Spectral Stability of the Robin Laplacian

We consider the Robin Laplacian in two bounded regions $\Omega_1$ and $\Omega_2$ of $\mathbb R^N$ with Lipschitz boundaries and such that $\Omega_2\subset\Omega_1$, and we obtain two-sided estimates for the eigenvalues $\lambda_{n,2}$ of the Robin Laplacian in $\Omega_2$ via the eigenvalues $\lambda_{n,1}$ of the Robin Laplacian in $\Omega_1$. Our estimates depend on the measure of the set difference $\Omega_1\!\setminus\Omega_2$ and on suitably defined characteristics of vicinity of the boundaries $\partial\Omega_1$ and $\partial\Omega_2$, and of the functions defined on $\partial\Omega_1$ and on $\partial\Omega_2$ that enter the Robin boundary conditions.