Graduate Texts in Mathematics 128 Editorial Board S Axler F.W Gehring P.R Halmos Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 T AKEUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra 2nd cd MAC LANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTJlZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules 2nd ed GOLUBlTSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKJlSAMUEL Commutative Algebra VoL I ZARISKJlSAMUEL Commutative Algebra VoU! JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 WERMER Banach Algebras and Several Complex Variables 2nd ed 36 KELLEY/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENY/SNELUKNAPP Denumerable Markov Chains 2nd ed 41 ApOSTOL Modular Functions and Dirichlet Series in N urn ber Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LOEVE Probability Theory I 4th ed 46 LOEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS/WU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVER/WATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELUFox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed continued after index Jeffrey Rauch Partial Differential Equations With 42 Illustrations , Springer Jeffrey Rauch Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA Editorial Board: S Axler Department of Mathematics Michigan State University East Lansing, MI 48824 USA F.W Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA P.R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classifications (1991): 35-01, 35AXXX Library of Congress Cataloging-in-Publication Data Rauch, Jeffrey Partial differential equations / Jeffrey Rauch p cm - (Graduate texts in mathematics ; 128) Includes bibliographical references and index ISBN 978-1-4612-6959-5 ISBN 978-1-4612-0953-9 (eBook) DOI 10.1007/978-1-4612-0953-9 Differential equations, Partial Title II Series QA374.R38 1991 515' 353-dc20 90-19680 CIP Printed on acid-free paper © 1991 by Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc.in 1991 Softcover reprint ofthe hardcover lst edition 1991 AII rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, 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 by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Typeset by Asco Trade Typesetting Ltd., Hong Kong (Corrected second printing, 1997) ISBN 978-1-4612-6959-5 Preface This book is based on a course I have given five times at the University of Michigan, beginning in 1973 The aim is to present an introduction to a sampling of ideas, phenomena, and methods from the subject of partial differential equations that can be presented in one semester and requires no previous knowledge of differential equations The problems, with hints and discussion, form an important and integral part of the course In our department, students with a variety of specialties-notably differential geometry, numerical analysis, mathematical physics, complex analysis, physics, and partial differential equations-have a need for such a course The goal of a one-term course forces the omission of many topics Everyone, including me, can find fault with the selections that I have made One of the things that makes partial differential equations difficult to learn is that it uses a wide variety of tools In a short course, there is no time for the leisurely development of background material Consequently, I suppose that the reader is trained in advanced calculus, real analysis, the rudiments of complex analysis, and the language offunctional analysis Such a background is not unusual for the students mentioned above Students missing one of the "essentials" can usually catch up simultaneously A more difficult problem is what to about the Theory of Distributions The compromise which I have found workable is the following The first chapter of the book, which takes about nine fifty-minute hours, does not use distributions The second chapter is devoted to a study of the Fourier transform of tempered distributions Knowledge of the basics about @(O), 0"(0), 9&'(0), and 0"'(0) is assumed at that time My experience teaching the course indicates that students can pick up the required facility I have provided, in an appendix, a short crash course on Distribution Theory From Chapter on, Distribution Theory is the basic language of the text, providing a good vi Preface setting for