One-dimensional symmetry of periodic minimizers for a mean field equation

Lin, Chang-Shou; Lucia, Marcello

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 2, pp. 269-290.

We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation

$$\Delta u+\rho \left(\frac{e^u}{{\int }_T{e}^u}-\frac{1}{\left|T\right|}\right)=0,{\int }_{T}u=0.$$

It is well-known that the associated energy functional admits a minimizer for each $\rho \le 8\pi$. The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting ${\lambda }_1\left(T\right)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $\rho \le min\{8\pi ,{\lambda }_1\left(T\right)|T\left|\right\}$. Our results hold more generally for solutions that are Steiner symmetric, up to a translation.

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