{
  "abstract": "In this paper, we study local minimizers of the energy functional$J(\\mathbf{u}) = \\int_D (|\\nabla \\mathbf{u}|^2 + Q^2(x)χ_{Ω_{\\mathbf{u}}})\\,dx$ which give rise to a singular cooperative system. Here \\(\\mathbf{u}=(u_1,\\dots,u_m): D\\to\\R_+^m\\) is a vector-valued unknown function, \\(Ω_{\\mathbf{u}}:=\\{|\\mathbf{u}|>0\\}\\) is the positive set, \\(χ_{Ω_{\\mathbf{u}}}\\) is the characteristic function of the positive set \\(Ω_{\\mathbf{u}}\\), and \\(Q(x)\\) is the Bernoulli weight function. This problem was first introduced by Caffarelli, Shahgholian, and Yeressian in the poineer work ({\\it Duke Math. J.} \\textbf{167}(10), 2018), where the regularity theory for minimizers and for the free boundary \\(\\partialΩ_{\\mathbf{u}}\\) was established for nondegenerate Bernoulli weights, i.e., \\(Q(x)\\ge Q_{\\rm min}>0\\). The present paper studies the same free boundary problem but for a degenerate Bernoulli weight. The main difficulty lies in the coupling between the degeneracy of \\(Q(x)\\) and the vector-valued nature of the problem. We show that the the free boundary set \\(\\partialΩ_{\\mathbf{u}}\\cap\\{Q(x)=0\\}\\) decomposes into a nondegenerate part and a degenerate part. Nondegenerate points admit nontrivial blow-up limits; degenerate points have zero weighted density. We classify nondegenerate points into single-phase, multi-phase non-branching, and multi-phase branching points, analyzing their geometric structure. The branching-point singular structure for multi-phase points is purely vectorial and new in this setting. Finally, following Naber--Valtorta (Ann. Math. 185, 2017) and Edelen--Engelstein (Trans. Amer. Math. Soc. 371, 2019), we prove the nondegenerate set has locally finite $\\mathcal{H}^{n-2}$-measure and is countably $(n-2)$-rectifiable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25544",
  "author_total": 3,
  "authors": [
    {
      "name": "Lili Du"
    },
    {
      "name": "Chunlei Yang"
    },
    {
      "name": "Jing Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/61rztmekeqt94.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25544",
  "fulltext_url": null,
  "id": "61rztmekeqt94",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25544",
  "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": "W7214245922",
  "pdf_url": "https://arxiv.org/pdf/2609.25544",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/61rztmekeqt94.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.25544",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25544"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A minimization problem for a cooperative system with a degenerate Bernoulli weight",
  "title_text": "A minimization problem for a cooperative system with a degenerate Bernoulli weight",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/61rztmekeqt94",
  "venue": "arXiv",
  "versions": []
}