Existence of C^1,1 critical subsolutions in discrete weak KAM theory

2010/04/01 by Maxime Zavidovique, Zavidovique, Maxime · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.AP #math.DS

paper · pdf · doi:10.48550/arxiv.1004.0086

28 pages

arxiv created 2010/04/01 · openalex publication_date 2010/04/01 · arxiv updated 2010/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, following a first work of the author, we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function c:M × M→ \R defined on a smooth connected manifold is locally semi-concave and verifies twist conditions, then there exists a C^1,1 critical subsolution strict on a maximal set (namely, outside of the Aubry set). We also explain how this applies to costs coming from Tonelli Lagrangians. Finally, following ideas introduced in the work of Fathi-Maderna and Mather, we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analogue of Mather's α function on the cohomology.

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• Weak KAM theorem on non compact manifolds

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