nonlinear analysis

1. Series in Mathematical Analysis and Applications Edited by Ravi P. Agarwal and Donal ORegan VOLUME 9 NONLINEAR ANALYSIS 2005 by Taylor & Francis Group, LLC

2. SERIES IN MATHEMATICAL ANALYSIS AND APPLICATIONS Series in Mathematical Analysis and Applications (SIMAA) is edited by Ravi P. Agarwal, Florida Institute of Technology, USA and Donal ORegan, National University of Ireland, Galway, Ireland. The series is aimed at reporting on new developments in mathematical analysis and applications of a high standard and or current interest. Each volume in the series is devoted to a topic in analysis that has been applied, or is potentially applicable, to the solutions of scientific, engineering and social problems. Volume 1 Method of Variation of Parameters for Dynamic Systems V. Lakshmikantham and S.G. Deo Volume 2 Integral and Integrodifferential Equations: Theory, Methods and Applications Edited by Ravi P. Agarwal and Donal ORegan Volume 3 Theorems of Leray-Schauder Type and Applications Donal ORegan and Radu Precup Volume 4 Set Valued Mappings with Applications in Nonlinear Analysis Edited by Ravi P. Agarwal and Donal ORegan Volume 5 Oscillation Theory for Second Order Dynamic Equations Ravi P. Agarwal, Said R. Grace, and Donal ORegan Volume 6 Theory of Fuzzy Differential Equations and Inclusions V. Lakshmikantham and Ram N. Mohapatra Volume 7 Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations V. Lakshmikantham, S. Koksal, and Raymond Bonnett Volume 8 Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems Leszek Gasinski and Nikolaos S. Papageorgiou Volume 9 Nonlinear Analysis Leszek Gasinski and Nikolaos S. Papageorgiou 2005 by Taylor & Francis Group, LLC

3. Series in Mathematical Analysis and Applications Edited by Ravi P. Agarwal and Donal ORegan VOLUME 9 NONLINEAR ANALYSIS Leszek Gasinski Nikolaos S. Papageorgiou Boca Raton London New York Singapore 2005 by Taylor & Francis Group, LLC

4. Published in 2005 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 2005 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-484-3 (Hardcover) International Standard Book Number-13: 978-1-58488-484-2 (Hardcover) Library of Congress Card Number 2005045529 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Gasinski, Leszek. Nonlinear analysis / Leszek Gasinski, Nikolaos S. Papageorgiou. p. cm. -- (Series in mathematical analysis and applications ; v. 9) Includes bibliographical references and index. ISBN 1-58488-484-3 1. Nonlinear functional analysis. 2. Nonlinear operators. I. Papageorgiou, Nikolaos Socrates. II. Title. III. Series. QA321.5.G37 2005 515'.7--dc22 2005045529 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Taylor & Francis Group is the Academic Division of T&F Informa plc. 2005 by Taylor & Francis Group, LLC

5. To Prof. Zdzislaw Denkowski 2005 by Taylor & Francis Group, LLC

6. Contents 1 Hausdor Measures and Capacity 1 1.1 Measure Theoretical Background . . . . . . . . . . . . . . . . 3 1.2 Covering Results . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Hausdor Measure and Hausdor Dimension . . . . . . . . . 22 1.4 Dierentiation of Hausdor Measures . . . . . . . . . . . . . 44 1.5 Lipschitz Functions . . . . . . . . . . . . . . . . . . . . . . . 52 1.6 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2 Lebesgue-Bochner and Sobolev Spaces 107 2.1 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . 108 2.2 Lebesgue-Bochner Spaces and Evolution Triples . . . . . . . 127 2.3 Compactness Results . . . . . . . . . . . . . . . . . . . . . . 150 2.4 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 179 2.5 Inequalities and Embedding Theorems . . . . . . . . . . . . . 213 2.6 Fine Properties of Functions and BV-Functions . . . . . . . 239 2.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 3 Nonlinear Operators and Young Measures 265 3.1 Compact and Fredholm Operators . . . . . . . . . . . . . . . 266 3.2 Operators of Monotone Type . . . . . . . . . . . . . . . . . . 303 3.3 Accretive Operators and Semigroups of Operators . . . . . . 343 3.4 The Nemytskii Operator and Integral Functions . . . . . . . 405 3.5 Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . 427 3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 4 Smooth and Nonsmooth Analysis and Variational Principles 467 4.1 Dierential Calculus in Banach Spaces . . . . . . . . . . . . 468 4.2 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . 488 4.3 Haar Null Sets and Locally Lipschitz Functions . . . . . . . 501 4.4 Duality and Subdierentials . . . . . . . . . . . . . . . . . . 512 4.5 Integral Functionals and Subdierentials . . . . . . . . . . . 558 4.6 Variational Principles . . . . . . . . . . . . . . . . . . . . . . 578 4.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 vii 2005 by Taylor & Francis Group, LLC

7. viii 5 Critical Point Theory 607 5.1 Deformation Results . . . . . . . . . . . . . . . . . . . . . . . 608 5.2 Minimax Theorems . . . . . . . . . . . . . . . . . . . . . . . 642 5.3 Structure of the Critical Set . . . . . . . . . . . . . . . . . . 654 5.4 Multiple Critical Points . . . . . . . . . . . . . . . . . . . . . 661 5.5 Lusternik-Schnirelman Theory and Abstract Eigenvalue Prob- lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 5.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 6 Eigenvalue Problems and Maximum Principles 707 6.1 Linear Elliptic Operators . . . . . . . . . . . . . . . . . . . . 708 6.2 The Partial p-Laplacian . . . . . . . . . . . . . . . . . . . . . 732 6.3 The Ordinary p-Laplacian . . . . . . . . . . . . . . . . . . . 759 6.4 Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . 775 6.5 Comparison Principles . . . . . . . . . . . . . . . . . . . . . . 788 6.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 7 Fixed Point Theory 803 7.1 Metric Fixed Point Theory . . . . . . . . . . . . . . . . . . . 804 7.2 Topological Fixed Point Theory . . . . . . . . . . . . . . . . 821 7.3 Partial Order and Fixed Points . . . . . . . . . . . . . . . . . 833 7.4 Fixed Points of Multifunctions . . . . . . . . . . . . . . . . . 877 7.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 Appendix 895 A.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 A.2 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . 899 A.3 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . 908 A.4 Calculus and Nonlinear Analysis . . . . . . . . . . . . . . . . 912 List of Symbols 915 References 925 2005 by Taylor & Francis Group, LLC

8. Preface Linear functional analysis deals with innite dimensional topological vector spaces (which mix in a fruitful way the linear (algebraic) structure with topological one) and the linear operators acting between them. The eort was to extend standard results of linear analysis to an innite dimensional context. The rst half of the twentieth century is marked by intensive theoretical investigations in this area, which were also accompanied by detailed treat- ment of linear mathematical models. With the exception of a short period during the 1930's (compact operators and Leray-Schauder degree), nonlinear operators were out of the emerging picture. However, mounting evidence from diverse other elds such as physics, engineering, economics, biology and others suggested that there should be an eort to extend the linear theory to various kinds of nonlinear operators. Systematic eorts in this direction started in the early 1960's and mark the beginning of what is known today as Nonlinear Analysis." Since then several theories have been developed in this respect and today some of them are well established approaching their limits, while others are still the object of intense research activity. It is not a co- incidence that simultaneously with the advent of nonlinear analysis, we have the appearance of nonsmooth analysis and of multivalued analysis, both of which were motivated by concrete needs in applied areas such as control theory, optimization, game theory and economics. Their development provided nonlinear analysis with new concepts, tools and theories that enriched the subject considerably. Today nonlinear analysis is a well established mathematical discipline, which is characterized by a remarkable mixture of analysis, topology and applications. It is exactly the fact that the subject combines in a beautiful way these three items that makes it attractive to mathematicians. The notions and techniques of nonlinear analysis provide the appropriate tools to develop more realistic and accurate models describing various phenomena. This gives nonlinear analysis a rather interdisciplinary character. Today the more theoretically inclined nonmathematician (engineer, economist, biologist or chemist) needs a working knowledge of at least a part of nonlinear analysis in order to be able to conduct a complete qualitative analysis of his models. This supports a high demand for books on nonlinear analysis. Of course the subject is big (vast is maybe a more appropriate word) and no single book can cover all its theoretical and applied parts. In this volume, we have fo- cused on those topics of nonlinear analysis which are pertinent to the theory of boundary value problems and their applications such as control theory and calculus of variations. ix 2005 by Taylor & Francis Group, LLC

9. x In Chapter 1 we deal with Hausdor measures and capacities, which provide the means to estimate the size" or dimension" of thin" or highly irregular" sets. The recent development of fractal geometry and its uses in a variety of applied areas (such as Brownian motion of particles, turbulence in uids, geographical coastlines and surfaces etc) renewed the interest on Hausdor measures, which for a long period were a topic of secondary importance within measure theory. In this chapter we also have our rst encounter with Lipschitz At this point we prove the celebrated Rademacher's theorem." Chapter 2 deals with certain classes of function spaces, which arise naturally in the study of boundary value problems. These are the Lebesgue-Bochner spaces (the suitable spaces for the analysis of evolution equations) and the Sobolev spaces (the suitable spaces for weak solutions of elliptic equations). We conduct a detailed study of these spaces with special emphasis on compactness and embedding results. Also using the tool of Hausdor measures and capacities, we investigate the ne properties of Sobolev functions and also introduce and study functionals of bounded variation which are useful in theoretical mechanics. In Chapter 3, we deal with certain large classes of nonlinear operators which arise often in applications. We examine compact operators for which we develop in parallel the corresponding linear theory, with one of the main results being the spectral theorem for compact self-adjoint operators on a Hilbert space. We also investigate nonlinear operators of monotone type which have their roots in the calculus of variations and exhibit remarkable surjectivity properties. Monotone operators lead to accretive operators, the two families being identical in the context of Hilbert spaces. Accretive operators are closely connected with the generation theory of semigroups of operators. We also examine both linear and nonlinear semigroups. Semigroups are basic tools in the study of evolution equations. In addition, we examine the Nemytskii operator which is a nonlinear operator encountered in almost all problems. Finally, in the last section of the chapter, we discuss Young measures which provide the right framework to examine the limit behavior of the minimizing sequence of variational problems which do not have a solution. Young measures are used in optimal control and in the calculus of variations in connection with the so-called relaxation method." nonsmooth functions. We start with the G^ateaux and Frechet derivatives. We discuss the generic dierentiability of continuous convex functions (Mazur's theorem) and extend Rademacher's theorem to locally Lipschitz functions between certain Banach spaces by using the notion of Haar-null sets. Then we pass to nondierentiable functions and develop the duality properties and subdierential theory of convex functions and the generalized subdierential of locally Lipschitz functions. We also examine integral functionals and discuss the celebrated Ekeland variational principle establishing its equivalence with some other geometric results of nonlinear analysis. and locally Lipschitz functionals which will be examined again in Chapter 4. Chapter 4 presents the calculus of smooth and of certain broad classes of 2005 by Taylor & Francis Group, LLC

