An introduction to nonlinear analysis

Martin Schecter

Cambridge Studies in Advanced Mathematics, Volume 95

This book is concerned with the solution of non-linear problems in analysis, and especially with non-linear differential equations and the calculus of variations.

It is well-known that a serious study of these matters requires a substantial background in many areas of modern mathematics, including the following: linear algebra, including eigenvalues; standard real analysis and elementary topology; Fouries series; Hilbert spaces, and maybe Sobolov spaces; the Lebesgue integral and measure theory; functional analysis, including Banach spaces and weak topologies on dual spaces; and maybe some distribution theory. Logically one should first study all these topics to some reasonable level, and then turn to applications in non-linear analysis. However, as the author discusses, this is a long and steep road for a student, and the point at which the applications is reached can be much delayed. As the author points out `students having no more background than second year calculus are often required by their disciplines to study such problems'.

The present book seeks to finesse this problem by starting with the applications, considering some specific non-linear differential equations. Further it explains where these equations come from, and why one would like to find a solution. It then provides a collection of tools to solve these equations, and claims, reasonably enough, that knowledge of these tools and their applications will allow one to deal with other equations that arise in particular applications.

Thus the author starts with succesively more challenging problems, and brings in the theoretical background when it is required. (The problems become more challenging because the hypotheses on the functions that arise are successively weakened.) Indeed one sees that quite a lot of modern functional analysis was developed precisely to give a framework in which linear and non-linear differential equations, and problems in the calculus of variations, could be solved; this aspect appears clearly in the text.

The background in analysis that is required is covered in about 40 pages of Appendices; here results are usually stated without proof. These appendices cover most of the background mentioned above, save that there is no explicit discussion of distribution theory. In general this seems to me to be a successful approach, and the book could be an exciting text from which students can learn at least the flavour of non-linear theory. However the phrase about `students having no more background than second year calculus' could be somewhat misleading. From the beginning the author implicitly assumes that the students have mastered, for example, the following topics from standard real analysis and topology: Cauchy sequences of functions in certain topologies; use of liminf and limsup, and semi-continuous functions; the fact that a continuous function on a compact space attains its supremum; Tychonoff's theorem on the product of compact spaces; recognition of a_n: sum |a_n|^2 < infinity as a Hilbert space. One would like to think that all of these matters would be covered by all those with a degree in mathematics, but `second year calculus' is not adequate. Further, the details of the calculations, which are given clearly, require a considerable level of manipulative competence, and some dedication. Thus, realistically, I feel that this is a book for beginning graduate students, rather than undergraduates.

Topics covered include the following: Frechet and Gateaux derivatives; saddle points in finite-dimensional and then Hilbert spaces; the contraction mapping theorem; Picard's, Brouwer's and Schauder's fixed point theorems; mollifiers and test functions; convexity and lower semi-continuity; implicit function theorems; Euler equations; Sard's theorem and Leray-Schauder degree; isoperimetric inequalities; Lagrange multipliers. The book concludes with a final, rather long, chapter tackling problems in higher dimensions.

The book seesm to be clearly written, and the `chat' around the formal solutions is cheerful. [However, I find the format of, say, Theorem 3.27 as `Assume that (1.78), (2.28), (2.37), and (3.60) hold. Then there is a non-trivial solution of (3.1), (3.2).' to be a little bleak.]

This is an attractive introductory text to non-linear analysis for well-prepared and diligent students.

Reviewed by H.G. Dales, University of Leeds.

UK Nonlinear News would like to thank the Cambridge University Press for providing a copy of this volume for review.