v. g. maz'ja sobolev spaces978-3-662-09922-3/1.pdf · in the present monograph we consider...

V. G. Maz'ja 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

<J I u Iq df1,)I/q ~ ell u Iisl , p

where S~ is the Sobolev space or its generalization, to hold? 2. What are the minimal assumptions on the domain for the Sobolev im

bedding theorem to remain valid? How do these theorems vary under the degeneration of the boundaries? How does the class of admissible domains depend on additional requirements placed upon the behavior of a function near the boundary?

3. How "massive" must a subset e of the domain Q be in order that "the Friedrichs inequality"

II U IlL (0) ~ c II \l[u IlL (0) q p

hold for all smooth functions that vanish in a neighborhood of e? The investigation of these and similar problems is not only of interest in its

own right. By virtue of well-known general considerations it leads to conditions for the solvability of boundary value problems for elliptic equations and to theorems on the structure of the spectrum of the corresponding operators. Such applications are also included.

The selection of topics was mainly influenced by my involvement in their study, so a considerable part of the text is a report of my work in the field.

The book has no essential intersection with the monographs by S. M. Nikol'skfi [202], Besov, Il'in, Nikol'skfi [27], R.A. Adams [12], Peetre [210] and Triebel [244, 245], which were published during the last decade and are also devoted to spaces of differentiable functions.

VIII Preface

Each of the twelve chapters of the book is divided into sections and most of the latter consist of subsections. The sections and subsections are numbered by two and three numbers, respectively (3.1 is Section 1 in Chapter 3, 1.4.3 is Subsection 3 in Section 4 in Chapter 1). Inside subsections we use an independent numbering of theorems, lemmas, propositions, corollaries, remarks, etc. H a subsection contains only one theorem or lemma then this theorem or lemma has no number. In references to the material from another section or subsection we first indicate the number of this section or subsection. For example, Theorem 1.2.1/1 means Theorem 1 in Subsection 1.2.1, (2.712) denotes formula (2) in Section 2.7.

The reader can obtain a general idea of the contents of the book from the Introduction. Most of the references to the literature are collected in Comments. The list of notation is given at the end of the book.

A part of this monograph was published in German in three volumes of Teubner-Texte zur Mathematik, Leipzig (Einbettungssatze fUr Sobolewsche Raume, Teil1, 1979; Teil2, 1980; Zur Theorie Sobolewscher Raume, 1981). In the present volume the material is essentially expanded and revised.

Acknowledgements

I gratefully remember my talented pupil A. L. Rosin who perished in 1976 in a mountain ascent and with whom I repeatedly discussed the project of this book.

I wish to express my gratitude to Ju. D. Burago who in spite of being busy with writing his own book found time for the joint work on Chapter 6.

I take pleasure in acknowledging the friendly help of S. P. Preobrazenskil and S. V. Poborcil for their contributions to the improvement of the text.

My cordial thanks are due Leo F. Boron who kindly agreed to prepare the manuscript for publication and reduced the quantity of obscurities and slips.

It is my pleasant duty to thank Springer-Verlag for the excellent quality of this edition.

The dedication of this book to its translator and my wife Dr. T.O. Saposnikova is a weak expression of my gratitude for her infinite patience, useful advice and constant assistance.

Leningrad, autumn 1985 V.G. Maz'ja

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1. Basic Properties of Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . 6

§ 1.1. The Spaces Lb(.Q), Vb(.Q) and wb(.Q) ..................... 6 1.1.1. Notation....................................... 6 1.1.2. Local Properties of Elements in the Space L b(.Q) ..... 7 1.1.3. Absolute Continuity of Functions inL b(.Q) . . . . . . . . . . 8 1.1.4. The Spaces Wb(.Q) and vb (.Q) .................... 10 1.1.5. Approximation of Functions in Sobolev Spaces by

Smooth Functions on.Q .......................... 11 1.1.6. Approximation of Functions in Sobolev Spaces by

Functions in COO(Q) ............................. 13 1.1.7. Transformation of Coordinates in Norms of Sobolev

Spaces ......................................... 15 1.1.8. Domains Starshaped with Respect to a Ball... ....... 17 1.1.9. Domains of the Class Co. lor Domains Having the

