Estimates for the deviation of solutions and eigenfunctions of second-order elliptic Dirichlet boundary value problems under domain perturbation

Estimates in suitable Lebesgue or Sobolev norms for the deviation of solutions and eigenfunctions of second-order uniformly elliptic Dirichlet boundary value problems subject to domain perturbation in terms of natural distances between the domains are given. The main estimates are formulated via certain natural and easily computable “atlas” distances for domains with Lipschitz continuous boundaries. As a corollary, similar estimates in terms of more “classical” distances such as the Hausdorff distance or the Lebesgue measure of the symmetric difference of domains are derived. Sharper estimates are also proved to hold in smoother classes of domains.