Convergence rates of eigenvalue problems in perforated domains: the case of small volume

TL;DR

This paper investigates the asymptotic behavior of Dirichlet eigenvalues and eigenfunctions for the Laplace operator in perforated domains with small volume, providing optimal error estimates and convergence rates.

Contribution

It introduces a new approach to derive optimal quantitative error estimates and convergence rates for eigenvalues and eigenfunctions in perforated domains with small volume.

Findings

Optimal error estimates for eigenvalues independent of spectral gaps

Convergence rates for eigenfunctions established

Asymptotic expansion with two leading terms

Abstract

This paper is concerned with the Dirichlet eigenvalue problem for Laplace operator in a bounded domain with periodic perforation in the case of small volume. We obtain the optimal quantitative error estimates independent of the spectral gaps for an asymptotic expansion, with two leading terms, of Dirichlet eigenvalues. We also establish the convergence rates for the corresponding eigenfunctions. Our approach uses a known reduction to a degenerate elliptic eigenvalue problem for which a quantitative analysis is carried out.