductivity

σ n (h)(x) = Σ n (h) x ε n , x ∈ Ω, (3.46)
and the associated homogenization problem:

P n -div σ n (h)∇u n = f in Ω u n = 0 on ∂Ω. (3.47)
By virtue of Theorem 3.1 and Proposition 3.1, we focus on the study of the limit of

σ 0 n * α ′ 1,n (h), α ′ 2,n (h) .
Proposition 3.2 Let σ n (h) be the conductivity defined by (3.45) and (3.46) its homogenization problem (3.47). We assume that:

2t n (ℓ + 1)α 2,n -→ n→∞ α 2 > 0 and 2t n (ℓ + 1)β 2,n -→ n→∞ β 2 > 0. (3.48)
Then, the homogenized conductivity is given by

σ * (h) =     α 1 + α 2 2 + β 2 2 h 2 (ℓ + 1)α 2 -hβ 1 hβ 1 α 1 + α 2 2 + β 2 2 h 2 ℓ(ℓ + 1)α 2     .
Remark 3.4 The previous proposition does not respect exactly the framework defined at the beginning of this section because the period cell is not the unit square Y = (0, 1) 2 : we can nevertheless extend all this section to any type of period cells.

Remark 3.5 The condition (3.48) is a condition of boundedness in L 1 (Ω) 2×2 of σ n because |ω n | = 2t n (ℓ + 1) -4t 2 n ∼ 2t n (ℓ + 1),
which will ensure the convergence of σ 0 n * .

Proof of Proposition 3.2. In order to apply Proposition 3.1, we consider two positive sequences

α ′ 1,n (h), α ′ 2,n (h) satisfying lim n→∞ α ′ 1,n (h) = α 1 and α ′ 2,n (h) ∼ n→∞ α 2 2 + h 2 β 2 2 α 2 2 α 2,n . (3.49)
We will study the homogenization of

σ ′ n := σ 0 n α ′ 1,n (h), α ′ 2,n (h)
. To this end, consider a corrector W λ n = λ • x -X λ n in the Murat-Tartar sense (see, e.g., [START_REF] Murat | Topics in the Mathematical Modelling of Composite Materials[END_REF]) associated with

Σ ′ n := α ′ 2,n (h) I 2 in ω n = {(x 1 , x 2 ) ∈ Y | |x 1 |, |x 2 | ≤ t n } α ′ 1,n (h) I 2 in Y \ ω n (3.50)
and defined by

         div Σ ′ n ∇X λ n = div Σ ′ n λ in D ′ (R 2 ) X λ n is Y -periodic ˆY X λ n dy = 0. (3.51)
On one hand, the extra diagonal coefficients of (σ ′ n ) * are equal to 0 because, as Σ ′ n is an even function on Y , we have, for i = 1, 2,

y i -→ W e i n (y) is an odd function, y i -→ W e j n (y) is an even function for i = j, which implies that y 1 -→ Σ ′ n ∇W e 1 n •∇W e 2
n is an odd function. Then, by symmetry of Y with respect to 0,

(σ ′ n ) * e i • e j = ˆY Σ ′ n ∇W e i n • ∇W e j
n dy = 0.

On the other hand, as Σ ′ n is isotropic, for the diagonal coefficients, we use the Voigt-Reuss inequalities (see, e.g., [START_REF] Jikov | Homogenization of Differential Operators and Integral Functionals[END_REF] p.44 or [START_REF] Mortola | Un problema di omogeneizzazione bidimensionale[END_REF]): for any i = 1, 2 and j = i,

(Σ ′ n e i • e i ) -1 -1 i j ≤ (σ ′ n ) * e i • e i ≤ Σ ′ n e i • e i -1 j -1 i (3.52)
where • i denotes the average with respect to y i at a fixed y j for j = i.

An easy computation gives, for the direction e 1 ,

(1 -2t n ) ℓ -2t n ℓα ′ 1,n (h) + 2t n ℓα ′ 2,n (h) -1 + 2t n ℓ ℓα ′ 2,n (h) -1 ≤ (σ ′ n ) * e 1 • e 1 and (σ ′ n ) * e 1 • e 1 ≤ ℓ ℓ -2t n (1 -2t n )α ′ 1,n (h) + 2t n α ′ 2,n (h) + 2t n α ′ 2,n (h) -1
.

By (3.48) and (3.49), we have the convergence

lim n→∞ (σ ′ n ) * e 1 • e 1 = α 1 + α 2 2 + β 2 2 h 2 (ℓ + 1)α 2 .
A similar computation on the direction e 2 gives the asymptotic behavior:

lim n→∞ (σ ′ n ) * = lim n→∞ σ 0 n * α ′ 1,n (h), α ′ 2,n (h) =     α 1 + α 2 2 + β 2 2 h 2 (ℓ + 1)α 2 0 0 α 1 + α 2 2 + β 2 2 h 2 ℓ(ℓ + 1)α 2     .
(3.53)

Moreover, the matrix σ n (h) clearly satisfies all the hypothesis of Theorem 3.1. By Theorem 3.1 and (3.53), we have

lim n→∞ (σ n ) * (h) = lim n→∞ σ 0 n * α ′ 1,n (h), α ′ 2,n (h) + β 1 hJ =     α 1 + α 2 2 + β 2 2 h 2 (ℓ + 1)α 2 -hβ 1 hβ 1 α 1 + α 2 2 + β 2 2 h 2 ℓ(ℓ + 1)α 2     .
We finally apply Theorem 3.1 to get that σ * (h) = lim n→∞ (σ n ) * (h).

A three-dimensional fibered microstructure

In this section we study a particular two-phase composite in dimension three. One of the phases is composed by a periodic set of high conductivity fibers embedded in an isotropic medium (figure 4.1a). The conductivity σ n (h) is not symmetric due to the perturbation of a magnetic field.

