Marc Briane
email: mbriane@insa-rennes.frmilton@math.utah.edu

Graeme W Milton

Bounds on strong field magneto-transport in three-dimensional composites

Keywords: bounds, homogenization, magneto-transport, strong field, multiphase composites AMS classification: 35B27, 74Q20

Bounds on strong field magneto-transport in three-dimensional composites

Introduction

It is known since the seminal discovery of Hall in the end of the 19th century [START_REF] Hall | On a new action of the magnet on electric currents[END_REF], that a low magnetic field perturbs the matrix resistivity (or equivalently the conductivity) of a conductor by inducing a small non-symmetric part characterized by the so-called Hall coefficient. In the 80's Bergman [START_REF] Bergman | Self-duality and the low field Hall effect in 2D and 3D metal-insulator composites[END_REF] first gave a general formula for the effective Hall coefficient involving currents that solve the symmetric conductivity equations in the absence of a magnetic field. However, there are few explicit formulas for the effective Hall coefficient except in very particular cases like two-phase 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 three-dimensional composite with a one-or two-dimensional microstructure[END_REF][START_REF] Briane | Homogenization of the two-dimensional Hall effect[END_REF], or 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] Strelniker | Exact relations 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 situation is still less favorable in the strong field case [START_REF] Bergman | Strong-field magnetotransport of conducting composites with a columnar microstructure[END_REF][START_REF] Bergman | Recent advances in strong field magneto-transport in a composite medium[END_REF], namely when the symmetric part and the antisymmetric part of the conductivity are of the same order. In three dimensions, only when the antisymmetric part is constant do we have an exact formula for the antisymmetric part of the effective tensor [START_REF] Stroud | New exact results for the Hall-coefficient and magnetoresistance of inhomogeneous two-dimensional metals[END_REF]. So, rather than trying to get explicit relations for the effective tensors it seems more practical to derive bounds. The theory of the bounds in homogenization has gone through a considerable development since the original variational approach of Hashin and Shtrikman [START_REF] Hashin | A variational approach to the theory of the effective magnetic permeability of multiphase materials[END_REF]. We refer to [START_REF] Milton | The Theory of Composites[END_REF] for a comprehensive survey. In fact, very little is known about bounds for strong field magneto-transport. Recently, we derived in [START_REF] Briane | New bounds on strong field magneto-transport in multiphase columnar composites[END_REF] optimal bounds for multiphase columnar composites. The aim of this paper is to extend, at least partially, the result of [START_REF] Briane | New bounds on strong field magneto-transport in multiphase columnar composites[END_REF] to two-phase three-dimensional composites.

In the present context we consider a three-dimensional conductor having a periodic structure (this is actually not a restrictive assumption) in the presence of a fixed vertical strong magnetic field. Under the transverse isotropy assumption the local conductivity of the conductor takes the general form

σ(y) =   a(y) -c(y) 0 c(y) a(y) 0 0 0 b(y)   , for y ∈ R 3 , (1.1)
where the coefficient c(y) is induced by the presence of the magnetic field parallel to the y 3 -axis, which also influences a(y) and b(y) and causes them to be non-equal in the case of a conductor that is isotropic in the absence of the magnetic field. Similarly, assuming an transversely isotropic microstructure, or at least one that is invariant under 90 • or 120 • rotations about the y 3 -axis, the constant effective conductivity of the composite is given by

σ * =   a * -c * 0 c * a * 0 0 0 b *   . (1.2)
Our goal is to derive bounds for the effective coefficients a * , b * , c * of σ * in terms of the coefficients a(y), b(y), c(y) of the local conductivity σ(y).

In Section 2, we derive elementary bounds (see Theorem 2.1) on the effective coefficients a * , b * and c * . These are obtained by taking uniform trial fields in a variational principle for nonsymmetric tensors deduced in [START_REF] Milton | On characterizing the set of possible effective tensors of composites: The variational method and the translation method[END_REF][START_REF] Fannjiang | Convection enhanced diffusion for periodic flows[END_REF] from a symmetrization of the constitutive law j = σe, and its adjoint, adapted from the variational approach performed in [START_REF] Cherkaev | Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli[END_REF] for complex tensors.

In Section 3, we show (see Theorem 3.1) that contrary to the columnar case of [START_REF] Briane | New bounds on strong field magneto-transport in multiphase columnar composites[END_REF], it is not possible to bound the antisymmetric part of the effective conductivity σ * only in terms of the coefficients c(y). Indeed, when σ(y) is independent of y 3 for a vertical columnar structure, the key ingredient for the derivation of the optimal bounds in [START_REF] Briane | New bounds on strong field magneto-transport in multiphase columnar composites[END_REF] is based on the positivity of the (2 × 2) determinant of the local electric field [START_REF] Alessandrini | Univalent σ-harmonic mappings[END_REF][START_REF] Alessandrini | Beltrami operators, non-symmetric elliptic equations and quantitative Jacobian bounds[END_REF]

, i.e. ∆ 1,2 (DU) := ∂ 1 u 1 ∂ 2 u 2 -∂ 1 u 2 ∂ 2 u 1 > 0 a.e. in R 3 , (1.3)
where the vector-valued potential U = (u 1 , u 2 , u 3 ) solves the conductivity problem

div (σDU) = 0 in R 3 U(y) -y is Y -periodic. (1.4)
Due to a suitably constructed rank-two laminate with high-contrast conductivity, we prove simultaneously that the inequality (1.3) does not hold, and that arbitrarily large effective coefficients c * can be obtained while the local coefficient c(y) is bounded. This negative result agrees with the pathologies obtained in [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] with different microstructures, related to bounds on the effective Hall coefficient in the low magnetic field regime. As a consequence, a bound for c * involves both the upper bound for |c(y)| and the bounds from below and above for a(y)

in (1.1) (see Theorem 3.1 and Remark 3.2). This bound shows c * → 0 when the upper bound on |c(y)| goes to zero, provided a(y) remains bounded from below and above. In Section 4, to improve the previous bounds we restrict ourselves to a two-phase local conductivity

σ(y) = χ 1 (y)   a 1 -c 1 0 c 1 a 1 0 0 0 b 1   + χ 2 (y)   a 2 -c 2 0 c 2 a 2 0 0 0 b 2   , for y ∈ R 3 , (1.5)
with prescribed volume fraction f i = χ i , for i = 1, 2, with f 1 + f 2 = 1. We then derive (see Theorem 4.1) Hashin-Shtrikman type bounds for the effective conductivity σ * , involving three intermediate coefficients a Y , b Y , c Y which are explicitly expressed in terms of the entries of σ i , for i = 1, 2, * . In particular, it is shown that the point (a Y , -c Y ) belongs to a disk which is tangent to the axis a = 0 at some point (0, c), and which contains the disk passing through the points (a 1 , c 1 ), (a 2 , c 2 ), and tangent to the axis a = 0 at the same point (0, c) (see Figure 1 below). The derivation of these new bounds is based on a combination of three main ingredients:

• the geometric isotropy of the phases defined in [START_REF] Willis | Bounds and self-consistent estimates for the overall properties of anisotropic composites[END_REF] for random composites,

• the variational principle for non-symmetric tensors,

• the use of Y -tensors similar to [START_REF] Gibiansky | On the effective viscoelastic moduli of two-phase media. I. Rigorous bounds on the complex bulk modulus[END_REF] (see also [START_REF] Berryman | Effective medium theory for elastic composites[END_REF]), giving relations between the averages of the fields in each phase.

Notations

• (e 1 , e 2 , e 3 ) denotes the canonic basis of R 3 .

• I denotes the unit matrix of R 3×3 , and J := 0 -1 0 1 0 0 0 0 0 .

• For any matrix M ∈ R 2×2 , M T denotes the transpose of M, M S := 1 2 M + M T the symmetric part of M, and M A := 1 2 M -M T the antisymmetric part of M. • Y denotes the unit cube [0, 1] 3 , and • the Y -average.

• For a function f defined on the unit sphere S 2 of R 3 , f S 2 denotes the average of f over S 2 , i.e.

f S 2 := - S 2 f (ξ) dξ = 1 4π 2π 0 dϕ π 0 f (sin θ cos ϕ, sin θ sin ϕ, cos θ) sin θ dθ. (1.6) • For α, β > 0, M ♯ (α, β; Y ) denotes the set of the Y -periodic invertible matrix-valued functions A : R 3 → R 2×2 such that ∀ ξ ∈ R 3 , A(y) ξ • ξ ≥ α |ξ| 2 and A -1 (y) ξ • ξ ≥ β -1 |ξ| 2 a.e. y ∈ Y. (1.7)
• L 2 ♯ (Y ) denotes the space of the Y -periodic functions, which are square integrable in Y . • H 1 ♯ (Y ) denotes the space of the Y -periodic functions, with gradient in L 2 ♯ (Y ) 3 . • For u : R 3 -→ R, ∇u := ∂u ∂x i 1≤i≤3 .

• For U : R 3 -→ R 3 , U = (u 1 , u 2 , u 3 ), DU := ∂u j ∂x i 1≤i,j≤3 .

• For Σ : R 3 -→ R 3×3 , div (Σ) :=

∂Σ ij ∂x i 1≤j≤3 .

Elementary bounds on magneto-transport

To derive bounds on composites we may assume that the associated microstructures are Yperiodic (see, e.g., [START_REF] Allaire | Shape Optimization by the Homogenization Method[END_REF] Theorem 1.3.23), where Y is any cube of R 3 , say Y = [0, 1] 3 . In this section and the next we consider a three-dimensional Y -periodic conductor in the presence of a strong magnetic field parallel to the y 3 -axis so that the resulting matrix-valued conductivity σ(y) is given by

σ(y) =   a(y) -c(y) 0 c(y) a(y) 0 0 0 b(y)   , for y ∈ R 3 , (2.1)
where the coefficients a(y), b(y), c(y) satisfy for prescribed positive numbers a, a, c > 0, with a ≤ a, the following bounds a ≤ a(y) ≤ a and |c(y

)| ≤ c, a.e. y ∈ R 3 . (2.2)
By virtue of the periodic homogenization formula (see, e.g., [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF]) the effective conductivity σ * associated with σ(y) is given by

σ * = σDU , where the potential U solves div (σDU) = 0 in R 3 U(y) -y is Y -periodic. (2.3)
Recall that σ * is also the homogenized conductivity obtained from the oscillating sequence σ( x ε ) as ε → 0 by a homogenization process (see, e.g., [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF]). Now consider a periodic electric field e ∈ L 2 ♯ (Y ) 3 and a periodic current field j ∈ L 2 ♯ (Y ) 3 that solves the conductivity equations j = σe, div j = 0, curl e = 0 (2.4) and another periodic electric field e ′ ∈ L 2 ♯ (Y ) 3 and another periodic current field j ′ ∈ L 2 ♯ (Y ) 3 that solves the adjoint equations j ′ = σ T e ′ , div j ′ = 0, curl e ′ = 0.

