{
  "abstract": "In this paper, we give a partial answer to Brezis' Open Problem~2.1, which concerns the uniqueness of solutions to the Ginzburg--Landau equation in the unit disc with the degree-one boundary condition. Let $λ_1$ be the first Dirichlet eigenvalue of $-Δ$ in the unit disc and set $\\varepsilon_*:=λ_1^{-1/2}$. We prove that there exists $δ>0$ such that the radial solution is the unique weak solution for every $\\varepsilon\\in(\\varepsilon_*-δ,\\infty)$. More generally, we establish the analogous uniqueness result for the Ginzburg--Landau system on bounded connected $C^{1,1}$ domains in $\\mathbb R^N$, $2\\leq N\\leq4$, with nontrivial boundary data. In particular, uniqueness persists slightly below the convexity threshold $\\varepsilon_*$, where the strict convexity argument is no longer available. The proof combines strict convexity for $\\varepsilon\\geq\\varepsilon_*$ with a compactness argument, nondegeneracy of the solution at $\\varepsilon=\\varepsilon_*$ and the implicit function theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.21896",
  "author_total": 2,
  "authors": [
    {
      "name": "Chao Ji"
    },
    {
      "name": "Kai Sheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqdfmt9fsyrkv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.21896",
  "fulltext_url": null,
  "id": "bqdfmt9fsyrkv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.21896",
  "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": "W7204207360",
  "pdf_url": "https://arxiv.org/pdf/2608.21896",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bqdfmt9fsyrkv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2608.21896",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.21896"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A partial answer to Brezis' Open Problem 2.1",
  "title_text": "A partial answer to Brezis' Open Problem 2.1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/bqdfmt9fsyrkv",
  "venue": "arXiv",
  "versions": []
}