Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 48, pp. 1-24.
Title: Dirichlet problem for degenerate
elliptic complex Monge-Ampere equation
Author: Saoussen Kallel-Jallouli (Campus Univ., Tunis, Tunisie)
Abstract:
We consider the Dirichlet problem
$$
\det \big({\frac{\partial^2u}{\partial z_i\partial \overline{z_j}}}
\big)=g(z,u)\quad\mbox{in }\Omega\,, \quad
u\big|_{ \partial \Omega }=\varphi\,,
$$
where $\Omega$ is a bounded open set of $\mathbb{C}^{n}$ with regular
boundary, $g$ and $\varphi$ are sufficiently smooth functions, and $g$ is
non-negative. We prove that, under additional hypotheses on $g$ and
$\varphi $, if $|\det \varphi _{i\overline{j}}-g|_{C^{s_{\ast}}}$ is
sufficiently small the problem has a plurisubharmonic solution.
Submitted May 15, 2003. Published April 6, 2004.
Math Subject Classifications: 35J70, 32W20, 32W05.
Key Words: Degenerate elliptic; omplex Monge-Ampere; Plurisubharmonic function.