On regularity of complex Monge-Ampere equation

TL;DR

This paper investigates the regularity of solutions to the complex Monge-Ampere equation, establishing interior $C^2$ estimates under certain conditions and constructing Pogorelov-type examples to illustrate solution behaviors.

Contribution

It provides new interior regularity estimates for solutions with specific bounds and constructs explicit examples demonstrating solution properties.

Findings

Interior $C^2$ estimates under $L^p$ bounds and Lipschitz conditions.

Construction of Pogorelov-type solutions for complex Monge-Ampere equations.

Examples include generalized entire and viscosity solutions.

Abstract

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