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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 \(J_\rho\).

35J60 Nonlinear elliptic equations
35B10 Periodic solutions to PDEs