Optimal boundary regularity for a singular Monge–Ampère equation

Abstract In this paper we study the optimal global regularity for a singular Monge–Ampere type equation which arises from a few geometric problems. We find that the global regularity does not depend on the smoothness of domain, but it does depend on the convexity of the domain. We introduce ( a , η ) type to describe the convexity. As a result, we show that the more convex is the domain, the better is the regularity of the solution. In particular, the regularity is the best near angular points.