Spectral boundary value problems for Laplace-Beltrami operator: moduli of continuity of eigenvalues under domain deformation

Let M be a compact Riemannian manifold, possibly with non-empty boundary partial M, let Cal{A} be a strongly elliptic operator with Lipschitz coefficients and consider the eigenvalue problem Cal{A} u= lambda u, quad uin ocirc{H}{}^1(Omega) on a domain Omegasubseteq M with overline{Omega}cap partial M=emptyset. par Under suitable assumptions on the domains, the authors establish the resolvent continuity of the boundary value problem with respect to domain perturbations (Theorem 4.2). As a consequence, they obtain an estimate for the moduli of continuity of the eigenvalues relative to different domains Omega_1 and Omega_2 in terms of their Hausdorff-Pompeiu distance (Theorem 4.1).