{
  "id": "2102.00923",
  "version": "v1",
  "published": "2021-02-01T15:47:48.000Z",
  "updated": "2021-02-01T15:47:48.000Z",
  "title": "$C^\\infty$ partial regularity of the singular set in the obstacle problem",
  "authors": [
    "Federico Franceschini",
    "Wiktoria Zatoń"
  ],
  "comment": "70 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We show that the singular set $\\Sigma$ in the classical obstacle problem can be locally covered by a $C^\\infty$ hypersurface, up to an \"exceptional\" set $E$, which has Hausdorff dimension at most $n-2$ (countable, in the $n=2$ case). Outside this exceptional set, the solution admits a polynomial expansion of arbitrarily large order. We also prove that $\\Sigma\\setminus E$ is extremely unstable with respect to monotone perturbations of the boundary datum. We apply this result to the planar Hele-Shaw flow, showing that the free boundary can have singular points for at most countable many times.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-01T15:47:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singular set",
      "partial regularity",
      "planar hele-shaw flow",
      "hausdorff dimension",
      "exceptional set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}