Hölder regularity of the solution to the complex Monge-Ampère equation with $L^p$ density

Abstract

On a smooth domain $\Omega \subset \subset {\mathbb{C}}^n$, we consider the Dirichlet problem for the complex Monge-Ampère equation $((d{d}^{c}u{)}^n=fdV,u{|}_{b\Omega }\equiv \varphi )$. We state the Hölder regularity of the solution $u$ when the boundary value $\varphi$ is Hölder 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 $\varphi \in C^{1,1}$ (in addition to assuming $\Omega$ strongly pseudoconvex).

final version

References in corpus (3)

• Hölder continuous solutions to Monge-Ampère equations
• The Dirichlet problem for degenerate complex Monge-Ampere equations
• Boundary behavior of the Kobayashi metric near a point of infinite type