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Sur l’équation de Monge-Ampère complexe dans la boule de $${\mathbb{C}}^ n$$. (On the complex Monge-Ampère equation in the ball of $${\mathbb{C}}^ n)$$. (French) Zbl 0678.35037
On considère le problème de Dirichlet: $(dd^ cu)^ n=0\quad dans\quad B;\quad u|_{\partial B}=\phi$ où B désigne la boule unité de $$C^ n.$$
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.
Reviewer: A.Dufresnoy

##### MSC:
 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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##### References:
 [1] E. BEDFORD & B.A. TAYLOR, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., 37 (1976), 1-44. · Zbl 0315.31007 [2] W. RUDIN, Function theory in the unit ball of cn, Grundlehren der mathematischen Wissenschaften, 241, Springer-Verlag. · Zbl 0495.32001
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