Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains

TL;DR

This paper advances the understanding of mean field equations by establishing non-degeneracy and uniqueness of solutions with singular data on non-smooth domains, using inequalities and eigenvalue analysis.

Contribution

It extends prior work by addressing general singular data and non-smooth domains, providing new results on solution properties.

Findings

Proves non-degeneracy of solutions under broad conditions

Establishes uniqueness of solutions with singular data

Develops new analytical techniques for singular Liouville problems

Abstract

The aim of this paper is to complete the program initiated in [50], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov-Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems.