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.

Submitted Bulletin of the Australian Mathematical Society November 2019. Accepted December 2019