One-dimensional symmetry of periodic minimizers for a mean field equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 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 Δu+ρe u T e u -1 |T|=0, T u=0. It is well-known that the associated energy functional admits a minimizer for each ρ8π. The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting λ 1 (T) to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever ρmin{8π,λ 1 (T)|T|}. 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, Série 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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