Plane Waves And Spherical Means Applied To Partial Differential Equations

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ISBN: 9780486438047 | 048643804X
Cover: Paperback
Copyright: 7/26/2004

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This collection of results on partial differential equations employs certain elementary identities for plane and spherical integrals of an arbitrary function, showing how a variety of results on fairly general differential equations follow from those identities. Explores the decomposition of arbitrary functions into functions of the type of plane waves; Radon transformation; more. 1955 edition.

Introduction

(6)

Decomposition of an Arbitrary Function into Plane Waves

Explanation of notation

(1)

The spherical mean of a function of a single coordinate

(2)

Representation of a function by its plane integrals

(6)

The Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients

Hyperbolic equations

(1)

Geometry of the normal surface for a strictly hyperbolic equation

(4)

Solution of the Cauchy problem for a strictly hyperbolic equation

(3)

Expression of the kernel by an integral over the normal surface

(6)

The domain of dependence

(3)

The wave equation

(4)

The initial value problem for hyperbolic equations with a normal surface having multiple points

(7)

The Fundamental Solution of a Linear Elliptic Differential Equation with Analytic Coefficients

Definition of a fundamental solution

(2)

The Cauchy problem

(4)

Solution of the inhomogeneous equation with a plane wave function as right hand side

(1)

The fundamental solution

(7)

Characterization of the fundamental solution by its order of magnitude

(4)

Structure of the fundamental solution

(4)

The fundamental solution for elliptic operators with constant coefficients

(7)

Fundamental solution of linear elliptic systems with analytic coefficients

(5)

Identities for Spherical Means

Symbolic expression for spherical means

(1)

The fundamental identity for iterated spherical means

(4)

Expression for a function in terms of its iterated spherical means

(6)

The differential equation of Darboux

(3)

The Theorems of Asgeirsson and Howard

Ellipsoidal means of a function

(2)

The mean value theorem of Asgeirsson

(2)

Applications to the equations of Darboux and the wave equation

(5)

The identity of Aughtum S. Howard

100

(5)

Applications of Howard's identy

105

(4)

Determination of a Function from its Integrals over Spheres of a Fixed Radius

Functions periodic in the mean

109

(5)

Functions determined by their integrals over spheres of radius 1

114

(9)

Determination of a field of forces from its effect on a mobile sphere

123

(4)

Differentiability Properties of Solutions of Elliptic Systems

Canonical systems of differential equations

127

(1)

Reduction of determined systems of differential equations to canonical form

128

(7)

The formula for integration by parts on a sphere

135

(1)

Spherical integrals of solutions of a canonical system

136

(1)

Differentiability of solutions of linear elliptic systems

137

(2)

Differentiability of solutions of non-linear elliptic systems

139

(3)

Analyticity of solutions of linear elliptic systems analytic coefficients

142

(3)

Differentiability of continuous weak solutions of linear elliptic equation

145

(8)

Explicit representations and estimates for the derivatives of a solution of a linear elliptic equation

153

(4)

Regularity Properties for Integrals of Solutions over Time-like Lines

Definition of ``time-like''

157

(1)

The corresponding canonical system

157

(1)

Derivatives of cylindrical integrals of a solution

158

(1)

Differentiability of integrals of solutions over time-like curves

159

(4)

Integrals of solutions over time-like curves with common endpoints

163

(2)

Bibliography

165

(6)

Index

171

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Plane Waves And Spherical Means Applied To Partial Differential Equations

Summary

Table of Contents

Supplemental Materials

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Plane Waves And Spherical Means Applied To Partial Differential Equations

Summary

Table of Contents

Supplemental Materials

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