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In this Chapter, we introduce the metric setting suitable to develop discrete weak KAM theoryweak KAMtheory. We define the discrete Lax-Oleinik operator, and its discounted counterpart and prove the discrete weak KAM Theorem. We go on to introduce the fundamental Aubry set and its links to subsolutions and strict subsolutions. Finally, we present, without proofs, the analogous results and notions coming from classical weak KAM theory. We prove new links between the classical and the discrete theories, when the cost function is given by the action functional.

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The Discrete Setting, Weak KAM Solutions and Subsolutions

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  • 发表日期：

    2025年

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  • Maxime Zavidovique

摘要

In this Chapter, we introduce the metric setting suitable to develop discrete weak KAM theoryweak KAMtheory. We define the discrete Lax-Oleinik operator, and its discounted counterpart and prove the discrete weak KAM Theorem. We go on to introduce the fundamental Aubry set and its links to subsolutions and strict subsolutions. Finally, we present, without proofs, the analogous results and notions coming from classical weak KAM theory. We prove new links between the classical and the discrete theories, when the cost function is given by the action functional.