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



Ly Kim Ha (Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, Ho Chi Minh City, Vietnam)

Tran Vu Khanh (School of Mathematics and Applied Statistics, University of Wollongong, New South Wales, Australia)


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].

Accepted 21 October 2014

Published 16 April 2015