Elliptic and Parabolic Equations Involving the Hardy-Leray Potential (de Gruyter Nonlinear Analysis and Applications)

514 Pages·2021·5.581 MB·English

by  Ireneo Peral Alonso, Fernando Soria de Diego

IreneoPeralAlonso,FernandoSoriadeDiego
EllipticandParabolicEquationsInvolvingtheHardy-LerayPotential
De Gruyter Series in Nonlinear
Analysis and Applications
|
Editor-inChief
JürgenAppell,Würzburg,Germany
Editors
CatherineBandle,Basel,Switzerland
AlainBensoussan,Richardson,Texas,USA
ManueldelPino,SantiagodeChile,Chile
AvnerFriedman,Columbus,Ohio,USA
MikioKato,Tokyo,Japan
WojciechKryszewski,Torun,Poland
UmbertoMosco,Worcester,Massachusetts,USA
SimeonReich,Haifa,Israel
VicenţiuD.Rădulescu,Krakow,Poland
Volume 38
Ireneo Peral Alonso, Fernando Soria de Diego
Elliptic and Parabolic
Equations Involving the
Hardy-Leray Potential
|
MathematicsSubjectClassification2010
35A15,35B25,35B30,35B33,35B40,35J58,35B99,35C15,35D05,35D10,35D30,35D35,35J20,
35J25,35J58,35J70,35K05,35K15,35K55,35K57,35K58,35R09,35R11,45K05,46E30,46E35,47G20
Authors
Prof.Dr.IreneoPeralAlonso Prof.Dr.FernandoSoriadeDiego
UniversidadAutonomadeMadrid UniversidadAutonomadeMadrid
DepartamentodeMatemáticas DepartamentodeMatemáticas
CalledeTomásyValiente7 CalledeTomásyValiente7
CiudadUniv.deCantoblanco CiudadUniv.deCantoblanco
28049Madrid 28049Madrid
Spain Spain
[email protected] [email protected]
ISBN978-3-11-060346-0
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Toourfamilies
ThisbookisdedicatedtoMagdalenaandAngelines,fortheirlove,supportandperse-
verance;toourdaughtersandson,IreneandMagdalenaandIrene,EvaandAlberto;
totheirspouses,DavidandCarlosandCarlos,CarlosandIrene;andtofiveimportant
persons,ourgrandchildren,Iria,Oscar,LauraandJaimeandEmma.

Preface
Thismonographcontainspartoftheworkdevelopedbythefirstauthorduringthe
last25yearsinconnectionwiththestudyofsolutionstocertainellipticandparabolic
partialdifferentialequationsassociatedtotheHardy–Leraypotential.So,theguest
starofthisstoryisnothingbut
1
N
V(x)= , x∈ℝ , N ≥3,
|x|2
apotentialfunctionthatiscloselyrelatedtotheuncertaintyprincipleofHeisenbergin
quantummechanics.
ThestartingpointofthetheoryisHardy’sinequalityinonedimension,theonly
oneproperlyattributedtoHardy.Ourinterestinthisfieldcamewhenreadingthesem-
inalworkofJeanLeray[225]ontheNavier–Stokesequations,whereonecanfindthe
proofofasimilar inequalitytothatbyHardy,butthistimeindimensionthree,the
naturaldimensionfortheseequations.
Inthepaper[257]thisinequalitywasusedtostudythestabilityoftheextremal
solutionofaproblemrelatedtoareactioncombustionmodel.Theideabehinditwas
toestablishablow-upresult.ButthefirstapplicationofwhatisnowcalledtheHardy–
LerayinequalitycanbeseeninthepaperbyJeanLerayabouttheNavier–Stokesequa-
tionsquotedabove.TheHardy–Lerayinequalityalsoappearsasatoolinthestudyof
theSchrödingerequation
(Zq−Q) Q
iut+Δu=V(x,t)u, V(x,t)=− 4π|x| − 4π|x−v0t|, Z ∈ℕ, t >0.
Thisdescribesthewavefunctionofanelectronundertheactionofanα-particlewith
charge Q that, at time t = 0, leaves the nucleus of an atom with velocity v0 ∈ ℝ3
(see[133,ChapterXVIII,page752–758]).
ThepeculiarbehavioroftheHardy–Leraypotentialismotivatedbythefactthat
ithasthesamehomogeneityastheDirichletintegraloftheLaplacianwhenrescaling
thespacevariables.Inthissense,asimilarbehaviorisobservedfor
1
N
V(x)= , x∈ℝ \{0}, N >p>1,
|x|p
withrespecttothep-Laplacian,i.e.,theoperatorΔp(u) = div(|∇u|p−2∇u).Seethepa-
per [179], where we began to study some nontrivial extensions of the Hardy–Leray
inequality.Inthismonographwedonotfollowthisdirection,butsuggestthereader
tosearchfortheinformationaboutthesubsequentresultsinthebibliography.
