Estimates for the deviation of solutions and eigenfunctions of second-order elliptic Dirichlet boundary value problems under domain perturbation
The paper under review deals with second-order uniformly elliptic operators of the form \[ Su=-\operatorname{div} (A(x)Du)+b(x)u\quad \text{in}\;\mathbb{R}^N \] with locally \(C^{1,\alpha}\) coefficients, \(\alpha\in(0,1].\) Estimates in suitable Lebesgue or Sobolev spaces are derived for the deviation of solutions and eigenfunctions of the Dirichlet problem for \(S\) subject to domain perturbation, in terms of the natural distances between the domains. 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.
scientific article; zbMATH DE number 6522103
scientific article