Variational Inequalities and Multivalued Convex and Nonconvex Problems in Mechanics

Abstract

In the first section of this chapter two general types of variational inequalities are studied and a general scheme is proposed for the derivation of variational inequality “principles”. The second section deals with the study of problems of coexistent phases, using the notion of subdifferentiability. Also minimum problems are derived for Gibbsian states. In the third section we attempt to generalize the notion of superpotential for nonconvex energy functionals through the concepts of the generalized gradient of Clarke and of the derivate container of Warga. Hemivariational inequalities are derived, substationarity properties are proved and certain classes of material laws and boundary conditions leading to such problems are discussed. This chapter closes with a section concerning the study of inequality problems in terms of multivalued integral equations.