Domain dependence of eigenvalues of elliptic type operators

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.