Cone Property .................................. 18 1.1.10. The Sobolev Integral Representation ............... 20 1.1.11. Generalized Poincare Inequality ................... 22 1.1.12. Completen~s/s of Wb(.Q) and Vb(.Q) ................ 24 1.1.13. The Space L p(.Q) and its Completeness ............. 24 1.1.14. Dual of a Sobolev Space. . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1.15. Equivalent Norms in Wb(.Q) ...................... 26 1.1.16. Extension of Functions in Vb(.Q) onto R n ••......... 27 1.1.17. Comments to § 1.1. .............................. 29

§ 1.2. Some Facts from Set Theory and Function Theory . . . . . . . . . . . 30 1.2.1. Two Theorems on Coverings ...................... 30 1.2.2. Theorem on Level Sets of a Smooth Function ........ 33 1.2.3 Representation of the Lebesgue Integral as a Riemann

Integral Along a Halfaxis ......................... 36 1.2.4. A Formula for the Integral of the Modulus of the

Gradient ....................................... 37 1.2.5. Comments to § 1.2. .............................. 39

§ 1.3. Some Inequalities for Functions of One Variable ............ 39 1.3.1 The Case p :s;; q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

x Contents

1.3.2. The Case p > q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.3.3. Three Inequalities for Functions on (0,00) ........... 49 1.3.4. Comments to § 1.3. .............................. 51

§ 1.4. Imbedding Theorems of Sobolev Type ..................... 51 1.4.1. D. R. Adams' Theorem on Riesz Potentials. . . . . . . . . . 51 1.4.2. An Estimate for the Norm in Lq(R n, /l) by the Integral

ofthe Modulus of the Gradient .................... 54 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 ........... 57 1.4.4. Corollaries of Previous Results .................... 59 1.4.5. Generalized Sobolev Theorem. . . . . . . . . . . . . . . . . . . . . 60 1.4.6. Compactness Theorems .......................... 62 1.4.7. A Multiplicative Inequality ....................... 65 1.4.8. Comments to § 1.4. .............................. 68

§ 1. 5. More on Extension of Functions in Sobolev Spaces .......... 70 1.5.1. Survey of Results and Examples of Domains ....... . . 70 1.5.2 Domains inEVb which are not Quasidisks ..... . . . . . . 73 1.5.3. Extension with Zero Boundary Data. . . . . . . . . . . . . . . . 77

§ 1.6. Inequalities for Functions that Vanish on the Boundary along with their Derivatives up to Some Order . . . . . . . . . . . . . . . . . . . . 79 1.6.1. Integral Representation for Functions of One

Independent Variable ............................ 79 1.6.2. Integral Representation for Functions of Several

Variables with Zero Boundary Data ................ 80 1.6.3. Imbedding Theorems for Functions that Satisfy Zero

Boundary Conditions ............................ 82 1.6.4. The Necessity of the Condition I :s;; 2k . . . . . . . . . . . . . . . 84

Chapter 2. Inequalities for Gradients of Functions that Vanish on the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

§ 2.1. Conditions for the Validity of Certain Integral Inequalities (The Case p = 1) ....................................... 88 2.1.1. A Condition in Terms of Arbitrary Admissible Sets ... 88 2.1.2 A Particular Case (The Condition in Terms of Balls) .. 90 2.1.3. One More Inequality Containing the Norms in Lq(Q, /l)

andLr(Q,v)(TheCasep=1) ..................... 91 2.1.4. Two Examples of Inequalities Containing Concrete

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.1.5. The Case of a Weighted Norm in the Right-Hand Side. 92 2.1.6. Inequalities of Hardy-Sobolev Type as Corollaries of

Theorem 2.1.5/1 ................................ 96

§ 2.2. On the (p, IP)-capacity .................................. 99 2.2.1. Definition and Some Properties of the (p, IP )-capacity . 99

Contents XI

2.2.2. Expression for the (p, «P)-Capacity Containing an Integral over Level Surfaces . . . . . . . . . . . . . . . . . . . . . . . 102

2.2.3. Lower Estimates for the (p, «P)-Capacity ............ 104 2.2.4. On the p-Capacity of a Ball ....................... 106 2.2.5. On the (p, «P)-Capacity for p = 1 . . . . . . . . . . . . . . . . . . . 107

§ 2.3. Conditions for the Validity of Integral Inequalities (The Case p ~ 1) ....................................... 107 2.3.1. Estimate for the Integral Containing the

(p, «P )-Capacity of the Set A"t . . . . . . . . . . . . . . . . . . . . . . 1 07 2.3.2. An Estimate for the Norm in Orlicz Space ........... 111 2.3.3. The Sobolev Type Inequalities as Corollaries of

