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

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 $c:M \times M\to \R{}$ 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.