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Nonlinear analysis. (English) Zbl 1086.47001

Series in Mathematical Analysis and Applications 9. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-484-3/hbk). xi, 971 p. (2006).
Nonlinear analysis is a rapidly developing field of contemporary applied mathematics covering a vast spectrum of topics that goes from classical analysis, measure theory and topology to partial differential equations, variational methods and multivalued analysis. This volume focuses on topics of nonlinear analysis related to the analysis of boundary value problems, control theory and the calculus of variations. It is mostly self-contained and, at the same time, it contains an impressive amount of material, organized in 7 chapters – each of which, without being exhaustive, surveys an area of nonlinear analysis, starting with a brief introduction and finishing with historical comments. At the end of the book, there is an appendix containing basic results on topology, measure theory and functional analysis.
Each chapter, which could have been a book by its own, moves rapidly from elementary definitions to involved results and generally presumes a degree of familiarity with the subject or requires a certain level of mathematical maturity. In this sense, this book is designated to be a reference book for working researchers or postgraduate students, rather than a first textbook for initiation into the subject.
The overall spirit is quiteoretical, aiming more at developing theories of abstract models rather than treating practical issues which are often raised by engineers or researchers of applied mechanics. Nevertheless, this volume can potentially be highly appreciated by readers without a pure theoretical background and orientation. Here one can find results previously contained only in scattered papers, not always easily accessible. In this sense, the current book complements perfectly the many other books of the literature on nonlinear analysis. The reader might also find extremely useful the index table at the end, and profit from the extensive – though not complete – bibliography.
Let us proceed with a chapter by chapter description of this volume:
Chapter 1: Hausdorff measures and Capacity. This chapter starts with the Carathéodory construction of a measure via the outer measure, defines the Hausdorff measure and the Hausdorff dimension, considers the question of differentiability of Hausdorff measures and gives a complete proof of the Rademacher differentiation theory for locally Lipschitz functions. The chapter finishes with some local properties of Sobolev spaces and the notion of capacity. The material treated in this chapter is influenced by and complements the classical books of L. Evans and R. Gariepy [“Measure theory and fine properties of functions” (Studies in Advanced Mathematics, CRC Press) (1992; Zbl 0804.28001)] and H. Federer [“Geometric measure theory” (Grundlehren 153, Springer) (1969; Zbl 0874.49001)].
Chapter 2: Lebesgue-Bochner and Sobolev spaces. In this chapter, the rich theory of vector measures and Bochner integration in the spirit of [J. Diestel and J. Uhl, “Vector measures” (Mathematical Surveys 15, American Mathematical Society, Providence) (1977; Zbl 0369.46039)] is revisited. The Lebesgue \(L^{p}(\Omega;X)\) spaces are considered and relations with the geometry of Banach spaces (Radon-Nikodým property) are examined. The last part contains results on inequalities and embedding of Sobolev spaces.
Chapter 3: Nonlinear Operators and Young Measures. This is the biggest chapter of the book, containing almost 200 pages, and dealing with the theory of nonlinear operators (compact operators, monotone and accretive operators, etc.) and the related theory of semigroups. In the last part, we find a useful introduction to Young measures (the interested reader might also want to consult [C. Castaing, P. Raynaud de Fitte and M. Valadier, “Young measures on topological spaces. With applications in control theory and probability theory” (Mathematics and its Applications 571, Kluwer Academic Publishers, Dordrecht) (2004; Zbl 1067.28001)] for further details).
Chapter 4: Smooth and nonsmooth analysis and Variational Principles. This chapter is also a large part of the book which deals with differential calculus in Banach spaces, Fréchet and Gâteaux differentiation of convex and locally Lipschitz functions, Asplund spaces, Fenchel duality and subdifferentials for convex functions, maximal cyclic monotonicity, variational principles, viscosity subdifferentials defined for a given bornology and, finally, applications to integral functionals. Let us mention for the interested reader the existence of two recent books complementing the topic and being closer to the spirit of nonsmooth optimization: [J. Borwein and Q. Zhu, “Techniques Of Variational Analysis” (Springer) (2005; Zbl 1076.49001) and B. Mordukhovich, “Variational Analysis and Generalized Differentiation”, Vol. I (Basic Theory) (2006; Zbl 02176472), Vol. II (Applications) (2005; Zbl 1100.49002)].
Chapter 5: Critical point theory. This chapter presents a first link to the theory of boundary value problems via the study of critical points of appropriate functionals. The chapter starts with the theory of deformation (going back to works of Palais, Palais-Smale and Rabinowitz) which requires implicitly a certain familiarity with differential topology, even if pure geometrical aspects such as the Ehresmann fibration theory are not treated. The presentation is oriented towards analysis, with the minimax theorems and the mountain pass lemma, and concludes with results on the existence and the structure of the critical points for smooth functionals. Finally, the Lusternik–Schnirelman theory for nonlinear eigenvalue problems is presented.
Chapter 6: Eigenvalue Problems and Maximum Principles. This chapter uses the results of the previous chapter to develop the spectrum of linear elliptic differential operators, of the partial \(p\)-Laplacian and of the scalar and vector \(p\)-Laplacian with Dirichlet, Neumann and periodic boundary conditions. The last part of the chapter establishes maximum principles for nonlinear differential operators involving partial \(p\)-Laplacians. For the interested reader, let us mention the related book of [L. Evans, “Partial differential equations” (Graduate Studies in Mathematics 19, American Mathematical Society, Providence) (1998; Zbl 0902.35002)].
Chapter 7: Fixed Point Theory. This last chapter treats the important topic of fixed point theorems in a metric (respectively, topological, order or multivalued analysis) framework and discusses applications. It is a valuable complement of the classical book of [A. Granas and J. Dugundji, “Fixed point theory” (Springer Monographs in Mathematics, Springer) (2003; Zbl 1025.47002)].
Overall, this Volume 9 of the Series in Mathematical Analysis and Applications is definitely not a book for beginners. It may serve as a valuable reference text for active researchers in the area of nonlinear analysis and/or partial differential equations treated with variational techniques.

MSC:

47-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to operator theory
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
28B05 Vector-valued set functions, measures and integrals
35A15 Variational methods applied to PDEs
35J30 Higher-order elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H10 Fixed-point theorems
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