Sobolev Spaces with Applications to Elliptic Partial Differential Eq
Sobolev spaces play an outstanding role in modern analysis, in particular, in the theory of partial differential equations and its applications in mathematical physics. They form an indispensable tool in approximation theory, spectral theory, differential
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Series editors M. Berger P. de la Harpe F. Hirzebruch N.J. Hitchin L. Hörmander A. Kupiainen G. Lebeau F.-H. Lin B.C. Ngô M. Ratner D. Serre Ya.G. Sinai N.J.A. Sloane A.M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan
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For further volumes: http://www.springer.com/series/138
Vladimir Maz’ya
Sobolev Spaces with Applications to Elliptic Partial Differential Equations 2nd, revised and augmented Edition
Professor Vladimir Maz’ya Department of Mathematics Sciences University of Liverpool Liverpool L69 7ZL, UK and Department of Mathematics Linköping University Linköping 581 83, Sweden [email protected]
The 1st edition, published in 1985 in English under Vladimir G. Maz’ja in the Springer Series of Soviet Mathematics was translated from Russian by Tatyana O. Shaposhnikova
ISSN 0072-7830 ISBN 978-3-642-15563-5 e-ISBN 978-3-642-15564-2 DOI 10.1007/978-3-642-15564-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011921122 Mathematics Subject Classification: 46E35, 42B37, 26D10 c Springer-Verlag Berlin Heidelberg 1985, 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Tatyana
Preface
Sobolev spaces, i.e., the classes of functions with derivatives in Lp , occupy an outstanding place in analysis. During the last half-century a substantial contribution to the study of these spaces has been made; so now solutions to many important problems connected with them are known. In the present monograph we consider various aspects of theory of Sobolev spaces in particular, the so-called embedding theorems. Such theorems, originally established by S.L. Sobolev in the 1930s, proved to be a useful tool in functional analysis and in the theory of linear and nonlinear partial differential equations. A part of this book first appeared in German as three booklets of TeubnerTexte f¨ ur Mathematik [552, 555]. In the Springer volume of “Sobolev Spaces” [556] published in 1985, the material was expanded and revised. As the years passed the area became immensely vast and underwent important changes, so the main contents of the 1985 volume had the pot
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