Mathematics – Classical Analysis and ODEs
Scientific paper
2010-04-30
Mathematics
Classical Analysis and ODEs
There have been a large number of changes made from the first version. They mostly consists of shortening the article and supp
Scientific paper
10.2140/pjm.2011.254.211
For any compact set $K\subset \mathbb{R}^n$ we develop the theory of Jensen measures and subharmonic peak points, which form the set $\mathcal{O}_K$, 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 $C(\mathcal{O}_K)\cong h(K)$ for compact sets with $\mathcal{O}_K$ 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 $C_b(\mathcal{O}_K)$ for all compact sets $K$.
No associations
LandOfFree
The Dirichlet Problem for Harmonic Functions on Compact Sets does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Dirichlet Problem for Harmonic Functions on Compact Sets, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Dirichlet Problem for Harmonic Functions on Compact Sets will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-388707