Introduction

Invariance of Laplace's Equation and the Dirichlet Problem

Let D be a domain whose boundary is made up of piecewise smooth contours joined end to end. The Dirichelet problem is to find a function that is harmonic in D such that takes on prescribed values at points on the boundary. Let us first study the problem in the upper half plane.

Theorem 10.1. (Invariance of Laplace's Equation) Let be harmonic in a domain G in the w plane. Then satisfies Laplace's equation at each point in G. If is a conformal mapping from a domain D in the z plane onto G, then the composition is harmonic in D, and satisfies Laplace's equation at each point in D.

Theorem 10.2. (N-Value Dirichlet Problem for the Upper Half Plane)
Let denote real constants. The function

is harmonic in the upper half plane and takes on the boundary values

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(c) John Mathews, 1998, 2006