{
  "id": "2308.13209",
  "version": "v1",
  "published": "2023-08-25T07:06:54.000Z",
  "updated": "2023-08-25T07:06:54.000Z",
  "title": "Generic properties in free boundary problems",
  "authors": [
    "Xavier Fernández-Real",
    "Hui Yu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work, we show the generic uniqueness of minimizers for a large class of energies, including the Alt-Caffarelli and Alt-Phillips functionals. We then prove the generic regularity of free boundaries for minimizers of the one-phase Alt-Caffarelli and Alt-Phillips functionals, for a monotone family of boundary data $\\{\\varphi_t\\}_{t\\in(-1,1)}$. More precisely, we show that for a co-countable subset of $\\{\\varphi_t\\}_{t\\in(-1,1)}$, minimizers have smooth free boundaries in $\\mathbb{R}^5$ for the Alt-Caffarelli and in $\\mathbb{R}^3$ for the Alt-Phillips functional. In general dimensions, we show that the singular set is one dimension smaller than expected for almost every boundary datum in $\\{\\varphi_t\\}_{t\\in(-1,1)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-25T07:06:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35",
      "35A02"
    ],
    "keywords": [
      "free boundary problems",
      "generic properties",
      "alt-phillips functional",
      "boundary datum",
      "minimizers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}