Spectral stability estimates for the eigenfunctions of second order elliptic operators

Stability of the eigenfunctions of nonnegative selfadjoint second-order linear elliptic operators subject to homogeneous Dirichlet boundary data under domain perturbation is investigated. Let omega, omega ' subset of R-n be bounded open sets. The main result gives estimates for the variation of the eigenfunctions under perturbations omega ' of omega such that omega(epsilon)={x epsilon omega:dist(x, R-n\omega)>epsilon}subset of omega 'subset of(omega)overbar ' in terms of powers of epsilon, where the parameter epsilon > 0 is sufficiently small. The estimates obtained here hold under some regularity assumptions on omega, omega '. They are obtained by using the notion of a gap between linear operators, which has been recently extended by the authors to differential operators defined on different open sets.