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Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities

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Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities COURANT 5 EMMANUEL HEBEY LECTURE NOTES Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities American Mathematical Society Courant Institute of Mathematical Sciences Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities Courant Lecture Notes in Mathematics Executive Editor Jalal Shatah Managing Editor Paul D. Monsour Production Editor Reeva Goldsmith Copy Editor Joe Shearer http://dx.doi.org/10.1090/cln/005 Emmanuel Hebey Université de Cergy-Pontoise 5 Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities Courant Institute of Mathematical Sciences New York University New York, New York American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 58E35; Secondary 53C21. Library of Congress Cataloging-in-Publication Data Hebey, Emmanuel, 1964– Nonlinear analysis on manifolds : Sobolev spaces and inequalities / Emmanuel Hebey. p. cm. — (Courant lecture notes ; 5) Originally published: New York : Courant Institute of Mathematical Sciences, New York University, c1999. Includes bibliographical references. ISBN 0-8218-2700-6 1. Sobolev spaces. 2. Inequalities (Mathematics) 3. Riemannian manifolds. I. Title. II. Series. QA323 .H43 2000 515.782—dc21 00-061835 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 acknowledgment of the source is given. 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 Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. c 1999 held by the author. All rights reserved. Printed in the United States of America. Reprinted by the American Mathematical Society, 2000 The American Mathematical Society retains all rights except those granted to the United States Government. ∞ 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 URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00 To Isabelle i Contents Preface xi Chapter 1. Elements of Riemannian Geometry 1 1.1. Smooth Manifolds 1 1.2. Riemannian Manifolds 7 1.3. Curvature and Topology 10 1.4. From Local to Global Analysis 11 1.5. Special Coordinates 13 Chapter 2. Sobolev Spaces: The Compact Setting 19 2.1. Background Material 19 2.2. Sobolev Spaces on Riemannian Manifolds 21 2.3. Sobolev Embeddings: General Results 25 2.4. The Case of the Euclidean Space 30 2.5. Sobolev Embeddings I 32 2.6. Sobolev Embeddings II 33 2.7. Compact Embeddings 37 2.8. Poincar´e and Sobolev-Poincar´e Inequalities 40 2.9. A Finiteness Theorem 44 Chapter 3. Sobolev Spaces: The Noncompact Setting 47 3.1. Density Problems 47 3.2. Sobolev Embeddings I 52 3.3. Sobolev Embeddings II 63 3.4. Disturbed Sobolev Inequalities 67 Chapter 4. Best Constants in the Compact Setting I 75 4.1. Program B,PartI 76 4.2. The Role of αq (M) 84 4.3. Program A,PartI 91 4.4. On the Scale of Optimal Inequalities 114 ix x CONTENTS Chapter 5. Best Constants in the Compact Setting II 119 5.1. The Case of the Standard Unit Sphere 120 5.2. Program B, Part II 124 5.3. Program A, Part II 132 5.4. The Role of B0(g) 150 5.5. One More Question 155 Chapter 6. Optimal Inequalities with Constraints 161 6.1. The Case of an Arbitrary Compact Manifold 161 6.2. The Case of the Sphere 166 6.3. Applications to the Nirenberg Problem 170 Chapter 7. Best Constants in the Noncompact Setting 181 7.1. Questions 1 and 2 182 7.2. Questions 3 and 4 186 7.3. Proof of Theorem 7.2 191 7.4. Explicit Inequalities 222 Chapter 8. Euclidean-Type Sobolev Inequalities 225 8.1. Euclidean-Type Generic Sobolev Inequalities 226 8.2. Euclidean-Type Optimal Sobolev Inequalities 235 8.3. Nash’s Inequality 244 Chapter 9. The Influence of Symmetries 249 9.1. Geometric Preliminaries 249 9.2. Compact Manifolds 251 9.3. Optimal Inequalities for Compact Manifolds 253 9.4. Compactness for Radially Symmetric Functions 260 9.5. A Main Lemma for Complete Manifolds 263 9.6. The Codimension 1 Case 266 9.7. The General Case 271 Chapter 10. Manifolds with Boundary 279 Bibliography 283 Preface These notes deal with the theory of Sobolev spaces on Riemannian manifolds. Though Riemannian manifolds are natural extensions of Euclidean space, the naive idea that what is valid for Euclidean space must be valid for manifolds is false. Sev- eral surprising phenomena appear when studying Sobolev spaces on manifolds. Questions that are elementary for Euclidean space become challenging and give rise to sophisticated mathematics, where the geometry of the manifold plays a cen- tral role. The reader will find many examples of this in the text. These notes