Boundary Hölder Gradient Estimates for the Monge–Ampère Equation

We investigate global Holder gradient estimates for solutions to the Monge–Ampere equation $$\begin{aligned} {\mathrm {det}}\;D^2 u=f\quad {\mathrm {in}}\;\Omega , \end{aligned}$$where the right-hand side f is bounded away from 0 and $$\infty $$. We consider two main situations when (a) the domain $$\Omega $$ is uniformly convex and (b) $$\Omega $$ is flat.