On the mean oscillation of the Hessian of solutions to the Monge–Ampère equation

Abstract

We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge–Ampère equation det D 2 u = f ( x ) with zero boundary values, where f ( x ) is a non-Dini continuous function. If the modulus of continuity of f ( x ) is φ ( r ) such that lim r → 0 φ ( r ) log ( 1 / r ) = 0 , then D 2 u ∈ VMO .