First, describe the geometry of the microstructure. Let Y := -1 2 , 1 2 3 be the unit cube centered at the origin of R 3 . For r n ∈ 0, 1 2 , consider the closed cylinder ω n parallel to the x 3 -axis, of radius r n and centered in Y :

ω n := y ∈ Y | y 2 1 + y 2 2 ≤ r 2 n . (4.1)
Let Ω = Ω × (0, 1) be an open cylinder of R 3 , where Ω is a bounded domain of R 2 with a Lipschitz boundary. For ε n ∈ (0, 1), consider the closed subset Ω n of Ω defined by the intersection with Ω of the ε n Y -periodic network in R 3 composed by the closed cylinders parallel to the x 3 -axis, centered on the points ε n k, k ∈ Z 2 , in the x 1 -x 2 plane, and of radius ε n r n , namely:

Ω n := Ω ∩ ν∈Z 3 ε n (ω n + ν). (4.2)
The period cell of the microstructure is represented in figure 4.1b.

(a) The fibers lattice

ε n ε n r n Ω n ∩ ε n (Y + ν) x 1 x 2
x 3 (b) The period cell We then define the two-phase conductivity by

σ n (h) = α 1 I 3 + β 1 E (h) in Ω \ Ω n α 2,n I 3 + β 2,n E (h) in Ω n , (4.3)
where α 1 > 0, β 1 ∈ R, α 2,n > 0 and β 2,n are real sequences, and

E (h) :=   0 -h 3 h 2 h 3 0 -h 1 -h 2 h 1 0   , for h =   h 1 h 2 h 3   ∈ R 3 .
Our aim is to study the homogenization problem

P Ω,n -div(σ n (h)∇u n ) = f in Ω u n = 0 on ∂Ω. (4.4)
Theorem 4.1 Let α 1 > 0, β 1 ∈ R, and let ε n , r n , α 2,n , β 2,n , n ∈ N, be real sequences such that ε n , r n > 0 converge to 0, α 2,n > 0, and

lim n→∞ ε 2 n | ln r n | = 0, lim n→∞ |ω n | α 2,n = α 2 > 0, lim n→∞ |ω n | β 2,n = β 2 ∈ R. (4.5)
Consider, for h ∈ R 3 , the conductivity σ n (h) defined by (4.3).

Then, there exists a subsequence of n, still denoted by n, such that, for any f ∈ H -1 (Ω) and any h ∈ R 3 , the solution u n of P Ω,n converges weakly in H 1 0 (Ω) to the solution u of

P Ω, * -div(σ * (h)∇u n ) = f in Ω u = 0 on ∂Ω, (4.6)
where σ * (h) is given by

σ * (h) = α 1 I 3 + α 3 2 + α 2 β 2 2 |h| 2 α 2 2 + β 2 2 h 2 3 e 3 ⊗ e 3 + β 1 E (h).
ω n := 3 j=1 y ∈ Y | i =j y 2 i ≤ r 2 n
and

Ω n := Ω ∩ ν∈Z 3 ε n (ω n + ν),
as represented in figure 4.2. Then, we derive the following homogenization conductivity: Remark 4. [START_REF] Bergman | Self-duality and the low field Hall effect in 2D and 3D metal-insulator composites. Percolation Structures and Processes[END_REF] We can check that when the volume fraction θ n = θ and the highly conducting phase of the conductivity α 2,n = α θ and β 2,n = β θ are independent of n, the explicit formula of [START_REF] Grabovsky | Exact relations for effective conductivity of fiber-reinforced conducting composites with the hall effect via a general theory[END_REF] denoted by σ * (θ, h), for the classical (since the period cell is now independent of n) periodically homogenized conductivity (see (3.4)) has a limit as θ → 0 when θα θ and θβ θ converge. Indeed, we may replace in the computations of [START_REF] Grabovsky | Exact relations for effective conductivity of fiber-reinforced conducting composites with the hall effect via a general theory[END_REF] the optimal Vigdergauz shape by the circular cross-section in the previous asymptotic regime. Therefore, Theorem 4.1 validates the double process characterized by the homogenization at a fixed volume fraction θ combined with the limit as θ → 0, by one homogenization process in which both the period and the volume fraction θ n = πr 2 n of the high conductivity phase tend to 0 as n → ∞.

σ * (h) = α 1 I 3 + 3 i=1 α 3 2 + α 2 β 2 2 |h| 2 α 2 2 + β 2 2 h 2 i e i ⊗ e i + β 1 E (h). ε n ε n r n Ω n ∩ ε n (Y + ν) x 1 x 2 x 3

Remark 4.4

The hypothesis on the convergence of ε 2 n | ln r n | (4.5) allows us to avoid nonlocal effects in dimension three (see [START_REF] Fenchenko | Asymptotic of solution of differential equations with strongly oscillating matrix of coefficients which does not satisfy the condition of uniform boundedness[END_REF][START_REF] Bellieud | Homogenization of elliptic problems in a fiber reinforced structure[END_REF]). These effects do not appear in dimension two as shown in [START_REF] Briane | Homogenization of non-uniformly bounded operators: Critical barrier for nonlocal effects[END_REF]. Therefore, we can make a comparison between dimension two and dimension three based on the strong field perturbation in the absence of nonlocal effects. which reduces to the simplified two-dimensional case when the symmetric part of the conductivity is independent of h 3 (i.e. σ 0 * in (2.40) does not depend on its second argument).

