{
  "id": "1101.1649",
  "version": "v2",
  "published": "2011-01-09T15:46:21.000Z",
  "updated": "2011-02-01T03:58:57.000Z",
  "title": "An overdetermined problem in Riesz-potential and fractional Laplacian",
  "authors": [
    "Guozhen Lu",
    "Jiuyi Zhu"
  ],
  "comment": "11 pages. This version replaces an earlier posting on the arxiv",
  "categories": [
    "math.AP"
  ],
  "abstract": "The main purpose of this paper is to address two open questions raised by W. Reichel on characterizations of balls in terms of the Riesz potential and fractional Laplacian. For a bounded $C^1$ domain $\\Omega\\subset \\mathbb R^N$, we consider the Riesz-potential $$u(x)=\\int_{\\Omega}\\frac{1}{|x-y|^{N-\\alpha}} \\,dy$$ for $2\\leq \\alpha \\not =N$. We show that $u=$ constant on the boundary of $\\Omega$ if and only if $\\Omega$ is a ball. In the case of $ \\alpha=N$, the similar characterization is established for the logarithmic potential. We also prove that such a characterization holds for the logarithmic Riesz potential. This provides a characterization for the overdetermined problem of the fractional Laplacian. These results answer two open questions of W. Reichel to some extent.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-02-01T03:58:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional laplacian",
      "overdetermined problem",
      "riesz-potential",
      "logarithmic riesz potential",
      "main purpose"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.1649L"
    }
  }
}