{
  "id": "2409.01260",
  "version": "v1",
  "published": "2024-09-02T13:53:47.000Z",
  "updated": "2024-09-02T13:53:47.000Z",
  "title": "Weak limits of Sobolev homeomorphisms are one to one",
  "authors": [
    "Ondřej Bouchala",
    "Stanislav Hencl",
    "Zheng Zhu"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove that the key property in models of Nonlinear Elasticity which corresponds to the non-interpenetration of matter, i.e. injectivity a.e., can be achieved in the class of weak limits of homeomorphisms under very minimal assumptions. Let $\\Omega\\subseteq \\mathbb{R}^n$ be a domain and let $p>\\left\\lfloor\\frac{n}{2}\\right\\rfloor$ for $n\\geq 4$ or $p\\geq 1$ for $n=2,3$. Assume that $f_k\\in W^{1,p}$ is a sequence of homeomorphisms such that $f_k\\rightharpoonup f$ weakly in $W^{1,p}$ and assume that $J_f>0$ a.e. Then we show that $f$ is injective a.e.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-02T13:53:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak limits",
      "sobolev homeomorphisms",
      "nonlinear elasticity",
      "minimal assumptions",
      "non-interpenetration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}