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A First Course in Sobolev Spaces
 
Giovanni Leoni Carnegie Mellon University, Pittsburgh, PA
A First Course in Sobolev Spaces
Now available in new edition: GSM/181
A First Course in Sobolev Spaces
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A First Course in Sobolev Spaces
Giovanni Leoni Carnegie Mellon University, Pittsburgh, PA
Now available in new edition: GSM/181
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 1052009; 607 pp
    MSC: 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 one-variable 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.

    Readership

    Graduate 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1052009; 607 pp
MSC: 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 one-variable 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.

Readership

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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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