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

We consider the mean field equation: (1) Δu+ρ\frac{e^u}{\int_Ωe^u}=0 & \hbox{in} \;Ω, u=0 & \hbox{on}\;\partialΩ, where $Ω\subset \mathbb{R}^2$ 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 $ρ\in[0,8π)$. 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 $ρ\in[0,8π]$. 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.