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

We prove that every nonnegative source $f\in L^1(\mathrm{\Omega })$ in the Monge–Ampère equation $$\det D^{2}u=f\text{in }\mathrm{\Omega },$$ on a bounded uniformly convex $C^{1,1}$ domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, 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 $+\mathrm{\infty }$. For each $\omega \in {\mathbb{S}}^{n-1}$, it suffices to know these slopes for the cylindrical Dirichlet data $$\frac{t_k^{\omega }}{2}\mathrm{\mid }{P}_{\omega }x{\mathrm{\mid }}^2,t_k^{\omega }\to \mathrm{\infty },$$ along any sequence of amplitudes, where $P_{\omega }$ projects onto ${\omega }^{\perp }$. 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 $L^{\mathrm{\infty }}$ 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 $$\vartheta (\mathrm{\nabla }u)\det D^{2}u=f,$$ 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.