An introduction to the geometrical analysis of vector fields : with applications to maximum principles and lie groups PDF
Preview An introduction to the geometrical analysis of vector fields : with applications to maximum principles and lie groups
An Introduction to the Geometrical Analysis of Vector Fields with Applications to Maximum Principles and Lie Groups TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk An Introduction to the Geometrical Analysis of Vector Fields with Applications to Maximum Principles and Lie Groups Stefano Biagi Università Politecnica delle Marche, Italy Andrea Bonfiglioli Alma Mater Studiorum - Università di Bologna, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Biagi, Stefano, author. | Bonfiglioli, Andrea, author. Title: An introduction to the geometrical analysis of vector fields : with applications to maximum principles and Lie groups / by Stefano Biagi (Università Politecnica delle Marche, Italy), Andrea Bonfiglioli (University of Bologna, Italy). Description: New Jersey : World Scientific, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2018043034 | ISBN 9789813276611 (hardcover : alk. paper) Subjects: LCSH: Vector fields. | Maximum principles (Mathematics) | Lie groups. Classification: LCC QA613.619 .B53 2018 | DDC 516/.182--dc23 LC record available at https://lccn.loc.gov/2018043034 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2019 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11165#t=suppl Printed in Singapore We dedicatethisbook toErmannoLanconelli ontheoccasionofhisbecomingEmeritusProfessor ofAlmaMaterStudiorum,UniversityofBologna Withourgratitudeforhisinvaluableteaching TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk Preface VECTOR fields, and many of their applications, are the main subjects of this book. Usually,studentsofscientificdisciplinesfirstmeetvectorfieldsduring theirundergraduatecourses,eitherinconnectionwithOrdinaryDifferentialEqua- tions (ODEs, in the sequel), or within physics, studying conservative forces and potentials. Laterintheirstudies,manystudentsencounterthegeometricmeaning ofvectorfieldsinthecontextofdifferentialgeometryandmanifoldtheory. Vector fieldsreappearinmoreadvancedstudies,suchasinLiegrouptheoryorthrough- outtheliteratureofPartialDifferentialEquations(PDEs,inthesequel). Itisman- ifest, owing to their interdisciplinarynature, that vector fields play a remarkable rolenotonlyinmathematics,butalsoinphysicsandinappliedmathematics. The aim of this book is to provide the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic propertiesofvectorfieldsandflows,andendingwithsomeoftheircountlessap- plications. Thebuilding blocksofrichbackgroundmaterial(Chaps. 1-to-7)com- prisethefollowingtopics,justtonameafewofthem: - commutators andLiederivatives; the semi-grouppropertyandtheequationof variation;globalvectorfields;C0,Ck,Cω-dependence; - relatedness,invarianceandcommutability;Hadamard-typeformulas; - compositionofflows: Taylorapproximationsandexactformulas; - the algebraic Campbell-Baker-Hausdorff-Dynkin (CBHD, in the sequel) Theo- rem; the CBHD series and its convergence; the CBHD operation and its local associativity;Poincaré-typeODEs; - iteratedcommutatorsandtheHörmanderbracket-generatingcondition;connec- tivity;sub-unitcurvesandthecontroldistance. Oncethebackgroundmaterialisestablished,theapplicationsmainlydeal,accord- ingtoourchoice,withthefollowingthreesettings(Chaps. 8-to-18): (I) ODEtheory; (II) maximumprinciples(weak,strongandpropagationprinciples); (III) Liegroups(withanemphasisontheconstructionofLiegroups). vii viii GeometricalAnalysisofVectorFieldswithApplications Beforedescribingtheseparatecontents,wemakesomeoverallcommentsonthese applications,thusprovidingageneralmotivationforthebook. (I). First, we would like to focus on the leading role played by ODEs in this monograph. Thisroleistwofold: - weshallobtainapplicationsofourtheorytoODEs;and - weshalluseODEsasatoolforthederivationofmanyresultsofourtheory. Thus ODEsintervene, in a manner of speaking, in a “circularway”: asanin- strument and, atthe same time, asanobject of