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DCDS

In this paper, we study the regularity of convex solutions to the Dirichlet problem
of the homogeneous Monge-Ampère equation $\det D^2 u=0$.
We prove that if the domain is a strip region and the boundary functions
are locally uniformly convex and $C^{k+2,\alpha}$ smooth,
then the solution is $C^{k+2,\alpha}$ smooth up to boundary.
By an example, we show the solution may fail to be $C^{2}$ smooth
if boundary functions are not locally uniformly convex.
Similar results have also been obtained for the Dirichlet problem
on bounded convex domains.

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