zbMATH — the first resource for mathematics

Degree theory in analysis and applications. (English) Zbl 0852.47030
Oxford Lecture Series in Mathematics and its Applications. 2. Oxford: Clarendon Press. viii, 211 p. (1995).
Starting from the pioneering surveys by J. Cronin [Fixed points and topological degree in nonlinear analysis (1964; Zbl 0117.34803)] and J. T. Schwartz [Nonlinear functional analysis (1969; Zbl 0203.14501)], topological degree theory has become an extremal prominent ingredient of nonlinear analysis. Now, the reader may choose among a large variety of excellent textbooks on both the theory and applications of topological degree; we just mention the monographs by K. Deimling [Nonlinear functional analysis (1985; Zbl 0559.47040)], M. A. Krasnosel’skij and P. P. Zabrejko [Geometric methods of nonlinear analysis (1980; Zbl 0326.47052)], and E. Zeidler [Nonlinear functional analysis and its applications. I (1986; Zbl 0583.47050)].
The title of the book under review is quite misleading: it is not at all a book on degree theory in analysis and applications, but just focusses on a special topic, namely the definition, properties, and applications of topological degree for Sobolev functions. In fact, in Chapters 1-3 the authors summarize the properties of the classical Brouwer degree in finite-dimensional spaces, while in Chapter 4 they recall some standard properties of Sobolev space functions. The only part which really deserved to be published is Chapters 5/6. Here, the authors discuss some properties of the degree for Sobolev functions, as well as a “change of variable formula” for such functions due to M. Marcus and V. Mizel [Bull. Am. Math. Soc. 79, 790-795 (1973; Zbl 0275.49041)], V. M. Gol’dshtejn and Yu. G. Reshetnyak [Quasiconformal mappings and Sobolev spaces (1990; Zbl 0687.30001)], and V. Šverák [Arch. Ration. Mech. Anal. 100, No. 2, 105-127 (1988; Zbl 0659.73038)]. Some contributions to this field, as well as to the local invertibility theorems treated in Chapter 6, are also due to the authors; this apparently was their motivation to write this book.
It is completely incomprehensible why the authors then added a final chapter on Leray-Schauder degree in infinite-dimensional spaces; apart from the usual definition of this degree through the Brouwer degree of finite-dimensional approximations, there is no connection between this chapter and any of the preceding ones. Summarizing, the reviewer thinks that the authors had better publish Chapters 5/6 as a survey paper in some suitable journal, rather than include them in a book containing much material which may be found equally well written, or even better, in the monographs cited above.

47H11 Degree theory for nonlinear operators
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
74B20 Nonlinear elasticity