Inverse Problems for the Monge-Ampère Equation: Linearization and Nonlinear Recovery
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 D2u=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.
Related
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BibTeX
@misc{uhlmann2026inverse,
title = {{Inverse Problems for the Monge-Ampère Equation: Linearization and Nonlinear Recovery}},
author = {Gunther Uhlmann and Philipp Zimmermann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.13939},
eprint = {2606.13939},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.13939},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Inverse Problems for the Monge-Ampère Equation: Linearization and Nonlinear Recovery
AU - Gunther Uhlmann
AU - Philipp Zimmermann
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2606.13939
UR - https://arxiv.org/abs/2606.13939
AB - 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.
ER -