Distinct solutions to generated Jacobian equations cannot intersect

Abstract

We prove that if two $C^{1,1}(\Omega)$ solutions of the second boundary value problem for the generated Jacobian equation intersect in $\Omega$ then they are the same solution. In addition we extend this result to $C^{2}(\overline{\Omega})$ solutions intersecting on the boundary, via an additional convexity condition on the target domain.