Weakly differentiable functions : Sobolev spaces and functions of bounded variation / William P. ZiemerType de document : MonographieCollection : Graduate texts in mathematics, 120Langue : anglais.Pays : Etats Unis.Éditeur : New York : Springer-Verlag, 1989Description : 1 vol. (XVI-308 p.) ; 24 cmISBN : 0387970177.ISSN : 0072-5285.Bibliographie : Bibliogr. p. 283-295. Index.Sujet MSC : 46E35, Linear function spaces and their duals, Sobolev spaces and other spaces of "smooth'' functions, embedding theorems, trace theorems
28A75, Classical measure theory, Length, area, volume, other geometric measure theory
46-02, Research exposition (monographs, survey articles) pertaining to functional analysisEn-ligne : Springerlink | Zentralblatt | MathSciNet
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This is an interesting and useful introduction to the theory of Sobolev spaces and to the theory of functions with bounded variation (especially the space BV). The main purpose of this book is to study local behavior of these classes of functions. The central part of the book consists of Chapters 3 (for Sobolev spaces) and 5 (for the space BV). They concern the following problems: continuity properties of functions in terms of Lebesgue points, behavior of integral averages, density of smooth functions in corresponding Banach spaces, approximate and fine continuity, description of dual spaces in terms of measures and others.
The final chapter (Chapter 5) contains a very interesting part about sets of finite diameter, reduced boundaries and a description of traces for the spaces BV. The background for the detailed investigation of the class BV is the generalized Poincaré inequality and its applications. This is the main concern in Chapter 4. Different sharp versions of this inequality based on detailed investigations of dual spaces and sharp capacity estimates are given.
The first chapter contains traditional topics of real analysis that are needed for the study of Sobolev spaces. Here one finds measures in Euclidean spaces, covering theorems in the sense of Vitali and Besicovitch, an introduction to the Hausdorff measures, some topics about distribution theory, and Lorentz spaces.
The second chapter concerns basic classical properties of Sobolev spaces. It contains the Rellich-Kondrashov embedding theorems, some topics about Bessel and Riesz potentials and capacities, and different versions of the Sobolev inequality. In addition the author makes remarks about the change of variable problem and the extension problem for Sobolev spaces.
There are historical notes at the end of each chapter. These notes are incomplete in the case of the change of variable and the extension problem and some other questions (MathSciNet).
Bibliogr. p. 283-295. Index