Homogenization in general periodically perforated domains by a spectral approach

Abstract

In this article we study the asymptotic behaviour as \(\epsilon\) tends to 0 of the Neumann problem $-\Delta u_\epsilon+u_\epsilon=\( in a \)\epsilon$-periodic bounded open set \(\Omega_\epsilon\) of \({\mathbb R}^d\). The period cell of \(\Omega_\epsilon\) is equal to \(\epsilon Y_\epsilon\) where \(Y_\epsilon\) is a regular open subset of the d-dimensional torus. We prove that if there exists a smallest integer \(n\geq 1\) such that the n-th non-zero eigenvalue \(\Lambda_n(\epsilon)\) of the spectral problem \(-\Delta V_\epsilon=\Lambda(\epsilon) V_\epsilon\) in \(Y_\epsilon\) satisfies \(\Lambda_n(\epsilon) >> \epsilon^2\), the limiting problem is a linear system of second order p.d.e.'s, of size n. By this spectral approach we extend in the periodic framework a result due to Khruslov without making strong geometrical assumptions on the perforated domain \(\Omega_\epsilon\).