III - Potential Theory

Summary

The goal of this chapter is to solve the Dirichlet problem on an arbitrary plane domain Ω. There are three traditional ways to solve this problem:

1. (i) The Wiener method is to approximate Ω from inside by subdomains Ω_n of the type studied in Chapter II and to show that the harmonic measures ω(z, E, Ω_n) converge weak-star to a limit measure on ∂Ω. With Wiener's method one must prove that the limit measure ω(z, E, Ω) does not depend on the approximating sequence Ω_n.

2. (ii) The Perron method associates to any bounded function f on ∂Ω a harmonic function P_f on Ω. The function P_f is the upper envelop of a family of subharmonic functions constrained by f on ∂Ω. Perron's method is elegant and general. With Perron's method the difficulty is linearity; one must prove that P_−f = −P_f, at least for f continuous.

3. (iii) The Brownian motion approach, originally from Kakutani [1944a], identifies ω(z, E, Ω) with the probability that a randomly moving particle, starting at z, first hits ∂Ω in the set E. This method has considerable intuitive appeal, but it leaves many theorems hard to reach.

We follow Wiener and use the energy integral to prove that the limit ω(z, E, Ω) is unique. This leads to the notions of capacity, equilibrium distribution, and regular point and to the characterization of regular points by Wiener series.