Regularization of subsolutions in discrete weak KAM theory

TL;DR

This paper explores various regularization techniques for subsolutions in discrete weak KAM theory, demonstrating the existence of smooth, strict subsolutions outside the Aubry set and their density in the space of subsolutions.

Contribution

It introduces new regularization methods that ensure the existence and density of $C^{1,1}$ subsolutions, advancing the understanding of their structure in discrete weak KAM theory.

Findings

Existence of $C^{1,1}$ subsolutions proven.

Density of $C^{1,1}$ subsolutions established.

Subsolutions can be made strict and smooth outside the Aubry set.

Abstract

We expose different methods of regularizations of subsolutions in the context of discrete weak KAM theory. They allow to prove the existence and the density of $C^{1,1}$ subsolutions. Moreover, these subsolutions can be made strict and smooth outside of the Aubry set.