{
  "id": "math/0703497",
  "version": "v1",
  "published": "2007-03-16T19:00:21.000Z",
  "updated": "2007-03-16T19:00:21.000Z",
  "title": "On some nonlinear partial differential equations involving the 1-Laplacian",
  "authors": [
    "Mouna Kraiem"
  ],
  "comment": "16pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we present an approximation result concerning the first eigenvalue of the 1-Laplacian operator. More precisely, for $\\Omega$ a bounded regular open domain, we consider a minimisation of the functional ${\\ds \\int_\\Omega}|\\nabla u|+n({\\ds \\int_\\Omega} |u|-1)^2 $ over the space $W_0^{1,1}(\\Omega)$. For $n$ large enough, the infimum is achieved in some sense on $BV(\\Omega)$, and letting $n$ go to infinity this provides an approximation of the first eigenfunction for the first eigenvalue, since the term $n({\\ds \\int_\\Omega} |u|^2-1)^2$ \"tends\" to the constraint $\\|u\\|_1=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-16T19:00:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J70",
      "35J60"
    ],
    "keywords": [
      "nonlinear partial differential equations",
      "first eigenvalue",
      "bounded regular open domain",
      "first eigenfunction",
      "approximation result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3497K"
    }
  }
}