The Mean Field Equation with Critical Parameter in a Plane Domain

2006/11/08 by Yilong Ni, Ni, Yilong
Mathematics · #35J20 #35J60 #49J10 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35J20 #msc:35J60 #msc:49J10

paper · pdf · doi:10.48550/arxiv.math/0611246

13 pages, to appear in Differential and Integral Equations

arxiv created 2006/11/08 · arxiv updated 2009/12/01

Abstract

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).

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