Proof of Theorem 4.1 The proof will be divided into four parts. We first prove the weak- * convergence in M(Ω) of σ n (h)∇u n in Ω n . Then we establish a linear system satisfied by the limits defined by

1 Ωn |ω n | ∂u n ∂x i -⇀ ξ i weakly- * in M(Ω).
Moreover, we deduce from Lemma 4.1 that

1 Ωn |ω n | ∂u n ∂x 3 -⇀ ∂u ∂x 3 weakly- * in M(Ω).
We finally calculate the homogenized matrix.

We first remark that, classically, the sequence of solutions u n of P Ω,n (see (4.4)) is bounded in

H 1 0 (Ω) because, since α 2,n diverges to ∞ : ||∇u n || 2 L 2 (Ω) 3 ≤ C ˆΩ(α 1 1 Ω\Ωn I 3 + α 2,n 1 Ωn I 3 )∇u n • ∇u n dx = ˆΩ σ n (h)∇u n • ∇u n dx.
By the Poincaré inequality, the previous inequality and (4.4) lead us to

||u n || 2 H 1 0 (Ω) ≤ C||∇u n || 2 L 2 (Ω) 3 ≤ C f, u n H -1 (Ω),H 1 0 (Ω) ≤ C||f || H -1 (Ω) ||u n || H 1 0 (Ω)
and then to

||u n || H 1 0 (Ω) ≤ C||f || H -1 (Ω)
. Thus, up to a subsequence still denoted by n, u n converges weakly to some function u in H 1 0 (Ω).

First step: Weak- * convergence in M(Ω) of the conductivity in the fibers 1 Ωn α 2,n I 3 +β 2,n E (h) ∇u n . We proceed as in [START_REF] Briane | Fibered microstructures for some nonlocal Dirichlet forms[END_REF] with a suitable oscillating test function. For R ∈ (0, 1/2), define the Y -periodic (independent of y 3 ) function V n by

V n (y 1 , y 2 , y 3 ) =        1 if y 2 1 + y 2 2 ≤ r n ln R -ln y 2 1 + y 2 2 ln R -ln r n if r n ≤ y 2 1 + y 2 2 ≤ R 0 if y 2 1 + y 2 2 ≥ R, for y ∈ Y,
and the rescaled function

v n (x) = V n x ε n , for x ∈ R 3 . (4.8)
In particular, by using the cylindrical coordinates and the fact that r n converges to 0, this function satisfies the inequalities

||v n || 2 L 2 (Ω) ≤ C||V n || 2 L 2 (Y ) = C ln R r n -2 πr 2 n + ˆ2π 0 ˆR rn r ln 2 R r drdθ = C ln R r n -2 π R 2 -r 2 n 2 -πr 2 n ln 2 R r n -π ln R r n ≤ C ln R r n -2 , ||∇v n || 2 L 2 (Ω) 3 ≤ C ε 2 n ||∇V n || 2 L 2 (Y ) 3 = C ε 2 n ln R r n -2 ˆ2π 0 ˆR rn 1 r drdθ ≤ C ε 2 n ln R r n -1
and, consequently

||v n || L 2 (Ω) + ε n ||∇v n || L 2 (Ω) 3 ≤ C ln R r n -1 -→ n→∞ 0. (4.9)
Let λ be a vector in R 3 perpendicular to the x 3 -axis. Define the Y -periodic function X n by ∇ X n = λ in ω n , such that X n ∈ D(Y ) and is Y -periodic, and the rescaled function X n by

X n (x) = ε n X n x ε n . (4.10)
In particular, X n satisfies

||X n || ∞ = ε n || X n || ∞ ≤ Cε n , ||∇X n || ∞ = ||∇ X n || ∞ ≤ C and ∇X n = λ in Ω n . (4.11)
We have, by (4.11) and (4.9),

||v n X n || H 1 (Ω) ≤ ||X n || ∞ ||v n || L 2 (Ω) + ||X n || ∞ ||∇v n || L 2 (Ω) 3 + ||∇X n || ∞ ||v n || L 2 (Ω) ≤ C ||v n || L 2 (Ω) + ε n ||∇v n || L 2 (Ω) 3 -→ n→∞ 0, which gives ∀ ϕ ∈ D(Ω), ϕ v n X n -→ n→∞ 0 strongly in H 1 0 (Ω). (4.12)
Let ϕ ∈ D(Ω). By the strong convergence (4.12), we have

ˆΩ σ n (h)∇u n • ∇ ϕ v n X n dx = f, ϕ v n X n H -1 (Ω),H 1 0 (Ω) -→ n→∞ 0. (4.13)
Let us decompose this integral which converges to 0, into the integral on the fibers set Ω n and the integral on its complementary:

ˆΩ σ n (h)∇u n • ∇ ϕ v n X n dx = ˆΩ\Ωn (α 1 I 3 + β 1 E (h))∇u n • ∇ ϕ v n X n dx (4.14a) + ˆΩn (α 2,n I 3 + β 2,n E (h))∇u n • ∇ ϕ v n X n dx. (4.14b)
The expression (4.14a) converges to 0 since, by the Cauchy-Schwarz inequality, the boundedness of u n in H 1 0 (Ω) and (4.12), we have

ˆΩ\Ωn (α 1 I 3 + β 1 E (h))∇u n • ∇ ϕ v n X n dx ≤ |α 1 I 3 + β 1 E (h)| ||∇u n || L 2 (Ω) 3 ||ϕ v n X n || H 1 0 (Ω) -→ n→∞ 0.
(4.15) Consequently, as v n = 1 and ∇X n = λ on Ω n , by (4.13), (4.14a), (4.14b) and (4.15), we have

