On the complex Monge-Ampère equation in the ball of \(\mathbb{C}^n)\). (Sur l’équation de Monge-Ampère complexe dans la boule de \(\mathbb{C}^n\).) (French) Zbl 0678.35037

On considère le problème de Dirichlet: \[ (dd^cu)^ n=0\quad\text{ dans }B;\quad u|_{\partial B}=\phi \] où \(B\) désigne la boule unité de \(\mathbb C^n\).
Nous donnons une démonstration simple du fait que si \(\phi \in C^{1,1}(\partial B)\), alors \(u\in C^{1,1}(B)\); de plus la croissance du coefficient de Lipschitz de la différentielle de \(u\) est contrôlée par l’inverse de la distance au bord.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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[1] [1] & , The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., 37 (1976), 1-44. · Zbl 0315.31007
[2] [2] , Function theory in the unit Ball of Cn, Grundlehren der mathematischen Wissenschaften, 241, Springer-Verlag. · Zbl 0495.32001
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