Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains

TL;DR

This paper develops new quantitative estimates for how the eigenvalues of the Laplacian change when the domain's shape is perturbed, applicable to rough domains like Lipschitz and Reifenberg-flat domains.

Contribution

It introduces an abstract lemma linking spectral stability to projection operator estimates, extending stability results to less regular domains.

Findings

Established stability estimates for eigenvalues in Lipschitz domains

Extended spectral stability results to Reifenberg-flat domains

Provided boundary decay estimates for eigenfunctions

Abstract

In this paper we establish new quantitative stability estimates with respect to domain perturbations for all the eigenvalues of both the Neumann and the Dirichlet Laplacian. Our main results follow from an abstract lemma stating that it is actually sufficient to provide an estimate on suitable projection operators. Whereas this lemma could be applied under different regularity assumptions on the domain, here we use it to estimate the spectrum in Lipschitz and in so-called Reifenberg-flat domains. Our argument also relies on suitable extension techniques and on an estimate on the decay of the eigenfunctions at the boundary which could be interpreted as a boundary regularity result.