Marc Briane
email: mbriane@insa-rennes.fr

David Manceau
email: dmanceau@univ-rennes1.fr

Graeme W Milton
email: milton@math.utah.edu

Homogenization of the two-dimensional Hall effect

Keywords: Homogenization, Hall effect, dimension two, two-phase composite

published or not. The documents may come

Introduction

Consider a conducting material with symmetric resistivity ρ. In electrodynamics it is well known (see e.g. [START_REF] Landau | Électrodynamique des Milieux Continus[END_REF]) that a magnetic field h induces a non-symmetric conductivity ρ(h) which corresponds to the Hall effect. In two dimensions and under the low field limit, h → 0, the modified resistivity reads as ρ(h) = ρ + rh J + o(h), (0.1)

where r is the Hall coefficient and J is the 90 • rotation matrix. Now, consider a highly heterogeneous material with resistivity ρ ε , where ε is a small parameter representing the scale of the microstructure.

According to the first-order expansion (0.1), a low magnetic field h induces a perturbed resistivity ρ ε (h) satisfying

ρ ε (h) = ρ ε + r ε h J + o(h), (0.2)
with a heterogeneous Hall coefficient r ε . The problem is to compute the effective Hall coefficient r * obtained from r ε in the homogenization process as ε → 0. Bergman [START_REF] Bergman | Self-duality and the low field Hall effect in 2D and 3D metal-insulator composite[END_REF] obtained for a periodic composite a formula for the effective Hall coefficient as an average-value only involving the local Hall coefficient and some local current fields in the absence of a magnetic field. His method is based on a small perturbation argument.

In this paper, we extend the Bergman approach in the theoretical framework of H-convergence due to Murat and Tartar [START_REF] Murat | H-convergence[END_REF]. To this end, we consider the general setting of a sequence of equicoercive and equi-bounded matrix-valued functions A ε (h) (not necessarily symmetric) in a bounded open set Ω of R N , N ≥ 1, and which depends on a vector h ∈ R n , n ≥ 1. We assume that A ε (h) satisfies the uniform Lipschitz condition (1.12) with respect to h. According to the H-convergence theory the sequence A ε (h) converges, up to a subsequence, in a suitable sense (see Definition 1.1 and the compactness Theorem 1.2) to some homogenized or effective matrix-valued A * (h). Then, if A ε (h) admits a first-order expansion of type (0.2), so does the homogenized matrix A * (h), hence

A ε (h) = A ε (0) + A ε 1 • h + o(h) H - ε→0 A * (h) = A * (0) + A * 1 • h + o(h). (0.3)
Then, we prove (see Theorem 1.7) that the effective first-order term A * 1 •h is deduced from a weak limit only involving the first-order term A ε 1 • h combined with the correctors (see Definition 1.3) associated with the unperturbed matrix-valued functions A ε (0) and A ε (0) T (and we do not necessarily assume that A ε (0) is symmetric).

We apply this homogenization process to the two-dimensional Hall effect with the conductivity σ ε (h) := ρ ε (h) -1 satisfying the uniform Lipschitz condition (2.4) with respect to h and the firstorder expansion (0.2). Therefore, the conductivity σ ε (h) H-converges to the homogenized conductivity σ * (h) so that the effective resistivity defined by ρ * (h) := σ * (h) -1 satisfies the expansion

ρ * (h) = ρ * + r * h J + o(h).
(0.4)

We then obtain the effective Hall coefficient r * in (0.4) by the the following process (see Theorem 2.3): the product r * det (σ * (0)) is the limit in the distributions sense of the local Hall coefficient r ε times the determinant of the unperturbed current field, i.e. the product of the conductivity σ ε (0) by the corrector associated with σ ε (0) in the absence of a magnetic field. This limit process allows us to prove the following positivity property (see Theorem 2.4): if the original Hall coefficient r ε is bounded (from below or above) by a continuous function independent of ε, so is the effective Hall coefficient r * . We illustrate this homogenization approach of the two-dimensional Hall effect with two examples. The first one is based on a explicit formula (see Theorem 3.1) obtained by the third author [START_REF] Milton | Classical Hall effect in two-dimensional composites: A characterization of the set of realizable effective conductivity tensors[END_REF] for an isotropic composite with two isotropic phases, which immediately gives the effective Hall coefficient and clearly shows the positivity property. The result of the second example seems new although it is also based on the same duality transformations due to Dykhne [START_REF] Dykhne | Conductivity of a two-dimensional two-phase system[END_REF]. It consists of a periodic two-phase material the phases of which are not necessarily isotropic but interchangeable from the point of view of the homogenization process. For this geometry we obtain an explicit formula for the determinant and for the antisymmetric part of the homogenized matrix. From this we deduce (see Corollary 3.9) an explicit formula for the effective Hall coefficient when the interchangeable phases have an unperturbed conductivity matrix σ ε (0) in proportion to one another. As a consequence of the explicit formulas in the former two-phase examples, we also derive (see Corollary 3.4 and Corollary 3.9) the limit value of the determinant of the corrector associated with σ ε (0) in each of the two phases.

The paper is organized as follows. In Section 1, we recall some results about H-convergence and the correctors, and we state a result of H-convergence with a parameter (Theorem 1.7). In Section 2 we show the homogenization process involving the Hall coefficient in a general two-dimensional microstructure, and the positivity property satisfied by the effective Hall coefficient. Section 3 is devoted to explicit formulas for the effective Hall coefficient for particular two-phase composites.

All along this article, we will use the following basic notations:

Notations • N ∈ N, N ≥ 1. • For x, y ∈ R N , x • y := N i=1 x i y i where x := (x 1 , . . . , x N ), y := (y 1 , . . . , y N ). • R M ×N is the set of the (M × N ) real matrices. • For A ∈ R M ×N , A = [A ij ], we denote by A T ∈ R N ×M its transpose defined by [A T ] := [A ji ].
• I 2 is the unit matrix of R 2×2 and J is the rotation matrix of 90 • .

• M + ⊂ R 2×2 is the set of (2 × 2) matrices with a positive quadratic form, and M s ⊂ R 2×2 is the set of (2 × 2) symmetric matrices (i.e. A T = A, ∀ A ∈ M s ). Then, any matrix A ∈ M + is uniquely decomposed into

A = A s + α(A) J, where A s ∈ M s and α(A) ∈ R. (0.5) • M s + := M + ∩ M s .
• D(Ω) denotes the space of functions of class C ∞ on Ω with compact support in Ω, and D (Ω) denotes the space of distributions on Ω.

• We denote by lim

D (Ω)
the weak limit in the distributions sense.

• M(Ω) denotes the space of Radon measures on Ω and we denote by lim

M(Ω)
the weak- * limit in the Radon measures sense.

• For u : R N → R, ∇u := ∂u ∂x i 1≤i≤N .

• For U :

R N → R N , U := (u 1 , • • • , u N ), DU := ∂u j ∂x i 1≤i,j≤N and div (U ) := N i=1 ∂u i ∂x i . (0.6) • For M : R N → R N ×N , Div(M ) := N i=1 ∂M ij ∂x i 1≤j≤N and Curl(M ) := ∂M ij ∂x k - ∂M kj ∂x i 1≤i,j,k≤N . (0.7)
1 A few results from homogenization theory

Review of H-convergence

We recall the definition and some properties of H-convergence theory for second-order elliptic scalar equations introduced by Murat and Tartar [START_REF] Murat | H-convergence[END_REF] in the general case and by De Giorgi and Spagnolo [START_REF] Spagnolo | Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche[END_REF] (under the name of G-convergence) in the symmetric case. Furthermore, we also give the definition of the correctors in homogenization.

Definition 1.1. (Murat-Tartar [START_REF] Murat | H-convergence[END_REF])

i) Let Ω be a bounded open set of R N . We define the space M(α, β; Ω) as the set of measurable matrix-valued functions A defined on Ω such that

∀ ξ ∈ R N , A(x)ξ • ξ ≥ α|ξ| 2 and A -1 (x)ξ • ξ ≥ β -1 |ξ| 2 , a.e. x ∈ Ω. (1.1) ii) A sequence A ε of M(α, β; Ω) is said to H-converge to A * if A * ∈ M(α, β; Ω), f ∈ H -1 (Ω) and the solution u ε of -div (A ε ∇u ε ) = f in Ω, u ε ∈ H 1 0 (Ω), (1.2)
satisfies the weak convergences

u ε u 0 in H 1 (Ω)-weak, A ε ∇u ε A * ∇u 0 in L 2 (Ω)-weak, (1.3)
where u 0 is the solution of -div (A * ∇u 0 ) = f in Ω,

u 0 ∈ H 1 0 (Ω). (1.4) The H-convergence of A ε to A * is denoted by A ε H A * .
An important result of H-convergence is the following "compactness theorem" due to Murat-Tartar [START_REF] Murat | H-convergence[END_REF]: Theorem 1.2. (Murat-Tartar [START_REF] Murat | H-convergence[END_REF]) If A ε is a sequence of M(α, β; Ω), then there exists a subsequence, still denoted by ε, and

A * ∈ M (α, β; Ω) such that A ε H A * .
Finally, we recall the definition of correctors in homogenization and a result about the convergence of the correctors (see [START_REF] Murat | H-convergence[END_REF]). Definition 1.3. Let A ε be a sequence of M(α, β; Ω). Any matrix-valued function P ε in L 2 (Ω) N ×N satisfying the properties

       P ε I N in L 2 (Ω) N ×N -weak, Curl(P ε ) is compact in H -1 (Ω) N ×N ×N , Div(A ε P ε ) is compact in H -1 (Ω) N , (1.5)
is called a corrector associated with A ε .

Example 1.4. Let A ε be a sequence of M(α, β; Ω) with H-limit A * and let

U ε ∈ H 1 (Ω) N be the solution of Div (A ε DU ε ) = Div (A * ) in Ω, U ε = I N on ∂Ω.
(1.6)

Then, the matrix-valued function defined by P ε := DU ε is a corrector associated with A ε .

We have the following result which is a consequence of the div-curl lemma of Murat-Tartar [START_REF] Murat | Compacité par compensation[END_REF], [START_REF] Murat | H-convergence[END_REF].

Proposition 1.5.

i) Assume that A ε H A * . Then, any corrector P ε associated with A ε satisfies the weak convergences

A ε P ε A * in L 2 (Ω) N ×N -weak, (P ε ) T A ε P ε A * in D (Ω) N ×N .
(1.7)

ii) Conversely, let A ε ∈ M(α, β; Ω) and let P ε be a sequence such that

             P ε I N in L 2 (Ω) N ×N -weak, Curl(P ε ) is compact in H -1 (Ω) N ×N ×N , Div(A ε P ε ) is compact in H -1 (Ω) N , A ε P ε A * in L 2 (Ω) N ×N -weak.
(1.8)

Then, A ε H A * . iii) If P ε and Q ε are two correctors associated with A ε , then P ε -Q ε strongly converges to 0 in L 2 loc (Ω) N ×N .

H-convergence with a parameter

In the sequel, we use the following notation:

Notation 1.6. Let n ∈ N, n ≥ 1, and let (E, || • ||) be a normed space. Let f 0 ∈ E and f, f 1 : R n → E. We set f (h) = f 0 + f 1 (h) + o E (h), (1.9)
whenever there exists δ : [0, +∞) → [0, +∞) such that, for any h ∈ R n with small enough norm, we have

||f (h) -f 0 -f 1 (h)|| ≤ |h| δ(|h|) with lim t→0 δ(t) = 0. (1.10) If E := R n , we will simply denote o E (h) = o(h). Moreover, when f = f ε , f 0 = f ε 0 , f 1 = f ε 1 depend on an additional small parameter ε > 0, the expansion f ε (h) = f ε 0 + f ε 1 (h) + o E (h), (1.11)
has the same sense as (1.9), the remainder o E (h) then being uniform with respect to ε.

