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

Abstract

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 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 supersolutions of the singular Liouville-type equation.

The sphere covering inequality and its applications

“…In particular, if we assume further that µ (M ) ≤ 4π(1−Θ) a , then (M, g) satisfies (1 − Θ, a)-isoperimetric inequality. In fact this is even true when g is singular, see [BC1,BC2,BGJM] and the references therein.…”

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

“…To this end, the first tool we need is an Alexandrov-Bol's inequality for solutions of (5) suitable for our setting. Such an inequality was first proved in the analytical framework in [1] and more recently generalized to the weak setting in [4,5]. However what we need here is a more general statement which allows one to push the inequality, still in this weak setting, up to the (non-smooth) boundary of the domain.…”

“…The following version of the Alexandrov-Bol inequality was first proved in the analytical framework in [1] and more recently generalized to the weak setting in [4,5]. Actually, if ω in the statement is a relatively compact subset of Ω 0 , then the result is just a particular case of Theorem 1.5 in [5].…”

“…Proof. The proof can be worked out by a step by step adaptation of the one provided in [5] with minor changes borrowed from Theorem 4.1 in [4].…”

“…Also, the needed estimates were first derived in [1], but only in the analytic framework, and in [4] for weak subsolutions, but only for the model case (3)-(4). We solve this problem here by proving some uniform estimates of independent interest for weak solutions of (5), in the same spirit of [1,4]. We finally remark that the uniqueness part concerning solutions to (5) was very recently obtained in [9] by a completely different argument (see also [37] for a similar application of the latter method).…”