Dirichlet Problem for Unbounded Regions

Abstract

In this chapter, a study of the Dirichlet problem on unbounded regions is undertaken that necessitates enlarging $R^n$ to $R^n_{\infty }$ to include the point at infinity. Harmonic and superharmonic functions are defined anew on $R^n_{\infty }$ to deal with the behavior of these functions at $\infty$. A Poisson type integral is developed to solve the Dirichlet problem on the exterior of a ball and then to show that the Perron-Weiner-Brelot method can be used to prove the existence of a solution to the Dirichlet problem on unbounded regions. It is shown that the Poincaré and Zaremba criteria for regularity is applicable to finite boundary points and it is shown that $\infty$ is always a regular boundary point for an unbounded region in $R^n_{\infty }, n \ge 3$. A concept of thinness is used to characterize regular boundary points. A topology based on the concept of thinness is defined which is more natural from the potential theory view than the usual metric topology.