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# Mathematical Research Letters

## Volume 22 (2015)

### Number 2

### Boundary regularity of the solution to the complex Monge-Ampère equation on pseudoconvex domains of infinite type

Pages: 467 – 484

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n2.a7

#### Authors

#### Abstract

Let $\Omega$ be a $C^2$-smooth, bounded, pseudoconvex domain in $\mathbb{C}^n$ satisfying the “$f$-property”. The $f$-property is a consequence of the geometric “type” of the boundary. All pseudoconvex domains of finite type satisfy the $f$-property as well as many classes of domains of infinite type. In this paper, we prove the existence, uniqueness, and “weak” Hölder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equation\[\begin{cases}\mathrm{det} \biggl[ \dfrac{\partial^2{(u)}}{\partial z_i \partial \overline{z}_j} \biggr] = h \geq 0 & \textrm{in} \; \Omega \\u = \phi & \textrm{on} \; b\Omega\end{cases}\]The idea of our proof goes back to Bedford and Taylor [1]. However, the basic geometrical ingredient is based on a recent result by Khanh [12].

Published 16 April 2015