TEUBNER-TEXTE zur Mathematik Band 137. V. I. Burenkov. Sobolev Spaces on Domains

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1 TEUBNER-TEXTE zur Mathematik Band 137 V. I. Burenkov Sobolev Spaces on Domains

2 TEUBNER-TEXTE zur Mathematik Herausgegeben von Prof. Dr. Jochen Bruning, Berlin Prof. Dr. Herbert Gajewski, Berlin Prof. Dr. Herbert Kurke, Berlin Prof. Dr. Hans Triebel, Jena Die Reihe soli ein Forum tor Beitrage zu aktuellen Problemstellungen der Mathematik sein. Besonderes Anliegen ist die Veroffentlichung von Darstellungen unterschiedlicher methodischer Ansatze, die das Wechselspiel zwischen Theorie und Anwendungen sowie zwischen Lehre und Forschung reflektieren. Thematische Schwerpunkte sind Analysis, Geometrie und Algebra. In den Texten sollen sich sowohl Lebendigkeit und Originalitat von Spezialvorlesungen und Seminaren als auch Diskussionsergebnisse aus Arbeitsgruppen widerspiegeln. TEUBNER-TEXTE erscheinen in deutscher oder englischer Sprache.

3 Sobolev Spaces on Domains By Prof. Dr. Victor I. Burenkov University of Wales, Cardiff Springer Fachmedien Wiesbaden GmbH 1998

4 Prof. Dr. Victor I. Burenkov Born in Russia (Yaroslavl region) in Studied mathematics and physics in Moscow Institute of Physics and Technology, gained Ph. D. in Mathematics in 1967 and D. Sc. in Mathematics in 1982 (both in Steklov Institute of Mathematics, Moscow). From 1984 to 1994 Professor of the Department of Differential Equations and Functional Analysis at the Peoples' Friendship University of Russia (Moscow). Since 1994 Professor of the School of Mathematics at the University of Wales, Cardiff. More than 100 publications. Fields of research: real and functional analysis, partial differential equations, especially the theory of function spaces (Sobolev spaces, spaces with fractional order of smoothness) and applications. Gedruckt auf chlorirei gebleichtem Papier. Die Deutsche Bibliothek- CIP-Einheitsaufnahme Burenkov, Viktor 1.: Sobolev spaces on domains I Viktor I. Burenkov. (Teubner-Texte zur Mathematik ; Bd. 137) ISBN ISBN (ebook) DOI / Das Werk einschlief31ich aller seiner Teile ist urheberrechtlich geschotzt. Jede Verwertung auf3erhalb der eng en Grenzen das Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzulassig und strafbar. Das gilt besonders fur Vervielfaltigungen, Obersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Springer Fachmedien Wiesbaden 1998 Originally published by B.G. Teubner Verlagsgesellschaft Leipzig in 1998 Umschlaggestaltung: E. Kretschmer, Leipzig

5 Preface The book is based on the lecture course "Function spaces", which the author gave for more than 10 years in the People's Friendship University of Russia (Moscow). The idea to write this book was proposed by Professors H. Triebel and H.-J. SchmeiBer in May-June 1993, when the author gave a short lecture course for post-graduate students in the Friedrich-Schiller University Jena. The initial plan to write a short book for post-graduate students was transformed to wider aims after the work on the book had started. Finally, the book is intended both for graduate and post-graduate students and for researchers, who are interested in applying the theory of Sobolev spaces. Moreover, the methods used in the book allow us to include, in a natural way, some recent results, which have been published only in journals. Nowadays there exist numerous variants and generalizations of Sobolev spaces and it is clear that this variety is inevitable since different problems in real analysis and partial differential equations give rise to different spaces of Sobolev type. However, it is more or less clear that an attempt to develop a theory, which includes all these spaces, would not be effective. On the other hand, the basic ideas of the investigation of such spaces have very much in common. For all these reasons we restrict ourselves to the study of Sobolev spaces themselves. However, we aim to discuss the main ideas in detail, and in such a way that, we hope, it will be clear how to apply them to other types of Sobolev spaces. We shall discuss the following main topics: approximation by smooth functions, integral representations, embedding and compactness theorems, the problem of traces and extension theorems. The basic tools of investigation will be mollifiers with a variable step and integral representations. Mollifiers with variable step are used both for approximation by smooth functions and for extension of functions (from open sets in IRn in Chapter 6 and from manifolds of lower dimensions in Chapter 5). All approximation and extension operators constructed in these chapters are the best possible in

