{
  "id": "1311.4856",
  "version": "v2",
  "published": "2013-11-19T20:00:54.000Z",
  "updated": "2014-10-27T09:03:05.000Z",
  "title": "Strong maximum principle for Schrödinger operators with singular potential",
  "authors": [
    "Luigi Orsina",
    "Augusto C. Ponce"
  ],
  "comment": "21 pages",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We prove that for every $p > 1$ and for every potential $V \\in L^p$, any nonnegative function satisfying $-\\Delta u + V u \\ge 0$ in an open connected set of $\\mathbb{R}^N$ is either identically zero or its level set $\\{u = 0\\}$ has zero $W^{2, p}$ capacity. This gives an affirmative answer to an open problem of B\\'enilan and Brezis concerning a bridge between Serrin-Stampacchia's strong maximum principle for $p > \\frac{N}{2}$ and Ancona's strong maximum principle for $p = 1$. The proof is based on the construction of suitable test functions depending on the level set $\\{u = 0\\}$ and on the existence of solutions of the Dirichlet problem for the Schr\\\"odinger operator with diffuse measure data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-19T20:00:54.000Z",
      "abstract": "We prove that for every $p > 1$ and for every potential $V \\in L^p$, any nonnegative function satisfying $-\\Delta u + V u \\ge 0$ in an open connected set of $\\mathbb{R}^N$ is either identically zero or its level set $\\{u = 0\\}$ has zero $W^{2, p}$ capacity. This gives an affirmative answer to an open problem of B\\'enilan and Brezis concerning a bridge between Trudinger's strong maximum principle for $p > \\frac{N}{2}$ and Ancona's strong maximum principle for $p = 1$. The proof is based on the construction of suitable test functions depending on the level set $\\{u = 0\\}$ and on the existence of solutions of the Dirichlet problem for the Schr\\\"odinger operator with diffuse measure data.",
      "comment": "19 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-27T09:03:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B05",
      "35B50",
      "31B15",
      "31B35"
    ],
    "keywords": [
      "schrödinger operators",
      "singular potential",
      "anconas strong maximum principle",
      "trudingers strong maximum principle",
      "level set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.4856O"
    }
  }
}