Nonsmooth Variational Problems and Their Inequalities

A powerful and fruitful tool for proving existence and comparison results for a wide range of nonlinear elliptic and parabolic boundary value problems is the method of sub- and supersolutions.

In this chapter, we provide the mathematical background as it will be used in later chapters.

This chapter deals with existence and comparison results for weak solutions of nonlinear elliptic and parabolic problems. The ideas and methods developed here will also be useful in the treatment of nonsmooth variational problems in later chapters. Section 3.1 deals with semilinear elliptic Dirichlet boundary value problems and may be considered as a preparatory section for Sect. 3.2 and Sect. 3.3, where general quasilinear elliptic and parabolic problems are treated. The purpose of Sect. 3.1 is to emphasize the basic ideas and to present various approaches without overburdening the presentation with too many technicalities. As an application of the general results of Sect. 3.2 combined with critical point theory, the existence of multiple and sign-changing solutions is proved in Sect. 3.4. Finally, in Sect. 3.5, the concept of sub-supersolutions is extended to some nonstandard elliptic boundary value problem, which in the one-space dimensional and semilinear case reduces to a second-order ordinary differential equation subject to periodic boundary conditions. The chapter concludes with bibliographical notes and further applications and extensions of the theory developed in the preceeding sections.

The subject of this chapter is boundary value problems for quasilinear differential inclusions of elliptic and parabolic type whose governing multivalued terms are of Clarke’s gradient type. We introduce concepts of sub- and supersolutions that are designed to obtain existence and comparison results and that generalize the notion of sub- and supersolutions of variational equations considered in Chap. 3 in a natural way. Thus, the least requirement of any notion of sub-supersolutions for inclusions is that to include the corresponding notion for equations as introduced in Chap. 3. In Sect. 4.1, we first provide some motivation for differential inclusions with the help of elementary examples and introduce the basic concept of sub- and supersolutions. Depending on the structure and growth assumptions imposed on the multivalued terms, the notion of sub- and supersolutions and the comparison principles related with them are further developed in Sect. 4.2, Sect. 4.3, and Sect. 4.5 for general quasilinear elliptic and parabolic inclusion problems. As an application of the theory presented in this chapter, an elliptic inclusion is considered whose multivalued term is given in Sect. 4.4 by the difference of Clarke’s generalized gradient and the usual subdifferential. An alternative notion of sub-supersolution existing in the literature and its relation to the one introduced here is considered in Sect. 4.6. The chapter concludes with comments and further bibliographical notes.

The goal of this chapter is starting a systematic study of the sub-supersolution method in variational inequalities. By using subsolutions or supersolutions, we can show the solvability and the existence of extremal solutions of noncoercive inequalities. Despite the nonsymmetric structure of variational inequalities, we show that both supersolutions and subsolutions can be defined in an appropriate manner, which naturally extends the corresponding concepts in equations.

Hemivariational inequalities have been introduced by P. D. Panagiotopoulos (see [179, 180]) to describe, e.g., problems in mechanics and engineering governed by nonconvex, possibly nonsmooth energy functionals (so-called superpotentials). This kind of energy functionals appear if nonmonotone, possibly multivalued constitutive laws are taken into account. Hemivariational inequalities are of the following abstract setting:

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.