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

TL;DR

This paper proves the existence and uniqueness of solutions for a mean field equation on multiply-connected domains at the critical parameter, extending prior results and solving longstanding open problems in mathematical physics.

Contribution

It establishes the first uniqueness result for the mean field equation on multiply-connected domains at the critical parameter and provides a necessary and sufficient condition for solution existence at this parameter.

Findings

Proved uniqueness of solutions for  at =8 on multiply-connected domains.

Derived a new version of Bol's inequality applicable to multiply-connected domains.

Solved the open problem of the equivalence of canonical and microcanonical ensembles in 2D turbulence.

Abstract

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.