*Accepted Paper*

**Inserted:** 12 may 2011

**Journal:** Ergodic Theory and Dynamical Systems

**Year:** 2011

**Abstract:**

We consider the functional
$$ \int \frac{

\nabla u^{2}{2}+F}(x,u)\,dx$$
in a periodic setting.
We discuss whether the minimizers or the stable solutions
satisfy some symmetry or monotonicity properties,
with special emphasis on the autonomous case
when $F$ is $x$-independent.

In particular, we give an answer to a question posed by Victor Bangert when $F$ is autonomous in dimension $n\le3$ and in any dimension for nonzero rotation vectors.

**Download:**