Global regularity for the Monge-Ampère equation with natural boundary condition

In this paper, we establish the global C2,α and W2,p regularity for the Monge-Ampère equation detD2u= f subject to boundary condition Du(Ω)= Ω*, where Ω and Ω* are bounded convex domains in the Euclidean space Rn with C1,1 boundaries, and f is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications