{
  "id": "0712.2147",
  "version": "v2",
  "published": "2007-12-13T13:11:31.000Z",
  "updated": "2008-06-05T19:34:00.000Z",
  "title": "Sobolev homeomorphisms and Poincare inequality",
  "authors": [
    "V. Gol'dshtein",
    "A. Ukhlov"
  ],
  "comment": "In the first version, there was an inaccuracy in Theorem 4. In the revised version added additional assumptions",
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "We study global regularity properties of Sobolev homeomorphisms on $n$-dimensional Riemannian manifolds under the assumption of $p$-integrability of its first weak derivatives in degree $p\\geq n-1$. We prove that inverse homeomorphisms have integrable first weak derivatives. For the case $p>n$ we obtain necessary conditions for existence of Sobolev homeomorphisms between manifolds. These necessary conditions based on Poincar\\'e type inequality: $$ \\inf_{c\\in \\mathbb R} \\|u-c\\mid L_{\\infty}(M)\\|\\leq K \\|u\\mid L^1_{\\infty}(M)\\|. $$ As a corollary we obtain the following geometrical necessary condition: {\\em If there exists a Sobolev homeomorphisms $\\phi: M \\to M'$, $\\phi\\in W^1_p(M, M')$, $p>n$, $J(x,\\phi)\\ne 0$ a. e. in $M$, of compact smooth Riemannian manifold $M$ onto Riemannian manifold $M'$ then the manifold $M'$ has finite geodesic diameter.}}",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-06-05T19:34:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "30C65"
    ],
    "keywords": [
      "sobolev homeomorphisms",
      "poincare inequality",
      "necessary condition",
      "compact smooth riemannian manifold",
      "study global regularity properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.2147G"
    }
  }
}