**Graduate Studies in Mathematics**

Volume: 105;
2009;
607 pp;
Hardcover

MSC: Primary 46;
Secondary 26

Print ISBN: 978-0-8218-4768-8

Product Code: GSM/105

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-1-4704-1169-5
Product Code: GSM/105.E**

List Price: $89.00

Individual Member Price: $71.20

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#### Supplemental Materials

# A First Course in Sobolev Spaces

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*Giovanni Leoni*

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.

#### Table of Contents

# Table of Contents

## A First Course in Sobolev Spaces

- Cover Cover11 free
- Title page iii5 free
- Contents v7 free
- Preface ix11 free
- Acknowledgments xv17 free
- Part I. Functions of one variable 119 free
- Monotone functions 321
- Functions of bounded pointwise variation 3957
- Absolutely continuous functions 7391
- Curves 115133
- Lebesgue–Stieltjes measures 155173
- Decreasing rearrangement 187205
- Functions of bounded variation and Sobolev functions 215233
- Part II. Functions of several variables 229247
- Absolutely continuous functions and change of variables 231249
- Distributions 255273
- Sobolev spaces 279297
- Sobolev spaces: Embeddings 311329
- Sobolev spaces: Further properties 349367
- Functions of bounded variation 377395
- Besov spaces 415433
- Sobolev spaces: Traces 451469
- Sobolev spaces: Symmetrization 477495
- Functional analysis 493511
- Measures 507525
- The Lebesgue and Hausdorff measures 543561
- Notes 581599
- Notation and list of symbols 587605
- Bibliography 593611
- Index 603621 free
- Back Cover Back Cover1626

#### Readership

Graduate students and research mathematicians interested in Sobolev spaces, particularly their applications to PDEs.