Strict convexity and $C^1$ regularity of solutions to generated Jacobian equations in dimension two

Abstract

We present a proof of strict $g$-convexity in 2D for solutions of generated Jacobian equations with a $g$-Monge-Ampère measure bounded away from 0. Subsequently this implies $C^1$ differentiability in the case of a $g$-Monge-Ampère measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge-Ampère case. Thus, like theirs, our argument is local and yields a quantitative estimate on the $g$-convexity. As a result our differentiability result is new even in the optimal transport case: we weaken previously required domain convexity conditions. Moreover in the optimal transport case and the Monge-Ampère case our key assumptions, namely A3w and domain convexity, are necessary.

minor modification of domain conditions, one proof moved to appendix, To appear in Calculus of Variations and Partial Differential Equations