{
  "abstract": "We study the Perron extremal solutions of the stationary one-phase Bernoulli problem. These solutions are important in several applications, but not much is known about the regularity of their free boundaries due to their non-variational construction. We show that, in fact, Perron extremal solutions retain some variational features; specifically, they are solutions in the sense of inner variations. We establish this property by showing that extremal solutions arise as infinite-time limits of monotone solutions of the parabolic Bernoulli problem. As a consequence, we are able to describe the structure of singularities for Perron extremal solutions in two dimensions: largest subsolutions have smooth free boundaries, while smallest supersolutions have free boundary points that are either smooth, or whose blow-up limits are two-plane wedges with equal slope.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14981",
  "author_total": 3,
  "authors": [
    {
      "name": "Farhan Abedin"
    },
    {
      "name": "William M. Feldman"
    },
    {
      "name": "Kerrek Stinson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ryn8ch6yt0xx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14981",
  "fulltext_url": null,
  "id": "7ryn8ch6yt0xx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14981",
  "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": "W7213254796",
  "pdf_url": "https://arxiv.org/pdf/2609.14981",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ryn8ch6yt0xx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.14981",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14981"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Variational properties of Perron's extremal solutions in the Bernoulli one-phase problem",
  "title_text": "Variational properties of Perron's extremal solutions in the Bernoulli one-phase problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/7ryn8ch6yt0xx",
  "venue": "arXiv",
  "versions": []
}