This paper is devoted to the asymptotic behaviour of quadratic forms defined on . More precisely we consider the -convergence of these functionals for the -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 to the conductivity equations. We show that if converges weakly to the identity, then for any laminate 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 to the conductivity equations. We show that if converges weakly to the identity, then for any laminate 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 estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on , or on a regular bounded open set of . 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 estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on , or on a regular bounded open set of . 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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