Vol. 254, No. 1, 2011

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The Dirichlet problem for harmonic functions on compact sets

Tony L. Perkins

Vol. 254 (2011), No. 1, 211–226

The main goal of this paper is to study the Dirichlet problem on a compact set K 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(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.

Harmonic measure, Jensen measures, subharmonic functions, potential theory, fine topology
Mathematical Subject Classification 2010
Primary: 31B05
Secondary: 31B10, 31B25, 31C40
Received: 25 February 2011
Revised: 8 July 2011
Accepted: 17 October 2011
Published: 7 February 2012
Tony L. Perkins
Deparment of Mathematics
215 Carnegie Building
Syracuse University
Syracuse, NY 13244-1150
United States