ˆΩn σ n (h)∇u n • λ ϕ dx + ˆΩn σ n (h)∇u n • ∇ϕ X n dx -→ n→∞ 0. (4.16)
To prove the convergence to 0 of the right term, we now show that 1 Ωn α 2,n I 3 + β 2,n E (h) ∇u n is bounded in L 1 (Ω) 3 . We have, by the Cauchy-Schwarz inequality, (4.5) and the classical equivalent

|Ω n | ∼ n→∞ |Ω| |ω n |, ˆΩn α 2,n I 3 + β 2,n E (h) ∇u n dx 2 ≤ I 3 + α -1 2,n β 2,n E (h) 2 |Ω n | α 2,n ˆΩn α 2,n |∇u n | 2 dx ≤ C ˆΩ σ n (h)∇u n • ∇u n dx ≤ C ||f || H -1 (Ω) ||u n || H 1 0 (Ω) .
This combined with the boundedness of u n in H 1 0 (Ω) implies that 1 Ωn α 2,n I 3 + β 2,n E (h) ∇u n is bounded in L 1 (Ω) 3 . This bound and the uniform convergence to 0 of X n (see (4.11)) imply the convergence to 0 of the right term of (4.16), hence ˆΩn α 2,n I 3 + β 2,n E (h) ∇u n • λ ϕ dx -→ n→∞ 0.

We rewrite this condition as Therefore, putting λ = e 1 , e 2 in this limit and using condition (4.5), we obtain the linear system

∀ λ ⊥ e 3 ,
α 2 ξ 1 + β 2 h 2 ξ 3 -β 2 h 3 ξ 2 = 0 α 2 ξ 2 + β 2 h 3 ξ 1 -β 2 h 1 ξ 3 = 0 in M(Ω), which is equivalent to          ξ 1 = β 2 2 h 1 h 3 -α 2 β 2 h 2 α 2 2 + β 2 2 h 2 3 ξ 3 ξ 2 = β 2 2 h 2 h 3 + α 2 β 2 h 1 α 2 2 + β 2 2 h 2 3 ξ 3 in M(Ω). ( 4

.19)

Third step: Proof of ξ 3 = ∂u ∂x 3 .

We need the following result which is an extension of the estimate (3.13) of [START_REF] Briane | Semi-strong convergence of sequences satisfying a variational inequality[END_REF]. The statement of this lemma is more general than necessary for our purpose but is linked to Remark 4.1. Combining this estimate and the first condition of (4.5) with

||∇(u n ϕ)|| L 2 (Ω) 3 ≤ ||∇u n || L 2 (Ω) 3 ||ϕ|| ∞ + ||u n || L 2 (Ω) ||∇ϕ|| ∞ ≤ C, it follows that 1 Ωn |ω n | u n -u n ⇀ 0 in D ′ (Ω).
This convergence does not hold true when ε 2 n | ln r n | converges to some positive constant. Under this critical regime, non-local effects appear (see Remark 4.4).

Finally, as 1 Ωn does not depend on the x 3 variable, we have

1 Ωn |ω n | ∂u n ∂x 3 = ∂ ∂x 3 1 Ωn |ω n | u n = ∂ ∂x 3 1 Ωn |ω n | u n -u n + ∂u n ∂x 3 ⇀ ∂u ∂x 3 = ξ 3 in D ′ (Ω).
Fourth step: Derivation of the homogenized matrix.

We now study the limit of σ n (h)∇u n in order to obtain σ * (h). We have

σ

Proposition 2 . 2

 22 Let Ω be a bounded open subset of R 2 such that |∂Ω| = 0. Let ω n , n in N, be a sequence of open subsets of Ω with characteristic function χ n , satisfying θ n := |ω n | < 1, θ n converges to 0, and

Theorem 2 . 2

 22 Let Ω be a bounded open subset of R 2 such that |∂Ω| = 0. Let ω n , n ∈ N, be a sequence of open subsets of Ω and denote by χ n their characteristic function. We assume that θ n = |ω n | < 1 converges to 0 and χ n θ n H( --⇀ a ∈ L ∞ (Ω) weakly- * in M(Ω).

Figure 3 . 1 :

 31 Figure 3.1: The period of the cross-like thin structure

Figure 4 . 1 :

 41 Figure 4.1: The fibered structure in dimension 3

(4. 7 ) 4 . 1 1 2Remark 4 . 2

 741142 Remark Theorem 4.1 can be actually extended to fibers with a more general cross-section. More precisely, we can replace the disk r n D of radius r n by the homothetic r n Q of any connected open set Q included in the unit disk D, such that the present fiber ω n is replaced by the new fiberr n Q × -1 2 ,in the period cell of the microstructure. On the one hand, this change allows us to use the same test function v n (4.8) defined in the proof of Theorem 4.1, since v n remains equal to 1 in the new fibers due to the inclusion Q ⊂ D. On the other hand, Lemma 4.1 allows us to replace the disk D by the open set Q ⊂ D. We can also extend the result of Theorem 4.1 to an isotropic fibered microstructure composed by three similar periodic fibers lattices arranged in the three orthogonal directions e 1 , e 2 , e 3 , namely

Figure 4 . 2 :

 42 Figure 4.2: The period cell of the isotropic fibered structure in dimension 3

Remark 4 . 5 3 0

 453 If h = h 3 e 3 , the homogenized conductivity becomes σ * (h) = α 1 I 3 + α 2 e 3 ⊗ e 3 + β 1 h

Lemma 4 . 1 ≤ 3 .

