Partial differential equations: A unified Hilbert space approach PDF
Preview Partial differential equations: A unified Hilbert space approach
De Gruyter Expositions in Mathematics 55 Editors Victor P.Maslov, Moscow, Russia Walter D.Neumann, New York, USA Markus J.Pflaum, Boulder, USA Dierk Schleicher, Bremen, Germany Raymond O.Wells, Bremen, Germany Rainer Picard Des McGhee Partial Differential Equations A Unified Hilbert Space Approach De Gruyter Mathematical Subject Classification 2010: 35-01, 35-02, 34G10, 44A10, 47A60, 47D06, 47E05, 47F05,47G10. ISBN 978-3-11-025026-8 e-ISBN 978-3-11-025027-5 ISSN 0938-6572 LibraryofCongressCataloging-in-PublicationData Picard,R.H.(RainerH.) Partialdifferentialequations:aunifiedHilbertspaceapproach/ byRainerPicard,DesMcGhee. p.cm.(cid:2)(DeGruyterexpositionsinmathematics;55) Includesbibliographicalreferencesandindex. ISBN978-3-11-025026-8(alk.paper) 1.Hillbertspace. 2.Differentialequations,Partial. I.McGhee,D.F. II.Title. QA322.4.P53 2011 5151.733(cid:2)dc22 2011004423 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableintheInternetathttp://dnb.d-nb.de. ”2011WalterdeGruyterGmbH&Co.KG,Berlin/NewYork Typesetting:Da-TeXGerdBlumenstein,Leipzig,www.da-tex.de Printingandbinding:Hubert&Co.GmbH&Co.KG,Göttingen (cid:3)Printedonacid-freepaper PrintedinGermany www.degruyter.com Dedication WededicatethisbooktothememoryofCliffBartlett(1923–2010),amathematician, artist and singer and a member of the Department of Mathematics (as it then was), UniversityofStrathclydeforoverthirtyyears,retiringinthemid-1980s. Cliffoffered great encouragement and support to both of us in the early parts of our careers, to DMasanewlyappointedlecturerandtoRPduringayearatStrathclydeasaVisiting Lecturer. RainerPicard,DesMcGhee Preface Giventhatthereisamultitudeofwell-writtenandusefultextbooksandmonographs on partial differential equations, one of the obvious concerns of prospective authors mightbewhythereshouldbeanotheronewritten. So, itisappropriatetoexplainat the outset some of the particular features which make this book different from other texts. It will be obvious to the reader that the present book is not a run-off-the-mill text book on linear partial differential equations. First of all, the whole approach – although(withsomeadditionalwork)extendabletoamoregeneralBanachspaceset- ting–isestablishedinaHilbertspacesetting,asthetitleofthismonographindicates. Of course, a Banach space setting is more general and sometimes more appropri- ate, but usually the core results rely nevertheless on a Hilbert space solution theory, a fact sometimes only tacitly acknowledged. We hope to show that, for presenting coreideas,ourfocusonaHilbertspacesettingisnotaconstraint,butratherahighly suitableapproachforprovidingamoretransparentandevenfairlyelementaryframe- work for presenting the main issues in the discussion of a solution theory for partial differentialequations. The reader may also find many topics, dealt with elsewhere, presented here in a slightlydifferentflavor. Indeed,thebuildingblocks(suchasextrapolationandinter- polationspaces,sumsofoperators,vector-valuedLaplacetransform)arelargelywell knownwithsomeoftheideasdatingbacktotheearly1960s,seee.g.[35],[26]forthe ideaofinterpolation/extrapolationspaces. Therefore,ithasbeenandstillissurprising tousthatthefullpoweroftheseconcepts,whichweutilizeinthisapproach,hasnot beenpreviouslyexploitedtotheextentwehavefoundsouseful. Thedifferencesare somewhatsubtleandamoresuperficialreadermayfailtoappreciatehowdifferentour perspective on the theory of linear partial differential equations is. Indeed, it is this perspectiveonourapproachwhichmaybeconsideredthemostinnovativefeatureof thismonograph. Incontrasttomanyotherbooks,whichareeitherfocusingonspecific typesofpartialdifferentialequationsoronacollectionoftoolsforsolvingavariety of problems associated with various specific linear partial differential equations, we areattemptingtoassumeamoreglobalpointofviewontheissuesinvolved. Ourapproachcanbeclassifiedasafunctionalanalyticalone,butthissaysverylit- tle,sincenowadaysitistheacceptedstandardtoemployfunctionalanalyticlanguage toformulatePDEproblems. Itmay,however,comeasasurprisethataHilbertspace setting is sufficiently general to cover the core issues of solving PDE problems. We focus on the case of linear partial differential equations, which is of course in one wayasevereconstraint,butgiventhatbyaruleofthumbnon-linearproblems,ifthey are at all well-posed, are frequently solvable by using a priori estimates and a fixed viii Preface point argument based on perturbations of the linear theory, we see the restriction to linearpartialdifferentialequationsmoreasfoundationlayingratherthananexclusion ofnon-linearissues. A natural guideline for approaching problem solving is provided by Hadamard’s celebratedcriteriaforwell-posedness:1 I Uniqueness: thereisatmostonesolution, I Existence: thereisatleastonesolution(atleastforadensesetofdata), I Continuous Dependence: the dependence of the