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

2007/02/25 by Gershon Wolansky, Wolansky, Gershon
Mathematics · Physics and Astronomy · #49K20 #65K10 #90C47 #Analysis of PDEs (math.AP) #Class (philosophy) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry #Geometry and complex manifolds #Invariant (physics) #Kolmogorov–Arnold–Moser theorem #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Measure (data warehouse) #Pure mathematics #Quantum chaos and dynamical systems #Rotation (mathematics) #Torus #math.AP #math.DS #msc:49K20 #msc:65K10 #msc:90C47

paper · pdf · doi:10.48550/arxiv.math/0702743

35 pages, no figures

openalex publication_date 2007/02/25 · arxiv created 2007/11/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Given a probability measure μ on the n-torus Tⁿ and a rotation vector k∈ Rⁿ, we ask wether there exists a minimizer to the integral ∫_Tⁿ |\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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