Gauge rigidity in an inverse problem for the prescribed Gaussian curvature equation

Abstract

We prove a uniqueness result for an inverse boundary problem associated with the prescribed Gaussian curvature equation \[ \det D^2u=K(x)(1+|\nabla u|^2)^2 \] for graphs over a planar domain. The comparison is made on a common open class of smooth boundary values for which both admissible Dirichlet problems are well posed. We show that, if the corresponding nonlinear Dirichlet-to-Neumann maps agree on this class and the two prescribed curvatures have the same first boundary jet, then the curvatures agree in the whole domain. The main difficulty comes from a gauge obstruction already present at the first linearization. In logarithmic form, the linearized equation is a two-dimensional non-divergence form elliptic equation with drift. Its boundary data determine the coefficients only up to a boundary-fixing change of variables and a scalar gauge factor. For the prescribed Gaussian curvature equation this gauge is not an artifact of the method: the gradient dependence leaves a residual gauge which cannot be removed by the first variation. The proof uses the nonlinear structure to remove this remaining gauge. We derive a second-linearized interaction identity in the first-linearized gauge. In this identity, the covariant Hessian of the gauge map appears in the leading part. Testing the identity with two complementary CGO families gives a pair of residual equations for the gauge variables. Combined with the drift and conductivity identities from the first linearization, these equations form a closed system for the residual gauge. A boundary unique continuation argument for this system gives the trivial gauge and hence determines the prescribed curvature.