Bloch waves homogenization of a Dirichlet problem in a periodically perforated domain

Abstract

In this paper we use the spectral method of Bloch waves to study the homogenization process of the Poisson equation in a periodically perforated domain, under homogeneous Dirichlet conditions on both exterior and interior boundaries, as the hole size goes to zero more rapidly than the micro-structure size. Using this method, we find the exact value of the critical hole size, which separates the different behaviors, where the classical strange term may or may not appear in the homogenized equation. This strange term is related to the asymptotic behavior of the first eigenvalue with respect to the hole radius.