Lyapunov-type Inequalities: With Applications to Eigenvalue Problems PDF
Preview Lyapunov-type Inequalities: With Applications to Eigenvalue Problems
SPRINGER BRIEFS IN MATHEMATICS Juan Pablo Pinasco Lyapunov-type Inequalities With Applications to Eigenvalue Problems 123 SpringerBriefs in Mathematics SeriesEditors KrishnaswamiAlladi NicolaBellomo MicheleBenzi TatsienLi MatthiasNeufang OtmarScherzer DierkSchleicher VladasSidoravicius BenjaminSteinberg YuriTschinkel LoringW.Tu G.GeorgeYin PingZhang SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. Forfurthervolumes: http://www.springer.com/series/10030 Juan Pablo Pinasco Lyapunov-type Inequalities With Applications to Eigenvalue Problems 123 JuanPabloPinasco DepartamentodeMatematica UniversidaddeBuenosAires BuenosAires,Argentina ISSN2191-8198 ISSN2191-8201(electronic) ISBN978-1-4614-8522-3 ISBN978-1-4614-8523-0(eBook) DOI10.1007/978-1-4614-8523-0 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013947680 MathematicsSubjectClassification(2010):34L15,34B05,34B15,34C10,35P30 ©JuanPabloPinasco2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Ceci, Fede, andSelva Preface I used to think that the Sturm–Liouvilletheory of second-orderordinarydifferen- tial equations was one of the most beautiful areas of mathematics. Its simplicity, togetherwiththepowerofthecomparisonandoscillationtheorems,shedadiffer- entlightonsecond-orderordinarydifferentialequations.However,whilereadinga transcriptionofatalkofG.C.Rota,Irealizedsomething:therearemanyinteresting problems,both of theoreticaland applied origin,that cannotbe analyzedwith the Sturmiantools. TaketheunitballinRN:justthesimplereductiontopolarcoordinatesintroduces the coefficient rN−1, which vanishes at the origin and is bounded above by 1, for allN. Moreover,Bessel, Hermite,Legendre,..., almostallthe specialfamiliesof functionsthatappearaseigenfunctionsofsecond-orderordinarydifferentialopera- tors,areindeedeigenfunctionsofsingularordegenerateoperators,andtheSturmian argumentsfail.Whatcanwedonow? ∞ IfwewritetheSturmianboundsinmodernnotation,weareusingtheL norm ofthe weight,andwhathappensif we changeitto anothernorm,say L1? Indeed, the answer is known,and it is related to the stability of solutionsof second-order differential equations, a problem studied by Lyapunov almost 150 years ago. He introducedan integralconditionthatthe weightmustsatisfy in orderto guarantee stability. However, he never proved Lyapunov’sinequality. Later, Borg, Hartman, Krein,andothermathematiciansworkingonstabilitygavehisnametothiskindof SturmianboundwithanL1 norm. However,unboundeddomainsstillpresentadifficulty,sinceLyapunov’sinequal- ityincludesthelengthoftheintervalonwhichtheproblemwasstudied.Wemight decidetoignorethisproblem,dismissingitasahifalutintheoreticalquestion.But not so fast! It was, in fact, a legitimate question, inspired by quantum mechanics and related to the number of bound states of the Schro¨dinger equation. Ordinary differentialequationsonunboundedintervalswerestudiedinthe1950sand1960s by Jost, Pais, Bargmann, Calogero, Cohn, and Nehari (the only one who was not thinking of quantum-mechanicalproblems), among several others. They obtained beautifulinequalities,involvingdifferentnormsofthecoefficients. vii viii Preface Andinthelasttwentyyears,manymathematicianshaveextendedthoseresultsto avarietyofsettings,includingp-Laplacianoperators,ordinarydifferentialequations inOrliczspaces,N-dimensionalproblems,andsystems. I designed this book as a guided tour throughthose results, together with their applicationstoeigenvalueproblems,presentingfullproofsandextensionsofthose inequalities, and showing the less-traveled paths, suggesting directions for future work.Itriedtoincludeinthereferencesalltherelevantpapersonthissubject,and Iapologizeherefortheinevitableomissions. Iwishtothankseveralpeoplewhocontributeddirectlyorindirectlytothisbook: P. Amster, J.M. Castro, P. De Na´poli, J. Ferna´ndezBonder,and A. Salort. Also, I wish to thank the people at UCo-CEMIC, Buenos Aires, for their hospitality, and thefinancialsupportfromUniversidaddeBuenosAiresandCONICET. BuenosAires,Argentina JuanPabloPinasco Contents 1 Introduction................................................... 1 1.1 AFewWordsAboutFourTheorems........................... 1 1.2 OrganizationoftheBook .................................... 7 2 Lyapunov’sInequality .......................................... 11 2.1 TheClassicalInequality..................................... 11 2.1.1 TheLinearCase ..................................... 11 2.1.2 AnInterestingExtension.............................. 16 2.2 QuasilinearProblems ....................................... 18 2.2.1 ASimpleProof...................................... 19 2.2.2 RelationshipwithIntegralComparisonTheorems ......... 20 2.3 SomeIncompleteGeneralizations............................. 23 2.3.1 Higher-OrderQuasilinearProblems..................... 23 2.3.2 NonconstantCoefficients.............................. 26 2.3.3 SingularCoefficients ................................. 27 2.3.4 OptimalityoftheConstants............................ 28 2.4 EigenvalueProblems:LowerBoundsofEigenvalues ............ 33 2.4.1 OptimalityoftheBound .............................. 34 2.4.2 ADifferentBound ................................... 36 3 Nehari–Calogero–CohnInequality ............................... 39 3.1 TheWorkofCalogeroandCohn.............................. 39 3.1.1 Cohn’sProof........................................ 40 3.1.2 Calogero’sProof..................................... 42 3.1.3 APartialConverse ................................... 43 3.2 Nehari’sProofandGeneralizations............................ 43 3.2.1 Nehari’sProofforSecond-OrderProblems............... 44 3.2.2 Nehari’sProof for LinearHigher-OrderDifferential Equations........................................... 50 3.3 TheInequalityfor p-LaplacianProblems....................... 54 3.3.1 AnExtensionforDifferentPowers...................... 58 ix
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