Ifweconsider
1
N
V(x)= , x∈ℝ \{0}, N >2s, 0<s<1,
|x|2s
https://doi.org/10.1515/9783110606270-201
VIII | Preface
wewillfindthesamebehaviorofthehomogeneitywithrespecttotheso-calledfrac-
tionalLaplacianthatwasintroducedmyM.Rieszin[272].Inrecentyearsthisoperator
hasattractedtheattentionofmathematiciansandisappearinginalargenumberof
problems.ThedefinitionandpropertiesofthefractionalLaplacianwillbeexplained
inChapter8.
Themaingoalsofthisworkaretodevelopthekeyingredientstostudyhowthe
singularpotentials
1 1
V(x)= |x|2 and Vs(x)= |x|2s, 0<s<1, N >2s,
affectsomeproblemsmodeledbypartialdifferentialequationsandintegro-differential
equations,respectively.Itisinthislastpartinwhichthecooperationoftheauthors
hasbeendefinitivelyusedinadeepwayinordertocompletethetotallandscapeof
boththelocalandthenonlocalsetting.
The Hardy–Leray potentials, V(x) and Vs(x), have something fascinating: they
complicatethecomputationsandaredifficulttohandle,butatthesametimethey
provideaveryrichsetofproperties.InthissenseV andV arebothamonsteranda
s
friend,aswewilltrytoshowinthismonograph.Wearereallyinthepresenceofanold
classictheme,especiallyinthecaseofV,butwithmanynewresultsinrecenttimes
forbothpotentials.
AnimportantfeatureaboutV(x)isitsdimensionlesscharacter.Thisisduetothe
factthatitcorresponds,moduloconstants,totheFouriertransformofthefundamental
solutionoftheLaplaceequationinanydimensiongreaterthanorequaltothree.
Thatis,inthesenseoftheFouriertransformdefinedon𝒮󸀠(ℝN),thedualofthe
Schwartzclass,wefind
a b
ℱ(|x|NN−2)(ξ)= |ξN|2.
ThesamepropertyclearlyholdsforthepotentialV .SeeChapters8and9tofindthe
s
details.
Therearemanyreasons,comingparticularlyfromphysics,abouttheimportance
of studying partial differential equations associated to the Hardy–Leray potential.
Fromapurelypointofview,wemustsaythatthepotentialgivenbythereciprocal
ofthesquarefunctionisaperfectlaboratorytostudyborderlinepropertiesorcritical
behaviorinmanysituations,liketheexistenceofsolutionstononlinearpartialdiffer-
entialequations,certaineigenvalueproblems,instantaneousandcompleteblowup
andmanyothers.
ThesamecanbesaidforthefractionalHardypotential,V ,forwhichtheframe-
s
workisthefractionalLaplacianthatinthelastyearshasreceivedalotofattentionof
researchersduetotheanalyticalinterestandthelargeamountofapplications.The
reader,again,canfindmanyinterestingreferencesaboutthisinthebibliography.
Preface | IX
The first seven chapters of the monograph are devoted to the classical Hardy–
LeraypotentialV(x)= |x1|2 indimensionN ≥3.
InChapter1wedescribeseveralsettingsinwhichtheHardy–Leraypotentialap-
pearsinanaturalway.InquantummechanicstheHardy–Lerayinequalityrepresents,
perhaps,themostelementarywaytoexpresstheuncertaintyprincipleandprovides
aswellathresholdformanyassociatedresults.Fromtheanalyticalpointofview,ob-
N
servingthatV ∉ L2 ,althoughV(x) ∈ Lp forp < N,itgivesagoodexampleforthe
loc loc 2
spectraltheoryoftheLaplacianinboundeddomains.
WewillseehowtheHardypotentialappearsinthestudyofthestabilityofthesin-
gularsolutionofareactionofcombustionbyFrank-KamenetskiiandbyGelfandand
collaborators.TheHardy–Lerayinequalityalsoappearsin[99]tostudythestability
oftheSimmonscones(see[289]).InSection1.3weintroduceaveryusefultool,the
Piconeinequality.
Chapter2dealswiththeelementarypropertiesoftheHardy–Leraypotential,the
proofoftheHardy–LerayinequalityinL2(ℝN)and,ingeneral,inLp(ℝN),1 < p < N.
Inparticulartheoptimalconstantisevaluatedanditsoptimalityisprovedaswellas
itsnonattainability.Tothisend,wewillalsopresentsomeresultsonclassicalsym-
metrization.ThelastsectionisdevotedtoobtainingtheHardy–Lerayinequalitywith
aremainderterm,whichdependsonaweightedL2-normofthegradient.
