Inverse Problems for the Monge--Ampère Equation: Linearization and Nonlinear Recovery

Abstract: We study inverse boundary value problems for the nonlinear Monge--Ampère equation [ \det D^2u=a(x,u,\nabla u) ] in a bounded domain. We introduce a nonlinear Cauchy data set and investigate the recovery of the nonlinearity from boundary measurements. Linearizing around a strictly convex background solution, we establish local well-posedness, smooth dependence on boundary data, and a higher-order linearization framework. We show that the first variation of the nonlinear Cauchy data is governed by a linear elliptic operator whose principal coefficient is the cofactor matrix of the background Hessian. An Alessandrini-type identity then yields a reduction principle from the nonlinear inverse problem to an anisotropic Calderón-type inverse problem. As a consequence, under suitable uniqueness assumptions for the associated linear problem, the nonlinear Cauchy data determine the first-order derivatives of the nonlinearity along the background jet. Higher-order linearization identities provide recovery of higher derivatives and, under suitable density assumptions, determine the full Taylor expansion of the nonlinearity along the background solution. We also discuss applications to semilinear equations and nonlinearities arising in optimal transport.