C2+α estimates for solutions of Monge-Ampere equations

Consider the Monge-Ampere equation det(D2u) = f(x) in a convex domain Ω is contained in Rn subject to the Dirichlet boundary condition u = φ on δ Ω. We prove that if δ Ω and φ are C3 smooth, and f(x) is an element of Cα (Ω-bar) for some α is an element of (0,1), then the solution u is an element of C2+α (Ω-bar). We also give examples to show that if δ Ω or φ is only C2,1 smooth, the solution may not be C2 smooth near the boundary. (author). 12 refs