{
  "abstract": "Let $0<κ<1$ and set $p_+=2/(1-κ)$ and $p_-=2/(1+κ)$. We construct a coefficient $κ\\leq a\\leqκ^{-1}$, smooth outside a compact set of Lebesgue measure zero, for which the $W^{1,2}_{p_+}$ estimate fails for the one-space dimensional nondivergence form parabolic equation. The corresponding solution has second spatial derivative in the weak $L_{p_+}$ space, but not in $L_{p_+}$. By duality, the $W^{1,2}_{p_-}$ a priori estimate also fails. Conversely, we demonstrate that the $W^{1,2}_p$ estimate holds and the equation is uniquely solvable for $|p-2|<cκ$, showing that the size of the solvability interval around $2$ has optimal order $κ$ as $κ\\downarrow0$, which addresses a question raised in [17]. These results imply that the Alexandrov maximum principle holds for $p>2-cκ$ and this order is sharp as $κ\\to 0$. The corresponding results for divergence form equations are also obtained.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25568",
  "author_total": 2,
  "authors": [
    {
      "name": "Hongjie Dong"
    },
    {
      "name": "Zongyuan Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e398mtbvgfbdq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25568",
  "fulltext_url": null,
  "id": "e398mtbvgfbdq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25568",
  "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": "W7214068073",
  "pdf_url": "https://arxiv.org/pdf/2609.25568",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e398mtbvgfbdq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25568",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25568"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle",
  "title_text": "On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/e398mtbvgfbdq",
  "venue": "arXiv",
  "versions": []
}