10. xi In Chapter 5 we present the critical point theory of C1-functions dened on a Banach space. This theory is in the core of the variational methods used in the study of boundary value problems. We follow the deformation approach which leads to minimax characterizations of the critical values. We also study the structure of the set of critical points and derive results on the existence of multiple critical points. Next we present the Lusternik-Schnirelman theory which extends to nonlinear eigenvalue problems the corresponding linear theory of R. Courant. Chapter 6 uses the abstract results of Chapter 5 as well as results from earlier chapters to develop the spectrum of linear elliptic dierential operators, of the partial p-Laplacian (with Dirichlet and Neumann boundary conditions) and of the scalar and vector ordinary p-Laplacian (with Dirichlet, Neumann and periodic boundary conditions). We also present linear and nonlinear maximum principles and comparison results, which are useful tools in the study of boundary value problems. Finally in Chapter 7 we have gathered some basic xed point theorems. We present results from metric xed point theory, from topological xed point theory and xed point results based on the partial order induced by a closed, convex pointed cone. We also indicate how many of these results can be extended to multifunctions (set-valued functions). We have tried to make the volume self-contained. For this reason at the end the general results used in the book. Nevertheless, within the test whenever we are in the need of using some results not proved in the book, we also give exact references where the interested reader can nd additional information. Now that the project has reached its conclusion, we would like to thank the good people of CRC Press (especially Mrs. Jessica Vakili) for their help and kind cooperation during the preparation of this book. We would like to thank the two editors of this series, Prof. R.P. Agarwal and Prof. D.O'Regan, for supporting this eort. of the book we have included a rather extended appendix for easy reference of 2005 by Taylor & Francis Group, LLC

11. Chapter 1 Hausdor Measures and Capacity During the golden era of measure theory (namely the rst two decades of the 20th century), Caratheodory was the rst to consider the notion of length" for sets in RN. Later, in 1919, Hausdor, motivated by the ideas of Caratheodory, introduced the measure and dimensional concepts that we shall discuss in this chapter. So in the modern language, the length" of a set A RN will be its Hausdor one-dimensional outer measure (denoted by (1)). Following the pioneering works of Caratheodory and Hausdor, signif- icant contributions to the subject were made by Besicovitch. In fact, in the rst decade of development of the subject, the main advances on the subject were made by Besicovitch and his students, since geometric measure theory was not part of the mainstream measure theory. However, since the early 70's, the subject attracted a large number of researchers, due to its fundamental importance in the study of the so-called Fractal Geometry." Fractal sets arise in many applications, such as turbulence in uids, geographical coastlines and surfaces, uctuation of prices in stock exchanges, the Brownian motion of particles and others. Mandelbrojt was the rst to emphasize their use to model a variety of phenomenona. There have been many ways to estimate the size" or dimension" of small (thin) sets and of highly irregular sets and to gen- eralize the idea that points, curves and surfaces have dimensions 0, 1 and 2 respectively. Hausdor measure has the advantage of being a measure and together with the notion of Hausdor dimension can provide a more delicate sense of the size of sets in RN than Lebesgue measure provides. To illustrate this, consider in R2 the set A df= t; sin 1 t : t 2 (0; 1) : Suppose we wish to measure the length of the curve A. A rst approximation can be based on the Caratheodory outer measure, which denes: 1(A) df= inf 1S n=1 A Cn 1X n=1 (Cn); i.e., the inmum is taken over all countable covers of A (by (A) we denote the diameter of the set A; see (1.1)). If we adopt this denition, we see that 1(A) < +1, while we know that the length of A is innite. The reason for 1 2005 by Taylor & Francis Group, LLC

12. 2 Nonlinear Analysis this is that in the denition of 1(A), the covers of A are not forced to follow the geometry of A. For this reason the Hausdor s-dimensional measures (s)(A) are dened as limits of outer measures (s) which follow the local geometry of A (see Denition 1.3.5). As another illustrative example, consider the unit square S in R2 (i.e., square of side length equal to 1) and dene 1(S) df= inf 1S n=1 S Cn 1X n=1 (Cn); i.e., again the inmum is taken over all countable covers of S. We observe that we can do no better than cover S itself. Indeed, if we cover S with smaller squares of diameter less or equal to 1 n, then we see that we need at least n2 squares to achieve the covering and so the approximation of 1(S) obtained this way exceeds n p 2. So the smaller the squares we use to cover, the bigger the estimate for 1(S). Therefore, small squares are irrelevant in the calculation of 1(S) and yet it is precisely them that should have an inuence on the evaluation of 1(S). We expect 1(S) = 0, since the diameter is a one-dimensional concept and it is used to measure a square in R2, which is a two dimensional concept. For this we need a denition which takes into account the local geometry of the set under consideration. In this chapter, in Section 1.1 we recall some basic denitions and facts from measure theory, which will be needed in what follows. In Section 1.2, we discuss some covering theorems." Covering results play a central role in geometric measure theory. In Section 1.3 we introduce and study Hausdor measures and the Hausdor dimension of sets. Among other things we calculate the Hausdor dimension of some classical irregular sets in R (Cantor-like sets). From these calculations, the reader will realize that the Hausdor measure and the Hausdor dimension of sets (even of simple ones) may be hard to calculate. For this reason sometimes other notions may be more suitable (such as capacity; see Section 1.6). In Section 1.4 we discuss the dierentiation of Hausdor measures and derive the Lebesgue-Besicovitch dierentiation theorem. In Section 1.5, using the tools of Hausdor measures, we study the geometry of Lipschitz continuous functions. Among other things, we obtain the area and coarea formulas" and the associated with them change of vari- ables formulas." Finally in Section 1.6, we present an alternative analytical notion measuring small sets in RN, namely the p-capacity. We derive some basic properties of the p-capacities and compare them to the Hausdor measures. 2005 by Taylor & Francis Group, LLC

13. 1. Hausdor Measures and Capacity 3 1.1 Measure Theoretical Background In this section we recall some basic denitions and facts from measure theory, which we shall need in the sequel. Let us start with the concept of outer measure, which, when restricted to a suitable -eld of sets, leads to a measure. DEFINITION 1.1.1 Let X be a set. A map : 2X ! [0;+1] is said to be an outer measure, if (a) (;) = 0; (b) A B =) (A) 6 (B) (monotonicity); (c) for any sequence of sets fAngn>1 2X, we have 1[ n=1 An 6 1X n=1 (An) (subadditivity). For a given outer measure on X and A 2 2X, we dene the restriction of on A, denoted by bA, by (bA)(B) df= (A B) 8 B 2 2X: We say that is a nite outer measure if (X) < +1 (i.e., has values in R+). REMARK 1.1.2 Note that bA is an outer measure on X, while we dene jA to be the restriction of (as a function) on 2A, i.e., jA : 2A ! [0;+1] is dened by jA (B) df= (B) 8 B 2 2A 2X: Outer measures are useful because they lead to measures when restricted to suitably dened -elds. These -elds can be quite large. DEFINITION 1.1.3 Let X be a set and an outer measure on X. A set A 2 2X is said to be -measurable, if (B) = (A B) + (B n A) 8 B 2 2X; i.e., A decomposes" every set B additively. 2005 by Taylor & Francis Group, LLC

14. 4 Nonlinear Analysis REMARK 1.1.4 Let X be a set and an outer measure on X. (a) By virtue of the subadditivity property of an outer measure, to show that A 2 2X is -measurable, it is enough to check that (B) > (A B) + (B n A) 8 B 2 2X: (b) Clearly, if A 2 2X and (A) = 0, then A is -measurable. (c) If A 2 2X, then any -measurable set is also bA-measurable. (d) A is -measurable if and only if Ac = X n A is -measurable. It is straightforward to check the following result. PROPOSITION 1.1.5 If X is a set and is an outer measure on X, then the collection of all -measurable sets is a -eld and restricted on is a measure. REMARK 1.1.6 While Denition 1.1.3 involves only additivity of , the conclusion in Proposition 1.1.5 is about -additivity of on . This reveals the power of Denition 1.1.3. Note that from Remark 1.1.4(b), it follows that the -eld is -complete. DEFINITION 1.1.7 Let X be a nonempty Hausdor topological space and let be an outer measure on X. (a) Let T be a family of 2X. We say that is T -regular, if (A) = inf B 2 T A B (B) 8 A 2 2X: If T = , then we simply say that is regular. (b) We say that is a Borel measure, if B(X) with B(X) being the Borel -eld of X. (c) We say that is a Borel regular measure, if is a Borel measure which is B(X)-regular. (d) We say that is a Radon measure, if is a Borel regular measure and (K) < +1 8 K X; K-compact: 2005 by Taylor & Francis Group, LLC

15. 1. Hausdor Measures and Capacity 5 REMARK 1.1.8 Let X be a Hausdor topological space and let be an outer measure on X. (a) Note that is regular if and only if 8A 2 2X 9B 2 : (A) = (B): (b) If is regular on X and fAngn>1 2X is increasing (i.e., An An+1 for n > 1), then 1[ n=1 An = sup n>1 (An): PROPOSITION 1.1.9 If X is a Hausdor topological space, is an outer measure on X which is Borel regular and A 2 with (A) < +1, then bA is a Radon measure. PROOF Let 1 df= bA: Evidently 1 and so 1 is a Borel measure. Also for every compact K X, we have 1(K) < +1: It remains to show that 1 is Borel regular. To this end note that since is Borel regular, for a given A 2 2X, we can nd B 2 B(X), A B, such that (A) = (B) < +1: Because A 2 , from Denition 1.1.3, we have (B n A) = (B) (A) = 0: Since A 2 , for every C 2 2X, we have (B C) n A (bB)(C) = (B C) = (B C A) + 6 (C A) + (B n A) = (C A) = (bA)(C): As A B, we infer that bB = bA: So without any loss of generality, we may assume that A 2 B(X). Let C 2 2X. Since is Borel regular, we can nd D 2 B(X), such that A C D and (A C) = (D) 2005 by Taylor & Francis Group, LLC

