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

TL;DR

This paper establishes spectral stability estimates for elliptic operators under domain deformations induced by locally Lipschitz homeomorphisms, including cases with non-uniformly bounded gradients, and applies these results to domains with cusps.

Contribution

It provides new stability estimates for eigenvalues and eigenfunctions under domain transformations without requiring uniform gradient bounds.

Findings

Stability estimates hold without uniform bounds on deformation gradients.

Convergence rates for eigenvalues and eigenfunctions are derived for cusp domains.

Results apply to Lipschitz domain approximations of irregular shapes.

Abstract

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain $\Omega $ in ${\mathbb{R}}^N$. We consider deformations $\phi (\Omega)$ of $\Omega $ obtained by means of a locally Lipschitz homeomorphism $\phi $ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of $\phi $. We prove general stability estimates without using uniform upper bounds for the gradients of the maps $\phi$. 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.