Modern Methods in the Calculus p of Variations:L Spaces Irene Fonseca Giovanni Leoni Modern Methods in the Calculus p of Variations: L Spaces Irene Fonseca Gi ovanni Leoni Department of Mathematical Sciences Department of Mathematical Sciences Carnegie Mellon University Carnegie Mellon University Pittsburgh, PA 15213 Pittsburgh, PA 15213 USA USA [email protected] [email protected] ISBN: 978-0-387-35784-3 e-ISBN : 978-0-387-69006-3 Library ofCongress Control Number:2007931775 Mathematics Subject Classification (2000): 49-00, 49-01, 49-02, 49J45, 28-01, 28-02, 28B20, 52A © 2007 Springer Science+Business Media,LLC All rights reserved.This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media,LLC,233 Spring Street, New York,NY,10013,USA),except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation,computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication oftrade names,trademarks,service marks,and similar terms,even if they are not identified as such,is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 springer.com To our families Preface In recent years there has been renewed interest in the calculus of variations, motivated in part by ongoing research in materials science and other disci- plines. Often, the study of certain material instabilities such as phase transi- tions, formation of defects, the onset of microstructures in ordered materials, fractureanddamage,leadstothesearchforequilibriathroughaminimization problem of the type min{I(v): v ∈V}, wheretheclassV ofadmissiblefunctionsvisasubsetofsomeBanachspace V. Thisistheessenceofthecalculusofvariations:theidentificationofneces- saryandsufficientconditionsonthefunctionalI thatguaranteetheexistence of minimizers. These rest on certain growth, coercivity, and convexity condi- tions,whichoftenfailtobesatisfiedinthecontextofinterestingapplications, thusrequiringtherelaxationoftheenergy.Newideaswereneeded,andthein- troduction of innovative techniques has resulted in remarkable developments in the subject over the past twenty years, somewhat scattered in articles, preprints, books, or available only through oral communication, thus making it difficult to educate young researchers in this area. This is the first of two books in the calculus of variations and measure theory in which many results, some now classical and others at the forefront of research in the subject, are gathered in a unified, consistent way. A main concern has been to use contemporary arguments throughout the text to re- visit and streamline well-known aspects of the theory, while providing novel contributions. The core of this book is the analysis of necessary and sufficient conditions for sequential lower semicontinuity of functionals on Lp spaces, followed by relaxation techniques. What sets this book apart from existing introductory textsinthecalculusofvariationsistwofold:Insteadoflayingdownthetheory in the one-dimensional setting for integrands f = f(x,u,u(cid:1)), we work in N dimensions and no derivatives are present. In addition, it is self-contained in VIII Preface the sense that, with the exception of fundamentally basic results in measure theory that may be found in any textbook on the subject (e.g., Lebesgue dominated convergence theorem), all the statements are fully justified and proved. This renders it accessible to beginning graduate students with basic knowledge of measure theory and functional analysis. Moreover, we believe thatthistextisuniqueasareferencebookforresearchers,sinceittreatsboth necessaryandsufficientconditionsforwell-posednessandlowersemicontinuity of functionals, while usually only sufficient conditions are addressed. The central part of this book is Part III, although Parts I and II contain original contributions. Part I covers background material on measure theory, integration,andLp spaces,anditcombinesbasicresultswithnewapproaches to the subject. In particular, in contrast to most texts in the subject, we do not restrict the context to σ-finite measures, therefore laying the basis for the treatment of Hausdorff measures, which will be ubiquitous in the setting of the second volume, in which gradients will be present. Moreover, we call attention to Section 1.1.4, on “comparison between measures”, which is com- pletelynovel:TheRadon–NikodymtheoremandtheLebesguedecomposition theoremareprovedforpositivemeasureswithoutourhavingfirsttointroduce signedmeasures,asisusualintheliterature.Thenewargumentsarebasedon an unpublished theorem due to De Giorgi treating the case in which the two measuresinplayarenotσ-finite.Here,asDeGiorgi’stheoremstates,adiffuse measuremustbeaddedtotheabsolutelycontinuousandsingularpartsofthe decomposition. Also, we give a detailed proof of the Morse covering theorem, which does not seem to be available in other books on the subject, and we derive as a corollary the Besicovitch covering theorem instead of proving it directly. PartIIstreamlinesthestudyofconvexfunctions,andthetreatmentofthe direct method of the calculus of variations introduces the reader to the close connection between sequential lower semicontinuity properties and existence of minimizers. Again here we present an unpublished theorem of De Giorgi, the approximation theorem for real-valued convex functions, which provides anexplicitformulafortheaffinefunctionsapproximatingagivenconvexfunc- tionf.Amajoradvantageofthischaracterizationisthatadditionalregularity hypotheses on f are reflected immediately on the approximating affine func- tions. In Part III we treat sequential lower semicontinuity of functionals defined on Lp, and we separate the cases of inhomogeneous and homogeneous func- tionals. The latter are studied in Chapter 5, where (cid:1) I(u):= f(v(x))dx E with E a Lebesgue measurable subset of the Euclidean space RN, f : Rm → (−∞,∞] and v ∈ Lp(E;Rm) for 1 ≤ p ≤ ∞. This material is intended for anintroductorygraduatecourseinthecalculusofvariations,sinceitrequires onlybasicknowledgeofmeasuretheoryandfunctionalanalysis.Wetreatboth