Strict $g$-convexity for generated Jacobian equations with applications to global regularity

TL;DR

This paper proves strict $g$-convexity and global $C^3$ regularity of solutions to generated Jacobian equations, extending domain conditions and utilizing a modified existence approach for smooth solutions.

Contribution

It establishes strict $g$-convexity and global regularity for generated Jacobian equations, improving domain conditions and linking convexity to regularity proofs.

Findings

Solutions are strictly $g$-convex under natural assumptions.

Global $C^3$ regularity is achieved for Aleksandrov solutions.

The approach modifies existing existence results to ensure smooth solutions.

Abstract

This article has two purposes. The first is to prove solutions of the second boundary value problem for generated Jacobian equations are strictly $g$-convex. The second is to prove the global $C^3$ regularity of Aleksandrov solutions to the same problem under stronger hypothesis. These are related because the strict $g$-convexity is essential for the proof of the global regularity. The assumptions for the strict $g$-convexity are the natural extension of those used by Chen and Wang in the optimal transport case. They improve the existing domain conditions though at the expense of requiring a $C^3$ generating function. We prove the global regularity under the hypothesis that Jiang and Trudinger recently used to obtain the existence of a globally smooth solution and an additional condition on the height of solutions. Our proof of global regularity is by modifying Jiang and Trudinger's existence result to construct a globally $C^3$ solution intersecting the Aleksandrov solution. Then the strict convexity yields the interior regularity to apply the author's uniqueness results.