{
  "abstract": "We consider the Dirichlet problem for the $p$-Laplacian on a bounded Lipschitz domain $Ω\\subset\\mathbb R^d$, with boundary data given by the indicator of a closed set. At a smooth boundary point where the data vanish, we study the upper and lower inward normal Dini derivatives as $p\\downarrow1$. We give sufficient geometric conditions for these derivatives to be of order $(p-1)^{-1}$, and other conditions under which they decay exponentially in $1/(p-1)$. Whether explosion or decay occurs is not determined locally: on one fixed planar domain with real-analytic boundary, changing the data on a set of arbitrarily small total length, uniformly separated from the observation point, changes explosion into exponential decay. We also exhibit a critical example of a cylinder in $\\mathbb R^{d+1}$ where the inward normal derivative is of order $\\sqrt{d/(p-1)}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.03322",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuval Peres"
    },
    {
      "name": "Han Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7kb3jx8qqkg2m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.03322",
  "fulltext_url": null,
  "id": "7kb3jx8qqkg2m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.03322",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7160381506",
  "pdf_url": "https://arxiv.org/pdf/2605.03322",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7kb3jx8qqkg2m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2605.03322",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.03322"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Explosion versus decay for boundary derivatives of $p$-harmonic functions as $p$ tends to 1: nonlocality",
  "title_text": "Explosion versus decay for boundary derivatives of p-harmonic functions as p tends to 1: nonlocality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/7kb3jx8qqkg2m",
  "venue": "arXiv",
  "versions": []
}