The main goal of this paper is
to study the Dirichlet problem on a compact set K ⊂ ℝn. 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(∂fK)≅H(K) for compact sets with ∂fK closed, where ∂fK is the fine
boundary of K. However, in general, a continuous solution cannot be expected, even
for continuous data on ∂fK. Consequently, we show that for any bounded continuous
boundary data on ∂fK, 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 ∂fK for all
compact sets K.
Keywords
Harmonic measure, Jensen measures, subharmonic functions,
potential theory, fine topology
