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

In this paper, we establish the global C 2,α and W 2,p regularity for the Monge-Ampère equation det D 2 u = f subject to boundary condition Du(Ω) = Ω*, where Ω and Ω* are bounded convex domains in the Euclidean space ℝ n with C 1,1 boundaries, and f is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.