Evolutionary Variational Inequalities

Abstract

In this chapter, we continue the study of evolutionary variational inequalities started in Chapter 4. The di.erence between the problems studied there and those studied here lies in the fact that the variational inequalities presented in this chapter do not involve a viscosity term. Their study is more complicated, since it cannot be done based on the unique solvability of time-dependent elliptic variational inequalities in velocities. We start with the study of evolutionary variational inequalities involving a di.erentiable functional j, for which we prove an existence and uniqueness result. The proof is based on the study of a sequence of evolutionary variational inequalities with viscosity, compactness, and lower semicontinuity arguments. Then, we extend this result to a class of evolutionary variational inequalities for which the function j can be approached, in a sense that will be described below, by a family of di.erentiable functionals. We complete our results with a convergence result that shows that the solution of the evolutionary variational inequality with viscosity converges to the solution of the corresponding inviscid evolutionary variational inequality, as the viscosity converges to zero. Finally, we present an existence result for evolutionary quasivariational inequalities, obtained by using a time discretization method. The results presented in this chapter will be applied in the study of quasistatic antiplane frictional contact problems with elastic materials. As usual, everywhere in this chapter X is a real Hilbert space with the inner product (·,·)X and the norm $$||\cdot||\ X$$ , and [0, T] denotes the time interval of interest, T > 0. Moreover, X is assumed to be separable everywhere in Section 5.4 of this chapter.