Lectures On Linear Partial Differential Equations
Lectures on LinearPartial DifferentialEquationsGregory EskinGraduate Studiesin MathematicsVolume 123American Mathematical Society
http://dx.doi.org/10.1090/gsm/123Lectures on LinearPartial DifferentialEquations
Lectures on LinearPartial DifferentialEquationsGregory EskinGraduate Studiesin MathematicsVolume 123American Mathematical SocietyProvidence, Rhode Island
EDITORIAL COMMITTEEDavid Cox (Chair)Rafe MazzeoMartin ScharlemannGigliola Staffilani2010 Mathematics Subject Classification. Primary 35J25, 35L40, 35K30, 35L05, 35L30,35P20, 35P25, 35S05, 35S30.For additional information and updates on this book, visitwww.ams.org/bookpages/gsm-123Library of Congress Cataloging-in-Publication DataEskin, G. I. (Grigorii Il ich)Lectures on linear partial differential equations / Gregory Eskin.p. cm. — (Graduate studies in mathematics ; v. 123)Includes bibliographical references.ISBN 978-0-8218-5284-2 (alk. paper)1. Differential equations, Elliptic.2. Differential equations, Partial.I. Title.QA372.E78 2011515 .3533—dc222010048243Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for suchpermission should be addressed to the Acquisitions Department, American Mathematical Society,201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made bye-mail to reprint-permission@ams.org.c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rightsexcept 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/10 9 8 7 6 5 4 3 2 116 15 14 13 12 11
In memory of my brother Michael Eskin
ContentsPrefacexvAcknowledgmentsChapter I. Theory of DistributionsIntroduction to Chapters I, II, IIIxvi11§1.1.1.1.2.1.3.1.4.Spaces of infinitely differentiable functionsProperties of the convolutionApproximation by C0 -functionsProof of Proposition 1.1Proof of property b) of the convolution22355§2.2.1.2.2.2.3.Definition of a distributionExamples of distributionsRegular functionalsDistributions in a domain6678§3.3.1.3.2.3.3.Operations with distributionsDerivative of a distributionMultiplication of a distribution by a C -functionChange of variables for distributions99910§4. Convergence of distributions4.1. Delta-like sequences1012§5. Regularizations of nonintegrable functions5.1. Regularization in R15.2. Regularization in Rn141517§6. Supports of distributions20vii
viiiContents6.1. General form of a distribution with support at 06.2. Distributions with compact supports2022§7.7.1.7.2.7.3.7.4.2424262728The convolution of distributionsConvolution of f D and ϕ C0 Convolution of f D and g E Direct product of distributionsPartial hypoellipticity§8. ProblemsChapter II. Fourier Transforms3033§9. Tempered distributions9.1. General form of a tempered r transforms of tempered distributionsFourier transforms of functions in SFourier transform of tempered distributionsGeneralization of Liouville’s theorem§11. Fourier transforms of distributions with compact supports42§12. Fourier transforms of 3.7.Sobolev spacesDensity of C0 (Rn ) in Hs (Rn )Multiplication by a(x) SSobolev’s embedding theoremAn equivalent norm for nonintegerRestrictions to hyperplanes (traces)Duality of Sobolev spacesInvariance of Hs (Rn ) under changes of variables4649505152535455§14. Singular supports and wave front sets of distributions14.1. Products of distributions14.2. Restrictions of distributions to a surface576163§15. Problems65Chapter III. Applications of Distributions to Partial 5.Partial differential equations with constant coefficientsThe heat equationThe Schrödinger equationThe wave equationFundamental solutions for the wave equationsThe Laplace equation69697072737478
Contentsix16.6. The reduced wave equation16.7. Faddeev’s fundamental solutions for ( Δ k 2 )8184§17. Existence of a fundamental solution85§18. Hypoelliptic equations18.1. Characterization of hypoelliptic polynomials18.2. Examples of hypoelliptic operators878990§19.19.1.19.2.19.3.19.4.19.5.The radiation conditionsThe Helmholtz equation in R3Radiation conditionsThe stationary phase lemmaRadiation conditions for n 2The limiting amplitude principle9191939598101§20. Single and double layer potentials20.1. Limiting values of double layers potentials20.2. Limiting values of normal derivatives of single layerpotentials102102§21. Problems107Chapter IV. Second Order Elliptic Equations in Bounded Domains106111Introduction to Chapter IV111§22. Sobolev spaces in domains with smooth boundaries11222.1. The spaces H s (Ω) and Hs (Ω)22.2. Equivalent norm in Hm (Ω)11211322.3. The density of C0 in H s (Ω)22.4. Restrictions to Ω22.5. Duality of Sobolev spaces in Ω114115116§23. Dirichlet problem for second order elliptic PDEs23.1. The main inequality11711823.2. Uniqueness and existence theorem in H 1 (Ω)23.3. Nonhomogeneous Dirichlet problem120121§24. Regularity of solutions for elliptic equations24.1. Interior regularity24.2. Boundary regularity122123124§25. Variational approach. The Neumann problem25.1. Weak solution of the Neumann problem25.2. Regularity of weak solution of the Neumann problem125127128§26. Boundary value problems with distribution boundary data129
