Fully Degenerate Monge Amp\'ere Equations

TL;DR

This paper studies a nonlinear eigenvalue problem for the Monge-Ampère equation, establishing existence and regularity of solutions with smooth free boundaries in convex domains for certain parameter ranges.

Contribution

It introduces a new approach to find classical solutions with smooth free boundaries for the degenerate Monge-Ampère equation in convex domains.

Findings

Existence of smooth solutions with convex free boundaries.

Solutions are classical and smooth on the positive set.

Free boundary is proven to be smooth.

Abstract

In this paper, we consider the following nonlinear eigenvalue problem for the Monge-Amp\'ere equation: find a non-negative weakly convex classical solution $f$ satisfying {equation*} {cases} \det D^2 f=f^p \quad &\text{in $\Omega$} f=\vp \quad &text{on $\partial\Omega$} {cases} {equation*} for a strictly convex smooth domain $\Omega\subset\R^2$ and $0<p<2$. When $\{f=0\}$ contains a convex domain, we find a classical solution which is smooth on $\bar{\{f>0\}}$ and whose free boundary $\partial\{f=0\}$ is also smooth.