 413 Let Q be a non-empty connected open subset of the unit disk D. Then, there exists a constant C > 0 such that any function U ∈ H 1 (Y ) satisfies the estimate1 |r n Q| ˆrnQ×(-1 2 , 1 2 ) U dy -ˆY U dy ≤ C | ln r n | ||∇U || L 2 (Y ) 3 . (4.20) Proof of Lemma 4.1. Let U ∈ H 1 (Y ). To prove Lemma 4.1, we compare the average value of U on r n Q and r n D. Denoting y = (y 1 , y 2 ), we have, for any y 3 ∈ -1 2 , 1 2 , rnQ U ( y, y 3 ) d y -rnD U ( y, y 3 ) d y = Q U (r n y, y 3 ) d y -D U (r n y, y 3 ) d y ≤ Q U (r n y, y 3 ) -D U (r n y, y 3 ) d y d y, and, since Q ⊂ D, rnQ U ( y, y 3 ) d y -rnD U ( y, y 3 ) d y ≤ |D| |Q| D U (r n y, y 3 ) -D U (r n y, y 3 ) d y d y y 3 ) d y, the last inequality being a consequence of the Poincaré-Wirtinger inequality. Hence, integrating the previous inequality with respect to y 3 ∈ -1 2 , 1 2 and applying the Cauchy-Schwarz inequality, we obtain that C||∇U || L 2 (Y ) This combined with the estimate (3.13) of [21], i.e. (4.20) for Q = D, and the fact that | ln r n | diverges to ∞ give the thesis. Let ϕ ∈ D(Ω). A rescaling of (4.20) with Q = D implies the inequality 1 |ω n | ˆΩn u n ϕ dx -ˆΩ u n ϕ dx ≤ Cε n | ln r n | ||∇(u n ϕ)|| L 2 (Ω) 3 .

  .37) Finally, by combining (2.31), (2.32), (2.34) and (2.37), we obtain, for any simply connected open subset ω of Ω such as ω ⊂⊂ Ω,

  Let Ω be a bounded open subset of R 2 with a Lipschitz boundary. Consider a highly oscillating sequence of matrix-valued functions σ n satisfying (3.1), (3.2), (3.3) and the constant matrix (σ n ) * defined by(3.4). We assume that

	3.1 A uniform convergence result
	Theorem 3.1
		)
	and
	w n := (w e 1 n , w e 2 n ) = (w 1 n , w 2 n ).	(3.8)

  Proposition 3.1 Let χ n be a sequence of characteristic functions of subsets of Y , α 1 , α 2 > 0, a positive sequence α 2,n , β 1 , β 2 ∈ R, and a sequence β 2,n such that lim

	n→∞	β 2,n α 2,n	=	β 2 α 2	.	(3.29)

n→∞ α 2,n = ∞, lim inf n→∞ |β 2,n -β 1 | > 0, and lim

  1 Ωn α 2,n I 3 + β 2,n E (h) ∇u n • λ -⇀ 0 weakly- * in M(Ω). | ˆΩn |∇u n | dx ≤ 1 α 2,n |ω n | |Ω n | |ω n | ˆΩn α 2,n |∇u n | 2 dxwhich leads us, by (4.5) and the asymptotic behavior|Ω n | ∼ Ωn |ω n | ∂u n ∂x i L 1 (Ω) ≤ C α 2,n |ω n | ˆΩ σ n (h)∇u n • ∇u n dx ≤ C f, u n H -1 (Ω),H 1 0 (Ω)which is bounded by the boundedness of u n in H 1 0 (Ω). This allows us to define, up to a subsequence, the following limits1 Ωn |ω n | ∂u n ∂x i ⇀ ξ i weakly- * in M(Ω), for i = 1, 2, 3.(4.18)Then, by (4.17) we have α 2,n I 3 + β 2,n E (h) 1 Ωn ∇u n • λ = α 2,n |ω n |I 3 + β 2,n |ω n |E (h) 1 Ωn |ω n | ∇u

					(4.17)
	Second step: Linear relations between weak- * limits of Thanks to the Cauchy-Schwarz inequality, we have	1 Ωn |ω n |	∂u n ∂x i	.
	1 Ωn |ω n |	∂u n ∂x i L 1 (Ω)	≤	1 |ω n

n→∞ |Ω| |ω n |, to 1 n • λ ⇀ 0 weakly- * in M(Ω).

  n (h)∇u n • e 1 = 1 Ω\Ωn α 1 ∂u n ∂x 1 -β 1 h 3 ∂u n ∂x 2 + β 1 h 2 ∂u n ∂x 3 +α 2,n |ω n | 1 Ωn |ω n | ∂u n ∂x 1 -β 2,n h 3 |ω n | 1 Ωn |ω n | ∂u n ∂x 2 + β 2,n h 2 |ω n | 1 Ωn |ω n |Hence, passing to the weak- * limit in M(Ω) this equality and using the linear system (4.[START_REF] Briane | An antisymmetric effective hall matrix[END_REF],σ n (h)∇u n • e 1 weakly- * converges in M(Ω) to + α 2 ξ 1 -β 2 h 3 ξ 2 + β 2 h 2 ξ 3 = α 1 I 3 + β 1 E (h) ∇u • e 1 + α 2 β 2 2 h 1 h 3 -α 2 β 2 h 2 α 2 2 + β 2 2 h 2 3 ξ 3 -β 2 h 3 β 2 2 h 2 h 3 + α 2 β 2 h 1 α 2 2 + β 2 2 h 2 3 ξ 3 + β 2 h 2 ξ 3 = α 1 I 3 + β 1 E (h) ∇u • e 1 + α 2 (β 2 2 h 1 h 3 -α 2 β 2 h 2 ) -β 2 h 3 (β 2 2 h 2 h 3 + α 2 β 2 h 1 ) + β 2 h 2 (α 2 (h)∇u n • e 1 -⇀ α 1 I 3 + β 1 E (h) ∇u • e 1 weakly- * in M(Ω). (4.22)The same calculus leads us toσ n (h)∇u n • e 2 -⇀ α 1 I 3 + β 1 E (h) ∇u • e 2 weakly- * in M(Ω).(4.23)We have, for the last direction e 3 ,σ n (h)∇u n • e 3 ⇀ α 1 ∂u ∂x 3 -β 1 h 2 ∂u ∂x 1 + β 1 h 1 ∂u ∂x 2 + α 2 ξ 3 + β 2 h 2 ξ 1 -β 2 h 1 ξ 2 weakly- * in M(Ω). + α 2 ξ 3 -β 2 h 2 ξ 1 + β 2 h 1 ξ 2 = α 1 I 3 + β 1 E (h) ∇u • e 3 + α 2 ξ 3 -β 2 h 2 β 2 2 h 1 h 3 -α 2 β 2 h 2

