{
  "id": "1211.2341",
  "version": "v1",
  "published": "2012-11-10T18:09:56.000Z",
  "updated": "2012-11-10T18:09:56.000Z",
  "title": "Sobolev regularity for Monge-Ampère type equations",
  "authors": [
    "Guido De Philippis",
    "Alessio Figalli"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly $c$-convex potentials arising in optimal transportation belong to $W^{2,1+\\kappa}_{\\rm loc}$ for some $\\kappa>0$. This generalizes some recents results concerning the regularity of strictly convex Alexandrov solutions of the Monge-Amp\\`ere equation with right hand side bounded away from zero and infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-10T18:09:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "monge-ampère type equations",
      "sobolev regularity",
      "right hand side bounded away",
      "optimal transportation belong",
      "necessary structural conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.2341D"
    }
  }
}