Boundary regularity for the second boundary-value problem of Monge-Amp\`ere equations in dimension two

In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Amp\`ere equation $\mbox{det } D^2 u =f$ in dimension two subject to the natural boundary condition $Du(\Omega) = \Omega^*$ is $C^{2,\alpha}$ smooth up to the boundary. We establish the estimate under the sharp conditions that the inhomogeneous term $f\in C^{\alpha}$ and the domains are convex and $C^{1,\alpha}$ smooth. When $f\in C^0$ (resp. $1/C<f<C$ for some positive constant $C$), we also obtain the global $W^{2,p}$ (resp. $W^{2,1+\epsilon}$) regularity.