Variational-Hemivariational Inequalities

Abstract

In the previous chapters, the BVPs we considered in the form of hemivariational inequalities were formulated on the whole space. We are now taking into account problems subject to constraints for hemivariational inequalities, which means dealing with variational-hemivariational inequalities. The aim of this chapter is three-fold: (a) to develop the method of sub- and supersolutions for quasilinear elliptic variational-hemivariational inequalities; (b) to treat an evolution variational-hemivariational inequality by the method of sub- and supersolutions; and (c) to study variational-hemivariational inequalities by minimax methods in the nonsmooth critical point theory viewing the (weak) solutions as critical points of the corresponding nonsmooth functionals. The two general methods, namely the sub-supersolutions approach and the nonsmooth critical point theory, are complementary and permit us to investigate various types of problems. Specifically, Sect. 7.1 and Sect. 7.2 deal with the method of sub- and supersolutions for hemivariational inequalities, whereas Sect. 7.3, Sect. 7.4, and Sect. 7.5 present applications of nonsmooth critical point results for this kind of problem emphasizing the treatment for corresponding eigenvalue problems. In both methods, an essential feature consists of the use of comparison arguments. They allow us to provide location information for the solutions.