							(4.21)
							∂u n ∂x 3	.
	α 1	∂u ∂x 1	-β 1 h 3	∂u ∂x 2	+ β 1 h 2	∂u ∂x 3
							α 2 2 + β 2 2 h 2 3	2 + β 2 2 h 2 3 )	ξ 3 ,
							=0
	that is
	σ n Hence, again with the linear system (4.19),
	α 1	∂u ∂x 3	-β 1 h 2	∂u ∂x 1	+ β 1 h 1	∂u ∂x 2
							α 2 2 + β 2 2 h 2 3	ξ 3 + β 2 h 1	β 2 2 h 2 h 3 + α 2 β 2 h 1 α 2 2 + β 2 3 2 h 2	ξ 3 .
	Finally, by the previous equality, (4.22) and (4.23), we get that
				σ * (h) = α 1 I 3 +	α 3 2 + α 2 β 2 2 |h| 2 α 2 2 + β 2 2 h 2

3 e 3 ⊗ e 3 + β 1 E (h).

Acknowledgments:

The authors wish to thank the referees for suggestions which led to improvement.


Michel Haritopoulos
email: michel.haritopoulos@univ-orleans.fr

Cécile Capdessus
email: cecile.capdessus@univ-orleans.fr

Asoke K Nandi
email: a.nandi@liverpool.ac.uk

Extraction of Foetal Contribution to ECG Recordings Using Cyclostationarity-Based Source Separation Method

Keywords: Cyclostationarity, Semi-Blind Source Extraction, Foetal Electrocardiogram

In this paper we propose a cyclostationary approach to the problem of the foetal electrocardiogram (FECG) extraction from a set of cutaneous potential recordings of an expectant mother. We adopted a semi-blind source separation (BSS) method for which the only necessary prior knowledge is that of the fundamental cyclic frequency of the cyclostationary process to estimate. The estimated cyclostationary FECG source of interest is found to be free from any interferences with the mother's ECG (MECG) signal. Experimental results and perspectives for future research conclude this paper.

Introduction

Electrocardiograms (ECG) are very often used as diagnostic tools for heart monitoring. In the case of pregnant women, the recorded ECGs contain information concerning both mother's and foetus' condition. Early diagnosis in the case of the mother's ECGs (MECG) can be helpful for medical assessment of possible diseases, while in the foetus case (FECG), any prenatal information that can be provided may be of great importance to prevent complications. ECGs acquired with non-invasive techniques do not give direct access to FECGs, because those are hidden by the maternal ECGs of higher amplitude and contaminated by various sources of disturbances (biological or other).

It is shown in the literature ( [START_REF] Lathauwer | Foetal Electrocardiogram Exctraction by Blind Source Subspace Separation[END_REF], [START_REF] Zarzoso | Maternal and Foetal ECG Separation Using Blind Source Separation Methods[END_REF], [START_REF] Bacharakis | Foetal ECG Extraction Using Blind Source Separation Methods[END_REF], [START_REF] Bacharakis | Foetal ECG Separation from Surface Measurements[END_REF]), that this problem can be formulated in blind source separation (BSS) terms. It consists of separating or extracting one or more signals of interest from a set of observations which contain a linear or non-linear mixing of the unknown source signals that one wants to estimate.

In the case of ECG signals, m electrodes are placed at several key locations on the mother's body and the observations carried out by these electrodes are modelled in terms of BSS as a vector x assumed to be a linear mixture of the unknown source vector s :

x(t) = As(t) [START_REF] Lathauwer | Foetal Electrocardiogram Exctraction by Blind Source Subspace Separation[END_REF] where s(t) are the n unknown sources [s 1 (t), s 2 (t), ..., s n (t)] † ∈ R n×1 , x(t) outputs the m linearly mixed observations [x 1 (t), x 2 (t), ..., x m (t)] † ∈ R m×1 , A stands for the m × n mixing matrix and † denotes the transpose operator. Equation ( 1) is the well known noiseless linear instantaneous model of the BSS problem.

Various approaches have been used to solve the FECG extraction problem by using BSS techniques. One can nd in the literature second-order statistics (SOS) based BSS methods [START_REF] Jafari | Sequential Blind Source Separation Based Exclusively on Second-Order Statistics Developed for a Class of Periodic Signals[END_REF] or adaptive noise cancelling methods compared with higher-order statistics (HOS) based BSS techniques [START_REF] Zarzoso | Noninvasive Foetal Electrocardiogram Extraction: Blind Separation Versus Adaptive Noise Cancellation[END_REF]. Sparse representations [START_REF] Blumensath | Blind Separation of Maternal and Foetal ECG Recordings using Adaptive Sparse Representations[END_REF] as well as wavelet based ICA [START_REF] Azzerboni | A New Approach Based on Wavelet-ICA Algorithms for Foetal Electrocardiogram Extraction[END_REF] applied to this particular problem have also been reported.

Here we choose to apply an algorithm that allows to extract one specic cyclostationary source whose cyclic frequency is a priori known. A preliminary study of the data indicates that the observations exhibit cyclostationarity at the frequency of the foetal heartbeat rate, which is estimated and then used for the extraction.

