Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Dirichlet Problem for Harmonic Functions on Compact Sets   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 $K$.

0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio