{
  "id": "1312.6495",
  "version": "v1",
  "published": "2013-12-23T09:40:28.000Z",
  "updated": "2013-12-23T09:40:28.000Z",
  "title": "Nonlinear elliptic equations with measures revisited",
  "authors": [
    "Haïm Brezis",
    "Moshe Marcus",
    "Augusto C. Ponce"
  ],
  "journal": "In: Mathematical Aspects of Nonlinear Dispersive Equations (J. Bourgain, C. Kenig, S. Klainerman, eds.), Annals of Mathematics Studies, 163, Princeton University Press, Princeton, NJ, 2007, pp. 55--110",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We study the existence of solutions of the nonlinear problem $$ \\left\\{ \\begin{alignedat}{2} -\\Delta u + g(u) & = \\mu & & \\quad \\text{in } \\Omega,\\\\ u & = 0 & & \\quad \\text{on } \\partial \\Omega, \\end{alignedat} \\right. $$ where $\\mu$ is a Radon measure and $g : \\mathbb{R} \\to \\mathbb{R}$ is a nondecreasing continuous function with $g(0) = 0$. This equation need not have a solution for every measure $\\mu$, and we say that $\\mu$ is a good measure if the Dirichlet problem above admits a solution. We show that for every $\\mu$ there exists a largest good measure $\\mu^* \\leq \\mu$. This reduced measure has a number of remarkable properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-23T09:40:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J15",
      "35J61",
      "31B15",
      "31B35"
    ],
    "keywords": [
      "nonlinear elliptic equations",
      "radon measure",
      "nonlinear problem",
      "dirichlet problem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.6495B"
    }
  }
}