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A First Course in Sobolev Spaces A First Course in Sobolev Spaces 'IOVANNI,EONI 'RADUATE3TUDIES IN-ATHEMATICS 6OLUME !MERICAN-ATHEMATICAL3OCIETY http://dx.doi.org/10.1090/gsm/105 A First Course in Sobolev Spaces A First Course in Sobolev Spaces Giovanni Leoni Graduate Studies in Mathematics Volume 105 American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 46E35; Secondary 26A24, 26A27, 26A30, 26A42, 26A45, 26A46, 26A48, 26B30. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-105 Library of Congress Cataloging-in-Publication Data Leoni, Giovanni, 1967– A first course in Sobolev spaces / Giovanni Leoni. p. cm. — (Graduate studies in mathematics ; v. 105) Includes bibliographical references and index. ISBN 978-0-8218-4768-8 (alk. paper) 1. Sobolev spaces. I. Title. QA323.L46 2009 515.782—dc22 2009007620 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgmentofthesourceisgiven. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents Preface ix Acknowledgments xv Part 1. Functions of One Variable Chapter 1. Monotone Functions 3 §1.1. Continuity 3 §1.2. Differentiability 8 Chapter 2. Functions of Bounded Pointwise Variation 39 §2.1. Pointwise Variation 39 §2.2. Composition in BPV (I)55 §2.3. The Space BPV (I)59 §2.4. Banach Indicatrix 66 Chapter 3. Absolutely Continuous Functions 73 §3.1. AC (I)VersusBPV (I)73 §3.2. Chain Rule and Change of Variables 94 §3.3. Singular Functions 107 Chapter 4. Curves 115 §4.1. Rectifiable Curves and Arclength 115 §4.2. Fr´echet Curves 130 §4.3. Curves and Hausdorff Measure 134 §4.4. Jordan’s Curve Theorem 146 v vi Contents Chapter 5. Lebesgue–Stieltjes Measures 155 §5.1. Radon Measures Versus Increasing Functions 155 §5.2. Signed Borel Measures Versus BPV (I) 161 §5.3. Decomposition of Measures 166 §5.4. Integration by Parts and Change of Variables 181 Chapter 6. Decreasing Rearrangement 187 §6.1. Definition and First Properties 187 §6.2. Absolute Continuity of u∗ 202 §6.3. Derivative of u∗ 209 Chapter 7. Functions of Bounded Variation and Sobolev Functions 215 §7.1. BV (Ω) Versus BPV (Ω) 215 §7.2. Sobolev Functions Versus Absolutely Continuous Functions 222 Part 2. Functions of Several Variables Chapter 8. Absolutely Continuous Functions and Change of Variables 231 §8.1. The Euclidean Space RN 231 §8.2. Absolutely Continuous Functions of Several Variables 234 §8.3. Change of Variables for Multiple Integrals 242 Chapter 9. Distributions 255 §9.1. The Spaces DK (Ω), D (Ω), and D (Ω) 255 §9.2. Order of a Distribution 264 §9.3. Derivatives of Distributions and Distributions as Derivatives 266 §9.4. Convolutions 275 Chapter 10. Sobolev Spaces 279 §10.1. Definition and Main Properties 279 §10.2. Density of Smooth Functions 283 §10.3. Absolute Continuity on Lines 293 §10.4. Duals and Weak Convergence 298 §10.5. A Characterization of W 1,p (Ω) 305 Chapter 11. Sobolev Spaces: Embeddings 311 §11.1. Embeddings: 1 ≤ p<N 312 §11.2. Embeddings: p = N 328 §11.3. Embeddings: p>N 335 Contents vii §11.4. Lipschitz Functions 341 Chapter 12. Sobolev Spaces: Further Properties 349 §12.1. Extension Domains 349 §12.2. Poincar´e Inequalities 359 Chapter 13. Functions of Bounded Variation 377 §13.1. Definition and Main Properties 377 §13.2. Approximation by Smooth Functions 380 §13.3. Bounded Pointwise Variation on Lines 386 §13.4. Coarea Formula for BV Functions 397 §13.5. Embeddings and Isoperimetric Inequalities 401 §13.6. Density of Smooth Sets 408 §13.7. A Characterization of BV (Ω) 413 Chapter 14. Besov Spaces 415 §14.1. Besov Spaces Bs,p,θ,0<s<1 415 §14.2. Dependence of Bs,p,θ on s 419 §14.3. The Limit of Bs,p,θ as s → 0+ and s → 1− 421 §14.4. Dependence of Bs,p,θ on θ 425 §14.5. Dependence of Bs,p,θ on s and p 429 §14.6. Embedding of Bs,p,θ into Lq 437 §14.7. Embedding of W 1,p into Bt,q 442 §14.8. Besov Spaces and Fractional Sobolev Spaces 448 Chapter 15. Sobolev Spaces: Traces 451 §15.1. Traces of Functions in W 1,1 (Ω) 451 §15.2. Traces of Functions in BV (Ω) 464 §15.3. Traces of Functions in W 1,p (Ω), p>1 465 § 1,p 15.4. A Characterization of W0 (Ω) in Terms of Traces 475 Chapter 16. Sobolev Spaces: Symmetrization 477 §16.1. Symmetrization in Lp Spaces 477 §16.2. Symmetrization of Lipschitz Functions 482 §16.3. Symmetrization of Piecewise Affine Functions 484 §16.4. Symmetrization in W 1,p and BV 487 Appendix A. Functional Analysis 493 §A.1. Metric Spaces 493 viii Contents §A.2. Topological Spaces 494 §A.3. Topological Vector Spaces 497 §A.4. Normed Spaces 501 §A.5. Weak Topologies 503 §A.6. Hilbert Spaces 506 Appendix B. Measures 507 §B.1. Outer Measures and Measures 507 §B.2. Measurable and Integrable Functions 511 §B.3. Integrals Depending on a Parameter 519 §B.4. Product Spaces 520 §B.5. Radon–Nikodym’s and Lebesgue’s Decomposition Theorems 522 §B.6. Signed Measures 523 §B.7. Lp Spaces 526 §B.8. Modes of Convergence 534 §B.9. Radon Measures 536 §B.10. Covering Theorems in RN 538 Appendix C. The Lebesgue and Hausdorff Measures 543 §C.1. The Lebesgue Measure 543 §C.2. The Brunn–Minkowski Inequality and Its Applications 545 §C.3. Convolutions 550 §C.4. Mollifiers 552 §C.5. Differentiable Functions on Arbitrary Sets 560 §C.6. Maximal Functions 564 §C.7. Anisotropic Lp Spaces 568 §C.8. Hausdorff Measures 572 Appendix D. Notes 581 Appendix E. Notation and List of Symbols 587 Bibliography 593 Index 603 Preface The Author List, I: giving credit where credit is due. The first author: Senior grad student in the project. Made the figures. — Jorge Cham, www.phdcomics.com There are two ways to introduce Sobolev spaces: The first is through the el- egant (and abstract) theory of distributions developed by Laurent Schwartz in the late 1940s; the second is to look at them as the natural development and unfolding of monotone, absolutely continuous, and BV functions1 of one variable. To my knowledge, this is one of the first books to follow the second approach. I was more or less forced into it: This book is based on a series of lecture notes that I wrote for the graduate course “Sobolev Spaces”, which I taught in the fall of 2006 and then again in the fall of 2008 at Carnegie Mellon University. In 2006, during the first lecture, I found out that half of the students were beginning graduate students with no background in functional analysis (which was offered only in the spring) and very little in measure theory (which, luckily, was offered in the fall). At that point I had two choices: continue with a classical course on Sobolev spaces and thus loose half the class after the second lecture or toss my notes and rethink the entire operation, which is what I ended up doing. I decided to begin with monotone functions and with the Lebesgue dif- ferentiation theorem. To my surprise, none of the students taking the class had actually seen its proof. I then continued with functions of bounded pointwise variation and abso- lutely continuous functions. While these are included in most books on real analysis/measure theory, here the perspective and focus are rather different, in view of their applications to Sobolev functions. Just to give an example, 1BV functions are functions of bounded variation. ix x Preface most books study these functions when the domain is either the closed in- terval [a, b]orR. I needed, of course, open intervals (possibly unbounded). This changed things quite a bit. A lot of the simple characterizations that hold in [a, b] fall apart when working with arbitrary unbounded intervals. After the first three chapters, in the course I actually jumped to Chapter 7, which relates absolutely continuous functions with Sobolev functions of one variable, and then started with Sobolev functions of several variables. In the book I included three more chapters: Chapter 4 studies curves and arclength. I think it is useful for students to see the relation between recti- fiable curves and functions with bounded pointwise variation. Some classical results on curves that most students in analysis have heard of, but whose proof they have not seen, are included, among them Peano’s filling curve and the Jordan curve theorem. Section 4.3 is more advanced. It relates rectifiable curves with the H1 Hausdorff measure. Besides Hausdorff measures, it also makes use of the Vitali–Besicovitch covering theorem.
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