Local Density of Solutions to Fractional Equations (De Gruyter Studies in Mathematics)

144 Pages·2019·2.439 MB·English

by  Alessandro Carbotti, Serena Dipierro, Enrico Valdinoci

AlessandroCarbotti,SerenaDipierro,andEnricoValdinoci
LocalDensityofSolutionstoFractionalEquations
De Gruyter Studies in
Mathematics
|
Editedby
CarstenCarstensen,Berlin,Germany
GavrilFarkas,Berlin,Germany
NicolaFusco,Napoli,Italy
FritzGesztesy,Waco,Texas,USA
NielsJacob,Swansea,UnitedKingdom
ZenghuLi,Beijing,China
Karl-HermannNeeb,Erlangen,Germany
Volume 74

Alessandro Carbotti, Serena Dipierro, and
Enrico Valdinoci
Local Density of
Solutions to Fractional
Equations
|
MathematicsSubjectClassification2010
Primary:26A33,34A08,35R11;Secondary:60G22
Authors
AlessandroCarbotti Prof.Dr.EnricoValdinoci
DipartimentodiMatematicaeFisica DepartmentofMathematicsandStatistics
UniversitàdelSalento UniversityofWesternAustralia
ViaPerArnesano 35StirlingHighway
73100Lecce Crawley,WA6009
Italy Australia
[email protected] [email protected]
Prof.Dr.SerenaDipierro
DepartmentofMathematicsandStatistics
UniversityofWesternAustralia
35StirlingHighway
Crawley,WA6009
Australia
[email protected]
ISBN978-3-11-066069-2
e-ISBN(PDF)978-3-11-066435-5
e-ISBN(EPUB)978-3-11-066131-6
ISSN0179-0986
LibraryofCongressControlNumber:2019946148
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2019WalterdeGruyterGmbH,Berlin/Boston
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Preface
Thestudyofnonlocaloperatorsoffractionaltypepossessesalongtradition,moti-
vatedbothbymathematicalcuriosityandbyreal-worldapplications.Thoughthisline
ofresearchpresentssomesimilaritiesandanalogieswiththestudyofoperatorsofin-
tegerorder,italsopresentsanumberofremarkabledifferences,oneofthegreatest
beingtherecentlydiscoveredphenomenonthatallfunctionsare(locally)fractionally
harmonic(uptoasmallerror).Thisfeatureisquitesurprising,sinceitisinsharpcon-
trastwiththecaseofclassicalharmonicfunctions,anditrevealsagenuinelynonlocal
peculiarity.
Moreprecisely,ithasbeenprovedin[25]thatgivenanyCk-functionfinabounded
domainΩandgivenanyϵ>0,thereexistsafunctionf whichisfractionallyharmonic
ϵ
inΩsuchthattheCk-distanceinΩbetweenf andf islessthanϵ.
ϵ
Interestingly,thiskindofresultscanbealsoappliedatanyscale,asshowninFig-
ures1,2,and3.Roughlyspeaking,givenanyfunction,withoutanyspecialgeomet-
ricprescription,inagivenboundeddomain(asinFigure1),onecan“complete”the
functionoutsidethedomaininsuchawaythattheresultingobjectisfractionallyhar-
monic.Thatis,onecanendowthefunctiongivenintheboundeddomainwithanum-
berofsuitableoscillationsoutsidethedomaininordertomakeanintegro-differential
operatoroffractionaltypevanish.ThisideaisdepictedinFigure2.Asamatteroffact,
Figure2mustbeconsideredjusta“qualitative”pictureofthismethod,andshouldnot
beregarded“realistic.”However,evenifFigure2doesnotprovideacorrectfractional
harmonicextensionofthegivenfunctionoutsidethegivendomain,theresultcanbe
repeatedatalargerscale,asinFigure3,addingfurtherremoteoscillationsinorderto
obtainafractionalharmonicfunction.
Figure1:Allfunctionsarefractionalharmonic,atdifferentscales(scaleoftheoriginalfunction).
https://doi.org/10.1515/9783110664355-201
VI | Preface
Figure2:Allfunctionsarefractionalharmonic,atdifferentscales(“first”scaleofexterioroscilla-
tions).
Figure3:Allfunctionsarefractionalharmonic,atdifferentscales(“second”scaleofexterioroscilla-
tions).
Inthissense,thistypeofresultsreallysaysthatwhatevergraphwedrawonasheet
ofpaper,itisfractionallyharmonic(morerigorously,itcanbeshadowedwithanar-
