Symmetry of solutions of a mean field equation on flat tori

TL;DR

This paper proves symmetry properties of solutions to a mean field equation on flat tori, showing solutions are symmetric about certain lines under specific conditions, and confirms a conjecture by Lin and Lucia.

Contribution

It establishes symmetry of solutions on flat tori under conditions on the function K and parameter ρ, and confirms a conjecture regarding one-dimensionality when K is constant.

Findings

Solutions are symmetric about lines through critical points of u+ln(K).

All solutions are one-dimensional if K≡1 and ρ ≤ 8π.

Results are sharp and extend to mean field equations on annuli.

Abstract

We study symmetry of solutions of the mean field equation \[ \Delta u +\rho(\frac{Ke^u}{\int_{T_\epsilon} Ke^u} -\frac{1}{|T_\epsilon|} )=0\] on the flat torus $T_\epsilon=[-\frac{1}{2\epsilon}, \frac{1}{2\epsilon}] \times [-\frac{1}{2}, \frac{1}{2}]$ with $0<\epsilon \leq 1$, where $K\in C^2({T}_\epsilon)$ is a positive function with $-\Delta \ln K \leq \frac{\rho}{|T_\epsilon|}$ and $\rho \leq 8\pi$. 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 $\rho \leq 8\pi$. 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.