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One-dimensional symmetry of periodic minimizers for a mean field equation. (English) Zbl 1150.35036
The authors consider the equation $\Delta u + \rho \left(\frac{e^u}{\int_T e^u}-\frac{1}{|T|}\right)=0,\qquad \int_T u=0, (*)$ on a two-dimensional flat torus $$T$$ defined by a rectangular cell $$\Omega=(-a,a)\times(-b,b)$$. Here $$\rho$$ is a real parameter and solutions are sought in the Sobolev space $${\overset \circ H}(T) =\{ u\in H^1(T):\int_T u=0 \}$$.
This problem has a variational formulation: $$(*)$$ is the Euler-Lagrange equation of the functional $$J_\rho:{\overset \circ H}(T)\rightarrow{\mathbb R}$$ defined by $J_\rho(u)=\frac{1}{2} \int_T |\nabla u|^2-\rho\log\left(\frac{1}{|T|}\int_T e^u \right).$ Various results have been proved about solutions of $$(*)$$ (see the paper for references). In particular, it is known that a minimizer exists for $$\rho\leq 8\pi$$ and that $$u\equiv 0$$ is the unique solution of $$(*)$$ if $$\rho$$ is sufficiently close to zero. If $$b/a\leq 1/2$$, the trivial solution $$u\equiv 0$$ is the unique solution of $$(*)$$ if and only if $$\rho\leq \lambda_1(T)|T|$$ where $$\lambda_1(T)$$ denotes the first nonzero eigenvalue of the Laplacian on $$T$$. If $$\Omega$$ is a square, $$(*)$$ has only the trivial solution when $$\rho\leq 8\pi$$. This result is optimal, as the existence of nontrivial solutions is known if $$\rho>8\pi$$.
Motivated by these uniqueness questions, the authors investigate whether solutions of $$(*)$$ are one dimensional in the sense that $\frac{\partial u}{\partial x_1} \equiv 0 \quad\text{in}\quad\Omega\qquad \text{or}\qquad \frac{\partial u}{\partial x_2} \equiv 0 \quad\text{in}\quad\Omega.$ C. Cabré, M. Lucia and M. Sanchón [Commun. Partial Differ. Equ. 30, 1315–1330 (2005; Zbl 1115.35041)] have shown that every solution is one dimensional if $$\rho\leq\rho^*(T)$$, where $$\rho^*(T)$$ is the maximum conformal radius of $$\Omega$$. Any such $$\rho^*(T)$$ must be strictly larger than $$4\pi$$, and it is expected that $$\rho^*(T)=8\pi$$. Here the authors confirm this conjecture. Furthermore, they show that if $$\rho\leq \min\{ 8\pi, \lambda_1(T)|T|\}$$, then $$u\equiv 0$$ is the unique global minimizer of $$J_\rho$$.

##### MSC:
 35J60 Nonlinear elliptic equations 35B10 Periodic solutions to PDEs