On the Generalized Dirichlet Problem

Abstract

<p> In this thesis, we shall solve the classical Dirichlet problem for a ball in n-dimensional Euclidean space, and then point out that the classical Dirichlet problem is not always solvable. Following Wiener and Brelot, we then introduce a generalized Dirichlet problem for any bounded region in n-dimensional Euclidean space and establish necessary and sufficient conditions for its solution. We show that the solution of the generalized Dirichlet problem coincides with the solution of the classical Dirichlet problem whenever the latter exists. Finally, we characterize those regions for which the classical Dirichlet problem is solvable by considering the boundary behaviour of those functions for which the generalized problem is solvable.</p>