On the regularity of the complex Monge-Ampère equations

He, Weiyong

doi:10.1090/S0002-9939-2011-11178-X

On the regularity of the complex Monge-Ampère equations
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by Weiyong He PDF

Proc. Amer. Math. Soc. 140 (2012), 1719-1727 Request permission

Abstract:

We shall consider the regularity of solutions for the complex Monge-Ampère equations in $\mathbb {C}^n$ or a bounded domain. First we prove interior $C^2$ estimates of solutions in a bounded domain for the complex Monge-Ampère equations with the assumption of an $L^p$ bound for $\triangle u$, $p>n^2$, and of a Lipschitz condition on the right-hand side. Then we shall construct a family of Pogorelov-type examples for the complex Monge-Ampère equations. These examples give generalized entire solutions (as well as viscosity solutions) of the complex Monge-Ampère equation $\det (u_{i\bar j})=1$ in $\mathbb {C}^n$.

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