{
  "abstract": "We prove a sharp quantitative version of the Talenti rearrangement inequality. Specifically, we show that, for any $p\\in [1;+\\infty]$, for any $f\\in L^\\infty(Ω),\\, f\\geq 0,\\, f\\not\\equiv 0$, for any open set $Ω\\subset \\mathbb{R}^d$ with finite volume, the solution $u_{f,Ω}$ to $-Δu_{f,Ω}=f,\\, u_{f,Ω} \\in W^{1,2}_0(Ω)$, satisfies, in dimension $d\\geq 1$, $\\Vert u_{f^\\sharp,Ω^\\sharp}\\Vert_{L^p(Ω^\\sharp)}-\\Vert u_{f,Ω}\\Vert_{L^p(Ω)}\\gtrsim α(Ω)^2+\\inf_{z\\in \\mathbb{R}^d}\\mathscr A(f,z)^2$ and, in dimension 1, $ \\Vert u_{f^\\sharp,Ω^\\sharp}\\Vert_{L^p(Ω^\\sharp)}-\\Vert u_{f,Ω}\\Vert_{L^p(Ω)}\\gtrsim α(Ω)+\\inf_{z\\in \\mathbb{R}}\\mathscr A(f,z)^2$ where $α(Ω)$ denotes the Fraenkel asymmetry of $Ω$, $\\mathscr A(f,z)$ a weighted asymmetry of $f$ with respect to some center $z$ and $Ω^\\sharp$ (resp. $f^\\sharp$) is the Schwarz rearrangement of $Ω$ (resp. of $f$). In dimension $d\\geq 2$, this estimate implies $ \\Vert u_{f^\\sharp,Ω^\\sharp}\\Vert_{L^p(Ω^\\sharp)}-\\Vert u_{f,Ω}\\Vert_{L^p(Ω)}\\gtrsim α(Ω)^2+\\inf_{z\\in \\mathbb{R}^d}\\Vert f-f^\\sharp(\\cdot-z)\\Vert_{L^1}^2.$ Our result sharpens and extends recent quantitative versions, and does not rely on shape differentiation arguments, but on a generalisation of the quantitative Saint-Venant inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31369",
  "author_total": 3,
  "authors": [
    {
      "name": "Paolo Acampora"
    },
    {
      "name": "Jimmy Lamboley"
    },
    {
      "name": "Idriss Mazari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bbpfyeq2ba0fy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31369",
  "fulltext_url": null,
  "id": "bbpfyeq2ba0fy",
  "issns": [],
  "landing_url": "https://hal.science/hal-05836411v1",
  "language": "en",
  "licence": "CC-BY-ND-4.0",
  "licence_label": "CC BY-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
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      "openalex",
      "2604|Nonlinear Partial Differential Equations"
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  "openalex": "W7223292196",
  "pdf_url": "https://hal.science/hal-05836411/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bbpfyeq2ba0fy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05836411",
      "source": "hal",
      "url": "https://hal.science/hal-05836411v1"
    },
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31369",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31369"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp stability for the Talenti inequality",
  "title_text": "Sharp stability for the Talenti inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/bbpfyeq2ba0fy",
  "venue": "HAL",
  "versions": []
}