Tony L. Perkins (PhD '11)
Abstract: 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.
Journal: Pacitic Journal of Mathematics, Vol. 254 (2011), No. 1, 211–226
URL: http://msp.berkeley.edu/pjm/2011/254-1/p11.xhtml