For any compact set K⊂Rn we develop the theory of Jensen
measures and subharmonic peak points, which form the set OK, to
study the Dirichlet problem on K. Initially we consider the space h(K) of
functions on K which can be uniformly approximated by functions harmonic in a
neighborhood of K as possible solutions. As in the classical theory, our
Theorem 8.1 shows C(OK)≅h(K) for compact sets with
OK closed. However, in general a continuous solution cannot be
expected even for continuous data on \rO_K as illustrated by Theorem 8.1.
Consequently, we show that the solution can be found in a class of finely
harmonic functions. Moreover by Theorem 8.7, in complete analogy with the
classical situation, this class is isometrically isomorphic to
Cb(OK) for all compact sets K.Comment: There have been a large number of changes made from the first
version. They mostly consists of shortening the article and supplying
additional reference