Uniform convergence of solutions to elliptic equations on domains with shrinking holes

Abstract

We consider solutions of the Poisson equation on a family of domains with holes shrinking to a point. Assuming Robin or Neumann boundary conditions on the boundary of the holes, we show that the solution converges uniformly to the solution of the Poisson equation on the domain without the holes. This is in contrast to Dirichlet boundary conditions where there is no uniform convergence. The results substantially improve earlier results on $L^p$-convergence. They can be applied to semi-linear problems.