Hölder continuous solutions to Monge-Ampère equations

Abstract

We study the regularity of solutions to the Dirichlet problem for the complex Monge–Ampère equation (ddc u)n=f dV on a bounded strongly pseudoconvex domain Ω⊂ℂn. We show, under a mild technical assumption, that the unique solution u to this problem is Hölder continuous if the boundary data ϕ is Hölder continuous and the density f belongs to Lp(Ω) for some p>1. This improves previous results by Bedford and Taylor and Kolodziej.