DOCSLIB.ORG

A First Course in Sobolev Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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.
Recommended publications
  • Sobolev Spaces, Theory and Applications

    Sobolev Spaces, Theory and Applications

    Sobolev spaces, theory and applications Piotr Haj lasz1 Introduction These are the notes that I prepared for the participants of the Summer School in Mathematics in Jyv¨askyl¨a,August, 1998. I thank Pekka Koskela for his kind invitation. This is the second summer course that I delivere in Finland. Last August I delivered a similar course entitled Sobolev spaces and calculus of variations in Helsinki. The subject was similar, so it was not posible to avoid overlapping. However, the overlapping is little. I estimate it as 25%. While preparing the notes I used partially the notes that I prepared for the previous course. Moreover Lectures 9 and 10 are based on the text of my joint work with Pekka Koskela [33]. The notes probably will not cover all the material presented during the course and at the some time not all the material written here will be presented during the School. This is however, not so bad: if some of the results presented on lectures will go beyond the notes, then there will be some reasons to listen the course and at the same time if some of the results will be explained in more details in notes, then it might be worth to look at them. The notes were prepared in hurry and so there are many bugs and they are not complete. Some of the sections and theorems are unfinished. At the end of the notes I enclosed some references together with comments. This section was also prepared in hurry and so probably many of the authors who contributed to the subject were not mentioned.
  • Notes on Partial Differential Equations John K. Hunter

    Notes on Partial Differential Equations John K. Hunter

    Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨olderspaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 7 1.7. Convolutions 7 1.8. Derivatives and multi-index notation 8 1.9. Mollifiers 10 1.10. Boundaries of open sets 12 1.11. Change of variables 16 1.12. Divergence theorem 16 Chapter 2. Laplace's equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack's inequality 31 2.5. Green's identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 47 3.3. Distributions 50 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 56 3.6. Approximation of Sobolev functions 57 3.7. Sobolev embedding: p < n 57 3.8. Sobolev embedding: p > n 66 3.9. Boundary values of Sobolev functions 69 3.10. Compactness results 71 3.11. Sobolev functions on Ω ⊂ Rn 73 3.A. Lipschitz functions 75 3.B. Absolutely continuous functions 76 3.C. Functions of bounded variation 78 3.D. Borel measures on R 80 v vi CONTENTS 3.E. Radon measures on R 82 3.F. Lebesgue-Stieltjes measures 83 3.G. Integration 84 3.H. Summary 86 Chapter 4.
  • Lecture Notes for MATH 592A

    Lecture Notes for MATH 592A

    Lecture Notes for MATH 592A Vladislav Panferov February 25, 2008 1. Introduction As a motivation for developing the theory let’s consider the following boundary-value problem, u′′ + u = f(x) x Ω=(0, 1) − ∈ u(0) = 0, u(1) = 0, where f is a given (smooth) function. We know that the solution is unique, sat- isfies the stability estimate following from the maximum principle, and it can be expressed explicitly through Green’s function. However, there is another way “to say something about the solution”, quite independent of what we’ve done before. Let’s multuply the differential equation by u and integrate by parts. We get x=1 1 1 u′ u + (u′ 2 + u2) dx = f u dx. − x=0 0 0 h i Z Z The boundary terms vanish because of the boundary conditions. We introduce the following notations 1 1 u 2 = u2 dx (thenorm), and (f,u)= f udx (the inner product). k k Z0 Z0 Then the integral identity above can be written in the short form as u′ 2 + u 2 =(f,u). k k k k Now we notice the following inequality (f,u) 6 f u . (Cauchy-Schwarz) | | k kk k This may be familiar from linear algebra. For a quick proof notice that f + λu 2 = f 2 +2λ(f,u)+ λ2 u 2 0 k k k k k k ≥ 1 for any λ. This expression is a quadratic function in λ which has a minimum for λ = (f,u)/ u 2. Using this value of λ and rearranging the terms we get − k k (f,u)2 6 f 2 u 2.
  • Introduction to Sobolev Spaces

