Schauder estimates for degenerate Monge-Amp\`ere equations and smoothness of the eigenfunctions

Abstract

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