The Dirichlet problem for the Monge-Ampere equation in convex (but not strictly convex) domains

Abstract

It is well-known that the Dirichlet problem for the Monge-Ampère equation det D2u = μ in a bounded strictly convex domain Ω in ℝn has a weak solution (in the sense of Aleksandrov) for any finite Borel measure μ on Ω and for any continuous boundary data. We consider the Dirichlet problem when Ω is only assumed to be convex, and give a necessary and sufficient condition on the boundary data for solvability.