    Introduction to Sobolev Spaces

    Introduction to Sobolev Spaces Lecture Notes MM692 2018-2 Joa Weber UNICAMP December 23, 2018 Contents 1 Introduction1 1.1 Notation and conventions......................2 2 Lp-spaces5 2.1 Borel and Lebesgue measure space on Rn .............5 2.2 Definition...............................8 2.3 Basic properties............................ 11 3 Convolution 13 3.1 Convolution of functions....................... 13 3.2 Convolution of equivalence classes................. 15 3.3 Local Mollification.......................... 16 3.3.1 Locally integrable functions................. 16 3.3.2 Continuous functions..................... 17 3.4 Applications.............................. 18 4 Sobolev spaces 19 4.1 Weak derivatives of locally integrable functions.......... 19 1 4.1.1 The mother of all Sobolev spaces Lloc ........... 19 4.1.2 Examples........................... 20 4.1.3 ACL characterization.................... 21 4.1.4 Weak and partial derivatives................ 22 4.1.5 Approximation characterization............... 23 4.1.6 Bounded weakly differentiable means Lipschitz...... 24 4.1.7 Leibniz or product rule................... 24 4.1.8 Chain rule and change of coordinates............ 25 4.1.9 Equivalence classes of locally integrable functions..... 27 4.2 Definition and basic properties................... 27 4.2.1 The Sobolev spaces W k;p .................. 27 4.2.2 Difference quotient characterization of W 1;p ........ 29 k;p 4.2.3 The compact support Sobolev spaces W0 ........ 30 k;p 4.2.4 The local Sobolev spaces Wloc ............... 30 4.2.5 How the spaces relate.................... 31 4.2.6 Basic properties { products and coordinate change.... 31 i ii CONTENTS 5 Approximation and extension 33 5.1 Approximation............................ 33 5.1.1 Local approximation { any domain............. 33 5.1.2 Global approximation on bounded domains.......
  • Advanced Partial Differential Equations Prof. Dr. Thomas

    Advanced Partial Differential Equations Prof. Dr. Thomas

    Advanced Partial Differential Equations Prof. Dr. Thomas Sørensen summer term 2015 Marcel Schaub July 2, 2015 1 Contents 0 Recall PDE 1 & Motivation 3 0.1 Recall PDE 1 . .3 1 Weak derivatives and Sobolev spaces 7 1.1 Sobolev spaces . .8 1.2 Approximation by smooth functions . 11 1.3 Extension of Sobolev functions . 13 1.4 Traces . 15 1.5 Sobolev inequalities . 17 2 Linear 2nd order elliptic PDE 25 2.1 Linear 2nd order elliptic partial differential operators . 25 2.2 Weak solutions . 26 2.3 Existence via Lax-Milgram . 28 2.4 Inhomogeneous bounday value problems . 35 2.5 The space H−1(U) ................................ 36 2.6 Regularity of weak solutions . 39 A Tutorials 58 A.1 Tutorial 1: Review of Integration . 58 A.2 Tutorial 2 . 59 A.3 Tutorial 3: Norms . 61 A.4 Tutorial 4 . 62 A.5 Tutorial 6 (Sheet 7) . 65 A.6 Tutorial 7 . 65 A.7 Tutorial 9 . 67 A.8 Tutorium 11 . 67 B Solutions of the problem sheets 70 B.1 Solution to Sheet 1 . 70 B.2 Solution to Sheet 2 . 71 B.3 Solution to Problem Sheet 3 . 73 B.4 Solution to Problem Sheet 4 . 76 B.5 Solution to Exercise Sheet 5 . 77 B.6 Solution to Exercise Sheet 7 . 81 B.7 Solution to problem sheet 8 . 84 B.8 Solution to Exercise Sheet 9 . 87 2 0 Recall PDE 1 & Motivation 0.1 Recall PDE 1 We mainly studied linear 2nd order equations – specifically, elliptic, parabolic and hyper- bolic equations. Concretely: • The Laplace equation ∆u = 0 (elliptic) • The Poisson equation −∆u = f (elliptic) • The Heat equation ut − ∆u = 0, ut − ∆u = f (parabolic) • The Wave equation utt − ∆u = 0, utt − ∆u = f (hyperbolic) We studied (“main motivation; goal”) well-posedness (à la Hadamard) 1.
  • Arxiv:0906.4883V2 [Math.CA] 4 Jan 2010 Ro O Olna Yeblcsses[4.W Hwblwta El’ T Helly’S That Theorem

