Partial Regularity for Stationary Solutions to Liouville-Type Equation in dimension 3

TL;DR

This paper proves that the singular set of stationary solutions to the Liouville equation in three dimensions has Hausdorff dimension at most one, revealing partial regularity properties of these solutions.

Contribution

It establishes a partial regularity result for stationary solutions to the Liouville equation in dimension three, showing the singular set's Hausdorff dimension is at most one.

Findings

Singular set has Hausdorff dimension at most 1

Stationary solutions belong to W^{1,2}

Partial regularity result for Liouville equation in 3D

Abstract

In dimension $n=3$, we prove that the singular set of any stationary solution to the Liouville equation $-\Delta u=e^u$, which belongs to $W^{1,2}$, has Hausdorff dimension at most 1.