Continuity of maps solutions of optimal transportation problems

In this paper we investigate the continuity of maps solutions of optimal transportation problems. These maps are expressed through the gradient of a potential for which we establish C1 and C1,α regularity. Our results hold assuming a condition on the cost function (condition A3 below), that was the one used for C2 a priori estimates in [5]. The optimal potential will solve a Monge-Ampère equation of the form det(M(x,∇φ) +Dφ) = f whereM depends on the cost function. One of the interesting outcome is that under the condition A3, the regularity obtained is better than the one obtained in the case of the ’usual’ Monge-Ampère equation detD2φ = f , in particular we will obtain here C1,α regularity for φ under the condition f ∈ Lp, p > n.