A sign-changing Liouville Equation

TL;DR

This paper investigates periodic solutions to a Liouville equation with sign-changing weights, deriving formulas for various boundary data types and analyzing the impact of singularities on solution regularity.

Contribution

It introduces a representation formula for solutions with singular and nonsingular boundary data, and establishes regularity criteria for a generalized Liouville equation.

Findings

Derived a representation formula for solutions with different boundary data

Analyzed the impact of boundary singularities on interior regularity

Established regularity criteria for a generalized Liouville equation

Abstract

We examine periodic solutions to an initial boundary value problem for a Liouville equation with sign-changing weight. A representation formula is derived both for singular and nonsingular boundary data, including data arising from fractional linear maps. In the case of singular boundary data we study the effects the induced singularity has on the interior regularity of solutions. Regularity criteria are also found for a generalized form of the equation.