Functional Analysis, Sobolev Spaces And Partial .
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Haim BrezisFunctional Analysis,Sobolev Spaces and PartialDifferential Equations1C
Haim BrezisDistinguished ProfessorDepartment of MathematicsRutgers UniversityPiscataway, NJ 08854USAbrezis@math.rutgers.eduandProfesseur émérite, Université Pierre et Marie Curie (Paris 6)andVisiting Distinguished Professor at the TechnionEditorial board:Sheldon Axler, San Francisco State UniversityVincenzo Capasso, Università degli Studi di MilanoCarles Casacuberta, Universitat de BarcelonaAngus MacIntyre, Queen Mary, University of LondonKenneth Ribet, University of California, BerkeleyClaude Sabbah, CNRS, École PolytechniqueEndre Süli, University of OxfordWojbor Woyczyński, Case Western Reserve UniversityISBN 978-0-387-70913-0e-ISBN 978-0-387-70914-7DOI 10.1007/978-0-387-70914-7Springer New York Dordrecht Heidelberg LondonLibrary of Congress Control Number: 2010938382Mathematics Subject Classification (2010): 35Rxx, 46Sxx, 47Sxx Springer Science Business Media, LLC 2011All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science Business Media, LLC, 233 Spring Street, New York, NY10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or bysimilar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.Printed on acid-free paperSpringer is part of Springer Science Business Media (www.springer.com)
To Felix Browder, a mentor and close friend,who taught me to enjoy PDEs through theeyes of a functional analyst
PrefaceThis book has its roots in a course I taught for many years at the University ofParis. It is intended for students who have a good background in real analysis (asexpounded, for instance, in the textbooks of G. B. Folland [2], A. W. Knapp [1],and H. L. Royden [1]). I conceived a program mixing elements from two distinct“worlds”: functional analysis (FA) and partial differential equations (PDEs). The firstpart deals with abstract results in FA and operator theory. The second part concernsthe study of spaces of functions (of one or more real variables) having specificdifferentiability properties: the celebrated Sobolev spaces, which lie at the heart ofthe modern theory of PDEs. I show how the abstract results from FA can be appliedto solve PDEs. The Sobolev spaces occur in a wide range of questions, in both pureand applied mathematics. They appear in linear and nonlinear PDEs that arise, forexample, in differential geometry, harmonic analysis, engineering, mechanics, andphysics. They belong to the toolbox of any graduate student in analysis.Unfortunately, FA and PDEs are often taught in separate courses, even thoughthey are intimately connected. Many questions tackled in FA originated in PDEs (fora historical perspective, see, e.g., J. Dieudonné [1] and H. Brezis–F. Browder [1]).There is an abundance of books (even voluminous treatises) devoted to FA. Thereare also numerous textbooks dealing with PDEs. However, a synthetic presentationintended for graduate students is rare. and I have tried to fill this gap. Students whoare often fascinated by the most abstract constructions in mathematics are usuallyattracted by the elegance of FA. On the other hand, they are repelled by the neverending PDE formulas with their countless subscripts. I have attempted to presenta “smooth” transition from FA to PDEs by analyzing first the simple case of onedimensional PDEs (i.e., ODEs—ordinary differential equations), which looks muchmore manageable to the beginner. In this approach, I expound techniques that arepossibly too sophisticated for ODEs, but which later become the cornerstones ofthe PDE theory. This layout makes it much easier for students to tackle elaboratehigher-dimensional PDEs afterward.A previous version of this book, originally published in 1983 in French and followed by numerous translations, became very popular worldwide, and was adoptedas a textbook in many European universities. A deficiency of the French text was thevii
viiiPrefacelack of exercises. The present book contains a wealth of problems. I plan to add evenmore in future editions. I have also outlined some recent developments, especiallyin the direction of nonlinear PDEs.Brief user’s guide1. Statements or paragraphs preceded by the bullet symbol are extremely important, and it is essential to grasp them well in order to understand what comesafterward.2. Results marked by the star symbol can be skipped by the beginner; they are ofinterest only to advanced readers.3. In each chapter I have labeled propositions, theorems, and corollaries in a continuous manner (e.g., Proposition 3.6 is followed by Theorem 3.7, Corollary 3.8,etc.). Only the remarks and the lemmas are numbered separately.4. In order to simplify the presentation I assume that all vector spaces are overR. Most of the results remain valid for vector spaces over C. I have added inChapter 11 a short section describing similarities and differences.5. Many chapters are followed by numerous exercises. Partial solutions are presented at the end of the book. More elaborate problems are proposed in a separatesection called “Problems” followed by “Partial Solutions of the Problems.” Theproblems usually require knowledge of material coming from various chapters.I have indicated at the beginning of each problem which chapters are involved.Some exercises and problems expound results stated without details or withoutproofs in the body of the chapter.AcknowledgmentsDuring the preparation of this book I received much encouragement from two dearfriends and former colleagues: Ph. Ciarlet and H. Berestycki. I am very grateful toG. Tronel, M. Comte, Th. Gallouet, S. Guerre-Delabrière, O. Kavian, S. Kichenassamy, and the late Th. Lachand-Robert, who shared their “field experience” in dealingwith students. S. Antman, D. Kinderlehrer, and Y. Li explained to me the backgroundand “taste” of American students. C. Jones kindly communicated to me an Englishtranslation that he had prepared for his personal use of some chapters of the originalFrench book. I owe thanks to A. Ponce, H.