Theorem 1.7. Let n ∈ N, n ≥ 1, let B κ be the open ball of R n of radius κ and let α, β > 0. Let A ε (h), for h ∈ B κ , be a sequence in M(α, β; Ω) which satisfies the uniform Lipschitz condition ∃ C > 0, ∀ h, k ∈ B κ , ||A ε (h) -A ε (k)|| L ∞ (Ω) N ×N ≤ C |h -k|, (1.12)
and the first-order expansion at h = 0

A ε (h) = A ε + A ε 1 • h + o L ∞ (Ω) N ×N (h). ( 1

.13)

where

A ε = A ε (0) and A ε 1 is a uniformly bounded sequence in L ∞ (Ω) n×N ×N . i) Then, there exists a subsequence of ε, still denoted by ε, such that A ε (h) H-converges to A * (h) in M(α, β; Ω) for any h ∈ B κ , and
A * (h) = A * + A * 1 • h + o L 2 (Ω) N ×N (h), (1.14)
where

A * = A * (0) and A * 1 ∈ L 2 (Ω) n×N ×N .
ii) Moreover, if P ε and Q ε are correctors associated respectively with A ε and (A ε ) T we get, for any

h ∈ B κ , (Q ε ) T (A ε 1 • h)P ε A * 1 • h in D (Ω) N ×N . (1.15)
Remark 1.8. Colombini and Spagnolo proved in [START_REF] Colombini | Sur la convergence de solutions d'équations paraboliques[END_REF] that the homogenized matrix A * (h) is of class C k with respect to the parameter h when all the derivatives D j h A ε (h), j = 0, • • • , k, satisfy the uniform Lipschitz condition in h. In Theorem 1.7 we show that the Lipschitz control (1.12) of A ε (h) in h allows us to obtain the differentiability (1.14) of A * (h) at zero. The price to pay is that the remainder in

(1.14) is only controlled in L 2 (Ω) N ×N and not in L ∞ (Ω) N ×N .
The proof of Theorem 1.7 which is based on classical H-convergence arguments is done in the appendix for the reader's convenience.

About duality transformations

We recall a few results about two-dimensional duality transformation in the framework of H-convergence (see e.g. [START_REF] Milton | The Theory of Composites[END_REF] Chapters 3, 4 for a general presentation and complete references). Notation 1.9. For any a, b, c ∈ R, we define for A ∈ M + , f (A) := (aA + bJ)(-aI 2 + cJA) -1 .

(1.16)

For fixed a, b, c, we call f the duality function associated with (a, b, c).

Lemma 1.10. For any

A ∈ M + , f (A) ∈ M + if and only if bc > a 2 . Moreover, f is an involution on M + .
The following result is due to Dykhne [START_REF] Dykhne | Conductivity of a two-dimensional two-phase system[END_REF] who extended the pioneering work of Keller [START_REF] Keller | A theorem on the conductivity of a composite medium[END_REF] on duality transformations. Here, the statement is written in terms of H-convergence: Theorem 1.11. (Dykhne [START_REF] Dykhne | Conductivity of a two-dimensional two-phase system[END_REF]) Let a, b, c ∈ R be such that bc > a 2 and let f be the duality function associated with (a, b, c).

If A ε ∈ M(α, β; Ω) H-converges to A * , then f (A ε ) H-converges to f (A * ).
Remark 1.12. The case a = 0, b = c = 1 corresponds to the following homogenization formula due to Mendelson [START_REF] Mendelson | A theorem on the effective conductivity of a two-dimensional heterogeneous medium[END_REF]:

A ε H A * =⇒ (A ε ) T det(A ε ) H (A * ) T det(A * ) . (1.17)
2 Homogenization of the Hall effect in dimension 2

Definition of the Hall coefficient

In dimension N , consider a conducting material with conductivity σ. Under the effect of a constant low magnetic field h, the resulting conductivity σ(h) depends on h and the corresponding resistivity ρ(h) := σ(h) -1 satisfies the first-order expansion

ρ(h) = ρ + ρ 1 • h + o(h), (2.1)
where ρ := σ -1 . Moreover, physical considerations (see e.g. [START_REF] Landau | Électrodynamique des Milieux Continus[END_REF]) imply that σ(h) T = σ(-h), or equivalently, ρ(h) T = ρ(-h), hence ρ is a symmetric matrix-valued function of x and ρ 1 • h is an antisymmetric matrix-valued function of x.

In dimension N = 2, the magnetic field h then reduces to a scalar and the first-order expansion of ρ(h) thus reads as

ρ(h) = ρ + rh J + o(h), where J := 0 -1 1 0 , (2.2)
and ρ = ρ(0) is symmetric and r is a scalar function.

In (2.1), (2.2) and in the text which follows, ρ(h), ρ, σ(h), σ, . . . are matrix-valued functions and r, s, . . . are scalar functions implicitly depending on spatial coordinates x.

Definition 2.1. The function r in (2.2) is called the Hall coefficient in presence of the magnetic field h. Now consider a heterogeneous material with conductivity σ ε . Under a low magnetic field h in (-κ, κ), κ > 0 small enough, the resulting conductivity σ ε (h) and resistivity ρ ε (h) satisfy the first-order expansions

σ ε (h) = σ ε + s ε h J + o L ∞ (Ω) 2×2 (h) ρ ε (h) = ρ ε + r ε h J + o L ∞ (Ω) 2×2 (h) where r ε , s ε ∈ L ∞ (Ω). (2.3)
We also assume that there exist α, β > 0 such that σ ε (h) ∈ M(α, β; Ω), and that σ ε (h) satisfies the uniform Lipschitz condition

∃ C > 0, ∀ h, k ∈ (-κ, κ), ||σ ε (h) -σ ε (k)|| L ∞ (Ω) 2×2 ≤ C |h -k|. (2.4)
Note that, since the remainders of (2.3) are uniform with respect to ε, estimate (2.4) implies that s ε and r ε are bounded sequences in L ∞ (Ω).

There is a link between the Hall coefficient r ε and the coefficient s ε for conductivity, given by the following result: Proposition 2.2. One has

s ε = -det (σ ε ) r ε .
(2.5)

Proof. Since ρ ε (h)σ ε (h) = I 2 and ρ ε σ ε = I 2 , we deduce from (2.3) that s ε (σ ε ) -1 J + r ε Jσ ε = 0. (2.6)
Taking into account the symmetry of σ ε , this leads us to

s ε I 2 = -r ε Jσ ε J -1 σ ε = -det (σ ε ) r ε I 2 , (2.7)
which gives equality (2.5).

Homogenization of the Hall effect

We have the following homogenization result:

Theorem 2.3. Let Ω be a bounded open set of R 2 . Let σ ε (h), for h ∈ (-κ, κ), be a sequence in M(α, β; Ω) satisfying (2.
3) and (2.4) with s ε , r ε two bounded sequences in L ∞ (Ω). Then, there exists a subsequence of ε, still denoted by ε, such that σ ε (h) H-converges to σ * (h) for any h ∈ (-κ, κ). The homogenized conductivity σ * (h) and the effective resistivity defined by ρ * (h

) := σ * (h) -1 , satisfy the expansions σ * (h) = σ * + s * h J + o L 2 (Ω) 2×2 (h) ρ * (h) = ρ * + r * h J + o L 2 (Ω) 2×2 (h), with s * = -det (σ * ) r * , (2.8)
where σ * is the H-limit of σ ε and ρ * := (σ * ) -1 . Moreover, s * and the effective Hall coefficient r * belong to L ∞ (Ω) and are given by

s * = lim D (Ω)
s ε det (P ε ) and det (σ * ) r * = lim

D (Ω) r ε det (σ ε P ε ) , (2.9)
for any corrector P ε associated with the matrix σ ε .

Proof. On the one hand, by Theorem 1.7 ii) σ ε (h) H-converges to σ * (h), up to a subsequence, for any h ∈ (-κ, κ), and

σ * (h) = ρ * + hσ * 1 + o L 2×2 (Ω)(h) , with σ * 1 = lim D (Ω) 2×2 r ε (P ε ) T JP ε , (2.10)
where P ε is a corrector associated with σ ε . Since by assumption σ ε (h) T = σ ε (-h) and by a classical property of H-convergence σ ε (h) T H-converges to σ * (h) T , we get σ * (h) T = σ * (-h). Hence, the matrix-valued function σ * 1 in (2.10) is antisymmetric. Therefore, there exists s * ∈ L 2 (Ω) such that σ * 1 = s * J. This combined with (2.10) yields the first-order expansion

σ * (h) = σ * + s * h J + o L 2×2 (Ω) (h), (2.11)
where s * ∈ L 2 (Ω) is given by

s * I 2 = lim D (Ω) 2×2 s ε J -1 (P ε ) T JP ε = lim D (Ω) 2×2 s ε det (P ε ) I 2 , (2.12)
which implies the first equality of (2.9). On the other hand, by the uniform Lipschitz condition (2.4) combined with the estimate of the difference of two H-limits (see e.g. [START_REF] Boccardo | Homogénéisation de problèmes quasi-linéaires[END_REF]) we have

σ * (h) -σ * L ∞ (Ω) 2×2 ≤ c |h| (2.13)
By the second part of Theorem 2.4 above and the boundedness of s ε in L ∞ (Ω), the function s * belongs to L ∞ (Ω). This combined with expansion (2.11) and estimate (2.13) implies that the effective resistivity ρ * (h) := σ * (h) -1 satisfies the second expansion of (2.8). Similarly to (2.5) we deduce from the expansions of (2.8) the equality s * = -det (σ * ) r * , which concludes the proof of (2.8). Finally, by the first equality of (2.9) and (2.5) we obtain det (σ * ) r * = -s * = -lim

D (Ω) 2×2 s ε det (P ε ) = lim D (Ω) 2×2 r ε det (σ ε P ε ) , (2.14)
which yields the second equality of (2.9).

Positivity property of the Hall effect

We have the following result:

Theorem 2.4. Under the assumptions of Theorem 2.3, let r 1 , r 2 be two continuous functions in Ω.

Then, if the effective Hall coefficient r ε satisfies the inequalities r 1 ≤ r ε ≤ r 2 a.e in Ω, so does the effective Hall coefficient r * .

Similarly and independently, let s 1 , s 2 be two continuous functions in Ω. Then, if the coefficient s ε satisfies the inequalities s 1 ≤ s ε ≤ s 2 a.e in Ω, so does the effective coefficient s * .

Remark 2.5. Let r be a continuous function in Ω. The particular case r ε = r a.e. in Ω implies that the effective Hall coefficient also satisfies r * = r a.e in Ω.

Proof. The proof of Theorem 2.4 is based on the result due to Raitums [START_REF] Raitums | On the local representation of G-closure[END_REF] (see also Theorem 1.3.23 of [START_REF] Allaire | Shape Optimization by the Homogenization Method[END_REF] page 60), that any H-limit is the pointwise limit of a sequence of periodic homogenized matrices, combined with the positivity of the determinant of the periodic correctors due to Alessandrini and Nesi [START_REF] Alessandrini | Univalent σ-harmonic mappings[END_REF] (see also [START_REF] Bauman | Univalent solutions of an elliptic system of partial differential equations arising in homogenization[END_REF]).

