H\"older regularity of the solution to the complex Monge-Amp\`ere equation with $L^p$ density

TL;DR

This paper proves that solutions to the complex Monge-Ampère equation are H"older continuous when the boundary data is H"older and the density is in L^p, relaxing previous regularity assumptions.

Contribution

It establishes H"older regularity of solutions under weaker assumptions on the density function, extending prior results that required smoother boundary data.

Findings

Solutions are H"older continuous with explicit regularity estimates.

Regularity holds for densities in L^p with p>1, even with less smooth boundary data.

Extends previous work by relaxing the smoothness conditions on the density and boundary data.

Abstract

On a smooth domain $\Omega\subset\subset\mathbb C^n$, we consider the Dirichlet problem for the complex Monge-Amp\`ere equation $((dd^cu)^n=fdV,\,u|_{b\Omega}\equiv\phi)$. We state the H\"older regularity of the solution $u$ when the boundary value $\phi$ is H\"older continuous and the density $f$ is only $L^p$, $p>1$. Note that in former literature (Guedj-Kolodziej-Zeriahi) the weakness of the assumption $f\in L^p$ was balanced by taking $\phi\in C^{1,1}$ (in addition to assuming $\Omega$ strongly pseudoconvex).