Spectral stability estimates for elliptic operators subject to domain transformations with non-uniformly bounded gradients

Abstract

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain ω in ℝ N. We consider deformations Ø•(ω) of ω obtained by means of a locally Lipschitz homeomorphism Ø• and we estimate the variation of the eigenfunctions and eigenvalues upon variation of Ø•. We prove general stability estimates without assuming uniform upper bounds for the gradients of the maps Ø•. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of Lipschitz domains. © 2012 Copyright University College London