Taking into account the continuity of the functions r 1 , r 2 and using a locality argument we can assume that r 1 , r 2 are two constants in the sequel. Following the approach of [START_REF] Allaire | Shape Optimization by the Homogenization Method[END_REF], consider for fixed ε, t, h > 0 and x ∈ Ω, the periodic homogenized matrix σ * ε,t,x (h) defined by

σ * ε,t,x (h) := Y σ ε (h)(x + ty) DW ε,t,x (h, y) dy (2.15)
where Y := (0, 1) 2 , σ ε (h)(x + t •) is extended by Y -periodicity in R 2 , and W ε,t,x (h, •) is the unique vector-valued function in

H 1 loc (R 2 ) 2 solution of the cell problem div σ ε (h)(x + ty) DW ε,t,x (h, y) = 0 in D (R 2 ) y -→ W ε,t,x (h, y) -y is Y -periodic with zero Y -average. (2.16)
Consider, for fixed ε, t, x, the oscillating sequence ρ ε (h) x + t y δ as δ tends to zero. For this resistivity the second expansion of (2.3) reads as

ρ ε (h)(x + t•) = ρ ε (x + t•) + r ε (x + t•) h J + o L ∞ (Ω) 2×2 (h), (2.17)
where r ε (x + t•) is Y -periodic. Then, by (2.8) the expansion of the effective resistivity is given by

ρ * ε,t,x (h) = ρ * ε,t,x (0) + r * ε,t,x h J + o(h), (2.18)
where the effective resistivity ρ * ε,t,x (h) is the inverse of the constant homogenized matrix σ * ε,t,x (h) defined by (2.15). Moreover, by (2.16) the sequence of gradients DW ε,t,x 0, y δ is a corrector associated with the sequence σ ε x + t y δ in the sense of Definition 1.3. Therefore, by the second limit of (2.9) where the scale δ replaces ε, the product of the effective Hall coefficient r * ε,t,x by det σ * ε,t,x (h) is the limit in the distributions sense of the sequence r ε (x + t y δ ) det σ ε (0)(x + t y δ ) DW ε,t,x (0, y δ ) as δ tends to zero. Hence, again by periodicity we get Furthermore, thanks to the positivity result of [START_REF] Alessandrini | Univalent σ-harmonic mappings[END_REF] we have det DW ε,t,x (0, y) > 0 a.e. y ∈ Y . Then, from (2.19) and (2.20) we deduce that r 1 ≤ r * ε,t,x ≤ r 2 . Therefore, considering the scalar product of the expansion (2.18) with the matrix J, we obtain Then, passing to the double limit ε → 0, t → 0 in (2.21) it follows

r * ε,t,x det σ * ε,t,x (0) = Y r ε (x + ty) det σ ε (0)(x +
2r 1 h ≤ ρ * ε,t,x (h) : J -ρ * ε,t,x (0) : J + o(h) ≤ 2r 2 h. ( 2
2r 1 h n ≤ ρ * (h n )(x) : J -ρ * (x) : J + o(h n ) ≤ 2r 2 h n , ∀ n ∈ N and a.e. x ∈ Ω. (2.24)
On the other hand, consider a Lebesgue point x 0 ∈ Ω of the function r * and let B(x 0 , δ) be the ball of center x 0 and of radius δ > 0. The limit expansion (2.8) satisfied by ρ * (h) yields

- B(x 0 ,δ) ρ * (h n )(x) dx : J = - B(x 0 ,δ) ρ * (x) dx : J + 2 - B(x 0 ,δ) r * (x) dx h n + o δ (h n ), (2.25)
where, by the Cauchy-Schwarz inequality in

L 2 (B(x 0 , δ)), |o δ (h n )| ≤ 1 √ πδ 2 o(h n ).
The former estimate combined with (2.24) implies that for any n ∈ N,

r 1 ≤ - B(x 0 ,δ) r * (x) dx + o δ (h n ) h n ≤ r 2 .
(2.26) Therefore, passing successively to the limits h n → 0 and δ → 0 in (2.26), we get the desired inequalities

r 1 ≤ r * (x 0 ) ≤ r 2 .
The proof of the inequalities for the coefficient s * is quite similar, replacing in the previous proof the current field σ ε (0)(x + t•) DW ε,t,x (0, •) with the electric field DW ε,t,x (0, •).

Computation of the effective Hall coefficient and applications

We will consider particular cases of two-phase composites where, under some assumptions, explicit formulas of the Hall coefficient can be derived without the use of formula (2.9). These results combined with formula (2.9) then allow us to obtain the weak limit of the corrector determinant associated with the resistivity matrix in each of the two phases.

First, we recall the formula for the effective Hall coefficient for isotropic two-phase composites, obtained by the third author in [START_REF] Milton | Classical Hall effect in two-dimensional composites: A characterization of the set of realizable effective conductivity tensors[END_REF]. Then, we prove a new (up to our knowledge) formula for anisotropic interchangeable two-phase composites, like those depicted in figures 1 and 2. The two results are based on the duality transformations (1.16).

The isotropic two-phase case

Let ρ 1 , ρ 2 , r 1 , r 2 be four continuous even functions on R, ρ 1 , ρ 2 being positive. We consider a two-phase material with resistivity

ρ ε (h) := ρ ε (h) I 2 + r ε (h)h J, where ρ ε (h) := χ ε ρ 1 (h) + (1 -χ ε ) ρ 2 (h) r ε (h) := χ ε r 1 (h) + (1 -χ ε ) r 2 (h). (3.1)
We assume that the symmetric part σ ε (h) s of the conductivity σ ε (h) := ρ ε (h) -1 , H-converges to the isotropic matrix σ * (h) I 2 , where σ * (h) is a positive function in L ∞ (Ω), which is continuous and even with respect to h. Then, the third author proved the following homogenization result:

Theorem 3.1. (Milton [12]) Up to a subsequence,

σ ε (h) H-converges to σ * (h) = ρ * (h) -1
, where the effective resistivity satisfies ρ * (h) = ρ * (h) I 2 + r * (h)h J, and the effective Hall coefficient r * (h) is given by

r 2 (h) -r * (h) r 2 (h) -r 1 (h) = ρ 2 (h) 2 -ρ * (h) 2 + r 2 (h) -r * (h) 2 h 2 ρ 2 (h) 2 -ρ 1 (h) 2 + r 2 (h) -r 1 (h) 2 h 2 . ( 3.2)
In the low-field limit h → 0, formula (3.2) reduces to the Shklovskii's formula [START_REF] Shklovskii | Critical behavior of the Hall coefficient near the percolation threshold[END_REF]

r 2 (0) -r * (0) r 2 (0) -r 1 (0) = ρ 2 (0) 2 -ρ * (0) 2 ρ 2 (0) 2 -ρ 1 (0) 2 , (3.3)
and r * = r * (0) in the expansion (2.8).

Remark 3.2. In the isotropic case of Theorem 3.1 the conductivity σ ε (h) H-converges, up to a subsequence, to σ * (h) with σ * (h) s = σ * (h) I 2 . Then, thanks to the isotropy of the symmetric parts σ ε (h) s , σ * (h) s , and the duality transformation (1.17) we have

ρ ε (h) = σ ε (h) -1 = σ ε (h) T det (σ ε (h)) H σ * (h) T det (σ * (h)) = σ * (h) -1 = ρ * (h). (3.4)
Therefore, the resistivity ρ ε (h) H-converges to the effective resistivity ρ * (h). Moreover, a relation like (3.2) also holds for the homogenized conductivity matrix σ * (h).

Remark 3.3. In the Section 4.3 of [START_REF] Milton | The Theory of Composites[END_REF] page 65, the third author also gives an explicit formula for the skew part of the effective matrix for ordinary checkerboards (or isotropic interchangeable two-phase composites). This leads us easily to an explicit formula for the effective Hall coefficient r * (h). We will extend this formula to anisotropic interchangeable two-phase composites in the next section.

By the classical bounds on the effective matrix ρ * (0) I 2 we have min (ρ 1 (0), ρ 2 (0)) ≤ ρ * (0) ≤ max (ρ 1 (0), ρ 2 (0)) a.e. in Ω, (3.5) which implies that the right hand side of (3.

ε := χ ε ρ 1 + (1 -χ ε ) ρ 2 I 2 . (3.8)
Assume that the conductivity σ ε := (ρ ε ) -1 H-converges to the isotropic matrix (ρ * ) -1 I 2 . Then, any corrector P ε associated with σ ε satisfies the formula lim

D (Ω) χ ε det (P ε ) = ρ -2 * -ρ -2 2 ρ -2 1 -ρ -2 2 . (3.9)
Proof. Take r 1 (0) := ρ 2 1 and r 2 (0) := 0 in Theorem 3.1, which yields the equality r ε (0) det (σ ε P ε ) = χ ε det (P ε ) .

(3.10)

Then, the second formula of (2.9) and formula (3.3) imply the desired result.

The anisotropic interchangeable two-phase case

Definition 3.5. Consider a two-phase material with phases A and B, the conductivity matrix A ε of which is given by

A ε := χ ε A + (1 -χ ε )B. (3.11)
Also consider the two-phase material obtained by exchanging the two phases A and B, the conductivity matrix of which is thus

B ε := χ ε B + (1 -χ ε )A. (3.12)
The material is said to be interchangeable if B ε and A ε have the same H-limit.

Example 3.6.

1. A checkerboard is a periodic microstructure whose period cell is a parallelogram shared in four equal 1/2-homothetic parallelograms (see Figure 1). Consider a checkerboard with clockwise phases A and B (A, B ∈ M + ). Then, the checkerboard of phases B and A must have the same effective matrix. Thus, the two-phase periodic checkerboard represented in Fig. 1 is a periodic interchangeable material.

2. The periodic material represented in Fig. 2 by two of its period cells, is also an interchangeable two-phase material but not of checkerboard type.

We have the following result for interchangeable two-phase composites: Theorem 3.7. Consider an interchangeable two-phase material with phases A and λ A+µ J, λ, µ ∈ R.

Assume that λ > 0 and

λ det(A) + µ µ + 2λ α(A) λ + 1 > µ + 2λ α(A) λ + 1 2
, where A -A T = 2α(A) J.

(3.13)

Then, the matrix-valued function A ε associated with this two-phase material H-converges to the constant matrix A * such that

det(A * ) = λ det(A) + µ µ + 2λ α(A) λ + 1 and α(A * ) = µ + 2λ α(A) λ + 1 . (3.14)
Remark 3.8. The determinant and the antisymmetric part of A * are explicit but not the whole matrix in general.

Applying this result to the conductivity of a two-phase microstructure with interchangeable, symmetric and proportional phases, and using Theorem 2.3, we get the following result: Corollary 3.9. Consider an interchangeable two-phase material with conductivity

σ ε := χ ε σ 1 + (1 -χ ε )λ σ 1 , with σ 1 ∈ M s + and λ > 0, (3.15)
and consider the conductivity ρ ε (h) under the low magnetic field h

σ ε (h) = σ ε + s ε h J, where s ε := χ ε s 1 + (1 -χ ε ) s 2 , s 1 , s 2 ∈ R. (3.16)
Then, the resistivity ρ ε (h) := σ ε (h) -1 satisfies the expansion

ρ ε (h) = ρ ε + r ε h J + o L ∞ (Ω) 2×2 , where ρ ε := χ ε (σ 1 ) -1 + (1 -χ ε ) (λ σ 1 ) -1 r ε := χ ε r 1 + (1 -χ ε ) r 2 , (3.17)
where the constants r 1 , r 2 are defined by

r 1 := - s 1 det (σ 1 )
and r 2 := -s 2 det (λ σ 1 ) .

(3.18)

The coefficient s * and the effective Hall coefficient r * in expansion (2.8) are given by the following formulas

s * = s 2 + λ s 1 1 + λ and r * = r 1 + λ r 2 1 + λ . ( 3

.19)

Moreover, for any corrector P ε associated with σ ε , we have lim Proof. On the one hand, from A = A s + α(A)J we deduce that

D (Ω) χ ε det (P ε ) = λ λ + 1 . ( 3
AJA = A s JA s -α(A) A s -α(A) A s -α(A) 2 J = -2α(A) A s + det(A s ) -α(A) 2 J, (3.22)
taking into account that A s JA s = det(A s ) J. Furthermore, it is easy to check that

det(A) = det(A s ) + α(A) 2 , (3.23) hence AJA = -2α(A) A s + det(A) -2α(A) 2 J. (3.24)
On the other hand, AJA = λ A + µ J is equivalent to Then, the condition bc > a 2 is equivalent to condition (3.13).

AJA = λ A s + λα(A) + µ J. ( 3
On the other hand, we have A ε := χ ε A + (1 -χ ε )f (A). Set B ε := χ ε f (A) + (1 -χ ε ) A. Since the phases are interchangeable, B ε H-converges to A * . Furthermore, by Lemma 1.10 we clearly have f (A ε ) = B ε , hence f (A ε ) H-converges to A * . The condition bc > a 2 being satisfied, we deduce from Theorem 1.11 and the uniqueness of the H-limit, that f (A * ) = A * . This equality also reads as We apply Theorem 3.7 to the interchangeable two-phase material with conductivity A ε := σ ε (h). In this case A := σ 1 + s 1 h J and µ := (s 2 -λ s 1 ) h. Hence, condition (3.13) reads as A Proof of Theorem 1.7

λ det σ 1 +
A.1 Proof of part i)

We follow the construction of the H-limit used by Murat-Tartar (see [START_REF] Murat | H-convergence[END_REF]) which depends on the vector parameter h.

