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

Chang-Shou Lin; Marcello Lucia

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 2, page 269-290
  • ISSN: 0391-173X

Abstract

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We consider on a two-dimensional flat torus T defined by a rectangular periodic cell the following equation Δ u + ρ e u ∫ T e u - 1 | T | = 0 , ∫ T u = 0 . It is well-known that the associated energy functional admits a minimizer for each ρ ≤ 8 π . 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 ρ ≤ min { 8 π , λ 1 ( T ) | T | } . Our results hold more generally for solutions that are Steiner symmetric, up to a translation.

How to cite

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Lin, Chang-Shou, and Lucia, Marcello. "One-dimensional symmetry of periodic minimizers for a mean field equation." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.2 (2007): 269-290. <http://eudml.org/doc/272285>.

@article{Lin2007,
abstract = {We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation\[ \Delta u + \rho \left( \frac\{e^u\}\{\int \_\{T\} e^u\} - \frac\{1\}\{|T|\} \right) = 0, \quad \int \_\{T\} u = 0. \]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 $\lambda _1 (T)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $\rho \le \min \lbrace 8 \pi , \lambda _1 (T) |T| \rbrace $. Our results hold more generally for solutions that are Steiner symmetric, up to a translation.},
author = {Lin, Chang-Shou, Lucia, Marcello},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {269-290},
publisher = {Scuola Normale Superiore, Pisa},
title = {One-dimensional symmetry of periodic minimizers for a mean field equation},
url = {http://eudml.org/doc/272285},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Lin, Chang-Shou
AU - Lucia, Marcello
TI - One-dimensional symmetry of periodic minimizers for a mean field equation
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 2
SP - 269
EP - 290
AB - We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation\[ \Delta u + \rho \left( \frac{e^u}{\int _{T} e^u} - \frac{1}{|T|} \right) = 0, \quad \int _{T} u = 0. \]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 $\lambda _1 (T)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $\rho \le \min \lbrace 8 \pi , \lambda _1 (T) |T| \rbrace $. Our results hold more generally for solutions that are Steiner symmetric, up to a translation.
LA - eng
UR - http://eudml.org/doc/272285
ER -

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