Uniform stability of the Dirichlet spectrum for rough outer perturbations

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.