Spectral Geometry of Partial Differential Operators

379 Pages·2020·3.867 MB·\379

by  Michael Ruzhansky (Author), Makhmud Sadybekov (Author), Durvudkhan Suragan (Author)

Spectral Geometry of
Partial Differential
Operators

Spectral Geometry of
Partial Differential
Operators
Michael Ruzhansky
Makhmud Sadybekov
Durvudkhan Suragan
Firsteditionpublished2020
byCRCPress
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LibraryofCongressCataloging-in-PublicationData
Names:Ruzhansky,M.(Michael),author.|Sadybekov,Makhmud,author.|Suragan,Durvudkhan,
author.
Title:Spectralgeometryofpartialdifferentialoperators/MichaelRuzhansky,MakhmudSadybekov,
DurvudkhanSuragan.
Description:BocaRaton,FL:CRCPress,Taylor&FrancisGroup,[2020]|Includesbibliographical
referencesandindex.|
Identifiers:LCCN2019051883|ISBN9781138360716(hardback)|ISBN9780429432965(ebook)
Subjects:LCSH:Spectralgeometry.|Partialdifferentialoperators.
Classification:LCCQA614.95.R892020|DDC516.3/62--dc23
LCrecordavailableathttps://lccn.loc.gov/2019051883
ISBN:978-1-138-36071-6(hbk)
ISBN:978-0-429-43296-5(ebk)
TypesetinNimbusRomNo9L
byNovaTechsetPrivateLimited,Bengaluru&Chennai,India
Contents
Preface ix
Authors xi
1 Functionalspaces 1
1.1 Normedspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Hilbertspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Examplesofbasicfunctionalspaces . . . . . . . . . . . . . . . . . 6
1.4 TheconceptofLebesgueintegral . . . . . . . . . . . . . . . . . . 7
1.5 Lebesguespaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Sobolevspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 Initialconceptofembeddingofspaces . . . . . . . . . . . . . . . 18
1.9 Separablespaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Foundationsofthelinearoperatortheory 25
2.1 Definitionofoperator . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Linearoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Linearboundedoperators . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Spaceofboundedlinearoperators . . . . . . . . . . . . . . . . . . 38
2.5 Extensionofalinearoperatorwithrespecttocontinuity . . . . . . 39
2.6 Linearfunctionals . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7 Inverseoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.8 Examplesofinvertibleoperators . . . . . . . . . . . . . . . . . . . 48
2.9 Thecontractionmappingprinciple . . . . . . . . . . . . . . . . . . 53
2.10 Normallysolvableoperators . . . . . . . . . . . . . . . . . . . . . 56
2.11 Restrictionsandextensionsoflinearoperators . . . . . . . . . . . 59
2.12 Closedoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.13 ClosureofdifferentialoperatorsinL2(a,b) . . . . . . . . . . . . . 69
2.14 Generalconceptofstrongsolutionsoftheproblem . . . . . . . . . 72
2.15 Compactoperators . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.16 Volterraoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.17 Structureofthedualspace . . . . . . . . . . . . . . . . . . . . . . 84
2.18 Adjointtoaboundedoperator . . . . . . . . . . . . . . . . . . . . 85
2.19 Adjointtounboundedoperators . . . . . . . . . . . . . . . . . . . 91
2.20 Twoexamplesofnonclassicalproblems . . . . . . . . . . . . . . . 105
v
vi Contents
3 Elementsofthespectraltheoryofdifferentialoperators 115
3.1 Spectrumoffinite-dimensionaloperators . . . . . . . . . . . . . . 115
3.2 Theresolventandspectrumofanoperator . . . . . . . . . . . . . . 121
3.3 Spectralpropertiesofboundedoperators . . . . . . . . . . . . . . 127
3.4 Spectrumofcompactoperators . . . . . . . . . . . . . . . . . . . 129
3.5 Hilbert-Schmidttheoremanditsapplication . . . . . . . . . . . . . 139
3.6 Spectralpropertiesofunboundedoperators . . . . . . . . . . . . . 142
3.7 Someordinarydifferentialoperatorsandtheirspectrum . . . . . . 150
3.8 SpectraltheoryoftheSturm-Liouvilleoperator . . . . . . . . . . . 157
3.9 SpectraltraceandHilbert-Schmidtoperators . . . . . . . . . . . . 168
