Uniform stability of the Dirichlet spectrum for rough outer perturbations

TL;DR

This paper investigates how the Dirichlet eigenvalues of a domain change under rough outer perturbations, providing explicit bounds based on geometric properties and the eigenvalues of the difference domain.

Contribution

It establishes explicit stability estimates for Dirichlet eigenvalues under rough outer domain perturbations, allowing for arbitrary bounded outer domains.

Findings

Eigenvalue differences are explicitly controlled by geometric constants.

The bounds depend on the first eigenvalue of the difference domain.

Results apply to arbitrary bounded outer domains.

Abstract

The goal of this paper is to study the Dirichlet eigenvalues of bounded domains $\Omega\subset \Omega'$. With a local spectral stability requirement on $\Omega$, we show that the difference of the Dirichlet eigenvalues of $\Omega'$ and $\Omega$ is explicitly controlled from above in terms of the first eigenvalue of $\Omega'\setminus\bar{\Omega}$ and of geometric constants depending on the inner domain $\Omega$. In particular, $\Omega'$ can be an arbitrary bounded domain.