Regularity for Monge-Ampere equation near the boundary

Research Report MRR96-049

Regularity for Monge-Ampère equation near the boundary

Xu-Jia Wang

Abstract: In this paper we consider the Monge-Ampère equation $\det(D^2u)=f(x)$ in a convex domain $\Om\subset \R^n$ subject to the Dirichlet boundary condition $u=\phi$ on $\pom$. We prove that if $\pom$ and $\phi$ are $C^3$ smooth, then the solution $u\in C^{2+\alpha}(\bom)$. We also give examples to show that if $\pom$ or $\phi$ is only $C^{2,1}$ smooth, the solution may fail to be $C^2$ smooth near the boundary.

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