{
  "id": "1801.07469",
  "version": "v1",
  "published": "2018-01-23T10:26:25.000Z",
  "updated": "2018-01-23T10:26:25.000Z",
  "title": "Non-local Torsion functions and Embeddings",
  "authors": [
    "Giovanni Franzina"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Given $s \\in (0,1)$, we discuss the embedding of $\\mathcal D^{s,p}_0(\\Omega)$ in $L^q(\\Omega)$. In particular, for $1\\le q < p$ we deduce its compactness on all open sets $\\Omega\\subset \\mathbb R^N$ on which it is continuous. We then relate, for all q up the fractional Sobolev conjugate exponent, the continuity of the embedding to the summability of the function solving the fractional torsion problem in $\\Omega$ in a suitable weak sense, for every open set $\\Omega$. The proofs make use of a non-local Hardy-type inequality in $\\mathcal D^{s,p}_0(\\Omega)$, involving the fractional torsion function as a weight.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-23T10:26:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "46E35",
      "34K37"
    ],
    "keywords": [
      "non-local torsion functions",
      "fractional sobolev conjugate exponent",
      "open set",
      "fractional torsion problem",
      "non-local hardy-type inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}