The dirichlet problem for the multidimensional monge-ampere equation

In this paper we study the n-dimensional version of the Monge-Ampere operator Mu = u n u 2 2 ~~ u\ 2 where %, = dhildXidXj. The analogue arises from observing that if H(u) is the Hessian matrix of u then Mu = det H(u). The main goal is to prove a comparison theorem for nonconvex solutions of the Dirichlet problem for M. In the process we will give a self-contained exposition of the Dirichlet problem for M in the class of convex u.