Geodesic
Dirichlet problem for degenerate elliptic complex Monge-Ampère equation
Electronic Journal of Differential Equations, Tome 2004 (2004).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 35J70, 32W20, 32W05
Keywords: degenerate elliptic, omplex Monge-Ampère, plurisubharmonic function 
@article{EJDE_2004__2004__a82,
     author = {Kallel-Jallouli, Saoussen},
     title = {Dirichlet problem for degenerate elliptic complex {Monge-Amp\`ere} equation},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2004},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a82/}
}
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Kallel-Jallouli, Saoussen. Dirichlet problem for degenerate elliptic complex Monge-Ampère equation. Electronic Journal of Differential Equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a82/