This monograph deals with the mathematical theory of variational inequalities that contain certain nonconvex functionals. Recently these inequalities have been termed “hemivariational inequalities” by the second author. This theory extends the standard theory of variational inequalities by replacing the subdifferentiation of convex functionals with the directional differentiation, in the sense of Clarke, of nonconvex functionals. The main mathematical interest lies in the conditions for the existence of weak solutions to hemivariational inequalities for pseudomonotone operators which involve, in addition, scalar, Lipschitz, and vector-valued nonconvex superpotentials. The mathematical theory, which is developed carefully and thoroughly, is applied to a number of important examples in nonsmooth mechanics. The book will be used to those who investigate variational inequalities derived from static models with nonconvex potentials. This is especially true for contact mechanics, where the theory originated, and where there is a current need to relax the convexity assumptions in the contact conditions. The mathematical theory itself is of particular interest, and indeed, the material presented raises as many new issues and questions as it settles.
Introductory information on convex and nonconvex analysis is presented in Chapter 1. This is followed by a review of the concepts of pseudo-monotonicity and generalized pseudo-monotonicity. Chapter 3 develops the theory of the existence of weak solutions to variational inequalities with scalar static nonconvex superpotentials. This is applied to the problems of adhesive contact on an elastic body with a rigid support and the adhesively connected sandwich plates. Chapter 4 deals with locally Lipschitz functionals, especially those connected with pointwise maxima or minima of Lipschitz functions. Coupled variational-hemivariational inequalities and quasi-hemivariational inequalities are also addressed. Problems of adhesive frictional contact, contact with locking support, and fuzzy contact are considered as applications. Similarly, problems in structural analysis with nonmonotone and multivalued constitutive laws, laminated von Kármán plates, and rigid-viscoplastic flows with adhesion and nonmonotone friction are discussed. In Chapter 5 the theory of vector-valued superpotentials is developed and existence results are obtained for coercive operators and for strongly monotone operators. The case with relaxed directional growth is similarly addressed. The applications include nonmonotone skin friction in plane elasticity, masonry materials, semipermeability problems, nonlinear elasticity and nonmonotone laws in networks. Chapter 6 describes noncoercive operators and their connection to free boundary problems. The last chapter deals with variational problems which are posed on nonconvex star-shaped sets. The applications are in plasticity where the yield condition is nonconvex, and in the Signorini problem for star-shaped closed sets.
This book is well written and organized. The steps in the proofs are well presented, and the results are clearly stated. Many of the examples are explained in detail, although a few are somewhat brief. The book is a very important addition to the field of nonsmooth mechanics. It not only summarizes recent developments, but also provides a broader and more unified perspective on what still needs to be developed. There is no doubt that these results will be applied in the fields of mechanics, elasticity and various branches of engineering. For example, they facilitate the consideration of nonmonotone friction laws, which are routinely used by engineers in practice, and in their computer software, but which have been considered in the mathematical literature only in a few instances. More applications of the theory will be found, once it is widely recognized that these tools are available.