{
  "id": "2007.11549",
  "version": "v1",
  "published": "2020-07-22T17:11:03.000Z",
  "updated": "2020-07-22T17:11:03.000Z",
  "title": "Variational approach to regularity of optimal transport maps: general cost functions",
  "authors": [
    "Felix Otto",
    "Maxime Prod'homme",
    "Tobias Ried"
  ],
  "comment": "44 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We extend the variational approach to regularity for optimal transport maps initiated by Goldman and the first author to the case of general cost functions. Our main result is an $\\epsilon$-regularity result for optimal transport maps between H\\\"older continuous densities slightly more quantitative than the result by De Philippis-Figalli. One of the new contributions is the use of almost-minimality: if the cost is quantitatively close to the Euclidean cost function, a minimizer for the optimal transport problem with general cost is an almost-minimizer for the one with quadratic cost. This further highlights the connection between our variational approach and De Giorgi's strategy for $\\epsilon$-regularity of minimal surfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-22T17:11:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q22",
      "35B65",
      "53C21"
    ],
    "keywords": [
      "optimal transport maps",
      "general cost functions",
      "variational approach",
      "euclidean cost function",
      "optimal transport problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}