Calculus of Variations and Geometric Measure Theory
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D. Bartolucci - D. Castorina

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

created by castorina on 22 Apr 2016
modified on 31 Oct 2018


Accepted Paper

Inserted: 22 apr 2016
Last Updated: 31 oct 2018

Journal: Communications in Contemporary Mathematics
Year: 2015
Doi: 10.1142/S0219199715500686

ArXiv: 1409.3135 PDF


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.

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