Meccanica 32: 585-588, 1997.
Functional Analysis: Applications in Mechanics and Inverse Problems
L.P. Lebedev, I.I. Vorovich and G.M.L. Gladwell Kluwer Academic Publishers, Dordrecht 1996, viii, 239 pp., ISBN 0-7923-3849-9.
The possibility of applying functional analysis to physics began to be realized about in the thirties of this century. It is then that the theory of Hilbert spaces had achieved enough consistency to render it an autonomous branch of mathematics. On the other hand, the progress of physics urgently raised new and fascinating mathematical problems. Thus it is not by accident dial, in those years, the works of Courant, Friedrichs, and Lewy had given to scholars in mathematical physics new tools very different from the traditional ones for analyzing their problems, these old tools where constructive, the new ones were abstract. In addition, in the same years, the results by Caccioppoli, Leray and Schauder provided a new procedure for treating nonlinear problems.
In the fifties, functional analysis became popular. Old questions of theory of elasticity were settled by Friedrichs, Fichera, and Campanato; other results were obtained for Navier-Stokes equation by Olga Ladizenskaya and Solonnikov; also the unilateral problems of theory of elasticity were correctly posed and solved by Stampacchia and Lions. It is thus natural that, in consequence of so promising results, new books on the applications of functional analysis to physics started to appear. The first, and most celebrated, was Methoden der Mathematischen Physik by Courant and Hilbert (1924, Vol. I, and 1937, Vol. II). Dozens of texts followed Courant-Hilbert's book, as, for example, four books by Lions, two of Mikhlin in the seventies, and, recently, five volumes by Zeidler, published at the end of the eighties. And the tradition is still alive, through the faith in functional analysis is less firm than before.
We then ask if another book on the applications of functional analysis to mechanics, like that of Lebedev, Vorovich, and Gladwell, presents innovative characteristics with respect to the previous ones. The author's program is not to propose a treatise, but simply to write an introduction to functional analysis for engineers and applied mechanicians wishing to apply it in their fields.
In this respect, the book attains its purpose, because, in contrast with other more systematic books, the authors introduce inductively the essential concepts, starting from particular examples and letting the need for more general concepts arise spontaneously as that of weak convergence, and so on. Under this respect, the book recalls the style of a beautiful book by B. Friedman (Principles and Techniques in Applied Mathematics) published in 1956. Another merit of the book is that of having dedicated the last chapter to inverse problems. These are very important in applications, where often data are prescribed in a nonclassical form. But these problems are customarily exposed in monographic texts, and not in introductory works. Unfortunately, there are two regrettable omissions. There is no mention of calculus of variation and bifurcation theory.
Since the book is destined for applications, the reader expects to find, in each of the eight chapters, several examples, from mechanics and physics, in which the treatment of a problem through functional analysis is essential and not merely elegant. Unfortunately, the examples are very scarce and restricted to the linear theories of rectilinear beams and plane plates. This is a severe limitation for a work that must be read by persons wishing to be enlightened about the problems with which they grapple. For instance, there is no mention of the equations of beams in large displacements, of the Föppl- v.Kármán plate, and of shells; nor is there an example of a unilateral problem, of a free boundary problem, or of an analysis of structures in post-critical range. The theory, which is presented by the authors with so much elegance, can well deal with these problems.
Piero Villaggio