Probleme de Neumann pour l'equation de Monge–Ampere sur les varietes Riemanniennes

Abstract

Let (M,g) be a C ∞ compact Riemannian manifold with strictly convex boundary. Let f∈C ∞(T ★M× R) and ϕ∈C ∞(∂M× R) . Under various hypothesis on f and ϕ, using either continuity method, iterative procedure or fixed point argument, combined with C 2,α a priori estimates, we solve Monge–Ampère equations, with nonlinear Neumann boundary condition, of the following form: log( det ∇ i ju)=f(x,du;u) in M, ∂u ∂ν =ϕ(x,u) on ∂M.