    Arxiv:0906.4883V2 [Math.CA] 4 Jan 2010 Ro O Olna Yeblcsses[4.W Hwblwta El’ T Helly’S That Theorem

    THE KOLMOGOROV–RIESZ COMPACTNESS THEOREM HARALD HANCHE-OLSEN AND HELGE HOLDEN Abstract. We show that the Arzelà–Ascoli theorem and Kolmogorov com- pactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly’s theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem. 1. Introduction Compactness results in the spaces Lp(Rd) (1 p < ) are often vital in exis- tence proofs for nonlinear partial differential equations.≤ ∞ A necessary and sufficient condition for a subset of Lp(Rd) to be compact is given in what is often called the Kolmogorov compactness theorem, or Fréchet–Kolmogorov compactness theorem. Proofs of this theorem are frequently based on the Arzelà–Ascoli theorem. We here show how one can deduce both the Kolmogorov compactness theorem and the Arzelà–Ascoli theorem from one common lemma on compactness in metric spaces, which again is based on the fact that a metric space is compact if and only if it is complete and totally bounded. Furthermore, we trace out the historical roots of Kolmogorov’s compactness theorem, which originated in Kolmogorov’s classical paper [18] from 1931. However, there were several other approaches to the issue of describing compact subsets of Lp(Rd) prior to and after Kolmogorov, and several of these are described in Section 4. Furthermore, extensions to other spaces, say Lp(Rd) (0 p < 1), Orlicz spaces, or compact groups, are described. Helly’s theorem is often≤ used as a replacement for Kolmogorov’s compactness theorem, in particular in the context of nonlinear hyperbolic conservation laws, in spite of being more specialized (e.g., in the sense that its classical version requires one spatial dimension).
  • Five Lectures on Optimal Transportation: Geometry, Regularity and Applications

    Five Lectures on Optimal Transportation: Geometry, Regularity and Applications

    FIVE LECTURES ON OPTIMAL TRANSPORTATION: GEOMETRY, REGULARITY AND APPLICATIONS ROBERT J. MCCANN∗ AND NESTOR GUILLEN Abstract. In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Amp`ere type equations. An ap- plication to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically. c 2010 by Robert J. McCann. All rights reserved. Contents Preamble 2 1. An introduction to optimal transportation 2 1.1. Monge-Kantorovich problem: transporting ore from mines to factories 2 1.2. Wasserstein distance and geometric applications 3 1.3. Brenier’s theorem and convex gradients 4 1.4. Fully-nonlinear degenerate-elliptic Monge-Amp`eretype PDE 4 1.5. Applications 5 1.6. Euclidean isoperimetric inequality 5 1.7. Kantorovich’s reformulation of Monge’s problem 6 2. Existence, uniqueness, and characterization of optimal maps 6 2.1. Linear programming duality 8 2.2. Game theory 8 2.3. Relevance to optimal transport: Kantorovich-Koopmans duality 9 2.4. Characterizing optimality by duality 9 2.5. Existence of optimal maps and uniqueness of optimal measures 10 3. Methods for obtaining regularity of optimal mappings 11 3.1. Rectifiability: differentiability almost everywhere 12 3.2. From regularity a.e.
  • L P and Sobolev Spaces

