{
  "id": "2209.01063",
  "version": "v1",
  "published": "2022-09-02T13:54:49.000Z",
  "updated": "2022-09-02T13:54:49.000Z",
  "title": "Generic regularity of free boundaries for the thin obstacle problem",
  "authors": [
    "Xavier Fernández-Real",
    "Damià Torres-Latorre"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The free boundary for the Signorini problem in $\\mathbb{R}^{n+1}$ is smooth outside of a degenerate set, which can have the same dimension ($n-1$) as the free boundary itself. In [FR21] it was shown that generically, the set where the free boundary is not smooth is at most $(n-2)$-dimensional. Our main result establishes that, in fact, the degenerate set has zero $\\mathcal{H}^{n-3-\\alpha_0}$ measure for a generic solution. As a by-product, we obtain that, for $n+1 \\leq 4$, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in $\\mathbb{R}^3$ and $\\mathbb{R}^4$ for the thin obstacle problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-02T13:54:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35",
      "35B65"
    ],
    "keywords": [
      "free boundary",
      "thin obstacle problem",
      "generic regularity",
      "degenerate set",
      "main result establishes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}