Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori
Keyword(s):
The Mean
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Abstract We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.
An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime
2013 ◽
Vol 143
(5)
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pp. 1021-1045
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Keyword(s):
The Mean
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2020 ◽
Vol 269
(11)
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pp. 10239-10276
1997 ◽
Vol 6
(11)
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pp. 829-836
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Keyword(s):
2021 ◽
Vol 60
(6)
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