bitraryprecisionbyanothergraph,whichcanbeappropriatelycontinuedoutsidethe
sheetofpaperinawaywhichmakesitfractionallyharmonic).
Thisbookcontainsanewresultinthislineofinvestigation,statingthateveryfunc-
tionliesinthekernelofeverylinearequationinvolvingsomefractionaloperator,upto
asmallerror.Thatis,anygivenfunctioncanbesmoothlyapproximatedbyfunctions
lyinginthekernelofalinearoperatorinvolvingatleastonefractionalcomponent.The
settinginwhichthisresultholdsisverygeneral,sinceittakesintoaccountanomalous
diffusion,withpossiblefractionalcomponentsinbothspaceandtime.Theoperators
takenintoaccountcomprisethecaseofthesumofclassicalandfractionalLaplacians,
possiblyofdifferentorders,inthespacevariables,andclassicalorfractionalderiva-
tivesinthetimevariables.Namely,theequationcanbeofanyorder,itdoesnotneed
anystructure(itneedsnoellipticityorparabolicityconditions),andthefractionalbe-
haviorisintime,space,orboth.
Inasense,thistypeofapproximationresultsrevealsthetruepoweroffractional
equations,independentlyofthestructural“details”ofthesingleequationundercon-
sideration,andshowsthatspace-fractionalandtime-fractionalequationsexhibitava-
rietyofsolutionswhichismuchricherandmoreabundantthaninthecaseofclassical
diffusion.
Preface | VII
Though space- and time-fractional diffusions can be seen as related aspects of
nonlocal phenomena, they arise in different contexts and present important struc-
turaldifferences.Theparadigmaticexampleofspace-fractionaldiffusionisembodied
bythefractionalLaplacian,thatis,afractionalrootoftheclassicalLaplaceoperator.
Thissettingoftensurfacesfromstochasticprocessespresentingjumpsanditexhibits
theclassicalspatialsymmetriessuchasinvarianceundertranslationsandrotations,
plusascaleinvarianceoftheintegralkerneldefiningtheoperator.Differentlyfrom
this,time-fractionaldiffusionistypicallyrelatedtomemoryeffects,andthereforeit
distinguishesverystronglybetweenthe“past”andthe“future,”andthearrowoftime
playsamajorrole(inparticular,sincethepastinfluencesthefuture,butnotviceversa,
time-fractionaldiffusiondoesnotpossessthesametypeofsymmetriesofthespace-
fractionalone).Inthesepages,wewillbeabletoconsideroperatorswhichariseas
superpositionsofbothspace-andtime-fractionaldiffusion,possiblytakingintoac-
countclassicalderivativesaswell(thecasesofdiffusionwhichisfractionaljustin
eitherspaceortimearecomprisedasspecialsituationsofourgeneralframework).
Interestingly, we will also consider fractional operators of any order, showing, in a
sense,thatsomepropertiesrelatedtofractionaldiffusionpersistalsowhenhigheror-
deroperatorscomeintoplay,differentlyfromwhathappensintheclassicalcase,in
whichthetheoryavailablefortheLaplacianoperatorpresentssignificantdifferences
withrespecttothecaseofpolyharmonicoperators.
Toachievetheoriginalresultpresentedhere,wedevelopabroadtheoryofsome
fundamentalfactsaboutspace-andtime-fractionalequations.Someoftheseaddi-
tionalresultsareknownfromtheliterature,atleastinsomeparticularcases,butsome
otherarenewandinterestinginthemselves,and,indevelopingtheseauxiliarythe-
ories,thismonographpresentsacompletelyself-containedapproachtoanumberof
basicquestions,suchas:
– boundarybehaviorforthetime-fractionaleigenfunctions;
– boundarybehaviorforthetime-fractionalharmonicfunctions;
– Greenrepresentationformulas;
– existenceandregularityforthefirsteigenfunctionofthe(possiblyhigherorder)
fractionalLaplacian;
– boundary asymptotics of the first eigenfunctions of the (possibly higher order)
fractionalLaplacian;
– boundarybehaviorof(possiblyhigherorder)fractionalharmonicfunctions.
Wenowdiveintothetechnicaldetailsofthismatter.