Methods

Extraction Principle

The extraction method was previously presented in [START_REF] Capdessus | Source Extraction Algorithm Based on Cyclic Properties[END_REF]. It aims at extracting a cyclostationary source of known cyclic frequency α 0 from a set of observations. We thus look for a 1 × m extraction vector B such that:

z(t) = Bx(t) (2)
is an estimate of the cyclostationary source. Hypotheses are as follows:

The observations are instantaneous and additive mixtures of the sources.

There are at least as many sensors as sources.

The sources are supposed to be zero mean and uncorrelated to each other.

The source to be extracted is cyclostationary at a frequency α 0 that is a priori known, i.e. it can either be measured or computed.

The other sources can be either stationary or cyclostionary, provided that none of them is cyclostationary at the same frequency α 0 .

The extraction is performed by minimising over B a criterion C(B) based on second-order statistics of the observations. Let us denote by R z (t, 0) the autocorrelation function of the estimate z(t) at zero time-lag. Then:

C(B) = R z (0) R α0 z (0) (3)
where R z (0) and R α0 z (0) are the coecients of the Fourier series decomposition of R z (t, 0) at respective frequencies 0 and α 0 . Let us denote by R x (0) = ⟨x(t)x † (t)⟩ θ and R α0 x (0) = ⟨x(t)x † (t)e -2πjα0t ⟩ θ the estimates of the covariance matrix of the observations and their cyclic covariance matrix at frequency α 0 , respectively. The ⟨.⟩ θ operator stands for the temporal averaging over θ seconds. The criterion is estimated from the observations as follows:

x (t) x (t) x (t) x (t) x (t) x (t) x (t) x (t)
C(B) = B R x (0)B † B R α0 x (0)B † . ( 4
)
It has been shown in [START_REF] Capdessus | Source Extraction Algorithm Based on Cyclic Properties[END_REF] that minimising eq. ( 4) over B leads to the extraction of the corresponding cyclostationary component.

Methodology and Results

The proposed algorithm is very well adapted to the problem of FECG extraction as shown below. We apply it to a commonly used dataset in relation to FECG extraction research [START_REF]DaISy; Database for the Identication of Systems[END_REF].

Experimental Dataset

The dataset we used is illustrated in 1. These are eight-channel cutaneous potential recordings from eight skin electrodes placed at dierent positions of an expectant mother's body. We performed envelope spectrum analysis of the observations, i.e. the Fourier transform of the squared signals [START_REF] Randall | The Relationship between Spectral Correlation and Envelope Analysis for Cyclostationary Machine Signals[END_REF]. All eight channels give substantially the same envelope spectra; but as on the temporal ECG representations, it is at the rst channel that the foetal heartbeat rate can be distinguished clearly. Fig. 2 is a zoom of the rst mixture x 1 (t) vector's envelope spectrum; mother's heartbeat frequency (MHF) and its harmonic components k × M HF, k ∈ [START_REF] Lathauwer | Foetal Electrocardiogram Exctraction by Blind Source Subspace Separation[END_REF][START_REF] Zarzoso | Noninvasive Foetal Electrocardiogram Extraction: Blind Separation Versus Adaptive Noise Cancellation[END_REF] are dominating the whole spectrum. The frequency intervals [min M ECG, max M ECG] plotted with dotted and dash-dotted lines, respectively, correspond to the [2.538, 2.958] Hz frequency range values, within which mother's heartbeats vary. As expected, MHF lies in this interval.

Foetus' heartbeat frequency (FHF) frequency and its second harmonic also appear in this gure. But, the widening of the MHF harmonics at higher frequencies, makes it dicult to point out with certainty which spectral line corresponds to FHF harmonic. The presence of spectral lines in the envelope spectrum supports the thesis of second order cyclostationarity characterizing the FECG component, with a fundamental cyclic frequency α 0 equal to 4.49 Hz, i.e. the value of the FHF. Fig. 3. From the top: four FECG estimations provided by the proposed algorithm with input mixture vectors x1(t),...,x4(t), respectively.

Experiments and Results

We carried out experiments by applying our algorithm to dierent sets of observations. We describe, hereafter, the results obtained by using four dierent observations vectors. The rst mixture vector x 1 (t) is composed of observations [x 1 (t), x 2 (t), x 3 (t)] † , the second one x 2 (t) = [x 1 (t), x 2 (t), x 5 (t)] † , the third one x 3 (t) is [x 1 (t), x 3 (t), x 5 (t)] † and the last one is made of four observations vectors x 4 (t) = [x 1 (t), x 2 (t), x 3 (t), x 5 ] † (see Fig. 1). Note, that, we did not include into our mixtures vectors recorded data from channel number 4, to which FECG contribution is very low. Furthermore, baseline wander is observed in this channel as this is the case in many other ECG datasets. This introduces amplitude variations to all separated FECG contributions after application of a BSS algorithm [START_REF] Blumensath | Blind Separation of Maternal and Foetal ECG Recordings using Adaptive Sparse Representations[END_REF]. One can nd in the literature (e.g., [START_REF] Sameni | A Deation Procedure for Subspace Decomposition[END_REF]) preprocessing methods for baseline wander removal and bandpass ltering of ECG recordings. After running the new cyclostationary criterion based BSS algorithm [START_REF] Capdessus | Source Extraction Algorithm Based on Cyclic Properties[END_REF], with fundamental FECG cyclic frequency α 0 = 4.49 Hz as estimated from the rst channel's envelope spectrum, we obtain for each mixture vector x i (t), i ∈ [START_REF] Lathauwer | Foetal Electrocardiogram Exctraction by Blind Source Subspace Separation[END_REF][START_REF] Bacharakis | Foetal ECG Separation from Surface Measurements[END_REF] an estimate of the foetal electrocardiogram signal. Extraction results are shown in Fig. 3. The extracted FECG that exhibits both very little baseline wandering and the foetus heartbeats, is the last estimate, i.e. the one that used information from 4 channels for minimising criterion of eq. ( 4). Envelope spectra of extracted FECGs (Hz) Amplitude Fig. 4. From top to bottom: zoom to the envelope spectra of the extracted FECGs by using 4 dierent mixture vectors.

