Functional methods in differential equations PDF
Preview Functional methods in differential equations
www.crcpress.com Dedicated to Professor Wolfgang L. Wendland on the occasion of his 65th birthday Contents Introduction 1 Preliminaries 1.1 Function and distribution spaces 1.2 Monotone operators, convex functions, and subdifferentials 1.3 Some elements of spectral theory 1.4 Linear evolution equations and semigroups 1.5 Nonlinear evolution equations 2 Elliptic boundary value problems 2.1 Nondegenerate elliptic boundary value problems 2.2 Degenerate elliptic boundary value problems 3Parabolicboundaryvalueproblemswit halgebraicboundary conditions 3.1 Homogeneous boundary conditions 3.2 Nonhomogeneous boundary conditions 3.3 Higher regularity of solutions 4Parabolicboundaryvalueproblem swithalgebraic-differential boundary conditions 4.1 Homogeneous algebraic boundary condition 4.2 Nonhomogeneous algebraic boundary condition 4.3 Higher regularity of solutions 5Hyper bolicboundar yvalueproblem swit halgebraicbound- ary conditions 5.1 Existence, uniqueness, and long-time behavior of solutions 5.2 Higher regularity of solutions 6Hyper bolicboundaryvalu eproblemswithalgebraic-differential boundary conditions 6.1 Existence, uniqueness, and long-time behavior of solutions 6.2 Higher regularity of solutions 7 The Fourier method for abstract differential equations 7.1 First order linear equations 7.2 Semilinear first order equations 7.3 Second order linear equations 7.4 Semilinear second order equations 8 The semigroup approach for abstract differential equations 8.1 Semilinear first order equations 8.2 Hyperbolic partial differential systems with nonlinear boundary conditions 9 Nonlinear nonautonomous abstract differential equations 9.1Firs torderdiffere ntialandfunctionalequationscontainin gsub- differentials 9.2 An application 10 Implicit nonlinear abstract differential equations 10.1 Existence of solution 10.2 Uniqueness of solution 10.3 Continuous dependence of solution 10.4 Existence of periodic solutions Introduction In recent years, functional methods have become central to the study of the- oretical and applied mathematical problems. An advantage of such an ap- proach is its generality and its potential unifying effect of particular results and techniques. Functional analysis emerged as an independent discipline in the first half of the 20th century, primarily as a result of contributions of S. Banach, D. Hilbert,andF.Riesz. Significantadvanceshavebeenmadeindifferentfields, such as spectral theory, linear semigroup theory (developed by E. Hille, R.S. Phillips,andK.Yosida),thevariationaltheoryoflinearboundaryvalueprob- lems, etc. At the same time, the study of nonlinear physical models led to thedevelopmentofnonlinearfunctionalanalysis. Today,thisincludesvarious independentsubfields, suchasconvexanalysis(whereH.Br´ezis, J.J.Moreau, andR.T.Rockafellarhavebeenmajorcontributors),theLeray-Schaudertopo- logicaldegreetheory,thetheoryofaccretiveandmonotoneoperators(founded byG.Minty,F.Browder,andH.Br´ezis),andthenonlinearsemigrouptheory (developedby Y. Komura, T. Kato, H.Br´ezis, M.G.Grandall, A. Pazy, etc.). Asaconsequence,therehasbeensignificantprogressinthestudyofnonlin- earevolutionequationsassociatedwithmonotoneoraccretiveoperators(see, e.g., the monographs by H. Br´ezis [Br´ezis1], and V. Barbu [Barbu1]). The mostimportantapplicationsofthistheoryareconcernedwithboundaryvalue problems for partial differential systems and functional differential equations, including Volterra integral equations. The use of functional methods leads, in some concrete cases, to better results as compared to the ones obtained by classical techniques. In this context, it is essential to choose an appropriate functional framework. As a byproduct of this approach, we will sometimes arrive at mathematical models that are more general than the classical ones, andbetterdescribeconcretephysicalphenomena;inparticular,weshallreach a concordance between the physical sense and the mathematical sense for the solution of a concrete problem. ©2002 Thepurposeofthismonograp histoemphasiz etheimportanceoffunctional methodsinthestudyof abroadrangeofboundar yvalueproblems ,aswell asthatofvariou sclasse sofabstractdifferentialequations. Chapte r 1isdedicatedtoareviewofbasicconceptsan dresultsthatare usedthroughoutthebook.Mos toftheresultsareliste dwithou tpr oofs.In som einstances,however,theproofsar eincluded,particularlywhenwecould notidentif yanappropriatereferenc einliterature. Chapter s 2through6areconcernedwithconcret eelliptic ,parabolic,or hyperbolicboundaryvalueproblem sthatcanbetreate dbyappropriatefunc- tionalmeth ods. InChapter2,weinvestigatevariousclassesof ,mainlyone-dimensional, ellipticboundar yvalueproblems .Th efirs tsectiondealswithnonlinearnon- degenerateboundaryvalueproblems,bot hinvariationalan dnon-variational cases .Th eapproachreliesonconvexanalysisan dth emonoton eoperator theory.I nthesecondsection,westar twithatwo-dimensionalcapillarity problem.Inthespecialcaseof acirculartu be,weobtain adegenerateone- dimensionalproblem.Amoregeneral,doublynonlinearmulti valuedvariant ofthisproble misthoroughlyanalyzedunderminimalrestrictionsonth edata. Chapte r3isconcerne dwithnonlinearparabolicproblems.Weconsidera so-calledalgebraicboundar yconditionthatincludes,asspecialcases ,condi- tionsofDirichlet,Neumann,an dRobin-Steklovtype,aswellasspaceperi odic boundaryconditions.Th eterm“algebraic”indicate sthattheboundarycon- ditionisanalgebraicrelationinvolvin gth evaluesoftheunkn ownandits spacederivativeonth eboundary.Thetheor ycoversvariou sphysicalmodels, suchasheatpropagationi nalinearconductoranddiffusionphenomena.We treatth ecasesofhomogeneou san dnonhomogeneousboundaryconditions separately,sinceinth esecondcasewehaveatime-dependentproblem.The basicideaofou rapproachistoreprese ntourboundar yvalueproblemasa Cauchyproble mforanordinar ydifferentialequationinth e L2-space.Asa specialtopic,weinvestigateinthelastsectionofthischapter ,theproble mof th ehigherregularityofsolutions. InChapter4weconsiderth esamenonlinearparabolicequationasinChap- ter3,bu twithalgebraic-differentialboundaryconditions .Thi smean sthat wehaveanalgebraicboundar yconditionasinth epreviou schapter,aswell as adiffere ntialboundaryconditionthatinvolvesthetim ederivativeofthe unknown.Thisproble misesse ntiallydiffere ntfromtheon einChapter3, and a new framework is needed in order to solve it. Specifically, we arrive at a Cauchy problem in the space L2(0,1)×IR (see (4.1.6)-(4.1.7)). Actu- ally, this Cauchy problem is a more general model, since it describes physical situations that are not covered by the classical theory. More precisely, if the Cauchy problem has a strong solution (u,ξ), then necessarily ξ(t) = u(1,t); in other words, the second component of the solution is the trace of the first oneontheboundary. Otherwise,ξ(t)6=u(1,t),butitstilldescribesanevolu- tion on the boundary. This is important in concrete cases, such as dispersion or diffusion in chemical substances. As in the preceding chapter, we study th ecaseofahomogeneou salgebraicboundar yconditionseparatelyfromthe nonhomogeneousone .Th ehigherregularityofsolution sisals odiscussed. Chapte r5isdedicatedto aclassofsemilinearhyper bolicpartialdiffer- entialsystemswith ageneralnonlinearalgebraicboundarycondition.We firs tstudytheexistence,uniqueness ,an dasymptoti cbehaviorofsolution sas t→∞,byusin gtheproductspace L2(0,1)2 asabasi cfunctionalsetup.The theoryhasapplication sinphysic san dengineerin g(e.g.,unsteadyfluidfl ow withnonlinearpipefriction ,electrica ltransmissionphenomena,etc.) .Unli ke theparaboliccase,wedonotseparateth ehomogeneousandnonhomogeneous cases ,sinc ewecanal wayshomogenizeth eproblem.Althoughthislead stoa time-dependentsystem,wecaneasilyhandl eitbyappealin gt oclassicalre- sultsonnonlinearnonautonomouse volutionequations.Inthesecondsection ofthischapter ,wediscus sthehighe rregulari tyofsolutions .Thi sisim por- tant,forinstance ,forthesingularperturbationanalysisofsuchproblems. Thenaturalfunctionalframewor kforthistheoryseemst obeth e Ck-space. Itisalsoworthnotingthatthemethodweus etoobtainregularityresultsis differentfromth eonei nChapters3an d4,andi nvolvessomeclassicaltools sucha sD’Alemberttypefor mulae,an dfixe dpointargume nts. InChapte r6,weconsiderthesam ehyperboli cpartialdiffere ntialsystems asi nth eprecedin gchapter,bu twithalgebraic-differe ntialboundar ycondi- tions .Suchconditionsar esuggestedbysom eapplicationsarisinginelectrical engineering.Asbefore,werestrictou ratte ntiontothehomogeneouscase only.Thi sproblemhasdistinctfeatures,ascomparedtotheonei nvolving jus talgebraicboundar yconditions .WenowconsideraCau chyproble min th eproductspace L2(0,1)2×IR.Inth ecaseofstrongsolutions ,werecover th eoriginalproblem,bu tingeneral,thisincorporatesawiderrangeofappli- cations.Moreover,theweaksolutionofthisCauchyproble mcanbeviewed asageneralizedsolutionoftheoriginalm odel. Theremainderofthebookisdedicate dtoabstrac tdifferentialandintegro- differentialequationstowhichfunctionalmethod scanbeapplied. InChapter7,th eclassicalFouriermeth