xContents26.1. Partial hypoellipticity property of elliptic equations26.2. Applications to nonhomogeneous Dirichlet and Neumannproblems129§27. Variational inequalities27.1. Minimization of a quadratic functional on a convex set.27.2. Characterization of the minimum point134134135§28. Problems137Chapter V. Scattering TheoryIntroduction to Chapter V132141141§29. Agmon’s estimates142§30. Nonhomogeneous Schrödingerequation 130.1. The case of q(x) On 1 α ε14814830.2. Asymptotic behaviorof outgoingsolutions (the case of 1, α 0)q(x) On 1(1 x ) 2 α ε 130.3. The case of q(x) O (1 x )1 ε149(1 x )2149§31. The uniqueness of outgoing solutions31.1. Absence of discrete spectrum for k 2 031.2. Existence of outgoingfundamentalsolution (the case of 1)q(x) On 1 δ151155§32. The limiting absorption principle157§33. The scattering problem133.1. The scattering problem (the case of q(x) O( (1 x )n α ))160160133.2. Inverse scattering problem (the case of q(x) O( (1 x n α ))162(1 x )156233.3. The scattering problem (the case of q(x) 33.4. Generalized distorted plane waves33.5. Generalized scattering amplitude1O( (1 x )1 ε ))163164164§34. Inverse boundary value problem34.1. Electrical impedance tomography168171§35. Equivalence of inverse BVP and inverse scattering172§36. Scattering by obstacles36.1. The case of the Neumann conditions36.2. Inverse obstacle problem175179179§37. Inverse scattering at a fixed energy18137.1. Relation between the scattering amplitude and the Faddeev’sscattering amplitudes181
Contentsxi37.2. Analytic continuation of Tr18437.3. The limiting values of Tr and Faddeev’s scattering amplitude 18737.4. Final step: The recovery of q(x)190§38. Inverse backscattering38.1. The case of real-valued potentials191192§39. Problems193Chapter VI. Pseudodifferential OperatorsIntroduction to Chapter VI197197§40. Boundedness and composition of ψdo’s40.1. The boundedness theorem40.2. Composition of ψdo’s198198199§41.41.1.41.2.41.3.Elliptic operators and parametricesParametrix for a strongly elliptic operatorThe existence and uniqueness theoremElliptic ess and the Fredholm propertyCompact operatorsFredholm operatorsFredholm elliptic operators in Rn207207208211§43.43.1.43.2.43.3.The adjoint of a pseudodifferential operatorA general form of ψdo’sThe adjoint operatorWeyl’s l property and microlocal regularityThe Schwartz kernelPseudolocal property of ψdo’sMicrolocal regularity215215217218§45. Change-of-variables formula for ψdo’s221§46.46.1.46.2.46.3.The Cauchy problem for parabolic equationsParabolic ψdo’sThe Cauchy problem with zero initial conditionsThe Cauchy problem with nonzero initial conditions223223225226§47. The heat kernel47.1. Solving the Cauchy problem by Fourier-Laplace transform47.2. Asymptotics of the heat kernel as t 0.228228230§48. The Cauchy problem for strictly hyperbolic equations48.1. The main estimate231233
xiiContents48.2. Uniqueness and parabolic regularization48.3. The Cauchy problem on a finite time interval48.4. Strictly hyperbolic equations of second order235237240§49. Domain of dependence243§50.50.1.50.2.50.3.Propagation of singularitiesThe null-bicharacteristicsOperators of real principal typePropagation of singularities for operators of real principaltype50.4. Propagation of singularities in the case of a hyperbolicCauchy problem247247247§51. Problems258Chapter VII. Elliptic Boundary Value Problems and ParametricesIntroduction to Chapter VII§52.52.1.52.2.52.3.Pseudodifferential operators on a manifoldManifolds and vector bundlesDefinition of a pseudodifferential operator on a manifoldElliptic ψdo’s on a manifold249255263263264264265266§53. Boundary value problems in the half-space53.1. Factorization of an elliptic symbol53.2. Explicit solution of the boundary value problem266266268§54.54.1.54.2.54.3.54.4.Elliptic boundary value problems in a bounded domainThe method of “freezing” coefficientsThe Fredholm propertyInvariant form of the ellipticity of boundary conditionsBoundary value problems for elliptic systems of 5.3.Parametrices for elliptic boundary value problemsPlus-operators and minus-operatorsConstruction of the parametrix in the half-spaceParametrix in a bounded e heat trace asymptoticsThe existence and the estimates of the resolventThe parametrix constructionThe heat trace for the Dirichlet LaplacianThe heat trace for the Neumann LaplacianThe heat trace for the elliptic operator of an arbitrary order285285286288293294§57. Parametrix for the Dirichlet-to-Neumann operator276296