Let Ω be a bounded open set of R N such that Ω ⊂ Ω. We extend A ε (h) in Ω\Ω by αI N (in order to have A ε ∈ M(α, β; Ω)). We define A ε (h) ∈ L H 1 0 ( Ω); H -1 ( Ω) by ∀ u ∈ H 1 0 ( Ω), A ε (h)u := -div (A ε (h)∇u) . (A.1)

We proceed in two steps.

3 )Figure 1 :Corollary 3 . 4 .

 3134 Figure 1: Two period cells of a generalized checkerboard.

Figure 2 :

 2 Figure 2: Two period cells of an interchangeable material with a herring-bone pattern.

.20) 3 . 3 7 First we prove the following result: Lemma 3 . 10 .

 337310 Proof of the results3.3.1 Proof of Theorem 3.Let A ∈ M + . Then, the following equivalence holds true for any λ, µ ∈ R: AJA = λ A + µ J ⇐⇒ λ = -2α(A) and µ = det(A).(3.21)

  .25) From the uniqueness of the decompositions (3.24) and (3.25) we deduce the desired result. Now, let us prove Theorem 3.7. First, let us show there exist a, b, c ∈ R with bc > a 2 , such that f (A) = λ A + µ J, where f (A) is given by (1.16) and λ > 0. We have λc AJA = (a + aλ + cµ) A + (b + aµ) J. (3.26) Since λc = 0 by assumption, we deduce from Lemma 3.10 that a + aλ + cµ = -2λc α(A) and b + aµ = λc det(

2 3. 3 . 2

 232 a A * + b J = -a A * + c A * JA * , (3.29) or equivalently c A * JA * = 2a A * + b J. (3.30) Therefore, Lemma 3.10 and formulas (3.28) imply thatα(A * ) = µ + 2λ α(A) 1 + λ and det(A * ) = λ det(A) + µ µ + 2λ α(A) 1 + λ ,(3.31)which concludes the proof. Proof of Corollary 3.9

+ λ = r 1 + λ r 2 1

 11 combined with (3.35) yields in the case s 1 := 1 and s 2 :3.20). On the other hand, by the formula (3.14) applied with A := σ 1 and µ := 0, we obtain det (σ * ) = det (σ * (0)) = λ det σ 1 . (3.39)Hence, by the third equality of (2.8), formulas (3.37) and (3.18) it follows thatr * = -s * det (σ * ) = 1 λ det (σ 1 ) det λ σ 1 r 2 + λ det σ 1 r 1 1 19) and concludes the proof. 2

  ty) DW ε,t,x (0, y) dy.(2.19) On the other hand, since det is a null Lagrangian and σ ε (0)(x + t•) DW ε,t,x (0, •) is Y -periodic and divergence free, by definition (2.15) we have

	det σ ε (0)(x + ty) DW ε,t,x (0, y) dy = det σ * ε,t,x (0) .	(2.20)
	Y

  This holds true for small enough |h|, since λ > 0 and σ 1 ∈ M s + . Then, condition (3.13) holds true without additional assumption for small enough |h|. Therefore, by the formula (3.14) of Theorem 3.7 we obtainα(σ * (h)) = (s 2 -λ s 1 ) h + 2λ α(σ 1 + s 1 h J)Furthermore, by Theorem 2.3 and for any h, σ ε (h) H-converges toσ * (h) = σ * + s * h J + o L 2 (Ω) 2×2 (h), with s * = lim

	s 2 1 h 2 +	(s 2 2 -λ 2 s 2 1 ) h 2 1 + λ	>	s 2 + λ s 1 1 + λ	2	h 2 ,	(3.32)
	which is equivalent to
	λ det σ 1 > O(h 2 ).					(3.33)
		1 + λ		=	s 2 + λ s 1 1 + λ	h.	(3.34)

D (Ω) s ε det (P ε ) ,

(3.35)

which implies that the antisymmetric part of σ * (h) satisfies

α(σ * (h)) = s * h + o(h).

(3.36)

Hence, by (3.34) we get

s * = s 2 + λ s 1 1 + λ .

Acknowledgments. The referees are thanked for their helpful comments. Indeed, the first referee was very careful and helpful. G.M. is grateful for support from the National Science Foundation through grant DMS 0411035, from the Institut de Recherche Mathématique de Rennes (IRMAR) and the Université de Rennes 1 for its invitation in May 2006, and thanks the Institut National des Sciences Appliquées de Rennes (INSA) for its hospitality. M.B. and D.M. are grateful for support from the ACI-NIM plan lepoumonvousdisje grant 2003-45.

First step. For any h ∈ B κ , A ε (h) is bounded by β and equi-coercive, i.e.

So, from the Lax-Milgram theorem, A ε (h) is invertible and, since A ε (h) admits a first-order expansion, so does A ε (h) and B ε (h) := A ε (h) -1 . Furthermore, B ε (h) is bounded by α -1 , hence there exist a subsequence, still denoted by ε, and a linear operator B * (h) from H -1 ( Ω) to H 1 0 ( Ω) such that, for any

for any countable dense set of h. Due to the condition (1.12) satisfied by A ε (h), A ε (h) and B ε (h) satisfy a uniform Lipschitz condition

Therefore, convergence (A.3) holds true for any h ∈ B κ . Moreover, there exists a linear operator

there exist a subsequence of ε, still denoted by ε, and a linear operator

Then, passing to the weak limit in (A.5) and using the semicontinuity of the H 1 0 ( Ω)-norm, we get

is thus invertible, which allows us to define

Then, A * (h) satisfies

Moreover, thanks to (A.4) we have

Second step. To obtain an expansion of the H-limit of A ε (h), we construct a corrector

Then, we define

The uniform Lipschitz assumptions (A.4), (A.11) satisfied by B ε and A ε and the first-order expansions (A.5) and (A.10) satisfied by B ε (h) and A * (h) yield

and ,β; Ω) we have (up to a subsequence) A ε H A * . From the definition (A.13) of P ε (h), it is clear that P ε := P ε (0) is a corrector associated with A ε in Ω, hence by Proposition 1.5 we have

we obtain, for any h ∈ B κ , P ε (h)

Moreover, by (1.12) and (A.14) A ε (h)P ε (h) satisfies the uniform Lipschitz condition in L 2 (Ω) N ×N for h ∈ B κ . Hence, there exist a new subsequence of ε, still denoted by ε, and

By Proposition 1.5 ii), the previous convergence combined with (A.14) and (A.18) implies that A ε (h) H-converges to A * (h). Finally, by (1.12) and (A.15) we have

. Therefore, passing to the limit in the previous equality, we get

The proof of the part i) of Theorem 1.7 is done. 2

Remark A.1. From (A.15) we deduce that if P ε (h), Q ε (h) are the correctors associated with A ε (h) and A ε (h) T respectively, then P ε (h) and Q ε (h) admit the first-order expansions

where P ε and Q ε are the correctors associated with A ε and (A ε ) T respectively. Since P ε (h) and P ε are curl-free, we have

hence P ε 1 • h is also curl-free for any h ∈ B κ . Moreover, since P ε (h) and P ε weakly converge to I N in L 2 (Ω) N ×N , for any weakly convergent subsequence

hence, for the whole sequence ε and for any h ∈ B κ , we have

A.2 Proof of part ii)

By the part i) we obtain that for any h ∈ B κ , A ε (h) H-converges to A * (h) where

, we obtain by Proposition 1.5 i) and the div-curl lemma

On the other hand, the expansion (1.13) of A ε (h) and Remark A.1 lead us to

Hence, by the div-curl lemma and convergences (1.5) and (A.25) we get lim

There exists a subsequence ε , which is actually independent of h (by linearity), such that the sequence (Q ε ) T (A ε

Since the limit is independent of the subsequence ε , the whole sequence (Q ε ) T (A ε 1 )P ε thus converges to A * 1 • h in D (Ω) N ×N . 2


Marc Briane
email: mbriane@insa-rennes.fr

Graeme W Milton
email: milton@math.utah.edu

Giant Hall Effect in Composites

Keywords: Hall effect, homogenization, multi-scale, composites AMS subject classifications. 35B27, 74Q15

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1. Introduction. In electrodynamics a low magnetic field h modifies the conductivity of a conductor. The leading, first-order in h, effect is called the Hall effect (see e.g. [START_REF] Landau | Électrodynamique des Milieux Continus[END_REF] and [START_REF] Kohn | Elementary Solid State Physics[END_REF]), and the second-order in h effect is called the magnetoresistance. Due to quantum effects, multilayer metal composites can have giant magnetoresistances and thus have a conductivity which is very sensitive to small magnetic fields. Using them has enabled the miniaturization of read-heads on magnetic recording devices. The importance of the discovery of such materials was recognized by the award of the 2007 Physics Nobel Prize (see e.g. [START_REF] Baibich | Giant Magnetoresistance (001)Fe/(001)Cr Magnetic Superlattices[END_REF]). This paper addresses the question of whether, within the framework of classical electrodynamics, composites can exhibit giant Hall effects, when the constituent materials do not.

At first-order in h the resistivity matrix is equal to the unperturbed resistivity in the absence of magnetic field plus an antisymmetric first-order term. In dimension two one coefficient arises in the first-order term, it is called the Hall coefficient. In dimension three the first-order term involves a whole 3 × 3 matrix which is called the Hall matrix. In the three-dimensional isotropic case the Hall matrix is proportional to the identity matrix and the coefficient of proportionality is again called the Hall coefficient.

Following the seminal work of Bergman [START_REF] Bergman | Self duality and the low field Hall effect in 2D and 3D metal-insulator composites[END_REF] it is interesting to study the homogenized (or effective) Hall effect in composites. The situation is radically different in dimension two and in dimension three. Indeed, we proved in [START_REF] Briane | Homogenization of the two-dimensional Hall effect[END_REF] that in dimension two the effective Hall coefficient satisfies the same bounds as the local Hall coefficient (i.e. the Hall coefficient in the original microstructure before homogenization). This result is similar to that of the homogenized conductivity which preserves the bounds of the local conductivity by virtue of the arithmetic-harmonic mean bounds. Curiously the bounds are not preserved for three-dimensional Hall coefficients. In particular, we built in [START_REF] Briane | Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient[END_REF] a cubic chain mail (of the Middle Age armor type) with perfectly conductive rings, and a suitable positive local Hall coefficient in such a way that the effective Hall matrix is isotropic but negative. Hence, the bound from below is not preserved by the homogenization process.

It is then natural to see the influence of homogenization on the bound from above on the Hall coefficient. So, is it possible to obtain arbitrarily large effective Hall coefficients from composites with local Hall coefficients satisfying given bounds? In this paper we show that the answer is positive. Previously Bergman, Kantor, Stroud, and Webman [START_REF] Bergman | Critical behavior of the low-field Hall conductivity at a percolation threshold[END_REF] had shown numerically that discrete random resistor networks could exhibit enormous Hall coefficients near percolation, and Rohde and Micklitz [START_REF] Rohde | Critical behavior of the Hall conductivity near the percolation threshold in granular Sn:Ar mixtures[END_REF] had obtained experimental evidence of huge Hall coefficient near percolation in granular mixtures. We mention, in passing, that there are also a number of interesting magnetotransport effects associated with composites in strong magnetic fields (see e.g. [START_REF] Bergman | Recent advances in strong field magneto-transport in a composite medium[END_REF] and [START_REF] Bergman | The self-consistent effective medium approximation (SEMA): New tricks from an old dog[END_REF], and references therein).

We first prove (see Theorem 2.9) a sufficient condition of boundedness which claims that the homogenized Hall matrix is bounded by a constant times the upper bound on the local Hall coefficient provided that the local conductivity is bounded from above (at least in the region where the Hall coefficient is nonzero) and the homogenized conductivity matrix is bounded from below by prescribed constants. If one of the two previous assumptions is not satisfied we can obtain arbitrarily large homogenized Hall coefficients. Specifically, we present three microstructures, each of quite different nature, but which all can have arbitrarily large effective Hall coefficients:

• The first microstructure (see subsection 3.1) acts like n batteries in series. We find that one of the effective Hall coefficients is of order n 1, while the local Hall coefficient is bounded above by 1.

• The second microstructure (see subsection 3.2) is an isotropic, two-phase, rank-three laminate (with 8 elementary layers) based on the Schulgasser construction [START_REF] Schulgasser | Bounds on the conductivity of statistically isotropic polycrystals[END_REF]. One of the phases has a low conductivity t 1 and occupies a high volume fraction 1-t, and the other one has conductivity 1. We find that the effective conductivity is of order t, and that the effective Hall matrix is isotropic and of order t - 1 1, while the local Hall coefficient is still bounded by 1.

• The third microstructure (see subsection 3.3) is an isotropic two-phase rankfive laminate (with 16 elementary layers) still based on the Schulgasser construction. But this time one of the two phases has a high conductivity t -1 occupying a small volume fraction t, and the other one has conductivity 1. We find that the effective conductivity is of order 1, while the effective Hall matrix is asymptotically isotropic and of order t -1 1. Surprisingly we obtain a composite with an effective conductivity of order 1 but inducing a giant Hall effect.

The paper is divided in two sections. In Section 2 we recall the Hall effect principle, a few results of the Murat-Tartar H-convergence and of the homogenization of the Hall effect, and we give a sufficient condition for preserving the bound from above of the three-dimensional homogenized Hall matrix. Section 3 is devoted to the study of the three composites which have unboundedly high effective Hall coefficients.

Notations.

• Y d denotes the cube (0, 1) d ;

• for any measurable set of R d , |E| denotes the Lebesgue measure of E;

• I d denotes the unit matrix of R d×d ;

• for any A ∈ R d×d , A T denotes the transpose of A, det(A) its determinant, tr(A) its trace, and Cof(A) its Cofactor matrix;

• | • | denotes the Euclidian norm in R d ; for any A ∈ R d×d , |A| := sup |Ax| : x ∈ R d with |x| = 1 ;
• for α, β > 0, M(α, β; Ω) denotes the set of the 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. x ∈ Ω; (1.1) • H 1 (Y d ) denotes the space of functions which are Y d -periodic in R d and belong to H 1 loc (R d ); • for u : R d -→ R, ∇u := ∂u ∂x i 1≤i≤d ; • for U : R d -→ R d , U = (u 1 , . . . , u d ), DU := ∂u j ∂x i 1≤i,j≤d ; • for Σ : R d -→ R d×d , Div (Σ) := ∂Σ ij ∂x i 1≤j≤d , Curl (Σ) := ∂Σ ik ∂x j - ∂Σ jk ∂x i 1≤i,j,k≤d ;
• for a sequence of functions

f ε : O -→ H, ε > 0, where O is a neighborhood of 0 in R d and (H, • ) a Banach space, we denote f ε (h) = o H (h) ⇐⇒ lim h→0 1 |h| sup ε>0 f ε (h) = 0, (1.2)
i.e. the o H (h) is uniform with respect to ε; • D (Ω) denotes the space of the distributions on Ω.

2. The Hall effect and Homogenization.

2.1. The three-dimensional Hall effect in a microstructure. Let Ω be a bounded open subset of R 3 , and let α, β > 0. Consider a sequence Σ ε (h), for ε > 0 and h ∈ R 3 , of matrix-valued functions in M(α, β; Ω), which represents the conductivity matrix of a heterogeneous conducting material in the presence of a constant magnetic field h, and the microstructure of which depends on the small parameter ε (which in the simplest case of a periodic composite may represent the length of the unit cell). We assume that Σ ε (h) belongs to M(α, β; Ω) and satisfies the uniform Lipschitz condition

∃ C > 0, ∀ h, h ∈ R 3 , Σ ε (h) -Σ ε (h ) L ∞ (Ω) 3×3 ≤ C |h -h |. (2.1)
From the physics of the problem it can be shown (see e.g. Section 21 of [START_REF] Landau | Électrodynamique des Milieux Continus[END_REF] pages 132-135) that the resistivity ρ ε (h) := Σ ε (h) -1 satisfies the property

∀ h ∈ R 3 , ρ ε (h) T = ρ ε (-h), (2.2)
which implies that the symmetric part of ρ ε (h) is even and the antisymmetric one is odd with respect to h. Then, in the presence of a low magnetic field h we assume that ρ ε (h) has a first-order expansion around h = 0. The zeroth-order term ρ ε := ρ ε (0) of the expansion is the unperturbed resistivity which is a symmetric matrix. By contrast, the first-order term is an antisymmetric matrix and is linear with respect to h. Therefore, the first-order expansion of ρ ε (h) reads as (see [START_REF] Briane | Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient[END_REF] for more details)

ρ ε (h) = ρ ε + E(R ε h) + o L ∞ (Ω) 3×3 (h), (2.3)
where R ε is the local Hall matrix which is bounded in L ∞ (Ω) 3×3 , and E is the Levi-Civita tensor which maps a vector to an antisymmetric matrix according to

E   a 1 a 2 a 3   =   0 a 3 -a 2 -a 3 0 a 1 a 2 -a 1 0   for a 1 , a 2 , a 3 ∈ R. (2.4) Remark 2.1.
In dimension two the first-order term in the expansion (2.3) reads as r ε h J, where r ε , h are scalar and J is the rotation matrix by an angle of 90 • . So, the Hall matrix reduces to the single Hall coefficient r ε (see [START_REF] Briane | Homogenization of the two-dimensional Hall effect[END_REF]).

Review of homogenization.

Let Ω be a bounded open set of R d , and let α, β > 0. We recall here the definition of the Murat-Tartar [START_REF] Murat | H-convergence[END_REF] H-convergence for a sequence of matrix-valued functions in M(α, β; Ω) (see the notation (1.1)), some properties of H-convergence, and the definition of a corrector: Definition 2.2. (Murat-Tartar [START_REF] Murat | H-convergence[END_REF])

A sequence A ε in M(α, β; Ω) is said to H-converge to A * in M(α, β; Ω) if, for any f ∈ H -1 (Ω), the solution u ε of problem -div (A ε ∇u ε ) = f in D (Ω) u ε ∈ H 1 0 (Ω), (2.5)
weakly converges in H 1 0 (Ω) to the solution u of

-div (A * ∇u) = f in D (Ω) u ∈ H 1 0 (Ω), (2.6)
and the sequence A ε ∇u ε weakly converges to A * ∇u in L 2 (Ω) d . Then, the matrixvalued function A * is called the homogenized matrix or the H-limit of A ε . Murat and Tartar proved that any sequence in M(α, β; Ω) admits a subsequence which H-converges in M(α, β; Ω). We will confine our attention to convergent subsequences in the sequel.

The periodic case provides a classical formula for the H-limit (see e.g. [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF]): Let A be a Y d -periodic matrix-valued function in M(α, β; R d ). Then, the oscillating sequence A ε (x) := A ( x ε ) H-converges on every bounded open subset of R d to the constant matrix A * defined by

A * := Y d A DU dy, (2.7)
where U is the unique (up to an additive constant) solution in

H 1 loc (R d ) d of the periodic cell problem Div A DU = 0 in D (R d ) U ε (y) -y is Y d -periodic.
(2.8)

In Definition 2.2 the electric field ∇u ε does not strongly converge to ∇u in L 2 (Ω) d due to the oscillations of the microstructure. Murat and Tartar introduced the notion of corrector in order to measure the oscillations of the electric field and to recover the H-limit: Definition 2.3. (Murat-Tartar [START_REF] Murat | H-convergence[END_REF]) Let A ε be a sequence in M(α, β; Ω) which H-converges to A * . Any matrix-valued function

P ε in L 2 (Ω) 3 satisfying        P ε -I d weakly in L 2 (Ω) d×d Curl (P ε ) -→ 0 strongly in H -1 (Ω) d×d×d Div (A ε P ε ) -→ Div (A * ) strongly in H -1 (Ω) d×d , (2.9)
is called a corrector associated with A ε . The interest in correctors comes from the following result: Proposition 2.4. Let A ε be a sequence in M(α, β; Ω) which H-converges to A * , and let P ε be a corrector associated with A ε . Then, the following convergence holds true

A ε P ε -A * weakly in L 2 (Ω) d×d .
(2.10)

Moreover, the potentials u ε and u which are respectively solutions of (2.5) and (2.6), satisfy the strong convergence

∇u ε -P ε ∇u -→ 0 in L 1 (Ω) d . (2.11)
The following result allows us to build correctors: Proposition 2.5. (Murat-Tartar [START_REF] Murat | H-convergence[END_REF]) Let A ε be a sequence in M(α, β; Ω) which H-converges to A * . Let U ε be the solution in H 1 (Ω) 3 of the problem

Div (A ε DU ε ) = Div (A * ) in Ω U ε (x) = x on ∂Ω.
(2.12)

Then, DU ε is a corrector associated with A ε . In the periodic case there is a more explicit corrector: Example 2.6. Let A be a Y d -periodic matrix-valued function in M(α, β; R d ). Then, the sequence P ε := DU ( x ε ), where the vector-valued function U solves (2.8), is a corrector associated with the oscillating sequence A ε (x) := A ( x ε ).

The homogenized Hall matrix.

Let Ω be a bounded open set of R 3 and let α, β > 0. We consider a conductivity matrix Σ ε (h) in M(α, β; Ω), with h ∈ R 3 , which satisfies the uniform Lipschitz condition (2.1) and which is such that the resistivity ρ ε (h) := Σ ε (h) -1 satisfies the first-order expansion (2.3) at h = 0. We assume that Σ ε (h) H-converges to some conductivity Σ * (h) for any h (this is not a restrictive assumption if h belongs to a countable set).

Since Σ ε (h) T = Σ ε (-h) H-converges to Σ * (h) T by a classical property of Hconvergence, we also get that Σ * (h) T = Σ * (-h) for any h. Therefore, one can prove (see [START_REF] Briane | Homogenization of the two-dimensional Hall effect[END_REF] and [START_REF] Briane | Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient[END_REF]) that there exists a matrix-valued S * in L 2 (Ω) 3×3 such that the Hlimit Σ * (h) satisfies the first-order expansion

Σ * (h) = Σ * + E(S * h) + o L 2 (Ω) 3×3 (h).
(2.13)

We also assume that S * belongs to L ∞ (Ω) 3×3 . Then, (2.13) leads us to the first-order expansion satisfied by the homogenized resistivity ρ * (h) := Σ * (h) -1 :

ρ * (h) = ρ * + E(R * h) + o L 2 (Ω) 3×3 (h), (2.14)
where R * ∈ L ∞ (Ω) 3×3 is the homogenized Hall matrix.

The homogenization problem is now to derive the pair (ρ * , R * ) of homogenized matrices from the sequence (ρ ε , R ε ) of local matrices. As suggested by the work of Bergman [START_REF] Bergman | Self duality and the low field Hall effect in 2D and 3D metal-insulator composites[END_REF] the result simply depends on the solutions for the current field without any magnetic field present. More precisely, in [START_REF] Briane | Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient[END_REF] we proved the following distributional convergence:

For any corrector P ε associated with the unperturbed conductivity Σ ε := Σ ε (0), the following convergence holds true

Cof (Σ ε P ε ) T R ε -Cof (Σ * ) T R * = Cof (Σ * ) R * in D (Ω) 3×3 . (2.15)
Remark 2.7. In the two-dimensional case of [START_REF] Briane | Homogenization of the two-dimensional Hall effect[END_REF], the convergence (2.15) reduces to

det (Σ ε P ε ) r ε -det (Σ * ) r * in D (Ω) 3×3 , (2.16)
where r ε is the local Hall coefficient and r * is the homogenized one.

