A mean field equation on a torus: one-dimensional symmetry 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<...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2003 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/909 |
| Online Access: | https://hdl.handle.net/2117/909 |
| Access Level: | Open access |
| Keyword: | 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 |
| Summary: | 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 $\epsilon\leq1/2$. |
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