Hausdorff Convergence of Domains and Their Boundaries in Shape Optimal Design

A question, which arises frequently in shape optimal design, is the convergence of domains. Usually the Hausdorff metric or Hausdorff complementary metric is used in the potential topology of QTR R^N. Unfortunately the convergence of domains does not include convergence of its boundary. This fact can cause difficulties if considering PDEs, defined on the domains and with boundary conditions on its boundary. In this case a series of solutions of the PDEs may not converge to the solution of the PDE in the limit domain. Various restrictions on the set of domains, like for example a cone property or Lipschitz boundaries have been used in order to avoid difficulties of this kind. In this paper a criterion for a set of domains is defined, such that from the convergence of the domains follows convergence of the boundaries, if one is restricting to this set of domains. Moreover it is proved that this criterion is sharp and a similar criterion for the convergence of the Lebesgue measure of the boundaries is given.