Springer Monographs in Mathematics Forfurther volumes: http://www.springer.com/series/3733 Leonid P. Lebedev • Iosif I. Vorovich Michael J. Cloud Functional Analysis in Mechanics Second Edition † Leonid P. Lebedev Iosif I. Vorovich Department of Mathematics Department of Mathematics & Mechanics National University of Colombia Rostov State University Bogotá, Colombia Rostov-on-Don, Russia Michael J. Cloud Department of Electrical Engineering Lawrence Technological University Southfield, MI, USA ISSN 1439-7382 ISBN 978-1-4614-5867-8 ISBN 978-1-4614-5868-5 (eBook) DOI 10.1007/978-1-4614-5868-5 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012950048 Mathematics Subject Classification (2010): 4601, 4602, 7401, 7402, 74B05, 74K25, 35JXX, 35A01, 35A02 © Springer Science+Business Media, LLC 2003, 2013 This work is subject to copyright. 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Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface InRussia,auniversityMechanicsdepartmentwilltypicallyexistwithina“Mathe- maticalFaculty.”Suchadepartmentisnotanengineeringdepartmentinthewestern sense,butissomethingintermediatebetweenamathematicsdepartmentandanen- gineeringdepartment.Itwilloffercoursesoncalculus,linearalgebra,analysis,dif- ferential geometry, differential equations, and so on, along with extensive courses onanalyticalmechanics,thestrengthofmaterials,continuummechanics,elasticity, fluidmechanics,andmorespecializedsubjects. Whenthefirstauthorofthisbookwasastudentofthesecondauthor,functional analysis was not in the curriculum for mechanicists. In 1971, Professor Vorovich offeredashortcourseonfunctionalanalysistoabroadaudienceconsistingofmath- ematiciansandmechanicists,studentsandprofessors.Itincludedasimpleandmin- imalintroductiontothetheoryofBanachandHilbertspacesthatopenedthedoorto understanding(withsomedifficultyonthepartofthenon-mathematicians)certain interesting applications in mechanics. The mathematicians were surprised at how abstracttheoremscouldbeappliedtomechanicsand,moreover,thatthesetheorems could actually be rooted inmechanics. Itwas emphasized thatstrainenergy isnot onlyaphysicalnotion,butameasurebywhichanormandinnerproductcanbeim- posedonthesetofdeformationsorvelocitiesofabody.Thisideawasdevelopedby Vorovichinhisdoctoraldissertationonthenonlineartheoryofelasticshallowshells in1957,butwasimbeddedinjustafewlongexamples.Manysubsequentpublica- tions of the idea were made in Doklady AN USSR, the central scientific journal of theUSSR,wheretheresultswerepresentedwithoutproof. Later, Professor Vorovich’s lectures were extended and became the basis for a regular course at Rostov State University and many other institutions across the USSR. The course contents, including the applications considered, continued to evolve,butthepresentbookpreservesthemainideasoftheoriginalcourse.Asthe coursewasjustonesemesterinlength,itcontainedonlyaminimalsubsetoftheab- stracttheorythatenabledstudentstounderstandtheapplications.Thefirsteditionof thisbookcontainedsomeabstractmaterialthatwasnotpresentedinthecourse.This secondeditionincludesmoreextendedcoverageoftheclassical,abstractportionsof functional analysis, as well as additional mechanics problems. Taken together, the v vi Preface firstthreechaptersnowconstitutearegulartextonappliedfunctionalanalysis.This potential use of the book is supported by a significantly extended set of exercises withhintsandsolutions. TheIntroduction(pages1–7)isanunnumberedchapterwithsingle-digitinternal numberingofitsequations,examples,andtheorems.Chapters1through4employa three-digitschemewherethefirsttwonumbersarethechapterandsectionnumbers (henceRemark1.3.2isthesecondlabeledremarkinSection1.3).TheIntroduction closeswithsomeadditionalremarksonnotation. Acknowledgments. We would like to thank Senior Editor Achi Dosanjh and As- sociateEditorDonnaChernyk forassistancewiththissecondedition.Wearealso grateful to Elena Vorovich and toour wives, Natasha Lebedeva and Beth Lannon- Cloud,fortheirunderstandingandsupport. Bogota,Colombia LeonidLebedev Okemos,Michigan, USA MichaelCloud August2012 Contents Introduction ....................................................... 1 1 Metric,Banach,andHilbertSpaces .............................. 9 1.1 Preliminaries .............................................. 9 1.2 Ho¨lder’sInequalityandMinkowski’sInequality................. 14 1.3 MetricSpacesofFunctions .................................. 18 1.4 SomeRelationsfortheMetricsinL˜p(Ω)and(cid:3)p ................. 22 1.5 MetricsinEnergySpaces.................................... 24 1.6 SetsinaMetricSpace....................................... 28 1.7 ConvergenceinaMetricSpace ............................... 29 1.8 Completeness.............................................. 