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 $\Omega$ in ${\mathbb{R}}^N$. We consider deformations $\varphi (\Omega )$ of $\Omega$ obtained by means of a locally Lipschitz homeomorphism $\varphi$ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of $\varphi$. We prove general stability estimates without using uniform upper bounds for the gradients of the maps $\varphi$. 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.

25 pages

References in corpus (2)

• Spectral stability of higher order uniformly elliptic operators
• Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains