Partial Differential Equations
These Notes grew out of a course given by the author in 1952-53. Though the field of Partial Differential Equations has changed considerably since those days, particularly under the impact of methods taken from Functional Analysis, the author feels that t
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F. John
Partial Differential Equations Second Edition
With 31 Illustrations
Springer-Verlag New York· Heidelberg· Berlin 1975
F.John New York University Courant Institute New York, New York
AMS Classifications: 35-02, 35A 10, 35EXX, 35L05, 35L 10 Library of Congress Cataloging in Publication Data John, Fritz, 1910Partial differential equations. (Applied mathematical sciences, v. 1) Bibliography: p. Includes index. 1. Differential equations, Partial. I. Series. QA1.A647 vol. 1, 1975 [QA374] 515'.353 74-26827
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1975 by Springer-Verlag New York Inc.
ISBN-13: 978-0-387-90111-4 001: 10.1007/978-1-4615-9979-1
e-ISBN-13: 978-1-4615-9979-1
v PREFACE These Notes grew out of a course given by the author in 1952-53.
Though
the field of Partial Differential Equations has changed considerably since those days, particularly under the impact of methods taken from Functional Analysis, the author feels that the introductory material offered here still is basic for an understanding of the subject.
It supplies the necessary intuitive foundation
which motivates and anticipates abstract formulations of the questions and relates them to the description of natual phenomena. Added to this second corrected edition is a collection of problems and solutions, which illustrate and supplement the theories developed in the text. Fritz John New York September, 1974
vii TABLE OF CONTENTS Introd uction
1
CHAPrER I - THE SINGLE FIRST ORDER EQUATION
6
1.
The linear and quasi-linear equations.
2.
The general first order equation for a function of two variables. • • • • • • • • •
15
3.
The general first order equation for a function of n independent variables. • • • • •
37
CHAPrER II - THE CAUCHY PROBLEM FOR HIGHER ORDER EQUATIONS
48
1.
Analytic functions of several real variables •
2.
Formulation of the Cauchy problem. of characteristics. • • •
3.
The Cauchy problem for the general non-linear equation • • •
71
4.
The Cauchy-Kowalewsky theorem.
76
The notion
54
CHAPTER III - SECOND ORDER EQUATIONS WITH CONSTANT COEFFICIENTS 1.
Equations in two independent variables. Canonical forms
87
2.
The one-dimensional wave equation. • •
92
3.
The wave equation in higher dimensions. Method of spherical means. Method of descent • • • • • • • 101
4. The inhomogeneous wave equation by Duhamel's
principle . . . . . . • . . . . . . . • .
• 110
5.
The potential equation in two dimensions •
6.
The Dirichlet problem. •
7.
The Green's function and the fundamental solution ••
8.
Equations related to the potential equation.
9.
Continuation of harmonic functions •
• 167
The heat equation. • • •
• 170
10.
•• 116 • • • • 12 7
• 145 • • 151
CHAPrER r! - THE CAUCHY PROBLEM FOR LINEAR HYPERIDLIC EQUATIONS IN GENERAL 1,
Riemann's method of integration • • • • • • • • • • • • • • 186
viii
2.
Higher order equations in two independent variables.
196
3.
The method of plane wa
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