16. 6 Nonlinear Analysis (see Remark 1.1.8(a)). Let us take E df= D [ (X n A): Evidently E 2 B(X) and C (A C) [ (X n A) E: Moreover, since E A = D A, we have 1(E) = (E A) = (D A) 6 (D) = (A C) = 1(C); so 1 = bA is Borel regular (see Remark 1.1.8(a)), hence Radon. We conclude this section, by recalling the following basic measure theoretic approximations. PROPOSITION 1.1.10 If X is a Hausdor topological space and is an outer measure on X which is Borel, then (a) if A 2 B(X), (A) < +1 and " > 0, then we can nd an open set U" A and a closed set C" A, such that (U" n C") < "; i.e., (A) = inf U-open A U (U) = sup C-closed C A (C): (b) if is Radon, then for every A 2 2X, we have (A) = inf U-open A U (U) and if A 2 , then (A) = sup K-compact K A (K): REMARK 1.1.11 Note that in the rst part of Proposition 1.1.10(b), the set A need not be -measurable. 2005 by Taylor & Francis Group, LLC

17. 1. Hausdor Measures and Capacity 7 1.2 Covering Results One of the main tools in geometric measure theory is the so called Vitali covering theorem. For a given suciently large family of sets that cover a given set A, Vitali's covering theorem allows us to select a countable subfamily consisting of distinct sets with exactly the desired approximation properties. The basic principle embodied in the proof of Vitali's covering theorem is illustrated in the next proposition. In what follows for any subset A of a metric space (X; dX), we dene (A) = diam (A) df= sup x;y2A dX(x; y); (1.1) the diameter of A (by convention diam ; df= 0). PROPOSITION 1.2.1 If T is a collection of nondegenerate balls in RN with sup B2T (B) < +1; then we can nd a nite or countable subfamily F of T consisting of disjoint balls, such that [ B2T B [ B2F b B; with b B being the ball concentric with B, but with radius ve times the radius of B. PROOF Let d0 df= sup B2T (B); Tn df= B 2 T : d0 2n < (B) 6 d0 2n1 8 n > 1: Inductively, we generate subfamilies Fn Tn for n > 1. Namely, let F1 be any maximal disjoint collection of balls in T1. Suppose we have selected F1; : : : ;Fm. We choose Fm+1 to be any maximal disjoint subfamily of B 2 Tm+1 : B B0 = ; for all B0 2 m[ i=1 Fk and nally set F df= 1[ m=1 Fm: 2005 by Taylor & Francis Group, LLC

18. 8 Nonlinear Analysis Evidently F T and consists of disjoint balls. Claim. For each B 2 T , we can nd B0 2 F, such that B B06= ; and (B) 6 2(B0) (so also B B0). b For some m > 1, we have B 2 Tm. By virtue of the maximality of Fm, we mS can nd B0 k=1 Fk with B B06= ;. We have that d0 2m 6 (B0) and (B) 6 d0 2m1 : So (B) 6 2(B0) and this proves the claim. From the claim it follows at once that S B2T B S B02F b B0. DEFINITION 1.2.2 Let A RN. S A collection T of sets in RN is said to be a Vitali cover of A, if A B2T B and for every x 2 A and every " > 0, there exists B 2 T , such that x 2 B and 0 < (B) < ". REMARK 1.2.3 Note that from the second requirement of the above denition it follows that inf B2T (B) = 0: So T is a Vitali cover of a set A, if every point x 2 A is contained in an arbitrary small element of T . As a straightforward consequence of Proposition 1.2.1 we obtain the following proposition. PROPOSITION 1.2.4 If A RN, T is a Vitali cover of A consisting of closed balls, such that sup B2T (B) < +1; then there exists a countable family F = fBngn>1 consisting of disjoint balls from T , such that for each m > 1, we have A m[ n=1 Bn [ 1[ n=m+1 b Bn; where b Bn is the closed ball cocentric with Bn and radius ve times the radius of Bn. 2005 by Taylor & Francis Group, LLC

19. 1. Hausdor Measures and Capacity 9 mn PROOF Let F be as in the proof of Proposition 1.2.1. Select fBng =1 mS F. If A n=1 Bn, then we are done. Otherwise let x 2 A n m[ n=1 Bn. Since T is a Vitali cover of A consisting of closed balls, then we can nd B 2 T , such that x 2 B and B Bn = ; 8 n 2 f1; : : : ;mg: But from the claim in the proof of Proposition 1.2.1, we can nd B0 2 F, such that B b B0 and B B06= ; (so B0 2 fBng1 n=m+1). Now we are ready to state and prove Vitali's covering theorem. In what follows by N we denote the N-dimensional Lebesgue outer measure. THEOREM 1.2.5 (Vitali Covering Theorem) If A RN with 0 < N(A) < +1 and T is a Vitali cover of A consisting of closed sets, then we can nd a sequence fCngn>1 of elements in T , such that CnCm = ; for n6= m and N A n 1[ n=1 Cn = 0: PROOF Without any loss of generality, we can assume that there exists an open set U RN with N(U) < +1 and C U 8 C 2 T : We construct the sequence fCnginductively. Let C1 2 T . Suppose that n>1 nS C1; : : : ;Cn are disjoint sets in T . If A k=1 Ck, then we are nished. If not, setting Vn df= U n [n k=1 Ck; we introduce Tn df= C 2 T : C Vn and n df= sup C2Tn N(C): Because A n nS i=1 Ck6= ; and T is a Vitali cover of A, we see that Tn6= ; and so n > 0. We select Cn+1 2 T with n 2 < N(Cn+1): 2005 by Taylor & Francis Group, LLC

20. 10 Nonlinear Analysis We continue this process. Then either at some nite step n > 1 we shall have nS A k=1 Ck, in which case the proof of the theorem is complete or otherwise we produce a sequence fCngn>1 T of disjoint sets. Then we have 1X n=1 N(Cn) = N 1[ n=1 Cn 6 N(U) < +1: (1.2) For each n > 1 let Bn be a ball with center in Cn and radius equal to 3(Cn). We claim that A n [n k=1 Ck 1[ k=n+1 Bk 8 n > 1: (1.3) Let x 2 A n nS k=1 Ck. Since T is a Vitali cover of A, we can nd a set Cx 2 Tn, such that x 2 Cx and N(Cx) > 0: We shall show that Cx Ck6= ; for some k > n: Indeed, if this is not the case, then N(Cx) 6 k for all k > 1, which contra- dicts the fact that 0 6 lim k!+1 k 6 lim k!+1 2N(Ck+1) = 0 (recall the choice of Ck+1 and see (1.2)). Let m > n be the smallest integer, such that Cx Cm6= ;: Since Cx 2 Tm1, we have N(Cx) 6 m1 < 2N(Cm) and recalling the choice of Bm, also Cx Bm. So we have proved (1.3). Then for any n > 1, we have N A n 1[ k=1 Ck 6 N A n [n k=1 Ck 6 1X k=n+1 N(Bk): (1.4) Recalling that Bk is a ball of radius 3(Ck) and combining (1.2) and (1.4), we conclude that N A n 1[ k=1 Ck = 0: 2005 by Taylor & Francis Group, LLC

21. 1. Hausdor Measures and Capacity 11 Vitali's covering theorem may be dicult to digest at rst and probably it is necessary to see the lemma in action several times before appreciating it. For this reason we present four simple applications from classical analysis of functions of one-variable. We start with a denition which establishes the notation for various limits of the dierence quotient that we shall use in the sequel. These derivatives are often more useful than the ordinary derivative, since they are dened at every point. DEFINITION 1.2.6 For a given function f : [a; b] ! R, the upper right and lower right derivates of f at x 2 [a; b) are dened by D+f(x) df= lim sup h!0+ f(x + h) f(x) h and D+f(x) df= lim inf h!0+ f(x + h) f(x) h respectively. Similarly the upper left and lower left derivates of f at x 2 (a; b] are dened by Df(x) df= lim sup h!0 f(x + h) f(x) h and Df(x) df= lim inf h!0 f(x + h) f(x) h respectively. REMARK 1.2.7 Evidently, the derivates of a function at a point may be innite. The function f is dierentiable at x 2 (a; b), if 1 < D+f(x) = D+f(x) = Df(x) = Df(x) < +1: The function f is dierentiable at x = a or at x = b, if the appropriate two derivates are nite and equal. Also the one-sided derivatives exist at a point x, if D+f(x) = D+f(x) and Df(x) = Df(x): The derivates are also called Dini derivates and clearly we always have D+f(x) 6 D+f(x) 8 x 2 [a; b) and Df(x) 6 Df(x) 8 x 2 (a; b]: 2005 by Taylor & Francis Group, LLC

22. 12 Nonlinear Analysis In the literature, sometimes we nd the notion of a derived number for a function f at x. So 2 R is a derived number for f at x, if there is a sequence fhngn>1 R; such that hn ! 0; hn6= 0 8 n > 1 and lim n!+1 f(x + hn) f(x) hn = : A function f may have many derived numbers at a point x. Of course f is dierentiable at x if and only if all derived numbers of f at x agree and are nite. EXAMPLE 1.2.8 Consider the function f : R ! R dened by f(x) df= ( x sin 1 x if x6= 0; 0 if x = 0: We can check that Df(0) = 1 < D+f(0) = 1 and every number in [1; 1] is a derived number for f. The function f is not of bounded variation (see Denition A.2.15(a)). LEMMA 1.2.9 If f : [a; b] ! R is nondecreasing, then all four derivates of f are nite almost everywhere on [a; b]. PROOF Clearly all derivates are nonnegative. So it suces to show that D+f(x) < +1 and Df(x) < +1 for a.a. x 2 [a; b]: Let A df= x 2 [a; b] : D+f(x) = +1 and suppose that (A) = > 0; where is the Lebesgue outer measure on R. Let M > 0 be such that f(b) f(a) < M 2 : 2005 by Taylor & Francis Group, LLC

