Abstract

We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation $$ \Delta u + \rho \left = 0, \quad \int _{T} u = 0. $$ It is well-known that the associated energy functional admits a minimizer for each $\rho \le 8 \pi $. The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting $\lambda _1 $ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $\rho \le \min \lbrace 8 \pi, \lambda _1 |T| \rbrace $. Our results hold more generally for solutions that are Steiner symmetric, up to a translation