{
  "abstract": "We study the singular set of the free boundary for solutions of the thin Stefan problem in $\\mathbb{R}^{n+1}\\times \\mathbb{R}$, which models the melting of a thin sheet of ice over water. After a suitable transformation, we reduce to studying the parabolic Signorini (thin obstacle) problem with a time-monotonicity condition. First, we prove that the free boundary is locally the graph of a differentiable melting function $t=τ(x)$ defined on the $n$-dimensional thin space, and that the singular set coincides with the critical set of $τ$. Next, we show that the singular set has parabolic Hausdorff dimension at most $n$. We then establish generic regularity up to dimension $n+1=4$, proving that the free boundary is regular for almost every time. Finally, under smoothness assumptions on the obstacle, we show that the singular set is contained in a $C^\\infty$-hypersurface outside a set of parabolic dimension at most $n-1$. Our analysis combines ideas developed for the classical Stefan problem with several new ingredients specific to the thin setting, including frequency formulas for second blow-ups, epiperimetric inequalities, higher order expansions, and a fine stratification of the singular set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10751",
  "author_total": 3,
  "authors": [
    {
      "name": "Matteo Carducci"
    },
    {
      "name": "Roberto Colombo"
    },
    {
      "name": "Clara Torres-Latorre"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/btn37xbr2weyb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10751",
  "fulltext_url": null,
  "id": "btn37xbr2weyb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10751",
  "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": "W7221144683",
  "pdf_url": "https://arxiv.org/pdf/2610.10751",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/btn37xbr2weyb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10751",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10751"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The singular set in the thin Stefan problem",
  "title_text": "The singular set in the thin Stefan problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/btn37xbr2weyb",
  "venue": "arXiv",
  "versions": []
}