The Mean Field Equation with Critical Parameter in a Plane Domain

Consider the mean field equation with critical parameter $8π$ in a bounded smooth domain $Ω$. Denote by $E_{8π}(Ω)$ the infimum of the associated functional $I_{8π}(Ω)$. We call $E_{8π}(Ω)$ the "energy" of the domain $Ω$. We prove that if the area of $Ω$ is equal to $π$, then the energy of $Ω$ is always greater or equal to the energy of the unit disk and equality holds if and only if $Ω$ is the unit disk. We also give a sufficient condition for the existence of a minimizer for $I_{8π}(Ω)$.