(2.5)

The average fields are related by the effective tensor σ * :

j = σ * e , j ′ = σ T * e ′ . (2.6)
Define the symmetric tensor

L(y) := (σ S ) -1 -(σ S ) -1 σ A σ A (σ S ) -1 σ S -σ A (σ S ) -1 σ A (y). (2.7)
Then, an easy computation yields

F = e S j A = L j S e A = LE, where      e S := 1 2 (e + e ′ ) , e A := 1 2 (e -e ′ ) j S := 1 2 (j + j ′ ) , j S := 1 2 (j + j ′ ) .
(2.8)

Moreover, mimicking the approach of [START_REF] Cherkaev | Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli[END_REF] for complex tensors, extended in [START_REF] Milton | On characterizing the set of possible effective tensors of composites: The variational method and the translation method[END_REF][START_REF] Fannjiang | Convection enhanced diffusion for periodic flows[END_REF] (see also [START_REF] Milton | The Theory of Composites[END_REF], p. 277) for real but non-symmetric tensors, the following variational principle holds

j 0 e 0 T L * j 0 e 0 = min j S e A T L j S e A : e A ∈ L 2 ♯ (Y ) 3 , curl (e A ) = 0, e A = e 0 j S ∈ L 2 ♯ (Y ) 3 , div (j S ) = 0, j S = j 0 . , (2.9)
with the symmetric effective tensor

L * := (σ S * ) -1 -(σ S * ) -1 σ A * σ A * (σ S * ) -1 σ S * -σ A * (σ S * ) -1 σ A * . (2.10)
By substituting constant trial fields e A = e 0 and j S = j 0 in the variational principle one immediately obtains the elementary bound

L * ≤ L . (2.11)
This elementary bound implies the following theorem:

Theorem 2.1. Assuming σ * and σ(y) have the forms (1.2) and (2.1) the constant b * must satisfy the arithmetic and harmonic mean bounds

1/ 1/b ≤ b * ≤ b , (2.12)
and the pair (a * , c * ) must satisfy the circle bounds (which confine (a * , c * ) to lie within a circle in the a * -c * plane) given by

(c * -c L ) 2 ≤ (a * -a L ) (d L -a * ) , (2.13)
where

a L := 1 a -1 , c L := c a a L , d L := a + c 2 a - c 2 L a L . (2.14)
Remark 2.2. Taking the minimum in (2.9) over all fields e A and j S with e A = e 0 and j S = j 0 , and ignoring the differential constraints that curl (e A ) = 0 and div (j S ) = 0 gives the elementary bound L -1 * ≤ L -1 . However this does not yield any new inequalities beyond (2.12) and (2.13) due to the structure of the matrices L * and L(y).

Proof of Theorem 2.1. The proof follows the proof of the elementary bounds in Proposition 3.1 of [START_REF] Briane | New bounds on strong field magneto-transport in multiphase columnar composites[END_REF]. Assuming σ * and σ(y) have the forms (1.2) and (2.1) we obtain

L * =                 1 a * 0 0 0 c * a * 0 0 1 a * 0 - c * a * 0 0 0 0 1 b * 0 0 0 0 - c * a * 0 a * + c 2 * a * 0 0 c * a * 0 0 0 a * + c 2 * a * 0 0 0 0 0 0 b *                 . ( 2.15)
and

L =                  1 a L 0 0 0 c L a L 0 0 1 a L 0 - c L a L 0 0 0 0 1 b 0 0 0 0 - c L a L 0 d L + c 2 L a L 0 0 c L a L 0 0 0 d L + c 2 L a L 0 0 0 0 0 0 b                  . ( 2

.16)

The matrix L -L * will then be positive semi-definite if and only if (2.12) holds and

a * ≥ a L , (2.17)
c L a L - c * a * 2 ≤ 1 a L - 1 a * d L + c 2 L a L -a * - c 2 * a * (2.18)
By multiplying the last inequality by a * a L and expanding (and using the fact that a * a L > 0) we get (2.13). Also the inequality (2.17) is superfluous as it is implied by (2.13) and the inequality

d L ≥ a L .
3 Bounds on magneto-transport: 2d versus 3d

From (2.2) and the non-negativity of the 4th (or 5th) diagonal element of L -L * we deduce the additional (superfluous) bound

2 |c * | ≤ a * + c 2 * a * ≤ a + c 2 a ≤ a + c 2 a , (3.1)
where to obtain the first inequality we have used the fact that x + 1/x ≥ 2 for all x > 0. Thus, we have obtained an upper bound on |c * |, but one that involves not only c but also a and a. This is in contrast to the case for a columnar conductivity σ(y) which is independent of the y 3variable, where using the positivity of the determinant of the electric field DU(y) = DU(y 1 , y 2 ) established by Alessandrini and Nesi [START_REF] Alessandrini | Univalent σ-harmonic mappings[END_REF][START_REF] Alessandrini | Beltrami operators, non-symmetric elliptic equations and quantitative Jacobian bounds[END_REF], we proved in [START_REF] Briane | New bounds on strong field magneto-transport in multiphase columnar composites[END_REF] that c * satisfies the same bound c as the local coefficient c(y

) (i.e., |c * | ≤ c).
Let us now relax the assumption that the effective tensor takes the form (1.2). The composite is said to be partially isotropic if the antisymmetric part of σ * satisfies (σ * ) A = c * J, where J :=

  0 -1 0 1 0 0 0 0 0   . (3.2)
Given a partially isotropic composite we can always subdivide it into square columns with edges parallel to the y 3 -axis and with side length much larger than the existing microstructure, and then rotate each square column about its center axis by either 0 • , 90 • , 180 • or 270 • with equal probability in an uncorrellated way. The resulting polycrystal is invariant under rotations of 90 • about the y 3 -axis and thus will have an effective tensor of the form (1.2) and by a lemma of Stroud and Bergman [START_REF] Stroud | New exact results for the Hall-coefficient and magnetoresistance of inhomogeneous two-dimensional metals[END_REF] will have the same constant c * as the original partially isotropic composite.

The question naturally arises as to whether for partially isotropic composites |c * | can be bounded solely in terms of c, like in the case of a columnar conductivity σ(y)? The answer is no, it cannot. Indeed, we have the following result: Theorem 3.1. Consider a periodic conductivity σ(y) given by (2.1) which satisfies the bounds (2.2). Assume that the composite is partially isotropic in the sense of (3.2). Then, the effective coefficient c * satisfies

|c * | ≤ c a (σ * e 1 • e 1 ) 1/2 (σ * e 2 • e 2 ) 1/2 . (3.3)
On the other hand, given any arbitrarily large constant κ > 0 there exist transversely isotropic conductivities σ θ,κ (y) depending on a parameter θ > 0, with c(y) ∈ {0, 1} a.e. y ∈ R 3 , such that as θ → 0 the effective conductivity σ * θ,κ is partially isotropic and satisfies lim θ→0

(σ * θ,κ ) A = -κ J or lim θ→0 (σ * θ,κ ) A = κ J. (3.4)
Remark 3.2. In the case when σ * is transversely isotropic, taking the form (1.2), the bound

(3.3) reduces to |c * | ≤ a * a c, (3.5)
and using (3.1) we obtain that

|c * | ≤ c a a + c 2 a - c 2 * a * ≤ c a a + c 2 a . (3.6)
In contrast to the bounds (3.1) and (2.13) this new bound shows that c * necessarily goes to zero as c goes to zero if a and a are held fixed. Also if we add the antisymmetric matrix c 0 J to σ(y) then the effective tensor will change to σ * + c 0 J, implying from (3.5) that the inequality

|c * + c 0 | ≤ a * a max |c + + c 0 |, |c -+ c 0 | (3.7)
holds for all constants c 0 , where

c + := sup y∈Y c(y), c -:= inf y∈Y c(y). (3.8)
Taking the optimum value c 0 = -(c + + c -)/2 gives the bounds

|2 c * -c + -c -| ≤ a * a (c + -c -). (3.9)
Remark 3.3. Theorem 3.1 proves that contrary to the columnar case of [START_REF] Briane | New bounds on strong field magneto-transport in multiphase columnar composites[END_REF] we cannot expect to bound the effective coefficient c * only in terms of the bound c of the local coefficient c(y). Actually, (3.4) shows that arbitrarily large (positive or negative) effective coefficients c * can be derived although the local coefficient c(y) only takes values in {0, 1}. Here, the contrast of the symmetric part of the conductivity plays a crucial role as suggested in the bound (3.3). This is strongly linked to the fact that the (2 × 2) determinant

∆ 1,2 (DU) := ∂ 1 u 1 ∂ 2 u 2 -∂ 1 u 2 ∂ 2 u 1 , for U = (u 1 , u 2 , u 3 ), (3.10)
does not always have a constant sign throughout the material (see the proof of Theorem 3.1 below) contrary to the columnar case.

Proof of Theorem 3.1.