-M. Nguyen, H. Castro, and H. Wang,who checked carefully parts of the book. I was blessed with two extraordinary assistants who typed most of this book at Rutgers: Barbara Miller, who is retired, andnow Barbara Mastrian. I do not have enough words of praise and gratitude for theirconstant dedication and their professional help. They always found attractive solutions to the challenging intricacies of PDE formulas. Without their enthusiasm andpatience this book would never have been finished. It has been a great pleasure, as
Prefaceixever, to work with Ann Kostant at Springer on this project. I have had many opportunities in the past to appreciate her long-standing commitment to the mathematicalcommunity.The author is partially supported by NSF Grant DMS-0802958.Haim BrezisRutgers UniversityMarch 2010
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xiiContents3.2Definition and Elementary Properties of the Weak Topologyσ (E, E ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3 Weak Topology, Convex Sets, and Linear Operators . . . . . . . . . . . . . .3.4 The Weak Topology σ (E , E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5 Reflexive Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6 Separable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.7 Uniformly Convex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Comments on Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57606267727678794Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.1 Some Results about Integration That Everyone Must Know . . . . . . . 904.2 Definition and Elementary Properties of Lp Spaces . . . . . . . . . . . . . . 914.3 Reflexivity. Separability. Dual of Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4 Convolution and regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.5 Criterion for Strong Compactness in Lp . . . . . . . . . . . . . . . . . . . . . . . . 111Comments on Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.1 Definitions and Elementary Properties. Projection onto a ClosedConvex Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.2 The Dual Space of a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.3 The Theorems of Stampacchia and Lax–Milgram . . . . . . . . . . . . . . . . 1385.4 Hilbert Sums. Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Comments on Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Exercises for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466Compact Operators. Spectral Decomposition of Self-AdjointCompact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.1 Definitions. Elementary Properties. Adjoint . . . . . . . . . . . . . . . . . . . . . 1576.2 The Riesz–Fredholm Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1596.3 The Spectrum of a Compact Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.4 Spectral Decomposition of Self-Adjoint Compact Operators . . . . . . . 165Comments on Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168Exercises for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707The Hille–Yosida Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817.1 Definition and Elementary Properties of Maximal MonotoneOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817.2 Solution of the Evolution Problem dudt Au 0 on [0, ),u(0) u0 . Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . 1847.3 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917.4 The Self-Adjoint Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Comments on Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Contentsxiii8Sobolev Spaces and the Variational Formulation of Boundary ValueProblems in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.2 The Sobolev Space W 1,p (I ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2021,p8.3 The Space W0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2178.4 Some Examples of Boundary Value Problems . . . . . . . . . . . . . . . . . . . 2208.5 The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2298.6 Eigenfunctions and Spectral Decomposition . . . . . . . . . . . . . . . . . . . . 231Comments on Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Exercises for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2359Sobolev Spaces and the Variational Formulation of EllipticBoundary Value Problems in N Dimensions . . . . . . . . . . . . . . . . . . . . . . . 2639.1 Definition and Elementary Properties of the Sobolev SpacesW 1,p ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2639.2 Extension Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2729.3 Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2781,p9.4 The Space W0 ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2879.5 Variational Formulation of Some Boundary Value Problems . . . . . . . 2919.6 Regularity of Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2989.7 The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3079.8 Eigenfunctions and Spectral Decomposition . . . . . . . . . . . . . . . . . . . . 311Comments on Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31210Evolution Problems: The Heat Equation and the Wave Equation . . . . 32510.1 The Heat Equation: Existence, Uniqueness, and Regularity . . . . . . . . 32510.2 The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33310.3 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335Comments on Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34011Miscellaneous Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34911.1 Finite-Dimensional and Finite-Codimensional Spaces . . . . . . . . . . . . 34911.2 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35311.3 Some Classical Spaces of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 35711.4 Banach Spaces over C: What Is Similar and What Is Different? . . . . 361Solutions of Some Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435Partial Solutions of the Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