Contentsxiii57.1. Construction of the parametrix57.2. Determination of the metric on the boundary296300§58.58.1.58.2.58.3.58.4.Spectral theory of elliptic operatorsThe nonselfadjoint caseTrace class operatorsThe selfadjoint caseThe case of a compact .5.The index of elliptic operators in RnProperties of Fredholm operatorsIndex of an elliptic ψdoFredholm elliptic ψdo’s in RnElements of K-theoryProof of the index theorem311311313316317321§60. ProblemsChapter VIII. Fourier Integral Operators324329Introduction to Chapter VIII329§61.61.1.61.2.61.3.Boundedness of Fourier integral operators (FIO’s)The definition of a FIOThe boundedness of FIO’sCanonical 4.Operations with Fourier integral operatorsThe stationary phase lemmaComposition of a ψdo and a FIOElliptic FIO’sEgorov’s theorem334334335337338§63. The wave front set of Fourier integral operators340§64.64.1.64.2.64.3.64.4.Parametrix for the hyperbolic Cauchy problemAsymptotic expansionSolution of the eikonal equationSolution of the transport equationPropagation of .4.65.5.Global Fourier integral operatorsLagrangian manifoldsFIO’s with nondegenerate phase functionsLocal coordinates for a graph of a canonical transformationDefinition of a global FIOConstruction of a global FIO given a global canonicaltransformation349349350353358360
xivContents65.6. Composition of global FIO’s65.7. Conjugation by a global FIO and the boundedness theorem365369§66.66.1.66.2.66.3.Geometric optics at largeGenerating functions and the Legendre transformsAsymptotic solutionsThe Maslov index370370374377Oblique derivative problemReduction to the boundaryFormulation of the oblique derivative problemModel problemFirst order differential equations with symbols dependingon x 67.5. The boundary value problem on Ω381381382384§68. aphy403Index407
PrefaceThis book is based on the lectures on partial differential equations that Ihave given for many years at UCLA. It does not assume any knowledgeof partial differential equations and can be considered as a first graduatecourse in linear PDE. However, some basic knowledge of the Fourier transform, Lebesgue integrals and elementary functional analysis is required. Itis organized as lecture notes with emphasis on clarity and accessibility.We shall briefly describe the content of the book. The first three chapters are the elementary theory of distributions and Fourier transforms ofdistributions with applications to partial differential equations with constant coefficients. It is similar to the first chapters of the books by Gelfandand Shilov [GSh] and Shilov [Sh]. Additional material includes the wavefront sets of distributions, Sobolev spaces, the stationary phase lemma, theradiation conditions, and potential theory.In Chapter IV the Dirichlet and the Neumann boundary value problemsare considered for second order elliptic equations in a smooth bounded domain. The existence, uniqueness, and regularity of solutions are proven. Anontraditional topic of this chapter is the proof of the existence and uniqueness of the solutions of the Neumann and Dirichlet problems for homogeneous equations in Sobolev spaces of negative order on the boundary.Chapter V is devoted to scattering theory including inverse scattering,inverse boundary value problem, and the obstacle problem.Chapter VI starts with the theory of pseudodifferential operators withclassical symbols. It is followed by the theory of parabolic Cauchy problemsbased on pseudodifferential operators with symbols analytic in the half-planeand heat kernel asymptotics.xv
xviPrefaceThe next topic of Chapter VI is the Cauchy problem for hyperbolic equations of order m 2, the domains of dependence of solutions to hyperbolicequations, and Hörmander’s theory [H1] of propagation of singularities forthe equations of real principal type with applications to hyperbolic equations.In Chapter VII the Fredholm property for elliptic boundary value problems and parametrices in smooth domains are studied following the approachof the author’s book [E1]. The main application of the parametrix is thestudy of heat trace asymptotics as t 0. The parametrix constructionallows one to compute explicitly two leading terms of the heat trace asymptotics for the cases of Dirichlet and Neumann boundary conditions. ChapterVII concludes with elements of the spectral theory of elliptic operators andthe proof of the index theorem for elliptic operators in Rn following theworks of Atiyah-Singer [AtS1], [AtS2] and Seeley [Se3].The last Chapter VIII is devoted to the theory of Fourier integral operators. Starting with the local theory of FIO, we proceed to the global