{
  "abstract": "We study the interior regularity of convex solutions to $$\\det D^2u = f(u)$$ in planar domains, with no regularity or monotonicity assumptions on $f$. In particular, when $f$ is bounded between positive constants, we show that $u$ is locally $C^{1,\\,1}$, and that its level sets near a critical point are asymptotically ellipses. Our analysis relies on a new monotone quantity for a second-order flow satisfied by the support functions of the level sets of $u$. As a byproduct of our methods, we obtain Liouville theorems that generalize a classical result of Jörgens, and include a complete classification of global solutions in 2D to an obstacle problem studied by Jin-Tu-Xiong. As another application, we obtain regularity and rigidity results for rotating-wave solutions of the semigeostrophic equations from meteorology.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06978",
  "author_total": 2,
  "authors": [
    {
      "name": "Connor Mooney"
    },
    {
      "name": "Aranya Sen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4tyb6wg91vs63.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06978",
  "fulltext_url": null,
  "id": "4tyb6wg91vs63",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06978",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06978",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4tyb6wg91vs63.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.06978",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06978"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Regularity Theory for the Equation $\\det(D^2u) = f(u)$ in Two Dimensions",
  "title_text": "Regularity Theory for the Equation det(D^2u) = f(u) in Two Dimensions",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/4tyb6wg91vs63",
  "venue": "arXiv",
  "versions": []
}