The Dirichlet Problem

Abstract

In Chapter X we used the Poisson kernel to solve the Dirichlet problem for the unit disk. In this chapter we study the Dirichlet problem for more general domains in the plane. The basic method, due to O. Perron, is to look for the solution of the Dirichlet problem as the upper envelope of a family of subsolutions. In Section 2 we introduce subharmonic functions, which play the role of the subsolutions. In Section 3 we derive Harnack’s inequality, which provides a compactness criterion for families of harmonic functions. Perron’s procedure for solving the Dirichlet problem is developed in Section 4. We apply the method to give another proof of the Riemann mapping theorem in Section 5. In Sections 6 and 7 we introduce Green’s function.