Itiswellknown,bynow,whatisthesummabilityofthesolutiontotheproblem
−Δu=f, inΩ, u=0, on𝜕Ω,
intermsofthesummabilityofthedatumf.HereweassumethatΩ⊂ℝNisabounded
domain and 0 ∈ Ω. This can be obtained either as a byproduct of the Calderón–
Zygmund theory or as a particular case of the results by Stampacchia about the
summability of the solution of the equation in divergence form with measurable
bounded coefficients in terms of the datum. Chapter 3 has as an objective to study
theeffectofalinearterminvolvingtheHardy–Leraypotential,thatis,toanalyzethe
problem
u
−Δu−λ =f, inΩ, u=0, on𝜕Ω,
|x|2
where0<λ<ΛN,2andf ∈Lm(Ω),m≥1.Summarizingwecansaythatifweconsider
thefunction
N(m−1)(N−2m)
λ(m)= ,
m2
then,for0<λ<λ(m),thesummabilityisexactlythesameasthatintheunperturbed
∞
problem.Moreover,theseresultsareoptimal.Forinstance,iff ∈ L (Ω)thesolution
isunboundedandtheproblemisnotwellposedinL1(Ω),theoppositeofwhathap-
pensintheunperturbedproblem.Anecessaryandsufficientconditionforexistence
inweightedL1-spaces,amongotherresults,isobtainedinSection3.4.
X | Preface
InthepaperbyBrezis,DupaigneandTesei[89]andbyDupaigne[145],theeffect
oftheHardy–Leraypotentialwasstudiedinsomesemilinearellipticproblemsofthe
type
u
p
−Δu−λ =u +f, u≥0inΩ, 0∈Ω, u=0on𝜕Ω.
|x|2
Chapter4collectstheseresultsandsomeadditionaltopicsofellipticequations
withmeasuredata,potentialtheoryandaproofofthemountainpasstheorembyAm-
brosettiandRabinowitz.Westartcomputingthecriticalexponentfortheexistence
andthenwegivetheexistenceresultfortheDirichletproblemforellipticequationsin
divergenceformandwithmeasuredata.Wealsopresentamotivationoftheconceptof
Newtoniancapacityandaproof,basedonargumentsbyI.Ekeland[147],ofthemoun-
tain pass theorem. Chapter 4 includes the proofs of existence under the critical ex-
ponent,p+(λ),andthenonexistenceresultoverthecriticalexponent.Tounderstand
betterthenatureofthenonexistenceresultweprovetheso-calledcompleteblow-up.
FollowingapathtobetterunderstandingoftheeffectoftheHardy–Leraypoten-
tial,wedevoteChapter5tostudyingthebehavioroftheheatequationwithareaction
termofpowertype.Aninspiringresultforthelinearheatequationperturbedbythe
Hardy–LeraypotentialisduetoP.BarasandJ.Goldsteinin[41].Allalongthemono-
graphonewillfinddifferentapproachestothistypeofresultbutindifferentsettings.
Thethresholdpowerfortheexistenceturnsouttobethesameastheonewehave
foundforthestationaryequationinChapter4.Somepreviousresultsabouttheexis-
tenceofweaksolutiontotheheatequationareintroducedinordertomaketheexpo-
sitionmoresystematic.Theproofofnonexistenceifp≥p+(λ)ismoreinvolvedthanin
theellipticcase.Moreovertheproofoftheinstantaneousandcompleteblow-upgives
precisemeaningtothenonexistencephenomenon.TheCauchyproblemisalsoana-
lyzedandoneofthenewfactsthatareworthtoemphasizeistheshifttotherightof
theFujitaexponentoftheheatequation.
WeexploreinChapter6somequasilinearproblems,thatis,thosewithanonlin-
eartermwhichisapowerofthegradient.Thesekindsofproblemsappearinseveral
contextsrelatedtoHamilton–Jacobiequations,somegrowthmodels,asthestation-
arypartoftheKardan–Parisi–Zhangmodel,someflamepropagation,etc.See[213]
forsomeotherphysicalmotivations.
These quasilinear problems have an extra degree of difficulty according to the
nonlineargradientterm.Becauseofthis,somecompactnessresultshavetobeused,
evenmorewhenthereisaperturbationbytheHardy–Leraypotential.Someanalytical
toolsthatwillbeusedalongthechapterarediscussedasastartingpoint.Thenwewill
findthecriticalpowerofthegradient,q+(λ),inordertohaveexistenceandweprove
nonexistenceifq ≥ q+(λ)aswellascompleteblow-up.Theargumentsinthequasi-
linearcasearemoreinvolvedthaninthesublinearcase.Finallytheexistenceresult
isobtainedfor0 < λ < ΛN,2and1 < q < q+(λ)byusingthecompactnessarguments
explainedinthecomplementaryanalyticalresultsatthebeginningofChapter6.