Preface IX boundedandunboundeddomainsE,andweaddressmosttypesofstrongand weak convergence. In particular, the setting in which the underlying conver- (cid:1) gence is that of (C (E)) is new. b Chapter 6 and Chapter 7 are devoted to integrands f = f(x,v) and f = f(x,u,v), respectively, and are significantly more advanced, since the proofs ofthenecessitypartsareheavilyhingedontheconceptofmultifunctions.An important tool here is selection criteria, and the reader will benefit from a comprehensive and detailed study of this subject. Finally, Chapter 8 describes basic properties of Young measures and how they may be used in relaxation theory. The bibliography aims at giving the main references relevant to the con- tents of the book. It is by no means exhaustive, and many important contri- butions to the subject may have failed to be listed here. To conclude, this text is intended as a graduate textbook as well as a ref- erence for more-experienced researchers working in the calculus of variations, and is written with the intention that readers with varied backgrounds may access different parts of the text. This book prepares the ground for a second volume, since it introduces and develops the basic tools in the calculus of variations and in measure the- ory needed to address fundamental questions in the treatment of functionals involving derivatives. Finally, in a book of this length, typos and errors are almost inevitable. The authors will be very grateful to those readers who will write to either [email protected] or [email protected] indicating those that theyhavefound.Alistoferrorsandmisprintswillbemaintainedandupdated at the web page http://www.math.cmu.edu/˜leoni/book1. Pittsburgh, Irene Fonseca month 2007 Giovanni Leoni Contents Part I Measure Theory and Lp Spaces 1 Measures .................................................. 3 1.1 Measures and Integration................................. 3 1.1.1 Measures and Outer Measures ...................... 3 1.1.2 Radon and Borel Measures and Outer Measures....... 22 1.1.3 Measurable Functions and Lebesgue Integration ....... 37 1.1.4 Comparison Between Measures...................... 55 1.1.5 Product Spaces ................................... 77 1.1.6 Projection of Measurable Sets....................... 83 1.2 Covering Theorems and Differentiation of Measures in RN .... 90 1.2.1 Covering Theorems in RN .......................... 90 1.2.2 Differentiation Between Radon Measures in RN .......103 1.3 Spaces of Measures ......................................113 1.3.1 Signed Measures ..................................113 1.3.2 Signed Finitely Additive Measures...................119 1.3.3 Spaces of Measures as Dual Spaces ..................123 1.3.4 Weak Star Convergence of Measures .................129 2 Lp Spaces..................................................139 2.1 Abstract Setting ........................................139 2.1.1 Definition and Main Properties......................139 2.1.2 Strong Convergence in Lp ..........................148 2.1.3 Dual Spaces ......................................156 2.1.4 Weak Convergence in Lp ...........................171 2.1.5 Biting Convergence................................184 2.2 Euclidean Setting........................................190 2.2.1 Approximation by Regular Functions ................190 2.2.2 Weak Convergence in Lp ...........................198 2.2.3 Maximal Functions ................................208 2.3 Lp Spaces on Banach Spaces ..............................218 XII Contents Part II The Direct Method and Lower Semicontinuity 3 The Direct Method and Lower Semicontinuity.............231 3.1 Lower Semicontinuity ....................................231 3.2 The Direct Method ......................................245 4 Convex Analysis...........................................247 4.1 Convex Sets ............................................247 4.2 Separating Theorems ....................................254 4.3 Convex Functions .......................................258 4.4 Lipschitz Continuity in Normed Spaces.....................262 4.5 Regularity of Convex Functions ...........................266 4.6 Recession Function ......................................288 4.7 Approximation of Convex Functions .......................293 4.8 Convex Envelopes .......................................300 4.9 Star-Shaped Sets ........................................318 Part III Functionals Defined on Lp 5 Integrands f = f (z).......................................325 5.1 Well-Posedness..........................................326 5.2 Sequential Lower Semicontinuity ..........................331 5.2.1 Strong Convergence in Lp ..........................331 5.2.2 Weak Convergence and Weak Star Convergence in Lp ..334 5.2.3 Weak Star Convergence in (cid:2)the S(cid:2)ense o(cid:3)f(cid:3)Measures......340 (cid:1) 5.2.4 Weak Star Convergence in C E;Rm .............350 b 5.3 Integral Representation ..................................354 5.4 Relaxation..............................................364 5.4.1 Weak Convergence and Weak Star Convergence in Lp, 1≤p≤∞........................................365 5.4.2 Weak Star Convergence in the Sense of Measures......369 5.5 Minimization ...........................................373 6 Integrands f = f (x,z).....................................379 6.1 Multifunctions ..........................................380 6.1.1 Measurable Selections..............................380 6.1.2 Continuous Selections..............................395 6.2 Integrands..............................................401 6.2.1 Equivalent Integrands..............................401 6.2.2 Normal and Carath´eodory Integrands ................404 6.2.3 Convex Integrands.................................413 6.3 Well-Posedness..........................................428 6.3.1 Well-Posedness, 1≤p<∞ .........................428 6.3.2 Well-Posedness, p=∞.............................435