{
  "abstract": "After Corollary 1.3 in Calculus of Variations and Partial Differential Equations 60 (2021), Paper 231, Djitte, Fall, and Weth asked whether, when $1<p<2$, a volume-constrained local minimizing domain for the best fractional Sobolev constant must still be a ball. This paper solves that problem. Let $0<s<1$, $1<p<2$, and let $Ω\\subset\\mathbb{R}^N$ be a bounded $C^3$ domain. If $Ω$ is a local minimizing domain for $λ_{s,p}(Ω)=\\inf{[u]_s^2:u\\in\\mathcal{H}*0^s(Ω),\\ |u|*{L^p(Ω)}=1}$ under smooth volume-preserving deformations, then $Ω$ is a ball. The proof first uses the fractional Hadamard formula to reduce shape minimality to the overdetermined boundary condition $u/δ^s=C_0$. To overcome the moving-plane obstruction caused by the failure of $u^{p-1}$ to be Lipschitz at zero, we establish a weighted singular narrow-domain maximum principle whose absorption factor is exactly the $sp/N$ power of the measure of the negative set. The difficulty at a corner is resolved by a finite boundary expansion: setting $ρ=sp$, the boundary quotient is composed of finitely many constant-coefficient normal powers $δ^{kρ}$ and a $C^{1,\\varepsilon}$ remainder; when $kρ=1$, the unique resonant correction is $δ\\logδ$. This expansion makes all lower-order normal terms on the two sides of an orthogonal corner cancel, thereby yielding the first-order tangential vanishing required by the moving-plane corner lemma. The method does not require the domain to be convex and covers the full range $0<s<1$ and $1<p<2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13215",
  "author_total": 1,
  "authors": [
    {
      "name": "Zikang Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7s2ds352e8pv9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13215",
  "fulltext_url": null,
  "id": "7s2ds352e8pv9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13215",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213359831",
  "pdf_url": "https://arxiv.org/pdf/2609.13215",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7s2ds352e8pv9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13215",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13215"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Ball Rigidity of Local Minimizing Domains for the Best Fractional Sobolev Constant: The Subquadratic Case",
  "title_text": "Ball Rigidity of Local Minimizing Domains for the Best Fractional Sobolev Constant: The Subquadratic Case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7s2ds352e8pv9",
  "venue": "arXiv",
  "versions": []
}