THE DIRICHLET PROBLEM FOR THE MONGE-AMPERE EQUATION IN CONVEX (BUT NOT STRICTLY CONVEX) DOMAINS

It is well-known that the Dirichlet problem for the Monge-Ampere equation det D(2)u = mu in a bounded strictly convex domain Omega in R-n has a weak solution (in the sense of Aleksandrov) for any finite Borel measure mu on Omega and for any continuous boundary data. We consider the Dirichlet problem when Omega is only assumed to be convex, and give a necessary and sufficient condition on the boundary data for solvability.