have their origin in a series of lectures given at the Courant Insti- tute of Mathematical Sciences in 1998. For the sake of clarity, I decided to deal only with manifolds without boundary. An appendix concerning manifolds with boundary can be found at the end of these notes. To illustrate some of the results or concepts developed here, I have included some discussions of a special family of PDEs where these results and concepts are used. These PDEs are generalizations of the scalar curvature equation. As is well known, geometric problems often lead to limiting cases of known problems in analysis. The study of Sobolev spaces on Riemannian manifolds is a field currently un- dergoing great development. Nevertheless, several important questions still puzzle mathematicians today. While the theory of Sobolev spaces for noncompact man- ifolds has its origin in the 1970s with the work of Aubin, Cantor, Hoffman, and Spruck, many of the results presented in these lecture notes have been obtained in the 1980s and 1990s. This is also the case for the applications already mentioned to scalar curvature and generalized scalar curvature equations. A substantial part of these notes is devoted to the concept of best constants. This concept appeared very early on to be crucial for solving limiting cases of some partial differential equations. A striking example of this was the major role that best constants played in the Yamabe problem. These lecture notes are intended to be as self-contained as possible. In partic- ular, it is not assumed that the reader is familiar with differentiable manifolds and Riemannian geometry. The present notes should be accessible to a large audience, including graduate students and specialists of other fields. The present notes are organized into nine chapters. Chapter 1 is a quick in- troduction to differential and Riemannian geometry. Chapter 2 deals with the general theory of Sobolev spaces for compact manifolds, while Chapter 3 deals with the general theory of Sobolev spaces for complete, noncompact manifolds. Best constants problems for compact manifolds are discussed in Chapters 4 and 5, while Chapter 6 deals with some special type of Sobolev inequalities under xi x ii PREFACE constraints. Best constants problems for complete noncompact manifolds are dis- cussed in Chapter 7. Chapter 8 deals with Euclidean-type Sobolev inequalities. The influence of symmetries on Sobolev embeddings is discussed in Chapter 9. An appendix at the end of these notes briefly discusses the case of manifolds with boundaries. It is my pleasure to thank my friend Jalal Shatah for encouraging me to write these notes. It is also my pleasure to express my deep thanks to my friends and col- leagues Tobias Colding, Zindine Djadli, Olivier Druet, Antoinette Jourdain, Michel Ledoux, Fr´ed´ericRobert, and Michel Vaugon for stimulating discussions and valu- able comments about the manuscript. Finally, I wish to thank Reeva Goldsmith, Paul Monsour, and Joe Shearer for the wonderful job they did in the preparation of the manuscript. Emmanuel Hebey Paris, September 1998 Bibliography [1] Adams, R. A. Sobolev spaces. Academic Press, San Diego, 1978. [2] Allendoerfer, C. B. The Euler number of a Riemann manifold. Amer. J. Math. 62: 243–248, 1940. [3] Allendoerfer, C. B., and Weil, A. The Gauss-Bonnet theorem for Riemannian polyhedra. Trans. Amer. Math. Soc. 53: 101–129, 1943. [4] Anderson, M. T. Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 2: 455–490, 1989. [5] . Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102: 429–445, 1990. α [6] Anderson, M. T., and Cheeger, J. C -compactness for manifolds with Ricci curvature and injectivity radius bounded below. J. Differential Geom. 35: 265–281, 1992. [7] Aubin, T. Probl`emes isop´erim´etriques et espaces de Sobolev. C. R. Acad. Sci. Paris S´er. A-B 280: A279–A281, 1975. [8] . Espaces de Sobolev sur les vari´et´es riemanniennes. Bull. Sci. Math. (2) 100: 149–173, 1976. [9] . Equations´ diff´erentielles non lin´eaires et probl`eme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. (9) 55: 269–296, 1976. [10] . Probl`emes isop´erim´etriques et espaces de Sobolev. J. Differential Geom. 11: 573– 598, 1976. [11] . Meilleures constantes dans le th´eor`emed’inclusion de Sobolev et un th´eor`eme de Fredholm non lin´eaire pour la transformation conforme de la courbure scalaire. J. Funct. Anal. 32: 148–174, 1979. [12] . Nonlinear analysis on manifolds. Monge-Amp`ere equations. Grundlehren der Math- ematischen Wissenschaften, 252, Springer, New York–Berlin, 1982.
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