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An inverse problem for the Monge-Ampère equation

Tony Liimatainen, Yi-Hsuan Lin

TL;DR

The paper solves a nonlinear inverse boundary value problem for the Monge–Ampère equation det $D^2u=F$ in a planar convex domain, proving unique recovery of a positive source $F$ from the Dirichlet-to-Neumann map. The authors reduce the problem to recovering a Hessian-derived metric $g=(D^2u)^{-1}$, establish boundary determination, and show that the DN map determines $g$ up to a conformal factor, which is then fixed by a second linearization and a nonlocal ∂̄-equation. They construct complex-geometric-optics solutions, develop isothermal coordinates, and prove a Carleman-based unique continuation principle for a nonlocal PDE to force the conformal factor to be identically one. The result is the first uniqueness theorem for an inverse problem for a fully nonlinear PDE, with techniques (CGO, higher-order linearization, UCP for nonlocal ∂̄) expected to impact broader nonlinear inverse problems. The approach converts the inverse MA problem into an anisotropic Calderón problem in the first linearization and then leverages a nonlocal-∂̄ framework to achieve full interior recovery of $F$.

Abstract

We extend the study of inverse boundary value problems to the setting of fully nonlinear PDEs by considering an inverse source problem for the Monge-Ampère equation \[ \det D^2 u = F. \] We prove that, on a convex Euclidean domain in the plane, the associated Dirichlet-to-Neumann (DN) map uniquely determines a positive source function $F$. The proof relies on recovering the Hessian of a solution to the equation, which is interpreted as a Riemannian metric $g$. Interestingly, although the equation is posed on a Euclidean domain, the inverse problem becomes anisotropic since the metric $g$ appears as a coefficient matrix in the linearized equation. As an intermediate step, we prove that the DN map of the non-divergence form equation \[ g^{ab} \partial_{ab} v = 0 \] uniquely determines the conformal class of the metric $g$ on a simply connected planar domain, without the usual diffeomorphism invariance. To address the challenges of full nonlinearity, we develop asymptotic expansions for complex geometric optics solutions in the planar setting and solve a resulting nonlocal $\overline{\partial}$-equation by proving a unique continuation principle for it. These techniques are expected to be applicable to a wide range of inverse problems for nonlinear equations.

An inverse problem for the Monge-Ampère equation

TL;DR

The paper solves a nonlinear inverse boundary value problem for the Monge–Ampère equation det in a planar convex domain, proving unique recovery of a positive source from the Dirichlet-to-Neumann map. The authors reduce the problem to recovering a Hessian-derived metric , establish boundary determination, and show that the DN map determines up to a conformal factor, which is then fixed by a second linearization and a nonlocal ∂̄-equation. They construct complex-geometric-optics solutions, develop isothermal coordinates, and prove a Carleman-based unique continuation principle for a nonlocal PDE to force the conformal factor to be identically one. The result is the first uniqueness theorem for an inverse problem for a fully nonlinear PDE, with techniques (CGO, higher-order linearization, UCP for nonlocal ∂̄) expected to impact broader nonlinear inverse problems. The approach converts the inverse MA problem into an anisotropic Calderón problem in the first linearization and then leverages a nonlocal-∂̄ framework to achieve full interior recovery of .

Abstract

We extend the study of inverse boundary value problems to the setting of fully nonlinear PDEs by considering an inverse source problem for the Monge-Ampère equation We prove that, on a convex Euclidean domain in the plane, the associated Dirichlet-to-Neumann (DN) map uniquely determines a positive source function . The proof relies on recovering the Hessian of a solution to the equation, which is interpreted as a Riemannian metric . Interestingly, although the equation is posed on a Euclidean domain, the inverse problem becomes anisotropic since the metric appears as a coefficient matrix in the linearized equation. As an intermediate step, we prove that the DN map of the non-divergence form equation uniquely determines the conformal class of the metric on a simply connected planar domain, without the usual diffeomorphism invariance. To address the challenges of full nonlinearity, we develop asymptotic expansions for complex geometric optics solutions in the planar setting and solve a resulting nonlocal -equation by proving a unique continuation principle for it. These techniques are expected to be applicable to a wide range of inverse problems for nonlinear equations.

Paper Structure

This paper contains 29 sections, 14 theorems, 355 equations.

Table of Contents

  1. Introduction
  2. Mathematical formulations and main results
  3. Organization of the article
  4. Preliminaries
  5. Notations, function spaces and some fundamental tools
  6. Notations in complex analysis
  7. Function spaces
  8. Matrices computations
  9. Well-posedness
  10. Higher order linearization
  11. The first linearization
  12. The second linearization
  13. Boundary determination
  14. Unique determination of the metric
  15. Determination up to isometry and conformal factor
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $\Omega\subset {\mathbb R}^2$ be a bounded, uniformly convex domain with $C^{\infty}$ boundary $\partial\Omega$. Let $F\in C^{\infty}(\overline{\Omega})$ be a source with $F\geq c_0>0$ for some positive constant $c_0$. Suppose that $F$ is known up to second order on the boundary, then the DN map for $j=1,2$. Suppose that $F_1$ and $F_2$ agree up to second order on the boundary, then for $\del

Theorems & Definitions (33)

  • Remark 1.1
  • Theorem 1.2: Unique determination
  • Remark 1.3
  • Theorem 1.4
  • Proposition 2.1: Well-posedness
  • proof
  • Lemma 3.1: Boundary determination
  • proof
  • Lemma 4.1
  • proof
  • ...and 23 more