E-Book Content
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
522 Clifford O. Bloom Nicholas D. Kazarinoff
Short Wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions
Springer-Verlag Berlin.Heidelberg. New York 1976
Authors Clifford O. Bloom Nicholas D. Kazarinoff Department of Mathematics State University of New York at Buffalo Amherst, N. Y. 14226/USA
Library of Congress Cataloging in Publieation Data
Bloom, Clifford O 1935The asymptotic solution of h~h-frequency radiationscattering problems in inhomogeneous media. (Lecture notes in mathematics ; 522) Includes index. I. Radiation. 2. Scattering (Physics) 3. Asymptotic expansions. I. Kazarinoff, Nicholas D., joint author. II. Title: The asymptotic solution of highfrequency radiation-scattering problems ... III. Series: Lecture notes in mathematics (Berlin) ; 522. QA3.L28 no. 522 tQC~753 510'.8s E539'.2~ 76-17818
AMS Subject Classifications (1970): 35B40, 35B45, 35J05, 53C25, 78A05, 78A40
ISBN 3-540-0?698-0 Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-387-07698-0 Springer-Verlag New York 9 Heidelberg 9 Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
PREFACE
These notes are based upon a series of lectures given at the University of Oxford, Spring,
1975 by the second author.
The authors
tunity for their joint work to be presented
thank Dr. J. B. McLeod for the oppor-
in his seminar.
These notes are primarily concerned with existence, and the rigorous
asymptotic
Au+~2n(x)u
(p)
for
= f(x)
l
large, Here 5V C R m
V
estimates by a variation of ness of the solution approximate construct
u
solution
(x 6 ~v),
m-I 2 rlUr- i~,u+--~--r u dS = 0
In Chapter
K 9 Friedriehs'
abc-method.
of the boundary-value
(in p o w e r s of
there an approximate
on
5V .
problem
L2
These estimates (P)
l-i ) to the problem
of
u
We apply the a priori point-wise
expansion of the exact solution as high-frequency
and
above.
(P)
a priori
imply unique-
We construct
in Chapter 3.
an
We also
solution to the more. general radiation-scattering
in Chapter 3 that the approximate
amplitude
convex or star-shaped
I we obtain new point-wise
problem where the values of a linear combination prescribed
(Ixl = r)
is the exterior of a not necessarily
(m = 2 or 3) .
a priori estimates,
problem:
(x 6 V) ,
u = u0(x) lira ~ R-k= r=R
body
uniqueness,
solution of the radiation-scattering
and its normal derivative estimate of Chapter
solution to the problem I 4 =
(P)
This asymptotic
are
I to prove
is an asymptotic
approximation
yields a
asymptotic expansion of the leading term of the far field scattering
a(x,~)
, where a(x~)k) = lira (re -i~r u) .
In Chapter