C regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two

We prove C 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 [7], that was shown in [5] to be necessary for C regularity. Such a condition holds in particular for the case “cost = distance squared” which leads to the usual Monge-Ampère equation detDu = f . Our result is in some sense optimal, both for the assumptions on the density (thanks to the regularity counterexamples of X.J.Wang [13]) and for the assumptions on the cost-function (thanks to the results of the second author [5]).