Chapter 1The Hahn–Banach Theorems. Introduction tothe Theory of Conjugate Convex Functions1.1 The Analytic Form of the Hahn–Banach Theorem: Extensionof Linear FunctionalsLet E be a vector space over R. We recall that a functional is a function definedon E, or on some subspace of E, with values in R. The main result of this sectionconcerns the extension of a linear functional defined on a linear subspace of E by alinear functional defined on all of E.Theorem 1.1 (Helly, Hahn–Banach analytic form). Let p : E R be a functionsatisfying1(1)(2)p(λx) λp(x) x E and λ 0,p(x y) p(x) p(y) x, y E.Let G E be a linear subspace and let g : G R be a linear functional such that(3)g(x) p(x) x G.Under these assumptions, there exists a linear functional f defined on all of E thatextends g, i.e., g(x) f (x) x G, and such that(4)f (x) p(x) x E.The proof of Theorem 1.1 depends on Zorn’s lemma, which is a celebrated andvery useful property of ordered sets. Before stating Zorn’s lemma we must clarifysome notions. Let P be a set with a (partial) order relation . We say that a subsetQ P is totally ordered if for any pair (a, b) in Q either a b or b a (or both!).Let Q P be a subset of P ; we say that c P is an upper bound for Q if a c forevery a Q. We say that m P is a maximal element of P if there is no element1A function p satisfying (1) and (2) is sometimes called a Minkowski functional.H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,DOI 10.1007/978-0-387-70914-7 1, Springer Science Business Media, LLC 20111
21 The Hahn–Banach Theorems. Introduction to the Theory of Conjugate Convex Functionsx P such that m x, except for x m. Note that a maximal element of P neednot be an upper bound for P .We say that P is inductive if every totally ordered subset Q in P has an upperbound. Lemma 1.1 (Zorn). Every nonempty ordered set that is inductive has a maximalelement.Zorn’s lemma follows from the axiom of choice, but we shall not discuss itsderivation here; see, e.g., J. Dugundji [1], N. Dunford–J. T. Schwartz [1] (Volume 1,Theorem 1.2.7), E. Hewitt–K. Stromberg [1], S. Lang [1], and A. Knapp [1].Remark 1. Zorn’s lemma has many important applications in analysis. It is a basictool in proving some seem
Springer is part of Springer Science Business Media (www.springer.com) . lack of exercises. The present book contains a wealth of problems. I plan to add even more in future editions. I have also outlined some recent developments, especially . and such that (4) in c for . Functional Analysis, .
Appendix A:Sample Parking Garage Operations Manual: Page A-4 1.4 Parking Facility Statistics Total Capacity: 2,725 Spaces Total Compact: 464 Spaces (17%) Total Full Size: 2,222 Spaces Total Handicapped Spaces: 39 Spaces Total Reserved Spaces: 559 Spaces Total Bank of America Spaces: 2,166 Spaces Total Boulevard Reserved
B. Pipe and pipe fitting materials are specified in Division 15 piping system Sections. 1.3 DEFINITIONS A. Finished Spaces: Spaces other than mechanical and electrical equipment rooms, furred spaces, pipe and duct shafts, unheated spaces immediately below roof, spaces above ceilings, unexcavated spaces, crawl spaces, and tunnels.
Confined spaces are generally classified in one of two ways: permit required confined spaces and non-permit required confined spaces. Spaces classified as non-permit confined spaces do not have the potential to contain serious hazards and no special procedures are required to enter them. Permit required confined spaces have the .
21 Nuclear Locally Convex Spaces 21.1 Locally Convex -Spaces 478 21.2 Generalities on Nuclear Spaces 482 21.3 Further Characterizations by Tensor Products 486 21.4 Nuclear Spaces and Choquet Simplexes 489 21.5 On Co-Nuclear Spaces 491 21.6 Examples of Nuclear Spaces 496 21.7 A
Finite difference techniques can be applied to the numerical solution of the initial-boundary value problem in S for the semilinear Sobolev or pseudo-parabolic equation (xiUt "-b b u q ru whereai, b i, q and are functions ofspaceandtime variables, q is a boundedlydifferentiable function ofu, andSis anopen,connecteddomainin [R". Undersuitable .
Isoperimetric inequality for Sobolev functions 5 higher dimensions was given first by Kohler‐Jobin [13] and subsequently strengthened by Chiti [6], using rearrangement techniques of Talenti [21], to more general elliptic operators. Chiti also dealt completely with the case of equality. In the present work, we return to the original Payn
6.6 Concentration of convex Lipschitz functions 174 6.7 Exponential inequalities for self-bounding functions 175 6.8 Symmetrized modi ed logarithmic Sobolev inequalities 178 6.9 Exponential Efron-Stein inequalities 179 6.10 A modi ed logarithmic Sobolev inequality for the Poisson dis-tribution 182 6.11 Weakly self-bounding functions 183
The “Agile Software Development Manifesto” was developed in February 2001, by representatives from many of the fledgling “agile” processes such as Scrum, DSDM, and XP. The manifesto is a set of 4 values and 12 principles that describe “What is meant by Agile". THE AGILE VALUES 1. Individuals and interactions over processes and tools 2. Working software over comprehensive .
