Rearrangements and Convexity of Level Sets in PDE
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1150 Bernhard Kawohl
Rearrangements and Convexity of Level Sets in PDE
Springer-verlag Berlin Heidelberg New York Tokyo
Author
Bernhard Kawohl Universitat Heidelberg, Sonderforschungsbereich 123 1m Neuenheimer Feld 294, 6900 Heidelberg, Federal Republic of Germany
Mathematics Subject Classification (1980): 26B25, 26010, 35A 15, 35B05, 35B50, 35J20, 35J25, 35J65, 49G05 ISBN 3-540-15693-3 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15693-3 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data. Kawohl, Bernhard, 1952- Rearrangements and some maximum principles in POE. (Lecture notes in mathematics; 1150) Bibliography: p. Includes index. 1. Differential equations, Partial. 2. Maximum principles (Mathematics) I. Title, II. Series: Lecture notes in mathematics (Springer-Verlag); 1150. QA3.L28 no, 1150 [QA374] 510 s [515.3'53] 85-20806 ISBN 0-387-15693-3 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 1985 Printed in Germany Printinq and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
dedicated to my family
CONTENTS
I. II.
III.
Introduction Rearrangements
7
1.
A catalogue of rearrangements
2.
Common properties of rearrangements
20
3.
Monotone decreasing and quasiconcave rearrangement
27
4.
Symmetric decreasing rearrangement
37
5.
Monotone decreasing rearrangement in direction
6.
Starshaped rearrangement
7.
Steiner symmetrization with respect to
8.
Schwarz symmetrization
89
9.
Circular and spherical symmetrization
94
7
y
45 62
{y
O}
82
Maximum principles
100
10.
The moving plane method
100
11.
Convexity of level sets
103
12.
Concavity or convexity of functions
112
References
123
Index of examples and assumptions
135
Subject index
136
I.
INTRODUCTION
These notes have their origin in a conjecture of J. Rauch. Lct a domain in
mn
and let
v
the Laplace operator in
2
be
Q
be the smallest positive eigenvalue of under homogeneous Neumann boundary condi-
tions. Then, so the conjecture, the associated eigenfunction attain its maximum and min i.mum on the boundary
dQ.
u
2
should
Hore on the con-
jecture can be found in § 5. The conjecture and attempts to prove it led the author of these notes to questions of the type: Suppose equation
= 0
D.U + f(u)
ical points of
u
solves the semilinear differential
in a domain
u, i.e.
Q
.
What is known about the crit-
the points in which
vanishes? What is known about the shape of Q
influence the shape of
dQ
? If
Q
u, if
is convex, does
symme t r i c , does
u
u
u
vu, the gradient of
u
L? How does the sha
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