{
  "abstract": "In this paper, we study the Liouville equation with a nonlinear Neumann boundary condition \\begin{equation*} \\begin{cases} -Δu = Ke^{2u} & \\text{in } \\mathbb{B}^{2},\\\\[2mm] \\dfrac{\\partial u}{\\partial ν}+λ= ke^{u} & \\text{on } \\partial\\mathbb{B}^{2}, \\end{cases} \\end{equation*} where $K,k\\in\\mathbb{R}$, $λ\\in(0,1]$ are constants, and $ν$ denotes the outward unit normal on $ \\partial\\mathbb{B}^{2}$. We establish a classification of all smooth solutions to the equation. Our approach is a boundary-adapted P-function method which treats all signs of $K$ and $k$ within a unified framework. As applications of the classification result, we establish a family of sharp Sobolev-trace-type inequalities encompassing the classical Lebedev--Milin inequality, together with a corresponding deficit estimate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18762",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaohan Cai"
    },
    {
      "name": "Abdolhakim Shouman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0fpepn5y7sv7v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18762",
  "fulltext_url": null,
  "id": "0fpepn5y7sv7v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18762",
  "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": "W7213526694",
  "pdf_url": "https://arxiv.org/pdf/2609.18762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0fpepn5y7sv7v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Classification of solutions to the Liouville equation with a nonlinear Neumann boundary condition and applications to sharp inequalities",
  "title_text": "Classification of solutions to the Liouville equation with a nonlinear Neumann boundary condition and applications to sharp inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/0fpepn5y7sv7v",
  "venue": "arXiv",
  "versions": []
}