{
  "abstract": "We prove that every nonnegative $L^1$ source in the Monge-Ampère equation \\[ \\det D^2u=f\\quad\\text{in }Ω, \\] on a bounded uniformly convex $C^{1,1}$ domain $Ω\\subset\\mathbb R^n$, $n\\geq2$, is uniquely determined by generalized boundary measurements of Alexandrov solutions. The measurement is the extremal boundary slope, which extends the classical outward normal derivative and may take the value $+\\infty$. For each direction $ω\\in\\mathbb S^{n-1}$, it suffices to know these slopes for cylindrical Dirichlet data $\\frac{t_k^ω}{2}|P_ωx|^2$ along any sequence $t_k^ω\\to\\infty$, where $P_ω$ projects onto $ω^\\perp$. The boundary value and extremal slope determine the interior subgradient image up to a Lebesgue null set. An exact change of variables formula then converts equality of these images into integral identities for the source, and a large-amplitude limit recovers its Euclidean X-ray transform. This yields uniqueness without a positive lower bound on the source. For nonnegative $L^\\infty$ sources, the extremal slopes are bounded and agree with the interior normal derivatives, giving a Dirichlet-to-interior-Neumann formulation. We also extend the inverse argument to the gradient-weighted equation \\[ \\vartheta(\\nabla u)\\det D^2u=f, \\] where the known weight $\\vartheta$ is locally integrable, Borel measurable, and bounded below by a positive constant. We prove existence of the corresponding Alexandrov solutions and show that the same cylindrical boundary-slope data uniquely determine every nonnegative $L^1$ source.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17519",
  "author_total": 2,
  "authors": [
    {
      "name": "Gunther Uhlmann"
    },
    {
      "name": "Philipp Zimmermann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3jxfd1f84rx39.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17519",
  "fulltext_url": null,
  "id": "3jxfd1f84rx39",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17519",
  "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",
      "2610|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7221813936",
  "pdf_url": "https://arxiv.org/pdf/2610.17519",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3jxfd1f84rx39.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.17519",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17519"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Recovery of Rough Sources in the Inverse Monge-Ampère Problem",
  "title_text": "Recovery of Rough Sources in the Inverse Monge-Ampère Problem",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/3jxfd1f84rx39",
  "venue": "arXiv",
  "versions": []
}