Spectral stability of Dirichlet second order uniformly elliptic operators

We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation. The main results concern estimates for the variation of the eigenvalues via the Hausdorff distance between the domains or the Lebesgue measure of their symmetric difference. Our analysis includes domains with Lipschitz boundaries as well as domains with boundary degenerations of power type.