On Monge-Ampere equations with homogeneous right hand side

TL;DR

This paper investigates the local behavior and regularity of solutions to a class of degenerate Monge-Ampere equations with homogeneous right hand sides in two dimensions, revealing stability and uniqueness properties depending on the degree alpha.

Contribution

It characterizes the possible behaviors of solutions near the origin for different alpha values and establishes stability results for the radial behavior when alpha<0.

Findings

Solutions exhibit only radial or non-radial behavior near the origin for alpha > 0.

Radial behavior is unstable for alpha > 0.

Solutions are exclusively radial near the origin for alpha < 0.

Abstract

We study the regularity and behavior at the origin of solutions to the two-dimensional degenerate Monge-Ampere equation with homogeneous right hand side of degree alpha, alpha>-2. We show that when alpha > 0, solutions admit only two possible behaviors near the origin, radial and non-radial. We also show that the radial behavior is unstable. For alpha<0 we prove that solutions admit only the radial behavior near the origin.