Remark 2.8. The norm of the homogenized Hall matrix R * is invariant by an orthogonal change of variables. Indeed, let O be an orthogonal matrix of R 3×3 , i.e. OO T = I 3 . Making the change of variables x = Ox, the homogenized resistivity ρ * O in the new coordinates x is given by the formula ρ * O (x ) = O ρ * (x) O T (see e.g. Lemma 38 of [START_REF] Tartar | An introduction to the homogenization method in optimal design[END_REF]). This also yields for the perturbed resistivities ρ * O (h )(x ) = O ρ * (h)(x) O T , where h := Oh is the magnetic field in the new coordinates. Then, by (2.14) the homogenized Hall matrix R * O and the magnetic field h in the new coordinates satisfy

E R * O (x )h = O E R * (x)h O T .
(2.17)

On the other hand, using Lemma 1 of [START_REF] Briane | Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient[END_REF] and the orthogonality of O the right-hand side of (2.17) is also equal to E OR * (x)h , which thus implies the equalities

R * O (x ) = O R * (x) O T and R * O (x ) = R * (x) . (2.18)
2.4. Sufficient conditions of boundedness of the homogenized Hall matrix. In dimension two we proved [START_REF] Briane | Homogenization of the two-dimensional Hall effect[END_REF] that the homogenized Hall coefficient r * (see Remark 2.7) preserves the bounds of the local Hall coefficient r ε . This is not the case in dimension three. The next Section 3 is devoted to three counterexamples. However, if the local conductivity of the microstructure is uniformly bounded from above, and the homogenized conductivity is bounded, then the homogenized Hall matrix is controlled by the bounds on the local Hall matrix as in dimension two. More precisely, we have the following result: Theorem 2.9. Let Ω be a bounded open subset of R 3 , and let α, β, β H , r H > 0. Consider a microstructure in Ω with a conductivity matrix Σ ε ∈ M(a, β; Ω), with 0 < a ≤ α, and an isotropic Hall matrix R ε := r ε I 3 in L ∞ (Ω) 3×3 . We assume that Σ ε H-converges to some

Σ * ∈ M(α, β; Ω), (2.19)
and that Proof. Let P ε := DU ε be the corrector defined by Proposition 2.5, with

|Σ ε | ≤ β H a.e. in Ω ∩ {r ε = 0}, ( 2
U ε := (u ε 1 , u ε 2 , u ε 3 )
. By (2.15) the sequence S ε := Cof (Σ ε P ε ) T R ε converges in the distributions sense to the matrix-valued function S * := Cof (Σ * ) T R * which belongs to L ∞ (Ω) 3×3 . Let us estimate the sequence S ε 11 . First, the Cauchy-Schwarz inequality implies that for any i, j = 1, 2, 3,

(Σ ε P ε ) ij ≤ Σ ε P ε e j ≤ |Σ ε | 1 2 (Σ ε ) 1 2 P ε e j = |Σ ε | 1 2 Σ ε ∇u ε j • ∇u ε j 1 2 . (2.23) Let ϕ ∈ C 1 c (Ω), ϕ ≥ 0.
Using successively (2.20), (2.23) and the Cauchy-Schwarz inequality for the integrals we have

Ω S ε 11 ϕ dx ≤ Ω |r ε | Cof Σ ε P ε T 11 ϕ dx = Ω |r ε | (Σ ε P ε ) 22 (Σ ε P ε ) 33 -(Σ ε P ε ) 23 (Σ ε P ε ) 32 ϕ dx ≤ Ω |r ε | (Σ ε P ε ) 22 (Σ ε P ε ) 33 ϕ dx + Ω |r ε | (Σ ε P ε ) 23 (Σ ε P ε ) 32 ϕ dx ≤ 2 β H r H Ω Σ ε ∇u ε 2 • ∇u ε 2 1 2 Σ ε ∇u ε 3 • ∇u ε 3 1 2 ϕ dx ≤ 2 β H r H Ω Σ ε ∇u ε 2 • ∇u ε 2 ϕ dx 1 2 Ω Σ ε ∇u ε 3 • ∇u ε 3 ϕ dx 1 2
.

(2.24)

Similarly, by (2.23) we have for

S ε 12 Ω S ε 12 ϕ dx ≤ Ω |r ε | Cof Σ ε P ε T 12 ϕ dx = Ω |r ε | (Σ ε P ε ) 12 (Σ ε P ε ) 33 -(Σ ε P ε ) 13 (Σ ε P ε ) 32 ϕ dx ≤ 2 β H r H Ω Σ ε ∇u ε 2 • ∇u ε 2 ϕ dx 1 2 Ω Σ ε ∇u ε 3 • ∇u ε 3 ϕ dx 1 2
.

(2.25)

Moreover, the Murat-Tartar div-curl Lemma [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] combined with the Definition 2.2 implies the convergences Then consider a Lebesgue point x common to the matrix-valued functions S * and Σ * . By making an orthogonal change of variables which preserves the norms |Σ * |, |R * | (see Remark 2.8 above), we can assume that Σ * (x) is a diagonal matrix without loss of generality. Using the test function ϕ := 1 B(x,δ) /|B(x, δ)| in (2.27) (where B(x, δ) denotes the ball of center x and radius δ) and passing to the limit as δ → 0, we obtain the inequality

Ω Σ ε ∇u ε i • ∇u ε i ϕ dx -→ ε→0 Ω Σ * ii ϕ dx, for i = 1, 2,
|S * 11 (x)| ≤ 2 β H r H Σ * 22 (x) Σ * 33 (x) 1 2 .
(2.28)

Similarly, we have for any i, j = 1, 2, 3,

|S * ij (x)| ≤ 2 β H r H Σ * i1i1 (x) Σ * i2i2 (x) 1 2
, where {i, i 1 , i 2 } = {1, 2, 3}.

(2.29)

Now, let us go back to the homogenized Hall matrix

R * = Cof (Σ * ) -T S * = det (Σ * ) -1 Σ * S * .
Since Σ * (x) is a positive definite diagonal matrix, we have for any i, j = 1, 2, 3,

|Σ * ij (x)| ≤ Σ * ii (x) Σ * jj (x) 1 2
.

This combined with the estimates (2.29) yields for any i, j = 1, 2, 3, Theorem 3.1. There exists a microstructure in Ω depending on the positive integer n, with an εY 3 -periodic conductivity Σ ε,n , and a εY 3 -periodic isotropic Hall matrix R ε,n := r ε,n I 3 , such that

|R * ij (x)| = 1 det Σ * (x) Σ * i1 (x) S * 1j (x) + Σ * i2 (x) S * 2j (x) + Σ * i3 (x) S * 3j (x) ≤ 6 β H r H det Σ * (x) Σ * ii (x) 1 2 Σ * 11 (x) Σ * 22 (x) Σ * 33 (x) 1 2 = 6 β H r H Σ * ii (x) Σ * 11 (x) Σ * 22 (x) Σ * 33 (x) 1 2 ≤ 6 β H r H α , (Σ * kk (x) ≥ α),
Σ ε,n = I 3 a.e. in Ω ∩ {r ε,n = 0}, (3.1)
r ε,n ∈ {0, 1} a.e. in Ω, (3.2)
Moreover, the homogenized conductivity matrix Σ * ,n and the homogenized Hall matrix R * ,n are constant and satisfy

lim n→∞ |R * ,n | = ∞. (3.3) Remark 3.2.
The assumptions (2.20) and (2.21) of Theorem 2.9 hold in Theorem 3.1 with β H = 1 and r H = 1, thanks to conditions (3.1) and (3.2). However, the sequence Σ * ,n is not bounded from below by α I 3 , where α is a constant independent of n. So, the arbitrarily large homogenized Hall coefficients, implied by (3.3), are not forbidden by Theorem 2.9. As a consequence, by virtue of (3.3) we can obtain arbitrarily large homogenized Hall coefficients.

3.1.1. Description of the microstructure. We consider in Ω a εY 3 -periodic columnar microstructure aligned along the x 3 -axis, the cross section of which is represented in figure 3.1. The period cell Y 2 = (0, 1) 2 of the cross section of the microstructure is divided into three regions as shown in figure 3.2:

• The 1-conductivity (grey) region Q n is composed of n rectangles (n = 3 in figure 3.2) Q n 1 , . . . , Q n n . Each rectangle Q n k has an x 1 -length equal to L1 n and a x 2 -length equal to L 2 . The distance between two consecutive rectangles Q n k and Q n k+1 is of order 1 n . • The high conductivity (black) region Q n s has n connected components in the two-dimensional torus which link successive pairs of the n rectangles Q n 1 , . . . , Q n n , and which link Q n n with Q n 1 across the boundary of the unit cell, as sketched in the figure .  • The low conductivity (white) region

Q n w is the complementary of Q n ∪ Q n s in Y 2 .
It contains the three rectangles R 0 , R 1 , R 2 the width of each of which is of order 1, as well as additional thin connecting regions, which serve to insulate the various components. On the other hand, the conductivity of the microstructure is defined by the two parameters κ 1 and n ∈ N \ {0} as follows. Let σ ,κ,n be the Y 2 -periodic function defined by its restriction on Y 2 σ ,κ,n (y) :=

         1 if y ∈ Q n = Q n 1 ∪ • • • ∪ Q n n κ if y ∈ Q n s 1 n 3 if y ∈ Q n w .
(3.4) Denote by χ : R → R the 1-periodic function which agrees on the interval (0, 1) with the characteristic function of (0, 1 2 ), and denote by χ Q : R 2 → R the Y 2 -periodic function which agrees on Y 2 with the characteristic function of any set Q ⊂ Y 2 . We consider the εY 3 -periodic and anisotropic conductivity Σ ε,κ,n associated with the microstructure of figure 3.1 and defined for x = (x , x 3 ) ∈ Ω, by

Σ ε,κ,n (x) = Σ ,κ,n x ε :=   σ ,κ,n 0 0 0 σ ,κ,n 0 0 0 χ Q n s a + (1 -χ Q n s ) σ ,κ,n   x ε . (3.5)
where a is a constant which will be chosen later. Remark 3.3.

The conductivity Σ ε,κ,n is anisotropic only in the high conductivity (black) region to avoid a strong conductivity along the x 3 -direction. We could have considered an isotropic conductivity by introducing in the black region a laminate along the x 3 -direction at a smaller scale than ε. Using the reiterated homogenization for a multi-scale microstructure (see e.g. [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF]), this two-scale isotropic conductivity does the same job as the anisotropic conductivity Σ ε,κ,n . For the sake of clarity we directly start from the anisotropic one. By the periodic homogenization formula (2.7) and since Σ ε,κ,n is independent of x 3 , the H-limit of the sequence Σ ε,κ,n is the constant matrix

Σ * ,κ,n :=   Σ * ,κ,n 11 Σ * ,κ,n 12 0 Σ * ,κ,n 21 Σ * ,κ,n 22 0 0 0 Σ * ,κ,n 33   , (3.6)
where Σ * ,κ,n

where the functions u κ,n i , i = 1, 2, solve the period cell problem

div (σ ,κ,n ∇u κ,n i ) = 0 in D (R 2 ) u κ,n i (y) -y i is Y 2 -periodic.
(3.9)

Finally, we consider in Ω the εY 3 -periodic x 3 -independent isotropic Hall matrix defined by

R ε,κ,n (x) := r ε,κ,n (x) I 3 where r ε,κ,n (x) := χ Q n x ε , for x = (x , x 3 ) ∈ Ω. (3.10)
Note that the local Hall coefficient r ε,κ,n is concentrated in the (grey) region of conductivity 1.

A physical approach.

