Self gravitating cosmic strings and the Alexandrov's inequality for Liouville-type equations

TL;DR

This paper extends classical inequalities and estimates for Liouville-type equations with singularities, motivated by cosmic string models, and establishes a minimal mass property for solutions.

Contribution

It generalizes known inequalities to solutions in the distribution sense and applies these to cosmic string equations, deriving new pointwise estimates and a minimal mass property.

Findings

Generalized Bol's inequality for distributional solutions.

Derived new pointwise estimates for singular cosmic string equations.

Established a minimal mass property for supersolutions.

Abstract

Motivated by the study of self gravitating cosmic strings, we pursue the well known method by C. Bandle to obtain a weak version of the classical Alexandrov's isoperimetric inequality. In fact we derive some quantitative estimates for weak subsolutions of a Liouville-type equation with conical singularities. Actually we succeed in generalizing previously known results, including Bol's inequality and pointwise estimates, to the case where the solutions solve the equation just in the sense of distributions. Next, we derive some \uv{new} pointwise estimates suitable to be applied to a class of singular cosmic string equations. Finally, interestingly enough, we apply these results to establish a minimal mass property for solutions of the cosmic string equation which are \uv{supersolutions} of the singular Liouville-type equation.