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 $T_{\epsilon }=[-\frac{1}{2\epsilon },\frac{1}{2\epsilon }]\times [-\frac{1}{2},\frac{1}{2}]$ with $0<\epsilon \le 1$, where $K\in C^2(T_{\epsilon })$ is a positive function with $-\Delta \ln K\le \frac{\rho }{|T_{\epsilon }|}$ and $\rho \le 8\pi$. 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 $\rho \le 8\pi$. 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.

References in corpus (1)

• The Sphere Covering Inequality and Its Applications