C¹ regularity of solutions of the Monge-Ampere equation for optimal transport in dimension two

We prove C¹ regularity of c-convex weak Alexandrov solutions of a Monge-Ampère type equation in dimension two, assuming only a bound from above on the Monge-Ampère measure. The Monge-Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151-183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C¹ regularity. Such a condition holds in particular for the case "cost = distance squared" which leads to the usual Monge-Ampère equation det D²u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841-845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].