An Inverse Problem for the Prescribed Mean Curvature

Technical Summary: An Inverse Problem for the Prescribed Mean Curvature

Problem Statement
This paper addresses the inverse source problem for the prescribed mean curvature (PMC) equation in two dimensions. The governing equation on a bounded domain $\Omega \subset \mathbb{R}^2$ is given by:
$$\nabla \cdot \left[ \frac{\nabla u}{(1 + |\nabla u|^2)^{1/2}} \right] = H(x),$$
where $H \in C^\alpha(\Omega)$ is the source function representing the prescribed mean curvature. The objective is to determine the source function $H$ uniquely from the Dirichlet-to-Neumann (DN) map $\Lambda_H$. The DN map assigns a boundary value $f$ to the normal derivative $\partial_\nu u|_{\partial\Omega}$ of the solution $u$. The authors establish that if the DN maps for two source functions $H$ and $\tilde{H}$ coincide for boundary data sufficiently close to a fixed background value $f_0$, and if $H$ and $\tilde{H}$ agree to high order on the boundary, then $H = \tilde{H}$ throughout $\Omega$.

Methodology
The proof relies on a combination of higher-order linearization techniques, the analysis of the resulting linearized equations, and complex geometrical optics (CGO) solutions. The methodology proceeds in several distinct stages:

1. First Linearization and Geometric Interpretation:
   The authors linearize the nonlinear PMC equation around a background solution $u_0$ corresponding to a fixed boundary value $f_0$. The first linearization yields an anisotropic conductivity equation:
   $$\nabla \cdot (g^{-1} \nabla v) = 0,$$
   where the metric tensor $g$ depends explicitly on the gradient of the background solution, $\nabla u_0$. Specifically, $g^{-1}$ corresponds to a Riemannian metric derived from the Euclidean metric and the gradient of $u_0$. This transforms the inverse problem for the nonlinear equation into an inverse problem for a linear equation with coefficients depending on the unknown background solution.

2. Boundary Information and Gauge Invariance:
   Using the assumption that the DN maps coincide, the authors deduce that the metrics $g$ and $\tilde{g}$ associated with $H$ and $\tilde{H}$ are related by a conformal diffeomorphism $\phi: \Omega \to \Omega$ such that $g = \lambda \phi^* \tilde{g}$. By analyzing the boundary behavior and utilizing unique continuation properties, they show that $\phi$ fixes the boundary to infinite order and that the magnitudes of the gradients of the background solutions are related by $|\nabla u_0| = |\nabla \tilde{u}_0| \circ \phi$.

3. Second Linearization and Integral Identities:
   To break the remaining gauge invariance and establish full gradient equality ($\nabla u_0 = \nabla \tilde{u}_0$), the authors perform a second linearization. This yields a coupled system involving an integral identity over $\Omega$ of the form:
   $$\int_\Omega T \cdot \mathcal{L}(v_0, v_1, v_2) \, dV_g = 0,$$
   where $T$ is a vector field defined by the difference between the transformed gradients of the background solutions, and $v_0, v_1, v_2$ are solutions to the first linearized equation. The term $\mathcal{L}$ involves gradients and Laplacians of these solutions.

4. Complex Geometrical Optics (CGO) Solutions:
   The authors construct specific CGO solutions for the linearized Schrödinger-type equation associated with the metric $g$. These solutions are of the form $v = e^{\Phi/h}(a + r_h)$, where $\Phi$ is a holomorphic Morse function, $a$ is a holomorphic amplitude, and $r_h$ is a correction term. By choosing solutions with specific phase behaviors (some with critical points, some without), they perform an asymptotic analysis as the parameter $h \to 0$. This analysis isolates the leading order terms of the integral identity, effectively probing the vector field $T$ at interior points.

5. Uniqueness via Liouville-Type Rigidity:
   The asymptotic analysis leads to a system of equations implying that the components of $T$ satisfy an elliptic partial differential equation. Combined with the fact that $T$ vanishes to high order on the boundary (derived from the boundary agreement of $H$ and $\tilde{H}$), unique continuation principles force $T \equiv 0$ in $\Omega$. This implies $\nabla u_0 = \nabla \tilde{u}_0$. Finally, a Liouville-type uniqueness result for conformal mappings in two dimensions is applied to show that the diffeomorphism $\phi$ must be the identity map, thereby proving $H = \tilde{H}$.

Key Contributions and Results

• First Treatment of Inverse Source Problems for Quasilinear Equations: The paper presents the first rigorous treatment of an inverse source problem for a quasilinear elliptic equation. Previous works in this domain largely focused on linear equations or inverse problems for nonlinear terms without source terms.
• Derivation of Coupled Nonlinear Systems: The authors derive a coupled system of algebraic and geometric partial differential equations from boundary measurements. This system links the source function to the background solution's gradient and the conformal factor of the metric.
• Uniqueness Theorem: The main result (Theorem 1.1) proves the unique determination of the source function $H$ in two dimensions from the DN map, provided the sources agree to high order on the boundary.
• Application of Anisotropic Calderón Theory: The work demonstrates a novel application of the theory of the anisotropic Calderón problem. Although the original PMC equation is posed on a Euclidean domain, its linearization naturally leads to an anisotropic conductivity equation with a metric dependent on the solution, bridging nonlinear inverse problems with geometric inverse problems.

Significance and Motivation
The authors position this work as a foundational step toward solving inverse problems for general quasilinear elliptic equations of the form $\nabla \cdot (A(\nabla u)\nabla u) = H(x)$. The prescribed mean curvature equation serves as a prototypical example.

The paper highlights the physical relevance of the PMC equation in modeling surfaces under balanced tension with external forces, such as soap bubbles with internal pressure, capillary surfaces, and biological membranes. Furthermore, the authors note the growing significance of inverse problems for minimal surfaces (the case $H=0$) in theoretical physics, specifically within the AdS/CFT duality framework, where traditional mathematical tools have sometimes been insufficient. By extending these techniques to the prescribed mean curvature case, the paper contributes to the mathematical toolkit available for such physical applications.

The methodological contribution lies in the successful integration of higher-order linearization with CGO solutions to handle the specific structure of quasilinear equations, overcoming the challenges posed by the nonlinearity and the source term. The authors emphasize that similar systems of coupled equations will likely arise in other inverse problems for quasilinear equations, suggesting the broader applicability of their approach.