Theorem2.3.2 .................................. 113 2.3.4. A Multiplicative Inequality (The Casep ~ 1) ......... 117 2.3.5. An Estimate for the Norm in Lq(Q, fJ) with q <p

(The Necessary and Sufficient Condition) ........... 120 2.3.6. An Estimate for the Norm in Lq(Q, fJ) with q <p

(Sufficient Condition) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 2.3.7. An Inequality with the Norms in Lq(Q, fJ) and Lr(Q, v)

(The Case p ~ 1) ................................ 124 2.3.8. An Estimate for the Integral S I u 1

2dO' with a Charge 0'. 125 Q

2.3.9. A Multiplicative Inequality with the Norms in Lq(Q, fJ) andLr(Q,v)(TheCasep~1) ..................... 127

§ 2.4. Continuity and Compactness of Imbedding Operators of i ~(Q) and W~(Q) into the Orlicz Space. . . . . . . . . . . . . . . . . . . . . . . . . . 128 2.4.1. Conditions for Boundedness of Imbedding Operators 128 2.4.2. Criteria for Compactness ......................... 130

§ 2.5. Application to the Spectral Theory of the Multi-dimensional Schr6dinger Operator ................................... 134 2.5.1. Preliminaries and Notation ....................... 134 2.5.2. Positiveness of the Form 81 [u, u] .................. 136 2.5.3. Semiboundedness ofthe Schr6dinger Operator. . . . . . . 137 2.5.4. Discreteness of the Negative Spectrum .............. 140 2.5.5. Discreteness of the Negative Spectrum of the Operator

Sh for all h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 2.5.6. Finiteness ofthe Negative Spectrum ................ 143 2.5.7. Infiniteness and Finiteness of the Negative Spectrum of

the Operator Sh for all h .......................... 146 2.5.8. Proofs of Lemmas 2.5.1 and 2.5.2 ................. 147

§ 2.6. On a Degenerate Quadratic Form ......................... 149

§ 2.7. On the Completion in the Metric of a Generalized Dirichlet Integral ............................................... 152

§ 2.8. Comments to Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

XII Contents

Chapter 3. On Summability of Functions in the SpaceL ~ (n) 160

§ 3.1. Preliminaries .......................................... 160 3.1.1. Notation....................................... 160 3.1.2. Lemmas on Approximation of Functions in Wp1 ,(.Q)

1 ' and Lp(Q) ..................................... 161

§ 3.2. Classes of Sets fa and the Imbedding L f(.Q) C Lq(.Q) ........ 162 3.2.1 Classes fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3.2.2. A Technical Lemma ............................. 165 3.2.3. The ImbeddingLf(Q) C Lq(.Q) ............ ... ..... 167 3.2.4. The Function AM ................................ 170 3.2.5. An Example ofa Domain infl .................... 171

§ 3.3. Subareal Mappings and the Classes fa. . . . . . . . . . . . . . . . . . . . . 172 3.3.1. SubareaIMappings.............................. 172 3.3.2. An Estimate for the Function A in Terms of Subareal

Mappings ...................................... 174 3.3.3. Estimates for the Function A for Concrete Domains. . . 175

§ 3.4. Two-sided Estimates for the Function A for the Domain in Nikodym's Example .................................... 179

§ 3.5. Compactnessofthe.ImbeddingLf(Q) CLq(.Q)(q~1) ....... 182 3.5.1. TheClassfa .................................... 182 3.5.2. A Criterion for Compactness. . . . . . . . . . . . . . . . . . . . . . 183

§ 3.6. The Imbedding wL,(Q, 8Q) C Lq(.Q) ..................... 185 3.6.1. The Class Jra,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 3.6.2. Examples of Sets in Jra,p ......................... 185 3.6.3. On the Continuity of the Imbedding Operator

wL,(Q, 8Q) -+Lq(Q) ... .......... ............. .. 187


Chapter 4. On Summability of Functions in the SpaceL;(n) ........ 191

§ 4.1. Conductivity .......................................... 191 4.1.1. The Equivalence of Certain Definitions of

Conductivity ................................... 191 4.1.2. Some Properties of Conductivity. . . . . . . . . . . . . . . . . . . 193 4.1.3. The Dirichlet Principle with Prescribed Level Surfaces

and its Corollaries ............................... 194

§ 4.2. A Multiplicative Inequality for Functions that Vanish on a Subset of Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

