A Mean Field Equation on a Torus: One-Dimensional Symmetry of Solutions
Xavier Cabré, Marcello Lucia, Manel Sanchón · Communications in Partial Differential Equations · 2005
We study the equation for u ∈ E, where E = {u ∈ H 1(Ωϵ): u is doubly periodic, ∈ t Ωϵ u = 0} and Ωϵ is a rectangle of ℝ2 with side lengths 1/ϵ and 1, 0 < ϵ ≤ 1. We establish that every solution depends only on the x-variable when λ ≤ λ*(ϵ), where λ*(ϵ) is an explicit positive constant depending on the maximum conformal radius of the rectangle. As a consequence, we obtain an explicit range of parameters ϵ and λ in which every solution is identically zero. This range is optimal for ϵ ≤ 1/2.