reinforcing the fundamentals My experience in teaching this course is that students have less difficulty with the distribution theory than with geometric ideas from advanced calculus (e.g dq> is a one-form which annihilates the tangent space to {q> = O}) There is a good deal more material here than can be taught in one semester This provides material for a more leisurely two-semester course and allows the reader to browse in directions which interest him/her The essential core is the following: Chapter Almost all A selection of examples must be made Chapter All but the U theory for p ¥- Some can be left for students to read Chapter The first seven sections One of the ill-posed problems should be presented Chapter Sections 1, 2, 5, 6, and plus a representative sampling from Sections and Chapter Sections 1, 2, 3, 10, and 11 plus at least the statements of the standard Elliptic Regularity Theorems These topics take less than one semester An introductory course should touch on equations of the classical types, elliptic, hyperbolic, parabolic, and also present some other equations The energy method, maximum principle, and Fourier transform should be used The classical fundamental solutions should appear These conditions are met by the choices above I think that one learns more from pursuing examples to a certain depth, rather than giving a quick gloss over an enormous range of topics For this reason, many of the equations discussed in the book are treated several times At each encounter, new methods or points of view deepen the appreciation of these fundamental examples I have made a conscious effort to emphasize qualitative information about solutions, so that students can learn the features that distinguish various differential equations Also the origins in applications are discussed in conjunction with these properties The interpretation of the properties of solutions in physical and geometric terms generates many interesting ideas and questions It is my impression that one learns more from trying the problems than from any other part of the course Thus I plead with readers to attempt the problems Let me point out some omissions In Chapter 1, the Cauchy-Kowaleskaya Theorem is discussed, stated, and much applied, but the proof is only indicated Complete proofs can be found in many places, and it is my opinion that the techniques of proof are not as central as other things which can be presented in the time gained The classical integration methods of Hamilton and Jacobi for nonlinear real scalar first-order equations are omitted entirely My opinion is that when needed these should be presented along with sym- Preface Vll plectic geometry There is a preponderance oflinear equations, at the expense of nonlinear equations One of the main points for nonlinear equations is their differences with the linear Clearly there is an order in which these things should be learned If one includes the problems, a reasonable dose of nonlinear examples and phenomena are presented With the exception of the elliptic theory, there is a strong preponderance of equations with constant coefficients, and especially Fourier transform techniques The reason for this choice is that one can find detailed and interesting information without technical complexity In this way one learns the ideas of the theory of partial differential equations at minimal cost In the process, many methods are introduced which work for variable coefficients and this is pointed out at the appropriate places Compared to other texts with similar level and scope (those of Folland, Garabedian, John, and Treves are my favorites), the reader will find that the present treatment is more heavily weighted toward initial value problems This, I confess, corresponds to my own preference Many time-independent problems have their origin as steady states of such time-dependent problems and it is as such that they are presented here A word about the references Most are to textbooks, and I have systematically referred to the most recent editions and to English translations As a result the dates not give a good idea of the original publication dates For results proved in the last 40 years, I have leaned toward citing the original papers to give the correct chronology Classical results are usually credited without