{
  "abstract": "We settle the planar sharp Hessian-integrability conjecture of Armstrong, Silvestre, and Smart [\\emph{Comm. Pure Appl. Math.} \\textbf{65} (2012), 1169--1184] for viscosity supersolutions of fully nonlinear uniformly elliptic equations. If $κ=Λ/λ$, then the optimal exponent in the $W^{2,ε}$ regularity theory in the plane is exactly $$ ε_2(κ)=\\frac{2}{κ+1}. $$ The same mechanism reaches the known upper obstruction in dimension three throughout the full range $1\\le κ\\le 4$ and therefore gives $$ ε_3(κ)=\\frac{3}{2κ+1} \\qquad(1\\leκ\\le4). $$ Beyond this threshold, it yields a closed algebraic lower bound and identifies the precise spectral configuration responsible for the remaining gap. The proof introduces a spectrally resolved continuum-in-opening mechanism for contact geometry. Throughout the contact evolution, the vertex Jacobian retains the negative index of the Hessian and its mean negative curvature---data erased by the classical pointwise reduction---and these variables are optimized only after integration over all openings. This reduces the analysis to a finite family of explicit one-dimensional kernels whose critical exponents recover the sharp planar threshold, identify the exact three-dimensional regime above, and expose a broader principle for extracting second-order regularity from the spectral geometry of contact sets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06817",
  "author_total": 2,
  "authors": [
    {
      "name": "Thialita M. Nascimento"
    },
    {
      "name": "Eduardo V. Teixeira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ec0gchnky258.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06817",
  "fulltext_url": null,
  "id": "2ec0gchnky258",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06817",
  "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": "W7212001699",
  "pdf_url": "https://arxiv.org/pdf/2609.06817",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ec0gchnky258.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06817",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06817"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions",
  "title_text": "Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/2ec0gchnky258",
  "venue": "arXiv",
  "versions": []
}