Asymptotic behaviour of equicoercive diffusion energies in dimension two
Résumé
In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F-n, n is an element of N, defined in L-2(Omega), for a bounded open subset Omega of R-2. We prove that, contrary to the dimension three (or greater), the Gamma-limit of any convergent subsequence of F-n is still a diffusion energy. We also provide an explicit representation formula of the Gamma-limit when its domains contains the regular functions with compact support in Omega. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence F-n, which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity.