Bipotentials for non monotone multivalued operators: fundamental results and applications

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 Drücker-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. At the end we prove a new reconstruction theorem for a bipotential from a convex lagrangian cover, this time using a convexity notion related to a minimax theorem of Fan. ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, PO BOX 1-764,014700 Bucharest, Romania, e-mail: [email protected] Laboratoire de Mécanique de Lille, UMR CNRS 8107, Université des Sciences et Technologies de Lille, Cité Scientifique, F-59655 Villeneuve d’Ascq cedex, France, e-mail: [email protected] Laboratoire de Mécanique des Solides, UMR 6610, UFR SFA-SP2MI, bd M. et P. Curie, téléport 2, BP 30179, 86962 Futuroscope-Chasseneuil cedex, France, e-mail: [email protected]