First and second derivative H\"older estimates for generated Jacobian equations

TL;DR

This paper establishes sharp H"older regularity results for solutions of generated Jacobian equations, including $C^{1,eta}$ and $C^{2,eta}$ estimates under specific conditions, advancing understanding of their regularity properties.

Contribution

It provides the first sharp $C^{1,eta}$ and $C^{2,eta}$ regularity results for generated Jacobian equations under the A3 condition and Dini continuity assumptions.

Findings

Solutions are $C^{1,eta}$ under A3 and $L^p$ data.

Solutions are $C^{2,eta}$ with positive Dini continuous data.

Equation is uniformly elliptic under the given conditions.

Abstract

We prove two H\"older regularity results for solutions of generated Jacobian equations. First, that under the A3 condition and the assumption of nonnegative $L^p$ valued data solutions are $C^{1,\alpha}$ for an $\alpha$ that is sharp. Then, under the additional assumption of positive Dini continuous data, we prove a $C^{2}$ estimate. Thus the equation is uniformly elliptic and when the data is H\"older continuous solutions are in $C^{2,\alpha}$.