Book:Fritz John/Plane Waves and Spherical Means

Fritz John: Plane Waves and Spherical Means Applied to Partial Differential Equations

Published $\text {1955}$, Dover Publications

ISBN 0-486-43804-X

Subject Matter

• Partial Differential Equations

Contents

Foreword

Introduction

Chapter I: Decomposition of an Arbitrary Function into Plane Waves

Explanaiton of notation

The spherical mean of a function of a single coordinate

Representation of a function by it plane integrals

Chapter II: The Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients

Hyperbolic equations

Geometry of the normal surface for a strictly hyperbolic equation

Solution of the Cauchy problem for a strictly hyperbolic equation

Expression of the kernel by an integral over the normal surface

The domain of dependence

The wave equation

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

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

Definition of a fundamental solution

The Cauchy problem

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

The fundamental solution

Characterization of the fundamental solution by its order of magnitude

Structure of the fundamental solution

The fundamental solution for elliptic operators with constant coefficients

Fundamental solution of linear elliptic systems with analytic coefficients

Chapter IV: Identities for Spherical Means

Symbolic expression for spherical means

The fundamental identity for iterated spherical means

Expression for a function in terms of its iterated spherical means

The differential equation of Darboux

Chapter V: The Theorems of Asgeirsson and Howard

Ellipsoidal means of a function

The mean value theorem of Asgeirsson

Applications to the equations of Darboux and the wave equation

The identity of Aughtum S. Howard

Applications of Howard's identy

Chapter VI: Determination of a Function from its Integrals over Spheres of a Fixed Radius

Functions periodic in the mean

Functions determined by their integrals over spheres of radius $1$

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

Chapter VII: Differentiability Properties of Solutions of Elliptic Systems

Canonical systems of differential equations

Reduction of determined systems of differential equations to canonical form

The formula for integration by parts on a sphere

Spherical integrals of solutions of a canonical system

Differentiability of solutions of linear elliptic systems

Differentiability of solutions of non-linear elliptic systems

Analyticity of solutions of linear elliptic systems analytic coefficients

Differentiability of continuous weak solutions of linear elliptic equation

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

Chapter VIII: Regularity Properties for Integrals of Solutions over Time-like Lines

Definition of "time-like"

The corresponding canonical system

Derivatives of cylindrical integrals of a solution

Differentiability of integrals of solutions over time-like curves

Integrals of solutions over time-like curves with common endpoints

Bibliography

Index