Fig. 4 shows the envelope spectra of the extracted FECG signals after application of the proposed estimation criterion algorithm to the mixture vectors x i (t), i ∈ [1, 4] (from top to bottom). In order to compare these results with the envelope spectra of Fig. 2, we plotted a zoom for each extracted FECG signature. It is worth pointing out that our algorithm works well for all dierent combinations of the mixture vector components, as well as for a dierent observations number m. Indeed, one can see in this gure that, the FHF fundamental frequency and its next three harmonics are not contaminated by any MHF spectral lines. Experiments have been carried out with all eight channels ECG recordings as input mixing vector to our extraction algorithm. A study of the FECG signal spectrum extracted after application of our algorithm to the whole ECG recordings set, conrms the existence of the rst MHF harmonic, while no MHF components can be found in the spectrum computed from the extracted FECG using x 4 (t). This could be justied by assuming that recordings from the mother's thoracic region bring very few new information concerning the foetus' heartbeats.

3.3

Robustness to Foetus's Cyclic Frequency

The only prior knowledge required for our estimation method to work, is that of the foetus's fundamental cyclic frequency value α 0 . As explained in section 3.2, we used a value of 4.49 Hz for α 0 as it is estimated from the envelope spectrum of the rst ECG channel recordings. Deviations from this value can be detected depending which channel's envelope spectrum the estimation is based on. Thus, we run some experiments with the previously described x 4 (t) mixing vector. At each run, we applied the proposed cyclostationary extraction method to x 4 (t) with a dierent α 0 value and we computed a separation quality criterion for the corresponding estimated FECG signal. This is a periodicity measure (PM ∈ [0, 100]) based on second-order statistical properties and introduced in [START_REF] Sameni | A Deation Procedure for Subspace Decomposition[END_REF]; it is computed at the mean mother's heartbeat rate without taking into account its variability:

P M = |E{x(t)x(t + τ µ )}| √ E{x(t) 2 }E{x(t + τ µ ) 2 } * 100%, (5)
where τ µ is the mean mother's heartbeat period. If the extracted FECG is clear of MHF components, then PM should be 0%.

Computed PMs for α 0 values over the frequency range of interest, i.e., from 4.4 Hz to 4.6 Hz by steps of 0.01 Hz, take values < 0.5%. PM values computed for every single ECG channel lie within [17, 29]%. This means that, even by introducing small variations on the estimated α 0 value, the extraction algorithm still provides estimate FECG signals with about three times better quality in therms of PM with respect to the best ECG channel. Therefore, for a given ECG channel, the estimation of the foetus fundamental cyclic frequency from its envelope spectrum seems to be robust for small deviations from the true value of α 0 .

Discussions and Conclusions

We have introduced in this paper a novel application of our cyclostationaritybased BSS algorithm to the problem of foetus heartbeat signals extraction from a pregnant's woman ECG recordings. We pointed out the main lines of the estimation principle and then successfully applied it to real ECG signals.

To use the proposed method, there is no need to prewhiten (by the principal component analysis, for example) the sensor signals; centering the observations (zero mean) and decorrelation assumption are sucient. We just have to know/estimate the fundamental cyclic frequency of the cyclostationary process (i.e. the FECG signal, in this study) to extract from the mixtures. It then estimates the source signal of interest. In the case presented in this paper, since the cyclostationary frequency chosen for the extraction was that of the FECG signal, the estimated source ts the heartbeats of the foetus. The indeterminacies inherent to BSS methods, i.e. the sign and the amplitude of the separated source signal, still remain in the proposed method. But post processing of the extracted FECG signal may be used to compute its contribution to each sensor.

Further experiments are under progress in order to be able to compare our algorithm's performance with that of similar work reported in the literature; its robustness to the foetus' cyclic frequency is also studied for dierent ECG channels and dierent number of mixtures. Taking into account the presence of baseline wander in some sensors, extracting the mothers's heartbeat signals and the characteristic waveforms associated with each heartbeat are of great interest in order to perform a robust monitoring of the foetus well-being.

Fig. 1 .

 1 Fig. 1. Eight skin electrodes recordings form a pregnant woman

  0

  The rst ve observations denoted [x 1 (t), ..., x 5 (t)] were recorded from the abdominal area, while the last three [x 6 (t), ..., x 8 (t)] correspond to electrodes placed at the mother's thoracic region. All signals were sampled at frequency f e = 500 Hz during 5 s, so that the observations vector x(t) is N = 2500 samples long. The foetal heartbeat component can be perceived together with MECG contribution and noise in the rst x 1-5 (t) observations subset, while in the second one x 6-8 (t), the FECG contributions are barely visible because of the much longer distance between the foetus and the electrodes located on the mother's chest.

		16	4 x 10		MHF	Envelope spectrum of mixture number 1		min MECG
												max MECG
		14
		12				2 x MHF
								3 x MHF
		10
								4 x MHF
	Amplitude	8								5 x MHF
		6										6 x MHF
		4			FHF			2 x FHF
		2
		0	0	2	4	6	8	10	12	14	16	18	20
								Frequency (Hz)

Fig. 2. Mother's (MHF) and foetus' (FHF) heartbeat frequencies and their harmonics. Theoretical evolution of the MHF within the k × [2.538, 2.958] Hz values range (k = 1, .., 6) drawn in dotted and dash-dotted lines (minimum and maximum value, respectively).