Continuity of maps solutions of optimal transportation problems
Grégoire Loeper · arXiv (Cornell University) · 2005
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 C 1 and C 1,α regularity. Our results hold assuming a condition on the cost function (condition A3 below), that was the one used for C 2 a priori estimates in [5]. The optimal potential will solve a Monge-Ampère equation of the form det(M(x, ∇φ) + D 2 φ) = f where M 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 det D 2 φ = f, in particular we will obtain here C 1,α regularity for φ under the condition f ∈ L p,p> n.