applications. On the one hand, it isnotsurprisingthatODEsprovideamajortoolinthetheoriesofvectorfieldsor ofLiegroups: justthinkthattheflowofavectorfield(hence,theexponentialmap for a Liegroup) isthe solution to anODE. Onthe other hand, some applications oftheanalysisofvectorfieldstoODEs,especiallyconcerningtheCBHDTheorem, arelessknownintheliterature,andarethereforeamongstourmainachievements. An example may help clarify this double role of ODEs: since the advent of modernLiegrouptheory,thetheoremexpressingthegroup-multiplicationoftwo group-exponentialsasanothergroup-exponential,say Exp(X)Exp(Y)=Exp(Z(X,Y)), (P.1) isknown,quintessentially,astheCBHDTheorem.1 Mostofitsproofs(seee.g.,the onein[Varadarajan(1984)])relyonanODEvalidintheassociatedLiealgebra(see (P.8)),whichwemaytracebacktononeotherthanPoincaré(inthecontextofLie groupsoftransformations). ItdoesnotescapethenoticeofexpertsthatthisODE is validnot onlyin the Lie algebraof a Liegroup, butalsoinother more general settings; however, buriedasittendstobeinLiegrouptheory, the beautifulODE argumentmayeasilyescapetheattentionofastudent,orofanon-expert. Most importantly, the ODE tool used in the proof of (P.1) can be formulated as an independent ODE theorem in a completely autonomous way, and so can be 1ThefactthatthisLiegroupresulthasapredecessorinnon-commutativealgebrawhichshould beconsideredastheCBHDTheoremparexcellencehasalreadybeendiscussedin[BonfiglioliandFulci (2012)]. Intheliterature, Dynkin’snameisnotoftenusedasalabelforthistheorem; ourchoice is insteadconsistentwiththehistoricalpresentationin[BonfiglioliandFulci(2012)](seealso[Achilles andBonfiglioli(2012)]formoredetails). Preface ix viewedindependentlyofanyLiegroupcontext. Itcanbethoughtof(andtaught) asaresultofODEtheory;itcanevenbeexploitedintheconstructionofLiegroups (aswedointhisbook),aprocedurethatispossibleonlyaftertheODEresulthas beenliberatedfromtheLiegroupframework. As a novelty with respect to the existing literature, this is our point of view withwhatweshallname‘theCBHDTheoremforODEs’.InthechoiceoftheODE prerequisitesaswell,weshalltakethelibertytomakesomenon-standardchoices. For example, even within the ODE theory itself there are results which are pre- sentedmoreandmorerarelyinODEtextbooks,andwehopethatthismonograph maybeagoodoccasion(duetotheirsubsequentapplications)tobringthemback totheattentionofstudents. Forinstance,thisisthecasewiththeCω-dependence results(seeApp.B),orwiththeintegralversionoftheequationofvariationinthe non-autonomouscase(Chap.12). (II).Weshouldnowdiscloseourintentinintroducingfourchaptersdevotedto Maximum Principles (as applications of the vector-fields-and-flows machinery) in theirmanydeclinations: WeakandStrongMaximumPrinciples;MaximumProp- agationalongprincipalordriftvectorfields. We clearly have in mind the possible applications of Maximum Principles to PDEs,forexampleinobtaininganexhaustivePotentialTheoryforclassesofpartial differential operators. The theory presented here is general enough to comprise wideclassesofoperators,moregeneral(forexample)thanthesub-Laplacianson Carnot groups considered in [Bonfiglioli et al. (2007)]. As a consequence, Chaps. 8–11ofthebookmaybeusefulalsotoyoungresearchersinPDEs. (III).AsregardstheLiegrouptheory,ourapplicationsarebasedonthemulti- facetedusethatonecanmakeofthepreviouslyestablishedmachineryrelatingthe CBHDTheoremandvectorfields/flows: (1) intheproofof(P.1),whichreducestoafewlinesoncetheODEversionofthe CBHDTheoremisavailable; (2) inthe study of the CBHDseriesand of the CBHDoperationon finite-dimen- sionalLiealgebras; (3) inimplementingLie’sThirdTheoreminsomespecialbutmeaningfulcases:inits localformulation,andalsointheglobalonewhendealingwithnilpotentalge- bras;thetoolsmentionedin(2)provetobeparticularlypowerfulandnatural touseinsolvingthisproblem; (4) intheconstructionofLiegroupsstartingfromtheexponentiationofLiealgebras ofvectorfields,withoutusingtheglobalversionofLie’sThirdTheorem; (5) inasimpleconstructionofCarnotgroups(equippedwiththeirhomogeneous structure),startingfromthestratifiedLiealgebras.
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