Quadrature Domains
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934 Makoto Sakai
Quadrature Domains
Springer-Verlag Berlin Heidelberg New York 1982
Author
Makoto Sakai Department of Mathematics, Tokyo Metropolitan University Fukasawa, Setagaya, Tokyo, 158 Japan
AMS Subject Classifications (1980): 30 E 99, 31 A 99 ISBN 3-540-11562-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11562-5 Springer-Verlag New York Heidelberg 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 § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
CONTENTS INTRODUCTION CHAPTER I.
1 CONSTRUCTION OF QUADRATURE DOMAINS
§l.
Elementary properties and examples
4
§2.
Domains with quasi-smooth boundaries
8
§3.
Modifications of positive measures
18
§4.
Modifications under restrictions
33
§5.
Construction of quadrature domains for harmonic and analytic functions
CHAPTER II.
43
PROPERTIES OF QUADRATURE DOMAINS
§6.
Basic properties of quadrature domains
48
§7.
Existence of minimal quadrature domains
57
§8.
Relations between
l, domains for classes SL
HL 1 and AL 1 §9.
. 62
Uniqueness in the strict sense
65
§lO.
Monotone increasing families of quadrature domains
70
§ll.
Quadrature domains with infinite area
90
CHAPTER III.
APPLICATIONS
§12.
Analytic functions with finite Dirichlet integrals
100
§13.
Hele-Shaw flows with a free boundary
105
§14.
Quadrature formulas
113
BIBLIOGRAPHY
126
LIST OF SYMBOLS
129
INDEX
132
INTRODUCTION
The main purpose of this paper is to show the existence and uniqueness theorems on quadrature domains of positive measures. These can be considered a new type of "the sweeping-out principle" of measures and there are many their applications. Let v be a positive Borel measure on the complex plane [. For a domain
in [, we denote by
the class of all real
valued Borel measurable functions on
which are integrable with
respect to the two-dimensional Lebesgue measure m. a subclass of every domain
such that
Let
be
for every f E
E
and
containing
A nonempty domain
is called a quadrature domain of v for
class F if (1)
v is concentrated in
( 2)
f+dv < +00
for every f E
and
where
If -f E
r"
f'dv ;
and
for every f E
Let
J
fdm
max{f,O}.
for every f E Ifldv < +00
O.
namely,
then, from (2), we obtain
J
fdv =
J
fdm
be the class of all complex
valued analytic integrable functions and set Re
= {Re fl
f E
Then
and only if
satisfies (I) and, J[f\dv < +00 and Jfdv =
for every f E
is a quadrature domain of v for Re ALI if
Therefore let us call this domain a
fdID
2
quadrature domain for class ALI.
This is nothing but a
"classic
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