Lectures on Cauchy's Problem in Linear Partial Differential Equations

Jacques Hadamard

• Publisher: Dover Publications
• Year: 2014
• ISBN:
  9780486781488
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Would well repay study by most theoretical physicists." — Physics Today
"An overwhelming influence on subsequent work on the wave equation." — Science Progress
"One of the classical treatises on hyperbolic equations." — Royal Naval Scientific Service
Delivered at Columbia University and the Universities of Rome and Zürich, these lectures represent a pioneering investigation. Jacques Hadamard based his research on prior studies by Riemann, Kirchhoff, and Volterra. He extended and improved Volterra's work, applying its theories relating to spherical and cylindrical waves to all normal hyperbolic equations instead of only to one. Topics include the general properties of Cauchy's problem, the fundamental formula and the elementary solution, equations with an odd number of independent variables, and equations with an even number of independent variables and the method of descent.

BOOK IV

THE EQUATIONS WITH AN EVEN NUMBER OF INDEPENDENT VARIABLES AND THE METHOD OF DESCENT

CHAPTER I

INTEGRATION OF THE EQUATION IN 2m₁ VARIABLES

1. GENERAL FORMULÆ

134. The first cases in which the solution of Cauchy’s problem was known in Analysis do not, as we have seen, belong to the above treated class: Riemann’s and Kirchhoff’s methods correspond respectively to m = 2 and m = 4.

We shall presently see that, in such cases, singularities such as we met with in the preceding Book no longer occur, every improper integral being even eliminated. This explains why the above-mentioned solutions were found first.

In the general case, nevertheless, even values of m must be considered as bringing in new difficulties. The above methods are no longer valid, and this for two reasons :

First, the elementary solution is no longer well determined (§ 65).

Next, we can no longer introduce the finite part of the integrals which we shall be led to use, as the exponent

with which Γ will appear in the denominator of the elementary solution or its derivatives, will be an integer.

It will actually follow from the very form of the expressions which we shall find, that they could not have been obtained by mere imitation of our former method.

But, as we have already mastered the case of 2m₁ + 1 independent variables, this will enable us to reach the same result when the number of variables is 2m₁ by using our method of descent (§ 29). The solution of the equation

m being equal to 2m₁, will be deduced from the corresponding one for the equation in 2m₁ + 1 variables

denoting by z an (m + l)th auxiliary variable.

If, as we still assume, the characteristic form

of (E) contains one positive and (m – 1) negative squares, the corresponding form

relating to (E′) will consist of one positive and m negative squares. We have already seen that the quantity Γ′, analogous to Γ and relating to (E′), is

denoting by (x₁, x₂, ... x_m, z) and (a₁, a₂, ... a_m, c) two points of the (m + 1)-dimensional space. We have also found, in § 70, what relations exist between the elementary solutions of both equations: we have seen that the coefficients of the successive powers of Γ in one of them differ by numerical factors from the coefficients of the corresponding powers of Γ′ in the other.

Considering the adjoint equations

of (E) and (E′), the formulæ of § 70 express that, if

be the elementary solution of

, with

then the elementary solution of

will be

(w being a regular function); and, if we use the coefficients C_n of

§ 95, formulæ (62), (62 b) of § 70, applied to the adjoint equation, can be written

Now, this can be obtained directly by operating on ν′, or rather on the similar quantity

which relates to Γ′ < 0, as said in § 73. We form the expression (again a so...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents Page
• Preface
• Book I. General Properties of Cauchy’s Problem
• Book II. The Fundamental Formula and The Elementary Solution
• Book III. The Equations With An Odd Number of Independent Variables
• BOOK IV. The Equations With An Even Number of Independent Variables And The Method of Descent
• Index