Uniform convergence of elliptic problems on varying domains

Abstract

We consider the Poisson problem on varying, possibly unbounded domains with bounded source term. Extending the solutions by zero outside the domain we get a uniformly bounded sequence of functions on R^N. We discuss conditions under which the sequence of solutions converge (locally) uniformly on R^N. The concept of regular convergence is introduced implying local uniform convergence of solutions. Regular convergence is rather general. For instance we can cut into a domain, or the difference in measure between the sequence of domains and the limit domain does not need to go to zero.