On the second boundary value problem for Monge-Ampere type equations and   optimal transportation
  This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for Monge-Ampere equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a weak form of our condition A3, under which we proved interior regularity in a recent paper with Xi-nan Ma. Consequently they include the quadratic cost function case of Caffarelli and Urbas as well as the various examples in the earlier work. The approach is through the derivation of global estimates for second derivatives of solutions. 
Neil S Trudinger, Xu-jia Wang