{
  "abstract": "The aim of this paper is to establish the existence of extremal functions for the Trudinger-Moser inequality with logarithmic weights $(-\\log |x|)^β$ in two dimensions throughout the full range $0\\le β<1$. We also extend recent results on the existence of local maximizers for the Trudinger-Moser functional beyond the critical threshold. Our approach uses polygonal profiles to establish a lower bound uniform in the weight parameter. An explicit thirty-node profile gives the lower bound $\\mathrm{TM}_{4π}>4.38421793217\\,π$ for the classical critical supremum on the unit disk, improving on the value $4.3556\\,π$ obtained in the computer experiments of Carleson and Chang. This lower bound has a purely analytic proof, which uses only elementary inequalities and exact rational arithmetic. Numerical experiments with polygonal profiles of up to $102400$ nodes suggest that this lower bound can be raised to $4.402732044\\,π$, which improves the recent computational lower bound by Li and Sun.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33439",
  "author_total": 1,
  "authors": [
    {
      "name": "José Francisco de Oliveira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/enkysf6897g6q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33439",
  "fulltext_url": null,
  "id": "enkysf6897g6q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33439",
  "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": "W7223784578",
  "pdf_url": "https://arxiv.org/pdf/2610.33439",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/enkysf6897g6q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.33439",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33439"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Extremals for Trudinger-Moser inequalities via polygonal profiles",
  "title_text": "Extremals for Trudinger-Moser inequalities via polygonal profiles",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/enkysf6897g6q",
  "venue": "arXiv",
  "versions": []
}