First and second derivative Hölder estimates for generated Jacobian equations

Abstract

We prove two Hölder 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 Hölder continuous solutions are in $C^{2,\alpha }$.

24 pages