Nonlocal diffusion and applications

165 Pages·2016·1.377 MB·English

by  Bucur C., Valdinoci E

Lecture Notes of the Unione Matematica Italiana
Claudia Bucur
Enrico Valdinoci
Nonlocal
Diffusion
and
Applications
Lecture Notes of 20
the Unione Matematica Italiana
Moreinformationaboutthisseriesathttp://www.springer.com/series/7172
Editorial Board
CiroCiliberto FrancoFlandoli
(EditorinChief) Dipartimentodi
DipartimentodiMatematica MatematicaApplicata
UniversitàdiRomaTorVergata UniversitàdiPisa
ViadellaRicercaScientifica ViaBuonarrotic
Roma,Italy Pisa,Italy
e-mail:[email protected].it e-mail:fl[email protected]
SusannaTerracini AngusMacIntyre
(Co-editorinChief) QueenMaryUniversityofLondon
UniversitàdegliStudidiTorino SchoolofMathematicalSciences
DipartimentodiMatematica“GiuseppePeano” MileEndRoad
ViaCarloAlberto LondonENS,UnitedKingdom
Torino,Italy e-mail:[email protected]
e-mail:[email protected]
GiuseppeMingione
AdolfoBallester-Bollinches DipartimentodiMatematicaeInformatica
Departmentd’Àlgebra UniversitàdegliStudidiParma
FacultatdeMatemàtiques ParcoAreadelleScienze,/a(Campus)
UniversitatdeValència Parma,Italy
Dr.Moliner, e-mail:[email protected]
Burjassot(València),Spain
MarioPulvirenti
e-mail:[email protected]
DipartimentodiMatematica
AnnalisaBuffa UniversitàdiRoma“LaSapienza”
IMATI–C.N.R.Pavia P.leA.Moro
ViaFerrata Roma,Italy
Pavia,Italy e-mail:[email protected].it
e-mail:[email protected]
FulvioRicci
LuciaCaporaso ScuolaNormaleSuperiorediPisa
DipartimentodiMatematica PiazzadeiCavalieri
UniversitàRomaTre Pisa,Italy
LargoSanLeonardoMurialdo e-mail:[email protected]
I-Roma,Italy
ValentinoTosatti
e-mail:[email protected].it
NorthwesternUniversity
FabrizioCatanese DepartmentofMathematics
MathematischesInstitut SheridanRoad
Universitätstraÿe Evanston,IL,USA
Bayreuth,Germany e-mail:[email protected]
e-mail:[email protected]
CorinnaUlcigrai
CorradoDeConcini ForschungsinstitutfürMathematik
DipartimentodiMatematica HGG.
UniversitàdiRoma“LaSapienza” Rämistrasse
PiazzaleAldoMoro Zürich,Switzerland
Roma,Italy e-mail:[email protected]
e-mail:[email protected].it
CamilloDeLellis
InstitutfürMathematik
UniversitätZürich
Winterthurerstrasse
CH-Zürich,Switzerland TheEditorialPolicycanbefound
e-mail:[email protected] atthebackofthevolume.
Claudia Bucur • Enrico Valdinoci
Nonlocal Diffusion
and Applications
123
ClaudiaBucur EnricoValdinoci
DipartimentodiMatematica DipartimentodiMatematica
FederigoEnriques FederigoEnriques
UniversitaJdegliStudidiMilano UniversitaJdegliStudidiMilano
Milano,Italy Milano,Italy
ConsiglioNazionaledelleRicerche
IstitutodiMatematicaApplicatae
TecnologieInformaticheEnricoMagene
Pavia,Italy
WeierstraßInstitutfürAngewandte
AnalysisundStochasitk
Berlin,Germany
UniversityofMelbourne
SchoolofMathematicsandStatistics
Victoria,Australia
ISSN1862-9113 ISSN1862-9121 (electronic)
LectureNotesoftheUnioneMatematicaItaliana
ISBN978-3-319-28738-6 ISBN978-3-319-28739-3 (eBook)
DOI10.1007/978-3-319-28739-3
LibraryofCongressControlNumber:2016934714
©SpringerInternationalPublishingSwitzerland2016
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland
Preface
The purpose of these pages is to collect a set of notes that are a result of several
talksandminicoursesdeliveredhereandtherein theworld(Milan,Cortona,Pisa,
Roma, Santiago del Chile, Madrid,Bologna, Porquerolles,and Catania to name a
few).Wewillpresentheresomemathematicalmodelsrelatedtononlocalequations,
providingsomeintroductorymaterialandexamples.
Ofcourse,thesenotesandtheresultspresenteddonotaimtobecomprehensive
and cannot take into account all the material that would deserve to be included.
Even a thorough introduction to nonlocal (or even just fractional) equations goes
waybeyondthepurposeofthisbook.
Using a metaphor with fine arts, we could say that the picture that we painted
here is not even impressionistic, it is just naïf. Nevertheless, we hope that these
pagesmaybeofsomehelptotheyoungresearchersofallageswhoarewillingto
have a look at the exciting nonlocal scenario (and who are willing to tolerate the
partialandincompletepointofviewofferedbythismodestobservationpoint).
Milano,Italy ClaudiaBucur
Milano,Italy EnricoValdinoci
November2015
v

Acknowledgments
ItisapleasuretothankSerenaDipierro,RupertFrank,RichardMathar,Alexander
Nazarov, Joaquim Serra, and Fernando Soria for very interesting and pleasant
discussions. We are also indebted with all the participants of the seminars and
minicoursesfromwhichthissetofnotesgeneratedforthenicefeedbackreceived,
andwe hopethatthis work,thoughsomehowsketchyand informal,canbe useful
tostimulatenewdiscussionsandfurtherdevelopthisrichandinterestingsubject.
vii

Contents
1 AProbabilisticMotivation.................................................. 1
1.1 TheRandomWalkwithArbitrarilyLongJumps..................... 2
1.2 APayoffModel......................................................... 4
2 AnIntroductiontotheFractionalLaplacian............................. 7
2.1 PreliminaryNotions.................................................... 7
2.2 FractionalSobolevInequalityandGeneralizedCoareaFormula .... 16
2.3 MaximumPrincipleandHarnackInequality.......................... 19
2.4 Ans-HarmonicFunction............................................... 24
2.5 AllFunctionsAreLocallys-HarmonicUptoaSmallError......... 29
2.6 AFunctionwithConstantFractionalLaplacianontheBall ......... 33
3 ExtensionProblems.......................................................... 39
3.1 WaterWaveModel ..................................................... 40
3.1.1 ApplicationtotheWaterWaves............................... 42
3.2 CrystalDislocation..................................................... 43
3.3 AnApproachtotheExtensionProblemviatheFourier
Transform............................................................... 56
4 NonlocalPhaseTransitions ................................................. 67
4.1 TheFractionalAllen-CahnEquation.................................. 70
4.2 ANonlocalVersionofaConjecturebyDeGiorgi ................... 85
5 NonlocalMinimalSurfaces................................................. 97
5.1 Graphsands-MinimalSurfaces ....................................... 101
5.2 Non-existenceofSingularConesinDimension2.................... 111
5.3 BoundaryRegularity ................................................... 119
6 ANonlocalNonlinearStationarySchrödingerTypeEquation......... 127
6.1 FromtheNonlocalUncertaintyPrincipletoaFractional
WeightedInequality.................................................... 136
ix