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. 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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