Book:Fritz John/Plane Waves and Spherical Means

Subject Matter

 * Partial Differential Equations

Contents

 * Foreword


 * Introduction


 * 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


 * 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


 * :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


 * 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


 * 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


 * 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


 * 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


 * 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