{
  "id": "1302.0949",
  "version": "v2",
  "published": "2013-02-05T07:16:08.000Z",
  "updated": "2013-02-06T12:49:22.000Z",
  "title": "Singular measure as principal eigenfunction of some nonlocal operators",
  "authors": [
    "Jerome Coville"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution $(\\lambda,\\phi)$ of a nonlocal operator. $$\\int_{\\O}K(x,y)\\phi(y)\\, dy +a(x)\\phi(x) =-\\lambda \\phi(x),$$ where $\\O\\subset\\R^n$ is an open bounded connected set, $K$ a nonnegative kernel and $a$ is continuous. We prove that for the generalised principal eigenvalue $\\lambda_p:=\\sup \\{\\lambda \\in \\R \\, |\\, \\exists \\, \\phi \\in C(\\O), \\phi > 0 \\;\\text{so that}\\; \\oplb{\\phi}{\\O}+ a(x)\\phi + \\lambda\\phi\\le 0\\}$ there exists always a solution $(\\mu, \\lambda_p)$ of the problem in the space of signed measure. Moreover $\\mu$ a positive measure. When $\\mu$ is absolutely continuous with respect to the Lebesgue measure, $\\mu =\\phi_p(x)$ is called the principal eigenfunction associated to $\\lambda_p$. In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-02-06T12:49:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlocal operator",
      "principal eigenfunction",
      "generalised principal eigenvalue",
      "explicit singular measures",
      "spectral problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.0949C"
    }
  }
}