Schauder estimates for degenerate Monge-Ampère equations and smoothness of the eigenfunctions

Abstract

We obtain $C^{2,\beta }$ estimates up to the boundary for solutions to degenerate Monge-Ampère equations of the type $$\det D^{2}u=f\text{in}\Omega ,f\sim {\text{dist}}^{\alpha }(\cdot ,\partial \Omega )\text{near}\partial \Omega ,\alpha >0.$$ As a consequence we obtain global $C^{\infty }$ estimates up to the boundary for the eigenfunctions of the Monge-Ampère operator $(\det D^{2}u{)}^{1/n}$ on smooth, bounded, uniformly convex domains in $R^n$.

arXiv commentv2: minor typos fixed; to appear in Invent. Math