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

Hanani, Abdellah

Canadian journal of mathematics, Tome 52 (2000) no. 4, pp. 757-788

Résumé

Let $\left( {{M}_{n}},\,g \right)$ be a strictly convex riemannian manifold with ${{C}^{\infty }}$ boundary. We prove the existence of classical solution for the nonlinear elliptic partial differential equation of Monge-Ampère: $\det (-u\delta _{j}^{i}\,+\,\nabla _{j}^{i}u)\,=\,F(x,\,\nabla u;\,u)$ in $M$ with a Neumann condition on the boundary of the form $\frac{\partial u}{\partial v}=\varphi (x,u)$ , where $F\in {{C}^{\infty }}(TM\times \mathbb{R})$ is an everywhere strictly positive function satisfying some assumptions, $v$ stands for the unit normal vector field and $\varphi \in {{C}^{\infty }}(\partial M\times \mathbb{R})$ is a non-decreasing function in $u$ .

Détail

DOI : 10.4153/CJM-2000-032-1

Mots-clés : 35J60, 53C55, 58G30, connexion de Levi-Civita, équations deMonge-Ampère, problème de Neumann, estimées a priori, méthode de continuité

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