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We consider the mean field equation:(1)    ∆u + ρ e u Ω e u = 0 in Ω, u = 0 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 equation (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 a obtain 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 equation (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.

We prove the uniqueness of blow up solutions of the mean field equation as ρ n → 8πm, m ∈ N. If u n,1 and u n,2 are two sequences of bubbling solutions with the same ρ n and the same (non degenerate) blow up set, then u n,1 = u n,2 for sufficiently large n. The proof of the uniqueness requires a careful use of some sharp estimates for bubbling solutions of mean field equations [24] and a rather involved analysis of suitably defined Pohozaev-type identities as recently developed in [51] in the context of the Chern-Simons-Higgs equations. Moreover, motivated by the Onsager statistical description of two dimensional turbulence, we are bound to obtain a refined version of an estimate about ρ n − 8πm in case the first order evaluated in [24] vanishes.