Abstract

In this paper, we introduce an iteration argument to prove that a convex
solution to the MongeAmp\`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
positive constant $C$), we also obtain the global $W^{2,p}$ (resp.
$W^{2,1+\epsilon}$) regularity.