A First Course in Sobolev Spaces
American Mathematical Society (Verlag)
978-1-4704-7702-8 (ISBN)
This book is about differentiation of functions. It is divided into two parts, which can be used as different textbooks, one for an advanced undergraduate course in functions of one variable and one for a graduate course on Sobolev functions. The first part develops the theory of monotone, absolutely continuous, and bounded variation functions of one variable and their relationship with Lebesgue-Stieltjes measures and Sobolev functions. It also studies decreasing rearrangement and curves. The second edition includes a chapter on functions mapping time into Banach spaces. The second part of the book studies functions of several variables. It begins with an overview of classical results such as Rademacher's and Stepanoff's differentiability theorems, Whitney's extension theorem, Brouwer's fixed point theorem, and the divergence theorem for Lipschitz domains. It then moves to distributions, Fourier transforms and tempered distributions. The remaining chapters are a treatise on Sobolev functions. The second edition focuses more on higher order derivatives and it includes the interpolation theorems of Gagliardo and Nirenberg. It studies embedding theorems, extension domains, chain rule, superposition, Poincare's inequalities and traces. A major change compared to the first edition is the chapter on Besov spaces, which are now treated using interpolation theory.
Giovanni Leoni, Carnegie Mellon University, Pittsburgh, PA.
Part 1. Functions of one variable: Monotone functions
Functions of bounded pointwise variation
Absolutely continuous functions
Decreasing rearrangement
Curves
Lebesgue-Stieltjes measures
Functions of bounded variation and Sobolev functions
The infinite-dimensional case
Part 2. Functions of several variables: Change of variables and the divergence theorem
Distributions
Sobolev spaces
Sobolev spaces: Embeddings
Sobolev spaces: Further properties
Functions of bounded variation
Sobolev spaces: Symmetrization
Interpolation of Banach spaces
Besov spaces
Sobolev spaces: Traces
Appendix A. Functional analysis
Appendix B. Measures
Appendix C. The Lebesgue and Hausdorff measures
Appendix D. Notes
Appendix E. Notation and list of symbols
Bibliography
Index.
| Erscheinungsdatum | 08.05.2024 |
|---|---|
| Reihe/Serie | Graduate Studies in Mathematics |
| Verlagsort | Providence |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Analysis | |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-4704-7702-5 / 1470477025 |
| ISBN-13 | 978-1-4704-7702-8 / 9781470477028 |
| Zustand | Neuware |
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