3.10 Schatten-vonNeumannclasses . . . . . . . . . . . . . . . . . . . . 177
3.11 Regularisedtraceforadifferentialoperator . . . . . . . . . . . . . 187
3.12 Eigenvaluesofnon-self-adjointordinarydifferentialoperatorsofthe
secondorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
3.13 BiorthogonalsystemsinHilbertspaces . . . . . . . . . . . . . . . 203
3.14 BiorthogonalexpansionsandRieszbases . . . . . . . . . . . . . . 207
3.15 ConvolutionsinHilbertspaces . . . . . . . . . . . . . . . . . . . . 221
3.16 Rootfunctionsofsecond-ordernon-self-adjointordinarydifferential
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
4 Symmetricdecreasingrearrangementsandapplications 249
4.1 Symmetricdecreasingrearrangements . . . . . . . . . . . . . . . . 250
4.1.1 Definitionsandexamples . . . . . . . . . . . . . . . . . . . 250
4.1.2 Someproperties . . . . . . . . . . . . . . . . . . . . . . . 252
4.2 Basicinequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
4.2.1 Hardy-Littlewoodinequality . . . . . . . . . . . . . . . . . 255
4.2.2 Rieszinequality. . . . . . . . . . . . . . . . . . . . . . . . 255
4.2.3 Po´lya-Szego˝ inequality . . . . . . . . . . . . . . . . . . . . 257
4.2.4 Talenti’scomparisonprinciples . . . . . . . . . . . . . . . 258
4.3 Propertiesofsymmetricrearrangementsequences . . . . . . . . . . 261
4.3.1 Burchard-Guotheorem . . . . . . . . . . . . . . . . . . . . 261
4.3.2 ProofofBurchard-Guotheorem . . . . . . . . . . . . . . . 269
4.4 Applicationsinmathematicalphysics . . . . . . . . . . . . . . . . 271
4.4.1 Brownianmotioninaball . . . . . . . . . . . . . . . . . . 271
4.4.2 Optimalmembraneshapeforthedeepestbassnote . . . . . 272
4.4.3 Maximiserbodyofthegravitationalfieldenergy . . . . . . 273
4.4.4 Dynamicalstabilityproblemofgaseousstars . . . . . . . . 274
4.4.5 Stabilityofsymmetricsteadystatesingalacticdynamics . . 276
5 Inequalitiesofspectralgeometry 279
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
5.2 Logarithmicpotentialoperator . . . . . . . . . . . . . . . . . . . . 282
5.2.1 Spectralgeometricinequalitiesandexamples . . . . . . . . 283
5.2.2 Isoperimetricinequalitiesoverpolygons . . . . . . . . . . . 289
5.3 Rieszpotentialoperators . . . . . . . . . . . . . . . . . . . . . . . 294
Contents vii
5.3.1 SpectralpropertiesofR . . . . . . . . . . . . . . . . . 295
α,Ω
5.3.2 SpectralgeometricinequalitiesforR . . . . . . . . . . 299
α,Ω
5.4 Besselpotentialoperators . . . . . . . . . . . . . . . . . . . . . . 304
5.4.1 SpectralpropertiesofB . . . . . . . . . . . . . . . . . 305
α,Ω
5.4.2 BoundarypropertiesofB . . . . . . . . . . . . . . . . 306
α,Ω
5.5 Riesztransformsinsphericalandhyperbolicgeometries . . . . . . 311
5.5.1 Geometricinequalitiesforthefirsteigenvalue . . . . . . . . 313
5.5.2 Geometricinequalitiesforthesecondeigenvalue . . . . . . 315
5.6 Heatpotentialoperators . . . . . . . . . . . . . . . . . . . . . . . 319
5.6.1 Basicproperties. . . . . . . . . . . . . . . . . . . . . . . . 320
5.6.2 Spectralgeometricinequalitiesfortheheatpotential . . . . 322
5.6.3 Thecaseoftriangularprisms . . . . . . . . . . . . . . . . . 327
5.7 Cauchy-Dirichletheatoperator . . . . . . . . . . . . . . . . . . . . 329
5.7.1 SpectralgeometricinequalitiesfortheCauchy-Dirichletheat
operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
5.7.2 Thecaseofpolygonalcylinders . . . . . . . . . . . . . . . 336
5.8 Cauchy-Robinheatoperator . . . . . . . . . . . . . . . . . . . . . 339
5.8.1 Isoperimetricinequalitiesforthefirsts-number . . . . . . . 339
5.8.2 Isoperimetricinequalitiesfortheseconds-number . . . . . 342
5.9 Cauchy-NeumannandCauchy-Dirichlet-Neumannheatoperators . 343
5.9.1 Basicproperties. . . . . . . . . . . . . . . . . . . . . . . . 344
5.9.2 OntheSzego˝-Weinbergertypeinequality . . . . . . . . . . 346
5.9.3 InequalitiesfortheCauchy-Dirichlet-Neumannoperator . . 348
Bibliography 353
Index 363

