Plane Waves and Spherical Means: Applied to Partial Differential Equations

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Springer Science & Business Media, 1 déc. 2013 - 172 pages
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The author would like to acknowledge his obligation to all his (;Olleagues and friends at the Institute of Mathematical Sciences of New York University for their stimulation and criticism which have contributed to the writing of this tract. The author also wishes to thank Aughtum S. Howard for permission to include results from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for their cooperation and support, and particularly Lipman Bers, who suggested the publication in its present form. New Rochelle FRITZ JOHN September, 1955 [v] CONTENTS Introduction. . . . . . . 1 CHAPTER I Decomposition of an Arbitrary Function into Plane Waves Explanation of notation . . . . . . . . . . . . . . . 7 The spherical mean of a function of a single coordinate. 7 9 Representation of a function by its plane integrals . CHAPTER II Tbe Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients Hyperbolic equations. . . . . . . . . . . . . . . . . . . . . . 15 Geometry of the normal surface for a strictly hyperbolic equation. 16 Solution of the Cauchy problem for a strictly hyperbolic equation . 20 Expression of the kernel by an integral over the normal surface. 23 The domain of dependence . . . . . . . . . . . . . . . . . . . 29 The wave equation . . . . . . . . . . . . . . . . . . . . . . 32 The initial value problem for hyperbolic equations with a normal surface having multiple points . . . . . . . . . . . . . . . . . . . . 36 CHAPTER III The Fundamental Solution of a Linear Elliptic Differential Equation witL Analytic Coefficients Definition of a fundamental solution . . . . . . . . . . . . . . 43 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . 45 Solution of the inhomogeneous equation with a plane wave function as right hand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 The fundamental solution. . . . . . . . . . . . . . . . . . . . . .
 

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Table des matières

Introduction
1
Representation of a function by its plane integrals
9
Geometry of the normal surface for a strictly hyperbolic equation
16
Expression of the kernel by an integral over the normal surface
23
The domain of dependence
29
The initial value problem for hyperbolic equations with a normal surface
36
CHAPTER III
43
Solution of the inhomogeneous equation with a plane wave function as right
49
The differential equation of Darboux
88
Ellipsoidal means of a function
91
The identity of Aughtum S Howard
100
CHAPTER VI
109
Determination of a field of forces from its effect on a mobile sphere
123
The formula for integration by parts on a sphere
135
Analyticity of solutions of linear elliptic systems analytic coefficients
142
Explicit representations and estimates for the derivatives of a solution of
153

Characterization of the fundamental solution by its order of magnitude
57
The fundamental solution for elliptic operators with constant coefficients
65
Fundamental solution of linear elliptic systems with analytic coefficients
72
The fundamental identity for iterated spherical means
78
Differentiability of integrals of solutions over timelike curves 1 59
159
Bibliography
165
Index
171
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