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Elliptic differential equations and obstacle problems. (English) Zbl 0655.35002
The University Series in Mathematics. New York-London: Plenum Press. XVI, 370 p.; $ 49.50 (US & Canada); $ 59.40 (outside US & Canada) (1987).
Since their inception by G. Stampacchia and G. Fichera in the mid- sixties, variational inequalities have become a rich source of inspiration both in pure and applied mathematics. Today variational inequalities are an indispensable tool in a large variety of models in sciences and engineering. Unlike the monograph “Obstacle problems in mathematical physics” by J.-F. Rodrigues (1987; Zbl 0606.73017), where a general account of the applicability of elliptic variational inequalities to free boundary problems is given, Troianiello’s book provides a compact presentation of new and deep results, dealing with partial differential operators, stimulated by the variational inequalities. An adequate functional framework is described in the first chapter. Lattice properties and associate boundary inequalities are detailed. Chapter 2 contains general existence and uniqueness results for the variational formulation of second order elliptic boundary value problems and the corresponding weak maximum principle. Campanato’s machinery is used to extend the De Giorgi-Nash Regularity theorem to nonhomogeneous equations with lower order coefficients. This procedure is extended to show \(H^{k,p}\)- and \(C^{k,\delta}\)-regularity results in Chapter 3. To do these, some elaborate estimates on hemispheres are established as well as a crucial inequality in Morrey-Campanato spaces and an interpolation theorem. Chapter 4 includes basic existence criteria on variational inequalities. New results on uniqueness and regularity are given for noncoercive bilinear forms and for quasilinear operators under natural growth hypotheses. The Lewy-Stampacchia inequalities for \(H^{2,p}\)-regularity in a more general case and a penalty method for \(H^{1,\infty}\)- and \(H^{2,\infty}\)-regularity are also considered. The theory of nonvariational boundary value problems, started in chapter 3, is done thoroughly in the final chapter 5 as obstacle problems. Here lower and upper solutions are treated as obstacles in a constrained problem. The presentation is based largely on the author’s contributions. The concept of a generalized solution is implemented to the study of implicit unilateral problems. Each chapter is followed by problems that are invitations to the current literature and to classified references. The author confines himself to an analytic treatment of variational problems inequalities and he is successful in a self-contained presentation of the most sophisticated techniques related to second order elliptic operators. The text is well written with a high quality style. The entire approach leads to simpler proofs compared to previous monographs in this area. The book is a refined reading for specialists and a competent prerequisite for those interested in numerical aspects and applications of variational problems. It can also be an excellent text for an up-to-date two-semester graduate course.
Reviewer: D.Pascali

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J20 Variational methods for second-order elliptic equations
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35J65 Nonlinear boundary value problems for linear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
Zbl 0606.73017