{
  "abstract": "We consider the problem of finding critical domains $Ω\\subset{\\mathbb R}^3$ for the energy functional $$ \\mathcal E (Ω) = {\\rm Per}\\,(Ω) + \\frac 12 \\iint_{Ω\\times Ω} \\frac{dx\\,dy}{|x-y|} $$ under the volume constraint $|Ω|=V$. We look for smooth, embedded, compact surfaces $\\partialΩ$ that solve this problem. We construct an axially symmetric, non-minimizing solution that, for a sufficiently small $V>0$, resembles the union of two balls with volume $V/2$ connected by a tiny, approximately catenoidal neck with a width of the order $V^{\\frac 43}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00108",
  "author_total": 3,
  "authors": [
    {
      "name": "Manuel del Pino"
    },
    {
      "name": "Rupert L. Frank"
    },
    {
      "name": "Monica Musso"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5xkkn6cnq9jb2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00108",
  "fulltext_url": null,
  "id": "5xkkn6cnq9jb2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00108",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7206197652",
  "pdf_url": "https://arxiv.org/pdf/2609.00108",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5xkkn6cnq9jb2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:03Z",
      "native_id": "2609.00108",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00108"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The double sphere solution in the liquid drop model",
  "title_text": "The double sphere solution in the liquid drop model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/5xkkn6cnq9jb2",
  "venue": "arXiv",
  "versions": []
}