{
  "id": "1201.4055",
  "version": "v2",
  "published": "2012-01-19T13:22:32.000Z",
  "updated": "2015-10-05T18:34:09.000Z",
  "title": "Geometric approach to nonvariational singular elliptic equations",
  "authors": [
    "Damião Araújo",
    "Eduardo V. Teixeira"
  ],
  "comment": "Paper from D. Araujo's Ph.D. thesis, distinguesed at the 2013 Carlos Gutierrez prize",
  "journal": "Archive for Rational Mechanics and Analysis 2013",
  "doi": "10.1007/s00205-013-0633-9",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter $(\\gamma -1)$, for $0 < \\gamma < 1$, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases. We establish existence as well sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and obtain fine geometric-measure properties of the free boundary $\\mathfrak{F} = \\partial \\{u > 0 \\}$. In particular we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region $\\{u > 0 \\}$ and $\\mathcal{H}^{n-1}$ a.e. weak differentiability property of $\\mathfrak{F}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-19T13:22:32.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-10-05T18:34:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35",
      "35B65",
      "35J60"
    ],
    "keywords": [
      "nonvariational singular elliptic equations",
      "geometric approach",
      "study fully nonlinear elliptic equations",
      "free boundary",
      "weak differentiability property"
    ],
    "tags": [
      "journal article",
      "dissertation"
    ],
    "publication": {
      "doi": "10.1007/s00205-013-0633-9",
      "journal": "Archive for Rational Mechanics and Analysis",
      "year": 2013,
      "month": "Sep",
      "volume": 209,
      "number": 3,
      "pages": 1019,
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013ArRMA.209.1019A"
    }
  }
}