Symmetry of solutions of a mean field equation on flat tori

Abstract

We study symmetry of solutions of the mean field equation \[ Δu +ρ(\frac{Ke^u}{\int_{T_ε} Ke^u} -\frac{1}{|T_ε|} )=0\] on the flat torus $T_ε=[-\frac{1}{2ε}, \frac{1}{2ε}] \times [-\frac{1}{2}, \frac{1}{2}]$ with $0<ε\leq 1$, where $K\in C^2({T}_ε)$ is a positive function with $-Δ\ln K \leq \fracρ{|T_ε|}$ and $ρ\leq 8π$. We prove that if $(x_0,y_0)$ is a critical point of the function $u+ln(K)$, then $u$ is evenly symmetric about the lines $x=x_0$ and $y=y_0$, provided $K$ is evenly symmetric about these lines. In particular we show that all solutions are one-dimensional if $K\equiv 1$ and $ρ\leq 8π$. The results are sharp and answer a conjecture of Lin and Lucia affirmatively. We also prove some symmetry results for mean field equations on annulus.