Domain dependence of eigenvalues of elliptic type operators

TL;DR

This paper investigates how the eigenvalues of elliptic operators depend on the domain shape, providing an asymptotic formula and stability estimates that accommodate non-smooth boundaries and topological differences.

Contribution

It introduces a novel approach using orthogonal projectors to compare eigenvalues across domains with different geometries and boundary regularities.

Findings

Asymptotic formula for eigenvalue variation with domain perturbation

Stability estimates for eigenvalues based on eigenfunction gradients

Implications for inequalities in perturbed eigenvalue problems

Abstract

The dependence on the domain is studied for the Dirichlet eigenvalues of an elliptic operator considered in bounded domains. Their proximity is measured by a norm of the difference of two orthogonal projectors corresponding to the reference domain and the perturbed one; this allows to compare domains that have non-smooth boundaries and different topology. The main result is an asymptotic formula in which the remainder is evaluated in terms of this quantity. As an application, the stability of eigenvalues is estimated by virtue of integrals of squares of the gradients of eigenfunctions for elliptic problems in different domains. It occurs that these stability estimates imply well-known inequalities for perturbed eigenvalues.