{
  "id": "1405.4765",
  "version": "v3",
  "published": "2014-05-19T15:33:19.000Z",
  "updated": "2014-09-02T00:37:47.000Z",
  "title": "Sobolev spaces of isometric immersions of arbitrary dimension and codimension",
  "authors": [
    "Robert L. Jerrard",
    "Mohammad Reza Pakzad"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove the $C^{1}$ regularity and developability of $W^{2,p}$ isometric immersions of $n$-dimensional flat domains into ${\\mathbb R}^{n+k}$ where $p\\ge \\min\\{2k, n\\}$. Another parallel consequence of our methods is a similar regularity and rigidity result for the $W^{2,n}$ solutions of the degenerate Monge-Amp\\`ere equations in $n$ dimensions. The analysis also applies to the situations when the degeneracy is extended to $(k+1)\\times (k+1)$ minors of the Hessian matrix and the solution is $W^{2,p}$, with $p\\ge \\min\\{2k, n\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-19T15:33:19.000Z",
      "comment": "32 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-31T21:37:38.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-02T00:37:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B99",
      "53C24",
      "58D10",
      "30L05"
    ],
    "keywords": [
      "isometric immersions",
      "sobolev spaces",
      "arbitrary dimension",
      "codimension",
      "dimensional flat domains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.4765J"
    }
  }
}