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 1. The linear and quasi-linear equations. 6 2. The general first order equation for a function of two variables. โ{128}ข โ{128}ข โ{128}ข โ{128}ข โ{128}ข โ{128}ข โ{128}ข โ{128}ข โ{128}ข 15 The general first order equation for a function 3. of n independent variables. โ{128}ข โ{128}ข โ{128}ข โ{128}ข โ{128}ข 37 CHAPrER II - THE CAUCHY PROBLEM FOR HIGHER ORDER EQUATIONS 1. Analytic functions of several real variables โ{128}ข 48 2. Formulation of the Cauchy problem. The notion of characteristics. โ{128}ข โ{128}ข โ{128}ข 54 3. The Cauchy problem for the general non-linear equation โ{128}ขโ{128}ขโ{128}ข 71 4. The Cauchy-Kowalewsky theorem. 76 CHAPTER III - SECOND ORDER EQUATIONS WITH CONSTANT COEFFICIENTS 1. Equations in two independent variables
CONTENT
I โ{128}{148} The Single First Order Equation -- 1. The linear and quasi-linear equations -- 2. The general first order equation for a function of two variables -- 3. The general first order equation for a function of n independent variables -- II โ{128}{148} The Cauchy Problem for Higher Order Equations -- 1. Analytic functions of several real variables -- 2. Formulation of the Cauchy problem. The notion of characteristics -- 3. The Cauchy problem for the general non-linear equation -- 4. The Cauchy-Kowalewsky theorem -- III โ{128}{148} Second Order Equations with Constant Coefficients -- 1. Equations in two independent variables. Canonical forms -- 2. The one-dimensional wave equation -- 3. The wave equation in higher dimensions. Method of spherical means. Method of descent -- 4. The inhomogeneous wave equation by Duhamelโ{128}{153}s principle -- 5. The potential equation in two dimensions -- 6. The Dirichlet problem -- 7. The Greenโ{128}{153}s function and the fundamental solution -- 8. Equations related to the potential equation -- 9. Continuation of harmonic functions -- 10. The heat equation -- IV โ{128}{148} The Cauchy Problem for Linear Hyperbolic Equations in General -- 1. Riemannโ{128}{153}s method of integration -- 2. Higher order equations in two independent variables -- 3. The method of plane waves -- List of Books Recommended for Further Study -- Problems
