{
  "abstract": "In this paper, we establish global $C^{1,1}$ solvability of the Dirichlet problem for the degenerate $2$-Hessian equation on bounded strictly mean convex $C^{3,1}$ domains $Ω$, with general $C^{3,1}$ boundary values and nonnegative right-hand sides $f\\in C^{1,1}(\\overlineΩ)$, resolving the $k=2$ case of a longstanding open problem. We uncover a global semiconvexity structure for general $2$-admissible solutions by proving a lower bound for $σ_{3}[D^{2}u]$ independent of $\\inf_Ωf$. For $3\\leq k\\leq n-1$, we also establish global semiconvexity for $k$-admissible solutions on the unit ball with $C^{3}$ boundary values and $f^{1/(k-1)}\\in C^{1,1}(\\overline{B}_{1})$, with a bound independent of $\\inf_{B_{1}}f$. A counterexample shows that the $C^{3,1}$ boundary value assumption for global $C^{1,1}$ solvability is sharp in the Hölder scale.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06853",
  "author_total": 1,
  "authors": [
    {
      "name": "Yasheng Lyu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b8jnq5y216kh4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06853",
  "fulltext_url": null,
  "id": "b8jnq5y216kh4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06853",
  "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": "W7220738731",
  "pdf_url": "https://arxiv.org/pdf/2610.06853",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b8jnq5y216kh4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06853",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06853"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Dirichlet problem for the degenerate $2$-Hessian equation",
  "title_text": "On the Dirichlet problem for the degenerate $2$-Hessian equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/b8jnq5y216kh4",
  "venue": "arXiv",
  "versions": []
}