§ 4.3. The Classes of Sets fp, a ................................. 198 4.3.1. The Definition and Simple Properties of fp, a .••..... 198 4.3.2. IdentityoftheClassesf1,aandfa................. 199

Contents XIII

4.3.3. A Necessary and Sufficient Condition for the Validity of a Multiplicative Inequality for Functions in W~,s(Q) 200

4.3.4. The Function VM,p and the Relationship of the Classes Jp,aand7a'" .... ......... .... ... ..... ...... ... 202

4.3.5. Estimates for VM,p for Certain Concrete Domains. . . . . 203

§ 4.4. The Imbedding W~,s(Q) C Lq.(Q) for q * < p . . . . . . . . . . . . . . . 207 4.4.1. An Estimate for the Norm in Lq.(Q) with q* <p for

Functions that Vanish on a Subset of Q ............. 207 4.4.2. The Class Jt'p, a and the Imbedding W~,S<Q) C Lq.(Q)

for q* <p ...................................... 209 4.4.3. The Imbedding L~(Q) C Lq.(Q) for a Domain with

Finite Volume .................................. 210 4.4.4. Sufficient Conditions for Belonging to Jt'p, a ......... 211 4.4.5. Necessary Conditions for Belonging to the Classes

J p, a' Jt'p, a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 214 4.4.6. Examples of Domains in Jt'p, a ..................... 215 4.4.7. Other Descriptions of the Classes J p, a and Jt'p, a .. . . . . 217 4.4.8. Integral Inequalities for Domains with Power Cusps .. 218

§ 4.5. More on the Nikodym Example. . . . . . . . . . . . . . . . . . . . . . . . . . . 220

§ 4.6. Some Generalizations ................................... 228

§ 4.7. The Inclusion W~,r(Q) C Lq(Q) (r> q) for Domains with Infinite Volume ........................................ 232

00 00

4.7.1. The Classes .% and! ~,a . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 4.7.2. The Imbedding W p, r(Q) C Lq(Q) (r > q) ........... 235

00

4.7.3. Example of.a Domain in the Class ~,a""""""" 237 4.7.4. The Space L ~(Q) and Its Imbedding into Lq(Q) ...... 238 4.7.5. On the Poincare Inequality for Domains with Infinite

Volume........................................ 239

§ 4.8. On the Compactness <;>f the !mbedding L ~(Q) C Lq(Q) ....... 242 4.8.1. The Classes J p, a, Jt'p, a ........................... 243 4.8.2. A Criterion for Compactness ...................... 243 4.8.3. Sufficient Conditions for Compactness of the

Imbedding L ~(Q) C Lq.(Q) ........ . . . . . . . . . . . . . . . 244 4.8.4. A Compactness Theorem for an Arbitrary Domain

with Finite Volume .............................. 245 4.8.5. Examples of Domains in the Class Jp,a ............. 246

§ 4.9. On the ImbeddingL~(Q) C Lq(Q) ........................ 247

§ 4.10. Application to the Neumann Problem for Strongly Elliptic Operators ............................................. 249 4.10.1. Second Order Operators. . . . . . . . . . . . . . . . . . . . . . . . . . 249 4.10.2. The Neumann Problem for Operators of Arbitrary

Order.......................................... 250

XIV Contents

4.10.3. The Neumann Problem for a Special Domain. . . . . . . . 253 4.10.4. A Counterexample to Inequality (4.10.214) .......... 257

§ 4.11. Inequalities Containing Integrals over the Boundary ......... 257 4.11.1. The Imbedding W;.r(Q, aQ) C Lq(Q) .............. 257 4.11.2. TheClasses..f~~~l,f<:-l) ........................ 260 4.11.3. ExamplesofDomainsin..f~~~l),f<:-l) ............. 261 4.11.4. Estimates for the Norm in Lq(aQ) ................. 262 4.11.5. The Class j~~~l) and Theorems on Compactness ..... 264 4.11.6. Application to Boundary Value Problems for Second

Order Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 266


Chapter 5. On Continuity and Boundedness of Functions in Sobolev Spaces............ .... ............... .............. 270

§ 5.1. On the Imbedding W~(Q) C C(Q) nLoo(Q) .. . . . . . . . . . . . . . . 270 5.1.1. Criteria for the Continuity of the Imbedding Operators

of W~(Q) and L ~(Q) into C(Q) n Loo(Q) ... . . . . . . .. 270 5.1.2. A Sufficient Condition in Terms of AM for the

