{
  "abstract": "Let $Ω\\subset\\mathbb R^n$, $n\\ge3$, be a bounded uniformly convex $C^{3,γ}$ domain, where $0<γ<1$. We study the limit as the positive Robin parameter $α\\to\\infty$. For the $2$-Hessian torsion problem, let $U$ be the admissible Dirichlet solution. For all sufficiently large $α$, we obtain a unique admissible Robin solution $u_α$ and establish the optimal-order estimate \\[ \\|u_α-U\\|_{C^{2,γ}(\\overlineΩ)} \\le Cα^{-1}. \\] Moreover, $-\\sqrt{-u_α}$ has a uniform positive lower bound on its Hessian, and hence $\\sqrt{-u_α}$ is strictly concave. For the normalized eigenvalue problem, let $(λ_D,φ_D)$ be the normalized negative admissible Dirichlet eigenpair, and assume $φ_D\\in C^{2,γ}(\\overlineΩ)$. For all sufficiently large $α$, there is a unique normalized negative admissible Robin eigenpair $(λ_α,φ_α)$, which satisfies \\[ \\|φ_α-φ_D\\|_{C^{2,γ}(\\overlineΩ)} +|λ_α-λ_D|\\le Cα^{-1}. \\] The Hessian of $-\\log(-φ_α)$ also has a uniform positive lower bound. The main analytic tool is a uniform Hölder estimate for linear Robin problems, with an $α^{-1}$ gain for $C^{2,γ}$ boundary data. A normalized augmented inverse removes the scaling kernel in the eigenvalue problem. A Schur-complement argument combines the $C^2$ convergence with boundary geometry to prove convexity up to the boundary and anisotropic Hessian lower bounds at the scale ${\\rm dist}(x,\\partialΩ)+α^{-1}$. The convergence rate $O(α^{-1})$ for the Robin solution functions is optimal even in $C^0$, as follows from their boundary conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00836",
  "author_total": 3,
  "authors": [
    {
      "name": "Shujun Shi"
    },
    {
      "name": "Shenghui Sun"
    },
    {
      "name": "Yuzhou Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1jkmnt1qa42gq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00836",
  "fulltext_url": null,
  "id": "1jkmnt1qa42gq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00836",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00836",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1jkmnt1qa42gq.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00836",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00836"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Power Concavity and Log-Concavity for Large-Parameter Robin 2-Hessian Problems",
  "title_text": "Power Concavity and Log-Concavity for Large-Parameter Robin 2-Hessian Problems",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/1jkmnt1qa42gq",
  "venue": "arXiv",
  "versions": []
}