On the singular set of the free boundary for a Monge-Amp\`ere obstacle problem

TL;DR

This paper investigates the structure and properties of the free boundary in a Monge-Ampère obstacle problem, focusing on the non-strictly convex part, and establishes bounds and principles related to its geometry and stability.

Contribution

It extends previous work by analyzing the non-strictly convex free boundary, providing optimal dimension bounds and exploring maximum principles and stability for the problem.

Findings

Established optimal dimension bounds for the flat portion of the free boundary.

Analyzed the strong maximum principle in the context of the Monge-Ampère obstacle problem.

Proved stability properties of solutions related to the free boundary.

Abstract

This is a continuation of our earlier work [14] on the Monge-Amp\`ere obstacle problem \[ \det D^2 v = v^q \chi_{\{v>0\}}, \quad v \geq 0 \text{ convex} \] with $q \in [0,n)$, where we studied the regularity of the strictly convex part of the free boundary. In this work, we examine the non-strictly convex part of the free boundary and establish optimal dimension bounds for its flat portion. Additionally, we investigate the strong maximum principle and a stability property for this Monge-Amp\`ere obstacle problem.