This paper is devoted to the asymptotic behaviour of quadratic forms defined on ${L}^{2}$. More precisely we consider the $\mathrm{\Gamma}$-convergence of these functionals for the ${L}^{2}$-weak topology. We give an example in which some limit forms are not Markovian and hence the Beurling-Deny representation formula does not hold. This example is obtained by the homogenization of a stratified medium composed of insulating thin-layers.

We study the corrector matrix ${P}^{\u03f5}$ to the conductivity equations. We show that if ${P}^{\u03f5}$ converges weakly to the identity, then for any laminate $det{P}^{\u03f5}\ge 0$ at almost every point. This simple property is shown to be false for generic microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear]. In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal. 158 (2001) 155-171]. We use this...

We study the corrector matrix ${P}^{\epsilon}$ to the conductivity equations. We show
that if ${P}^{\epsilon}$ converges weakly to the identity, then for any laminate
$det{P}^{\epsilon}\ge 0$ at almost every point. This simple property is shown to be false for
generic microgeometries if the dimension is greater than two in the work Briane [, to appear].
In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [
(2001) 155-171]. We use this property of laminates to prove that, in any...

In this paper, a ${W}^{-1,{N}^{\text{'}}}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on ${\mathbb{R}}^{N}$, or on a regular bounded open set of ${\mathbb{R}}^{N}$. The proof is based partially on the Strauss inequality [Strauss, 23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [ 9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of a measure, and to prove an...

In this paper we prove a H-convergence type result for the homogenization of systems the coefficients of which satisfy a functional ellipticity condition and a strong equi-integrability condition. The equi-integrability assumption allows us to control the fact that the coefficients are not equi-bounded. Since the truncation principle used for scalar equations does not hold for vector-valued systems, we present an alternative approach based on an approximation result by Lipschitz functions due to...

In this paper, a ${W}^{-1,{N}^{\text{'}}}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on ${\mathbb{R}}^{N}$, or on a regular bounded open set of ${\mathbb{R}}^{N}$. The proof is based partially on the Strauss inequality [Strauss,
(1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [
(2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of...

In this paper we study the realizability of a given smooth periodic gradient field ∇ defined in R, in the sense of finding when one can obtain a matrix conductivity such that ∇ is a divergence free current field. The construction is shown to be always possible locally in R provided that ∇ is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both...

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