23. 1. Hausdor Measures and Capacity 13 For every x 2 A, we can nd a decreasing sequence fhx ngn>1 with hx n & 0; hx n6= 0 8 n > 1; such that M 6 f(x + hx n) f(x) hx n : The collection [x; x + hx n] x2A;n>1 is a Vitali cover of A. By virtue of Vitali's covering theorem (see Theo- rem 1.2.5), we can nd a family of disjoint intervals [xn; xn + hn] m n=1; such that Xm n=1 hn > 2 : Therefore Xm n=1 f(xn + hn) f(xn) > Xm n=1 Mhn > M 2 > f(b) f(a); a contradiction. This proves that (A) = 0 and so D+f(x) < +1: Analogously we can prove that Df(x) < +1. Using this lemma and Vitali's covering theorem, we can now prove that a nondecreasing function is dierentiable almost everywhere on [a; b]. THEOREM 1.2.10 If f : [a; b] ! R is nondecreasing, then f is dierentiable almost everywhere on [a; b]. PROOF For f to be dierentiable at x, we must have that all four derivates at x are nite and equal. By virtue of Lemma 1.2.9, it suces to show that all four derivates are equal almost everywhere. Let A df= x 2 (a; b) : D+f(x) < D+f(x) : 2005 by Taylor & Francis Group, LLC

24. 14 Nonlinear Analysis We show that A is Lebesgue-null. The proof for the other combinations of derivates is similar. Suppose that (A) > 0: We can nd rational numbers r; s, such that the set B df= x 2 A : D+f(x) < r < s < D+f(x) satises (B) = > 0: Let " 2 (0; ). From the regularity of the Lebesgue outer measure , we know that there exists an open set U (a; b), such that B U and 1(U) " < : For each x 2 B and n > 1, we can nd hx n > 0; with hx n & 0; such that x; x + hx n U and f(x + hx n) f(x) hx n < r: The family [x; x + hx x2B;n>1 n] is a Vitali cover of B. By virtue of Vitali's covering theorem (see Theo- rem 1.2.5), for a given " > 0, we can nd a disjoint subfamily [xn; xn + hn] m n=1 of the Vitali cover, such that B n m[ [xn; xn + hn] n=1 < ": We have Xm n=1 f(xn + hn) f(xn) < r Xm n=1 hn 6 r1(U) < r( + "): (1.5) Let us set C df= B m[ [xn; xn + hn] n=1 : We have that " < (C): 2005 by Taylor & Francis Group, LLC

25. 1. Hausdor Measures and Capacity 15 For every y 2 C and k > 1, we can nd uy k 2 y; y + 1 k , such that f(uy k) f(y) uy k y > s and [y; uy k] (xn; xn + hn); for some n 2 f1; : : : ;mg: The family [y; uy y2C;k>1 k] is a Vitali cover of C. Invoking Vitali's covering theorem (see Theorem 1.2.5), we can nd a disjoint subfamily [yk; uk] l k=1; such that (C) " < Xl k=1 (uk yk): Hence, we have Xl (C) " k=1 f(uk) f(yk) > s Xl k=1 (uk yk) > s > s( 2"): (1.6) For each 1 6 n 6 m, let Jn df= k 2 f1; : : : ; lg : [yk; uk] (xn; xn + hn) : Since f is nondecreasing, using (1.6) and (1.5), we have s( 2") < Xl k=1 f(uk) f(yk) = Xm n=1 X k2Jn f(uk) f(yk) 6 Xm n=1 f(xn + hn) f(xn) < r( + "): Let " & 0, to conclude that s 6 r, a contradiction. COROLLARY 1.2.11 If f : [a; b] ! R is of bounded variation, then f is dierentiable almost everywhere. PROOF Recall that f = f1 f2 with f1 and f2 nondecreasing and apply Theorem 1.2.10. 2005 by Taylor & Francis Group, LLC

26. 16 Nonlinear Analysis THEOREM 1.2.12 If f : [a; b] ! R is absolutely continuous (see Denition A.2.15(b)) and f0(x) = 0 for a.a. x 2 [a; b]; then f is constant on [a; b]. PROOF We show that f(u) = f(a) 8 u 2 (a; b]: So x a u 2 (a; b] and let A df= x 2 (a; u) : f0(x) = 0 and "; > 0. Because f is absolutely continuous, we can nd > 0, such that, if (rn; sn) m n=1 is a nite family of disjoint subintervals of [a; b] with Xm n=m (sn rn) < ; then we have Xm n=1 f(sn) f(rn) < ": We introduce the family T df= [x; y] : x 2 A; x < y < u and f(y) f(x) y x < : Clearly T is a Vitali cover of A. So by Vitali's covering theorem (see Theo- rem 1.2.5), we can nd a nite subfamily [xn; yn] m n=1 of disjoint sets in T , such that 1 A n m[ [xn; yn] n=1 < : We may assume that a < x1 < y1 < x2 < y2 < : : : < xm < ym < u: Since 1 (a; u) n A = 0; we have 1 (a; u) n m[ [xn; yn] n=1 < 2005 by Taylor & Francis Group, LLC

27. 1. Hausdor Measures and Capacity 17 and 1 (a; u) n m[ [xn; yn] n=1 = (x1 a) + Xm n=2 (xn yn1) + (u ym): Then, it follows that f(u) f(a) 6 Xm n=1 f(yn) f(xn) + f(x1) f(a) + Xm n=2 f(xn) f(yn1) + f(u) f(ym) < Xm (yn xn) + " 6 (u a) + ": n=1 Let "; & 0, to obtain that f(u) = f(a). COROLLARY 1.2.13 If f : [a; b] ! R is absolutely continuous, then f0 is Lebesgue integrable on [a; b] and for all x 2 [a; b], we have f(x) f(a) = Zx a f0(s) ds: PROOF From the fundamental theorem of Lebesgue calculus (see Theo- rem A.2.20), we know that '(x) = Zx a f0(s) ds is absolutely continuous on [a; b] and '0(t) = f0(t) for a.a. t 2 [a; b]: Then Theorem 1.2.12 implies that f = ' + , with a real constant . Evalu- ating at x = a, we have f(a) = '(a) + = : Therefore Zx a f0(s) ds = f(x) f(a) 8 x 2 [a; b]: 2005 by Taylor & Francis Group, LLC

28. 18 Nonlinear Analysis As a third application of Vitali's covering theorem (see Theorem 1.2.5), we show that a very weak local growth condition on a strictly increasing function f leads to a strong local growth condition. More precisely, we prove the following result. THEOREM 1.2.14 If f : [a; b] ! R is a strictly increasing function, r > 0, A [a; b] and at each point x 2 A there exists a derived number (see Remark 1.2.7), such that < r, then f(A) 6 r(A). PROOF If (A) = +1, then the inequality is obvious. So assume that (A) < +1: For a given " > 0, we can nd a bounded open set U R, such that A U and 1(U) " < (A): If x 2 A, then by hypothesis we can nd a sequence fhngn>1 R n f0g, such that hn ! 0; [x; x + hn] U 8 n > 1 (or [x + hn; x] U in the event hn < 0; but in the sequel for simplicity we shall write [x; x + hn] for both cases) and f(x + hn) f(x) hn < r 8 n > 1: (1.7) For all n > 1 and x 2 A, let Dn(x) df= [x; x + hn]; En(x) df= f(x); f(x + hn) : Because f is strictly increasing En(x) is a nondegenerate, closed interval and f Dn(x) En(x) 8 n > 1; x 2 A: Since 1 Dn(x) = jhnj and 1 En(x) = f(x + hn) f(x) ; from (1.7), we have 1 En(x) < r1 : (1.8) Dn(x) 2005 by Taylor & Francis Group, LLC

29. 1. Hausdor Measures and Capacity 19 Passing to the limit as n ! +1, we have jhnj ! 0 and so from (1.8), we obtain that lim n!+1 1 = 0: En(x) Let T df= En(x) x2A;n>1: Then T is a Vitali cover of the set f(A). So Vitali's covering theorem (see Theorem 1.2.5) implies the existence a disjoint sequence Enk (xk) k>1 T ; such that 1 f(A) n 1[ k=1 Enk (xk) = 0: (1.9) Using (1.9) and (1.8), it follows that f(A) 6 1 1[ k=1 Enk (xk) = 1X k=1 1 (Enk (xk)) < r 1X k=1 1 : (1.10) Dnk (xk) Since f is strictly increasing, we see that Dnk (xk) k>1 are pairwise disjoint too. So we have 1X k=1 1 Dnk (xk) = 1 1[ k=1 Dnk (xk) (1.11) 6 1(U) 6 (A) + ": (1.12) From (1.10) and (1.11), we infer that f(A) 6 r (A) + " : Let " & 0, to conclude that f(A) 6 r(A): In a similar fashion, we can have the following comparison result. 2005 by Taylor & Francis Group, LLC

30. 20 Nonlinear Analysis THEOREM 1.2.15 If f : [a; b] ! R is a strictly increasing function, s > 0, A [a; b] and at each point x 2 A there exists a derived number , such that > s, then > s(A): f(A) The nal application of Vitali's covering theorem (see Theorem 1.2.5) is the following criterion for measurability of sets in R. THEOREM 1.2.16 If F is any collection of intervals in R and A = [ D2F D; then A is Lebesgue measurable. PROOF Let T be a collection of all intervals E; such that E D for some D 2 F. Evidently T is a Vitali cover of A and so by Vitali's covering theorem (see Theorem 1.2.5), we can nd a sequence fEngn>1 of disjoint elements in T , such that A n 1[ n=1 En = 0: Because each En A, the set A df= 1[ n=1 En [ A n 1[ n=1 En is Lebesgue measurable. REMARK 1.2.17 Theorem 1.2.16 can be used to show that the upper and lower derivates of an arbitrary function are measurable. In particular then the four derivates of a measurable function are measurable and so is the derivative of a measurable function. We will not go into that here. 2005 by Taylor & Francis Group, LLC

31. 1. Hausdor Measures and Capacity 21 When N is replaced by an arbitrary Radon measure on RN, there is no systematic way to control ( b B) in terms of (B). So the proof of Vitali's covering theorem (see Theorem 1.2.5) which uses the principle involved in Proposition 1.2.1, namely the use of suitable expansions of balls, does not work. So we need an analog of Proposition 1.2.1, which does not require enlarging the balls, though. This is done by the so-called Besicovitch covering theorem." THEOREM 1.2.18 (Besicovitch Covering Theorem) If F is any collection of closed balls in RN, sup B2F (B) < +1 and A is the set of centers of all balls B 2 F; then there exist a positive integer k = k(N) > 1 and Tn F 8 n 2 f1; : : : ; kg; such that each Tn is a countable collection of disjoint balls in F and A [k n=1 [ B2Tn B: Using the above theorem, we can have the following counterpart of Vitali's covering theorem (see Theorem 1.2.5). THEOREM 1.2.19 If is a Borel measure on RN, T is a family of nondegenerate closed balls in RN, A is the set of centers of balls in T ; (A) < +1, inf Br(a)2F r = 0 8 a 2 A and U RN is an open set, then there exists a countable collection of disjoint balls F from T , such that [ B2F (A U) n B U and [ B2F B = 0: 2005 by Taylor & Francis Group, LLC

32. 22 Nonlinear Analysis 1.3 Hausdor Measure and Hausdor Dimension Hausdor measures were introduced as certain lower dimensional measures on RN which allow us to measure small" subsets in RN. The Hausdor measure and the associated Hausdor dimension of the set provide a more delicate sense of the size of a set in RN than the Lebesgue measure provides. We start with the introduction of a special class of outer measures, known as metric outer measures. DEFINITION 1.3.1 Let (X; dX) be a metric space (d is the metric function). (a) If A;B X, then we say that A and B are separated sets, if dX(A;B) df= inf a 2 A b 2 B dX(a; b) > 0: (b) If is an outer measure on X, then we say that is a metric outer measure, if (A [ B) = (A) + (B) 8 A;B X; A and B separated: We show that if is a metric outer measure, then B(X) (), i.e., is Borel. To this end we need the following auxiliary result, known as Caratheodory's lemma. In what follows (X; d) is a metric space. LEMMA 1.3.2 (Caratheodory Lemma) If is a metric outer measure on X, U X is an open subset, U6= X, A U and An df= x 2 A : d(x;Uc) > 1 n 8 n > 1; (1.13) then (A) = lim n!+1 (An). PROOF Note that the sequence fAngn>1 is an increasing sequence and so lim n!+1 (An) exists. Moreover, since An A for n > 1, we have lim n!+1 (An) 6 (A): So we need to show that (A) 6 lim n!+1 (An): (1.14) 2005 by Taylor & Francis Group, LLC

33. 1. Hausdor Measures and Capacity 23 Because U is open, we have d(x;Uc) > 0 8 x 2 A and so we can nd n0 > 1 large enough so that x 2 An0 . Therefore, we have A = 1[ n=1 An: For each n > 1, we introduce the set Cn df= An+1 n An = x 2 A : 1 n + 1 6 d(x;Uc) < 1 n : We have A = A2n [ 1[ k=2n Ck = A2n [ 1[ k=n C2k [ 1[ k=n C2k+1 and from the subadditivity of , it follows that (A) 6 (A2n) + 1X k=n (C2k) + 1X k=n (C2k+1): (1.15) If both series are convergent, then we obtain (1.14). So suppose that this is not true and, say, we have 1X k=1 (C2k) = +1: (1.16) Note that d C2k;C2k+2 > 1 2k + 1 1 2k + 2 8 k > 1 and so the sets fCkgk>1 are separated. Therefore, we have n[1 k=1 C2k = nX1 k=1 (C2k) 8 n > 1: (1.17) Note that n[1 k=1 C2k A2n 8 n > 1 and so n[1 k=1 C2k 6 (A2n) 8 n > 1: (1.18) 2005 by Taylor & Francis Group, LLC

34. 24 Nonlinear Analysis From (1.17) and (1.18), it follows that nX1 k=1 (C2k) 6 (A2n): Combining this with (1.16), we infer that lim n!+1 (A2n) = +1 and so (A) 6 lim n!+1 (A2n); as desired. Similarly, if 1P k=1 (C2k+1) = +1. THEOREM 1.3.3 If is an outer measure on X, then B(X) () (i.e., is Borel) if and only if is a metric outer measure. PROOF =)": Let A1;A2 X be separated sets and let us set df= d(A1;A2) > 0: For every x 2 A1, we dene U(x) df= B2 (x) = y 2 X : d(y; x) < 2 and U df= [ x2A1 U(x): Evidently U is open, A1 U and A2 U = ;. Since by hypothesis U 2 (), we have that (A1 [ A2) = (A1 [ A2) U + (A1 [ A2) Uc : (1.19) Because A1 U and A2 U = ;, from (1.19), it follows that (A1 [ A2) = (A1) + (A2); i.e., is metric outer measure. f= (=": It suces to show that () contains all closed sets. So let C X be closed and let us set dU Cc. Let D X, A df= D n C and let fAngn>1 be an increasing sequence of subsets of A as in Lemma 1.3.2. Then d(An;C) > 1 n 8 n > 1 2005 by Taylor & Francis Group, LLC

35. 1. Hausdor Measures and Capacity 25 and, from Lemma 1.3.2, we have (D n C) = (A) = lim n!+1 (An): (1.20) Since by hypothesis is a metric outer measure and the sets fAngn>1 are separated from C, we have (D C) [ An (D) > = (D C) + (An) 8 n > 1: Passing to the limit as n ! +1 and using (1.20), we obtain (D) > (D C) + (D n C): The reverse inequality is always true (subadditivity). So we obtain (D) = (D [ C) + (D n C) 8 D X: Thus C 2 () and hence B(X) (). To introduce the concept of Hausdor measure, we shall need the following notion. Recall that by (X; d) we denote a metric space. DEFINITION 1.3.4 A sequence fAngn>1 of subsets of X is a -cover of a set C, if C 1[ n=1 An and (An) 6 8 n > 1: By T(C) we denote the family of all -covers of the set C. Using this notion, we can introduce the Hausdor s-dimensional measure, s > 0. As usual, for any A X, (A) = diam (A) df= sup x;y2A d(x; y); the diameter of A (by convention diam ; df= 0). DEFINITION 1.3.5 For any s > 0, 0 < 6 +1 and C X, we dene (s) (C) df= inf fAngn>12T(C) 1X n=1 (An)s (as always we use the convention that inf ; = +1). The Hausdor s- dimensional outer measure (s) is dened by (s)(C) df= lim &0 (s) (C) = sup >0 (s) (C): 2005 by Taylor & Francis Group, LLC

36. 26 Nonlinear Analysis REMARK 1.3.6 It is easily seen that (s) is an outer measure. More- over, it is a metric outer measure. Indeed, if > 0 is less than the positive distance of two separate sets A and C, then no set in T(A[ C) can intersect both A and C and so it follows that (A) + (s) (s) (A [ C) = (s) (C): Letting & 0, we can obtain the same equality for (s). In addition by Theorem 1.3.3, (s) is Borel. The restriction of (s) on (s) is called the Hausdor s-dimensional measure. Sometimes it is convenient to consider -covers consisting of open or alternatively closed sets. In these cases, although a dierent value of (s) may be attained for > 0, the limit (s) as (s) is dierent, if & 0 is the same (see Davies (1970)). However, the limit we restrict ourselves to -covers by balls (see Besicovitch (1928)). In this case the resulting Hausdor measure is called the spherical Hausdor measure. Finally, if X = RN, it is easy to see that (s) remains the same if we consider -covers consisting only of convex sets. Next we show that for any set C X, there is a critical value s0, such that for s > s0, the corresponding Hausdor s-dimensional measure of C is zero, while for s < s0 the Hausdor s-dimensional measure of C is innite. THEOREM 1.3.7 If A RN and 0 6 s < t < +1, then (a) if (s)(A) < +1, then (t)(A) = 0; (b) if (t)(A) > 0, then (s)(A) = +1. PROOF (a) Let (s)(A) < +1 and t > s. Let fAngn>1 2 T 1 m (A). Then for any n > 1, we have (An)t (An)s = (An)ts 6 1 m ts ; so (t) 1 m (A) 6 1X n=1 (An)t 6 1 m ts 1X n=1 (An)s and thus (t) 1 t (A) 6 1 m ts (s) 1 m (A): Letting m ! +1, we obtain (t)(A) = 0. (b) Let (t)(A) > 0 and s < t. Assuming that (s)(A) < +1, from (a), we get that (t)(A) = 0, a contradiction. 2005 by Taylor & Francis Group, LLC

37. 1. Hausdor Measures and Capacity 27 f= This theorem leads to the following denition. DEFINITION 1.3.8 Let C X. If there is no s > 0, such that (s)(C) = +1, then dimC d0. Otherwise, let dimC df= sup s > 0 s: (s)(C) = +1 Then dimC is called the Hausdor dimension of C. Consider the Cantor ternary set C. It is well known that C is a nonempty, bounded, nowhere dense, perfect set in R which has Lebesgue measure zero. So the Lebesgue measure can contribute no additional information concerning the size of C. On the other hand, as we shall see the Hausdor dimension provides a more delicate sense of size. PROPOSITION 1.3.9 If C [0; 1] is the Cantor ternary set, then dimC = ln 2 ln 3 . PROOF We start with two simple observations concerning the Hausdor s-dimensional outer measure (s) on R. First note that (s) is translation invariant, namely (s)(A) = (s)(A + x) 8 A R; x 2 R (here A+x df= a+x : a 2 A ). Second, (s) is s-positive homogeneous, i.e., for every # > 0, (s)(#A) = #s(s)(A) 8 # > 0: In the construction of C we start by removing from [0; 1] the open middle third 1 3 ; 2 3 . The resulting set consists of two closed intervals 0; 1 3 and 2 3 ; 1 . Let C1 df= C 0; 1 3 and C2 df= C 2 3; 1 : Evidently C1 and C2 are translates of a multiple (by 1 3 ) of C. So we have (s)(C) = (s)(C1 [ C2) = (s)(C1) + (s)(C2) = 2(s) C2 = 2 1 3 s (s)(C) (1.21) (see Remark 1.3.6 and the observations in the beginning of this proof). From (1.21), it follows that (s)(C) = 0 or (s)(C) = +1 or 2 1 3 s = 1: 2005 by Taylor & Francis Group, LLC

