Plane Waves and Spherical Means: Applied to Partial Differential EquationsSpringer Science & Business Media, 1 déc. 2013 - 172 pages The author would like to acknowledge his obligation to all his (;Olleagues and friends at the Institute of Mathematical Sciences of New York University for their stimulation and criticism which have contributed to the writing of this tract. The author also wishes to thank Aughtum S. Howard for permission to include results from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for their cooperation and support, and particularly Lipman Bers, who suggested the publication in its present form. New Rochelle FRITZ JOHN September, 1955 [v] CONTENTS Introduction. . . . . . . 1 CHAPTER I Decomposition of an Arbitrary Function into Plane Waves Explanation of notation . . . . . . . . . . . . . . . 7 The spherical mean of a function of a single coordinate. 7 9 Representation of a function by its plane integrals . CHAPTER II Tbe Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients Hyperbolic equations. . . . . . . . . . . . . . . . . . . . . . 15 Geometry of the normal surface for a strictly hyperbolic equation. 16 Solution of the Cauchy problem for a strictly hyperbolic equation . 20 Expression of the kernel by an integral over the normal surface. 23 The domain of dependence . . . . . . . . . . . . . . . . . . . 29 The wave equation . . . . . . . . . . . . . . . . . . . . . . 32 The initial value problem for hyperbolic equations with a normal surface having multiple points . . . . . . . . . . . . . . . . . . . . 36 CHAPTER III The Fundamental Solution of a Linear Elliptic Differential Equation witL Analytic Coefficients Definition of a fundamental solution . . . . . . . . . . . . . . 43 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . 45 Solution of the inhomogeneous equation with a plane wave function as right hand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 The fundamental solution. . . . . . . . . . . . . . . . . . . . . . |
Table des matières
2 | |
Representation of a function by its plane integrals | 9 |
Geometry of the normal surface for a strictly hyperbolic equation | 16 |
Expression of the kernel by an integral over the normal surface | 23 |
The domain of dependence | 29 |
The initial value problem for hyperbolic equations with a normal surface | 36 |
CHAPTER III | 43 |
Solution of the inhomogeneous equation with a plane wave function as right | 49 |
The fundamental identity for iterated spherical means | 78 |
The differential equation of Darboux | 88 |
Ellipsoidal means of a function | 91 |
The identity of Aughtum S Howard | 100 |
CHAPTER VI | 109 |
Determination of a field of forces from its effect on a mobile sphere | 123 |
The formula for integration by parts on a sphere | 135 |
Analyticity of solutions of linear elliptic systems analytic coefficients | 142 |
Characterization of the fundamental solution by its order of magnitude | 61 |
Fundamental solution of linear elliptic systems with analytic coefficients | 72 |
Explicit representations and estimates for the derivatives of a solution of | 153 |
Autres éditions - Tout afficher
Plane Waves and Spherical Means: Applied to Partial Differential Equations F. John Affichage d'extraits - 1955 |
Plane Waves and Spherical Means: Applied to Partial Differential Equations F. John Affichage d'extraits - 1955 |
Expressions et termes fréquents
analytic coefficients analytic functions Asgeirsson assume bounded set C₁ C₂ canonical system Cauchy problem Chapter characteristic form class Cm constant coefficients continuous function Courant-Hilbert defined denote depend derivatives of order differential operator dwę elliptic differential equation elliptic equation elliptic operator equation L[u expression follows formula function f(x function of class fundamental solution given grad Q hence hyperbolic equation identity initial value problem integral sign intersects iterated spherical mean Let u(x linear elliptic differential linear elliptic equation linear elliptic system Math matrix neighborhood normal surface obtain origin partial differential equations plane wave plane wave function polynomial R₁ regular analytic respect right hand side satisfies sphere of radius spherical means system of differential tangential theorem vanishes variables vector wave equation x-derivatives Ωη