On the Continuous Dependence of the Exterior Dirichlet Problem on the Boundary
Peter Hähner · Mathematical Methods in the Applied Sciences · 1997
We consider a sequence of exterior domains Dj,j∈ℕ0, and assume that the boundaries ∂Dj converge to ∂D0 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 Dj⊂D0, j∈ℕ, then, by essentially using the method of Perron, we show that the solutions in the domains Dj converge to the solution in the domain D0 with respect to the maximum norm. We prove the same result in case that the requirement Dj⊂D0,j∈ℕ, is replaced by an equicontinuity property of all barrier functions to all boundary points. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 707–716 (1997)