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Mathematical analysis and numerical methods for science and technology. Volume 1: Physical origins and classical methods. With the collaboration of Philippe Bénilan, Michel Cessenat, André Gervat, Alain Kavenoky, Hélène Lanchon. Transl. from the French by Ian N. Sneddon. (English) Zbl 0683.35001
Berlin etc.: Springer-Verlag. xvii, 695 p. (1990).
This volume is the first of six in a series of translations from the French original “Analyse mathématique et calcul numérique pour les sciences et les techniques” (1985; Zbl 0642.35001). It is stated in the introduction that the object of the work is to put recent work of mathematicians and numerical analysts within the grasp of engineers, physicists and workers in mechanics. There are two chapters. In the first, entitled “Physical Examples” a brief survey is made of seven physical fields giving rise to the mathematics which will be treated subsequently - Classical Fluids, Linear Elasticity, Linear Viscoelasticity, Electromagnetism (it is surprising that a work intended for engineers does not use SI units), Neutronics, Quantum Mechanics and Mechanics. There is no modelling, however, the fundamental equations of the various fields of physics being stated, rather than derived, and it is indicated that the treatment will in general be linear.
The second chapter “The Laplace Operator” occupies about two thirds of the book. In this, topics such as Harmonic Functions, Newtonian Potentials, Dirichlet’s Problem, Capacities, Regularity of Solutions, and Boundary Conditions are studied, and it is indicated how some of the properties of Laplace’s and Poisson’s equations may be generalised to be relevant to elliptic equations of the second order. The book closes with a substantial bibliography, a table of notations and a table of contents of the remaining five volumes.
The syndicate of authors concerned has produced a well written substantial work of considerable depth and scholarship, and the translator is to be congratulated on producing a book which gives the impression of having been originally written in English. Nevertheless, the question must be asked as to whether the book will be of interest to those for whom it is stated to be intended. My feeling is that the present volume will not appeal to those who are interested in using Mathematics rather than in studying Mathematics. There are very few concrete examples, the approach of the second chapter is far too abstract (e.g. Radon measures, Herglotz’s Theorem and statements such as \(u\in B'(\Omega)\cap {\mathcal C}^ 0({\bar \Omega}))\); and engineers and scientists will just be scared off by the frequent use of a variety of abstract spaces. The tone of this chapter seems in fact to be that of a functional analysis treatment of the Laplacian operator suitable for very theoretical numerical analysts, and appears to be inspired more by Bourbaki than by Courant and Hilbert.
As is usual with this publisher, the printing and presentation are excellent. As suggested, the book will be of more interest to mathematicians than to engineers and scientists, but it can be recommended as a source book for the deep properties of Laplace’s equation.
Reviewer: L.G.Chambers

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35J15 Second-order elliptic equations
35Qxx Partial differential equations of mathematical physics and other areas of application
31Bxx Higher-dimensional potential theory
74B05 Classical linear elasticity
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy
35Dxx Generalized solutions to partial differential equations