Hardy Inequalities and Applications: Inequalities with Double Singular Weight

158 Pages·2022·3.931 MB·English

by  Nikolai Kutev, Tsviatko Rangelov

NikolaiKutev,TsviatkoRangelov
HardyInequalitiesandApplications
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Nikolai Kutev, Tsviatko Rangelov
Hardy Inequalities
and Applications
|
Inequalities with Double Singular Weight
MathematicsSubjectClassification2020
26D10,35P15,35K20,35Q93,35B44
Authors
Prof.Dr.NikolaiKutev Prof.Dr.TsviatkoRangelov
InstituteofMathematicsandInformatics InstituteofMathematicsandInformatics
MathematicalPhysicsDepartment MathematicalPhysicsDepartment
BulgarianAcademyofSciences BulgarianAcademyofSciences
Acad.G.Bonchevstr.8 Acad.G.Bonchevstr.8
1113Sofia 1113Sofia
Bulgaria Bulgaria
[email protected] [email protected]
ISBN978-3-11-099230-4
e-ISBN(PDF)978-3-11-098037-0
e-ISBN(EPUB)978-3-11-098049-3
LibraryofCongressControlNumber:2022943736
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
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Preface
TheaimofthebookistoderivenewHardytypeinequalitieswithoptimalconstant
andwithadditionalterms.Attainabilityoftheoptimalconstantforcertainfunctions
isinvestigatedtoo.
Thestudyisbasedonmathematicalanalysis,partialdifferentialequations,inte-
gralinequalitiesandspectraltheory.
The main results and contributions are applied for an estimate from below of
thefirsteigenvalueofthep-LaplacianinboundeddomainswithDirichletboundary
conditions and for the study of global existence, finite time blow-up and null non-
controllability for initial boundary value problems of singular or degenerate heat
equations.
Themainideas,mathematicaltoolsandapplicationsaredesignedformasterde-
greestudents,PhDstudentsandresearcherswholiketospecializeinthefieldofinte-
gralinequalitiesanddifferentialequations.
Sofia,Bulgaria,August2022 NikolaiKutev
TsviatkoRangelov
https://doi.org/10.1515/9783110980370-201

Contents
Preface|V
1 Introduction|1
2 PreliminaryremarksonHardyinequalities|7
2.1 Inequalitieswithgeneralweights|8
2.2 Inequalitieswithweightssingularontheboundary|9
2.3 Inequalitieswithweightssingularatapoint|11
2.4 Hardyinequalitieswithanadditionalterm|12
3 Hardyinequalitiesinabstractform|19
3.1 Generalweights|19
3.2 Hardyinequalitieswithweightthe firsteigenfunctionofthe
p-Laplacian|22
3.3 Hardyinequalitieswithadditionallogarithmicterm|25
3.4 One-parametricfamilyofHardyinequalities|28
3.5 Comparisonwithsomeexistingresults|34
4 Hardyinequalitiesinsphericalareas|37
4.1 Inequalitiesinanannulus|37
4.2 Inequalitiesinaball|44
4.3 Inequalitiesinanexteriorofaball|48
4.4 InequalityinBcforu∈W1,p(Bc)|55
r 0 r
4.5 Concludingnotes|57
5 GeneralHardyinequalitieswithoptimalconstant|59
5.1 WeightedHardyinequality|59
5.2 OptimalityoftheHardyconstant|65
5.3 Examplesandcomments|68
6 Hardyinequalitieswithweightssingularataninteriorpoint|73
6.1 Preliminaries|73
6.2 Mainresult|74
6.3 Star-shapeddomains|77
6.4 Generaldomains|79
6.5 Examplesandcomments|81
7 Hardyinequalitiesinstar-shapeddomainswithdoublesingular
weights|85
7.1 Preliminaries|85
VIII | Contents
7.2 Hardyinequalitieswithadditionalboundaryterm|86
7.3 AttainabilityoftheHardyconstant|89
7.4 Comments|95
8 Estimatesfrombelowforthefirsteigenvalueofthep-Laplacian|97
8.1 Existingestimatesofthefirsteigenvalueofthep-Laplacian|97
8.2 Estimatesfrombelowofλp,nusingHardyinequalities|100
8.2.1 EstimatesbymeansofHardyinequalitieswithdoublesingular
weights|101
8.2.2 EstimatesbymeansofHardyinequalitieswithadditionallogarithmic
term|103
8.2.3 Estimatesbymeansofaone-parametricfamilyofHardy
inequalities|107
8.3 Comparisonbetweendifferentanalyticalestimatesofλp,n|113
8.4 Numericalcomparison|119
9 ApplicationofHardyinequalitiesforsomeparabolicequations|123
9.1 Nullnoncontrollability|123
9.2 Globalexistence|130
9.3 Finitetimeblow-up|134
9.4 Comments|138
Bibliography|141
Index|149
1 Introduction
ThisbookisdevotedtotheclassicalHardyinequality,itsgeneralizationsandappli-
cations for estimates from below of the first eigenvalue λp,n(Ω) of the p-Laplacian,
p > 1,inaboundeddomainΩ ⊂ ℝn,n ≥ 2,andglobalsolvability,finitetimeblow-
upandnullnoncontrollabilityforsingularparabolicproblemswithHardypotential.
Onlythemultidimensionalcasen ≥ 2isconsideredbecauseforn = 1thereexistde-
tailedliteratureandsatisfactoryresults;seeforinstanceHardy[98,99,100],Neĉas
[146],Maz’ja[142],OpicandKufner[147],Hoffmann-Ostenhofetal.[104].Unlikethe
one-dimensionalcase,thetheoryforn≥2isfarfrombeingcompletelysolved.
ThisbookcanberegardedasaworkintheseriesofworksofMaz’ja[142],Opic
and Kufner [147], Ghoussoub and Moradifam [88], Balinsky et al. [22], Kufner et al.
[118],RuzhanskyandSuragan[163].WefocusontheoptimalityoftheHardyconstant
andonitsattainability.FurtheronintheworkwesaythattheHardyconstantisop-
timalifforagreateronethecorrespondingHardyinequalityfailsforallfunctionsof
theadmissibleclass.TheoptimalHardyconstantisattainableintheHardyinequality
whenanequalityisachievedforsomeadmissiblefunction.FortheHardyinequality
withsingularweightsataninteriorpointofΩ,orontheboundary𝜕Ω,wealwaysprove
theoptimalityoftheHardyconstant.AsfortheattainabilityoftheHardyconstant,we
showthatanequalityisachievedonlyforHardyinequalitywithadditional“nonlin-
ear”term.Itiswellknownthatforinequalitieswithoptimalconstantandadditional
“linear”term,anequalityisnotachieved.OnlyinthecasewhentheHardyconstant
isgreaterthantheoptimalone,byavariationaltechniqueitisprovedthattheHardy
constantisattainable;seeforexamplePinchoverandTintarev[151]andthereferences
therein.
IntheliteraturemainlyHardyinequalitieswithsingularweightsatapoint,oron
theboundary,𝜕Ω,oronsomek-dimensionalmanifold,1 ≤ k ≤ n−1,arestudied.
ThesubjectoftheinvestigationsinthisbookisHardytypeinequalitiesinbounded
domainsΩ∈ℝn,n≥2,withdoublesingularweights,inaninteriorpointofΩandon
theboundary𝜕Ω.
OuraimistoderivenewHardyinequalities,whichhaveanoptimalconstantand
suitableadditionaltermssuchthattheHardyconstantisattainable.Forexample,the
Hardyconstantisoptimalforconvexandstar-shapeddomainsandtheHardyconstant
isattainableasaresultofa“nonlinear”additionalterm.
The background of the theory of Hardy inequalities is mathematical and func-
tionalanalysis,differentialequations,spectralanalysisandtheuncertaintyprinciple
inphysics.
AfterthepioneeringpapersHardy[98,99,100]andthebookHardyetal.[101]
thereweremanygeneralizationsinthetwentiethcenturyoftheclassicalHardyin-
equalityaswellassomenewfunctionalinequalitiesmotivatedbyHardy’sworks;for
example,seeLieb[133]andRuzhanskyandSuragan[163],Chapters3and7:
https://doi.org/10.1515/9783110980370-001
2 | 1 Introduction
(i) theHardy–Littlewoodinequality(seeHardyandLittlewood[102,103])
󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫u(x)|x−y|−rv(y)dxdy󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨≤Cp(1,r),n‖u‖p‖v‖q,
ℝn
foreveryu∈Lp(ℝn),v∈Lq(ℝn),1<p,q<∞,0<r <n, 1 + 1 + r =2;
p q n
(ii) theHardy–Littlewood–Sobolevinequality(seeSobolev[170])
󵄩󵄩󵄩󵄩|x|−r⋆u󵄩󵄩󵄩󵄩q ≤Cp(2,r),n‖u‖p,
foreveryu∈Lp(ℝn),1<p,q,nr <∞, p1 + nr =1+ q1 andoptimalconstantCp(2,r),n;
(iii)the double weighted Hardy–Littlewood–Sobolev inequality of Stein and Weiss
[171]:
󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∫V(x,y)u(y)dy󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩q ≤Cα(3,β),r,n‖u‖p,
ℝn
withV(x,y) = |x|−β|x−y|−r|y|−α,foreveryu ∈ Lp(ℝn),0 ≤ α < pn−p1,0 ≤ β < qn,
1 + r+α+β =1+ 1;
p n q
(iv) theOkikiolu–Glaser–Martin–Grosse–Thirringinequality(seeGlaseretal.[89])
󵄩󵄩󵄩󵄩|x|−bu󵄩󵄩󵄩󵄩p ≤Cn(4,p)‖∇u‖2, n≥3, 0≤b<1, p= 2b+2nn−2,
foreveryu∈W1,2(ℝn),whichisageneralizationoftheSobolevinequality
(5)
‖u‖2n ≤Cn ‖∇u‖2, n≥3;
n−2
(v) theHardy–Rellichinequality(seeRellich[160])
(6) u2 2
Cn ∫ |x|4dx≤ ∫|Δu| dx, n≥5,
ℝn ℝn
foreveryu∈C0∞(ℝn\{0})andoptimalconstantCn(6) =(n(n4−4))2;
(vi) the Caffarelli–Kohn–Nirenberg inequality (see Caffarelli et al. [43], Catrina and
Wang[48])
p
|u|q q (7) |∇u|p
(∫ |x|δqdx) ≤Cn,p,γ,q ∫ |x|γp dx
ℝn ℝn
foreveryu∈C0∞(ℝn)andδq<n,γp<n,γ≤δ≤γ+1, q1 − nδ = p1 − γn+1.
ThelimitcaseoftheCaffarelli–Kohn–Nirenberginequalityforp = q,δ = γ+1is
theHardyinequalityinDall’Aglioetal.[52],