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

TL;DR

This paper proves the existence of smooth critical subsolutions in discrete weak KAM theory under certain conditions, extending the theory to invariant costs and covering spaces with applications to Mather's alpha function.

Contribution

It establishes the existence of $C^{1,1}$ critical subsolutions for locally semi-concave, twist condition cost functions, and extends the theory to invariant costs and covering spaces, introducing a discrete Mather alpha function.

Findings

Existence of $C^{1,1}$ critical subsolutions under twist and semi-concavity conditions.

Application of results to costs from Tonelli Lagrangians.

Development of a discrete analogue of Mather's alpha function on covering spaces.

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 $c:M \times M\to \R{}$ 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 $\alpha$ function on the cohomology.