zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 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:
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46-01Textbooks (functional analysis)
35J25Second order elliptic equations, boundary value problems
35K20Second order parabolic equations, initial boundary value problems
34D20Stability of ODE
35J65Nonlinear boundary value problems for linear elliptic equations
28A75Length, area, volume, other geometric measure theory
46N99Miscellaneous applications functional analysis