{
  "abstract": "We construct smooth, uniformly convex planar domains that admit minimal, strictly stable solutions of a semilinear Dirichlet problem whose superlevel sets are nonetheless nonconvex. The class of admissible nonlinearities includes, in particular, two prototypical cases: the Gelfand-type nonlinearity $e^u$ and the family of shifted power-type nonlinearities $(a+u)^p$, where $a>0$ and $p>1$. By applying the elementary scaling properties of the Dirichlet problem, we also show that the same lack of convexity of superlevel sets holds for the corresponding parameter-dependent equations. These results provide a negative answer to a question posed by Brezis, who inquired whether the stability of a solution necessarily entails quasiconcavity for these prototypical stable configurations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.06031",
  "author_total": 1,
  "authors": [
    {
      "name": "Yi Ru-Ya Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fcf13vwzerc7e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.06031",
  "fulltext_url": null,
  "id": "fcf13vwzerc7e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.06031",
  "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": "W7167682275",
  "pdf_url": "https://arxiv.org/pdf/2607.06031",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fcf13vwzerc7e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2607.06031",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.06031"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets",
  "title_text": "Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/fcf13vwzerc7e",
  "venue": "arXiv",
  "versions": []
}