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 $\det D^{2}u=a(x,u,\nabla u)$ 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 $\det D^{2}u=f(x)$, 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 $\Phi (x)=|x{|}^2/2$, 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.

47 pages