{
  "abstract": "We study the quantitative codimension-two estimate for the singular set in the boundary neighborhoods of the solutions of \\begin{equation*} Δu+V(x)u=0\\qquad\\text{in }Ω, \\qquad u=0\\qquad\\text{on }\\partialΩ, \\end{equation*} where $Ω\\subset\\mathbb R^n$ is a bounded $C^{1,\\mathrm{Dini}}$ domain and $V\\in L^p(Ω)$ for some $p>n$. We first prove the explicit upper bound for the doubling index is given by $C(n,p,Ω)(1+\\|V\\|_{L^p(Ω)}^{\\frac{2p}{3p-2n}})$.The analytic input is an interior volume estimate for solutions of second order elliptic equations with uniformly elliptic Dini leading coefficients and $V\\in L^p$. Using the quantitative doubling index bound, boundary flattening, we show an explicit upper bound for singular sets in the neighborhood of the boundary of the $C^{1,\\mathrm{Dini}}$ domain.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21988",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhiwei Wang"
    },
    {
      "name": "Jiuyi Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4csncmy8da01a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21988",
  "fulltext_url": null,
  "id": "4csncmy8da01a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21988",
  "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": "W7213922236",
  "pdf_url": "https://arxiv.org/pdf/2609.21988",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4csncmy8da01a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2609.21988",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21988"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Bounds of singular sets for elliptic equations in $C^{1, Dini}$ domains with singular potentials",
  "title_text": "Bounds of singular sets for elliptic equations in C^1, Dini domains with singular potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/4csncmy8da01a",
  "venue": "arXiv",
  "versions": []
}