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An Inverse Problem for the Prescribed Mean Curvature

Tony Liimatainen, Janne Nurminen

TL;DR

This work solves the inverse source problem for the two-dimensional prescribed mean curvature equation by proving that the source term $H$ is uniquely determined from the Dirichlet-to-Neumann map. The authors develop a two-step linearization framework: the first linearization yields an anisotropic conductivity problem with a metric $g$ tied to the background solution, while the second linearization produces a coupled nonlinear integral identity involving a gauge transform $\phi$. Employing complex geometrical optics solutions on a Riemannian surface and a Liouville-type rigidity result for conformal maps, they show that the gauge must be trivial, forcing $\nabla u_0 = \nabla\tilde u_0$ and hence $H=\tilde H$ in $\Omega$. The approach highlights how boundary measurements in a Euclidean setting induce an anisotropic Calderón problem, enabling a rigorous recovery of the source through conformal rigidity, with potential applicability to broader quasilinear inverse problems.

Abstract

We extend the recent study of inverse problems for minimal surfaces by considering the inverse source problem for the prescribed mean curvature equation \begin{equation*} \nabla \cdot \left[ \frac{\nabla u}{(1 + |\nabla u|^2)^{1/2}} \right] = H(x). \end{equation*} This work also represents the first treatment of inverse source problems for quasilinear equations. We prove that in two dimensions, the source function $H$ is uniquely determined by the associated Dirichlet-to-Neumann map. A notable feature of this problem is that although the equation is posed on an Euclidean domain, its linearization yields an anisotropic conductivity equation where the coefficient matrix corresponds to a Riemannian metric $g$ depending on the background solution. The main methodological contribution is the derivation of a coupled nonlinear system of algebraic and geometric partial differential equations from boundary measurements. Similar systems will naturally appear in other inverse problems for quasilinear equations. We solve the system using a Liouville type uniqueness result for conformal mappings, which recovers the source function uniquely.

An Inverse Problem for the Prescribed Mean Curvature

TL;DR

This work solves the inverse source problem for the two-dimensional prescribed mean curvature equation by proving that the source term is uniquely determined from the Dirichlet-to-Neumann map. The authors develop a two-step linearization framework: the first linearization yields an anisotropic conductivity problem with a metric tied to the background solution, while the second linearization produces a coupled nonlinear integral identity involving a gauge transform . Employing complex geometrical optics solutions on a Riemannian surface and a Liouville-type rigidity result for conformal maps, they show that the gauge must be trivial, forcing and hence in . The approach highlights how boundary measurements in a Euclidean setting induce an anisotropic Calderón problem, enabling a rigorous recovery of the source through conformal rigidity, with potential applicability to broader quasilinear inverse problems.

Abstract

We extend the recent study of inverse problems for minimal surfaces by considering the inverse source problem for the prescribed mean curvature equation \begin{equation*} \nabla \cdot \left[ \frac{\nabla u}{(1 + |\nabla u|^2)^{1/2}} \right] = H(x). \end{equation*} This work also represents the first treatment of inverse source problems for quasilinear equations. We prove that in two dimensions, the source function is uniquely determined by the associated Dirichlet-to-Neumann map. A notable feature of this problem is that although the equation is posed on an Euclidean domain, its linearization yields an anisotropic conductivity equation where the coefficient matrix corresponds to a Riemannian metric depending on the background solution. The main methodological contribution is the derivation of a coupled nonlinear system of algebraic and geometric partial differential equations from boundary measurements. Similar systems will naturally appear in other inverse problems for quasilinear equations. We solve the system using a Liouville type uniqueness result for conformal mappings, which recovers the source function uniquely.

Paper Structure

This paper contains 12 sections, 9 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Earlier works
  4. Method of proof
  5. First linearization
  6. Boundary information
  7. Second linearization
  8. Integral identity
  9. Showing that $\phi$ is the identity
  10. Boundary terms
  11. Complex geometrical optics solutions
  12. Proving that $T=0$

Key Result

Theorem 1.1

Let $H,\tilde{H}\in C^{\infty}(\overline{\Omega})$ be such that $H|_{\partial\Omega}=\tilde{H}|_{\partial\Omega}$ to high order and $f_0\in C^{\infty}(\partial\Omega)$. Assume that $u_0$ and $\tilde{u}_0$ solve for $H$ and $\tilde{H}$ on the right hand side respectively, and with $u_0|_{\partial\Omega}=\tilde{u}_0|_{\partial\Omega}= f_0$. Suppose that the DN maps of PMC satisfy with $\lVert f_0

Theorems & Definitions (16)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • proof
  • Lemma 6.3
  • ...and 6 more