reference I welcome comments, critiques, suggestions, corrections, etc from users of this book, so that later editions may benefit from experience with the first So many people have contributed in so many different way to my appreciation of partial differential equations that it is impossible to list and thank them all individually However, specific influences on the structure ofthis book have been P.o Lax and P Garabedian from whom I took courses at the level of this book; Joel Smoller who teaches the same course in a different but related way; and Howard Shaw whose class notes saved me when my own lecture notes disappeared inside a moving van The integration of problems into the flow of the text was much influenced by the Differential Topology text of Guillemin and Pollack I have also benefited from having had exceptional students take this course and offer their criticism In particular, I would like to thank Z Xin whose solutions, corrections, and suggestions have greatly improved the problems Chapters of a preliminary version of this text were read and criticized by M Beals, lL Joly, M Reed, J Smoller, M Taylor, and M Weinstein Their advice has been very helpful My colleagues and coworkers in partial differential equations have taught me much and in many ways I offer a hearty thank you to them all The love, support, and tolerance of my family were essential for the writing of this book The importance of these things to me extends far beyond professional productivity, and I offer my profound appreciation Contents Preface v CHAPTER Power Series Methods §l.l The Simplest Partial Differential Equation §1.2 The Initial Value Problem for Ordinary Differential Equations §1.3 Power Series and the Initial Value Problem for Partial Differential Equations §1.4 The Fully Nonlinear Cauchy-Kowaleskaya Theorem §1.5 Cauchy-Kowaleskaya with General Initial Surfaces §1.6 The Symbol of a Differential Operator §1.7 Holmgren's Uniqueness Theorem §1.8 Fritz John's Global Holmgren Theorem §1.9 Characteristics and Singular Solutions 11 17 22 25 33 41 52 CHAPTER Some Harmonic Analysis §2.1 §2.2 §2.3 §2.4 §2.5 §2.6 The Schwartz Space 9"([Rd) The Fourier Transform on Y([Rd) The Fourier Transform on U([Rd): :0:; p :0:; Tempered Distributions Convolution in ,'l'([Rd) and Y"([Rd) L2 Derivatives and Sobolev Spaces 61 61 64 70 74 83 87 CHAPTER Solution of Initial Value Problems by Fourier Synthesis §3.1 Introduction §3.2 Schrodinger's Equation 95 95 96 x §3.3 §3.4 §3.5 §3.6 §3.7 §3.8 §3.9 §3.1O §3.11 Contents Solutions of Schrodinger's Equation with Data in g'(~d) Generalized Solutions of Schrodinger's Equation Alternate Characterizations of the Generalized Solution Fourier Synthesis for the Heat Equation Fourier Synthesis for the Wave Equation Fourier Synthesis for the Cauchy-Riemann Operator The Sideways Heat Equation and Null Solutions The Hadamard-Petrowsky Dichotomy Inhomogeneous Equations, Duhamel's Principle 99 103 108 112 117 121 123 126 133 CHAPTER Propagators and x-Space Methods §4.1 §4.2 §4.3 §4.4 §4.5 §4.6 §4.7 §4.8 §4.9 Introduction Solution Formulas in x Space Applications of the Heat Propagator Applications of the Schrodinger Propagator The Wave Equation Propagator for d = Rotation-Invariant Smooth Solutions of D +3U = The Wave Equation Propagator for d = The Method of Descent Radiation Problems 137 137 137 140 147 151 152 158 163 167 CHAPTER The Dirichlet Problem 172 §5.1 §5.2 §5.3 §5.4 §5.5 §5.6 §5.7 §5.8 §5.9 §5.1O §5.11 172 177 181 188 197 200 Introduction Dirichlet's Principle The Direct Method of the Calculus of Variations Variations on the Theme fII and the Dirichlet Boundary Condition The Fredholm Alternative Eigenfunctions and the Method of Separation of Variables Tangential Regularity for the Dirichlet Problem Standard Elliptic Regularity Theorems Maximum Principles from Potential Theory E Hopf's Strong Maximum Principles 207 214 225 234 239 APPENDIX A Crash Course in Distribution Theory 247 References 259 Index 261 251 Appendix A Crash Course in Distribution Theory difference quotients converge to zero almost everywhere Since H is not a constant, zero is surely not the desired derivative The pointwise limit gives the wrong answer and the distribution derivative is the right answer The operations on distributions discussed above are special