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

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