Asymptotics for eigenvalues of the Laplacian in higher dimensional periodically perforated domains

Abstract

This paper considers the periodic spectral problem associated with the Laplace operator written in \({\mathbb{R}^N}\) (N = 3, 4, 5) periodically perforated by balls, and with homogeneous Dirichlet condition on the boundary of holes. We give an asymptotic expansion for all simple eigenvalues as the size of holes goes to zero. As an application of this result, we use Bloch waves to find the classical strange term in homogenization theory, as the size of holes goes to zero faster than the microstructure period.