7R13. Functional Analysis in Mechanics. - LP Lebedev (Dept of Math, Univ Nacional de Colombia, Bogota, Colombia) and II Vorovich (Deceased). Springer-Verlag, New York. 2003. 238 pp. ISBN 0-387-95519-4. $59.95.
Reviewed by I Andrianov (Inst fur Allgemeine Mechanik, RWTH, Templergraben 64, Aachen, D-52056, Germany).
This textbook is based on the course of lectures on functional analysis delivered to students of the Department of Mathematics and Mechanics (division of Mechanics) at Rostov State University (USSR, Russia). Outstanding scientist and tutor Professor II Vorovich initiated this course about 30 years ago. Many known scientists in the field of Mechanics were graduated from Rostov State University, so we can consider that the course itself has gotten a good approval. It is very good news that from now on, it may be used by Western students and scientists.
The book is divided into three parts: metric spaces, theory of operators, and nonlinear functional analysis. A brief description of the book’s layout is given below.
Part 1 is devoted to energy; Banach, Hilbert, and Sobolev spaces; convergence, weak convergence, completeness, compactness, and separability; weak and generalized solutions in Mechanics; Hausdorf criterion; Arzela`’s and decomposition theorems; Riesh representations; Ritz and Bubnov-Galerkin methods; and the Bramble-Hilbert lemma. A variant of elastico-plasticity proposed by Il’yushin is considered, and the method of elastic solutions for corresponding boundary value problems is justified.
Part 2 treats the Banach-Shteinhause principle; inverse, closed, and compact operators; spectrum and resolvent of linear operators; some applications of spectral theory; and the Courant minimax principle.
In Part 3, Fre´chet and Gateaux derivatives, Liapunov-Shmidt method, critical points of a functional, von Ka´rma´n equations of a plate, buckling of a thin elastic shell and equilibrium of elastic shallow shell, degree theory and steady-state flow of viscous liquid are discussed.
The purpose of the book is to offer quick access to the principal facts so the reader could rapidly gain familiarity with this valuable tool. General ideas and algorithms are clear and understandable. There is a good index attached.
In this reviewer’s opinion, the book has two shortcomings. First, there are no exercises to enable readers to check their understanding of the techniques employed and, in some cases, to amplify what has been described in the text. Second, the book does not contain a list of symbols. Such a list facilitates reading in cases where the symbol is not defined on the page being read.
As the authors put it, a knowledge of mechanics is not necessary. Generally, it is true, but my advice to newcomers familiar with mechanics, but not so good in mathematics, is to start with Functional Analysis in Mechanics—just for deeper mathematical understanding of problems considered.
The authors also tried to make the course self-contained and to cover the foundations of functional analysis. They have succeeded indeed, but for readers unexperienced in Functional Analysis, it would be better to also use a good mathematical course, for example, Introductory Real Analysis, by AN Kolmogorov and SV Fomin (Dover, New York, 1975).
This skillfully-written book is a reader-friendly and well-organized textbook in the field of Mathematical Mechanics. It can be highly recommended for students of technology universities as well as for researchers in Mechanics. In addition, this reviewer would like to recommend Functional Analysis in Mechanics for technology university lecturers as the basis for a lecture course. The book is recommended for purchase by university libraries and for individuals interested in the mathematical problems of Mechanics.
