DCDS
Regularity of the homogeneous Monge-Ampère equation
Qi-Rui Li Xu-Jia Wang
Discrete & Continuous Dynamical Systems - A 2015, 35(12): 6069-6084 doi: 10.3934/dcds.2015.35.6069
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.
keywords: regularity. Degenerate Monge-Ampère equation

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