{
  "abstract": "Let $u,v$ be two nontrivial functions that solve $$ \\mathrm{div}(A\\nabla u)=\\mathrm{div}(A\\nabla v)=0\\qquad\\mathrm{in \\ }B_r\\subset\\mathbb R^n,\\quad n\\geq2,\\quad r>0, $$ with $A$ symmetric, uniformly elliptic and Lipschitz continuous, and assume the zero set inclusion $Z(u)\\subseteq Z(v)$. We prove that the ratio $w=v/u$ continuously extends, together with its gradient, across $Z(u)=\\{u=0\\}$. In particular, $w\\in C^{1,α}_{\\mathrm{loc}}(B_{r})$ for any $α\\in(0,1)$. Furthermore, given any integer $k\\geq2$ and $α\\in(0,1)$, if $A\\in C^{k-1,α}(B_r)$, then $w\\in C^{k,α}_{\\mathrm{loc}}(B_{r})$. The estimates are uniform in classes of coefficients $A$ with prescribed ellipticity and regularity bounds, and in normalized classes of denominators $u$ with uniformly bounded doubling constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28077",
  "author_total": 1,
  "authors": [
    {
      "name": "Stefano Vita"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cyy35hccj9mv1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28077",
  "fulltext_url": null,
  "id": "cyy35hccj9mv1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28077",
  "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": "W7223159582",
  "pdf_url": "https://arxiv.org/pdf/2610.28077",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cyy35hccj9mv1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28077",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28077"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Schauder estimates for ratios of elliptic solutions under zero set inclusion",
  "title_text": "Schauder estimates for ratios of elliptic solutions under zero set inclusion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/cyy35hccj9mv1",
  "venue": "arXiv",
  "versions": []
}