Spectral Stability of the Robin Laplacian

We consider the Robin Laplacian in two bounded regions Ω1 and Ω2 of ℝ N with Lipschitz boundaries and such that Ω2 ⊂ Ω1, and we obtain two-sided estimates for the eigenvalues λ n,2 of the Robin Laplacian in Ω2 via the eigenvalues λ n, 1 of the Robin Laplacian in Ω1. Our estimates depend on the measure of the set difference Ω\Ω2 and on suitably defined characteristics of vicinity of the boundaries ∂Ω1 and ∂Ω2, and of the functions defined on ∂Ω1 and on ∂Ω2 that enter the Robin boundary conditions.