Dirichlet Problem for Unbounded Regions

Abstract

The principle problem associated with unbounded regions is the lack of uniqueness of the solution to the Dirichlet problem. To achieve uniqueness, the point at infinity ∞ will be adjoined to R n with the enlarged space denoted by Rn∞. This will require redefinition of harmonic and superharmonic functions. The Dirichlet problem for the exterior of a ball will be solved by a Poisson type integral. Using this result, it will be shown that the Perron- Wiener-Brelot method can be used to solve the Dirichlet problem for unbounded regions. Poincaré's exterior ball condition and Zaremba's exterior cone condition are sufficient conditions for a finite boundary point to be a regular boundary point for the Dirichlet problem. Both conditions preclude the boundary point from being “too surrounded” by the region. On the other hand, the Lebesgue spine is an example of a region that does “surround” a boundary point too much; in some sense, the complement of the region is “thin” at the boundary point. A concept of thinness will be explored and related to a topology on Rn∞ finer than the metric topology which is more natural from the potential theoretic point of view. The words “open,” “neighborhood,” “continuous,” etc., will be prefixed by “fine” or “finely” when used in this context.