Uniform stability of the Dirichlet spectrum for rough outer perturbations

Abstract

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