{
  "abstract": "A classical consequence of Caffarelli's fully nonlinear regularity theory [L. A. Caffarelli, Ann. of Math. (2) 130 (1989), no. 1, 189-213] is that viscosity solutions of uniformly elliptic equations $F(D^2u)=f$, with $f\\in L^p$, $p > n$, are locally $C^{1,γ}$ for every $γ< \\min\\{α_H,σ_p\\}$. Here $α_H=α_H(n,λ,Λ)\\in(0,1)$ denotes the universal Hölder exponent for gradient regularity of $F(D^2h)=0$, while $σ_p=1-n/p$ is the scaling exponent of the source term. In the source-limited regime $σ_p < α_H$, the singularity of $f$ is the decisive obstruction and the endpoint $γ=σ_p$ is attainable. In the homogeneous-limited regime $α_H\\leσ_p$, however, classical theory only yields $γ< α_H$, leaving the limiting differentiability estimate unquantified. This is the endpoint gap addressed here. When $α_H < σ_p$, we prove that solutions admit pointwise Taylor expansions satisfying $|u(x)-u(x_0)-Du(x_0)\\cdot(x-x_0)|\\lesssim |x-x_0|^{1+α_H}\\left(1+\\log\\frac{1}{|x-x_0|}\\right)^m$. Thus the homogeneous differentiability scale is reached up to an explicit logarithmic defect. At the critical threshold $α_H=σ_p$, finite logarithmic powers no longer close the iteration; nevertheless, a slower selection of scales yields $|u(x)-u(x_0)-Du(x_0)\\cdot(x-x_0)|=O\\left(|x-x_0|^{1+α_H}\\exp\\left(A\\sqrt{1+\\log\\frac{1}{|x-x_0|}}\\right)\\right)$. Both estimates improve the full family of classical sub-endpoint $C^{1,γ}$ bounds, $γ< α_H$, by quantifying differentiability at the limiting homogeneous exponent. The proof introduces a new scale-selection mechanism for endpoint Campanato-type recurrences, suggesting a flexible tool whenever the limiting smoothness is dictated by the homogeneous theory itself. We also discuss the role and possible optimality of the resulting logarithmic defects.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16326",
  "author_total": 2,
  "authors": [
    {
      "name": "Aelson Sobral"
    },
    {
      "name": "Eduardo V. Teixeira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d6qkem50z5xyg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16326",
  "fulltext_url": null,
  "id": "d6qkem50z5xyg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16326",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213434397",
  "pdf_url": "https://arxiv.org/pdf/2609.16326",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d6qkem50z5xyg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16326",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16326"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Endpoint Differentiability Moduli for Fully Nonlinear Elliptic Equations",
  "title_text": "Endpoint Differentiability Moduli for Fully Nonlinear Elliptic Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/d6qkem50z5xyg",
  "venue": "arXiv",
  "versions": []
}