Bound for c * : The div-curl lemma of Murat-Tartar (see [START_REF] Tartar | Cours Peccot[END_REF][START_REF] Murat | Séminaire d'Analyse Fonctionnelle et Numérique[END_REF][START_REF] Murat | H-convergence[END_REF]) and the formula (2.3) for

σ * yield (DU) T σDU = (DU) T σDU = DU T σ * DU = σ * . (3.11)
Hence passing to the antisymmetric part it follows that

(σ * ) A = (DU) T σ A DU = c (DU) T JDU = c 0 -∆ 1,2 (DU ) -∆ 1,3 (DU ) ∆ 1,2 (DU ) 0 -∆ 2,3 (DU ) ∆ 1,3 (DU ) ∆ 2,3 (DU ) 0 , (3.12)
where ∆ i,j (DU

) := ∂ 1 u i ∂ 2 u j -∂ 1 u j ∂ 2 u i .
Therefore, since σ * is partially isotropic, we obtain the following formula for the effective coefficient c * ,

c * = c ∆ 1,2 (DU) . (3.13)
Using the Cauchy-Schwartz inequality we have

|c * | ≤ c |∂ 1 u 1 | |∂ 2 u 1 | • |∂ 2 u 2 | |∂ 1 u 2 | ≤ c |∂ 1 u 1 | 2 + |∂ 2 u 1 | 2 1/2 |∂ 1 u 2 | 2 + |∂ 2 u 2 | 2 1/2 . (3.14)
On the other hand (3.11) also implies for i = 1, 2, Derivation of arbitrarily large coefficients c * : Let θ, κ be two positive numbers, let ξ 1 , ξ 2 θ be the vectors defined by

σ * e i • e i = σ∇u i • ∇u i ≥ a |∂ 1 u i | 2 + |∂ 2 u i | 2 . ( 3
ξ 1 θ := 0, θ √ 1 + θ 2 , 1 √ 1 + θ 2 , ξ 2 := 0, 1 √ 2 , 1 √ 2 , (3.16)
and let σ 1 θ,κ , σ 2 , σ 3 be the (transversely isotropic) phases defined by

σ 1 θ,κ :=   κ θ -2 0 0 0 κ θ -2 0 0 0 1   , σ 2 := I, σ 3 := 2 I + J. (3.17)
Consider the rank-two laminate mixing in the direction ξ 1 θ the phase σ 1 θ,κ , with volume fraction 1 -θ, and the rank-one laminate, with volume fraction θ, composed of the mixture in the direction ξ 2 of the phases σ 2 and σ 3 with volume fraction 1 2 . The two-scale conductivity σ θ,κ is defined by

σ θ,κ (y, z) := χ θ (ξ 1 θ • y) σ 1 θ,κ + 1 -χ θ (ξ 1 θ • y) χ(ξ 2 • z) σ 2 + 1 -χ(ξ 2 • z) σ 3 , (3.18)
where y = x ε , z = x ε 2 are the ordered fast variables, χ θ is the 1-periodic function which agrees with the characteristic function of [0, 1 -θ] in [0, 1], and χ is the 1-periodic function which agrees with the characteristic function of [0, 1 2 ] in [0, 1]. By [START_REF] Milton | Modelling the properties of composites by laminates[END_REF][START_REF] Briane | Corrector for the homogenization of a laminate[END_REF] the local electric field E θ,κ associated with the conductivity σ th,κ has the same laminate structure as (3.18), and thus can be written as

E θ,κ (y, z) := χ θ (ξ 1 θ • y) E 1 θ,κ + 1 -χ θ (ξ 1 θ • y) χ(ξ 2 • z) E 2 θ,κ + 1 -χ(ξ 2 • z) E 3 θ,κ . (3.19)
The constant matrices E 1 θ,κ , E 2 θ,κ , E 3 θ,κ are the solutions of the linear system

               (1 -θ) E 1 θ,κ + θ 2 (E 2 θ,κ + E 3 θ,κ ) = I average-value E 2 θ,κ -E 3 θ,κ = ξ 2 ⊗ η 2 jump of the curl at the scale ε 2 E 1 θ,κ -1 2 (E 2 θ,κ + E 3 θ,κ ) = ξ 1 θ ⊗ η 1 jump of the curl at the scale ε (σ 2 E 2 θ,κ -σ 3 E 3 θ,κ ) T ξ 2 = 0 jump of the div at the scale ε 2 σ 1 θ,κ E 1 θ,κ -1 2 (σ 2 E 2 θ,κ + σ 3 E 3 θ,κ )
T ξ 1 θ = 0 jump of the curl at the scale ε.

(3.20)

We refer to [START_REF] Briane | Corrector for the homogenization of a laminate[END_REF] for more details. Similarly to (3.11) and taking into account the two-scale structure (3.18) the effective conductivity σ * θ,κ is given by

σ * θ,κ = (1 -θ) (E 1 θ,κ ) T σ 1 θ,κ E 1 θ,κ + θ 2 (E 2 θ,κ ) T σ 2 E 2 θ,κ + (E 3 θ,κ ) T σ 3 E 3 θ,κ . (3.21)
Taking into account the values (3.17) of the matrix conductivities we deduce that

(σ * θ,κ ) A = θ 2 (E 3 θ,κ ) T JE 3 θ,κ . (3.22)
Using Maple to compute explicitly the solutions E 1 θ,κ , E 2 θ,κ , E 3 θ,κ of the linear system (3.20), we get the following asymptotics as θ → 0,

(σ * θ,κ ) A = θ 2 ∆ 1,2 E 3 θ,κ J + O(θ) = - κ 17 J + O(θ) (3.23)
Therefore, σ * θ,κ is asymptotically partially isotropic, and the effective coefficient

c * θ,κ := θ 2 ∆ 1,2 E 3 θ,κ = - κ 17 + O(θ), (3.24)
is both negative and arbitrarily large when κ is arbitrarily large. Moreover, the (2 × 2) determinant ∆ 1,2 of the electric field satisfies

∆ 1,2 E 3 θ,κ = -∆ 1,2 E 2 θ,κ + O(1) = - 2 κ 17 θ + O(1), (3.25)
and thus for large κ has not the same sign throughout the material, contrary to the columnar case.

On the other hand, if we replace in (3.17) the matrix σ 3 by

σ 3 :=   2 -1 0 1 2 0 0 0 1 2   , (3.26)
then the previous procedure leads us to the asymptotics

(σ * θ,κ ) A = θ 2 ∆ 1,2 E 3 θ,κ J + O(θ) = κ 13 J + O(θ). (3.27)
Hence, the effective conductivity σ * θ,κ is still asymptotically partially isotropic, and the effective coefficient

c * θ,κ = θ 2 ∆ 1,2 E 3 θ,κ = κ 13 + O(θ), (3.28)
is arbitrarily large but positive. As before, the minor ∆ 1,2 of the electric field satisfies

∆ 1,2 E 3 θ,κ = -∆ 1,2 E 2 θ,κ + O(1) = 2 κ 13 θ + O(1), (3.29)
and does not have the same sign throughout the material when κ is large.

Hashin-Shtrikman type bounds under geometric isotropy 4.1 Y -tensors, Γ-operator, and geometric isotropy

For given α, β > 0, consider a periodic two-phase composite with local conductivity

σ(y) = χ 1 (y) σ 1 + χ 2 (y) σ 2 ∈ M ♯ (α, β; Y ), (4.1)
where χ i is the characteristic function of the phase i with volume fraction f i , for i = 1, 2.

Denote by σ * its effective conductivity. Following [START_REF] Gibiansky | On the effective viscoelastic moduli of two-phase media. I. Rigorous bounds on the complex bulk modulus[END_REF] (see also [START_REF] Milton | The Theory of Composites[END_REF], Chapter 19) there exists an effective tensor Y * associated with the conductivity σ, defined by (e ∈ L 2 ♯ (Y ) 3 is the electric field and j ∈ L 2 ♯ (Y ) 3 is the current field) P (j) = -Y * P (e), where j = σ e and P (g) := χ 1 (g -g ) . (4.2)

In some sense P is the projection on phase 1 of the fluctuating component of the field. Also we have for the adjoint problem

P (j ′ ) = -Y T * P (e ′ ), where j ′ = σ T e ′ . ( 4
where similarly to (2.10),

Y * = (Y S * ) -1 (Y S * ) -1 Y A * -Y A * (Y S * ) -1 Y S * -Y A * (Y S * ) -1 Y A * . (4.5)
Now, we will derive a Hashin-Shtrikman type variational inequality associated with the variational principle (2.9). To this end, let us consider for a given reference tensor L 0 , the nonlocal operator Γ defined for periodic vector-valued functions A, B ∈ L 2 ♯ (Y ) 6 , by

B = ΓA if Γ 1 B = B and Γ 1 (A -L 0 B) = A -L 0 B, (4.6)
where Γ 1 represents the projection on the space of fields which satisfy the same differential constraints as E ∈ L 2 ♯ (Y ) 6 in (2.8). Since E is composed by a divergence free field j S ∈ L 2 ♯ (Y ) 3 , and a curl free field e A ∈ L 2 ♯ (Y ) 3 , the operator Γ 1 in Fourier space is given by

Γ 1 (k) = Γ 1 (ξ) = I -ξ ⊗ ξ 0 0 ξ ⊗ ξ , where ξ := k |k| , for k ∈ Z 3 \ {0}. (4.7)
Under the conditions L i > L 0 ≥ 0, for i = 1, 2, the Hashin-Shtrikman type variational inequality associated with the variational principle (2.9) is given by the formula (13.30) of [START_REF] Milton | The Theory of Composites[END_REF], which reads as

(L * -L 0 ) -1 F : F ≤ Γ + (L * -L 0 ) -1 F : F , for any F ∈ L 2 ♯ (Y ) 6 . (4.8)
Following the computations of [START_REF] Milton | The Theory of Composites[END_REF] (Section 23.6) this inequality implies the bound

Y * + L 0 ≥   1 f 1 f 2 k∈Z 3 \{0} χ1 (k) 2 Γ(k)   -1 , (4.9)
which also holds for the enlarged inequalities L i ≥ L 0 ≥ 0, for i = 1, 2. Note that by virtue of the Plancherel equality the Fourier coefficients χ1 (k) of the characteristic function χ 1 satisfy the equality 1

f 1 f 2 k∈Z 3 \{0} χ1 (k) 2 = 1 f 1 f 2 (χ 1 -f 1 ) 2 = 1. (4.10)
So, the series in (4.9) can be regarded of an average of the operator Γ. Finally, consider the case of a two-phase random composite. According to [START_REF] Willis | Bounds and self-consistent estimates for the overall properties of anisotropic composites[END_REF] (see also [START_REF] Milton | The Theory of Composites[END_REF], Section 15.6) the composite is said to have a geometric isotropy if all correlation functions associated with the geometry represented by the characteristic function χ 1 are invariant by rotation (or reflection). Then, under geometric isotropy the average series of (4.9) reduces to an average of Γ over all directions of the unit sphere S 2 . Therefore, we get the bound (see [START_REF] Milton | The Theory of Composites[END_REF], Section 23.6)

Y * + L 0 ≥ 1 Γ S 2 . (4.11)

Hashin-Shtrikman type bounds

Consider a periodic two-phase composite with non-symmetric positive definite conductivities

σ i :=   a i -c i 0 c i a i 0 0 0 b i   , with b 1 ≥ b 2 , (4.12)
of respective volume fractions f i , for i = 1, 2. The effective conductivity σ * of the composite is assumed to be transversely isotropic, i.e.

