Inserted: 28 nov 2008
Last Updated: 3 feb 2010
Journal: Calc. Var. Partial Differential Equations
We prove $C^1$ 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 by Ma-Trudinger-Wang, that was shown to be necessary for $C^1$ regularity. Such a condition holds in particular for the case ``cost = distance squared'' which leads to the usual Monge-Ampère equation $\det D^2u = f$. Our result is in some sense optimal, both for the assumptions on the density (thanks to the regularity counterexamples of X.J.Wang) and for the assumptions on the cost-function (thanks to the results of the second author).