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Bipotentials for Non-monotone Multivalued Operators: Fundamental Results and Applications

type de publication      article dans une revue internationale avec comité de lecture
date de publication 2010
auteur(s) Buliga Marius; De Saxce Géry; Vallee Claude
journal (abréviation) Acta Applicandae Mathematicae (Acta Applicandae Mathematicae)
volume (numéro) 110 (2)
  
pages 955 – 972
résumé 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 between 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.
mots clés Bipotentials - Non-monotone multivalued operators - Implicit normality rules
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