{
  "abstract": "In this paper, we study the nonlinear boundary value problem \\begin{equation*} \\begin{cases} -Δu=e^{2u},& \\mbox{in } {\\mathbb{D}},\\\\ \\frac{\\partial u}{\\partialν}+λ=e^u ,& \\mbox{on } {\\mathbb{S}^{1}}, \\end{cases} \\end{equation*} where $\\mathbb D$ is the unit disk, $λ$ is a constant and $ν$ denotes the outer unit normal on $\\mathbb S^1$. For $0<λ\\le2$, we establish a complete classification of smooth solutions. For $2<λ\\leq3$, we prove a dichotomy between radial solutions and nonradial solutions. We develop a Hardy-Wronskian boundary rigidity method that transforms the nonlinear boundary value problem into a spectral rigidity problem for normalized holomorphic frames. This method exploits the complex-analytic structure of the Liouville equation with a Robin boundary condition and provides a new rigidity framework for related two-dimensional elliptic boundary value problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12847",
  "author_total": 3,
  "authors": [
    {
      "name": "Jingbo Dou"
    },
    {
      "name": "Yunyun Hu"
    },
    {
      "name": "Keqing Peng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5j2z80yrtn63f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12847",
  "fulltext_url": null,
  "id": "5j2z80yrtn63f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12847",
  "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": "W7213027910",
  "pdf_url": "https://arxiv.org/pdf/2609.12847",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5j2z80yrtn63f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12847",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12847"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Classification of solutions to the Liouville equation with a nonlinear Robin boundary condition on the unit disk",
  "title_text": "Classification of solutions to the Liouville equation with a nonlinear Robin boundary condition on the unit disk",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/5j2z80yrtn63f",
  "venue": "arXiv",
  "versions": []
}