{
  "abstract": "In this paper, we first apply the implicit function theorem to find nontrivial solutions to the following Bernoulli's overdetermined problem \\begin{equation} -Δu=γ\\,\\, \\text{in}\\,\\,Ω\\setminus\\overline{D}, \\,\\, u=1,\\,\\,\\partial_\\mathbf{n} u=Q\\,\\,\\text{on}\\,\\,\\partial D, \\,\\,u=0\\,\\,\\text{on}\\,\\,\\partialΩ\\nonumber \\end{equation} for a fixed boundary $\\partialΩ$ and a free boundary $\\partial D$ in $\\mathbb{R}^{N}$ with $N\\geq2$, where $γ$ is a sufficiently small constant, the Bernoulli value $Q$ is determined by the construction, and $Ω\\setminus\\overline{D}$ is a bounded ring-shaped domain. In dimension $N=n+1$ with $n\\geq2$, we further study the existence of a nontrivial unbounded domain $Ω\\setminus\\overline{D}$ in $\\mathbb{R}^N$ such that the Bernoulli's overdetermined problem admits a solution. This unbounded domain we construct is periodic in one variable and radial in the other variables by bifurcation analysis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14742",
  "author_total": 2,
  "authors": [
    {
      "name": "Yong Zhang"
    },
    {
      "name": "Jialong Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6vmdymganeqpp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14742",
  "fulltext_url": null,
  "id": "6vmdymganeqpp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14742",
  "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": "W7221765247",
  "pdf_url": "https://arxiv.org/pdf/2610.14742",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6vmdymganeqpp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14742",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14742"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Nontrivial domains for the Bernoulli's overdetermined problem with small constant vorticity",
  "title_text": "Nontrivial domains for the Bernoulli's overdetermined problem with small constant vorticity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/6vmdymganeqpp",
  "venue": "arXiv",
  "versions": []
}