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The complex Monge-Ampère equation. (English) Zbl 0913.35043

L’article commence par l’énumération de résultats bien connus déjà anciens concernant l’équation de Monge-Ampère et les fonctions \(psh\) sur les ouverts de \(\mathbb{C}^n\). Il s’agit principalement des travaux de Bedford et Taylor. Puis l’auteur passe aux variétés kählériennes compactes avec les résultats d’Aubin et de Yau. Enfin il rappelle ses propres résultats concernant les fonctions \(psh\) sur les ouverts strictement pseudoconvexes de \(\mathbb{C}^n\).
Dans la deuxième partie de l’article, l’auteur met en évidence des conditions sur le deuxième membre de telle sorte que l’équation de Monge-Ampère admette une solution faible continue. Sur une variété kählérienne compacte, sous une certaine hypothèse l’auteur montre qu’il existe une solution continue de l’équation lorsque le second membre \(F\in L_1\) ou si \(F\) appartient à certains espaces d’Orlitz. Sur un ouvert strictement pseudo-convexe de \(\mathbb{C}^n\), lorsque le second membre est une mesure \(d\mu= fd\lambda\) (avec \(d\lambda\) la mesure de Lebesgue), l’équation de Monge-Ampère admet une solution continue si \(f\) est continue d’après Bedford et Taylor. L’auteur généralise ce résultat lorsque \(f\) appartient à certains espaces d’Orlitz. Enfin l’équation admet une solution si le problème de Dirichlet admet une sous-solution.

MSC:

35J60 Nonlinear elliptic equations
32W20 Complex Monge-Ampère operators
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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