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# Le problème de Neumann pour certaines équations du type de Monge-Ampère sur une variété riemannienne

Published:2000-08-01
Printed: Aug 2000
• Abdellah Hanani
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## Abstract

Let $(M_n,g)$ be a strictly convex riemannian manifold with $C^{\infty}$ boundary. We prove the existence\break of classical solution for the nonlinear elliptic partial differential equation of Monge-Amp\`ere:\break $\det (-u\delta^i_j + \nabla^i_ju) = F(x,\nabla u;u)$ in $M$ with a Neumann condition on the boundary of the form $\frac{\partial u}{\partial \nu} = \varphi (x,u)$, where $F \in C^{\infty} (TM \times \bbR)$ is an everywhere strictly positive function satisfying some assumptions, $\nu$ stands for the unit normal vector field and $\varphi \in C^{\infty} (\partial M \times \bbR)$ is a non-decreasing function in $u$.
 Keywords: connexion de Levi-Civita, équations de Monge-Ampère, problème de Neumann, estimées a priori, méthode de continuité
 MSC Classifications: 35J60 - Nonlinear elliptic equations 53C55 - Hermitian and Kahlerian manifolds [See also 32Cxx] 58G30 - unknown classification 58G30