    L P and Sobolev Spaces

    NOTES ON Lp AND SOBOLEV SPACES STEVE SHKOLLER 1. Lp spaces 1.1. Definitions and basic properties. Definition 1.1. Let 0 < p < 1 and let (X; M; µ) denote a measure space. If f : X ! R is a measurable function, then we define 1 Z p p kfkLp(X) := jfj dx and kfkL1(X) := ess supx2X jf(x)j : X Note that kfkLp(X) may take the value 1. Definition 1.2. The space Lp(X) is the set p L (X) = ff : X ! R j kfkLp(X) < 1g : The space Lp(X) satisfies the following vector space properties: (1) For each α 2 R, if f 2 Lp(X) then αf 2 Lp(X); (2) If f; g 2 Lp(X), then jf + gjp ≤ 2p−1(jfjp + jgjp) ; so that f + g 2 Lp(X). (3) The triangle inequality is valid if p ≥ 1. The most interesting cases are p = 1; 2; 1, while all of the Lp arise often in nonlinear estimates. Definition 1.3. The space lp, called \little Lp", will be useful when we introduce Sobolev spaces on the torus and the Fourier series. For 1 ≤ p < 1, we set ( 1 ) p 1 X p l = fxngn=1 j jxnj < 1 : n=1 1.2. Basic inequalities. Lemma 1.4. For λ 2 (0; 1), xλ ≤ (1 − λ) + λx. Proof. Set f(x) = (1 − λ) + λx − xλ; hence, f 0(x) = λ − λxλ−1 = 0 if and only if λ(1 − xλ−1) = 0 so that x = 1 is the critical point of f. In particular, the minimum occurs at x = 1 with value f(1) = 0 ≤ (1 − λ) + λx − xλ : Lemma 1.5.
  • The Logarithmic Sobolev Inequality Along the Ricci Flow

    The Logarithmic Sobolev Inequality Along the Ricci Flow

    The Logarithmic Sobolev Inequality Along The Ricci Flow (revised version) Rugang Ye Department of Mathematics University of California, Santa Barbara July 20, 2007 1. Introduction 2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow 6. The κ-noncollapsing estimate Appendix A. The logarithmic Sobolev inequalities on the euclidean space Appendix B. The estimate of e−tH Appendix C. From the estimate for e−tH to the Sobolev inequality 1 Introduction Consider a compact manifold M of dimension n 3. Let g = g(t) be a smooth arXiv:0707.2424v4 [math.DG] 29 Aug 2007 solution of the Ricci flow ≥ ∂g = 2Ric (1.1) ∂t − on M [0, T ) for some (finite or infinite) T > 0 with a given initial metric g(0) = g . × 0 Theorem A For each σ > 0 and each t [0, T ) there holds ∈ R n σ u2 ln u2dvol σ ( u 2 + u2)dvol ln σ + A (t + )+ A (1.2) ≤ |∇ | 4 − 2 1 4 2 ZM ZM 1 for all u W 1,2(M) with u2dvol =1, where ∈ M R 4 A1 = 2 min Rg0 , ˜ 2 n − CS(M,g0) volg0 (M) n A = n ln C˜ (M,g )+ (ln n 1), 2 S 0 2 − and all geometric quantities are associated with the metric g(t) (e.g. the volume form dvol and the scalar curvature R), except the scalar curvature Rg0 , the modified Sobolev ˜ constant CS(M,g0) (see Section 2 for its definition) and the volume volg0 (M) which are those of the initial metric g0.
  • A Practical Guide to Compact Infinite Dimensional Parameter Spaces

