The very interesting monograph under review provides a systematic, unified, and relatively self-contained study of a large class of multi-valued variational differential inequalities and inclusions of nonpotential type of the form \[u\in X\cap D(\partial\Psi)\colon\ 0\in \mathcal{A}(u)+\mathcal{F}(u)+\partial \Psi(u)\quad \text{in}\ X^*,\tag{1} \] with a second-order Leray-Lions leading differential operator \(\mathcal{A}\colon\ X\to 2^{X^*},\) while the multi-valued lower order operator \(\mathcal{F}\colon\ X\to 2^{X^*}\) may depend on \(u\) and its gradient \(\nabla u\) and is supposed to be only upper semicontinuous with respect to \(u\) and \(\nabla u.\) The constraint is given by the subdifferential of a convex, lower semicontinuous and proper functional \(\Psi\colon\ X\to \mathbb{R}\cup \{+\infty\},\) and detailed discussion is proposed on constraints defined by closed and convex subsets \(K\subset X\) which amount to \(\Psi=I_K\) with the indicator function \(I_K\) of \(K.\)
The problem \((1)\) is studied in various function classes \(X\) such as Sobolev spaces, Orlicz-Sobolev spaces, and Sobolev spaces with variable exponents, depending on the growth conditions imposed on the operators \(\mathcal{A}\) and \(\mathcal{F}.\) Moreover, the authors provide a systematic study of \((1)\) also on unbounded domains, for which an appropriate setting is in Beppo-Levi spaces and weighted Lebesgue spaces. As is well known, the unboundedness of the domain under consideration causes a number of additional difficulties to the investigation, and therefore the analysis of (1) in this case requires new techniques and is not a straightforward extension of the study of its corresponding problem on bounded domains.
The authors present a rich exposition of the existence and comparison principles applied to \((1)\) based on a generalized method of sub-supersolution that preserves the characteristic features of the commonly known sub-supersolution method for quasilinear elliptic problems. This method is established not only for the stationary multi-valued variational inequality \((1)\) but also for its evolutionary counterpart \[ u\in L^p(0,\tau;X)\cap D(\partial\Psi)\colon\ 0\in u'+\mathcal{A}(u)+\mathcal{F}(u)+\partial \Psi(u)\quad \text{in}\ L^{p'}(0,\tau;X^*),\tag{2} \] where \(u'=du/dt\) stands for the generalized derivative in the sense of vector-valued distributions.
This monograph is an outgrowth of the authors’ research on the subject during the last ten years, but a great deal of the material presented has been obtained only recently so it appears for the first time in book form.
The monograph is very well written and will be of interest to graduate students in mathematics and researchers in pure and applied mathematics, physics, and theoretical mechanics.
Apart from the Introduction, the book consists of seven Chapters and provides a very rich bibliography to the readers.
In Chapter 1 the authors present some interesting motivating examples and outline the topics studied.
Chapter 2 is of auxiliary character and it presents the various mathematical techniques and methods from nonlinear and nonsmooth analysis, partial differential equations, and function spaces employed later.
Chapter 3 is devoted to coercive and noncoercive single- and multi-valued elliptic and parabolic equations and inclusions governed by general Leray-Lions operators of nonpotential type and single- and multi-valued lower order terms for which the Clarke generalized gradient is only a special case. The noncoercive problems are treated by the sub-supersolution method established in this chapter.
In Chapter 4 general multi-valued elliptic variational inequalities (MVIs) of the form \((1)\) are studied, where the constraints are given by closed convex sets \(K,\) that is, \(\Psi=I_K.\) Here the sub-supersolution method is developed in its full generality to treat problems that lack coercivity and to prove existence of extremal solutions which requires the elaboration of new and subtle techniques. The sub-supersolution method established allows to verify the equivalence between generalized variational-hemivariational inequalities and a particular class of multi-valued elliptic variational inequalities. Further, MVIs with discontinuously perturbed lower order multi-valued terms are investigated. Finally, the sub-supersolution method is extended to systems of MVIs in Orlicz-Sobolev spaces.
Chapter 5 deals with evolutionary MVIs of the type \((2)\) and related systems with \(\Psi=I_K.\) The difficulties due to the evolutionary character of the problem studied are overcomed by employing various machineries ranging from the Rockafellar theorem about sums of maximal monotone operators to the application of penalty techniques allowing to treat general obstacle problems.
Chapter 6 provides a study of MVIs \((1)\) and \((2)\) with \(\Psi=I_K\) over unbounded domains. The Beppo-Levi and the weighted Lebesgue spaces are the natural frameworks now and this allow to overcome a number of difficulties arising in the functional analytic treatment of such problems. New techniques, such as the Kelvin transform, are developed to treat problems in exterior domains.
In the final Chapter 7 the sub-supersolution method is extended to \((1)\) and \((2)\) with general convex, lower semicontinuous and proper functionals \(\Psi.\) The convex functionals are seen here as characterizations of various constraints imposed on the problems, as well as potential functionals of possibly multi-valued leading operators. Compared to the case of MVIs on closed and convex sets, this more general situation is not a direct extension and requires the introduction of new concepts and implementation of new techniques in both classes of stationary and evolutionary MVIs. Stationary MVIs are also investigated in this chapter that are formulated in Sobolev spaces with variable exponents, in which the lower order terms may depend on both the unknown function \(u\) and on its gradient \(\nabla u.\)