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Analysis in classes of discontinuous functions and equations of mathematical physics. (English) Zbl 0564.46025
Mechanics: Analysis, 8. Dordrecht - Boston - Lancaster: Martinus Nijhoff Publishers, a member of the Kluwer Academic Publishers Group. XVIII, 678 p. Dfl. 340.00; $ 117.50; £86.50 (1985).
This book is the result of an effort, to apply on the one hand the modern techniques and facilities of distribution theory to partial differential equations and mathematical physics, and to avoid on the other hand the problems one has with multiplication of distributions. The authors study BV-spaces, i.e. spaces of (in general discontinuous) functions, whose first generalized derivatives are measures, and Sobolev type spaces constructed from them. Boundary value problems are considered on open sets in \({\mathbb{R}}^ n\) whose characteristic functions belong to BV; these are exactly the sets of finite perimeter, sets whose essential boundary has finite (n-1)-dimensional Hausdorff measure. No smoothness properties at the boundary are required! Another typical feature of this book is to substitute missing continuity and smoothness properties by approximate ”averaging” methods. Typical notions in this context are: approximate limit of a measurable function, average value of a measurable function in a regular point, averaged superposition (leading to a Leibniz type formula for differentiation of products), essential boundary of a set (leading to traces and Green’s formula), to mention a few.
This apparatus, which is developed very carefully and at a moderate pace, to make the material accessible to a wide circle of readers, is then applied very successfully to the equations of mathematical physics and even chemical physics. To be more precise, the following applications are treated: Ordinary differential equations, elliptic (linear and quasilinear) and parabolic partial differential equations, the equations of chemical kinetics and mathematical problems of macrokinetics - these applications cover more than one half of the book.
It is admirable, how the authors managed to display such a wealth of material including the necessary prerequisites of functional analysis and measure theory, so that a reader with a knowledge of only the fundamentals of higher mathematics can follow the exposition. A welcome addition to the literature!
Reviewer: J.Lorenz

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
35J25 Boundary value problems for second-order elliptic equations
35K20 Initial-boundary value problems for second-order parabolic equations
34D20 Stability of solutions to ordinary differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
28A75 Length, area, volume, other geometric measure theory
46N99 Miscellaneous applications of functional analysis