Imbedding W~(Q) C C(Q) n Loo(Q) ............... 273

§ 5.2. On a Multiplicative Estimate for the Modulus of a Function in W~(Q) ............................................... 274 5.2.1. Conditions for the Validity of a Multiplicative

Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 5.2.2. On the Multiplicative Inequality in the Limit Case

r = (p- n)/n ................................... 276

§ 5.3. OntheModulusofContinuityofFunctionsinL~(Q)........ 279 § 5.4. On the Boundedness of Functions with Derivatives in Orlicz

Classes ............................................... 281

§ 5.5. On the Compactness of the Imbedding W~(Q) C C(Q) n Loo(Q) .................... . . . . . . . . . . . . 284 5.5.1. A Criterion for Compactness. . . . . . . . . . . . . . . . . . . . . . 284 5.5.2. A Sufficient Condition for Compactness in Terms of

the Function AM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 5.5.3. A Domain for Which the Imbedding Operator of

W~(Q) into C(Q) n Loo(Q) is Bounded but not Compact ....................................... 286

§ 5.6. Generalizations to Sobolev Spaces of an Arbitrary Integer Order................................................. 288 5.6.1. On the (p, I)-conductivity. . . . . . . . . . . . . . . . . . . . . . . .. 288 5.6.2. TheImbeddingL~(Q) C C(Q) nLoo(Q) ............ 289 5.6.3. The Imbedding V1(Q) C C(Q) nLoo(Q) .... ........ 290 5.6.4. The Compactness of the Imbedding

L 1(Q) C C(Q) n Loo(Q) ......................... 291

Contents

5.6.5. Sufficient Conditions for the Continuity and the Compactness of the Imbedding

xv

L~(Q) C C(Q) nL",,(Q) ......................... 292 5.6.6. On Imbedding Operators for the Space

I • k Wp(Q) n Wp(Q). I> 2k ......................... 293

Chapter6. On Functions in the SpaceBV(n) ...................... 296

§ 6.1. Properties ofthe Set Perimeter and of Functions in B V(Q) . . . . 296 6.1.1. Definitions of the Space B V(Q) and of the Relative

Perimeter ...................................... 296 6.1.2. Approximation of Functions in B V(Q) ............. 297 6.1.3. Approximation of Sets with Finite Perimeter. . . . . . . .. 299 6.1.4. Compactness of the Family of Sets with Uniformly

Bounded Relative Perimeters .. . . . . . . . . . . . . . . . . . . . . 300 6.1.5. Isoperimetric Inequality .......................... 301 6.1.6. An Integral Formula for the Norm in B V(Q) ........ 301 6.1.7. On the Imbedding BV(Q) C LiQ) ... ....... ... .... 302

§ 6.2. The Gauss-Green Formula for Lipschitz Functions .......... 303 6.2.1. The Normal in the Sense of Federer and the Reduced

Boundary ...................................... 303 6.2.2. The Gauss-Green Formula ........................ 304 6.2.3. Several Auxiliary Assertions ...................... 304 6.2.4. The Study ofthe Set N ........................... 306 6.2.5. The Relations Betweenvar \i'Xc and s on 8t ......... 309

§ 6.3. On the Extension of Functions in B V(Q) onto the Whole Space 313 6.3.1. Proof ofthe Necessity of Condition (6.3/2) . . . . . . . . . . 314 6.3.2. ThreeLemmasonPc.a(t) ........................ 314 6.3.3. Proof ofthe Sufficiency of Condition (6.312) ........ 316 6.3.4. An Equivalent Statement of Theorem 6.3 ........... 318 6.3.5. One More Extension Theorem. . . . . . . . . . . . . . .. . . . . . 319

§ 6.4. Certain Exact Constants for Convex Domains .............. 320 6.4.1. Lemmas on Approximation by Polyhedra ........... 320 6.4.2. On a Property of Pc.a ............................ 322 6.4.3. An Expression for the Set Function r.a(t) for a Convex

Domain Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 6.4.4. The Function IQ I for a Convex Domain. . . .. . .. . . .. . 323

§ 6.5. The Rough Trace of Functions in BV(Q) and Certain Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 6.5.1. The Definition of the Rough Trace and its Properties. . 325 6.5.2. On Summability of the Rough Trace. . . . . . . . . . . . . . . . 326 6.5.3. Exact Constants in Certain Integral Estimates for the

