A diffusion equation through a highly heterogeneous medium

Abstract

We study an elliptic degenerate equation in the framework of the periodic homogenization in a highly heterogeneous medium. Denoting by ϵ the size of the period, let F ϵ be the set of the cylindrical fibres with thickness of order ϵ and let M ϵ be the matrix in such a way that the cube Ω can be written as Ω = F ϵ ∪ M ϵ. We assume that the ratio between the magnitude of the diffusion outside and inside the fibres behaves as with some α > 0. The diffusion equation is completed by Dirichlet condition on the lower and upper faces of Ω and by Neumann condition on the lateral faces. We prove that when α > 0, α ≠ 2, the structure of the homogenized problem is analogous to the microscopic one since it does not involve nonlocal effects and involves only the diffusion through the axis of the fibres which play the role of holes. This is in contrast to the case α = 2 for which the limit problem involves nonlocal terms as it was proved in several papers.