solution on the data is locally (orweaklylocally)uniformlycontinuous. Comparedtothesefundamentalrequirements,qualitativepropertiesofasolutionare asecondaryconsideration. Thisremarkappliesinparticulartotheissueofregularity inconnectionwithsolvinglinearpartialdifferentialequations. The historical focus on regularity issues has fostered a number of guiding ideas, which in the light of our approach appear as occasionally misleading. Among these arethenotionsthat I partial differential equations are best classified according to the regularizing propertiesoftheirsolutionoperators, I one type of space should be used for solving “all” problems associated with partialdifferentialequations, I problemsinvolvingellipticpartialdifferentialequationsare“easier”thanthose involvingparabolicpartialdifferentialequations,whichareagain“easier”than thoseinvolvinghyperbolicpartialdifferentialequations. Thesystematicapproachpresentedherewillshedadifferentandhopefullymoreillu- minatinglightontheseandotherissuesby I proposingadifferentclassificationscheme, I advocatingtheconstructionof“tailor-made”distributionspacesadaptedtothe particularequationathand, 1 These requirements are straightforwardly illustrated, if we consider the “problem” of finding a solutionoftheequationF.x/ D f,whereF denotesamappingbetween–say–metricspaces. The propertiesarethatF mustbeinjective,withdenserangeandwithF(cid:2)1being(atleastweakly)locally uniformlycontinuous. Wenoteinparticularthatthe(weakly)localuniformcontinuityofF(cid:2)1 allows theextensionofF toitsclosureF givenbyF.x/WDlimF.y/foranysequenceyconvergingtoxsuch thatF.y/alsoconverges.Indeed,fortwosequencesy.k/,kD0;1,convergingtoxsuchthatF.y.k//, kD0;1,arealsoconvergent,wehavethatF.y.0//,F.y.1//mustconvergetothesamelimit,sothatF iswell-defined,i.e.F isclosable. Weaklylocaluniformcontinuityisindeedcharacterizedbymapping CauchysequencestoCauchysequences. Preface ix I showingthatfromourpointofviewparabolicandhyperbolicpartialdifferential equationsare,inaway,“easier”thanellipticpartialdifferentialequations. Itwillalsobeseenthat ourgeneral frameworkissufficiently powerful tobeapplied to general evolution equations. However, to flesh out the ideas presented, we shall illustratebyexamplesandconsiderparticularsystemsofpartialdifferentialequations frommathematicalphysics. Accordingly,thebookisdividedintoseveralnaturalparts. InChapter1wesupply some additional material on functional analysis2 in Hilbert space which may be dif- ficult to find elsewhere. Chapter 2 introduces the idea of what we shall call Sobolev lattices. InChapter3,asafirstapplication,weconsiderpartialdifferentialequations with constant coefficients in RnC1, n 2 N. The results are extended to tempered distributions(setinasuitablyextendedSobolevlattice). TheninChapter4theideas presented are transferred to a more general framework covering a large class of ab- stractevolutionequations. InChapter5thisgeneralsettingisexemplifiedbyapplica- tionstoavarietyofinitial-boundaryvalueproblemsfrommathematicalphysics. The concluding Chapter 6 offers a new approach to initial boundary value problems by expandingontheideasandconceptspresented. The material is based on lecture notes developed for introductory and advanced graduate level courses on partial differential equations and functional analysis given bythefirstauthoroverthepastthreedecadesattheRheinischeFriedrich-Wilhelms- UniversitätatBonn,Germany,attheUniversityofWisconsin-Milwaukee,Wisconsin, USA,andattheTechnischeUniversitätDresden,Germany,andonaseriesoflectures given at the Strathclyde University, Glasgow, Scotland, UK. This development from lecture notes has led to proofs being given in more detail than one might normally expect in a monograph. This may slow the readers progress, but it should, however, make the text not only useful as a resource for courses on the topic but also make it suitableasatextforareadingcourseorforself-study. Apart from the novel approach the material presented in this monograph may in manywaysbeconsideredelementary,however,researcherswillneverthelessfindnew resultsforparticularevolutionarysystemfrommathematicalphysicsinlaterpartsof this monograph as well as a very different perspective on seemingly familiar evolu- tionaryproblems. Thisbookhasbeeninpreparationforanumberofyearsandmanycolleagueshave contributed to the effort either directly through discussion and comment on research papers or at seminar or indirectly through general support and encouragement. The firstofus(RP)wouldliketoparticularlyacknowledgethehospitalityoftheDepart- ment and Mathematics and Statistics, University of Strathclyde, that he has enjoyed firstasayoungVisitingResearcherandmorerecently asaVisitingProfessor, while DMwouldmentionallcolleagues,pastandpresent,intheDepartmentandMathemat- icsandStatistics, UniversityofStrathclyde, butparticularlyAdamMcBride, Wilson 2WehavechosentoassumethatthereaderisfamiliarwithbasicfunctionalanalysisinHilbertspace.
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