{
  "abstract": "We establish a principle that converts qualitative regularity into uniform interior estimates for degenerate equations. For the $p$-Laplace equation, $p>2$, we prove that local $C^{1,γ}$ regularity of every solution, without any uniform bound on the corresponding seminorms, implies uniform $C^{1,β}$ estimates for every $0<β<γ$. Thus, the optimal qualitative and quantitative gradient Hölder exponents coincide as suprema in every dimension. In the plane, this yields estimates controlled by the $L^p$ norm of the solution for every exponent below the sharp Iwaniec--Manfredi threshold. Applied to Xu's recent qualitative $C^{1,1/3}$ regularity for planar infinity-harmonic functions, we obtain uniform interior $C^{1,β}$ estimates for every $0<β<1/3$. For the nonlocal superconductivity equation $|Du|(-Δ)^s u=0$, with $1/2<s<1$, we prove that pointwise differentiability of all solutions with integrable tails implies uniform interior $C^{1,α}$ estimates for bounded solutions, for some $α>0$. The proofs combine compactness for shifted equations with an improvement-of-flatness iteration at variable scales.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13137",
  "author_total": 3,
  "authors": [
    {
      "name": "Aelson Sobral"
    },
    {
      "name": "Eduardo V. Teixeira"
    },
    {
      "name": "José Miguel Urbano"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dqher323fn6pq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13137",
  "fulltext_url": null,
  "id": "dqher323fn6pq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13137",
  "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": "W7221571807",
  "pdf_url": "https://arxiv.org/pdf/2610.13137",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dqher323fn6pq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13137",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13137"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Qualitative-to-quantitative regularity for degenerate equations",
  "title_text": "Qualitative-to-quantitative regularity for degenerate equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/dqher323fn6pq",
  "venue": "arXiv",
  "versions": []
}