odi susedi nth estud yoffirstand secondorde rlineardifferentialequation si n aHilber tspac e H.Th eoperator appearingintheequation sisassumedtobelinear,symmetric,andc oercive. Inorde rtouse amoregeneralconcep tofsolution,wereplac eth eabstract operatori ntheequationbyits“energetic ”extension .Abasicassumptionis thatth ecorrespondin genergeticspac eiscompactl ye mbeddedinto H.This guaranteestheexistenceoforthonormalbasesofeige nvectors,andenables ustoemployFourie rtypemethods .Existenc ean dregularityresultsforthe solutionareestablished .Inthecaseofpartia ldifferentialequations,oursolu- tion sreducetogeneralized(Sobolev)solutions .Finally,nonlinearfunctional perturbationsar ehandledbyafixed-pointapproach.Asapplicationsvarious parabolican dhyper bolicpartialdifferentia lequationsareconsidered.Since theperturbation sarefunctional,alargeclas sofintegro-differentialequations isals ocovered. InChapter8,wediscus sth eexistenceandregularityofsolution sforfirst orde rlineardifferentialequation sinBanachspaceswithnonlinearfunctional perturbations.Th emai nmethodsar ethevariationofconstantsfor mulafor linea rsemigroup sandth eBanachfixed-pointtheorem.Thetheoryisap- plie dtoth estudyof aclassofhyperbolicpartialdifferentialequationswith nonlinearboundaryconditions. InChapte r9,weconsiderfirstordernonlinear,nonautonomou sdifferen- tialequation sinHilber tspaces .Th eequationsinvolve atime-dependent unboundedsubdifferentialwithtime-de pendentdomain,perturbe dbytime- dependentmaximalmonoton eoperatorsan dfunctionalsthatcanbetypically integralsoftheunkn ownfunction.Thetreatme ntofth eproble mwithout functionalperturbationrelie sonthemethod sofH.Br´ezis[Br´ezis1];theprob- lemwithfunctionalperturbationishandledbyafixed-pointreasoning.Asan application ,anonlinearparabolicpartialdifferentialequationwithnonlinear boundarycondition sisstudied. Chapter10isconcernedwithimplicitdifferentialequationsinHil bertspaces. Resultsonth eexistence,uniqueness ,an dcontinuousde pendenceofsolutions forrelatedinitialvalueproblemsar epresented.Thestud yofimplicitdif- ferentialequationsismotivatedbythetwophaseStefanproblem,whichhas recentlyattracte datte ntionbecauseofitsimportanc efortheoptimalcontrol ofcontinuouscastin gofsteel. Wecontinuewithsom egeneralremarksregardingthestructureofthebook. Thematerialisdividedintochapters,whi ch,inturn ,aredividedintosections. Themaindefinitions ,theorems,propositions ,etc.ar edenotedbythreedigits: th efirstindicatesth echapter,th esecon dth ecorrespondingsection,and th ethirdthepositionofth eres pectiveite minth esection.Forexample, Proposition1.2.3denote sthethirdpropositionofSection 2inChapter1. Eachchapterhasitsownbibliographybutth elabelsareuniquethroughout th ebook. Wealsonotethatmanyresultsareonlys ketched,inorde rtokee pthebook lengthwithinreasonablelimits.O ntheotherhand,thisrequire sanacti ve participationofth ereader. Wit htheexceptionofChapter1,thebookcontainsmaterialmainlydue to the authors, as considerably revised or expanded versions of earlier works. An earlier book by one of the authors must be here quoted [Moro6]. We would like to mention that the contribution of the former author was partly accomplished at Ohio University in Athens, Ohio, USA, in the winter of2001. TheworkofthelatterauthorwascompletedduringhisvisitsatOhio University in Athens, Ohio, USA (fall 2000) and the University of Stuttgart, Germany (2001). We are grateful to Professor Klaus Kirchg¨assner (University of Stuttgart) andDr. AlexandruMurgu(UniversityofJyv¨askyl¨a)fortheirnumerous com- ments on the manuscript of our book. Special thanks are due to Professor Sergiu Aizicovici (Ohio University, Athens) for reading a large part of the manuscript and for helpful discussions. We also express our gratitude to Professors Ha¨ım Br´ezis, Eduard Feireisl, Jerome A. Goldstein, Weimin Han, Andreas M. Hinz, Jon Kyu Kim, Enzo Mitidieri, Dumitru Motreanu, Rainer Nagel, Eckart Schnack, Wolfgang L. Wendland and many others, for their kind appreciation of our book. Last but not least, we dearly acknowledge the kind cooperation of Alina Moro¸sanu who has contributed to the improvement of the language style. References [Barbu1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. [Br´ezis1] H.Br´ezis,Op´erateurs maximaux monotones et semi-groupes de contrac- tions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973. [Moro6] Gh. Moro¸sanu, Metode func¸tionaleˆın studiul problemelor la limit˘a, Ed- itura Universit˘a¸tii de Vest din Timi¸soara, Timi¸soara, 1999. ©2002
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