The Mean Field Equation with Critical Parameter in a Plane Domain

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