σ * :=   a * -c * 0 c * a * 0 0 0 b *   . (4.13)
Let g : (0, ∞) → R be the function defined by

g(r) := 1 2 π 0 cos 2 θ sin θ cos 2 θ + r -1 sin 2 θ dθ ∈ (0, 1), for r > 0. (4.14) Consider the coefficients α ± , t ± 1 , s ± 1 , a Y , b Y , c Y defined by α ± := a 1 c 2 -a 2 c 1 ± a 1 a 2 (a 1 -a 2 ) 2 + (c 1 -c 2 ) 2 a 1 -a 2 , (4.15) t ± 1 := a 1 a 2 1 + (c 1 -α ± ) 2 , s ± 1 := 2 t ± 1 1 + g(b 1 t ± 1 ) -t ± 1 , (4.16
)

a Y + i c Y := -f 2 (a 1 + i c 1 ) -f 1 (a 2 + i c 2 ) + f 1 f 2 (a 1 + i c 1 -a 2 -i c 2 ) 2 f 1 (a 1 + i c 1 ) + f 2 (a 2 + i c 2 ) -a * -i c * , (4.17) b Y := -f 2 b 1 -f 1 b 2 + f 1 f 2 (b 1 -b 2 ) 2 f 1 b 1 + f 2 b 2 -b * . (4.18)
Then, we have the following result:

a 2 Y + (c Y + α ± ) 2 - a Y s ± 1 ≤ 0, (4.19) a c (a 1 , c 1 ) (a 2 , c 2 ) (a Y , -c Y ) α + α - Figure 1:
The circles HS ± surrounding (a Y , -c Y ) and the (dashed) circles ± passing through (a i , c i ), for i = 1, 2, assuming α -≤ α + .

while the points (a 1 , c 1 ), (a 2 , c 2 ) solve the equations

a 2 + (c -α ± ) 2 - a t ± 1 = 0. (4.20)
Moreover, the coefficient b * satisfies the bounds ) is the same as the circle ± of equation (4.20) passing through the points (a i , c i ), i = 1, 2, when s ± 1 = t ± 1 . Moreover, since g(r) ∈ (0, 1) for r > 0, we have 0 < s ± 1 < t ± 1 in (4.16). This implies that the radius (2s ± 1 ) -2 of HS ± is larger than the radius (2t ± 1 ) -2 of ± . The circles HS ± and ± are also tangent at the same point of the c -axis. The geometrical picture is given by Figure 1 in the a-c plane.

1 b Y + 1 b 1 ≥ 1 b 1 1 -g(b 1 t ± 1 ) , b Y ≥ 0. ( 4
Remark 4.4. The inequalities (4.19), (4.21) do not allow us to show that c * tends to zero when c 1 and c 2 approach zero, while keeping a 1 , a 2 , b 1 , and b 2 fixed. To this end, we only have the bound (3.9) which reads as

|2 c * -c 1 -c 2 | ≤ a * min (a 1 , a 2 ) |c 1 -c 2 |. (4.22)
Proof of Theorem 4.1. The proof is divided in four steps. In the first step we determine a suitable reference tensor L 0 . In the second step we compute the tensor Γ(ξ) involved in the Y -tensor approach. In the third step we compute the average Γ S 2 . The fourth step is devoted to the derivation of the bounds.

First step : Determination of L 0 . Similarly to (2.7), let L i , for i = 1, 2, * , be the symmetric tensor defined by

L i := (σ S i ) -1 -(σ S i ) -1 σ A i σ A i (σ S i ) -1 σ S i -σ A i (σ S i ) -1 σ A i =                 1 a i 0 0 0 c i a i 0 0 1 a i 0 - c i a i 0 0 0 0 1 b i 0 0 0 0 - c i a i 0 a i + c 2 i a i 0 0 c i a i 0 0 0 a i + c 2 i a i 0 0 0 0 0 0 b i                 . (4.23)
Now, let L 0 be the symmetric tensor defined by

L 0 := C 1 C 2 C T 2 C 3 =         t 1 0 0 0 t 2 0 0 t 1 0 -t 2 0 0 0 0 t 4 0 0 0 0 -t 2 0 t 3 0 0 t 2 0 0 0 t 3 0 0 0 0 0 0 t 5        
, where

C j ∈ R 3×3 , C T 2 = -C 2 . (4.24)
The condition L 0 ≥ 0 is equivalent to

t 1 ≥ 0, t 4 ≥ 0, t 3 ≥ 0, t 5 ≥ 0, t 1 t 3 ≥ t 2 2 . (4.25)
We also need L i ≥ L 0 , which is equivalent to

1 a i ≥ t 1 , det     1 a i -t 1 c i a i -t 2 c i a i -t 2 a i + c 2 i a i -t 3     ≥ 0, for i = 1, 2, (4.26) 1 b i ≥ t 4 , b i ≥ t 5 , for i = 1, 2. (4.27)
From now on, assume that b 1 ≥ b 2 , and set

t 4 := 1 b 1 , t 5 = b 2 , (4.28)
in order to make the inequalities (4.27) as sharp as possible.

On the other hand, the inequalities (4.26) show that the points (a 1 , c 1 ) and (a 2 , c 2 ) belong to the disk in the a-c plane,

t 1 (a 2 + c 2 ) + (t 2 2 -t 1 t 3 -1) a -2t 2 c + t 3 ≤ 0, (4.29)
which lies in the half-plane a ≥ 0. To make these bounds as tight as possible, we consider the two circles which are tangent to the c-axis, and which pass through the points (a 1 , c 1 ) and (a 2 , c 2 ). This requires

t 3 = t 2 2 t 1 , (4.30)
and the two circle equations

t 2 1 (a 2 i + c 2 i ) -t 1 a i -2t 1 t 2 c i + t 2 2 = 0, for i = 1, 2, (4.31)
which can be written as

t 2 1 a 2 (a 2 1 + c 2 1 ) -t 1 a 1 a 2 -2t 1 t 2 a 2 c 1 + t 2 2 a 2 = 0 t 2 1 a 1 (a 2 2 + c 2 2 ) -t 1 a 1 a 2 -2t 1 t 2 a 1 c 2 + t 2 2 a 1 = 0. (4.32)
Subtracting and dividing by t 2 1 , we get that α := t 2 /t 1 solves

(a 1 -a 2 ) α 2 -2 (a 1 c 2 -a 2 c 1 ) α + a 1 (a 2 2 + c 2 2 ) -a 2 (a 2 1 + c 2 1 ) = 0, (4.33)
the discriminant of which is

(a 1 c 2 -a 2 c 1 ) 2 + (a 1 -a 2 ) a 2 (a 2 1 + c 2 1 ) -a 1 (a 2 2 + c 2 2 ) = a 1 a 2 (a 1 -a 2 ) 2 + (c 1 -c 2 ) 2 ≥ 0. (4.34)
Hence, equation (4.33) has two real solutions (one for each circle) α ± which are given by (4.15). Moreover, putting t 2 = α t 1 in (4.31) we obtain that

t 1 = a i a 2 i + (c i -α) 2 ≤ 1 a i , for i = 1, 2, (4.35)
which implies that the (2 × 2) matrix in (4.26) is non-negative. Therefore, the choice of the coefficients t 1 , t 2 , t 3 given by

t 1 = a 1 a 2 1 + (c 1 -α) 2 , t 2 = α t 1 , t 3 = α 2 t 1 , for α = α ± , (4.36)
combined with (4.28), implies the desired inequalities L i ≥ L 0 , for i = 1, 2. Making this choice in (4.29) the points (a 1 , c 1 ) and (a 2 , c 2 ) belong to the two circles of equation (4.20) which are tangent to the line a = 0.

Second step : Computation of Γ(ξ). Let ξ ∈ S 2 , ξ = (sin θ cos ϕ, sin θ sin ϕ, cos θ). By virtue of Section 4.1 the tensor Γ(ξ) is defined from the tensor L 0 (4.24), by

B 1 B 2 = Γ(ξ) A 1 A 2 , for A 1 , A 2 , B 1 , B 2 ∈ R 3×3 , (4.37)
if and only if

I -ξ ⊗ ξ 0 0 ξ ⊗ ξ B 1 B 2 = B 1 B 2 , (4.38)
and

I -ξ ⊗ ξ 0 0 ξ ⊗ ξ A 1 A 2 - C 1 C 2 C T 2 C 3 B 1 B 2 = 0. (4.39)
By (4.38) we have B T 1 ξ = 0, and B 2 = ξ ⊗ η for some vector η. From (4.39) it follows that

