{
  "abstract": "In this paper, we study rigidity phenomena associated with a class of overdetermined two-phase problems on the $n$-dimensional sphere $\\mathbb{S}^n$. Specifically, we analyze the solutions of coupled elliptic equations defined on a domain $Ω\\subset \\mathbb{S}^n$ and its complement, subject to Dirichlet boundary conditions on the interface $\\partialΩ$ and a compatibility condition on the gradient. We utilize two different methods. The method of moving planes works when $\\partialΩ$ is contained inside an open hemisphere. The other method works when $Ω$ is simply connected in $\\mathbb{S}^2$. Both lead to the same conclusion. Namely, the existence of a solution to such overdetermined problems necessarily implies that $Ω$ is a geodesic ball. In the latter, we prove this rigidity property for three distinct scenarios: a baseline problem with piecewise constant source terms, which is the two-phase version of the torsion problem; a generalization including a class of positive and regular nonlinearities; and finally an eigenvalue problem involving the first Dirichlet eigenvalue of the respective phases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15103",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrea Bisterzo"
    },
    {
      "name": "Shigeru Sakaguchi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4e3zpst5wqpfb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15103",
  "fulltext_url": null,
  "id": "4e3zpst5wqpfb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15103",
  "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"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213345325",
  "pdf_url": "https://arxiv.org/pdf/2609.15103",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4e3zpst5wqpfb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15103",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15103"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Rigidity for two-phase overdetermined problems in $\\mathbb{S}^n$",
  "title_text": "Rigidity for two-phase overdetermined problems in 𝕊^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/4e3zpst5wqpfb",
  "venue": "arXiv",
  "versions": []
}