An Inverse Problem for the Prescribed Mean Curvature

Abstract: We extend the 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*} 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. This work represents the first treatments of inverse source problems for quasilinear equations. The proof relies on the higher order linearization method. The main methodological contribution is the development of an approach to decouple the resulting system of nonlinear algebraic and geometric equations, which enables the complete determination of the source term from boundary measurements. The decoupling uses a Liouville type uniqueness result for conformal mappings.