Bipotentials for non monotone multivalued operators: fundamental results and applications

TL;DR

This paper surveys recent advances in bipotentials, a mathematical tool extending Fenchel's inequality, for representing multivalued operators, with applications in non smooth mechanics and connections to Fitzpatrick functions and selfdual lagrangians.

Contribution

It introduces new ways to construct bipotentials for non monotone operators and explores their relation to other mathematical concepts like Fitzpatrick functions and selfdual lagrangians.

Findings

Bipotentials can represent some non monotone operators.

Relations between bipotentials and Fitzpatrick functions are established.

Methods for constructing bipotentials for specific operators are described.

Abstract

This is a survey of recent results about bipotentials representing multivalued operators. The notion of bipotential is based on an extension of Fenchel's inequality, with several interesting applications related to non associated constitutive laws in non smooth mechanics, such as Coulomb frictional contact or non-associated Drucker-Prager model in plasticity. Relations betweeen bipotentials and Fitzpatrick functions are described. Selfdual lagrangians, introduced and studied by Ghoussoub, can be seen as bipotentials representing maximal monotone operators. We show that bipotentials can represent some monotone but not maximal operators, as well as non monotone operators. Further we describe results concerning the construction of a bipotential which represents a given non monotone operator, by using convex lagrangian covers or bipotential convex covers.