To simplify the physical understanding, let us slightly change the problem and suppose the isotropic white region has zero conductivity, and the black region has infinite transverse conductivity in the x 1 -x 2 plane. Next suppose the average current is directed along the x 3 axis. Since the black and grey regions have finite conductivities a and 1 in this direction, a fixed portion of the total current will flow in the grey material in the direction x 3 . If the magnetic field h is directed in the x 1 direction this current will induce a Hall voltage V H (independent of n) in the x 2 direction across each grey region Q n k , for k = 1, . . . , n. Thus, each of these grey regions will act in the transverse plane as a battery, and the black connecting regions serve to connect them in series. Since the black regions are perfectly conducting in the transverse direction there will be zero voltage drop in the transverse direction across each connected black segment. Therefore, the total voltage drop in the x 2 direction across the unit cell will be nV H . This implies the Hall matrix coefficient R * 11 is proportional to n, which becomes unboundedly large as n → ∞. 3.1.3. A mathematical proof. Taking into account the Example 2.6, the convergence (2.15) combined with the εY 3 -periodicity of Σ ε,κ,n (3.5) and R ε,κ,n (3.10), imply that the homogenized Hall matrix R * ,κ,n is given by

Cof (Σ * ,κ,n ) R * ,κ,n = Q n Cof Σ ,κ,n DU κ,n T dy, (3.11)
where U κ,n := (u κ,n 1 , u κ,n 2 , u κ,n 3 ) is defined by (3.9), and u κ,n 3 = y 3 (up to an additive constant) due to the y 3 -independence of Σ ,κ,n in (3.5). Since Σ ,κ,n = I 3 in Q n and U κ,n := (u κ,n 1 , u κ,n 2 , y 3 ), we also have

R * ,κ,n = Σ * ,κ,n det (Σ * ,κ,n ) Q n Cof DU κ,n T dy, (3.12)
where the transpose of the corrector Cofactor matrix reads as We will estimate R * ,κ,n 11 passing successively to the limits κ → ∞ and n → ∞.

Cof DU κ,n T =      ∂u κ,n 2 ∂y2 - ∂u κ,n 2 ∂y1 0 - ∂u κ,n 1 ∂y2 ∂u κ,n 1 ∂y1 0 0 0 ∂u κ,n 1 ∂y1 ∂u κ,n 2 ∂y2 - ∂u κ,n 1 ∂y2 ∂u κ,n 2 ∂y1      . ( 3
First step : Passage to the limit κ → ∞.

For a fixed positive integer n and i = 1, 2, by equicoercivity of Σ κ,n the sequence u κ,n i weakly converges, as κ → ∞, in H 1 loc (R 2 ) to the function u n i which solves the periodic cell problem

           Y2 σ ,n ∇u n i • ∇v dy = 0 ∀ v ∈ H 1 (Y 2 ), ∇v = 0 in Q n s , ∇u n i = 0 in Q n s , u n i (y) -y i is Y 2 -periodic, (3.16)
where the conductivity σ ,n is the Y 2 -periodic function defined by

σ ,n (y) :=    1 if y ∈ Q n 1 n 3 if y ∈ Q n w .
(3.17)

By equation (3.9) and the weak convergence of ∇u κ,n i to ∇u n i in L 2 (Y 2 ), we also have

Y2 σ ,κ,n ∇u κ,n i -∇u n i 2 dy = Y2 σ ,n ∇u n i • ∇(u n i -u κ,n i ) dy -→ κ→∞ 0. (3.18)
This combined with (3.6) yields for any i, j = 1, 2,

Σ * ,κ,n ij = Y2 σ ,κ,n ∇u κ,n i • ∇u κ,n j dy -→ κ→∞ Σ * ,n ij := Y2 σ ,n ∇u n i • ∇u n j dy, (3.19)
hence we deduce that the coefficient R * ,κ,n of (3.12) satisfies the following limit

* ,n ii = Y2 σ ,n |∇u n i | 2 dy = min Y2 σ ,n |∇v| 2 dy : v(y) -y i ∈ H 1 (Y 2 ), ∇v = 0 in Q n s . (3.21) Consider a test function v 1 = v 1 (y 1 ) in (3.21) with v 1 (1) = 1, such that v 1 is equal to zero in Y 2 \ R 1 (see figure 3.2).
Then, by the definition (3.17) of σ ,n we have

Σ * ,n 11 ≤ 1 n 3 R1 |∇v 1 | 2 dy ≤ c 1 n 3 . (3.22)
Moreover, by the Cauchy-Schwarz inequality and the 1-periodicity of (y 1 → u 1 (y) -y 1 ), we get

Σ * ,n 11 ≥ 1 n 3 Y2 ∂u n 1 ∂y 1 2 dy ≥ 1 n 3 Y2 ∂u n 1 ∂y 1 dy 2 = 1 n 3 . (3.23)
Therefore, there exists a constant c 1 > 0 such that

1 n 3 ≤ Σ * ,n 11 ≤ c 1 n 3 . (3.24)
The estimate of Σ * ,n 22 is more delicate. On the one hand, we can build a test function v n 2 of (3.21) satisfying the following properties (see figure 3.2):

             v n 2 (y 1 , 0) = 0, v n 2 (y 1 , 1) = 1, v n 2 = 1 -k n in each connected component of Q n s joining Q n k and Q n k+1 , ∇v n 2 = 1 nL2 e 2 in Q n , i.e. v n 2 is an affine function of y 2 in each set Q n k , |∇v n 2 | is bounded in Y 2 by a constant independent of n. (3.25)
To deduce the fourth property of (3.25) from the values of v n 2 given by the first three ones, we first make an interpolation in the rectangles between two consecutive sets Q n k and Q n k+1 , the width of these rectangles being of order 1 n . Then, we make a second interpolation in the (white) rectangles R 0 , R 1 , R 2 the width of which is of order 1 (see figure 3.2).

Putting the test function v n 2 in (3.21) we get by the third and fourth properties of (3.25),

Σ * ,n 22 ≤ Y2 σ ,n |∇v n 2 | 2 dy = Q n |∇v n 2 | 2 dy + 1 n 3 Q n w |∇v n 2 | 2 dy = L 1 n 2 L 2 + O 1 n 3 . (3.26)
On the other hand, let c n,i 0 be the value of u n i in the (black) upper connected component of Q n s in figure 3.2, let c n,i n be the one in the lower connected component, and let c n,i k be the value of the connected component of Q n s which joins two consecutive sets Q n k and Q n k+1 , for i = 1, 2 and k = 1, . . . , n -1. Then, by the Cauchy-Schwarz inequality we have

Σ * ,n 22 ≥ Q n |∇u n 2 | 2 dy ≥ Qn ∂u n 2 ∂y 2 2 dy ≥ 1 |Q n | Q n ∂u n 2 ∂y 2 dy 2 . (3.27)
Moreover, since c n,2 0 -c n,2 n = 1 by the 1-periodicity of (y 2 → u n 2 (y) -y 2 ), we obtain owing to an integration by parts

1 |Q n | Q n ∂u n 2 ∂y 2 dy 2 = 1 L 1 L 2 L 1 n n k=1 c n,2 k-1 -c n,2 k 2 = L 1 n 2 L 2 . (3.28)
This combined with (3.27) and (3.26) yields

L 1 n 2 L 2 ≤ Q n |∇u n 2 | 2 dy ≤ Σ * ,n 22 ≤ L 1 n 2 L 2 + O 1 n 3 , (3.29)
which implies that

Σ * ,n 22 = L 1 n 2 L 2 + O 1 n 3 , 1 n 3 Q n w |∇u n 2 | 2 dy = Σ * ,n 22 - Q n |∇u n 2 | 2 dy = O 1 n 3 .
(3.30)

Moreover, using successively (3.30), (3.25), the first property of (3.16) with v := v n 2 -u n 2 , and again (3.30) we get

Q n |∇u n 2 -∇v n 2 | 2 dy = Q n |∇u n 2 | 2 dy + Q n |∇v n 2 | 2 dy -2 Q n ∇u n 2 • ∇v n 2 dy = 2 L 1 n 2 L 2 + O 1 n 3 -2 Y2 σ ,n ∇u n 2 • ∇v n 2 dy + O 1 n 3 = 2 L 1 n 2 L 2 -2 Q n σ ,n ∇u n 2 • ∇u n 2 dy + O 1 n 3 = 2 L 1 n 2 L 2 -2 Σ * ,n 22 + O 1 n 3 = O 1 n 3 . ( 3
= Y2 σ ,n ∇u n 1 • ∇u n 2 dy = Q n ∇u n 1 • ∇u n 2 dy + O 1 n 3 = Q n ∇u n 1 • ∇v n 2 dy + Q n ∇u n 1 • (∇u n 2 -∇v n 2 ) dy + O 1 n 3 . (3.32)
On the one side, by the definition of the constants c 1,n k (defined after (3.26)) and the 1-periodicity of (y 2 → u n 1 (y)), it follows that

Q n ∇u n 1 • ∇v n 2 dy = 1 nL 2 n k=1 Q n k ∂u n 1 ∂y 2 dy = L 1 n 2 L 2 n k=1 c n,1 k-1 -c n,1 k = L 1 n 2 L 2 c n,1 0 -c n,1 n = 0. (3.33)
On the other side, the Cauchy-Schwarz inequality combined with estimates (3.24) and (3.31) yields First of all, by proceeding as in (3.27) and (3.24) we obtain Finally, we consider the microstructure of figure 3.1 associated with the local conductivity matrix Σ ε,n := Σ ε,κn,n of (3.5) and the local Hall coefficient r ε,n := r ε,κn,n of (3.10). Therefore, (3.38) implies that the homogenized Hall matrix R * ,n := R * ,κn,n of (3.12) satisfies the desired result (3.3).

Q n ∇u n 1 • (∇u n 2 -∇v n 2 ) dy ≤ Σ * ,n 11
1 2 Q n |∇u n 2 -∇v n 2 | 2 dy 1 2 = O 1 n 3 .
Q n ∂u n i ∂y 2 dy = L 1 n n k=1 c n,i k-1 -c n,i k = L 1 n c n,i 0 -c n,i n =    0 if i = 1 L 1 n if i = 2. ( 3

3.

2. An isotropic laminate with a low effective conductivity. The microstructure we have just studied achieves a large Hall effect by combining transverse Hall voltages like batteries in series. By contrast, the laminate materials analyzed in this subsection and the subsequent subsection achieve large Hall voltages through large local electrical currents flowing through a small volume of material having a Hall coefficient of order 1, and conductivity much greater than the surrounding material.

We have the following result: Theorem 3.4. There exists a laminate microstructure associated with an isotropic conductivity Σ ε,t depending on a small parameter t > 0, and an isotropic Hall matrix R ε = r ε,t I 3 , with r ε,t ∈ {0, 1}, such that the homogenized conductivity Σ * ,t = σ * ,t I 3 and the homogenized Hall matrix R * ,t = r * ,t I 3 are isotropic, with Proof. The proof of Theorem 3.4 is based on a two-phase rank-one laminate in the direction ξ ∈ {e 1 , e 2 , e 3 }, where (e 1 , e 2 , e 3 ) is the canonical basis of R 3 . Let Σ 1 , Σ 2 be the conductivity phases of the laminate with the respective volume fractions θ, 1 -θ. By virtue of [START_REF] Milton | Modelling the properties of composites by laminates[END_REF] and [START_REF] Briane | Corrector for the homogenization of a laminate[END_REF] there exists an associated corrector (electric field) which has the same laminate structure with components P 1 , P 2 in the two phases satisfying the relation Moreover, the effective conductivity matrix of the laminate is given by

P 1 = M P 2 where M := I 3 + 1 Σ 1 ξ • ξ (ξ ⊗ ξ) (Σ 2 -Σ 1 ) . ( 3
Σ * := Σ 1 +(1-θ) (Σ 2 -Σ 1 ) N -1 where N := I 3 + θ Σ 1 ξ • ξ (ξ ⊗ ξ) (Σ 2 -Σ 1 ) , (3.41)
in such a way that θ Σ 1 P 1 + (1 -θ) Σ 2 P 2 = Σ * P where P := θ P 1 + (1 -θ) P 2 .

(3.42)

To obtain an isotropic composite we will apply the Schulgasser construction [START_REF] Schulgasser | Bounds on the conductivity of statistically isotropic polycrystals[END_REF] to a two-phase rank-one laminate of the previous type, considered as a single crystal with the effective conductivity matrix (3.41). The construction will provide a rank-three laminate based on 4 elementary two-phase rank-one laminates (that is to say 8 layers at the smallest scale) as shown in figure 3.3. Following the principle stated in [START_REF] Milton | Modelling the properties of composites by laminates[END_REF] and proved in [START_REF] Briane | Corrector for the homogenization of a laminate[END_REF] there exists a corrector which is constant in each layer of the laminate. This combined with (2.15) will allow us to derive an explicit formula for the effective Hall matrix.

