Integral Operators in Potential Theory
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823 Josef Kr,~l
Integral Operators in Potential Theory
Springer-Verlag Berlin Heidelberg New York 1980
Author Josef Kral Matematick~, Ostav ~itn& 25 11567 Praha 11 (~SSR
AMS Subject Classifications (1980): 31 B 10, 31 B 20, 35 J 05, 35 J 25, 45 B05, 45P05 ISBN 3-540-10227-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10227-2 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in Publication Data. Kr&l, Josef, DrSc. Integral operators in potential theory. (Lecture notes in mathematics; 823) Bibliography: p. Includes indexes.1. Potential,Theory of. 2. Integral operators. I. Title. II. Series: Lecture notes in mathematics (Berlin); 823. O~3.L28. no. 823. [QA404.?]. 510s. [515.7] 80-23501 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 § 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. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
CONTENTS
§I
§2 §3
§4
§5 §6
Introductory Weak
normal
Double
remark
.............................
derivatives
layer potentials
Contractivity
of p o t e n t i a l s
...........
.........................
of N e u m a n n ' s
operator
73
Boundary
value
problems
.........................
125
Comments
and references
.........................
150
Subject
index
.........
28
radius
index
operator
6
Fredholm
Symbol
of the N e u m a n n
.............
I
102
....................................
169
...................................
171
Introducto~j remark We shall be concerned with relations of analytic properties of classical potential theoretic operators to the geometry of the corresponding domain in the Euclidean m-space
Rm ,
m ~ 2 .
Let us recall that a function in an open set rentiable in
G CR m G
h
is termed harmonic
if it is twice continuously diffe-
and satisfies there the so-called Laplace
equation m
~h= where
~i
ih=0
i=I
,
denotes the partial derivative with respect to
the i-th variable.
(In fact, such a function
h
is necessa-
rily infinitely differentiable and even real-analytic; this is usually proved in elementary theory of harmonic functions on account of the Poisson integral which will be derived in the example following theorem 2.19.) If we try to determine a harmonic function in where
c~
rivative in
Rm ~ ( 0 )
of the form
h(x) = ~ ( [ x [ )
is an imknown fmnction with a continuous second de~0,+~°E C R I , we obtain an ordinary differential
equation d 2 ~ (r) dr 2
÷
m-1 d oo(r) r
dr
0
,
whose
solutions are
~(r) = where
~, p
Rm
in case
m > 2 ,
log r + #
in case
m = 2 ,
are a r b i t r a r y
Let us
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