Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb's dry friction law

TL;DR

This paper explores the connection between cyclically monotone graphs and bipotentials, extending previous results to convex covers and applying them to Coulomb's dry friction law.

Contribution

It introduces a new link between inf convolution properties and bipotential construction, generalizes bipotential convex covers, and applies these to Coulomb's friction law.

Findings

The intersection of maximal cyclically monotone graphs forms a critical set of a bipotential.

Extended bipotential convex cover concept to include non-maximal cyclically monotone graphs.

Established a specific bipotential convex cover for Coulomb's dry friction law.

Abstract

We show a surprising connexion between a property of the inf convolution of a family of convex lower semicontinuous functions and the fact that the intersection of maximal cyclically monotone graphs is the critical set of a bipotential. We then extend the results from arXiv:math/0608424v4 to bipotentials convex covers, generalizing the notion of a bi-implicitly convex lagrangian cover. As an application we prove that the bipotential related to Coulomb's friction law is related to a specific bipotential convex cover with the property that any graph of the cover is non maximal cyclically monotone.