A Note For Plane Laplace's Exterior Boundary Value Problems

摘要

The solution of conventional boundary integral equations (CBIEs) sometimes does not exist or is not unique, which has been demonstrated in a large number of numerical experiments. According to the authors' opinion, there exist two reasons which can lead to this phenomenon. One reason is that the solution of the CBIEs can not describe the behavior of the solution of the corresponding boundary value problem at infinity accurately; the other one is that the form of exterior boundary value problem has deficiency, which is still a problem to be solved but has not attracted adequate attention. In this paper, a sufficient and necessary condition with respect to the Dirichlet exterior boundary value problem, which can ensure the existence and uniqueness of the solution, is provided and fully proved. Based on the proposed condition, equivalent boundary integral equations (EBIEs) for exterior problems are established. In addition, an extremum principle on the exterior domain is introduced in this paper.