# 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.

### 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