Preface
Thisbookisanattempttocollectanumberofpropertiesemerginginrecentresearch
describingcertainfeaturesofthetheoryofpartialdifferentialequationsthatcanbe
attributedtothefieldofspectralgeometry.Bothbeingvastfields,ourattemptisnot
togiveacomprehensiveaccountofthewholetheory,buttoprovidethereaderwith
aquickintroductiontoanumberofitsimportantaspects.
The topic of spectral geometry is a broad research area appearing in different
mathematicalsubjects.Assuch,itallowsonetocomparespectralinformationasso-
ciatedwithvariousobjectsoverdifferentdomainswithselectedgeometricproperties.
Forexample,whentheareaofthedomainisfixed,oneoftentalksoftheisoperimetric
inequalitiesinsuchcontext.Thepurposeofthisbookistohighlightonedirectionof
suchresearchaimedattheunderstandingofspectralpropertiesofpartialdifferential
operatorsaswellasoftherelatedintegraloperators.
An indispensable language of the area is that of functional analysis and, aimed
alsoatthestudentreadership,wegiveabasicintroductiontothetheory.Ingeneral,
thefunctionalanalysiscanbeviewedasapowerfulcollectionofmathematicaltools
allowing one to obtain significant generalisations of various effects detected in the
investigation of one concrete problem. At present, this is a vast mathematical field
withnumerousinvestigationsandexcellentmonographsreadilyavailable.
Thus, in the first chapters of this book we give a brief account of the basics of
functional analysis aiming at consequent applications in the theory of differential
equations. Chapter 1 deals with the basic notions of function spaces, Chapter 2 is
devotedtothefoundationofthetheoryoflinearoperators,andChapter3discusses
thebasicsofthespectraltheoryofdifferentialoperators.Theseareaimedtoprovide
the reader with a quick introduction to the subject. As there are many detailed and
comprehensivemonographsalreadyavailable,weomitmanyproofsofbasicresults
that can be easily found in a variety of sources. From this point of view, this book
isonlya“guidebook”indicatingthemaindirectionsandnecessarybasicfacts.The
general course of the functional analysis contains a large number of various defi-
nitions and facts. It is clear that it is impossible to cover and understand all these
concepts even briefly. Therefore, in the present book we introduce only those con-
ceptswhicharenecessary(but,ofcourse,notsufficient)forabeginnerwantingtodo
researchonspectralproblemsforthedifferentialoperators.
We introduce only the main concepts of the functional analysis, and only those
on which we will lean in the further exposition. This presentation is therefore far
from being complete, and there are many other concepts and ideas widely used in
the theory. We also do not dwell on detailed justifications of introduced concepts
and their general properties, as the proof of those (standard) facts goes beyond the
ix