Rough Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 327 6.5.4. More on Summability of the Rough Trace ........... 329

XVI Contents

6.5.5. Extension of a Function in BV(.Q) to CQby a Constant 330 6.5.6. Multiplicative Estimates for the Rough Trace ........ 331 6.5.7. An Estimate for the Norm in L nl(n-l)(Q) of a Function

in B V(Q) with Summable Rought Trace ............ 333

§ 6.6. Traces of Functions in B V(Q) on the Boundary and the Gauss-Green Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 6.6.1. The Definition of the Trace ....................... 334 6.6.2. Coincidence of the Trace and the Rough Trace ....... 335 6.6.3. The Trace of the Characteristic Function ............ 338 6.6.4. On summability of the Trace of a Function in B V(Q). 338 6.6.5. The Gauss-Green Formula for Functions in B V(.Q) ... 339


Chapter 7. Certain Function Spaces, Capacities and Potentials . . . . . . . 342

§ 7.1. The Spaces of Functions Differentiable of Arbitrary Positive Order................................................. 342 7.1.1. The Spaces w~, W~, b~, B~ for I > 0 ............... 342 7.1.2. The Riesz and Bessel Potential Spaces .............. 346 7.1.3. Some Other Properties of the Introduced Function

Spaces.... . .... ................................ 349

§ 7.2. Some Facts from Potential Theory ........................ 351 7.2.1. The Capacity cap(e, S~) and Its Properties .......... 351 7.2.2. Nonlinear Potentials ............................. 353 7.2.3. Metric Properties of the Capacity .................. 356 7.2.4. Refined Functions ............................... 358

Chapter 8. On Summability with Respect to an Arbitrary Measure of Functions with Fractional Derivatives .................. 360

§ 8.1. Description of Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

§ 8.2. An Estimate for the Integral of the Capacity of a Set Bounded by a Level Surface ...................................... 363 8.2.1. The Case of Second Order Derivatives .............. 363 8.2.2. A Proof Based on the Smooth Truncation of a

Potential Near its Level Surfaces ...... . . . . . . . . . . . . . 366 8.2.3. A Proof Based on the Maximum Principle for

Nonlinear Potentials ............................. 368

§ 8.3. Conditions for the Validity ofImbedding Theorems in Terms of Isoperimetric Inequalities .................. . . . . . . . . . . . . . . 370

§ 8.4. The Imbedding into Lq(u) for p > q > 0 . . . . . . . . . . . . . . . . . . . . 372 8.4.1. An Auxiliary Estimate ........................... 372 8.4.2. The Main Theorem .............................. 373

Contents XVII

8.4.3. A Sufficient Condition (p > q > 0) ................. 375 8.4.4. Two Simple Cases ............................... 376

§ 8.5. A Cartan Type Theorem and Estimates for Capacities. . . . . . . . 377

§ 8.6. Imbedding Theorems (Conditions in Terms of Balls) ......... 381

§ 8.7. Imbedding Theorems for p = 1 ........................... 384

§ 8.8. Applications........................................... 386 8.8.1. Criteria for Compactness ......................... 386 8.8.2. Applications to the Theory of Elliptic Operators. . . . . . 387

§ 8.9. Comments to Chapter 8 ................................. 388

Chapter 9. A Variant of Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

§ 9.1. The Capacity Cap ...................................... 390 9.1.1. SimplePropertiesoftheCapacityCap(e,i~(.Q)) ..... 390 9.1.2. The Capacity of a Continuum ..................... 392 9.1.3. The Capactiy of a Cylinder. . . . . . . . . . . . . . . . . . . . . . . . 395 9.1.4. The Sets of Zero Capacity Cap(·, W~) . . . . . . . . . . . . . . 395

§ 9.2. On (p, I)-polar Sets ..................................... 396

§ 9.3. The Equivalence of Two Capacities ....................... 397 9.3.1. An Auxiliary Multiplicative Inequality. . . . . . . . . . . . . . 397 9.3.2. The Relation Cap - cap for p > 1 .................. 399

§ 9.4. Comments to Chapter 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 401

Chapter 10. An Integral Inequality for Functions on a Cube ......... 402

§ 10.1. The Connection Between the Best Constant and the Capacity (The Case k = 1) ....................................... 402 10.1.1. Definition of a (p, I)-Unessential Set. . . . . . . . . . . . . . . . 402 10.1.2. The Main Theorem............ .......... ........ 403 10.1.3. A Variant of Theorem 10.1.2 and Its Corollaries ..... 406

