Some maximum principles and symmetry results for a class of boundary valueproblems involving the Monge-Ampere equation

In this paper we investigate a class of boundary value problems for the Mon ge-Ampere equation det (partial derivative (2)u/partial derivativex(i)partial derivativex(k)) = f(u,/delu/(2), r) > 0 in Omega, where Omega is a strictly convex bounded domain in R-N, N greater than or e qual to 2. When f = g(u)h(/delu(2)) with g and h satisfying the differentia l inequality h(-1/N)g ' /g + 2g(1/N)h ' /h greater than or equal to 0 we show in Sec. 2 that the function Phi := integral (/delu/2)(0) h(-1/N) (s)ds-2 integral (u)(0) g(1/N) (s)ds takes its maximum value on the boundary partial derivative Omega. This maxi mum principle generalizes a recent result of Ma who investigated the case f = const in R-2. In Sec. 3 we investigate symmetry properties of u under sp ecific boundary conditions or geometry of Omega.