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PrefaceNonsmooth variational problems have their origin in the study of nondifferentiable energy functionals, and they arise as necessary conditions of critical points of such functionals. In this way, variational inequalities are related with convex energy or potential functionals, whereas the new class of hemivariational inequalities arise in the study of nonconvex potential functionals that are, in general, merely locally Lipschitz. The foundation of variational inequalities is from Fichera, Lions, and Stampacchia, and it dates back to the 1960s. Hemivariational inequalities were first introduced by Panagiotopoulos about two decades ago and are closely related with the development of the new concept of Clarke's generalized gradient. By using this new type of inequalities, Panagiotopoulos was able to solve various open questions in mechanics and engineering. This book focuses on nonsmooth variational problems not necessarily related with some potential or energy functional, which arise, e.g., in the study of boundary value problems with nonsmooth data and/or nonsmooth constraints such as multivalued elliptic problems with multifunctions of Clarke's subgradient type, variational inequalities, hemivariational inequalities, and their corresponding evolutionary counterparts. The main purpose is to provide a systematic and unified exposition of comparison principles based on a suitably extended sub-supersolution method. This method manifests as an effective and flexible technique to obtain existence and comparison results of solutions. Moreover, it can be employed for the investigation of various qualitative properties such as location, multiplicity, and extremality of solutions. In the treatment of the problems under consideration, a wide range of methods and techniques from nonlinear and nonsmooth analysis are applied; a brief outline of which has been provided in a preliminary chapter to make the book self-contained. The book is an outgrowth of the authors' research on the subject during the past 10 years. A great deal of the material presented here has been obtained only in recent years and appears for the first time in book form.
