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

We consider the mean field equation: (1) \Delta u+\rho\frac{e^u}{\int_\Omega e^u}=0 & \hbox{in} \;\Omega, u=0 & \hbox{on}\;\partial\Omega, where $\Omega\subset \mathbb{R}^2$ is an open and bounded domain of class $C^1$. In his 1992 paper, Suzuki proved that if $\Omega$ is a simply-connected domain, then equation (1) admits a unique solution for $\rho\in[0,8\pi)$. This result for $\Omega$ a simply-connected domain has been extended to the case $\rho=8\pi$ by Chang, Chen and the second author. However, the uniqueness result for $\Omega$ a multiply-connected domain has remained a long standing open problem which we solve positively here for $\rho\in[0,8\pi]$. 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 $\rho=8\pi$. We a obtain necessary and sufficient condition for the solvability of the mean field equation at $\rho=8\pi$ which is expressed in terms of the Robin's function $\gamma$ for $\Omega$. For example, if equation (1) has no solution at $\rho=8\pi$, then $\gamma$ 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.