A T 2 ξ -B T 1 C 2 ξ -B T 2 C T 3 ξ = A T 2 ξ -B T 1 C 2 ξ -(C 3 ξ • ξ) η = 0, (4.40) A 1 -C 1 B 1 -C 2 (ξ ⊗ η) = ξ ⊗ A T 1 ξ -B T 1 C T 1 ξ -B T 2 C T 2 ξ = ξ ⊗ A T 1 ξ -B T 1 C T 1 ξ -(C 2 ξ • ξ) η = ξ ⊗ A T 1 ξ -B T 1 C T 1 ξ since C T 2 = -C 2 , = ξ ⊗ k where k := A T 1 ξ -B T 1 C T 1 ξ. (4.41) Noting that C -1 1 C 2 is antisymmetric, this implies that 0 = B T 1 ξ = (C -1 1 A 1 ) T ξ -(η⊗ξ) (C -1 1 C 2 ) T ξ -(k⊗ξ) (C -1 1 ) T ξ = (C -1 1 A 1 ) T ξ -(C -1 1 ξ •ξ) k, (4.42) so we have k = (C -1 1 A 1 ) T ξ C -1 1 ξ • ξ . (4.43)
Moreover, replacing B 1 given by (4.41) in (4.40) and using that (C -1 1 ) T C 2 is antisymmetric, we get that

0 = A T 2 ξ -(C -1 1 A 1 ) T C 2 ξ + (η ⊗ ξ) (C -1 1 C 2 ) T C 2 ξ + (k ⊗ ξ) (C -1 1 ) T C 2 ξ -(η ⊗ ξ) C T 3 ξ = A T 2 ξ -(C -1 1 A 1 ) T C 2 ξ -(C 3 -C T 2 C -1 1 C 2 ) ξ • ξ η, (4.44)
hence η = A T 2 ξ -(C T 2 C -1 1 A 1 ) T ξ Dξ • ξ , where D := C 3 -C T 2 C -1 1 C 2 . (4.45)
Again using (4.41) combined with (4.43) and (4.45) we deduce that

B 1 = C -1 1 A 1 - C -1 1 C 2 (ξ ⊗ ξ) Dξ • ξ A 2 -C T 2 C -1 1 A 1 - C -1 1 (ξ ⊗ ξ) C -1 1 ξ • ξ C -1 1 A 1 B 2 = ξ ⊗ ξ Dξ • ξ A 2 - ξ ⊗ ξ Dξ • ξ C T 2 C -1 1 A 1 . (4.46)
Hence, from definition (4.37) it follows that

Γ(ξ) =      C -1 1 + C -1 1 C 2 (ξ ⊗ ξ) C T 2 C -1 1 Dξ • ξ - C -1 1 (ξ ⊗ ξ) C -1 1 C -1 1 ξ • ξ C -1 1 C T 2 (ξ ⊗ ξ) Dξ • ξ (ξ ⊗ ξ) C 2 C -1 1 Dξ • ξ ξ ⊗ ξ Dξ • ξ      (4.47) which is a symmetric matrix since C T 1 = C 1 and C T 2 = -C 2 . Third step : Computation of ( Γ S 2 ) -1 .
Note that the computation of Γ(ξ) can be carried out if the matrix D of (4.45)

D =   d 1 0 0 0 d 1 0 0 0 d 2   =       t 3 - t 2 2 t 1 0 0 0 t 3 - t 2 2 t 1 0 0 0 t 5       , ( 4.48)
is positive definite, i.e. t 1 t 3 > t 2 2 and t 5 > 0. Let us assume these conditions for the moment. We shall be able pass to the limit as d 1 → 0 in the expression of ( Γ S 2 ) -1 . Set

P =   p 1 0 0 0 p 1 0 0 0 p 2   := ξ ⊗ ξ Dξ • ξ S 2 , (4.49) Q =   q 1 0 0 0 q 1 0 0 0 q 2   := ξ ⊗ ξ C -1 1 ξ • ξ S 2 , R =   r 1 0 0 0 r 1 0 0 0 r 2   := C -1 1 -C -1 1 QC -1 1 . (4.50)
By definition (1.6) we have

p 1 = ξ 2 1 d 1 + (d 2 -d 1 ) ξ 2 3 S 2 -→ d 1 →0 1 4 π 0 sin 3 θ d 2 cos 2 θ dθ = ∞, (4.51)
p 2 = ξ 2 3 d 1 + (d 2 -d 1 ) ξ 2 3 S 2 -→ d 1 →0 1 d 2 = 1 t 5 . ( 4

.52)

Moreover, the matrix

C -1 2 1 Q C -1 2 1 =       q 1 t 1 0 0 0 q 1 t 1 0 0 0 q 2 t 4       = C -1 2 1 (ξ ⊗ ξ) C -1 2 1 C -1 1 ξ • ξ S 2 (4.53)
has the property that its trace is 1. This combined with definitions (1.6) and (4.14) yields

         q 2 = t 4 ξ 2 3 (t 4 /t 1 )(ξ 2 1 + ξ 2 2 ) + ξ 2 3 S 2 = t 4 g(t 1 /t 4 ) q 1 = t 1 2 1 - q 2 t 4 = t 1 2 1 -g(t 1 /t 4 ) , (4.54)
which also implies that

       r 1 = 1 t 1 - q 1 t 2 1 = 1 2t 1 1 + g(t 1 /t 4 ) r 2 = 1 t 4 - q 2 t 2 4 = 1 t 4 1 -g(t 1 /t 4 ) . (4.55)
On the other hand, by definition (4.49) we have Fourth step : Derivation of the bounds. On the one hand, the Appendix of [START_REF] Gibiansky | On the effective viscoelastic moduli of two-phase media. I. Rigorous bounds on the complex bulk modulus[END_REF] (see also formula (19.3) of [START_REF] Milton | The Theory of Composites[END_REF]) yields the following formula for the Y -tensor defined by (4.2) we deduce from the first (2 × 2) block of (4.60) the relation (4.17).

C -1 1 C 2 P C T 2 C -1 1 =       t 2 2 t 2 1 p 1 0 0 0 t 2 2 t 2 1 p 1 0 0 0 0       , C -1 1 C T 2 P =      0 - t 2 t 1 p 1 0 t 2 t 1 p 1 0 0 0 0 0      . ( 4
Y * = -f 2 σ 1 -f 1 σ 2 + f 1 f 2 (σ 1 -σ 2 ) (f 1 σ 1 + f 2 σ 2 -σ * ) -1 (σ 1 -σ 2 ) .
On the other hand, by (4.61) the formula (4.5) for Y * reads as

Y * =                 1 a Y 0 0 0 - c Y a Y 0 0 1 a Y 0 c Y a Y 0 0 0 0 1 b Y 0 0 0 0 c Y a Y 0 a Y + c 2 Y a Y 0 0 - c Y a Y 0 0 0 a Y + c 2 Y a Y 0 0 0 0 0 0 b Y                 . ( 4

. 15 )

 15 Combining(3.14) and(3.15) gives the desired bound (3.3).

Theorem 4 . 1 .

 41 Assume that the composite is geometrically isotropic. Then, in view of definitions (4.15)-(4.18) the coefficients a * , c * of the effective conductivity σ * (4.13) satisfy the Hashin-Shtrikman type bounds

. 21 ) 4 . 2 .Remark 4 . 3 .

 214243 RemarkIn the a Y -c Y plane the bounds (4.[START_REF] Hall | On a new action of the magnet on electric currents[END_REF]) correspond to the intersection of two disks parametrized by α ± , which are tangent to the c Y -axis. Due to definition (4.17) these bounds remain the same if we replace c 1 , c 2 , -c Y by c 1 + c 0 , c 2 + c 0 , -c Y + c 0 . This reflects the fact if we add a antisymmetric matrix to the local conductivity σ, then the same antisymmetric matrix is added to σ * . Also note that if c 1 = c 2 = c * = c, then c Y = -c. With the change c Y to -c Y , the circle HS ± satisfying the equality in (4.19

  due to the transverse isotropy of σ i , for i = 1, 2, * , we haveY * =   a Y -c Y 0 c Y a Y 0 0 0 b Y   . (4.61)This relation also separates into blocks, so that we obtain for the 33 entry of Y * the relation(4.18). Moreover, making the correspondence a -c c a ←→ a + i c, (4.62)

  Therefore, passing to the limit as d 1 → 0, or equivalently t 3 → t 2 2 /t 1 , (4.51) and (4.52) imply that

	Then, putting (4.50) and (4.56) in the S 2 -average of (4.47) we get that
		Γ S 2 =	               r 1 + 0 t 2 2 t 2 1 0 0 -0 t 1 p 1 p 1 t 2	0 r 1 + t 2 2 t 2 1 0 0 0 t 1 p 1 t 2	0 p 1 0 r 2 0 0 0	0 t 2 p 1 t 1 0 0 0 p 1	-	t 2 t 1 0 0 0 p 1 0	p 1 0 0 0 p 2 0 0	              	,	(4.57)
	which gives	1 Γ S 2	=	                 	1 r 1 0 0 0 -t 2 t 1 r 1 0	0 1 r 1 0 t 2 t 1 r 1 0 0	0 0 1 r 2 0 0 0	0 t 2 t 1 r 1 0 + -1 p 1 t 2 2 t 2 0 0 1 r 1	1 p 1	t 2 t 1 r 1 0 0 + t 2 0 1 r 1 2 t 2 0	0 0 0 p 2 1 0 0	                 	(4.58)
			1 Γ S 2	=	                	1 r 1 0 0 0 -t 2 t 1 r 1 0	0 1 r 1 0 t 2 t 1 r 1 0 0	0 0 -1 r 2 0 t 2 0 t 2 t 1 r 1 0 2 0 0 0 0 1 r 1 t 2	t 2 t 1 r 1 0 0 2 t 2 0 1 r 1 t 2 0	0 0 0 t 5 0 0	                	.	(4.59)
														.56)

  .63) Then, the bound (4.11) applied with the formulas (4.63) for Y * , (4.24) for L 0 and (4.59) for ( Γ S 2 ) -1 , combined with (4.28), (4.54), and (4.55), implies that Due to (4.30) we have s 1 s 3 = s 2 2 . Therefore, similarly to (4.26) and (4.29) the inequality (4.65) can be written as

					1 b Y	+	1 b 1	≥	1 r 2	=	1 b 1 1 -g(b 1 t 1 )	, b Y ≥ 0,	(4.64)
	and					det	   	1 a Y c Y a Y	-s 1 + s 2 a Y + c Y a Y	+ s 2 c 2 Y a Y -s 3	    ≥ 0,	(4.65)
	where
	     	s 1 :=	1 r 1	-t 1	= t 1			2 1 + g(b 1 t 1 )	-1 ≥ 0 (since 0 < g(b 1 t 1 ) < 1)
	
	   	s 2 :=	t 2 t 1 r 1	-t 2 = t 2			2 1 + g(b 1 t 1 )	-1	(4.66)
	     	s 3 :=	t 2 2 t 2 1 r 1	-t 3 .
								a 2 Y + c Y +	s 2 s 1	2	-	a Y s 1	≤ 0.	(4.67)
	Finally, taking into account (4.16), (4.36), (4.66) the inequalities (4.67) and (4.64) correspond
	respectively to the desired bounds (4.19), (4.21). Theorem 4.1 is proved.

