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Minimizers of Dirichlet functionals on the n-torus and the Weak KAM Theory

Gershon Wolansky
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Abstract

Given 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".

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Gershon Wolansky. 2007-11-17. Minimizers of Dirichlet functionals on the n-torus and the Weak KAM Theory. https://arxiv.org/abs/math/0702743

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