
Book DetailsGraduate Studies in MathematicsVolume: 105; 2009; 607 ppMSC: Primary 46; Secondary 26
Now available in Second Edition: GSM/181
Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable. In this way, the majority of the text can be read without the prerequisite of a course in functional analysis.
The first part of this text is devoted to studying functions of one variable. Several of the topics treated occur in courses on real analysis or measure theory. Here, the perspective emphasizes their applications to Sobolev functions, giving a very different flavor to the treatment. This elementary start to the book makes it suitable for advanced undergraduates or beginning graduate students. Moreover, the onevariable part of the book helps to develop a solid background that facilitates the reading and understanding of Sobolev functions of several variables.
The second part of the book is more classical, although it also contains some recent results. Besides the standard results on Sobolev functions, this part of the book includes chapters on BV functions, symmetric rearrangement, and Besov spaces.
The book contains over 200 exercises.
ReadershipGraduate students and research mathematicians interested in Sobolev spaces, particularly their applications to PDEs.

Table of Contents

Part 1. Functions of one variable

Chapter 1. Monotone functions

Chapter 2. Functions of bounded pointwise variation

Chapter 3. Absolutely continuous functions

Chapter 4. Curves

Chapter 5. Lebesgue–Stieltjes measures

Chapter 6. Decreasing rearrangement

Chapter 7. Functions of bounded variation and Sobolev functions

Part 2. Functions of several variables

Chapter 8. Absolutely continuous functions and change of variables

Chapter 9. Distributions

Chapter 10. Sobolev spaces

Chapter 11. Sobolev spaces: Embeddings

Chapter 12. Sobolev spaces: Further properties

Chapter 13. Functions of bounded variation

Chapter 14. Besov spaces

Chapter 15. Sobolev spaces: Traces

Chapter 16. Sobolev spaces: Symmetrization

Appendix A. Functional analysis

Appendix B. Measures

Appendix C. The Lebesgue and Hausdorff measures

Appendix D. Notes

Appendix E. Notation and list of symbols


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Now available in Second Edition: GSM/181
Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable. In this way, the majority of the text can be read without the prerequisite of a course in functional analysis.
The first part of this text is devoted to studying functions of one variable. Several of the topics treated occur in courses on real analysis or measure theory. Here, the perspective emphasizes their applications to Sobolev functions, giving a very different flavor to the treatment. This elementary start to the book makes it suitable for advanced undergraduates or beginning graduate students. Moreover, the onevariable part of the book helps to develop a solid background that facilitates the reading and understanding of Sobolev functions of several variables.
The second part of the book is more classical, although it also contains some recent results. Besides the standard results on Sobolev functions, this part of the book includes chapters on BV functions, symmetric rearrangement, and Besov spaces.
The book contains over 200 exercises.
Graduate students and research mathematicians interested in Sobolev spaces, particularly their applications to PDEs.

Part 1. Functions of one variable

Chapter 1. Monotone functions

Chapter 2. Functions of bounded pointwise variation

Chapter 3. Absolutely continuous functions

Chapter 4. Curves

Chapter 5. Lebesgue–Stieltjes measures

Chapter 6. Decreasing rearrangement

Chapter 7. Functions of bounded variation and Sobolev functions

Part 2. Functions of several variables

Chapter 8. Absolutely continuous functions and change of variables

Chapter 9. Distributions

Chapter 10. Sobolev spaces

Chapter 11. Sobolev spaces: Embeddings

Chapter 12. Sobolev spaces: Further properties

Chapter 13. Functions of bounded variation

Chapter 14. Besov spaces

Chapter 15. Sobolev spaces: Traces

Chapter 16. Sobolev spaces: Symmetrization

Appendix A. Functional analysis

Appendix B. Measures

Appendix C. The Lebesgue and Hausdorff measures

Appendix D. Notes

Appendix E. Notation and list of symbols