Blow-up analysis for a cosmic strings equation

Abstract

In this paper we develop a blow-up analysis for solutions of an elliptic PDE of Liouville type over the plane. Such solutions describe the behavior of cosmic strings (parallel in a given direction) for a W-boson model coupled with Einstein's equation. We show how the blow-up behavior of the solutions is characterized, according to the physical parameters involved, by new and surprising phenomena. For example in some cases, after a suitable scaling, the blow-up profile of the solution is described in terms of an equations that bares a geometrical meaning in the context of the "uniformization" of the Riemann sphere with conical singularities.

Symmetry and uniqueness of solutions to some Liouville-type equations and systems

“…Many results concerning (9) have been established especially for the full plane case. We refer to [12,13,56] for existence results, to [42,43] for what concerns symmetry issues, and to [54] for blow-up analysis. In particular, in [42,43] the authors provide necessary and sufficient conditions for the solvability of (9) in the full plane in the context of radially symmetric solutions, depending on the values of the total mass β =´R 2 e au + |x| 2N e u dx.…”

Sharp estimates for solutions of mean field equations with collapsing singularity

“…At first, we point out a fact which will be frequently used in the sequel. It was used first in [4] and more recently in [44] to carry out the blow-up analysis of cosmic strings. It gives us a criterion to recognise when "concentration" is bound to occur.…”

On Non-Topological Solutions for Planar Liouville Systems of Toda-Type

“…Equations of this type arise in the construction of Self-gravitating Cosmic Strings (cfr. [71]), and have been analysed in [14,58,59,69]. In particular, in this case, for i = 1 we may complete (3.10), by using (3.19) and (3.20) and obtain that every solutions of (3.21) satisfies:…”