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

Technical Summary: Inverse Problems for the Monge–Ampère Equation: Linearization and Nonlinear Recovery

Problem Statement
This work investigates inverse boundary value problems for the fully nonlinear Monge–Ampère equation:
$$\det D^2 u = a(x, u, \nabla u) \quad \text{in } \Omega,$$
where $\Omega \subset \mathbb{R}^n$ is a bounded domain and $a$ is an unknown nonlinearity. Motivated by the classical Calderón problem (electrical impedance tomography), the authors ask whether the "jet" of the nonlinearity $a$ along a given strictly convex background solution $\Phi$ can be recovered from boundary measurements. Specifically, the goal is to determine the derivatives of $a$ with respect to its arguments $(x, z, p)$ along the trajectory $(x, \Phi(x), \nabla \Phi(x))$ using the nonlinear Cauchy data set.

Unlike semilinear or quasilinear equations, fully nonlinear equations generally lack a canonical Dirichlet-to-Neumann (DN) map and do not naturally connect to linear inverse problems. The Monge–Ampère equation presents a unique challenge due to its fully nonlinear nature, yet it possesses a divergence structure via the cofactor matrix of the Hessian, which the authors exploit to establish a systematic framework.

Methodology
The authors develop a systematic higher-order linearization framework around a strictly convex background solution $\Phi$. The approach proceeds in three main stages:

1. Local Well-Posedness and Smooth Dependence:
   The authors first establish the local well-posedness of the Dirichlet problem for boundary data close to $\Phi|_{\partial \Omega}$. They prove that the solution operator $S: \eta \mapsto u_\eta$ is smooth ($C^k$) with respect to the boundary data. This allows for the derivation of explicit formulas for the higher-order Fréchet derivatives of the solution operator.

2. Linearization of the Cauchy Data:
   A nonlinear Cauchy data set is defined as the collection of pairs $(\eta, \partial_\nu u_\eta|_{\partial \Omega})$ for boundary data $\eta$ near $\Phi|_{\partial \Omega}$. The authors show that the first variation of this nonlinear data set coincides with the Cauchy data set of a linear elliptic operator $L_\Phi$ governing the linearized equation:
   $$L_\Phi v = \text{div}(C_\Phi \nabla v) - b_\Phi \cdot \nabla v - q_\Phi v = 0,$$
   where the coefficients are determined by the background solution and the nonlinearity:

   • $C_\Phi = \text{cof } D^2 \Phi$ (the cofactor matrix of the Hessian),
   • $b_\Phi = \nabla_p a(x, \Phi, \nabla \Phi)$,
   • $q_\Phi = \partial_z a(x, \Phi, \nabla \Phi)$.

   Crucially, the principal part of this linearized operator is anisotropic, determined by the cofactor matrix $C_\Phi$.

3. Reduction and Higher-Order Identities:
   Using an Alessandrini-type identity, the authors establish a reduction principle: equality of the nonlinear Cauchy data sets implies equality of the corresponding linearized DN maps. This reduces the nonlinear inverse problem to an anisotropic Calderón-type inverse problem for the operator $L_\Phi$.

   For higher-order derivatives, the authors derive integral identities involving products of solutions to the linearized equation and its adjoint. Under suitable density assumptions for these products, these identities allow for the recovery of higher-order derivatives of $a$ along the background jet.

Key Results

• First-Order Recovery: If two nonlinearities $a_1$ and $a_2$ generate the same nonlinear Cauchy data near a background solution $\Phi$, then the DN maps of their associated linearized operators coincide. Consequently, the quantities $\partial_z a(x, \Phi, \nabla \Phi)$ and $\nabla_p a(x, \Phi, \nabla \Phi)$ are determined along the background jet, up to the natural gauge invariance of the anisotropic Calderón problem (boundary-fixing diffeomorphisms).
• Higher-Order Recovery: If the background solution is fixed ($\Phi_1 = \Phi_2 = \Phi$) and the lower-order derivatives of $a$ are already known to agree, higher-order linearization identities allow for the recovery of higher derivatives $d^k_{(z,p)} a(x, \Phi, \nabla \Phi)$. This relies on a density property of products of solutions to the linearized equation and its adjoint.
• Specific Applications:
  • Semilinear Case: If $a$ is independent of $\nabla u$ (semilinear), the linearized operator reduces to a Schrödinger equation. In dimensions $n \ge 3$, this yields uniqueness for the first-order derivative $\partial_z a$ along quadratic backgrounds ($\Phi(x) = |x|^2/2$).
  • Optimal Transport: For nonlinearities arising in optimal transport, specifically in the equilibrium configuration $\Phi(x) = |x|^2/2$, the linearized equation reduces to a standard isotropic conductivity equation. This establishes a direct link to the classical Calderón problem, allowing for the recovery of transport densities.
  • Inverse Source Problems: In the case where $a(x, u, \nabla u) = f(x)$, the problem reduces to recovering the source term $f(x)$, which is linked to the rigidity properties of the class of cofactor matrices of strictly convex functions.

Significance and Claims
The paper claims to provide the first systematic study of inverse problems for fully nonlinear Monge–Ampère equations from the perspective of Calderón-type boundary measurements. Its primary contribution is demonstrating that despite the fully nonlinear nature of the equation, the inverse problem can be systematically reduced to a family of linear anisotropic inverse problems through higher-order linearization.

The authors emphasize that their approach bridges the gap between fully nonlinear PDEs and linear inverse problems, leveraging the specific divergence structure of the Monge–Ampère operator. While the recovery of the full nonlinearity depends on the solvability of the associated anisotropic Calderón problem (which remains open in general dimensions without additional geometric assumptions), the paper establishes that the nonlinear data determines the linearized coefficients up to the natural gauge. Furthermore, it demonstrates that under specific geometric settings (such as quadratic backgrounds) and structural assumptions (semilinearities or optimal transport equilibria), unconditional uniqueness results can be obtained, recovering the full Taylor expansion of the nonlinearity along the background trajectory.