Spectral Convergence of the Laplace Operator with Robin Boundary Conditions on a Small Hole

Abstract

In this paper, we study a bounded domain with a small hole removed. Our main result concerns the spectrum of the Laplace operator with the Robin conditions imposed at the hole boundary. Moreover, we prove that under some suitable assumptions on the parameter in the boundary condition, the spectrum of the Laplacian converges in the Hausdorff distance sense to the spectrum of the Laplacian defined on the unperturbed domain.