Acknowledgements. GWM is grateful for support from the Mathematical Sciences Research Institute and from National Science Foundation through grant DMS-0707978.


F James

N Vauchelet
email: vauchelet@ann.jussieu.fr

A remark on duality solutions for some weakly nonlinear scalar conservation laws

We investigate existence and uniqueness of duality solutions for a scalar conservation law with a nonlocal interaction kernel. Following [3], a notion of duality solution for such a nonlinear system is proposed, for which we do not have uniqueness. Then we prove that a natural definition of the flux allows to select a solution for which uniqueness holds.

Introduction

At a continuous level, many physical or biological systems are modelled thanks to scalar conservation laws. In this note we will focus on a weakly nonlinear system of the kind :

∂ t ρ + ∂ x (a(u)ρ) = 0, ∂ x u = ρ, (1.1)
where a is a given smooth function, a ∈ C 0 (R). This system is complemented with the initial data ρ(t = 0) = ρ 0 . We notice that we can rewrite (1.1) as a single equation since we have u = H * ρ where H is the Heaviside function and we recover the so-called non local aggregation equation. This model arises in several applications in physics and biology where a self-consistant interaction field u governs the evolution of a density of population ρ. Then u is defined as u = -∂ x φ where φ is the interaction potential. For instance, in the modelling of cell movement by chemotaxis, φ is the concentration of some chemical called chemo-attractant (when a is non-increasing) or chemo-repellent (when a is non-decreasing) which drives the dynamics of individuals (bacteria). In gas dynamics, this model can be derived thanks to a high-field limit from the Vlasov-Poisson-Fokker-Planck system [START_REF] Nieto | High field limit for Vlasov-Poisson-Fokker-Planck equations[END_REF], a nonincreasing (resp. nondecreasing) a corresponds the to the repulsive (resp. attractive) case. From a mathematical viewpoint, it is well-known that in the attractive case, i.e. when a is non-increasing, finite time blow-up of regular solutions for such system occurs (see e.g. [START_REF] Bertozzi | L p theory for the multidimensional aggregation equation[END_REF] and references therein). Therefore one has to look for solutions ρ which are measure-valued in space, which generates several difficulties, because the velocity a(u) turns out to be discontinuous, so that the product in the divergence term is not well defined, and the corresponding flow has to be defined cautiously. A recent approach consists in using techniques from optimal transport, see [START_REF] Carrillo | Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations[END_REF]. Another possibility is to define a priori the product. For the Vlasov-Poisson system, this has been done in [START_REF] Nieto | High field limit for Vlasov-Poisson-Fokker-Planck equations[END_REF].

The aim of this note is to interpret (2.2) as a linear conservation equation solved in the duality sense [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF], the product being defined afterwards, following the strategy introduced in [START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservation laws and uniqueness[END_REF] for pressureless gases. Therefore we recall in the next section the notion of duality solutions and some useful results. In Section 3 we state and prove the main result concerning existence and uniqueness of duality solutions of system (1.1). Section 4 is devoted to some examples of applications of this result.

Duality solutions for linear equations

The notion of duality solutions was introduced in [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF] to give a sense to linear conservation equations

∂ t ρ + ∂ x (bρ) = 0, (2.2)
when the coefficient b can be discontinuous but satisfies the so-called one-sided Lipschitz (OSL) condition

∂ x b(t, .) ≤ β(t) for β ∈ L 1 (0, T ) in the distribution sense. (2.3)
Duality solutions are defined as weak solutions, the test functions being specific Lipschitz solutions to the backward linear transport equation

∂ t p + b(t, x)∂ x p = 0, p(T, .) = p T ∈ Lip(R).
(2.4)

Definition 2.1 1.
We say that a Lipschitz solution p to (2.4) is a reversible solution if p is locally constant on the set

V e = (t, x) ∈ [0, T ] × R; ∃ p e ∈ E, p e (t, x) = 0 . 2. We say that ρ ∈ C([0, T ]; M loc (R) -σ(M loc , C c )) is a duality solution to (2.2) if for any 0 < τ ≤ T , and any reversible solution p to (2.4) with compact support in x, the function t → R p(t, x)ρ(t, dx) is constant on [0, τ ].
The most important facts for our purpose concerning duality solutions are gathered in the following theorem.

Theorem 2.2 (Bouchut, James [2]) 1. Given ρ • ∈ M loc (R), under the assumptions (2.3), there exists a unique ρ ∈ C([0, +∞[, M loc (R)), duality solution to (2.2), such that ρ(0, .) = ρ • . Moreover, if ρ • is nonnegative, then ρ(t, •) is nonnegative for a.e. t ≥ 0. And we have the mass conservation |ρ(t, •)|(R) = |ρ • |(R), for a.e. t ∈]0, T [.
2. Backward flow and push-forward: the duality solution satisfies

∀ t ∈ [0, T ], ∀ φ ∈ C c (R), R φ(x)ρ(t, dx) = R φ(X(t, 0, x))ρ 0 (dx), (2.5)
where the backward flow X is defined as the unique reversible solution to

∂ t X + b(t, x)∂ x X = 0 in ]0, s[×R, X(s, s, x) = x.
3. There exists a bounded Borel function b, called universal representative of b, such that b = b almost everywhere, and for any duality solution ρ,

∂ t ρ + ∂ x ( bρ) = 0 in the distributional sense. (2.6)
Remark 2.3 A similar notion of duality solution for the transport equation is available ∂ t u + b∂ x u = 0, and ρ is a duality solution of (2.2) iff u = x ρ is a duality solution to transport equation (see [START_REF] Bouchut | One-dimensional transport equations with discontinuous coefficients[END_REF]).

We shall need also the following result whose proof can be found in [START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservation laws and uniqueness[END_REF] (Theorems 3.1 and 3.2)

Theorem 2.4 Let f ∈ C 1 (R).
Let M be an entropy solution to the conservation equation

∂ t M + ∂ x f (M) = 0,
with nondecreasing initial datum M 0 . Then ρ := ∂ x M is a duality solution to

∂ t ρ + ∂ x (bρ) = 0
where we can choose b = f ′ (M) a.e. Moreover, for all t ∈]0, +∞[, ∂ x b ≤ 1/t and its universal representative b satisfies ∂ x f (M) = b∂ x M.

Duality solutions for weakly nonlinear equations

We introduce the following notion of duality solution for the coupled system (1.1), inspired by the strategy used in [START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservation laws and uniqueness[END_REF] for pressureless gases (see also section 4.2 below). We underline at once the fact that this definition does not lead to uniqueness as it stands. Indeed, assume that a is a non-increasing C 1 function on R and take for initial data ρ 0 = δ x 0 , a Dirac measure in x 0 ∈ R. Looking for a solution as a Dirac mass δ x 1 (t) , thanks to Remark 2.3 we solve the transport equation with coefficient a H(x -x 1 (t)) , where H denotes the Heaviside function. Then δ x 1 (t) , H(x -x 1 (t)) is a duality solution of (1.1) in the sense of Definition 3.1, provided that x 1 (0) = x 0 and that the admissibility condition a(1) < x ′ 1 (t) < a(0) holds. Thus we have an infinite family of duality solutions.

Definition 3.1 We say that (ρ, u) is a duality solution of (1.1) on ]0, T [ if there exists a bounded Borel function b with ∂ x b ≤ α ∈ L 1 loc (0, T ) such that 1. for all 0 < t 1 < t 2 < T , ∂ t ρ + ∂ x (bρ) = 0 in the sense of duality on ]t 1 , t 2 [,
Therefore the main result of this note is to explain how a more precise description of the product bρ in the scalar conservation equation allows to recover uniqueness. It is actually given in a very naive way by writing a(u)ρ = a(u)∂ x u = ∂ x A(u), where A is an antiderivative of a. This choice can be justified in a more rigourous way when the system (1.1) is obtained as the hydrodynamic limit of a kinetic system, as it is the case both in [START_REF] James | On the hydrodynamical limit for a one dimensional kinetic model of cell aggregation by chemotaxis[END_REF] and [START_REF] Nieto | High field limit for Vlasov-Poisson-Fokker-Planck equations[END_REF]. It turns out that the previous formal computation is correct at the kinetic level, so that the flux J ε := ξf ε (ξ) dξ, where f ε is the distribution function of particles, actually converges to J = ∂ x A(u), which defines the flux of the conservation equation. The point now is to justify that this can be used to solve the conservation equation in the duality sense. Theorem 3.2 Let ρ 0 ∈ M loc (R), ρ 0 ≥ 0. There exists a unique duality solution (ρ, u) to the non local interaction equation (1.1) in the sense of Definition 3.1, which satisfies bρ = ∂ x (A(u)) where A is an antiderivative of a.

Moreover, if a is a non-increasing function (i.e. in the attractive case), there exists a flow associated to a(u), in other words, there exists a Lipschitz function X such that ρ(t) = X(t) # ρ 0 .

Proof. Let us denote by u the entropy solution of equation

∂ t u + ∂ x A(u) = 0, (3.7)
with initial data u 0 := x ρ 0 (dx). From Theorem 2.4, ρ := ∂ x u is a duality solution of ∂ t ρ + ∂ x (bρ) = 0 where we can choose b = a(u) almost everywhere and it is the unique solution satisfying bρ = ∂ x A(u). Indeed if we have two such nonnegative solutions ρ 1 = ∂ x u 1 and ρ 2 = ∂ x u 2 , then u 1 and u 2 are monotonous solutions of (3.7). Thus they are entropy solutions of this scalar equation and u 1 = u 2 .