30 1.9 CompletionTheorem ....................................... 31 1.10 LebesgueIntegralsandtheSpaceLp(Ω)........................ 34 1.11 BanachSpaces............................................. 39 1.12 Normsonn-DimensionalSpaces.............................. 42 1.13 OtherExamplesofBanachSpaces ............................ 44 1.14 HilbertSpaces ............................................. 48 1.15 FactorSpaces.............................................. 50 1.16 Separability ............................................... 53 1.17 Compactness,HausdorffCriterion............................. 58 1.18 Arzela`’sTheoremandItsApplications......................... 61 1.19 TheoryofApproximationinaNormedSpace ................... 66 1.20 DecompositionTheorem,RieszRepresentation ................. 70 1.21 LinearOperatorsandFunctionals ............................. 72 1.22 SpaceofLinearContinuousOperators ......................... 76 1.23 UniformBoundednessTheorem .............................. 79 1.24 Banach–SteinhausPrinciple.................................. 82 1.25 ClosedOperatorsandtheClosedGraphTheorem................ 83 1.26 InverseOperator ........................................... 89 1.27 Lax–MilgramTheorem...................................... 92 1.28 OpenMappingTheorem..................................... 93 vii viii Contents 1.29 DualSpaces ............................................... 94 1.30 Hahn–BanachTheorem ..................................... 98 1.31 ConsequencesoftheHahn–BanachTheorem ...................102 1.32 ContractionMappingPrinciple ...............................103 1.33 Topology,WeakandWeak*Topologies ........................109 1.34 WeakTopologyinaNormedSpaceX .........................111 1.35 ConclusionandFurtherReading ..............................113 2 MechanicsProblemsfromtheFunctionalAnalysisViewpoint .......115 2.1 IntroductiontoSobolevSpaces ...............................115 2.2 OperatorofImbedding ......................................119 2.3 SomeEnergySpaces........................................120 2.4 GeneralizedSolutionsinMechanics...........................135 2.5 ExistenceofEnergySolutionstoSomeMechanicsProblems ......141 2.6 OperatorFormulationofanEigenvalueProblem.................142 2.7 ProblemofElastico-Plasticity;SmallDeformations..............144 2.8 BasesandCompleteSystems;FourierSeries....................151 2.9 WeakConvergenceinaHilbertSpace .........................157 2.10 RitzandBubnov–GalerkinMethods...........................166 2.11 CurvilinearCoordinates,Nonhomogeneous BoundaryConditions .......................................168 2.12 Bramble–HilbertLemmaandItsApplications...................171 3 SomeSpectralProblemsofMechanics............................177 3.1 Introduction ...............................................177 3.2 AdjointOperator ...........................................178 3.3 CompactOperators .........................................185 3.4 CompactOperatorsinHilbertSpace...........................190 3.5 FunctionsTakingValuesinaBanachSpace ....................192 3.6 SpectrumofaLinearOperator................................195 3.7 ResolventSetofaClosedLinearOperator......................199 3.8 SpectrumofaCompactOperatorinHilbertSpace ...............201 3.9 AnalyticNatureoftheResolventofaCompactLinearOperator ...209 3.10 SpectrumofaHolomorphicCompactOperatorFunction..........211 3.11 Self-AdjointCompactLinearOperatorinHilbertSpace ..........213 3.12 SomeApplicationsofSpectralTheory .........................219 3.13 Courant’sMinimaxPrinciple.................................222 4 ElementsofNonlinearFunctionalAnalysis........................225 4.1 Fre´chetandGaˆteauxDerivatives..............................225 4.2 Liapunov–SchmidtMethod ..................................231 4.3 CriticalPointsofaFunctional ................................233 4.4 VonKa´rma´nEquationsofaPlate .............................236 4.5 BucklingofaThinElasticShell ..............................243 4.6 NonlinearEquilibriumProblemforanElasticShallowShell ......251 Contents ix 4.7 DegreeTheory.............................................255 4.8 Steady-StateFlowofaViscousLiquid.........................258 SummaryofInequalitiesandImbeddings .............................265 A.1 Inequalities................................................265 A.2 Imbeddings................................................268 HintsforSelectedProblems .........................................269 References.........................................................299 InMemoriam:IosifI.Vorovich ......................................301 Index .............................................................303