  Given a probability measure $\mu$ on the $n-$torus $T^n$ and a rotation
vector $k\in R^n$, we ask wether there exists a minimizer to the integral
$\int_{T^n} |\grad\phi+k|^2 d\mu$. This problem leads, naturally, to a class of
elliptic PDE and to an optimal transportation (Monge-Kantorovich) class of
problems on the torus. It is also related to higher dimensional Aubry-Mather
theory, dealing with invariant sets of periodic Lagrangians, and is known as
the "Weak-KAM theory".
