Recovery of Rough Sources in the Inverse Monge-Ampère Problem
Abstract
We prove that every nonnegative L1 source in the Monge-Ampère equation det D2u=f in Ω, on a bounded uniformly convex C1,1 domain Ω⊂ℝn, n≥2, 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 +∞. For each direction ω∈𝕊n−1, it suffices to know these slopes for cylindrical Dirichlet data tkω/2|Pωx|2 along any sequence tkω→∞, where Pω projects onto ω⊥. 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∞ 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 ϑ(∇ u)det D2u=f, where the known weight ϑ 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 L1 source.
Related
- Inverse Problems for the Monge-Ampère Equation: Linearization and Nonlinear Recovery
- Stability Estimates for the Inverse Recovery of Drift and Diffusion Point Sources in Stochastic Parabolic Equations
- The inverse Monge--Ampère flow on klt Fano varieties
- Regularity of Free Boundary for Monge-Ampère Obstacle Problem up to the Boundary
- A recovery based linear finite element method for the Monge--Ampère equation
BibTeX
@misc{uhlmann2026recovery,
title = {{Recovery of Rough Sources in the Inverse Monge-Ampère Problem}},
author = {Gunther Uhlmann and Philipp Zimmermann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17519},
eprint = {2610.17519},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17519},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Recovery of Rough Sources in the Inverse Monge-Ampère Problem
AU - Gunther Uhlmann
AU - Philipp Zimmermann
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.17519
UR - https://arxiv.org/abs/2610.17519
AB - We prove that every nonnegative L^1 source in the Monge-Ampère equation det D^2u=f in Ω, on a bounded uniformly convex C^1,1 domain Ω⊂ℝ^n, n≥2, 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 +∞. For each direction ω∈𝕊^n−1, it suffices to know these slopes for cylindrical Dirichlet data t_k^ω/2|P_ωx|^2 along any sequence t_k^ω→∞, where P_ω projects onto ω^⊥. 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^∞ 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 ϑ(∇ u)det D^2u=f, where the known weight ϑ 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.
ER -