{
  "abstract": "For each nonnegative integer $m$, we construct smooth symmetric $3\\times 3$ coefficient matrices $A_m$ satisfying the fixed ellipticity bound \\[ I\\leq A_m\\leq 2^{81}I \\] for which the smooth solutions of uniformly elliptic equations in nondivergence form \\[ \\text{tr}(A_m(x)D^2 u_m)=A_m(x):D^2u_m=0\\qquad\\text{in }B_2\\subset {\\mathbb R}^3 \\] have common Dirichlet data, satisfy $\\|u_m\\|_{L^\\infty(B_2)}\\leq1$, but \\[ \\lim_{m\\to \\infty}\\|Du_m\\|_{L^1(B_1)}=\\infty. \\] Thus, there is no interior $W^{1,1}$ estimate depending only on ellipticity in dimension three, and consequently no such $W^{1,p}$ estimate for any $p\\geq1$. This resolves in the negative an open question raised by Nadirashvili, Tkachev, and Vlăduţ. The construction also gives a uniformly convergent limit $u\\notin \\text{BV}_{\\rm loc}(B_1)$ for a measurable uniformly elliptic coefficient matrix obtained as an $L^1$ limit of the $A_m$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.13380",
  "author_total": 3,
  "authors": [
    {
      "name": "Nam Q. Le"
    },
    {
      "name": "Qi Sun"
    },
    {
      "name": "Hung V. Tran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8cxerdm1bytta.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.13380",
  "fulltext_url": null,
  "id": "8cxerdm1bytta",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.13380",
  "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": "W7203502269",
  "pdf_url": "https://arxiv.org/pdf/2608.13380",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8cxerdm1bytta.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.13380",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.13380"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three",
  "title_text": "Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/8cxerdm1bytta",
  "venue": "arXiv",
  "versions": []
}