{
  "id": "2001.02226",
  "version": "v1",
  "published": "2020-01-07T18:56:36.000Z",
  "updated": "2020-01-07T18:56:36.000Z",
  "title": "Trace theory for Sobolev mappings into a manifold",
  "authors": [
    "Petru Mironescu",
    "Jean Van Schaftingen"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP",
    "math.DG",
    "math.FA"
  ],
  "abstract": "We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (\\mathbb{B}^{m-1}\\times (0,1), \\mathcal{N})$ of Sobolev mappings with values into a compact manifold $\\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $\\pi_p (\\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $\\pi_1 (\\mathcal{N})$ is finite and $\\pi_2 (\\mathcal{N}) \\simeq \\dotsb \\simeq \\pi_{\\lfloor p - 1 \\rfloor} (\\mathcal{N}) \\simeq \\{0\\}$. We present several open problems connected to the extension problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-07T18:56:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46T10",
      "46E35",
      "58D15"
    ],
    "keywords": [
      "sobolev mappings",
      "trace theory",
      "extension problem",
      "open problems",
      "current state"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}