An Integral Inequality for Functions on a Cube
Let Q d be an open n-dimensional cube with edge length d and with sides parallel to coordinate axes. Let p ≥ 1, and k, l be integers, 0 ≤ k ≤ l. We denote a function in W l p (Q d ), p ≥ 1, by u.
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Sobolev Spaces
Vladimir G. Maz'ja
Sobolev Spaces Translated from the Russian by T. o. Saposnikova
With 25 Figures
Springer-Verlag Berlin Heidelberg GmbH
Professor Vladimir G. Maz'ja Leningrad University Faculty of Mathematics and Mechanics 198904 Leningrad, USSR
This volume is part of the Springer Series in Soviet Mathematics Advisers: L. D. Faddeev (Leningrad), R. V. Gamkrelidze (Moscow) Mathematics Subject Classification (1980): 46E35, 35J, 35P, 31B15, 26B ISBN 978-3-662-09924-7 ISBN 978-3-662-09922-3 (eBook) DOI 10.1007/978-3-662-09922-3 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 Originally published by Springer-Verlag Berlin Heidelberg New York in 1985. Softcover reprint of the hardcover I st edition 1985 Typesetting: K + V Fotosatz GmbH, Beerfelden. Offsetprinting: Mercedes-Druck, Berlin
2141/3020-543210
To Tatyana
Preface The Sobolev spaces, i.e. the classes of functions with derivatives in L p , occupy an outstanding place in analysis. During the last two decades 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 Sobolev space theory. Attention is paid mainly to the so called imbedding 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. We list some questions considered in this book. 1. What are the requirements on the measure f1, for the inequality
q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three Inequalities for Functions on (0,00) ........... Comments to § 1.3. ..............................
45 49 51
Imbedding Theorems of Sobolev Type ..................... 1.4.1. D. R. Adams' Theorem on Riesz Potentials. . . . . . . . . . 1.4.2. An Estimate for the Norm in Lq(R n, /l) by the Integral ofthe Modulus of the Gradient .................... 1.4.3. An Estimate for the Norm in Lq(R n, /l) by the Integral of the Modulus of the l-th Order Gradient ........... 1.4.4. Corollaries of Previous Results .................... 1.4.5. Generalized Sobolev Theorem. . . . . . . . . . . . . . . . . . . . . 1.4.6. Compactness Theorems .......................... 1.4.7. A Multiplicative Inequality ....................... 1.4.8. Comments to § 1.4. ..............................
51 51
57 59 60 62 65 68
§ 1. 5.
More on Extension of Functions in Sobolev Spaces .......... 1.5.1. Survey of Results and Examples of Domains ....... . . 1.5.2 Domains inEVb which are not Q
