{
  "id": "2210.04730",
  "version": "v2",
  "published": "2022-10-10T14:37:53.000Z",
  "updated": "2022-10-11T06:52:50.000Z",
  "title": "Weak and strong $L^p$-limits of vector fields with finitely many integer singularities in dimension $n$",
  "authors": [
    "Riccardo Caniato",
    "Filippo Gaia"
  ],
  "comment": "67 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "For every given $p\\in [1,+\\infty)$ and $n\\in\\mathbb{N}$ with $n\\ge 1$, the authors identify the strong $L^p$-closure $L_{\\mathbb{Z}}^p(D)$ of the class of vector fields having finitely many integer topological singularities on a domain $D$ which is either bi-Lipschitz equivalent to the open unit $n$-dimensional cube or to the boundary of the unit $(n+1)$-dimensional cube. Moreover, the authors prove that $L_{\\mathbb{Z}}^p(D)$ is weakly sequentially closed for every $p\\in (1,+\\infty)$ whenever $D$ is an open domain in $\\mathbb{R}^n$ which is bi-Lipschitz equivalent to the open unit cube. As a byproduct of the previous analysis, a useful characterisation of such class of objects is obtained in terms of existence of a (minimal) connection for their singular set.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-10-11T06:52:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector fields",
      "integer singularities",
      "dimensional cube",
      "bi-lipschitz equivalent",
      "open unit cube"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 67,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}