{
  "abstract": "For $p>1$, with $p'=p/(p-1)$, we prove the sharp dual inequality $$ \\int_0^1-\\log\\left(1-\\frac{\\left|\\int_0^xf(y) dy\\right|^{p'}}{x}\\right) dx\\leq\\mathsf{K}_p, \\qquad \\|f\\|_{L^p(0,1)}\\leq1, $$ where $\\mathsf{K}_p$ is the logarithm of the optimal value in Moser's one-dimensional formulation of the Moser-Trudinger inequality. The dual functional follows from the Gibbs variational principle, monotone transport and the Legendre transform. Its supremum is attained, giving a dual proof of the existence theorems of Carleson-Chang (1986) and Hudson-Leckband (2002); the classical concentration threshold is realized by equality cases in Hoelder's inequality. For $p=2$ we prove, with rigorous computer-assisted estimates, that the nonnegative dual maximizer is unique and nondegenerate. Hence the extremal of the Moser-Trudinger inequality on a disk $D$ is unique up to sign, confirming conjectures of Adimurthi-Karthik-Giacomoni (2016) and of Chen-Lu-Xue-Zhu (2026). The positive extremal $u_*$ is nondegenerate, and, writing $\\mathcal{M}_D(u)=|D|^{-1}\\int_De^{4πu^2} dx$, we obtain $$ \\mathcal{M}_D(u_*)-\\mathcal{M}_D(u) \\geq c \\min_{\\pm}\\|\\nabla(u\\mp u_*)\\|_{L^2(D)}^2 $$ for all $u\\in H_0^1(D)$ with $\\|\\nabla u\\|_{L^2(D)}\\leq1$. Here $c>0$, and the exponent two is optimal, giving a quantitative stability theorem of Bianchi-Egnell type.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25375",
  "author_total": 2,
  "authors": [
    {
      "name": "Jungang Li"
    },
    {
      "name": "Wen Sun"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6menwgvh1hb13.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25375",
  "fulltext_url": null,
  "id": "6menwgvh1hb13",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25375",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223033770",
  "pdf_url": "https://arxiv.org/pdf/2610.25375",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6menwgvh1hb13.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25375",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25375"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals",
  "title_text": "A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/6menwgvh1hb13",
  "venue": "arXiv",
  "versions": []
}