First step: construction of 4 elementary two-phase rank-one laminates.

We start from a two-phase rank-one laminate in the direction e 1 composed of the hard (high conducting) phase Σ 1 := I 3 and the soft (low conducting) phase Σ 2 := t I 3 , with the respective volume fractions θ := t, 1 -t, where t 1. So, the soft phase occupies the major part of the volume. The effective conductivity matrix of the laminate is given by

Σ * 11 := I 3 -(1 -t) 2 I 3 + t (t -1) (e 1 ⊗ e 1 ) -1 =   λ 0 0 0 µ 0 0 0 µ   , (3.43)
where λ := t 1 -t + t 2 and µ := 2t -t 2 .

(3.44)

Thanks to (3.40) the correctors P h 1 (h for the hard phase) and P s 1 (s for the soft phase) associated with this first rank-one laminate satisfy

P h 1 = M 11 P s 1 where M 11 := I 3 + (t -1) (e 1 ⊗ e 1 ) . (3.45)
The second rank-one laminate is similar but in the direction e 2 , with the effective conductivity matrix The third rank-one laminate is a copy of the first one with the effective conductivity matrix Σ * 13 := Σ * 11 and the associated correctors P h 3 , P s 3 satisfying

Σ * 12 := I 3 -(1 -t) 2 I 3 + t (t -1) (e 2 ⊗ e 2 ) -1 =   µ 0 0 0 λ 0 0 0 µ   , ( 3
P h 3 = M 11 P s 3 . (3.48)
The fourth rank-one laminate is similar to the first one in the direction e 3 , with the effective conductivity matrix The previous results show that the effective conductivity is isotropic and tends to zero as t, and that the effective Hall matrix is isotropic and blows up as t -1 . The proof of Theorem 3.4 is thus done.

Σ * 14 := I 3 -(1 -t) 2 I 3 + t (t -1) (e 3 ⊗ e 3 ) -1 =   µ 0 0 0 µ 0 0 0 λ   , ( 3

3.

3. An isotropic laminate with an effective conductivity of order 1. In the two previous examples (subsection 3.1 and subsection 3.2) the effective conductivity is not bounded from below and only the assumption (2.19) of Theorem 2.9 does not hold. By contrast in the following example the condition (2.19) is satisfied but not the condition (2.20). We obtain a composite with an isotropic effective conductivity of order 1, while the effective Hall matrix is asymptotically isotropic and unbounded with respect to a small parameter.

Theorem 3.5. There exists a laminate associated with an isotropic conductivity Σ ε,t ≥ I 3 depending on a small parameter t > 0, and an isotropic Hall matrix R ε,t = r ε,t I 3 , with r ε,t ∈ {0, 1}, such that the homogenized conductivity Σ * ,t = σ * ,t I 3 is isotropic and the homogenized Hall matrix R * ,t = r * ,t I 3 + o(1) is asymptotically isotropic, with Proof. The microstructure is still based on the isotropic Schulgasser construction. However, the role of the single crystal is now played by a rank-three laminate defined as follows (this role is played by the rank-one laminate with the effective conductivity (3.43) in the Schulgasser construction of subsection 3.2):

• At the first level of lamination we construct a rank-one laminate in the direction e 1 := (1, 0, 0) built from two phases with conductivities t -1 I 3 , I 3 with the respective volume fractions t, 1 -t. Its effective conductivity matrix is given by • At the third level of lamination we construct a rank-one laminate in the direction e 3 := (0, 0, 1) composed by the two phases Σ * 12 , I 3 with the respective volume fractions This yields the second part of (3.67) and concludes the proof.

Σ * 11 :=   1 1-t+t 2

( 2 .

 2 30) which implies (2.22) since |R * (x)| ≤ 3 max 1≤i,j≤3 |R * ij (x)|.

3 .

 3 Unbounded homogenized Hall coefficients. This section is devoted to three examples of three-dimensional microstructures which have unbounded effective Hall coefficients due to the presence of low or/and high conductivity regions.3.1.A battery of Hall materials. In this Section, Ω = ω × (0, 1) is an open bounded cylinder of height 1 and of cross section ω ⊂ R 2 .

Fig. 3 . 1 .Fig. 3 . 2 .

 3132 Fig. 3.1. The cross section of the columnar microstructure

33 is the average value of the Y 2 -

 2 periodic function Σ ,κ,n 33 , i.e. Σ * ,κ,n 33 = a |Q n s | + |Q n |

. 13 ) 11 . 11 =

 131111 From now on, we choose a in (3.7) in such a way that Σ a being then of order 1 as n → ∞.Let us focus on the Hall coefficient R * ,κ,n The formula (3.12) combined with (3.14), (3.6) and (3.13), yields R * ,κ,n

  Second step : Estimate of Σ * ,n ij , for i, j = 1, 2. Let us start by Σ * ,n 11 . By (3.16) the function u n i , i = 1, 2, solves the minimization problem Σ

  Third step : Estimate of the homogenized Hall coefficient R * ,n 11 .

Fig. 3 . 3 .

 33 Fig. 3.3. The Schulgasser laminate

•, 1 5 .

 5 At the second level of lamination we construct a rank-one laminate in the direction e 2 := (0, 1, 0) composed by the two phases Σ * 11 , I 3 with the respective volume fractions4 5 Its effective conductivity matrix is given by

  Remark 2.10. The homogenized Hall matrix is bounded, up to a multiplicative constant, by the product of the contrast β H α of the local conductivity times the bound r H of the local Hall coefficient r ε . Note that this contrast is less or equal to the global contrast β α of the conductivity, since by (2.20) β H is the bound obtained in the region where the Hall coefficient is nonzero.

				.20)
	and
	|r ε | ≤ r H a.e. in Ω.	(2.21)
	Then, the homogenized Hall matrix R * satisfies the bound
	|R * | ≤ 18	β H α	r H a.e. in Ω.	(2.22)

  Third step: construction of the isotropic rank-three laminate. The final composite is a two-phase rank-one laminate in the direction e 1 of the composite phases Σ * 1 , Σ * 2 given by the second step, with the volume fractions 1 2 , 1 2 . By (3.41) its effective conductivity is the isotropic matrix Σ * ,t = σ * ,t I 3 := Σ * Moreover, using Maple the formulas (3.61) and (3.64) combined with (3.56) imply that R * ,t = r * ,t I 3 where r * ,t := 3 (1 + t + t 2 ) (1 -t + t 2 ) t (5 -6t + 6t 2 -2t 3 ) 2 =

	The first rank-two laminate is a rank-one laminate in the direction e 3 of the composite
	phases Σ * 11 , Σ * 12 , with the respective volume fractions 1 3 , 2 3 . By (3.41) the effective
	conductivity of this laminate is given by Σ * 1 := Σ * 11 + 2 3 (Σ * 12 -Σ * 11 ) I 3 + 3 Σ * 11 e 3 • e 3 1	(e 3 ⊗ e 3 ) (Σ * 12 -Σ * 11 ) 3 25 t -1 1 + O(t) . (3.66) -1
		=	 	λ+2µ 0 3		3 2λ+µ 0		0 0	 . 	(3.51)
				0		0		µ
	We set
						Pi := t P h i + (1 -t) P s i , for i = 1, . . . , 4.	(3.52)
	By [12] and [7] this rank-two laminate can be regarded as a two-phase rank-one
	laminate the correctors of which are (averaging at the smallest scale of the rank-three
	laminate) P1 , P2 satisfying
	P1 = M 1 P2 where M 1 := I 3 +	1 11 e 3 • e 3 Σ *	(e 3 ⊗ e 3 ) (Σ * 12 -Σ * 11 ) = I 3 .	(3.53)
	The second rank-two laminate is a rank-one laminate in the direction e 2 of the com-
	posite phases Σ * 13 , Σ * 14 , with the respective volume fractions 1 3 , 2 3 . By (3.41) and by the equality Σ * 13 = Σ * 11 its effective conductivity is given by
	Σ * 2	:= Σ * 11 +	2 3	(Σ * 14 -Σ * 11 ) I 3 +	3 Σ * 11 e 2 • e 2 1	(e 2 ⊗ e 2 ) (Σ * 14 -Σ * 11 )	-1
				λ+2µ 3		0	0		(3.54)
		=		0 0		µ 0 2λ+µ 0 3	 ,
	and the associated correctors P3 , P4 satisfy
	P3 = M 2 P4 where M 2 := I 3 +	1 11 e 2 • e 2 Σ *	(e 2 ⊗ e 2 ) (Σ * 14 -Σ * 11 ) = I 3 .	(3.55)
						1 +	1 2	(Σ * 2 -Σ * 1 ) I 3 +	2 Σ * 1 e 1 • e 1 1	(e 1 ⊗ e 1 ) (Σ * 2 -Σ * 1 )	-1	.49)
	and the associated correctors P h 4 , P s 4 satisfying P h 4 = M 14 P s 4 where M 14 := I 3 + (t -1) (e 3 ⊗ e 3 ) . = λ + 2µ I 3 . 3	(3.50) (3.56)
	We set with (3.52)
	Second step: construction of two rank-two laminates with the composite phases of the first step. P1 := 1 3 P1 + 2 3 P2 and P2 := 1 3 P3 + 2 3 P4 . (3.57)

  3 4 ,1 4 . Therefore, the effective conductivity of the rank-three

	or equivalently,
	R * ,t =	Σ * ,t det (Σ * ,t )	1 10t	Cof P 1 +	1 5t	Cof P 2 +	1 10t	Cof P 3 +	1 5t	Cof P 4 . (3.76)
	Therefore, the formula (3.76) combined with (3.73), (3.71) and (3.72) implies that
	R * ,t is diagonal with
				R * ,t 11 =	R * ,t 22 + R * ,t 33 2	,		(3.77)
	and satisfies the asymptotic expansion
				R * ,t =	15 418	t -1 I 3 + O(t) .	(3.78)

* Graeme Milton is grateful for support from the National Science Foundation through grants DMS-0411035 and DMS-070978. The authors wish to thank an unknown referee for relevant comments.

By [START_REF] Milton | Modelling the properties of composites by laminates[END_REF] and [START_REF] Briane | Corrector for the homogenization of a laminate[END_REF] this isotropic composite can be considered as a two-phase rank-one laminate the correctors of which are (averaging at the meso-scale of the rank-three laminate) P1 , P2 satisfying Moreover, by the first convergence of (2.9) the whole average of the rank-three laminate correctors is equal to I 3 , hence

Now, we consider the isotropic local Hall matrix R ε,t = r ε,t I 3 defined by Note that the effective matrix Σ * ,t 1 is similar to the one of (3.43). But this time Σ * ,t 1 is of order 1 with respect to t since the hard phase of conductivity t -1 is surrounded by the three soft phases of conductivity 1 in this rank-three laminate. Also note that the specific choice of the prescribed volume fractions 4 5 , 3 4 allows us to obtain an effective matrix with two equal eigenvalues and so to simplify a lot the next step. Now, we proceed to the Schulgasser construction from the previous four-phase rank-three laminate regarded as a three-dimensional single crystal with conductivity (3.70). On the one hand, this leads us to a rank-five laminate (i.e. with five ordered micro-scales) composed of 4 × 4 = 16 elementary layers (there are 2 × 4 = 8 elementary layers in the laminate of subsection 3.2 since the starting crystal is a twophase rank-one laminate). Resting on the lamination formulas (3.41) and (3.42) we obtain 16 associated correctors. The computations are quite similar to the ones of subsection 3.2, hence we do not give the details. In particular, using Maple we get that the correctors P 1 , P 2 , P 3 , P 4 associated with the 4 hard layers of conductivity t -1 , are given by

1 in the layers of P 1 of volume fraction 4 Then, since the conductivity is equal to t -1 in the layers of the correctors P i , the formula (2.15)