2026-04-17T21:23:56Zhttps://recercat.cat/oai/request

oai:recercat.cat:2117/9092025-07-17T06:39:19Zcom_2072_1033col_2072_452950

A mean field equation on a torus: one-dimensional symmetry of solutions Cabré Vilagut, Xavier Lucia D'Agostino, Marcello Sanchón Rodellar, Manuel Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions Partial differential equations mean field equation torus Equacions en derivades parcials Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions We study the equation $$-\Delta u=\lambda\left(\frac{e^u}{\int_\oep e^u}- \frac{1}{|\oep|}\right)\quad \text{in }\oep$$ for $u\in E$, where $E = \{ u \in H^1(\oep): u \hbox{ is doubly periodic}, \int_{\oep} u = 0 \}$ and $\oep$ is a rectangle of $\R^2$ with side lengths $1/\epsilon$ and $1$, $0< \epsilon \leq 1$. We establish that every solution depends only on the $x$--variable when $\lambda \leq \lambda^*(\epsilon)$, where $\lambda^*(\epsilon)$ is an explicit positive constant depending on the maximum conformal radius of the rectangle. As a consequence, we obtain an explicit range of parameters $\epsilon$ and $\lambda$ in which every solution is identically zero. This range is optimal for $\epsilon\leq1/2$. 2003 Article https://hdl.handle.net/2117/909 eng http://creativecommons.org/licenses/by-nc-nd/2.5/es/ Open Access Attribution-NonCommercial-NoDerivs 2.5 Spain 19 application/pdf