THE DIRICHLET PROBLEM FOR HARMONIC FUNCTIONS ON COMPACT SETS

The main goal of this paper is to study the Dirichlet problem on a compact set K subset of R-n. Initially we consider the space H(K) of functions on K that can be uniformly approximated by functions harmonic in a neighborhood of K as possible solutions. As in the classical theory, we show C(partial derivative(f) K) congruent to H(K) for compact sets with partial derivative(f) K closed, where partial derivative(f) K is the fine boundary of K. However, in general, a continuous solution cannot be expected, even for continuous data on partial derivative(f) K. Consequently, we show that for any bounded continuous boundary data on partial derivative(f) K, the solution can be found in a class of finely harmonic functions. Also, in complete analogy with the classical situation, this class is isometrically isomorphic to the set of bounded continuous functions on partial derivative(f) K for all compact sets K.