2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142336Given a probability measure $μ$ on the $n-$torus $T^n$ and a rotation vector $k\in R^n$, we ask wether there exists a minimizer to the integral $\int_{T^n} |\gradϕ+k|^2 dμ$. 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".35 pages, no figuresDynamical SystemsAnalysis of PDEs49K20, 65K10, 90C47text