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

TL;DR

This paper proves strict convexity and $C^1$ regularity of solutions to generated Jacobian equations in two dimensions, weakening previous assumptions and providing new quantitative estimates, especially in optimal transport scenarios.

Contribution

It establishes strict $g$-convexity and $C^1$ regularity for solutions in 2D under less restrictive conditions than before, extending results in optimal transport and Monge-Ampère equations.

Findings

Proves strict $g$-convexity in 2D for generated Jacobian equations.

Shows $C^1$ regularity under bounded $g$-Monge-Ampère measure.

Weakens domain convexity assumptions in optimal transport cases.

Abstract

We present a proof of strict $g$-convexity in 2D for solutions of generated Jacobian equations with a $g$-Monge-Amp\`ere measure bounded away from 0. Subsequently this implies $C^1$ differentiability in the case of a $g$-Monge-Amp\`ere measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge-Amp\`ere 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-Amp\`ere case our key assumptions, namely A3w and domain convexity, are necessary.