{
  "abstract": "In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations $$σ_k(D^2u)=0\\quad\\text{in}~\\mathbb{R}^n\\setminus\\overlineΩ$$ in three dimensional regimes. For $2\\le k<n$ and smooth strictly star-shaped domain $Ω\\Subset\\R^n$, we establish ball rigidity results in all three dimensional regimes. If $2\\leq k<\\frac{n}{2}$, the solution with boundary value $-1$ and limit zero at infinity is treated under strict $(k-1)$-convexity of $\\partialΩ$. If $k=\\frac{n}{2}$, the solution with boundary value $0$ and logarithmic growth at infinity is considered under strict $(k-1)$-convexity of $\\partialΩ$; if $k>\\frac{n}{2}$, the solution with boundary $1$ and fundamental power-growth at infinity is considered under strict $(k-1)$-convexity of $\\partialΩ$. In each case, for a positive constant $c$, $|Du|=c>0$ on $\\partialΩ$ forces $Ω$ to be a Euclidean ball and determines the solution explicitly. The three arguments are organized by the single parameter $q=(n-k)/k$ and the scale function \\[ F_q(t)=\\frac{t^{1-q}-1}{1-q},\\qquad F_1(t)=\\log t. \\] After normalization, $U=F_q(v)$ and $M_q[v]=vD^2v-qDv\\otimes Dv$ satisfies $σ_k(M_q[v])=0$. A unified contact-point calculation gives the sharp bound $|Dv|\\le b$;a boundary viscosity contact then yields $H_k\\ge qbH_{k-1}$. The integral mechanism that closes the argument changes at the critical exponent: Rellich--Pohozaev identities are used for $q>1$, a scale-invariant Wronskian--Newton current for $q=1$, and a renormalized Newton--Jacobi mass for $q<1$. When $q\\leq 1,$ we also obtain a global strict k-convex defining function by Minkowski gauge, this help us extend the exterior construction from convex domains to strictly star-shaped domains.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03339",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhihui Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bq9nd1wjee1p2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03339",
  "fulltext_url": null,
  "id": "bq9nd1wjee1p2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03339",
  "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": "W7208730633",
  "pdf_url": "https://arxiv.org/pdf/2609.03339",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bq9nd1wjee1p2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03339",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03339"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation",
  "title_text": "Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/bq9nd1wjee1p2",
  "venue": "arXiv",
  "versions": []
}