Closure of the set of diffusion functionals with respect to the Mosco-convergence.

M. CAMAR-EDDINE AND P. SEPPECHER

Laboratoire d'Analyse Non Linéaire Appliquée et Modélisation

Université de Toulon et du Var

BP 132 - 83957 La Garde Cedex

Email: seppecher@univ-tln.fr

Abstract:

We characterize the functionals which are Mosco-limits, in the ${{\rm L}^2(\Omega )}$ topology, of some sequence of functionals of the kind

$$F_n(u):=\int_\Omega \alpha _n(x) \vert\nabla u(x)\vert^2 dx ,$$

where $\Omega$ is a bounded domain of $\mathbb{R}^N$ ($N\geq3$). It is known that this family of functionals is included in the closed set of Dirichlet forms. Here, we prove that the set of Dirichlet forms is actually the closure of the set of diffusion functionals. A crucial step is the explicit construction of a composite material whose effective energy contains a very simple non-local interaction.

keywords : Homogenization, Mosco-convergence, $\Gamma$-convergence, Dirichlet forms, composite materials.