{
  "id": "2004.14506",
  "version": "v1",
  "published": "2020-04-29T22:29:32.000Z",
  "updated": "2020-04-29T22:29:32.000Z",
  "title": "Kernel of Trace Operator of Sobolev Spaces on Lipschitz Domain",
  "authors": [
    "I-Shing Hu"
  ],
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We are going to show that \\[ \\overline{C_0^\\infty \\left(D\\right)} = \\overline{C_c^\\infty \\left(D\\right)} \\] in $W^{1,p}\\left(D\\right)$, $p\\in[1,\\infty)$, on Lipschitz domain $D$ by showing both sides are kernel of trace operator \\[ T:\\,W^{1,p}(D)\\rightarrow L^{p}(\\partial D). \\] In Grisvard's book \\cite{key-1}, Corollary 1.5.1.6 states a much more general result which covers above. But we cannot find a complete proof in literature. Fortunately, we apply some change of variables formulas from Evans and Gariepy's book (Theorem 3.9 and 3.11, \\cite{key-3}), for Lipschitz coordinate transformation, in to improve the proof of \\[ \\overline{C_0^\\infty \\left(D\\right)} = \\overline{C_c^\\infty \\left(D\\right)} \\] in $W^{1,p}\\left(D\\right)$, $p\\in[1,\\infty)$, from $\\mathcal{C}^{1}$ (Theorem 2 in {\\S}5.5 of Evans' book \\cite{key-2}) to Lipschitz domain.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-29T22:29:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35"
    ],
    "keywords": [
      "lipschitz domain",
      "trace operator",
      "sobolev spaces",
      "lipschitz coordinate transformation",
      "complete proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}