Ioffe’s Fans and Unilateral Problems: A New Conjecture

Abstract

The present paper deals with a general theory of variational “principles”. First we consider laws expressed by convex and nonconvex superpotentials. Then a new type of superpotentials, the F-superpotentials, is proposed by using the notion of “fans” introduced recently by Ioffe. Then a conjecture is made which permits, first, the formulation of general variational principles in inequality form for systems with or without dissipation, and secondly, the derivation of new generalizations of the classical derivative. To this end we define the V-superpotentials whose definition is based on the notion of virtual work (or power). By means of this conjecture a general class of “generalized standard materials” is defined. To achieve it a new “Hypothesis of Dissipation” is introduced. The respective variational expressions are derived and one type of them, a variational-hemivariational inequality, is studied concerning the existence and the approximation of the solution. First the variational-hemivariational inequality is regularized and, simultaneously, by means of Galerkin’s method, its formulation in a finite dimensional space is given. Using appropriate a priori estimates, a compactness argument, and the well-known monotonicity argument, we prove that the solution of the finite dimensional regularized problem converges to the solution of the continuous problem.