Besov Spaces and Applications to Difference Methods for Initial Value Problems
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434 Philip Brenner Vidar Thornee Lars B. Wahlbin
Besov Spaces and Applications to Difference Methods for Initial Value Problems
Springer-Verlag Berlin· Heidelberg· New York 1975
Dr. Philip Brenner Prof. Vidar Thornee Department of Mathematics Chalmers University of Technology and University of Goteborq, Fack 8-40220 GOteborg 5/8weden Prof. Lars B. Wahlbin Department of Mathematics Cornell University White Hall Ithaca, NY 14850/U8A
Library of Congress Cataloging in Publication Data
Brenner, Philip, 1941Besoy spaces and applications to difference methods for initial value problems. (Lecture notes in mathematics ; 434) Includes bibliographies and index. 1. Differential equations, Partial. 2. Initial value problems. 3. Besov spaces. I. 'l'homee, Vidar, 1933joint author. II. Wahlbin, Lars BertH, 1945joint author. III. Title.
IV.
Series.
QAJ.L28 no. 434
[QAJ77,
510'.8s [515'.353,
74-32455
AMS Subject Classifications (1970): 35E15, 35L45, 42A18, 46E35, 65MlO,65M15 ISBN 3-540-07130-X Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-07130-XSpringer-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 the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1975. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE The purpose of these notes is to present certain Fourier techniques for analyzing finite difference approximations to initial value problems for linear partial differential equations with constant coefficients. In particular, we shall be concerned with stability and convergence estimates in the
L
norm of such approxima-
P
tions; the main theme is to determine the degree of approximation of different methods and the precise dependence of this degree upon the smoothness of the initial data as measured in
L In p'
L
2
the analysis generally depends on Parseval's rela-
tion and is simple; it is to overcome the difficulties present in order to obtain estimates in the maximum-norm, or more generally in
with
P
+2,
which is the
aa m of this study. The main tools which we shall use are some simple results on Fourier TIillltipliers based on inequalities by Carlson and Beurling and by van der Corput. Many results are Bs,q
expressed in terms of norms in Besov spaces the degree of smoothness with respect to
L
p
where
essentially describes
s
P
first two chapters contain the prerequisits on Fourier multipliers and on Besov spaces, respectively, needed for our applications. The purpose of these two chapters is only to make these notes self-contained and not to give an extensive treatment of their topics. Chapters 3 through 6 the
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