# Book:Fritz John/Plane Waves and Spherical Means

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## Fritz John:

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

Published $\text {1955}$, **Dover Publications**

- ISBN 0-486-43804-X

### Subject Matter

### 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