By Fritz John (auth.)
The writer want to recognize his legal responsibility to all his (;Olleagues and neighbors on the Institute of Mathematical Sciences of latest York collage for his or her stimulation and feedback that have contributed to the writing of this tract. the writer additionally needs to thank Aughtum S. Howard for permission to incorporate effects from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for his or her cooperation and aid, and especially Lipman Bers, who steered the booklet in its current shape. New Rochelle FRITZ JOHN September, 1955 [v] CONTENTS creation. . . . . . . 1 bankruptcy I Decomposition of an Arbitrary functionality into airplane Waves rationalization of notation . . . . . . . . . . . . . . . 7 The round suggest of a functionality of a unmarried coordinate. 7 nine illustration of a functionality via its aircraft integrals . bankruptcy II Tbe preliminary worth challenge for Hyperbolic Homogeneous Equations with consistent Coefficients Hyperbolic equations. . . . . . . . . . . . . . . . . . . . . . 15 Geometry of the traditional floor for a strictly hyperbolic equation. sixteen answer of the Cauchy challenge for a strictly hyperbolic equation . 20 Expression of the kernel via an vital over the traditional floor. 23 The area of dependence . . . . . . . . . . . . . . . . . . . 29 The wave equation . . . . . . . . . . . . . . . . . . . . . . 32 The preliminary worth challenge for hyperbolic equations with a standard floor having a number of issues . . . . . . . . . . . . . . . . . . . . 36 bankruptcy III the basic answer of a Linear Elliptic Differential Equation witL Analytic Coefficients Definition of a primary answer . . . . . . . . . . . . . . forty three The Cauchy challenge . . . . . . . . . . . . . . . . . . . . . forty five answer of the inhomogeneous equation with a airplane wave functionality as correct hand aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . forty nine the elemental answer. . . . . . . . . . . . . . . . . . . . . .
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Extra info for Plane Waves and Spherical Means: Applied to Partial Differential Equations
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27) for r > 0. 31) e c: I y I <=6' I X I <=6' X -1- y --r-' l (x- y) · 'YJ > -. 22). 30) their values r = I x ~ y I, C = (x- y)/1 x- y I, and differentiate with respect to X;, we find for W x i an expression of the same form Wx 1 = rm+k-1 (A 1 (x, y, r, C)+ Bl(x, y, r, C) logr) where Al(x, y, r, C)= (m+k)AC; + rAx 1 + BC; + Ac1 - 2: Ac3iCk+rArC; B (x, y, r, C)= (m 1 + k)BC; +rEx,+ Bc k 1 - 2: BckC;Ck + rBrC;· k A and B have the same regularity properties as A and B. • 1 1 . 54 PLANE WAVES AND SPHERICAL MEA:::-
CHAPTER III The Fundamental Solution of a Linear Elliptic Differential Equation with Analytic Coefficients 29 Definition of a fundamental solution Denote by L[u] a differential operator of the form (:u) L[u] = m ~ k=O A. (x) '1·· ·'k il, .. ,ik ~ =l, ... ,n a aku X;l • • • a . X;k A function K = K(x, y) = K(xv . , Yv ... 2) L[JK(x, y) f(y)dyJ= f(x). 3) L[K(x, y)] = <5(x- y) where <5 denotes the so-called Dirac function. 9) that a fundamental solution is given by X y 12-n for n > 2 K(x, y) = (2- n)ro,.
In the special case under consideration g(p) {l _ p2)