Properties of the solutions to the Monge-Ampère equation

Abstract

We consider solutions to the equation detD2φ=μ when μ has a doubling property. We prove new geometric characterizations for this doubling property (by means of the so-called engulfing property) and deduce the quantitative behaviour of φ. Also, a constructive approach to the celebrated C1,β-estimates proved by L. Caffarelli is presented, settling one of the open questions posed by Villani (Amer. Math. Soc. 58 (2003)).