Abstract
Given a probability measure μ on the n-torus Tn and a rotation vector P∈Rn, we ask whether there exists a minimizer to the integral ∫Tn|∇ϕ+P|2dμ. 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”.
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