Spectral Stability of the Robin Laplacian

We consider the Robin Laplacian in two bounded regions Omega(1) and Omega(2) of R(N) with Lipschitz boundaries and such that Omega(2) subset of Omega(1), and we obtain two-sided estimates for the eigen-values 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) \ Omega(2) and on suitably defined characteristics of vicinity of the boundaries partial derivative Omega(1) and partial derivative Omega(2), and of the functions defined on partial derivative Omega(1) and on partial derivative Omega(2) that enter the Robin boundary conditions.