W2,1 regularity for solutions of the Monge-Ampère equation

Abstract

In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Ampère equation, with right-hand side bounded away from zero and infinity, is W2,1loc. This is obtained by showing higher integrability a priori estimates for D2u, namely D2u∈LlogkL for any k∈ℕ. © 2012 Springer-Verlag.