    A Practical Guide to Compact Infinite Dimensional Parameter Spaces

    A Practical Guide to Compact Infinite Dimensional Parameter Spaces∗ Joachim Freybergery Matthew A. Mastenz May 24, 2018 Abstract Compactness is a widely used assumption in econometrics. In this paper, we gather and re- view general compactness results for many commonly used parameter spaces in nonparametric estimation, and we provide several new results. We consider three kinds of functions: (1) func- tions with bounded domains which satisfy standard norm bounds, (2) functions with bounded domains which do not satisfy standard norm bounds, and (3) functions with unbounded do- mains. In all three cases we provide two kinds of results, compact embedding and closedness, which together allow one to show that parameter spaces defined by a k · ks-norm bound are compact under a norm k · kc. We illustrate how the choice of norms affects the parameter space, the strength of the conclusions, as well as other regularity conditions in two common settings: nonparametric mean regression and nonparametric instrumental variables estimation. JEL classification: C14, C26, C51 Keywords: Nonparametric Estimation, Sieve Estimation, Trimming, Nonparametric Instrumental Variables ∗This paper was presented at Duke, the 2015 Triangle Econometrics Conference, and the 2016 North American and China Summer Meetings of the Econometric Society. We thank audiences at those seminars as well as Bruce Hansen, Kengo Kato, Jack Porter, Yoshi Rai, and Andres Santos for helpful conversations and comments. We also thank the editor and the referees for thoughtful feedback which improved this paper. yDepartment of Economics, University of Wisconsin-Madison, William H. Sewell Social Science Building, 1180 Observatory Drive, Madison, WI 53706, USA. [email protected] zCorresponding author.
  • Sobolev Met Poincaré

    Sobolev Met Poincaré

    MaxPlanckInstitut fur Mathematik in den Naturwissenschaften Leipzig Sob olev met Poincare by Piotr Hajlasz and Pekka Koskela PreprintNr Sob olev met Poincare Piotr Ha jlasz and Pekka Koskela Contents Intro duction What are Poincare and Sob olev inequalities Poincare inequalities p ointwise estimates and Sob olev classes Examples and necessary conditions Sob olev typ e inequalities by means of Riesz p otentials Trudinger inequality A version of the Sob olev emb edding theorem on spheres RellichKondrachov Sob olev classes in John domains John domains Sob olev typ e inequalities Poincare inequality examples Riemannian manifolds Upp er gradients Top ological manifolds Gluing and related constructions Further examples CarnotCaratheo dory spaces CarnotCaratheo dory metric Upp er gradients and Sob olev spaces Carnot groups Hormander condition Further generalizations Graphs Applications to PDE and nonlinear p otential theory Admissible weights Sob olev emb edding for p App endix Measures Uniform integrability p p spaces L
  • Classics in Mathematics Herbert Federer Geometric Measure Theory Springer-Verlag Berlin Heidelberg Gmbh Herbert Federer Was Born on July 23, 1920, in Vienna

    Classics in Mathematics Herbert Federer Geometric Measure Theory Springer-Verlag Berlin Heidelberg Gmbh Herbert Federer Was Born on July 23, 1920, in Vienna

    Classics in Mathematics Herbert Federer Geometric Measure Theory Springer-Verlag Berlin Heidelberg GmbH Herbert Federer was born on July 23, 1920, in Vienna. After emigrating to the US in 1938, he studied mathematics and physics at the University of California, Berkeley. Affiliated to Brown University, Providence since 1945, he is now Professor Emeritus there. The major part of Professor Federer's scientific effort has been directed to the development of the subject of Geometric Measure Theory, with its roots and applications in classical geometry and analysis, yet in the functorial spirit of modern topology and algebra. His work includes more than thirty research papers published between 1943 and 1986, as well as this book. Herbert Federer Geometric Measure Theory Reprint of the 1969 Edition " Springer Herbert Federer (Professor Emeritus) Department of Mathematics Brown University Providence, RI 02912 USA Originally published as VoI. 153 of the Grundlehren der mathematischen Wissenschaften Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Federer, Herbert: Geometric measure theory / Herbert Federer. - Reprint of the 1969 ed. - Berlin; Heidelberg ; New York; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 (Grundlehren der mathematischen Wissenschaften ; VoI. 153) (Classies in mathematics) NE: 1. GT Mathematics Subject Classification (1991): 53C65, 46AXX ISBN 978-3-540-60656-7 ISBN 978-3-642-62010-2 (eBook) DOI 10.1007/978-3-642-62010-2 This work is subject to copyright. AlI rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustration, recitation, broadcasting, reproduction on microfIlm or in any other way, and storage in data banks.