Two-Scale Convergence of Stekloff Eigenvalue Problems in Perforated Domains

Hermann Douanla

doi:10.1155/2010/853717

Abstract

By means of the two-scale convergence method, we investigate the asymptotic behavior of eigenvalues and eigenfunctions of Stekloff eigenvalue problems in perforated domains. We prove a concise and precise homogenization result including convergence of gradients of eigenfunctions which improves the understanding of the asymptotic behavior of eigenfunctions. It is also justified that the natural local problem is not an eigenvalue problem.

Highlights

We are interested in the spectral asymptotics as ε → 0 of the linear elliptic eigenvalue problem i,j 1 ∂xi x aij x, ε
A sequence uε ε∈E ⊂ L2 Sε is said to two-scale converge to some u0 ∈ L2 Ω × S if as follows
In the case when uε ε∈E is the sequence of traces on Sε of functions in H1 Ω, a link can be established between its usual and surface two-scale limits

Summary

Introduction

We are interested in the spectral asymptotics as ε → 0 of the linear elliptic eigenvalue problem. Boundary Value Problems the symmetry condition aji aij, the periodicity hypothesis: for each x ∈ Ω and for every k ∈ N one has aij x, y k aij x, y almost everywhere in y ∈ ÊNy , and the ellipticity condition: there exists α > 0 such that for any x ∈ Ω. Homogenization of eigenvalue problems in a fixed domain goes back to Kesavan 1, 2. Concerning homogenization of eigenvalue problems in perforated domains, we mention the work of Conca et al 6 , Douanla and Svanstedt 7 , Kaizu 8 , Ozawa and Roppongi 9 , Roppongi , and Pastukhova and the references therein.

Two-Scale Convergence on Periodic Surfaces

Preliminaries

Homogenization Result