On the continuous dependence of the exterior Dirichlet problem on the boundary

We consider a sequence of exterior domains D-j,j is an element of N-0, and assume that the boundaries partial derivative D-j converge to partial derivative D-0 with respect to the Hausdorff distance. We investigate solutions to the exterior Dirichlet problem for the Laplace equation and for the Helmholtz equation in these domains. Assuming convergence of the boundary data and D-j subset of D-0, j is an element of N, then, by essentially using the method of Perron, we show that the solutions in the domains D-j converge to the solution in the domain D-0 with respect to the maximum norm. We prove the same result in case that the requirement D-j subset of D-0, j is an element of N, is replaced by an equicontinuity property of all barrier functions to ail boundary points. (C) 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.