{
  "abstract": "In this note, we give an explicit construction of global solutions to the prescribed Jacobian equation \\[ \\det(\\nabla u)=f\\mbox{ in }\\mathbb{R}^2, \\] for a class of data. For every $f\\in C_c^1(\\mathbb{R}^2)$ and $p>1$, we construct a solution $u\\in \\dot W^{1,2p}(\\mathbb{R}^2)\\cap L^\\infty(\\mathbb{R}^2)$. In particular, no sign condition or integral constraint is imposed on the compactly supported data. For $f\\in C_c^\\infty(\\mathbb{R}^2)$, the construction yields a smooth solution. We also consider rapidly decaying data and prove the existence of global solutions with bounded gradient for every $f\\in\\mathcal{S}(\\mathbb{R}^2)$. Finally, we extend the construction to compactly supported data that are measurable in one variable and $C^1$ in the other. Our proof relies on a similar idea as in [Moser, Trans. Amer. Math. Soc. 1965].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14407",
  "author_total": 1,
  "authors": [
    {
      "name": "Animesh Jana"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dgfh6v5aw3re9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14407",
  "fulltext_url": null,
  "id": "dgfh6v5aw3re9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14407",
  "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": "W7213286685",
  "pdf_url": "https://arxiv.org/pdf/2609.14407",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dgfh6v5aw3re9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.14407",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14407"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A note on existence of smooth solution to Jacobian equation for compactly supported smooth data in $\\mathbb{R}^2$",
  "title_text": "A note on existence of smooth solution to Jacobian equation for compactly supported smooth data in ℝ^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/dgfh6v5aw3re9",
  "venue": "arXiv",
  "versions": []
}