Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains

TL;DR

This paper investigates how multiple eigenvalues of the Dirichlet Laplacian behave in perforated domains, revealing their asymptotic splitting depends on eigenfunction expansions and capacity calculations, especially in planar cases.

Contribution

It provides a detailed analysis of eigenvalue ramification in perforated domains, extending the understanding of eigenvalue asymptotics and capacity in two dimensions.

Findings

Multiple eigenvalues split based on convergence rates or expansion coefficients.

Asymptotic behavior depends on eigenfunction expansion and capacity.

Detailed treatment of planar domains with explicit capacity calculations.

Abstract

Taking advantage from the so-called "Lemma on small eigenvalues" by Colin de Verdi\`ere, we study ramification for multiple eigenvalues of the Dirichlet Laplacian in bounded perforated domains. The asymptotic behavior of multiple eigenvalues turns out to depend on the asymptotic expansion of suitable associated eigenfunctions. We treat the case of planar domains in details, thanks to the asymptotic expansion of a generalization of the so-called u-capacity which we compute in dimension 2. In this case multiple eigenvalues are proved to split essentially by different rates of convergence of the perturbed eigenvalues or by different coefficients in front of their expansion if the rate of two eigenbranches turns out to be the same.