{
  "id": "1304.2706",
  "version": "v3",
  "published": "2013-04-09T19:48:28.000Z",
  "updated": "2013-08-01T15:36:33.000Z",
  "title": "Partial regularity for singular solutions to the Monge-Ampere equation",
  "authors": [
    "Connor Mooney"
  ],
  "comment": "Final version, to appear in Comm. Pure Appl. Math",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that solutions to the Monge-Ampere inequality $$\\det D^2u \\geq 1$$ in $\\mathbb{R}^n$ are strictly convex away from a singular set of Hausdorff $n-1$ dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to $\\det D^2u = 1$ with singular set of Hausdorff dimension as close as we like to $n-1$. As a consequence we obtain $W^{2,1}$ regularity for the Monge-Ampere equation with bounded right hand side and unique continuation for the Monge-Ampere equation with sufficiently regular right hand side.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-08-01T15:36:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "monge-ampere equation",
      "singular solutions",
      "partial regularity",
      "sufficiently regular right hand side",
      "singular set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.2706M"
    }
  }
}