Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter

We consider the mean field equation: Δu + ρ u = 0 e u e u Ω = 0 in Ω, (1) on ∂Ω, where Ω ⊂ R 2 is an open and bounded domain of class C 1 . 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.