{
  "id": "1004.5575",
  "version": "v2",
  "published": "2010-04-30T17:03:01.000Z",
  "updated": "2011-01-21T18:13:38.000Z",
  "title": "The Dirichlet Problem for Harmonic Functions on Compact Sets",
  "authors": [
    "Tony Perkins"
  ],
  "comment": "There have been a large number of changes made from the first version. They mostly consists of shortening the article and supplying additional references",
  "doi": "10.2140/pjm.2011.254.211",
  "categories": [
    "math.CA"
  ],
  "abstract": "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 $K$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-01-21T18:13:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31B05",
      "31B10",
      "31B25",
      "31C40"
    ],
    "keywords": [
      "compact set",
      "dirichlet problem",
      "subharmonic peak points",
      "jensen measures",
      "functions harmonic"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.5575P"
    }
  }
}