In the attractive case, the entropy solution u is nondecreasing. Therefore ρ := ∂ x u is nonnegative and ∂ x a(u) = a ′ (u)∂ x u ≤ 0 since in the attractive case a is non-increasing. Thus the velocity field a(u) satisfies the OSL condition (2.3) and from Theorem 2.2 (ii) there exist a backward flow X such that (2.5) is satisfied.

In the general case, we can apply the classical Oleinik entropy condition and get that ∂ x b ≤ 1/t. Then the solution is defined on all ]t 1 , t 2 [ for 0 < t 1 < t 2 < T and the flow cannot be defined up to 0.

Examples

Positive chemotaxis

Equation (1.1) for a non-increasing function a can be obtained from a hydrodynamical limit of a kinetic model describing positive chemotaxis (see e.g. [START_REF] Dolak | Kinetic models for chemotaxis : Hydrodynamic limits and spatiotemporal mechanisms[END_REF][START_REF] James | On the hydrodynamical limit for a one dimensional kinetic model of cell aggregation by chemotaxis[END_REF]). Thus from Theorem 3.2, there exists a flow X such that ρ = X # ρ 0 . Let us first come back to the example in subsection 3.1: we assume that a is a given non-increasing C 1 function and take ρ 0 = δ x 0 . Then we solve the Riemann problem

∂ t u + ∂ x A(u) = 0, u(t = 0, x) = H(x -x 0 ),
where A is a concave function. Then the entropy solution is given by u(t, x) = H(x -x 1 (t)) where the Rankine-Hugoniot condition implies x ′ 1 (t) = A(1) -A(0). Thus the unique duality solution in the sense of Theorem 3.2 is given by (ρ, u) = (δ x 1 (t) , H(x -x 1 (t))) where x 1 (t) = x 0 + (A(1) -A(0))t.

On the other hand, if we look for a solution in the form ρ(t

) = δ x 1 (t) , then u(t, x) = H(x -x 1 (t)). Integrating equation (2.6) we get that bρ = -∂ t u = x ′ 1 (t)δ x 1 (t)
. By deriving in the distribution sense A(u), we get that the definition of the product in Theorem 3.

2 bρ = ∂ x A(u) is satisfied if and only if x ′ 1 (t) = A(1) -A(0)
, thus we recover the Rankine-Hugoniot condition. Hence the definition of the product allows to select one solution among those found in subsection 3.1. It gives more generally the dynamic of aggregates, which are modelled by a sum of Dirac masses ρ 0 = m i δ x i . A similar computation gives the velocity x ′ i (t) = A( i m j ) -A( i-1 m j ) /m i . Notice that the velocity of each aggregate is defined by a local equation, despite the fact that the initial equation is non local. In the particular case where A is strictly concave, aggregates collapse in finite time.

High field limit of Vlasov-Poisson-Fokker-Planck

In [START_REF] Nieto | High field limit for Vlasov-Poisson-Fokker-Planck equations[END_REF], the authors prove that solutions to the Vlasov-Poisson-Fokker-Planck system converge in the high field limit to solutions of (1.1) where a(u) = u in the repulsive case and a(u) = -u in the attractive case. To do so, the authors define a weak product ρu, which can be proved to coincide with the one used here. Applying Theorem 3.2 we can recover the result stated in Theorem 2 of [START_REF] Nieto | High field limit for Vlasov-Poisson-Fokker-Planck equations[END_REF] : there exists a unique global in time solution of (1.1) in the distribution sense such that the product ρu = ±u 2 /2. Moreover, in the attractive case, there exists a flow X such that ρ = X # ρ 0 , and the dynamics of aggregates is similar to the one of chemotaxis. In the general case, the Oleinik entropy condition gives that ρ ≤ 1/t. Finally, we notice that the result of [START_REF] Nieto | High field limit for Vlasov-Poisson-Fokker-Planck equations[END_REF] has been extended in two dimensions by Poupaud in [START_REF] Poupaud | Diagonal defect measures, adhesion dynamics and Euler equations[END_REF] by using defect measures to define the product of ρ by u. However, there is no uniqueness of solutions.

To conclude, we focus on the connection between pressureless gases and the Vlasov-Poisson-Fokker-Planck limit which is mentioned in [START_REF] Nieto | High field limit for Vlasov-Poisson-Fokker-Planck equations[END_REF]. The pressureless gases system reads

∂ t ρ + ∂ x (ρv) = 0, ∂ t (ρv) + ∂ x (ρv 2 ) = 0. ( 4.8)
Bouchut and James in [START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservation laws and uniqueness[END_REF] introduced the notion of duality solution to (4.8):

Definition 4.1 We say that a couple (ρ, q), ρ, q ∈ C([0, T [; M loc (R)), ρ ≥ 0, is a duality solution to (4.8) if there exists b ∈ L ∞ (]0, T [×R) and α ∈ L 1 loc (]0, T [) satisfying ∂ x b ≤ α in ]0, T [×R such that 1. For all 0 < t 1 < t 2 < T , we have in the sense of duality on ]t 1 , t 2 [×R ∂ t ρ + ∂ x (bρ) = 0, ∂ t q + ∂ x (bq) = 0; 2. bρ = q.
The existence result for the Cauchy problem with initial data (ρ 0 , q 0 ) strongly exploits the relationships between (4.8) and the conservation law ∂ t u + ∂ x A(u) = 0, where ρ = ∂ x u, q = ∂ x A(u) and A is determined by ρ and q. Uniqueness follows if A can be defined by the initial data, which enforces additional conditions on (ρ 0 , q 0 ). In the context of Vlasov-Poisson-Fokker-Planck, the function A is given: A(u) = -u 2 /2 (attractive case) or A(u) = u 2 /2 (repulsive case). Therefore we propose the following variant to the results of [START_REF] Bouchut | Duality solutions for pressureless gases, monotone scalar conservation laws and uniqueness[END_REF]. Theorem 4.2 Let ρ 0 ∈ M loc (R), ρ 0 ≥ 0 and A ∈ C 1 (R). Define q 0 = ∂ x A(u 0 ), where u 0 = x ρ 0 . Then there exists a duality solution (ρ 0 , q 0 ) of the pressureless gases system (4.8) in the sense of Definition 4.1. Moreover this solution is the unique duality solution which satisfy the relation bρ = ∂ x [A(u)], where u =

x ρ(dx).

This theorem is proved in the same way as Theorem 3.2. The solution to the high-field limit of the Vlasov-Poisson-Fokker-Planck system obtained in [START_REF] Nieto | High field limit for Vlasov-Poisson-Fokker-Planck equations[END_REF] is therefore the unique duality solution to (4.8) given by Theorem 4.2 for the corresponding A.

2 .

 2 we have ∂ x u = ρ in the weak sense, 3. b = a(u) almost everywhere.


Sandrine Larue

Thierry Maisonobe

Olivier Benveniste

Catherine Chapelon-Abric

Olivier Lidove

Thomas Papo

Bruno Eymard

Odile Dubourg
email: odile.dubourg@gmail.com

DISTAL MUSCLE INVOLVEMENT IN GRANULOMATOUS MYOSITIS CAN MIMIC INCLUSION BODY MYOSITIS

Keywords: inflammatory myopathy, inclusion body myositis, granulomatous myositis, sarcoidosis

We report on four aged patients with chronic myopathy suggestive of sporadic inclusion body myositis. They presented progressive and selective weakness of the quadriceps femoris muscles. Asymmetric and selective atrophy of the forearm muscles was noted in all, with more severe involvement of the flexors than the extensors. Biopsy revealed granulomatous myositis. Histological features of s-IBM were lacking. Evidence for systemic sarcoidosis was found in one patient. Corticosteroid treatment was associated with partial but significant improvement in two patients. Granulomatous myositis may mimic inclusion body myositis and may be steroid-responsive.

Introduction

Sporadic IBM (s-IBM) is the most common acquired myopathy in patients over the age of 50, and involvement of the quadriceps and deep finger flexors are clues to its early diagnosis. [START_REF] Dalakas | Sporadic inclusion body myositis-diagnosis, pathogenesis and therapeutic strategies[END_REF] In contrast, granulomatous myositis (GM) is rare and found only in 0.4% of all muscle biopsy specimens. [START_REF] Prayson | Granulomatous myositis. Clinicopathologic study of 12 cases[END_REF] We report on four aged patients with chronic myopathy suggestive of s-IBM, in whom muscle biopsy showed GM. Our four patients presented with progressive and relatively selective weakness of the quadriceps femoris muscles. Asymmetric and selective atrophy of the forearm muscles was also noted in all these patients, with more severe involvement of the flexors than the extensors. A diagnosis of s-IBM was made prior to muscle biopsy. Surprisingly, granulomas were found in muscle biopsy specimens obtained from these four patients, whereas no classical features of s-IBM could be identified.

Patients and Methods

Patients. The patients were three women and one man, aged between 76 and 84 at the time of muscle biopsy. None of them had a past history of sarcoidosis. Muscle biopsy was performed on extensor carpi radialis in all patients. This muscle is easily accessible and often informative, in our experience, for a diagnosis of s-IBM, which was first suspected.

Muscle Pathology. Muscle samples were conventionally processed for light microscopy using standard procedures. [START_REF] Dubowitz | Muscle biopsy: A practical approach[END_REF] Frozen sections were stained with hematoxylin-eosin (H&E), modified Gomori trichrome, Oil-red-O, periodic acid-Schiff (PAS), Congo Red and histoenzymatic reactions including NADH-tetrazolium reductase, succinate dehydrogenase, cytochrome C oxidase were performed. Several muscle fragments were paraffin-embedded and sections were stained with H&E.

Expression of major histocompatibility complex (MHC)-1 (Dakocytomation, Glostrup, Denmark), CD3, CD4, CD8, CD20, CD68 (Dakocytomation, Glostrup, Denmark), and SMI-31 (SMI-31R, Covance) were evaluated by immunoperoxidase assay performed on frozen sections, using an automated immunostainer (Ventana®, Tucson, Az). Sporadic IBM was excluded in all patients by the absence of rimmed vacuoles in muscle fibers, the absence of amyloid deposits on Congo Red staining viewed under a Rhodamine red filter, and the absence of tubulo-filamentous inclusions with SMI-31 immunohistochemistry. [START_REF] Engel | Inclusion-body myositis: clinical, diagnostic, and pathologic aspects[END_REF] Biopsy findings for all patients are described in the table and illustrated in the figure.

Case Reports

Patient 1: A 77-year-old woman presented with a six-year history of progressive difficulty in climbing stairs and steppage of the left foot. On examination, she had marked atrophy of both quadriceps and weakness of the left ankle dorsiflexor. Asymmetric and selective atrophy of her forearm flexor muscles was also noted. She had no dysphagia, myalgia, or skin rash.

There was no family history of neurological disease and no relevant prior disease. A full workup, including complete blood cell count, erythrocyte sedimentation rate, antinuclear antibodies, biochemistry profile, immunoelectrophoresis, B12 vitamin, folate, angiotensinconverting enzyme, and thyroid-stimulating hormone was carried out to exclude possible associated systemic disorders, and was completely normal. The serum creatine kinase (CK) level was also normal. Electromyographic studies (EMG) revealed asymptomatic sensory neuropathy, and upon needle detection, myogenic motor unit potentials (MUPs) in affected muscles. Muscle biopsy revealed numerous giganto-epithelioid noncaseating granulomas invading the endomysial tissue. Muscle tissue was variably affected, either normal in some areas or very remodeled in others by a fibro-adipose involution containing atrophic and necrotic fibers and granulomatous lesions. Pulmonary sarcoid disease was suspected on chest computerized tomography (CT), since hilar adenopathies were identified. Bronchial biopsy confirmed the presence of granulomas. After starting prednisone therapy (1mg/kg/day), muscular strength in the proximal lower limbs progressively improved during the first four months, after which her clinical condition stabilized. One year later, while prednisone had been progressively reduced to 10 mg/day, weakness reoccurred. The prednisone dose was then increased to the previous level and methotrexate introduced, which restored the patient's clinical status to what it was prior to prednisone reduction. However, review of the patient's medical records revealed that she presented bladder polyps eight years before, for which she had been treated with Bacillus Calmette Guérin (BCG) therapy for two years. While under treatment, bilateral uveitis developed, for which antituberculosis therapy was administered during one year, with no efficacy. A sarcoidosis workup was then carried out and uveitis remained isolated. Interferon alpha-2A treatment was then tried and effective against the uveitis. However, two months after the start of interferon therapy, the patient developed myalgia and muscular weakness, as mentioned above. Muscle biopsy was performed 12 months after interferon introduction, revealing muscle fiber necrosis, inflammation, and giganto-epithelioid granulomas. This led to the assumption that interferon treatment had caused the GM. [START_REF] Leclerc | Sarcoidosis and interferon therapy: report of five cases and review of the literature[END_REF] Indeed, shortly after interferon was withdrawn, the myalgia disappeared and muscular strength slowly improved. Corticosteroids were not applied.

Discussion

As opposed to other inflammatory myopathies in which weakness is mostly proximal, in s-IBM, distal muscles are involved. [START_REF] Briani | Update on idiopathic inflammatory myopathies[END_REF] Early in the course of the disease, weakness and atrophy is typically observed in the quadriceps femoris and forearm flexors, often asymmetric. Sporadic IBM is not only the most frequent inflammatory myopathy encountered in patients aged 50 and over, but also the most likely diagnosis when a patient over 50 complains of falls and has atrophic thighs. [START_REF] Dalakas | Sporadic inclusion body myositis-diagnosis, pathogenesis and therapeutic strategies[END_REF] GM is clearly not as frequent and usually present with bilateral and symmetrical weakness of proximal lower limbs. [START_REF] Roux | Granulomatous myositis: a clinical study of thirteen cases[END_REF] Distal involvement may be observed as the disease develops, but is rarely present at the outset. [START_REF] Roux | Granulomatous myositis: a clinical study of thirteen cases[END_REF] It is therefore not surprising that our patients were first diagnosed with s-IBM. Histological diagnosis of s-IBM is challenging, since the distribution of muscle abnormalities is patchy and the complete biopsy criteria may be missing even from clinically affected muscles. [START_REF] Chahin | Correlation of muscle biopsy, clinical course, and outcome in PM and sporadic IBM[END_REF] Similarly, granuloma in muscle may be easily overlooked unless numerous paraffin sections are examined. Thorough histological investigation of frozen and paraffin sections of several muscle fragments are therefore essential for diagnosis in both situations.

Few cases of association between sarcoidosis or other autoimmune disorders and s-IBM have been described. [START_REF] Bouillot | Inclusion body myositis associated with sarcoidosis: a report of 3 cases][END_REF][START_REF] Vattemi | Sarcoidosis and inclusion body myositis[END_REF][START_REF] Danon | Inclusion body myositis associated with systemic sarcoidosis[END_REF][START_REF] Badrising | Associations with autoimmune disorders and HLA class I and II antigens in inclusion body myositis[END_REF] In our muscle biopsy specimen, we did not detect amyloid or tubulofilamentous inclusions, which are markers of degenerative changes in s-IBM, [START_REF] Engel | Inclusion-body myositis: clinical, diagnostic, and pathologic aspects[END_REF] leading us to conclude that our patients just had GM.

The classically held view of s-IBM is that it clinically differs from other types of inflammatory myopathies by its male preponderance, muscle weakness, atrophy topography, and protracted course. Resistance to treatment with conventional forms of immunotherapy is also a feature of s-IBM. [START_REF] Dalakas | Sporadic inclusion body myositis-diagnosis, pathogenesis and therapeutic strategies[END_REF] On the other hand, GM may be significantly improved by steroids, although such response is unpredictable. Discrepancies among various studies may be due to the different proportions of acute and chronic myositis cases included, since the former has a better response to steroids and a more benign course. [START_REF] Roux | Granulomatous myositis: a clinical study of thirteen cases[END_REF] In our patients 1 and 3, who suffered from chronic myopathy, steroids were efficient.

The prevalence of granulomas in muscle biopsies is low, and sarcoidosis is known to be the single most commonly identifiable cause. [START_REF] Prayson | Granulomatous myositis. Clinicopathologic study of 12 cases[END_REF] The prevalence of asymptomatic GM in sarcoidosis has been reported to be as high as 50-80%. [START_REF] Silverstein | Muscle involvement in sarcoidosis. Asymptomatic, myositis, and myopathy[END_REF] Symptomatic involvement is clearly less common, and includes various types: nodular, acute myositis, or the more common, chronic myopathy, as in patient 1, which is slower in onset and occurs later in life. [START_REF] Roux | Granulomatous myositis: a clinical study of thirteen cases[END_REF] GM may also occur without systemic involvement. Although muscle pathology is similar to that seen in sarcoidosis, isolated GM is usually considered a separate clinicopathological entity. However, in most series and case reports, it remains unclear whether GM simply reflects an incomplete sarcoidosis work-up. [START_REF] Roux | Granulomatous myositis: a clinical study of thirteen cases[END_REF] It is possible to speculate why some patients with GM present with a similar clinical pattern to s-IBM. It could be related to the level of expression of IL-1beta, which is high in sarcoid myopathy granuloma and s-IBM. [START_REF] Authier | Interleukin-1 expression in inflammatory myopathies: evidence of marked immunoreactivity in sarcoid granulomas and muscle fibres showing ischaemic and regenerative changes[END_REF][START_REF] Schmidt | Interrelation of inflammation and APP in sIBM: IL-1 beta induces accumulation of beta-amyloid in skeletal muscle[END_REF] In spite of the age of the patients with GM in this report, the mechanism of ageing could also contribute to the unique pattern of affected muscles in both diseases.

Regardless whether granulomatosis is systemic or limited to muscles, it can occur late in life and manifests as a chronic myopathy strongly suggestive of s-IBM. Corticosteroids should be

Patient 2 :Patient 4 :

 24 An 84-year-old woman with a two-year history of progressive atrophy of both quadriceps and the left deep finger flexors. The blood workup was normal, except for a slightly elevated CK level (twofold the normal value.) There was no family history of neurological disease or relevant prior disease. Nerve conduction study revealed discrete sensory neuropathy and needle examination demonstrated myogenic MUPs in the atrophied muscles. Muscle biopsy showed signs of GM with variation in fiber size, necrosis and regeneration, diffuse inflammatory cells infiltrate invading non-necrotic fibers, and typical giganto-epithelioid granulomas. However, a sarcoidosis work-up was not carried out, since the patient refused further clinical investigation and declined follow-up. Patient 3: An 81-year-old man who complained of frequent falls since age 74 related to weakness of his right anterior tibial muscle. At age 77, he presented with weakness and atrophy of both quadriceps muscles and the left forearm flexor muscles. Nerve conduction study demonstrated asymptomatic sensory neuropathy, and needle examination revealed myogenic MUPs in the atrophied muscles. Muscle biopsy revealed multiple and voluminousgranulomatous lesions invading the endomysial connective tissue. The complete blood workup was normal, as were the chest CT, pulmonary function tests, and cerebral and lumbar magnetic resonance imaging (MRI).The patient was treated with corticosteroids (1mg/kg/day), with significant muscular strength improvement during the first six months of treatment, at the end of which he presented with acute hyponatremia related to a strict low sodium diet and diuretic treatment for hypertension, leading to severe weakness of all four limbs. Three months after adding methotrexate to the corticosteroids, the patient recovered the strength he had prior to electrolyte disequilibrium. His clinical condition has since remained stable. Over the past ten months, a 76-year-old woman presented progressive atrophy and weakness of both quadriceps muscles, anterior tibial muscles, and forearm flexors, suggestive of s-IBM. The complete blood workup was normal except for a moderate elevation of CK (1.3 the normal value.) The EMG demonstrated myogenic MUPs in the atrophied muscles.

Figure legend

 legend Figure legend

Acknowledgements: This work was supported by the Association Française contre les Myopathies (AFM), by the Assistance Publique des Hôpitaux de Paris (APHP), and the Université Pierre et Marie Curie.

tried whenever GM is symptomatic. Although it is rare, recognizing GM remains important from both the prognostic and therapeutic points of view.