{
  "abstract": "We study inverse boundary value problems for the fully nonlinear Monge-Ampère equation $ \\det D^2u=a(x,u,\\nabla u) $ in a bounded domain. Motivated by Calderón-type problems, we introduce nonlinear Cauchy data and investigate the recovery of $a$ from boundary measurements. Around a strictly convex background solution, under a sign condition ensuring invertibility of the linearized Dirichlet problem, we establish local well-posedness and differentiable dependence on boundary data. The first variation yields a linear elliptic operator with principal coefficient given by the cofactor of the background Hessian and lower-order coefficients determined by derivatives of $a$. An Alessandrini-type identity reduces equality of nonlinear measurements to an anisotropic Calderón problem. Conditional on uniqueness for this linear problem, we recover the first derivatives of $a$ along the background jet. Higher-order linearization yields integral identities whose inversion depends on a product-density property. We also establish unconditional uniqueness results in special settings. For $\\det D^2u=f(x)$, local nonlinear Cauchy data determine $f$ in dimension two, and in higher dimensions under real-analytic anisotropic hypotheses. We compare this result with a planar small-data method and an all-dimensional large-data approach based on the Euclidean X-ray transform. For semilinear nonlinearities around $Φ(x)=|x|^2/2$, Schrödinger inverse results recover the accessible $z$-jet. For equilibrium optimal-transport nonlinearities, linearization yields an isotropic conductivity equation and uniquely determines the transport density.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.13939",
  "author_total": 2,
  "authors": [
    {
      "name": "Gunther Uhlmann"
    },
    {
      "name": "Philipp Zimmermann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cz514acq90jb6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.13939",
  "fulltext_url": null,
  "id": "cz514acq90jb6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.13939",
  "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": "W7164804921",
  "pdf_url": "https://arxiv.org/pdf/2606.13939",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cz514acq90jb6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2606.13939",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.13939"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Inverse Problems for the Monge-Ampère Equation: Linearization and Nonlinear Recovery",
  "title_text": "Inverse Problems for the Monge-Ampère Equation: Linearization and Nonlinear Recovery",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/cz514acq90jb6",
  "venue": "arXiv",
  "versions": []
}