38. 28 Nonlinear Analysis From the last possibility, it follows that s = ln 2 ln 3: If we can show that 0 < (s)(C) < +1, then s = ln 2 ln 3 is the Hausdor dimension of C (see Theorem 1.3.7). First we show that (s)(C) > 0. Note that d(C1;C2) > 1 3: Let 6 1 3 . Then any collection fAngn>1 2 T(C) (which can be taken to consist of open intervals; see Remark 1.3.6) can be decomposed into two subcollections of intervals fAn;1gn>1 2 T(C1) and fAn;2gn>1 2 T(C2), such that 1X n=1 (An)s = 1X n=1 (An;1)s + 1X n=1 (An;2)s: (1.22) In the right hand side of (1.22) suppose that the rst sum is smaller than the second. Because C2 is a translate of C1, the same translation when applied to the intervals fAn;1ggives a subcollection A0 n>1 n;1 n>1 2 T(C2). Also from fAn;1gn>1 we can produce in a similar way a collection fA0 ngn>1 covering C, such that (A0 n) = 3(A0 n;1) 8 n > 1: (1.23) Then, from (1.23) and the choice of s, we have 1X n=1 (An)s > 1X n=1 (An;1)s + 1X n=1 (A0 n;1)s = 2 1X n=1 (A0 n;1)s = 2 1X n=1 1 3 s (A0 n)s = 1X n=1 (A0 n)s: If any one of the intervals fA0 ngn>1 has length bigger or equal to 1 3 , we have 1X n=1 (An)s > 1 3 s = 1 2: Because C is compact, we can use only nite coverings and so min n>1 (An) > 0: The intervals fA0 ngn>1 are multiples (by (1.23)) of a subfamily of the intervals fAngn>1, hence we have 3 min n>1 (A0 n) > min n>1 (An): 2005 by Taylor & Francis Group, LLC

39. 1. Hausdor Measures and Capacity 29 If every interval A0 n has length (diameter) less than 1 3 , we can apply the same process to the cover fA0 ngn>1. After a nite number of such steps, we produce a cover fA00 ngn>1, such that max n>1 (A00 n) > 1 3 and 1X n=1 (An)s > 1X n=1 (A00 n)s; so 1X n=1 (An)s > 1 3 s = 1 2 and thus 0 < (s)(C): Next we show that (s)(C) < +1: Let fAngn>1 2 T(C) consist of open intervals. From this family, as above, we obtain covers fAn;kgn>1 of Ck for k 2 f1; 2g, such that (An;k) 6 3 8 n > 1: Again from the choice of s, we have (An)s = (An;1)s + (An;2)s; so (s) (C) > (s) 3 (C): Because (s) is nondecreasing in > 0, we infer that (s) is independent of > 0. So we can take an open interval of length greater than 1 as an open cover of C and conclude that (s)(C) 6 1. This proves that dimE = ln 2 ln 3: One can show that for every 2 [0; 1], there exists a set A R, such that dimA = . This can be done using Cantor-like sets. These are sets which share most of the properties of the Cantor ternary set, but need not be Lebesgue-null. We can construct a Cantor-like set as follows. We start with the interval [0; 1] and proceed inductively. We remove an open interval B1;1 centered at 1 2 with length less than 1. We are left with closed intervals 2005 by Taylor & Francis Group, LLC

40. 30 Nonlinear Analysis D1;1 and D1;2 each with length less than 1 2 . At the n-th step of this process we are left with closed intervals Dn;1;Dn;2; : : : ;Dn;2n each with length less than 1 2n . In the (n + 1)-st step, from each closed interval Dn;k we remove an open interval En+1;k having the same center as Dn;k and length less than the length of Dn;k. We set Sn df= 2n [ k=1 Dn;k and S df= 1 n=1 Sn: The set S is a Cantor-like set. Cantor ternary set). However, unlike the Cantor ternary set, S need not be Lebesgue-null. More precisely, consider a sequence f#ngn>1 of positive numbers, such that 1 > 2#1 > 4#2 > : : : > 2n#n > : : : : Following the construction of S above, we remove from [0; 1] an open interval centered at 1 2 and having length 12#1. The remaining closed intervals D1;1 and D1;2 each have length #1. Then from each of the intervals D1;1 and D1;2 we remove cocentric open intervals each of length #1 2#2. We are left with closed intervals D2;1, D2;2, D2;3 and D2;4 each of length #2. We continue this way. In the n-th step we are left with 2n closed intervals each with length #n. Then we have 1(S) = lim n!+1 2n#n (1 being the Lebesgue measure on R). If #n = 1 3n , then S = C is the Cantor ternary set. Although S is nowhere dense, we can have 1(S) as close to 1 as we choose. Indeed, for a given 2 (0; 1), let #n df= 1 2n n + 1 n + 1 8 n > 1: Then we have 1(S) = . Suppose that in the construction of the Cantor-like set at each step the closed subintervals are divided in the same proportions as the original, namely (D1;1) = (D1;2) = # (D2;1) = (D2;2) = (D2;3) = (D2;4) = #2 and in general (Dn;k) = #k 8 k 2 f1; : : : ; 2ng: Then the resulting Cantor-like set is denoted by S#. Arguing as in the proof of Proposition 1.3.9, we obtain the following Proposition. 2005 by Taylor & Francis Group, LLC It is known (see Hewitt & Stromberg (1975, p. 71)) that S is nonempty, compact, nowhere dense and perfect (just as the

41. 1. Hausdor Measures and Capacity 31 PROPOSITION 1.3.10 If # 2 0; 1 2 , then dim S# = ln 2 ln #. REMARK 1.3.11 If # = 1 3 , then S = C is the Cantor ternary set and Propositions 1.3.9 and 1.3.10 coincide. COROLLARY 1.3.12 For each 2 [0; 1], there exists A R, such that dimA = . PROOF If = 0, then we take A to be a singleton. If 0 < < 1, then take # = exp ln 2 < 1 2 and use Proposition 1.3.10. If = 1, let A = I = [0; 1]. Then we can easily check that (s)(A) = 8< : +1 if 0 < s < 1; 1 if s = 1; 0 if s > 1: Therefore dimA = 1. REMARK 1.3.13 An alternative way to dene the Hausdor dimension of a set A X is by dimA df= inf s > 0 s: (s)(A) = 0 In general the Hausdor dimension of a set may be any number in [0;+1] and need not be an integer. Even if dimA is an integer and k = dimA > 0, the set A need not be a k-dimensional surface" in any sense (see Federer (1969)). Next we turn our attention to the case X = RN. Let us begin by recalling the denition of the N-dimensional outer measure N. DEFINITION 1.3.14 (a) We say that Q RN is a closed N-cube, NQ if there exist ak < bk for k = 1; : : : ;N, such that Q = [ak; bk]. We set k=1 jQj df= NY (bk ak): k=1 (b) The Lebesgue N-dimensional outer measure N, for all A RN, is dened by N(A) df= inf 1X k=1 jQkj : A 1[ k=1 : Qk; Qk is closed N-cube 2005 by Taylor & Francis Group, LLC

42. 32 Nonlinear Analysis REMARK 1.3.15 Clearly the denitions of 1 and (1) on R coincide. We shall show that for any N > 1 the outer measures N and (N) are closely related. In fact they dier by a multiplicative constant. This is not easy to establish and requires some preparation which culminates to the so-called isodiametric inequality," which says that the set of maximal volume for a given diameter is the sphere. LEMMA 1.3.16 If f : RN ! [0;+1] is Lebesgue measurable, then the set H df= (x; #) 2 RN R : 0 6 # 6 f(x) is Lebesgue measurable in RN+1. PROOF Let A df= x 2 RN : f(x) = +1 : Then A is Lebesgue measurable. Let g : Ac R+ ! R+ be dened by g(x; #) df= f(x) # 8 (x; #) 2 Ac R+: Evidently g is a Caratheodory function (i.e., it is Lebesgue measurable in x 2 RN and continuous in # 2 R). Therefore g is Lebesgue measurable on Ac R+ and so H0 df= (x; #) 2 Ac R+ : # 6 f(x) is Lebesgue measurable in RN+1. Finally note that H = H0 [ (A R+): In what follows for a; b 2 RN, kakRN = 1, we introduce the following objects: L(a; b) df= b + ta : t 2 R - the line passing from b in the direction of a and P(a) df= x 2 RN : (x; a)RN = 0 - the plane passing from the origin, perpendicular to a. 2005 by Taylor & Francis Group, LLC

43. 1. Hausdor Measures and Capacity 33 DEFINITION 1.3.17 Let a 2 RN with kakRN = 1 and A RN. We dene the Steiner symmetrization of A with respect to the plane P(a) to be the set S(a;A) df= [ b 2 P(a) A L(a; b)6= ; b + ta : jtj 6 1 2(1) A L(a; b) : REMARK 1.3.18 The above dened Steiner symmetrization with respect to an (N 1)-dimensional subspace Y of RN is the operation which associates to each A RN, the set V RN, such that for every L perpendicular to Y either L A = ; and L V = ;; or L A6= ; and L V is a closed segment centered in Y and (1)(L A) = (1)(L V ): If A is compact, then V is compact too and N(A) = N(V ): Also if A is convex, then V is convex too. The next Proposition summarizes the properties of the Steiner symmetrization. PROPOSITION 1.3.19 Let A RN and a 2 RN. (a) S(a;A) 6 (A). (b) If A RN is Lebesgue measurable, then so is S(a;A) and N S(a;A) = N(A). PROOF (a) Assume that (A) < +1 or otherwise the result is trivial. Also we may assume that A is closed. For a given " > 0, let x; y 2 S(a;A) be such that S(a;A) " 6 kx ykRN : Let b df= x (x; a)RN a and c df= y (y; a)RN a: 2005 by Taylor & Francis Group, LLC

44. 34 Nonlinear Analysis Then b; c 2 P(a). Let us set r df= inf t 2 R : b + ta 2 A ; s df= sup t 2 R : b + ta 2 A ; u df= inf t 2 R : c + ta 2 A ; v df= sup t 2 R : c + ta 2 A : We may assume that without any loss of generality that v r > s u: So v r > 1 2 (v r) + 1 2 (s u) = 1 2 (s r) + 1 2 (v u) > 1 2(1) + A L(a; b) 1 2(1) : A L(a; c) Note that (x; a)RN 6 1 2(1) A L(a; b) and (y; a)RN 6 1 2(1) A L(a; c) (recall that x; y 2 S(a; b)). It follows that v r > (x; a)RN + (y; a)RN > (x y; a)RN : Hence we have S(a;A) " 2 6 kx yk2 RN 6 kb ck2 RN + (x y; a)RN 2 6 kb ck2 RN + (v r)2 = 2 (b + ra) (c + va) RN 6 (A)2 (note that A is closed and so b + ra; c + va 2 A). It follows that S(a;A) " 6 (A): Let " & 0, to conclude that S(a;A) 6 (A). (b) Recall that the Lebesgue measure N is rotation invariant. So we may take a = eN = 2 6664 0...01 3 7775 : 2005 by Taylor & Francis Group, LLC

