Uniform bound for the volume of the solutions to Liouville type equations on the annulus

TL;DR

This paper establishes a uniform bound on the volume of solutions to Liouville type equations on an annulus, addressing variational problems with specific weights and boundary conditions.

Contribution

It provides the first known uniform volume bounds for solutions to Liouville equations under these boundary and weight conditions.

Findings

Bounded volume of solutions established

Applicable to equations with H{"o}lderian weights

Addresses boundary singularities

Abstract

We consider variational problems with regular H{\"o}lderian weight or with weight and boundary singularity and, Dirichlet condition. We prove the boundedness of the volume of the solutions to these equations on the annulus.