On C~(1 ,α) Regularity for Monge- Ampre Equation

In this paper,we consider the well known problem concerning global regularity ofsolutions of the Dirichlet problems for the Monge-Ampère equations.　Theorem1　LetΩ be a uniformly convex domain in Rn with the boundary Ω∈ C3,1 ,φ∈ C2 ,1 ( Ω ) and letf be a nonnegative function in Rnsuch thatf1 / n∈ C1 ,1 (Rn) . Then thereexistsa unique convex solution u∈C1 ,1 / 4 (Ω)∩C1 ,1 (Ω) of the Dirichletproblemdet D2 u =f(x) ,　　 inΩ (1 )u =φ(x) ,　　 on Ω , (2 )where D2 u=[Diju] …