An Inverse Problem for the Prescribed Mean Curvature

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.

💡 Research Summary

The paper addresses the inverse source problem for the prescribed mean curvature (PMC) equation in two dimensions, namely
∇·(∇u/√{1+|∇u|²}) = H(x) in Ω⊂ℝ²,
with Dirichlet data on ∂Ω. The central question is whether the source term H can be uniquely recovered from the Dirichlet‑to‑Neumann (DN) map Λ_H that sends a boundary function f to the normal derivative of the corresponding solution u_f. The authors prove a global uniqueness theorem: if two smooth source functions H and Ĥ agree to high order on the boundary and generate the same DN map for all sufficiently small perturbations of a fixed boundary datum f₀, then H≡Ĥ in Ω.

The proof proceeds via higher‑order linearization of the nonlinear PDE. First, the authors linearize the PMC equation around a smooth background solution u₀ (corresponding to f₀). The linearized operator takes the form
∂_a(g^{ab}∂_b v)=0, g^{ab}=∂_bF_a(∇u₀),
where F(p)=p√{1+|p|²}. The matrix g^{ab} is positive definite and can be interpreted as a Riemannian metric g on Ω that depends on u₀. Consequently, the first linearization is an anisotropic conductivity equation, or equivalently a Schrödinger equation (Δ_g+q)v=0 after a suitable gauge transformation.

Assuming Λ_H=Λ_Ĥ, the DN maps of the linearized equations coincide. By invoking recent results on the anisotropic Calderón problem (