{
  "id": "1606.05173",
  "version": "v1",
  "published": "2016-06-16T13:07:26.000Z",
  "updated": "2016-06-16T13:07:26.000Z",
  "title": "Partial $W^{2,p}$ regularity for optimal transport maps",
  "authors": [
    "Shibing Chen",
    "Alessio Figalli"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W^{2,p}_{loc}$ for any $p \\geq 1$ outside of a closed singular set of measure zero. We also establish global $W^{2,p}$ estimates when the cost is a small perturbation of the quadratic cost. The latter result is new even when the cost is exactly the quadratic cost.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-16T13:07:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal transport maps",
      "regularity",
      "quadratic cost",
      "optimal transportation problem",
      "potential function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}