Journal of Modern Dynamics (JMD)

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

Pages: 693 - 714, Issue 4, October 2010      doi:10.3934/jmd.2010.4.693

       Abstract        References        Full Text (285.8K)       Related Articles

Maxime Zavidovique - Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, siteMonod, UMR CNRS 5669, 46, allée d’Italie, 69364 LYON Cedex 07, France (email)

Abstract: In this article, following [29], we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function $c: M \times M\to \R$ defined on a smooth connected manifold is locally semiconcave and satisfies 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 [18] and [26], we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analog of Mather's $\alpha$ function on the cohomology.

Keywords:  Discrete weak KAM theory, critical subsolutions, Ilmanen's lemma, Aubry Mather theory, twist maps, Hamiltonian and Lagrangian dynamics.
Mathematics Subject Classification:  Primary: 37J50; Secondary: 37E40, 37B25.

Received: April 2010;      Revised: October 2010;      Available Online: January 2011.