Strong Localized Perturbations of Eigenvalue Problems

Abstract

This paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded or unbounded domains. The perturbations are the addition of a function of small support and large magnitude to the differential operator, the removal of a small subdomain from the domain of a problem with the imposition of a boundary condition on the boundary of the resulting hole, and a large alteration of the boundary condition on a small region of the boundary of the domain. For each of these perturbations, the eigenvalues and eigenfunctions for the perturbed problem are constructed by the method of matched asymptotic expansions for $\epsilon $ small, where $\epsilon $ is a measure of the extent of the perturbation. In some special cases, the asymptotic results are shown to agree well with exact results. The asymptotic theory is then applied to determine the exit time distribution for a particle undergoing Brownian motion inside a container having reflecting walls perforated by many small holes.