*Accepted Paper*

**Inserted:** 28 nov 2008

**Last Updated:** 3 feb 2010

**Journal:** Calc. Var. Partial Differential Equations

**Year:** 2008

**Abstract:**

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).

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