The Dirichlet Problem

Abstract

In its simplest form the Dirichlet problem may be stated as follows: for a given function $f \in C\left( {{\partial ^\infty }\Omega } \right)$, determine, if possible, a function h ∈ H(Ω) such that h(x) → f(y) as x → y for each $y \in {\partial ^\infty }\Omega$. Such a function h is called the (classical) solution of the Dirichlet problem on Ω with boundary function f, and the maximum principle guarantees the uniqueness of the solution if it exists. For example, if Ω is either a ball or a half-space and f ∈ C(δ^∞Ω), then the solution of the Dirichlet problem certainly exists and is given by the Poisson integral of f. This follows immediately from Theorems 1.3.3 and 1.7.5. On the other hand, there are quite simple examples in which there is no such solution.