The Mean Field Equation with Critical Parameter in a Plane Domain

Details The Mean Field Equation with Critical Parameter in a Plane Domain The Mean Field Equation with Critical Parameter in a Plane Domain

2006-11-08

arxiv.org/abs/math/0611246v1

Mathematics

Analysis of PDEs

13 pages, to appear in Differential and Integral Equations

Scientific paper

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 $I_{8\pi}(\Omega)$.

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