{
  "abstract": "We prove that the Dirichlet $p$-torsion function $u_p$ converges to the distance to the boundary $d$ at the rate $O(1/p)$ in the uniform norm, on every bounded domain and with a constant depending only on the dimension and the diameter. The upper bound comes from a radial barrier with its pole at a boundary point, which is an admissible comparison function for $p>n$ and needs no boundary regularity; the lower bound comes from an inscribed ball. On the ball the error equals $(1+\\log n)/p+O(p^{-2})$, so the order $1/p$ cannot be improved in general. Under a uniform exterior ball condition an annular barrier gives $u_p\\le K_Ω^{1/(p-1)}d$ for every $p>1$, with $K_Ω$ explicit in the dimension, the diameter and the exterior radius; on convex sets the constant is the diameter, and a multiple of $d$ is a supersolution whenever $-Δd$ has a positive distributional lower bound. On $C^2$ domains, the same barrier and the classical gradient maximum principle give $\\norm{\\nabla u_p}_{L^\\infty}^{p-1}\\le K_Ω$. When $-Δd$ is a measure of finite total variation, which we prove for $C^1$ domains with a uniform exterior ball, the gradients converge in $L^q$ at the rate $O(p^{-1/2})$ for $q\\le2$ and $O(p^{-1/q})$ for $q\\ge2$; these exponents are not claimed to be sharp. The explicit ball profile shows that the gradients do not converge uniformly.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06534",
  "author_total": 3,
  "authors": [
    {
      "name": "Farid Bozorgnia"
    },
    {
      "name": "Masoud Bayrami-Aminlouee"
    },
    {
      "name": "Khudoyor Mamayusupov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8d0ymjkz78kr6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06534",
  "fulltext_url": null,
  "id": "8d0ymjkz78kr6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06534",
  "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": "W7212000880",
  "pdf_url": "https://arxiv.org/pdf/2609.06534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8d0ymjkz78kr6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.06534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A quantitative convergence rate for torsional creep to the distance function as p tends to infinity",
  "title_text": "A quantitative convergence rate for torsional creep to the distance function as p tends to infinity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/8d0ymjkz78kr6",
  "venue": "arXiv",
  "versions": []
}