Recovery of Rough Sources in the Inverse Monge--Ampère Problem

Gunther Uhlmann; Philipp Zimmermann

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arXiv · 2610.17519

Recovery of Rough Sources in the Inverse Monge--Ampère Problem

Gunther Uhlmann·Philipp Zimmermann

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Abstract

We prove that every nonnegative source $f\in L^1(Ω)$ 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 outward normal derivative and may equal $+\infty$. For each $ω\in\mathbb S^{n-1}$, it suffices to know these slopes for the cylindrical Dirichlet data \[ \frac{t_k^ω}{2}|P_ωx|^2,\qquad t_k^ω\to\infty, \] along any sequence of amplitudes, 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 and a large-amplitude limit then recover the Euclidean X-ray transform of $f$, yielding uniqueness without a positive lower bound on the source. For nonnegative $L^\infty$ sources, the extremal slopes are bounded and coincide with the interior normal derivatives, giving a Dirichlet-to-interior-Neumann formulation. The argument also extends to \[ \vartheta(\nabla u)\det D^2u=f, \] where the known weight $\vartheta$ is locally integrable, Borel measurable, and bounded below by a positive constant. Conditional on existence of the corresponding Alexandrov solutions, the same cylindrical boundary-slope data determine every nonnegative $L^1$ source.

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BibTeX RIS

Gunther Uhlmann, Philipp Zimmermann. 2026-10-13. Recovery of Rough Sources in the Inverse Monge--Ampère Problem. https://arxiv.org/abs/2610.17519

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