# Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two

Journal Article

### Abstract

• 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. $1/C positive constant$C$), we also obtain the global$W^{2,p}$(resp.$W^{2,1+\epsilon}$) regularity. ### UOW Authors ### Publication Date • 2018 ### Web Of Science Accession Number ### Abstract • In this paper, we introduce an iteration argument to prove that a convex solution to the Monge-Amp\ere equation$\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

positive constant $C$), we also obtain the global $W^{2,p}$ (resp.

$W^{2,1+\epsilon}$) regularity.

• 2018