cases of a general algorithm The following version appears in unpublished notes of P.D Lax Proposition Suppose that L is a linear map from £Zl(01) to £Zl(02)' which is sequentially continuous in the sense that == (I, Idet DI1- 111/1 11- > This example is important if one wants to define distributions on a manifold Convolution Suppose that Q = ~d and cp E £!l(~d) Let L be the operator L(I/I) = cp * 1/1 Leibniz' rule for differentiating under the integral implies that L maps £!l(~d) continuously to itself Fubini's Theorem shows that the transpose of L is convolution with ~cp (exercise) Thus cp * I makes sense for any I E £!l'(~d) and is given by (cp * 1,1/1> == (I, (~cp) * 1/1> As an example, we compute cp * b, (cp * b, 1/1 >== (b, (~cp) * 1/1> = «~cp) * 1/1)(0) = f cp(y)l/I(y) dv = (cp, 1/1> Thus cp * b = cpo The definitions yield (aa(cp * I), 1/1> == (cp * I, ( _a)al/l> == (I, (!J€cp) * (- a)al/l >= (I, ( - a)a«~cp) * 1/1» A similar sequence of computations shows that the last term is equal to (cp * aal, 1/1 On the other hand, applying the derivative in the last term to the ~cp term ofthe convolution, and then unraveling, yields «aacp) * 1,1/1> In this way, we prove that in the sense of distributions > aa(cp * I) = cp * aal = (aacp) * I Applied when I = b, we find that cp * cab = aacp Convolution on the right, L(I/I) = 1/1 * cp, is equal to cp * 1/1 and also extends In particular, 1* cp = cp * I 253 Appendix A Crash Course in Distribution Theory Proposition If IE g'(lR d ) and cp E@(lR d ), then 1* cp is equal to the whose value at x is (I, TA~cp» PROOF First observe that if COO function x, then Tx (.~cp) -> TA911cp) in g(lR d ), which suffices to show that the function y(x) == (I, ~A~cp» is continuous on IRd Similarly, if L1~ is the forward difference operator ('-hej - l)/h approximating OJ, then Xn -> and as h -> O This suffices to show that y E C (lRd) and Ojy = (I, TA(~cp» By induction on n it follows that for all n E N, y E C"(lRd), and oay = (I, Txaa(~cp» for all letl ~ n Next we show that the distribution defined by y is equal to the convolution 1* cp Toward that end, write (y, t/J) as a limit of Riemann sums (y,t/J)=lim n~':.tJ L t/J(~)(I'Taln(.~cp»n-d=lim L 11,t/J(et)TalnUJfCP»)n-d n n d n- L t/J(!Xln)Taln(~CP) -> (.!Jicp) * t/J in g(lR (exercise), so the IlEl d Note that limit on the right is equal to as distributions n"'"'""""OO aeLd \ d) (1,(~cp)*t/J)=(cp*I,t/J) Thus, y=cp*1 Proposition Suppose that x,j E g(lR d ), fj(x) dx = 1, x(O) = 1,jeCx) == !'.-dj(xlF.), and Xe(x) == X(F.x) Then for any 1E 0;I'(lR d ), Xel,je * I, and XeUe * I) converge to I in g' (lRd) as [; tends to zero In particular, any such is the limit in 0;1' (lRd) of elements of 0;I(lR d ) The function Xe is a vast plateau of height very close to lover a diameter of order lie Thus multiplication by Xe is nearly the identity operator Convolution by je is close to convolution with the Dirac delta which is the identity operator These two approximation processes, plateau multiplication and convolution with an approximate delta, are simple but remarkably useful methods in analysis PROOF OF PROPOSITION We treat only XeUe * I) The definitions yield (XeUe * I), cp) == (I, (~je) * (XeCP» for any cp E g(lRd ) The result is then a consequence of the fact that (.~je) * (X,cp) -> cp in f0(lRd) The verification of that is an exercise in advanced calculus which is left to the reader If is a distribution on and w is an open subset of 0, then is equal to zero on w means that for all cp E E:0(w), (I, cp) = o Definition The support of lEg' (0) is the complement of {X E 0: I is equal to zero on a neighborhood of x} The support is denoted supp(l) The set of all IE E:0'(O) such that supp(l) is compact in is denoted 0'''(0) 254 Appendix A Crash Course in Distribution Theory EXAMPLES The support of every I is a closed subset of O supp (5~ = {~} If I is equal to a continuous function f, then supp(l) = cl {x:f(x) i= O} supp(aal) c supp(l) Proposition If I E £0'(0) and (Xl, cp) == c Since c = j * (Xl) I 2t(w) this suffices to show that I = c in w o Finally, we take a second look at convolutions to show that C'(IRd) * ~'(IRd) makes sense From the definitions it is not difficult to show that supp(cp * I) c supp(cp) u supp(l) Thus cp * I E f2(IR d) when IE C'(IRd) For such an I consider the map