Symmetry of Solutions of a Mean Field Equation on Flat Tori
Symmetry of Solutions of a Mean Field Equation on Flat Tori
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 …