theory.We consider only a subclass of Hörmander’s FIOs (see [H1]), assuming thatthe Lagrangian manifold of the FIO corresponds to the graph of a canonical transformation. In particular, having a global canonical transformation,we construct a global FIO corresponding to this canonical transformation.Next, following Maslov [M1], [M2], [MF], we construct a global geometric optic solution for a second order hyperbolic equation on arbitrary timeinterval [0, T ].Chapter VIII concludes with a section on the oblique derivative problem. The oblique derivative problem is a good example of nonelliptic boundary value problem, and it attracted the attention of many mathematicians:Egorov-Kondrat’ev [EgK], Malutin [Mal], Mazya-Paneah [MaP], Mazya[Ma], and others. The section is based on the author’s paper [E3], and ituses the FIOs to greatly simplify the problem. Similar results are obtainedindependently by Sjöstrand [Sj] and Duistermaat-Sjöstrand [DSj].At the end of each chapter there is a problem section. Some problems arerelatively simple exercises that help to study the material. Others are moredifficult problems that cover additional topics not included in the book. Inthose cases hints or references to the original sources are given.AcknowledgmentsI want to thank my friend and collaborator Jim Ralston for many fruitfuldiscussions and advice. I am very grateful to my former students Joe Bennish, Brian Sako, Carol Shubin, Borislava Gutarz, Xiaosheng Li and otherswho took notes during my classes. These notes were the starting point ofthis book. I express my deep gratitude to the anonymous referees whose
Acknowledgmentsxviiremarks and suggestions greatly improved the book. I am very thankfulto Neelesh Tiruviluamala for pointing out a mistake in an earlier draft.I am also grateful to Bernard Lascar and Ciprian Manolescu for valuablecomments. I am especially grateful to my wife Marina who prepared themanuscript. Without her help this book would not be written.My deepest gratitude to the editor Natalya Pluzhnikov for the outstanding work of improving and correcting the manuscript. It is impossible tooverestimate her contribution.Finally, I am very grateful to Sergei Gelfand for encouragement to writea book.
Bibliography[A1] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand, New York,1964.[A2] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann.Scuola Norm. Sup. Pisa 2 (1975), 151–218.[ADN] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic differential equations satisfying general boundary conditions, CPAM12 (1959), 623–727; II, CPAM 17 (1964), 35–92.[AH] S. Agmon and L. Hörmander, Assymptotic properties of solutions of differentialequations with simple characteristics, J. Analyse Math. 30 (1976), 1–38.[AK] S. Agmon and Y. Kannai, On the asymptotic behaviour of spectral functions andresolvent kernels of elliptic operators, Israel J. Math. 5 (1967), 1–30.[Ar1] V. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, NewYork, 1978.[Ar2] V. Arnold, On the characteristic class entering in the quantization condition, Funct.Anal. Appl. 1 (1967), 1–13.[At] M. Atiyah, K-theory, Lecture notes, Benjamin, Inc., New York-Amsterdam, 1967.[AtB] M. Atiyah and R. Bott, The index problem for the manifolds with boundary, 1964,Differential Analysis, Bombay Colloq. 1964, pp. 175–186.[AtS1] M. Atiyah and I. M. Singer, The index of elliptic operators I, Anal. of Math. (2)87 (1968), 484–530.[AtS2] M. Atiyah and I. M. Singer, The index of elliptic operators III, Anal. of Math. (2),87 (1968), 546–604.[B] R. Beals, A general calculus of pseudodifferential operators, Duke Math. J. 42 (1975),1–42.[BF] R. Beals and C. Feferman, Spatially inhomogeneous pseudodifferential operators I,Comm. Pure Appl. Math 27 (1974), 585–639.[CH] R. Courant and D. Hilbert, Methods of mathematical physics, vol. II, New York,1962.[CK] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory,Springer-Verlag, 1992.403
404Bibliography[ChP] J. Chazarain and A. Pirion, Introduction à la théorie des équations aux dérivéepartielle linéaire, Gauthier Villars, Paris, 1981.[D] J. J. Duistermaat, Fourier integral operators, Birkhäuser, Boston, 1996.[DH] J. J. Duistermaat and L. Hörmander, Fourier integral operators II, Acta Math. 128(1972), 183–269.[DL] G. Duvaut and J.-P. Lions, Les inéquations en méchanique et en physique, Dunod,Paris, 197
Partial Differential Equations Gregory Eskin American Mathematical Society Providence, Rhode Island Graduate Studies in Mathematics Volume 123. EDITORIAL COMMITTEE DavidCox(Chair) RafeMazzeo MartinScharlemann GigliolaStaffilani 2010 Mathematics Subject Classification. Primary 35J25, 35L40, 35K30, 35L05, 35L30,
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