{
  "id": "1911.10790",
  "version": "v1",
  "published": "2019-11-25T09:50:02.000Z",
  "updated": "2019-11-25T09:50:02.000Z",
  "title": "Regularity of free boundaries in optimal transportation",
  "authors": [
    "Shibing Chen",
    "Jiakun Liu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,\\alpha}$ regularity of the free boundary for optimal partial transport between convex domains for densities $f, g$ bounded from below and above. When $f, g \\in C^\\alpha$, and $\\partial\\Omega, \\partial\\Omega^*\\in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Amp\\`ere equations \\cite{CLW1}, our second result shows that the free boundaries are $C^{2,\\alpha}$. As an application, in the last we also obtain these regularities of the free boundary in an optimal transport problem with two separate targets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-25T09:50:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free boundary",
      "optimal transportation",
      "optimal partial transport",
      "optimal transport problem",
      "quadratic cost"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}