45. 1. Hausdor Measures and Capacity 35 Then P(a) = P(eN) = RN1: Note that the function f : RN1 ! R, dened by f(b) df= (1) A L(a; b) 8 b 2 RN1; is measurable (Fubini's theorem) and N(A) = Z A f(b)dN1(b) (since 1 = (1); see Remark 1.3.15). So by virtue of Lemma 1.3.16, we have that S(a; b) df= (b; #) 2 RN1 R : f(b) 2 6 # 6 f(b) 2 n (b; 0) 2 RN1 R : A L(a; b) = ; is Lebesgue measurable in RN and, moreover, N S(a;A) = Z RN1 f(b) dN1(b) = N(A): Now we are properly equipped to prove the so-called isodiametric inequality," which states that, if in RN we consider the family of all sets with given diameter, the one with maximum Lebesgue N-dimensional outer measure (N- volume) is the sphere. THEOREM 1.3.20 (Isodiametric Inequality) For all A RN, we have (A) N(A) 6 a(N) 2 N ; where a(N) df= N2 N 2 ! is the volume of the unit ball in RN. PROOF If (A) = +1, then there is nothing to prove. So suppose that (A) < +1: 2005 by Taylor & Francis Group, LLC

46. 36 Nonlinear Analysis Let fekgNk =1 be the standard basis of RN. We introduce A1 = S(e1;A); A2 = S(e2;A1); : : : ; AN = S(eN;AN1): Let us set A = AN. Claim 1. A is symmetric with respect to the origin. By virtue of the denition of the Steiner symmetrization, we have that A1 is symmetric with respect to the plain P(e1). Let 1 6 k 6 N 1 and suppose that Ak is symmetric with respect to P(e1); : : : ; P(ek). Again Ak+1 is symmetric with respect to P(ek+1). Let us x 1 6 m 6 k and let Rm: RN ! RN be reection with respect to P(em). Let b 2 P(ek+1). Because Rm(Ak) = Ak, we have (1) Ak L(ek+1; b) = (1) Ak L(ek+1;Rm(b)) ; so t 2 R : b + tek+1 2 Ak+1 = t 2 R : Rm(b) + tek+1 2 Ak+1 and thus Rm(Ak+1) = Ak+1; i.e., Ak+1 is symmetric with respect to P(em). It follows that A = AN is symmetric with respect to P(e1); : : : ; P(eN), hence it is symmetric with respect to the origin. Claim 2. N(A) 6 N2 N 2 ! (A) 2 N . Let x 2 A. Then because of Claim 1, we have x 2 A and so 2 kxkRN 6 (A): Hence A B(A) 2 (0) = y 2 RN : kykRN 6 (A) 2 and so N(A) 6 N B(A) 2 (0) 6 N2 N 2 ! (A) 2 N : Using Claim 2, we can have the isodiametric inequality. Note that A RN is Lebesgue measurable and so by Proposition 1.3.19, we have N A = N A and A 6 A : 2005 by Taylor & Francis Group, LLC

47. 1. Hausdor Measures and Capacity 37 Using Claim 2, it follows that N(A) 6 N = N A A 6 N2 N 2 ! (A ) 2 N 6 N2 N 2 ! (A) 2 N = N2 N 2 ! (A) 2 N : THEOREM 1.3.21 If A RN, then N(A) = cN(N)(A), with cN df= N2 2N N 2 ! . PROOF For a given " > 0, we can nd a cover fCngn>1 of A consisting of closed, convex sets, such that 1X n=1 (Cn)N 6 (N)(A) + ": By virtue of Theorem 1.3.20, we have N(Cn) 6 cN(Cn)N 8 n > 1: So N(A) 6 1X n=1 N(Cn) 6 cN 1X n=1 (Cn)N 6 cN(N)(A) + cN": Let " & 0 to conclude that N(A) 6 cN(N)(A): (1.24) To prove the opposite inequality, rst we show that (N) is absolutely continuous with respect to N (see Denition A.2.22). Note that for any N-cube Q, we have N(Q) = jQj 6 (Q) p N N : So for a given > 0, we have (N) (A) 6 inf Qn -N-cube A 1S n=1 Qn (Qn) 6 1X n=1 (Qn) 6 p N N N(A): 2005 by Taylor & Francis Group, LLC

48. 38 Nonlinear Analysis Let & 0, to conclude that (N) is absolutely continuous with respect to N (see Denition A.2.22). Next for a given "; > 0, we can nd a cover fQngn>1 of A consisting of N-cubes, such that (Qn) < 8 n > 1 and 1X n=1 N(Qn) 6 N(A) + ": (1.25) We may suppose that N-cubes are open by expanding them slightly so that the above inequality remains valid. Invoking Vitali's covering theorem (see Theorem 1.2.5), for every n > 1 we can nd disjoint balls fBn;kgk>1 contained in Qn, such that (Bn;k) 6 and N Qn n 1[ k=1 Bn;k = 0: By virtue of the absolute continuity of (N) with respect to N, we have (N) Qn n 1[ k=1 Bn;k = 0 and (N) Qn n 1[ k=1 Bn;k = 0: Therefore, using (1.25), we have (N) (A) 6 1X k=1 (N) (Qn) 6 1X n=1 1X k=1 (N) (Bn;k) + 1X n=1 (N) Qn n 1[ k=1 Bn;k 6 1X n=1 1X k=1 (Bn;k)N = 1X n=1 1X k=1 1 cN N(Bn;k) 6 1 cN 1X n=1 N(Qn) 6 1 cN N(A) + " cN : Let "; & 0, to conclude that cN(N)(A) 6 N(A): (1.26) From (1.24) and (1.26), we conclude that N = cN(N): 2005 by Taylor & Francis Group, LLC

49. 1. Hausdor Measures and Capacity 39 REMARK 1.3.22 Some authors, in order to get rid of the multiplicative constant cN, normalize the denition of the Hausdor measures on RN. So if C RN, 0 6 s < +1, 0 < 6 +1, they set (s) (C) df= inf C 1S n=1 An (An) 6 1X n=1 (An) a(s) 2 s ; where a(s) df= s2 ( s 2 + 1) . Here (s) df= Z+1 0 xs1ex dx is the gamma Euler function. The Hausdor s-dimensional outer measure (s) is dened by (s)(C) = lim &0 (s) (C) = sup >0 (s) (C) (cf., e.g., Evans & Gariepy (1992, p. 60)) . Recall that N B(x; r) = a(N)rN 8 x 2 RN: In this case Theorem 1.3.21 says that N = (N): Note that (0) is the counting measure. Let us prove some further properties of the Hausdor measures on RN. PROPOSITION 1.3.23 Let 0 6 s < +1. We have (a) (s)(A) = 0 for all A RN and all s > N. (b) (s)(A) = s(s)(A) for all A RN and all > 0. (c) (s) K(A) = (s)(A) for all A RN and for any ane isometry K: RN ! RN. PROOF (a) Let Q = (0; 1)N and let m > 1 be an integer. For k = (ki)Ni =1 2 K df= f0; : : : ;m 1gN; we set Qk df= NY i=1 ki m ; ki + 1 m : 2005 by Taylor & Francis Group, LLC

50. 40 Nonlinear Analysis Note that Q = [ k2K Qk and (Qk) = p N m : So we have (s) p N m (Q) 6 X k2K (Qk)s = mNs p N s : Letting m ! +1, since s > N, we obtain (s)(Q) = 0; from which it follows that (s)(RN) = 0: (b) Note that for all C RN, we have (C) = (C): So the result follows at once from Denition 1.3.5. (c) Note that for all C RN, we have K(C) = (C): Again the result follows from Denition 1.3.5. The next Proposition suggests a convenient way to check that (s) vanishes on a set. PROPOSITION 1.3.24 If A RN, 0 < 6 +1 and 0 6 s < +1 are such that (s) (A) = 0, then (s)(A) = 0. PROOF If s = 0, then (0) (A) = 0 implies that A = ; and so (0)(A) = 0. So suppose that s > 0. For a given " > 0, we can nd fCngn>1, such that A 1[ n=1 Cn; (Cn) 6 and 1X n=1 (Cn)s 6 ": Evidently (Cn)s 6 " 8 n > 1 and so (s) " (A) 6 ". Let " & 0, to conclude that (s)(A) = 0: 2005 by Taylor & Francis Group, LLC

51. 1. Hausdor Measures and Capacity 41 Taking into account that for a Lipschitz continuous function with constant c > 0, for every A RN, we have f(A) 6 c(A); and we obtain the following result. PROPOSITION 1.3.25 If f : RN ! RM is a Lipschitz continuous function with Lipschitz constant c > 0 (see Denition 1.5.1), A RN and 0 6 s < +1, then (s) f(A) 6 cs(s)(A). We conclude this section by returning to the notion of Hausdor dimension (see Denition 1.3.8) and having a second look at this concept. The Hausdor dimension has an intuitive appeal when familiar objects are under consideration. So for example dimRN = N (see Theorem 1.3.21). Suppose we want to determine the Hausdor dimension of a curve C R3. Our rst guess will be that dimC = 1. But recall that there are curves in R3 which ll the unit cube. Such a curve must have Hausdor dimension 3. Therefore we must proceed with caution. DEFINITION 1.3.26 Let (X; d) be a metric space. (a) By a curve in X we mean the image f [0; 1] of a continuous function f : [0; 1] ! X. (b) The length of a curve C = f [0; 1] is dened by l(C) df= sup Xm k=1 d f(xk1); f(xk) ; where the supremum is taken over all partitions 0 = x0 < x1 < : : : < xm = 1 of [0; 1]: (c) The curve C is said to be rectiable, if l(C) < +1. REMARK 1.3.27 A curve C is a continuum, i.e., a compact and connected set in X. In particular then a curve is a Borel set; hence it is also (s)-measurable. Moreover, if in Denition 1.3.26(a) f is injective, then f1 exists and is continuous and so C is the homeomorphic image of [0; 1]. Also in Denition 1.3.26(a), we can replace [0; 1] by any closed bounded interval [a; b]. Some authors require f to be injective. 2005 by Taylor & Francis Group, LLC