L(cp) == cp * l This is a sequentially continuous map of f2(IR d) to itself and it has transpose L'(t/J) = t/J * (Bll) (exercise) The transpose being sequentially continuous, it follows that L extends to an operator from ~'(IRd) to itself For u E !Zl'(IRd) and 1E C'(IRd), u * is given by (u*l, cp) == (u, cp*(BlI» Symmetrically, left convolution 1* u is defined as an extension of I * One has 1* u = u * I for all I E C'(IRd) and u E E0'(IRd) PROBLEMS Many details in this Appendix were left as exercises Working them out is good practice There are two things one needs to to learn distributions One must manipulate the Appendix A Crash Course in Distribution Theory 257 definitions in simple proofs and one must gain familiarity with computations with simple distributions The previous pages provide many exercises of the first sort The next problems have the second skill as goal Compute (d/d4Ixl j for j, k = 1,2, Compute (d/d4Isin xl for k = 1,2, (i) Let u(x) be the function which is equal to In(x) for x > and zero for x :5: o Then u is locally integrable Compute the distribution derivative du/dx Answer and in x < O This computes candidates for T which are correct up to a distribution supported at (0) Let /1 be the distribution which integrates a test function over the unit ball in IRd Compute (i) o,t/OXj; (ii) 1l/1 where Il == (%xf L For j E E0(lR d), j ;::: 0, j, == e-dj(x/e), sketch j, * /1 with /1 as in Problem 5, and e converging to zero With n = 1, define f by f(x) = sin I/x if x > and zero otherwise Sketch j, * f discussing the behavior as e tends to zero Letf be the characteristic function of the positive quadrant, {x E IR z: Xj > 0, i Compute (i) oxf; (ii) ox! ox,! = 1,2} Let f be the characteristic function of the set XI X z > O Perform the same calculations as in Problem 10 Define f E L "'(lRz) by f = X I xz/(xi + xD for x "# O Perform the calculations as in Problem Hint Be careful about x = O The answer must identify the distribution derivative on all of IRz Away from x = the derivatives are given by elementary calculus The formal second derivatives are not even locally integrable near the origin Problems 12 and 13 have similar difficulties 11 Let f be the characteristic function of a nonempty infinite wedge in IR x IR with vertex at the origin Find a constant coefficient second-order partial differential operator P(D) such that P(D)f = b 12 Prove that (!j(O!ox + i%y)(l/z) = nb Hint Write (l/z, Pt(D)q» = limJlzl>,z-1 Pl(D)q> dx dy Use Green's Theorem for the integral on Izl > e DISCUSSION The differential operator appearing,here is denoted %z The fact that h-1/oz = for z "# expresses the fact that Z l is holomorphic away from the origin The formula of this problem is equivalent to the Cauchy integral formula 258 Appendix A Crash Course in Distribution Theory 13 Take an example from a calculus text illustrating the inequality of mixed partials and compute the distribution derivatives showing how the mixed partials end up being equal 14 Prove that the principal value of I/x is a distribution of order 15 Show that the functional I(lp) = ~ ~(lp G) - lp(O») defines a distribution of order Find supp(I) Show that there does not exist a constant c such that 1(1, lp)1 :s; c max • upp(l) (llp(x)1 + IIp'(x)l) DISCUSSION This example of L Schwartz shows that a reasonable conjecture connecting support and order of distributions of compact support is not true References [A] S Agmon, Lectures on Elliptic Boundary Value Problems, with B.F Jones and G.W Batten, Van Nostrand, Princeton, NJ, 1965 [BJS] L Bers, F John, and M Schechter, Partial Differential Equations, American Mathematical Society, Providence, RI, 1964 [CH] R Courant and D Hilbert, Methods of Mathematical Physics, Interscience, New York, 1953 (vol 1), 1962 (vol 2) [Dolp] C Dolph, Nonlinear integral equations of Hamerstein type, Trans Amer Math Soc 66 (1949),289-307 [Doll] J Dollard, Scattering into cones I: Potential scattering, Comm Math Phys 12 (1969), 193-203 [Don] W Donoghue, Distributions and Fourier Transforms, Academic Press, New York,1969 [Fi] R Finn, Remarks relevant to minimal surfaces and to surfaces of prescribed mean curvature, J d'Analyse Math 14 (1965), 139-160 [Fo] G.B Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, NJ, 1976 [Fr] F.G Friedlander, On the radiation field