Global Lipschitz and Sobolev estimates for the Monge-Amp\`ere eigenfunctions of general bounded convex domains

TL;DR

This paper proves that Monge-Ampère eigenfunctions and certain degenerate solutions on bounded convex domains are globally Lipschitz, leading to new Sobolev regularity results within a sharp parameter threshold.

Contribution

It establishes the first global Lipschitz estimates for Monge-Ampère eigenfunctions and extends regularity results to degenerate equations with sharp parameter conditions.

Findings

Monge-Ampère eigenfunctions are globally Lipschitz on convex domains.

Global $W^{2,1}$ estimates are obtained for degenerate Monge-Ampère solutions.

Results hold within the sharp threshold $p>n-2$ for degeneracy.

Abstract

We show that the Monge-Amp\`ere eigenfunctions of general bounded convex domains are globally Lipschitz. The same result holds for convex solutions to degenerate Monge-Amp\`ere equations of the form $\det D^2 u =M|u|^p$ with zero boundary condition on general bounded convex domains in ${\mathbb R}^n$ within the sharp threshold $p>n-2$. As a consequence, we obtain global $W^{2, 1}$ estimates for these solutions.