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Mathematical analysis and numerical methods for science and technology. Volume 2: Functional and variational methods. With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean-Michel Combes, Hélène Lanchon, Bertrand Mercier, Claue Wild, Claude Zuily. Transl. from the French by Ian N. Sneddon. 2nd printing. (English) Zbl 0944.47001

Berlin: Springer. xvi, 592 p. (2000).
The book under review is the second volume of the series of six fundamental ones devoted to the thorough presentation of various problems, methods and structures of modern mathematics that forms the principal mathematical apparatus in a number of fields of physics such as electromagnetism, mechanics, quantum physics, etc. The volume consists of five chapters (III–VII) and a voluminous appendix. Its material should be considered as “purely” mathematical and concerned mainly with the theory of solvability of various types of differential and integral equations [for the first edition see (1988; Zbl 0664.47001)].
Chapter III contains a description of a number of fundamental mathematical techniques – functional transformations. It starts with the consideration of Fourier series and Fourier transform. Further, the Mellin and Hankel transforms are presented. All these transforms are introduced in a unified manner as certain procedures of diagonalization of some important differential operators thus the reader feels immediately the deepness of the subject. As applications of these transformation techniques solutions to a number of Dirichlet problems in various domains are given. For its importance in numerical calculation a considerable part of the chapter is devoted to the discrete and fast Fourier transforms with applications to boundary value problems, regularization and smoothing functions, practical calculation of the Fourier transform and some other.
Chapters IV and V contain a description of a series of principal objects, structures, ideas and results associated with the study of linear differential (mainly partial differential) operators. In particular, Chapter IV is devoted to the presentation of the “natural” domains of differential operators – Sobolev spaces. It contains a thorough picture of their properties including the embedding theorems, density theorems, trace theorems, the result on compactness and the fundamental inequalities. Chapter V presents the main important results on the theory of solvability of various classes of differential equations and the corresponding study of differential operators. It starts with the locality property as a characteristic property of differential operators and Cauchy-Kowalewskaya and Holmgren theorems for the operators with analytic coefficients. The main part of the chapter deals with operators with constant coefficients and here the reader will find the principal results of the theory of elliptic, hypoelliptic, hyperbolic and parabolic operators including the elementary solutions, discussion of the well posedness of the Cauchy problem, local reglarity properties and the maximum principle.
As the modern theory of differential operators involves heavily a number of fundamental structures and results of functional analysis the necessary information concerning mainly the general theory of operators in Banach spaces is given in Chapter VI. Chapter VII gives a description of the principal ideology of the variational method of solving boundary value problems involving elliptic operators. The problems of existence and uniqueness are treated here with the help of Lax-Milgram’s theorem. As applications of the method developed the Dirichlet and Neumann type problems for the Laplacian and more general operators are studied as well as some physical examples such as statical problems of elasticity are considered. Since the theory of distributions is a vital instrument when treating linear differential operators its main definitions and results are gathered in a separate appendix the material of which is exploited significantly throughout the whole of the volume.
The book will be of use for scientists and engineers as an advanced text presenting the general functional analysis methods and fundamental techniques and principles of the local ideology in the theory of partial differential equations.
[For the French version (1985) of volumes 1 to 3 see Zbl 0642.35001].
Reviewer: P.Zabreiko (Minsk)

MSC:

47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47F05 General theory of partial differential operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46F10 Operations with distributions and generalized functions
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
46F12 Integral transforms in distribution spaces
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35A15 Variational methods applied to PDEs
47A10 Spectrum, resolvent
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