Proceedings of the Japan Academy, Series A, Mathematical Sciences

Note on a general complex Monge-Ampère equation on pseudoconvex domains of infinite type

Ly Kim Ha

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Abstract

Let $\Omega$ be a smoothly bounded domain in $\mathbf{C}^{n}$, for $n\ge 2$. For a given continuous function $\phi$ on $b\Omega$, and a non-negative continuous function $\Psi$ on $\mathbf{R}\times \overline{\Omega}$, the main purpose of this note is to seek a plurisubharmonic function $u$ on $\Omega$, continuous on $\overline{\Omega}$, which solves the following Dirichlet problem of the complex Monge-Ampère equation \begin{equation*} \begin{cases} \det\left[\dfrac{\partial^{2}(u)}{\partial z_{i}\partial\bar{z}_{j}}\right](z)=\Psi(u(z),z)\geqslant 0 & \text{in}\quad\Omega,\\ u=\phi & \text{on}\quad b\Omega. \end{cases} \end{equation*} In particular, the boundary regularity for the solution of this complex, fully nonlinear equation is studied when $\Omega$ belongs to a large class of weakly pseudoconvex domains of finite and infinite type in $\mathbf{C}^{n}$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 10 (2016), 136-140.

Dates
First available in Project Euclid: 2 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.pja/1480669221

Digital Object Identifier
doi:10.3792/pjaa.92.136

Mathematical Reviews number (MathSciNet)
MR3579196

Zentralblatt MATH identifier
1368.32025

Subjects
Primary: 32W20: Complex Monge-Ampère operators
Secondary: 32T15: Strongly pseudoconvex domains 32T25: Finite type domains 32U05: Plurisubharmonic functions and generalizations [See also 31C10]

Keywords
Pseudoconvexity D’Angelo type complex Monge-Ampère operator Perron-Bremermann family

Citation

Ha, Ly Kim. Note on a general complex Monge-Ampère equation on pseudoconvex domains of infinite type. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 10, 136--140. doi:10.3792/pjaa.92.136. https://projecteuclid.org/euclid.pja/1480669221


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