{
  "id": "1207.6480",
  "version": "v2",
  "published": "2012-07-27T08:05:41.000Z",
  "updated": "2017-03-23T07:51:27.000Z",
  "title": "Flat solutions of the 1-Laplacian equation",
  "authors": [
    "Luigi Orsina",
    "Augusto C. Ponce"
  ],
  "comment": "Dedicated to Jean Mawhin. Revised and extended version of a note written by the authors in 2012",
  "categories": [
    "math.AP"
  ],
  "abstract": "For every $f \\in L^N(\\Omega)$ defined in an open bounded subset $\\Omega$ of $\\mathbb{R}^N$, we prove that a solution $u \\in W_0^{1, 1}(\\Omega)$ of the $1$-Laplacian equation ${-}\\mathrm{div}{(\\frac{\\nabla u}{|\\nabla u|})} = f$ in $\\Omega$ satisfies $\\nabla u = 0$ on a set of positive Lebesgue measure. The same property holds if $f \\not\\in L^N(\\Omega)$ has small norm in the Marcinkiewicz space of weak-$L^{N}$ functions or if $u$ is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-27T08:05:41.000Z",
      "title": "Nonexistence of solutions for the 1 Laplacian with L^N data",
      "abstract": "We prove that the 1 Laplacian equation - \\Delta_1 u = f in an open set of R^N cannot have a solution in the Sobolev space W_0^{1, 1} for any datum in L^N",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2017-03-23T07:51:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J70",
      "35J25",
      "35J62",
      "35J92"
    ],
    "keywords": [
      "nonexistence",
      "open set",
      "laplacian equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.6480O"
    }
  }
}