{
  "abstract": "Let $N\\ge1$, $p\\in[1,\\infty)$, and $Ω\\subset\\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\\ge2$. For any given $γ\\in\\mathbb R\\setminus\\{0\\}$ and $λ\\in(0,\\infty)$, consider the weak-type nonlocal functional \\begin{align*} F_{λ,p,γ}(u;Ω):=λ^p \\underset{|u(x)-u(y)|>λ|x-y|^{1+\\fracγ{p}}} {\\int_Ω\\int_Ω}|x-y|^{γ-N}\\,dy\\,dx. \\end{align*} In this article, we prove that, for $γ\\in(0,\\infty)$ as $λ\\to\\infty$ or for $γ\\in(-\\infty,-1]$ as $λ\\to0^+$, the family $F_{λ,p,γ}(\\cdot;Ω)$ converges, in the sense of $Γ$-convergence in $L^p(Ω)$, to the functional \\begin{align*} Ψ_{p,γ}^Ω(u) :=\\begin{cases} C_{N,p,γ}^{\\rm cell}\\displaystyle\\int_Ω|\\nabla u|^p\\,dx, &p\\in(1,\\infty)\\ \\hbox{and}\\ u\\in W^{1,p}(Ω),\\\\[2mm] C_{N,1,γ}^{\\rm cell}|Du|(Ω), &p=1\\ \\hbox{and}\\ u\\in BV(Ω),\\\\[1mm] \\infty,&\\hbox{otherwise}. \\end{cases} \\end{align*} where the positive constants $C_{N,p,γ}^{\\mathrm{cell}}$ are independent of $Ω$. As applications, we further show that, for $γ\\in(0,\\infty)$ as $λ\\to\\infty$ or for $γ\\in(-\\infty,-1]$ as $λ\\to0^+$, the weak-type nonlocal functional $F_{λ,p,γ}(\\cdot;\\mathbb R^N)$ converges, in the sense of $Γ$-convergence in $L^1_{\\rm loc}(\\mathbb R^N)$, to a similar functional; this gives an complete answer to the problem posed by Brezis et al. in [Section~7C, Anal. PDE 2024].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07001",
  "author_total": 5,
  "authors": [
    {
      "name": "Xiaosheng Lin"
    },
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Sibei Yang"
    },
    {
      "name": "Wen Yuan"
    },
    {
      "name": "Yangyang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8ww66e6jk2pz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07001",
  "fulltext_url": null,
  "id": "e8ww66e6jk2pz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07001",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.07001",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8ww66e6jk2pz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07001",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07001"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Full-Range $Γ$-Convergence of Weak-Type Nonlocal Functionals",
  "title_text": "Full-Range Γ-Convergence of Weak-Type Nonlocal Functionals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/e8ww66e6jk2pz",
  "venue": "arXiv",
  "versions": []
}