of pulse solutions of the wave equation II, Proc Roy Soc 279A (1964),386-394 [Gara] P Garabedian, Partial Differential Equations, Wiley, New York, 1964 [Gard] L Garding, Linear hyperbolic partial differential equations with constant coefficients, Acta Math 85 (1950), 1-62 [GS] I.M Gelfand and G.E Shilov, Generalized Functions, vol 1, trans by E Saletan, Academic Press, New York, 1964 [GT] D Gilbarg and N Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, New York, 1983 [He] D Henry, Geometric Theory of Semi linear Parabolic Equations, SpringerVerlag, New York, 1981 [HI] L Hormander, Linear Partial Differential Operators, 3rd printing, SpringerVerlag, New York, 1969 [H2] L Hormander, The Analysis of Linear Partial Differential Operators, SpringerVerlag, New York, 1982 (vol 1), 1983 (vol 2),1985 (vol 3),1985 (vol 4) [J] F John, Partial Differential Equations, 4th ed., Springer-Verlag, New York, 1982 260 References [LL] E Landesman and A Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J Math Mech 19 (1970),609-623 [La] p.o Lax, The formation and decay of shock waves, Amer Math Soc Monthly 79 (1972),227-241 [LP] P.D Lax and R Phillips, Scattering Theory, Academic Press, New York, 1967 [Lio] J.L Lions, Equations Differentielles Operationelles et Problemes aux Limites, Springer-Verlag, Berlin, 1963 [Lit] W Littman, The wave equation and Lp norms, J Math Mech 12 (1963), 55-68 [LS] L Loomis and S Sternberg, Advanced Calculus, Addison-Wesley, Reading, MA,1968 [Ma] A Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984 [Me] A Messiah, Quantum Mechanics, vol 1, trans G Temmer, Interscience, New York,1961 [N] L Nirenberg, An application of generalized degree to a class of nonlinear problems, Colloque sur I'Analyse Functionelle Tenu a Liege du 14 a 16 Septembre 1970, Vander, Louvain, 1971,57-74 [P] E Picard, Traite d'Analyse, 3ed., t 2, Gauthier-Villars, Paris, 1925 [PW] M Protter and H Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967 [RS] M Reed and B Simon, Methods of Mathematical Physics, vols and 2, Academic Press, New York, 1972 (vol 1), 1975 (vol 2) [RSzN] F Riesz and B Sz-Nagy, Functional Analysis, trans L Boron, Ungar, New York,1955 [R] W Rudin, Functional Analysis, McGraw-Hill, New York, 1973 [Sc1] L Schwartz, Theorie des Distributions, nouvelle ed., Hermann, Paris, 1966 [Sc2] L Schwartz, Mathematics for the Physical Sciences, Addison-Wesley, Reading, MA, 1966 [Sm] Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983 [Sp] M Spivak, Calculus on Manifolds, a Modern Approach to the Classical Theorem of Advanced Calculus, W.A Benjamin, New York, 1965 [Ta] M Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981 [Tr] F Treves, Basic Linear Partial Differential Equations, Academic Press, New York,1975 [Wh] G.B Whitham, Linear and Nonlinear Waves, Wiley New York, 1974 [W] D.V Widder, Positive temperatures on an infinite rod, Trans Amer Math Soc 55 (1944), 85-95 Index acoustic waves 117 See also sound waves Airyequation 139ff, 150 B;'an 216 Burgers' equation CO'(Q) 248 C""(x) 227 C'''([R : g'([Rd)) '@(Q) 248 @'(Q) 248 b~ 220 20ff, 31, 55ff, 140 130, 157 101 Cauchy-Kowaleskaya, theorem §1.3-§1.8,230 Cauchy-Riemann equations D'Alembert's formula 43tT, §4.5 descent, method of §4.8 dilation O'A 66,77, 139 Dirac's delta function 249 Dirichlet problem 118, Ch Dirichlet's principle 178ff discontinous dependence 121, 124, 4,15,81, dispersion 103, 105, 107, 149, 150 dissipation 120,215 distribution 248tT, Ch domain of determinacy 2, 3, 43tT, 152, §3.8, 124, 144 characteristic(s) 30,40, §1.9, 125,215 curves 3,19,21 lines 2,30 surfaces 30, 31 variety 31,95 coercive 189, 193, 195,225 comparison theorem 141,243 conormal 30,216 space 30 variety 30 convolution §2.5, 252, 256 cotangent space 29 critical point 178, 180 D 26 0" 12 161 domain of influence 2,3, 43tT, 152, 161 Duhamel's principle §3.11, §4.9 6"(Q) 248 6"'(Q) 253 metric for 254 eigenfunction expansions 70,201, §5.7 eikonal equation 30 elliptic equations and operators 1,6, 31, 42,54,81,96,106,172,191, Ch elliptic regularity theorem 80-81,201, 226 boundary 229ff interior 227tT, 239 tangential §5.8 ellipticity constant 189 262 Index energy method 191 Euler equation 114, 116, 120, 139, 140, 174, 190, 193 g; 64 '¥'* 67 fixed point equation 10, 205 Fourier transform §2.2ff Frechetspace 