Existence and uniqueness for mean field equations on multiply connected domains at the critical parameter
Journal
Mathematische Annalen
Journal Volume
359
Journal Issue
1-2
Pages
1-44
Date Issued
2014
Author(s)
Bartolucci, D.
Abstract
We consider the mean field equation: (Formula presented.) where Ω ⊂ℝ2 is an open and bounded domain of class C1. In his 1992 paper, Suzuki proved that if Ω is a simply-connected domain, then Eq. (1) admits a unique solution for ρ ∈ [0,8π). This result for Ω a simply-connected domain has been extended to the case ρ =8π by Chang, Chen and the second author. However, the uniqueness result for Ω a multiply-connected domain has remained a long standing open problem which we solve positively here for ρ ∈ [0,8π]. To obtain this result we need a new version of the classical Bol's inequality suitable to be applied on multiply-connected domains. Our second main concern is the existence of solutions for (1) when ρ =8π. We obtain a necessary and sufficient condition for the solvability of the mean field equation at ρ =8π which is expressed in terms of the Robin's function γ for Ω. For example, if Eq. (1) has no solution at ρ =8π, then γ has a unique nondegenerate maximum point. As a by product of our results we solve the long-standing open problem of the equivalence of canonical and microcanonical ensembles in the Onsager's statistical description of two-dimensional turbulence on multiply-connected domains. © 2013 Springer-Verlag Berlin Heidelberg.
Type
journal article