One-dimensional symmetry of periodic minimizers for a mean field equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 2, p. 269-290

We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation $\Delta u+\rho \left(\frac{{e}^{u}}{{\int }_{T}{e}^{u}}-\frac{1}{|T|}\right)=0,\phantom{\rule{1em}{0ex}}{\int }_{T}u=0.$ 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 $\rho \le min\left\{8\pi ,{\lambda }_{1}\left(T\right)|T|\right\}$. Our results hold more generally for solutions that are Steiner symmetric, up to a translation.

Classification:  35J60,  35B10
@article{ASNSP_2007_5_6_2_269_0,
author = {Lin, Chang-Shou and Lucia, Marcello},
title = {One-dimensional symmetry of periodic minimizers for a mean field equation},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {2},
year = {2007},
pages = {269-290},
zbl = {1150.35036},
mrnumber = {2352519},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_2_269_0}
}

Lin, Chang-Shou; Lucia, Marcello. One-dimensional symmetry of periodic minimizers for a mean field equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 2, pp. 269-290. http://www.numdam.org/item/ASNSP_2007_5_6_2_269_0/

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