63,123 Fredholm alternative, for elliptic boundary value problems §5.6, 244 fundamental solution 85, 86, 104, §4.2 generalized Schwartz inequality 90, 108,111 generalized solution 104, §3.5, 119, 212 Green's function 85, 86, §4.2 group velocity 99, 102, 149, 150 W(\Rd) Hl(Q) W(w) H-l(Q) Hl(Q) Hl(X) W(x) 89 192 225 191 184 228 165, 187,225 Hadamard~ Petrowsky condition §3.1O, 134, 139, 177 harmonic function 17,81, 157, 175, 176, 238~239 Hausdorff~ Young inequality 86 heat equation 12, 13, 16,40,81, §3.6, §3.7, 131, 133, 134, §4.2, §4.3, 172ff, 177ff, 196, 211ff, 215 Heisenberg's uncertainty principle 97, 98,99,105 Hodge theory 194ff, 216, 228 Holmgren's uniqueness theorem 16, §1.7, 98, 111, 124 global §1.8, 99, 126 semiglobal 39 hoi om orphic function 5,17,81,144, 236 Huygen's principle 160, 162, 164, 165 generalized 165 hyperbolic equations 1,96, 127, 162,228 initial value problem Ch 1, Ch 3, Ch operator 188,216 Laplace equation 12,15,81,131, 157 Lax duality theorem 90, 111 linearization 20,22,23,26, 175 loss of derivatives 131,156,157,161 Laplace~Beltrami maximum principles Cor 4.3.4, 172, §5.1O, §5.11 measure(s) 141, 164 Fourier transform of 82 mean curvature 174, 176, 246 mean value property 176, 236, 238ff method of characteristics 19,131 minimal surface 174,246 monotonicity 161, 165 natural boundary condition(s) 194, 196 Neumann problem 188, 191, 193,206, 224~225,244 noncharacteristic 18, 19,23,25,30,32, 40, §1.8, 124, 127 nonstationary phase 149,150, 166 null solutions 41,98, 111, §3.9 order, of a distribution 249 orthogonal invariance 86, 153, 154 transformation 70 Paley~ Wiener theorem 86, 122 parabolic equations 13, 96, 106, 107, 163,211ff Peter~Paul inequality 182 phase velocity 99, 102 Poisson kernel 176 propagator 134, Ch heat 113, §4.2, §4.3 Schrodinger 103, §4.2, §4.3 wave 118, §4.2, §4.5, §4.7, §4.8, §4.9 push forward 216,220 quasi-linear trick 116,246 69 radiation problems §4.9 rarefaction wave 56, 140 real analytic function 5,8, 11, 17,24, 144 ~ 263 Index reflection operator 69, 70, 84, 251 reflection of waves 152 regular boundary point 228 Rellich compactness theorem 202 Riesz Thorin Convexity Theorem 72, 142 Robin problem/boundary condition 196,244 g'([Rd) 61 ,'I"([Rd) 74 (fA 66 Schrodinger's equation §3.2, §3.3, §3.4, §3.5, 126, 131, 134, §4.2, §4.4, 212 self-similar 58,60, 139, 140 separation of variables 207,211-214 shock waves 6,54,59 sing supp 165 smoothing property 143, 212, 215 Sobolev embedding theorem 93, 184, 187,210,228,231 Sobolev spaces §2.6 sound waves 21,57, 117 subharmonic function 179, 235 support 248, 253 symbol of a ditTerential operator §1.6 Tx, Tx* 'h 28 66,250 tangentspace 28,30 Taylor's theorem 136 tempered distributions §2.4tT convergence 75 trace, at boundary 196, 198,200 translation, Th 66, 77, 250 transpose of a ditTerential operator 34, 252 of a linear operator 34, 76, 251 wave equation, wave operator 12,31, 43tT, §3.7, 131, 133, 136, §4.2, §4.5, §4.6, §4.7, §4.8, §4.9, 212 wave equation, damped or with friction 120,180,212,214 weak well-posed 131, 132 Graduate Texts in Mathematics continued from page ii 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 WHITEHEAD Elements of Homotopy Theory KARGAPOLOV/MERLZJAKOV Fundamentals of the Theory of Groups BOLLOBAS Graph Theory EDWARDS Fourier Series Vol I 2nd cd WELLS Differential Analysis on Complex Manifolds 2nd cd WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields n MASSEY Singular Homology Theory FARKAS/KRA Riemann Surfaces 2nd ed STILLWELL Classical Topology and Combinatorial Group Theory 2nd ed HUNGERFORD Algebra DAVENPORT Multiplicative Number Theory 2nd ed HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras lITAKA Algebraic Geometry HECKE Lectures on the Theory of Algebraic Numbers BURRIS/SANKAPPANAVAR A Course in Universal Algebra WALTERS An Introduction to Ergodic Theory ROBINSON A Course in the Theory of Groups 2nd ed FORSTER Lectures on Riemann Surfaces BOTT/Tu Differential Forms in Algebraic Topology WASHINGTON Introduction to Cyclotomic Fields 2nd ed IRELAND/RoSEN A Classical Introduction to Modem Number Theory 2nd ed EDWARDS Fourier Series Vol II 2nd ed v AN LINT Introduction to Coding Theory 2nd ed BROWN Cohomology of Groups PIERCE Associative Algebras LANG Introduction to Algebraic and Abelian Functions 2nd ed