Homogenization in perforated domains with rapidly pulsing perforations

Abstract

The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period of the holes goes to zero. Since standard conservation laws do not hold in this model, a rst diculty is to get ap rioriestimates of the solutions. We obtain them in a weighted space where the weight is the principal eigenfunction of an \adjoint periodic time-dependent eigenvalue problem. This problem is not a classical one, and its investigation is an important part of this work. Then, by using the multiple scale method, we construct the leading terms of a formal expansion (with respect to ) of the solution and give the limit \homogenized problem. An interesting peculiarity of the model is that, depending on the geometry of the holes, a large convection term may appear in the limit equation. Mathematics Subject Classication. 35B27, 74Q10, 76M50.