Electron. J. Diff. Eqns., Vol. 2004(2004), No. 48, pp. 1-24.

Dirichlet problem for degenerate elliptic complex Monge-Ampere equation

Saoussen Kallel-Jallouli

We consider the Dirichlet problem
 \det \big({\frac{\partial^2u}{\partial z_i\partial \overline{z_j}}}
 \big)=g(z,u)\quad\hbox{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.

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Saoussen Kallel-Jallouli
Faculte des Sciences, Campus Universitaire
1060 Tunis, Tunisie
email: Saoussen.Kallel@fst.rnu.tn

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