Introduction to the Douglas Problem

Abstract

In this final chapter the general problem of Plateau, usually called the Douglas problem, is studied for minimal surfaces defined on schlicht (i.e. planar) domains which span a given boundary configuration consisting of k closed Jordan curves. By a modification of a method due to Courant it is shown that there exists a solution of Douglas’s problem which minimizes both area and Dirichlet’s integral, if the sufficient Douglas condition is satisfied by the boundary configuration. In particular, this solves the Douglas problem for two linked boundary curves. Finally the connection to Koebe’s mapping theorem is pointed out.