§ 10.2. A Connection Between Best Constant and the (p, I)-inner Diameter (The Case k = 1) ............................... 408 10.2.1. The Set Function A~,q(G) '" . . . . . . . . . . . . . . . . . . . . .. 408 10.2.2. Definition of the (p, I)-inner Diameter .............. 409 10.2.3. Estimates for the Best Constant by the (p, I)-inner

Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 409

§ 10.3. Estimates for the Best Constant in the General Case. . . . . . . . . . 411 10.3.1. A Necessary and Sufficient Condition for the Validity

of the Basic Inequality ........................... 412 10.3.2. Capacities of Function Classes. . . . . . . . . . . . . . . . . . . .. 412 10.3.3. Estimates for the Best Constant in the Basic Inequality 414

• I 10.3.4. The Class Q:o(e) and the Capacity Capk(e, L P(Q2d) . . . . 416

XVIII Contents

10.3.5. A Lower Bound for Capk ......................... 418 10.3.6. Estimates for the Best Constant in the Case of Small

(p, I)-inner Diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 10.3.7. Application to the Boundary Uniqueness Theorem for

Analytic Functions in the Class L ~(U) .............. 421

§ 10.4. Comments to Chapter 10 ................................ 422

Chapter 11. Imbedding of the SpaceL~(.Q) into Other Function Spaces 424

§ 11.1. Preliminaries .......................................... 424

§ 11.2. Theimbeddingi~(.Q) C ~' (Q) .......................... 425 11.2.1. Auxiliary Assertions.. . ... .... .... ...... ....... .. 425 11.2.2. The Case Q = R n .•••..••..•......••.....••.•••.. 427 11.2.3. The Case n = pi, p < 1 ........................... 428 11.2.4. The Case n < pi and Fractional nip ................ 428 11.2.5. The Case n <pi, 1 <p < 00 and Integer nip ......... 429 · , § 11.3. The Imbedding Lp(Q) C Lq(Q, loc) ....................... 431 · , § 11.4. The Imbedding L p(Q) C Lq(Q) (The Case p ~ q) . . . . . . . . . . . . 434 11.4.1. A Condition in Terms of the (p, I)-inner Diameter .... 434 11.4.2. A Condition in Terms of Capacity ................. 434 · , § 11.5. The Imbedding L p(Q) C Lq(Q) (The Case p > q ~ 1) ........ 437 11.5.1. Definitions and Lemmas .......................... 437 11.5.2. The Basic Result ................................ 440 11.5.3. The Imbedding i~(Q) C Lq(Q) for an

"Infinite Funnel" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 442 . , § 11.6. The Compactness of the Imbedding L p(Q) C Lq(Q) ......... 444

11.6.1. The Casep ~ q . ...... . ..... ... ...... ... ......... 444 11.6.2. The Casep > q . ....... ..... ... ...... ... . ... ..... 446

§ 11.7. Application to the Dirichlet Problem for a Strongly Elliptic Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 446 11.7.1. The Dirichlet Problem with Nonhomogeneous

Boundary Data ................................. 447 11.7.2. The Dirichlet Problem with Homogeneous Boundary

Data .......................................... 448 11.7.3. The Discreteness of the Spectrum of the Dirichlet

Problem ....................................... 449 11.7.4. The Dirichlet Problem for a Nonselfadjoint Operator. 449


Chapter 12. The ImbeddingL ~(n, v) C W~ (n) ................... 453

§ 12.1. Auxiliary Assertions .................................... 453

§ 12.2. The Continuity of the Imbeddding Operator ., m

Lp(Q, v) -+ Wr (Q) ..................................... 455

Contents XIX

§ 12.3. The Compactness of the Imbedding Operator i~(Q, v) -+ W7'(Q) ..................................... 458 12.3.1. The Essential Norm of the Imbedding Operator ...... 458 12.3.2. Criteria for Compactness ......................... 460

§ 12.4. On the Closability of Imbedding Operators ................. 462

§ 12.5. Application: Positive Definiteness and Discreteness of the Spectrum of a Strongly Elliptic Operator ................... 464


References ................................................... 469

List of Symbols ............................................... 480

Author Index

Subject Index

483

485

v. g. maz'ja sobolev spaces978-3-662-09922-3/1.pdf · in the present monograph we consider...

Documents