A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

Abstract

We derive a singular version of the Sphere Covering Inequality which was recently introduced in [42], suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in [56]. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.2000 Mathematics Subject Classification. 35J61, 35R01, 35A02, 35B06.

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

“…Now, if |J| = N = 1, p 1 = 0 and Ω = B 1 (0), then solutions to (3) (which are radial and well known in this particular case) exist if and only if ρ < 8π(1 − α) ≡ 8π(1 + α 1 ), showing that our existence and uniqueness result is sharp in this case. On the other side, we stress that both our result and the one in [9] yield to the same uniqueness threshold which, for |J| ≥ 2, is lower than the subcritical existence threshold 8π 1 + min j {α j , 0} . This motivates the following interesting open problem:…”

“…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). From this point of view, we come up with a new proof of the uniqueness based on the non-degeneracy of (5).…”

“…Let Ω be an open, bounded, simply-connected and regular (according to Definition 1.1) domain. Let α < 1 be given as in (9). Then, for any ρ ≤ 8π(1 − α), there exists at most one weak H 1 0 (Ω) solution of (3) and the first eigenvalue of the corresponding linearized problem is strictly positive.…”

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.…”

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]).…”