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Plane Waves and Spherical Means

Applied to Partial Differential Equations

  • Textbook
  • © 1981

Overview

Authors:
  1. Fritz John

    1. Courant Institute of Mathematical Sciences, New York University, New York, USA

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-viii
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  2. Introduction

    • Fritz John
    Pages 1-5

  3. Decomposition of an Arbitrary Function into Plane Waves

    • Fritz John
    Pages 7-13

  4. The Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients

    • Fritz John
    Pages 15-41

  5. The Fundamental Solution of a Linear Elliptic Differential Equation with Analytic Coefficients

    • Fritz John
    Pages 43-76

  6. Identities for Spherical Means

    • Fritz John
    Pages 77-89

  7. The Theorems of Asgeirsson and Howard

    • Fritz John
    Pages 91-108

  8. Determination of a Function from Its Integrals over Spheres of a Fixed Radius

    • Fritz John
    Pages 109-125

  9. Differentiability Properties of Solutions of Elliptic Systems

    • Fritz John
    Pages 127-156

  10. Regularity Properties for Integrals of Solutions over Time-like Lines

    • Fritz John
    Pages 157-164

  11. Back Matter

    Pages 165-172
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About this book

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 The author also wishes to thank Aughtum S. Howard for permission to include results from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for their cooperation and support, and particularly Lipman Bers, who suggested the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. . . . . . . . . . . . . . . . . . . . . .

Authors and Affiliations

  • Courant Institute of Mathematical Sciences, New York University, New York, USA

    Fritz John

Bibliographic Information

  • Book Title: Plane Waves and Spherical Means

  • Book Subtitle: Applied to Partial Differential Equations

  • Authors: Fritz John

  • DOI: https://doi.org/10.1007/978-1-4613-9453-2

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1981

  • Softcover ISBN: 978-0-387-90565-5Published: 17 April 1981

  • eBook ISBN: 978-1-4613-9453-2Published: 01 December 2013

  • Edition Number: 1

  • Number of Pages: VIII, 172

  • Additional Information: Originally published by Interscience Publishers Inc., New York 1955

  • Topics: Analysis, Mathematical Methods in Physics, Numerical and Computational Physics, Simulation

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