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

TL;DR

This paper establishes boundary regularity estimates for degenerate Monge-Ampère equations and demonstrates smoothness of eigenfunctions of the Monge-Ampère operator on convex domains, advancing understanding of their boundary behavior.

Contribution

It provides new boundary regularity results for degenerate Monge-Ampère equations and proves smoothness of eigenfunctions on convex domains, extending classical regularity theory.

Findings

Boundary $C^{2,eta}$ estimates for solutions with degenerate right-hand side.

Global $C^ ext{infty}$ regularity of eigenfunctions up to the boundary.

Extension of regularity results to degenerate Monge-Ampère equations.

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 $R^n$.