52. 42 Nonlinear Analysis PROPOSITION 1.3.28 If (X; d) is a metric space, f : [0; 1] ! X is a nonconstant curve with length l and C = f [0; 1] , then (a) 0 < (1)(C) 6 l; (b) if f is injective, then (1)(C) = l. Therefore, if l is rectiable (i.e., l < +1), then dimC = 1. PROOF (a) First we show that (1)(C) 6 l: Assume that l < +1 or otherwise there is nothing to prove. Let fAkgmk =1 be a collection of closed subarcs of C, such that C = m[ k=1 Ak; (Ak) 6 1 n and (1) 1 n (C) 6 Xm k=1 (Ak): (1.27) Let us explicitly construct the subarcs Ak for k 2 f1; : : : ;mg. Note that f is uniformly continuous and so we can nd > 0, such that d f(x); f(y) < 1 n 8 x; y 2 [0; 1]; jx yj < : Consider a partition 0 = x0 < x1 < : : : < xn = 1 of [0; 1]; such that jxk xk1j < 8 k 2 1; : : : ;m : Let Ak df= f [xk1; xk] ; 8 k 2 1; : : : ;m : Evidently the subarcs fAkgmk =1 cover C and d f(xk1); f(xk) 6 (Ak) < 1 n 8 k 2 f1; : : : ;mg: Note that every Ak is compact and so we can nd points yk; zk 2 [xk1; xk], yk 6 zk, such that d f(yk); f(zk) = (Ak): We generate the ner partition 0 6 y1 6 z1 6 y2 6 z2 6 : : : 6 ym 6 zm 6 1: 2005 by Taylor & Francis Group, LLC

53. 1. Hausdor Measures and Capacity 43 From (1.27), we have (1) 1 n (C) 6 Xm k=1 (Ak) = Xm k=1 d f(yk); f(zk) 6 l: Passing to the limit as n ! +1, we obtain that (1)(C) 6 l. Next we show that 0 < (1)(C). To this end note that if 0 6 a < b 6 1, then d f(a); f(b) 6 (1) : (1.28) f([a; b]) To see this let h: E df= f([a; b]) ! R be the function h(u) df= d u; f(a) : Evidently h is a Lipschitz continuous function with Lipschitz constant 1 and J df= 0; h(b) = 0; d f(a); f(b) h(E): So, from Proposition 1.3.25, we have d f(a); f(b) = 1(J) = (1)(J) 6 (1) 6 (1)(E): h(E) This proves inequality (1.28). But from (1.28) and since for appropriately chosen a; b we have d f(a); f(b) > 0 (recall that the curve is nonconstant), we conclude that 0 < (1)(C). (b) Now suppose that f is injective. Let 0 = x0 < x1 < : : : < xm = 1 be a partition of [0; 1]. The sets Ak df= f [xk1; xk] are pairwise disjoint Borel subsets of X. Using inequality (1.28) on each subarc, we obtain Xm k=1 d f(xk1); f(xk) 6 Xm k=1 (1) f [xk1; xk] = (1) m[ k=1 f [xk1; xk] = (1) f [0; 1] = (1)(C): Since the partition of [0; 1] was arbitrary, it follows that l 6 (1)(C). Com- bining this with (a), we obtain that l = (1)(C). 2005 by Taylor & Francis Group, LLC

54. 44 Nonlinear Analysis 1.4 Dierentiation of Hausdor Measures From the general measure theory, we know that the dierentiation theory of real functions can be extended to a theory of dierentiation for measures, which has many similar features and interesting problems. For the Lebesgue measures N, N > 1, one of the basic results of this theory is the so-called Lebesgue density theorem, which we recall here. THEOREM 1.4.1 (Lebesgue Density Theorem) If A RN is a Lebesgue measurable set, then lim r&0 N(Br(x) A) N(Br(x)) = 8< : 1 for N-a.a. x 2 A; 0 for N-a.a. x 2 RN n A: DEFINITION 1.4.2 Let A RN and x 2 RN. We say that: (a) x is a point of density of A, if lim r&0 N(Br(x) A) N(Br(x)) = 1; (b) x is a point of dispersion of A, if lim r&0 N(Br(x) A) N(Br(x)) = 0: REMARK 1.4.3 According to Theorem 1.4.1, we see that N-almost every point of A is a point of density of A and N-almost every point of RN nA is a point of dispersion of A. We can think that the point of density of a set A form a kind of measure theoretic interior of A, while the points of dispersion of A form a kind of measure theoretic exterior of A. The purpose of this section is to establish analogs of Theorem 1.4.1 for lower dimensional Hausdor measures. In what follows we work in RN and 1 < s < N. THEOREM 1.4.4 If A RN is (s)-measurable and (s)(A) < +1, then lim r&0 (s)(Br(x) A) (2r)s = 0 for (s)-a.a. x 2 RN n A: 2005 by Taylor & Francis Group, LLC

55. 1. Hausdor Measures and Capacity 45 PROOF For every t > 0, let Ct df= x 2 RN n A : lim sup r&0 (s)(Br(x) A) : (2r)s > t To nish the proof it is enough to show that (s)(Ct) = 0 8 t > 0: Fix " > 0. We know that (s)bA is a Radon measure (see Proposition 1.1.9). So we can nd K A compact, such that (s)(A n K) 6 " (see Proposition 1.1.10(b)). Let U df= RN n K: Then U is open and Ct U: For xed > 0, we consider the family of closed balls T df= Br(x) : Br(x) U; 0 < r < ; (s)(Br(x) A) (2r)s > t : Without any loss of generality we may assume that T6= ; or otherwise Ct = ; and so (s)(Ct) = 0: Invoking Proposition 1.2.1, we can nd a sequence Brn(xn) n>1 of disjoint elements in T , such that Ct 1[ n=1 B5rn(xn): Then we have (s) 10(Ct) 6 1X n=1 (10rn)s 6 5s t 1X n=1 (s) Brn(xn) A 6 5s t (s)(U A) = 5s t (s)(A n K) 6 5s" t : Let & 0, to obtain (s)(Ct) 6 5s" t : Since " > 0 was arbitrary, we conclude that (s)(Ct) = 0. 2005 by Taylor & Francis Group, LLC

56. 46 Nonlinear Analysis To have a complete analog of Theorem 1.4.1, we need to check and see if something can be said about the density of A at its points. To do this we will make use of Proposition 1.2.4. THEOREM 1.4.5 If A RN is (s)-measurable and (s)(A) < +1, then 1 2s 6 lim sup r&0 (s)(Br(x) A) (2r)s 6 1 for (s)-a.a. x 2 A: PROOF First we show that lim sup r&0 (s)(Br(x) A) (2r)s 6 1 for (s)-a.a. x 2 A: (1.29) To this end, for every t > 1, we introduce the set Ct A dened by Ct df= x 2 A : lim sup r&0 (s)(Br(x) A) (2r)s > t : Fix " > 0. Again (s)bA is a Radon measure (see Proposition 1.1.9). We can nd an open set U RN, such that Ct U and (s)(U A) " 6 (s)(Ct) (1.30) (see Proposition 1.1.10(b)). We introduce the family T of closed balls dened by T df= Br(x) : Br(x) U; 0 < r < ; (s)(Br(x) A) (2r)s > t : By virtue of Proposition 1.2.4, we can nd a sequence Brn(xn) n>1 of disjoint balls in T , such that Ct m[ n=1 Brn(xn) [ 1[ n=m+1 B5rn(xn) 8 m > 1: Then for > 0, we have (s) 10(Ct) 6 Xm n=1 (2rn)s + 1X n=m+1 (10rn)s 6 1 t Xm n=1 (s) + Brn A 5s t 1X n=m+1 (s) Brn(xn) A 6 1 t (s)(U A) + 5s t (s) 1[ n=m+1 Brn(xn) A 8 m > 1: 2005 by Taylor & Francis Group, LLC

57. 1. Hausdor Measures and Capacity 47 Using (1.30) and letting m ! +1, we obtain (s) 10(Ct) 6 1 t (s)(U A) 6 1 t (s)(Ct) + " : Letting & 0, we see that (s)(Ct) 6 1 t (s)(Ct) + " : Since " > 0 was arbitrary, we nally have that (s)(Ct) 6 1 t (s)(Ct); i.e., (s)(Ct) = 0 (recall that t > 1). This proves (1.29). Next we show that 1 2s 6 lim sup r&0 (s)(Br(x) A) (2r)s for (s)-a.a. x 2 A: (1.31) For a given ; 2 (0; 1), we introduce the set A(; ) A, dened by A(; ) df= x 2 A : (s) (C A) 6 (C)s for all C RN; with (C) 6 and x 2 C : Let fCngn>1 be a -cover of A(; ), such that A(; ) 1[ n=1 Cn and (Cn) 6 ; and Cn A(; )6= ; 8 n > 1: So (s) A(; ) 6 1X n=1 (s) Cn A(; ) 6 1X n=1 (s) (Cn A) 6 1X n=1 (Cn)s and from Denition 1.3.5, we see that (s) A(; ) 6 (s) A(; ) : 2005 by Taylor & Francis Group, LLC

58. 48 Nonlinear Analysis Since 0 < < 1 and (s) A(; ) < +1; we have (s) A(; ) = 0: In particular, from Proposition 1.3.24, we see that (s) = 0: (1.32) A(; 1 ) Set D1 df= x 2 A : lim sup r&0 (s)(Br(x) A) (2r)s < 1 2s : If x 2 D1, then we can nd > 0, such that (s)(Br(x) A) (2r)s 6 1 2s 8 r 2 (0; ]: (1.33) For any C RN, with x 2 C A and (C) 6 ; from (1.33), we have (s) (C A) 6 (s)(C A) 6 (s) 6 (1 )(C)s: B(C)(x) A So it follows that x 2 A(; 1 ). Therefore, we have D1 1[ n=1 1 n A ; 1 1 n ; and, using also (1.32), we have (s)(D1) = 0: Thus we infer that (1.31) is true. For a given locally integrable function, we can establish the Hausdor measure of the set where the function is locally large. To do this we shall need the so-called Lebesgue dierentiation theorem or Lebesgue-Besicovitch dierentiation theorem THEOREM 1.4.6 (Lebesgue-Besicovitch Dierentiation Theorem) If f 2 L1 loc RN;RM , then lim r&0 1 N(Br(x)) Z Br(x) f(y) f(x) RM dN(y) = 0 for N-a.a. x 2 RN: 2005 by Taylor & Francis Group, LLC

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