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

Abstract

The aim of this paper is to complete the program initiated in [50], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov-Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems.2000 Mathematics Subject Classification. 35J61, 35R01, 35A02, 35B06.

Non-degeneracy and uniqueness of solutions to general singular Toda systems on bounded domains

“…Here, by non-degeneracy we mean that the first eigenvalue of the corresponding linearized problem is strictly positive. Up to know, this topic has been investigated only for the standard Liouville case (1.4) in various settings [4,5,6,12,33]. In our context, by using Alexandrov-Bol inequality and by a quite delicate eigenvalues analysis of linearized singular Liouville-type problems it was proven that if λ ≤ 4π (1.4) admits at most one solution which is non-degenerate, see in particular [5].…”

Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions

“…To simplify our notation, we shall always assume |M | = 1. Equation (1.3) and its counterpart on bounded planar domains arise in several areas of mathematics and physics and there are by now many results concerning existence ( [2,15,9,10,11,24,29,30,41,42]), uniqueness of solutions ( [4,12,13,14,26,47,48,49,60,65]) and blow-up analysis ( [3,5,16,17,19,28,31,57,58]). On one hand, they are derived as a mean field limit in the statistical mechanics description of two dimensional turbulent Euler flows ( [20,21]) and selfgravitating systems ( [54,56,72]).…”

Local uniqueness and non-degeneracy of blow up solutions of mean field equations with singular data

“…The mean field equation (P ρ n ) (and its counterpart on compact surfaces) have been widely discussed in the last decades because of their several applications in Mathematics and Physics, such as Electroweak and Chern-Simons self-dual vortices [47,49,53], conformal metrics on surfaces with [50] or without conical singularities [35], statistical mechanics of two-dimensional turbulence [20] and of selfgravitating systems [52] and cosmic strings [45], and the theory of hyperelliptic curves [22] and of the Painlevé equations [24]. There are by now many results concerning existence [1,3,4,5,15,21,26,28,30,31,32,36,42], multiplicity [5,29], uniqueness [6,7,10,11,12,13,14,23,33,34,40,41,48] and blow up analysis [2,9,16,18,17,19,25,27,…”

“…Equation (1.1) arises in many branches of Mathematics and Physics, such as conformal geometry ([3, 36, 43, 47, 74, 76]), Electroweak and Self-Dual Chern-Simons vortices ( [1,67,70,71,81]), the statistical mechanics description of turbulent Euler flows, plasmas and self-gravitating systems ([9, 24, 66, 75, 79]), Cosmic Strings ( [21,64]), the theory of hyperelliptic curves and modular forms ( [25]) and CMC immersions in hyperbolic 3-manifolds ( [73]). Among many other things which we cannot list here, these were some of the motivations for the huge effort made in the study of (1.1), including existence ( [5,6,7,15,18,22,37,26,30,31,33,34,38,39,40,55,55,60,61,62,63]), uniqueness ( [8,10,11,12,13,14,16,17,27,45,49,52,54,68...…”