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 $det{D}^{2}u=f$ in dimension two subject to the natural boundary condition $Du(\mathrm{\Omega })={\mathrm{\Omega }}^{\ast }$ 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. $1\mathrm{/}C<f<C$ for some positive constant $C$), we also obtain the global $W^{2,p}$ (resp. $W^{2,1+ϵ}$) regularity.