Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Juan Pablo Pinasco
Lyapunov-type
Inequalities
With Applications
to Eigenvalue
Problems
123
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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
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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
