Multivalued Variational Equations

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.