{
  "abstract": "Brezis, Merle, and Riviére [Arch. Rational Mech. Anal. 1994] proved that, if a smooth solution $u:\\mathbb{R}^2\\to\\mathbb{C}$ of the planar entire Ginzburg--Landau equation $$ -Δu = u(1-|u|^2)\\quad \\text{in}\\quad \\mathbb{R}^2 $$ satisfies the finite potential energy estimate that $$ \\int_{\\mathbb{R}^2}\\left[1-|u(x)|^2\\right]^2\\,dx < \\infty, $$ then it has the asymptotic property that $$ |u(x)|\\to 1\\quad \\text{as}\\quad |x|\\to\\infty. $$ The converse problem whether the asymptotic property implies the finite potential energy estimate was originally posed in their work and later formulated by Brezis as \\emph{Open Problem 2.5} in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 2023]. In this article, we give an affirmative answer to this question. Our proof relies on the Kelvin inversion, a Morrey-type energy decay for the translation Jacobi system, the $L^4$-integrability of the phase form, and a linearised amplitude equation; all these tools together imply the $L^2$-integrability of the function $1-|u|$ on an exterior domain, without any a priori integrability assumption.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19020",
  "author_total": 5,
  "authors": [
    {
      "name": "Xiaosheng Lin"
    },
    {
      "name": "Dachun Yang"
    },
    {
      "name": "Sibei Yang"
    },
    {
      "name": "Wen Yuan"
    },
    {
      "name": "Yangyang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d9g8ej56vwwa8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19020",
  "fulltext_url": null,
  "id": "d9g8ej56vwwa8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19020",
  "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.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213563868",
  "pdf_url": "https://arxiv.org/pdf/2609.19020",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d9g8ej56vwwa8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.19020",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19020"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Solution of Problem 2.5 by Brezis on the Planar Ginzburg--Landau Equation",
  "title_text": "A Solution of Problem 2.5 by Brezis on the Planar Ginzburg--Landau Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/d9g8ej56vwwa8",
  "venue": "arXiv",
  "versions": []
}