The optimal transportation problem was introduced by Monge in 1781. Monges cost is proportional to the distance the mass is transported, namely .
Since then the problem has been extensively studied and more general cost functions are allowed. The existence of optimal mappings was obtained by Brenier, Caffarelli, and Gangbo-McCann under a certain condition (A1), using the duality discovered by Kantorovich. The regularity of optimal mappings was then developed by Ma-Trudinger-Wang for transport costs satisfying a structural condition (A3). However Monges cost function does not satisfy the conditions (A1) and (A3), and one cannot introduce a proper PDE for this particular case. By the approximation, the existence for Monges problem was obtained by Trudinger-Wang, and Caffarelli-Feldman-McCann independently; and also by Evans-Gangbo using a different approach.
In this talk we will discuss the regularity for Monges problem. In two dimensional case, we shall show that the optimal mapping is continuous under very natural conditions. In higher dimensions, we will present uniform a priori estimates for eigenvalues of the Jacobian of the optimal mapping in the approximation, and give a counter-example to show that the Jacobian itself is not uniformly bounded in general. Our example also implies that the optimal mapping fails to be Lipschitz continuous in general.
This is joint work with F. Santambrogio and X.-J. Wang.