The Dirichlet problem for domains with multiple boundary points

Abstract

Several years ago Professor Kellogg called my attention to the desirability of extending the theory of the Dirichlet problem so as to include the case in which the domain has multiple boundary points with boundary values depending upon the manner of approach. In the two-dimensional case conformal mapping may sometimes be used. Also, a paper by Perront contains results related to this subject. It seems desirable, however, to develop a general theory for this extended form of the problem. Professor Kellogg noted that a spatial analogue of Caratheodory's4 theory of prime ends would be of value here, since this would render possible in some cases the definition of functions corresponding to barriers.? He communicated his ideas on this topic to me, and invited me to collaborate with him on the problem. Later he suggested that I develop the subject alone, a procedure which unfortunately was made necessary by his death. This paper contains the results of the ensuing study of the problem. The discussion is formulated for a general finite domain of three-dimensional space, except in the case of a few theorems where special restrictions are imposed. It is readily seen that corresponding results are valid in the plane. In Part I we introduce the notions of component and boundary element. These correspond to Caratheodory's ends and prime ends, respectively,