Recovery of Rough Sources in the Inverse Monge-Ampère Problem

Abstract

We prove that every nonnegative $L^1$ source in the Monge-Ampère equation \[ \det D^2u=f\quad\text{in }Ω, \] on a bounded uniformly convex $C^{1,1}$ domain $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, is uniquely determined by generalized boundary measurements of Alexandrov solutions. The measurement is the extremal boundary slope, which extends the classical outward normal derivative and may take the value $+\infty$. For each direction $\omega \in {\mathbb{S}}^{n-1}$, it suffices to know these slopes for cylindrical Dirichlet data $\frac{t_k^{\omega }}{2}|P_{\omega }x{|}^2$ along any sequence $t_k^{\omega }\to \infty$, 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 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 $L^{\infty }$ 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 \[ \vartheta(\nabla u)\det D^2u=f, \] where the known weight $\vartheta$ 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.

arXiv comment70 pages