@MISC{Perkins_harmonicfunctions, author = {Tony L. Perkins}, title = {HARMONIC FUNCTIONS ON COMPACT SETS IN Rn}, year = {} }

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Abstract

Abstract. 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 con-sider 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 con-tinuous data on OK as illustrated by Theorem 8.1. Consequently, we show that the solution can be found in a class of finely har-monic 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. 1.