Asymptotic behaviour of equicoercive diffusion energies in dimension two

Abstract

In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies Fn, n ∈ N, defined in L2(Ω), for a bounded open subset Ω of R2. We prove that, contrary to the three dimension (or greater), the Γ-limit of any convergent subsequence of Fn is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence Fn, 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.