Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains

TL;DR

This paper provides quantitative stability estimates for the resolvents, eigenvalues, and eigenfunctions of second-order elliptic operators on variable domains, linking domain perturbations to spectral variations.

Contribution

It introduces new Sobolev norm-based estimates for spectral stability under domain deformations, applicable to a broad class of elliptic operators and open sets.

Findings

Spectral variations are controlled by Sobolev norm differences of domain parametrizations.

Conditions for stability are satisfied for Lipschitz boundary domains.

Applications include stability analysis of solutions to the Poisson problem.

Abstract

We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by Lipschitz homeomorphisms $\phi $ defined on a fixed reference domain $\Omega$. Given two open sets $\phi (\Omega)$, $\tilde \phi (\Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde \phi -\phi \|_{W^{1,p}(\Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.