BR0NDSTED An Introduction to Convex Polytopes BEARDON On the Geometry of Discrete Groups 92 DIESTEL Sequences and Series in Banach 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 lID III 112 113 114 115 116 117 118 Spaces DUBROVIN/FoMENKO/NoVIKOV Modem Geometry-Methods and Applications Part I 2nd ed WARNER Foundations of Differentiable Manifolds and Lie Groups SHIRYAEV Probability 2nd ed CONWAY A Course in Functional Analysis 2nd ed KOBLITZ Introduction to Elliptic Curves and Modular Forms 2nd ed BROCKERiToM DIECK Representations of Compact Lie Groups GRovE/BENSON Finite Reflection Groups 2nd ed BERG/CHRISTENSEN/RESSEL Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions EDWARDS Galois Theory V ARADARAJAN Lie Groups, Lie Algebras and Their Representations LANG Complex Analysis 3rd ed DUBROVIN/FoMENKO/NoVIKOV Modem Geometry-Methods and Applications Part n LANG SL,(R) SILVERMAN The Arithmetic of Elliptic Curves OLVER Applications of Lie Groups to Differential Equations 2nd ed RANGE Holomorphic Functions and Integral Representations in Several Complex Variables LEHTO Univalent Functions and Teichmiiller Spaces LANG Algebraic Number Theory HUSEMOLLER Elliptic Curves LANG Elliptic Functions KARATLAS/SHREVE Brownian Motion and Stochastic Calculus 2nd ed KOSUTL A Course in Number Theory and Cryptography 2nd ed BERGERiGOSTIAUX Differential Geometry: Manifolds, Curves, and Surfaces KELLEy/SRINIVASAN Measure and Integral Vol I SERRE Algebraic Groups and Class Fields PEDERSEN Analysis Now 119 ROTMAN An Introduction to Algebraic Topology 120 ZIEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation 121 LANG Cyclotomic Fields I and II Combined 2nd ed 122 REMMERT Theory of Complex Functions Readings in Mathematics 123 EBBINGHAUs/HERMES et al Numbers Readings in Mathematics 124 DUBROVIN/FoMENKO/NoVIKOV Modem Geometry-Methods and Applications Part III 125 BERENSTEIN/GAY Complex Variables: An Introduction 126 BOREL Linear Algebraic Groups 127 MASSEY A Basic Course in Algebraic Topology 128 RAUCH Partial Differential Equations 129 FULTON/HARRIS Representation Theory: A First Course Readings in Mathematics 130 DODSON/POSTON Tensor Geometry 131 LAM A First Course in Noncommutative Rings 132 BEARDON Iteration of Rational Functions 133 HARRIS Algebraic Geometry: A First Course 134 ROMAN Coding and Information Theory 135 ROMAN Advanced Linear Algebra 136 ADKINS/WEINTRAUB Algebra: An Approach via Module Theory 137 AXLERiBoURDON/RAMEY Harmonic Function Theory 138 COHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKERiWEISPFENNING/KREDEL Grobner Bases A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 DOOB Measure Theory 144 DENNIS/FARB Noncommutative Algebra 145 VICK Homology Theory An Introduction to Algebraic Topology 2nd ed 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K-Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds 150 EISENBUD Commutative Algebra with a View Toward Algebraic Geometry 151 SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 fuLTON Algebraic Topology: A First Course 154 BROWN/PEARCY An Introduction to Analysis 155 KASSEL Quantum Groups 156 KECHRIS Classical Descriptive Set Theory 157 MALLIAYIN Integration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable II 160 LANG Differential and Riemannian Manifolds 161 BORWEIN/ERDELYI Polynomials and Polynomial Inequalities 162 ALPERIN/BELL Groups and Representations 163 DIXON/MORTIMER Permutation Groups 164 NATHANSON Additive Number Theory: The Classical Bases 165 NATHANSON Additive Number Theory: Inverse Problems and the Geometry of Sumsets 166 SHARPE Differential Geometry: Cartan's Generalization of Klein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd cd ... in Classical Mechanics 2nd ed continued after index Jeffrey Rauch Partial Differential Equations With 42 Illustrations , Springer Jeffrey Rauch Department of Mathematics University of Michigan... (1991): 35-01, 35AXXX Library of Congress Cataloging-in-Publication Data Rauch, Jeffrey Partial differential equations / Jeffrey Rauch p cm - (Graduate texts in mathematics ; 128) Includes bibliographical... Problem for Partial Differential Equations 11 §1.3 Power Series and the Initial Value Problem for Partial Differential Equations Our goal is to investigate through two examples the partial differential