6 6 PREFACE the sense that the derivatives of higher orders of approximating and extending functions have the minimal possible growth on approaching the boundary. Sobolev's integral representation is discussed in detail in Chaper 3. It is used in the proofs of the embedding theorems (Chapter 4) and some essential estimates in Chapter 6. An alternative proof of the embedding theorems, without application of Sobolev's integral representation, is also given. The direct trace theorems (Chapter 5) are proved on the basis of some elementary identities for the differences of higher orders and the definition of Nikol'ski'f-Besov spaces in terms of differences only. The author pays particular attention to all possible 'limiting" cases, includiug the cases f1 = x in approximation theorems, p = 1 in embedding theorems and The 1, oc in extension theorems. There are no references to the literature in the main text (Chapters 1-6): all relevant references are to be found in Chapter 7, which consists of brief notes and comments on the results presented in the earlier chapters. The proofs of all statements in the book consist of two parts: the idea of the proof and the proof itself. In some simple or less important cases the proofs are omitted. On the other hand, the proofs of the main results are given in full detail and sometimes alternative proofs are also given or at least discussed. The one-dimensional case is often discussed separately to provide a better understanding of the origin of multi-dimensional statements. Also sharper results for this case are presented. It is expected that the reader has a sound basic knowledge of functional analysis, the theory of Lebesgue integration and the main properties of the spaces Lp(fl). It is desirable, in particular, that he/she is accustomed to applying Holder's and Minkowski's inequalities for sums and integrals. The book is otherwise self-contained: all necessary references are given in the text or footnotes. Each chapter has its own numeration of theorems, corollaries, lemmas, etc. If you are reading, say, Chapter 4 and Theorem 2 is mentioned, then Theorem 2 of Chapter 4 is meant. If we refer to a theorem in another chapter, we give the number of that chapter, say, Theorem 2 of Chapter 3. For more than 30 years the author participated in the famous seminar "The theory of differentiable functions of several variables and applications" in the Stcklov Institute of Mathematics (Moscow) headed at different times by Professors S.L. Sobolev, V.I. Kondrashov, S.M. Nikol'ski'f, L.D. Kudryavtsev and O.V. Besov. He was much influenced by ideas discussed during its work and, in particular, by his personal talks with Professors S.M. Nikol'skil and S.L. Sobolev. It is a pleasure for the author to express his deepest gratitude to the partie-

7 PREFACE 7 ipants of that seminar, to his friends and co-authors, with whom he discussed the general plan and different parts of the book. I am grateful to my colleagues in the University of Wales Cardiff: Professor W.D. Evans, with whom I have had many discussions, and Mr. D.J. Harris. who has thoroughly read the manuscript of the book. I would also like to mention Dr. A.V. Kulakov who has actively helped in typing the book in 'I"&'<. Finally, I express my deepest love, respect and gratitude to my wife Dr. T.V. Tararykova who not only typed in TEX a considerable part of the book but also encouraged me in all possible ways. Moscow /Cardiff, ;.;ovember 1997 V.I. Burenko,

8 Contents Notation and basic inequalities 1 Preliminaries 1.1 :\Iollifiers Weak derivatives Sobolev spaces (basic properties) 2 Approximation by infinitely differentiable functions 201 Approximation by C0-functions on Rn Nonlinear mollifiers with variable step Approximation by C 00 -functions on open sets l Approximation with preservation of boundary values 205 Linear mollifiers with variable step The best possible approximation with preservation of boundary values Sobolev's integral representation 301 The one-dimensional case Star-shaped sets and sets satisfying the cone condition 303 Multidimensional Taylor's formula Sobolev's integral representation Corollaries Embedding theorems 401 Embeddings and inequalities The one-dimensional case Open sets with quasi-resolvable, quasi-continuous, smooth and Lipschitz boundaries Estimates for intermediate derivatives Hardy-Littlewood-Sobolev inequality for integral of potential type 177

9 10 CONTENTS 4.6 Embeddings into the space of continuous functions 4. 7 Embeddings into the space Lq Trace theorems 5.1 Notion of the trace of a function. 5.2 Existence of the traces on subspaces. 5.3 Nikol'ski'l-Besov spaces Description of the traces on subspaces 5.5 Traces on smooth surfaces Extension theorems 6.1 The one-dimensional case. 6.2 Pasting local extensions Extensions for sufficiently smooth boundaries 6A Extensions for Lipschitz boundaries Comments Bibliography Index l 269 2/

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