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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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