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Elliptic Regularity Theory by Approximation Methods (London Mathematical Society Lecture Note Series) PDF

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LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor:ProfessorEndreSüli,MathematicalInstitute,UniversityofOxford, WoodstockRoad,OxfordOX26GG,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat www.cambridge.org/mathematics 367 Randommatrices:Highdimensionalphenomena, G.BLOWER 368 GeometryofRiemannsurfaces, F.P.GARDINER,G.GONZÁLEZ–DIEZ&C.KOUROUNIOTIS(eds) 369 Epidemicsandrumoursincomplexnetworks, M.DRAIEF&L.MASSOULIÉ 370 Theoryofp-adicdistributions, S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH 371 Conformalfractals, F.PRZYTYCKI&M.URBAN´SKI 372 Moonshine:Thefirstquartercenturyandbeyond, J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds) 373 Smoothness,regularityandcompleteintersection, J.MAJADAS&A.G.RODICIO 374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction, S.ALINHAC 375 Triangulatedcategories, T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds) 376 Permutationpatterns, S.LINTON,N.RUŠKUC&V.VATTER(eds) 377 AnintroductiontoGaloiscohomologyanditsapplications, 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B.E.A.NUCINKIS(eds) 445 Introductiontohiddensemi-Markovmodels, J.VANDERHOEK&R.J.ELLIOTT 446 Advancesintwo-dimensionalhomotopyandcombinatorialgrouptheory, W.METZLER&S.‘ROSEBROCK (eds) 447 Newdirectionsinlocallycompactgroups, P.-E.CAPRACE&N.MONOD(eds) 448 Syntheticdifferentialtopology, M.C.BUNGE,F.GAGO&A.M.SANLUIS 449 Permutationgroupsandcartesiandecompositions, C.E.PRAEGER&C.SCHNEIDER 450 Partialdifferentialequationsarisingfromphysicsandgeometry, M.BENAYEDetal(eds) 451 Topologicalmethodsingrouptheory, N.BROADDUS,M.DAVIS,J.-F.LAFONT&I.ORTIZ(eds) 452 Partialdifferentialequationsinfluidmechanics, C.L.FEFFERMAN,J.C.ROBINSON&J.L.RODRIGO (eds) 453 Stochasticstabilityofdifferentialequationsinabstractspaces, K.LIU 454 Beyondhyperbolicity, M.HAGEN,R.WEBB&H.WILTON(eds) 455 GroupsStAndrews2017inBirmingham, C.M.CAMPBELLetal(eds) 456 Surveysincombinatorics2019, A.LO,R.MYCROFT,G.PERARNAU&A.TREGLOWN(eds) 457 Shimuravarieties, T.HAINES&M.HARRIS(eds) 458 IntegrablesystemsandalgebraicgeometryI, R.DONAGI&T.SHASKA(eds) 459 IntegrablesystemsandalgebraicgeometryII, R.DONAGI&T.SHASKA(eds) 460 Wigner-typetheoremsforHilbertGrassmannians, M.PANKOV 461 Analysisandgeometryongraphsandmanifolds, M.KELLER,D.LENZ&R.K.WOJCIECHOWSKI 462 ZetaandL-functionsofvarietiesandmotives, B.KAHN 463 Differentialgeometryinthelarge, O.DEARRICOTTetal(eds) 464 Lecturesonorthogonalpolynomialsandspecialfunctions, H.S.COHL&M.E.H.ISMAIL(eds) 465 ConstrainedWillmoresurfaces, Á.C.QUINTINO 466 Invarianceofmodulesunderautomorphismsoftheirenvelopesandcovers, A.K.SRIVASTAVA, A.TUGANBAEV&P.A.GUILASENSIO 467 ThegenesisoftheLanglandsprogram, J.MUELLER&F.SHAHIDI 468 (Co)endcalculus, F.LOREGIAN 469 Computationalcryptography, J.W.BOS&M.STAM(eds) 470 Surveysincombinatorics2021, K.K.DABROWSKIetal(eds) 471 Matrixanalysisandentrywisepositivitypreservers, A.KHARE 472 FacetsofalgebraicgeometryI, P.ALUFFIetal(eds) 473 FacetsofalgebraicgeometryII, P.ALUFFIetal(eds) 474 Equivarianttopologyandderivedalgebra, S.BALCHIN,D.BARNES,M.KE˛DZIOREK&M.SZYMIK(eds) 475 EffectiveresultsandmethodsforDiophantineequationsoverfinitelygenerateddomains, J.-H.EVERTSE& K.GYO˝RY 476 Anindefiniteexcursioninoperatortheory, A.GHEONDEA Published online by Cambridge University Press LondonMathematicalSocietyLectureNoteSeries:477 Elliptic Regularity Theory by Approximation Methods EDGARD A. PIMENTEL UniversityofCoimbra Published online by Cambridge University Press UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781009096669 DOI:10.1017/9781009099899 ©EdgardA.Pimentel2022 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 PrintedintheUnitedKingdombyTJBooksLimited,PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-009-09666-9Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Published online by Cambridge University Press (ToEmilia WhoItrulylovewithallmyheart. Andtowhomitmightsuffice.) Published online by Cambridge University Press Published online by Cambridge University Press Contents Preface pageix 1 EllipticPartialDifferentialEquations 1 1.1 BasicDefinitionsandFacts 1 1.2 Krylov–SafonovTheory 30 1.3 Lin’sIntegralEstimates 43 1.4 GradientHölderContinuity 48 1.5 Evans–KrylovTheory 50 1.6 Caffarelli’sRegularityTheory:ApproximationMethods 59 1.7 CounterexamplesandOptimalRegularity 70 BibliographicalNotes 74 2 FlatSolutionsAreRegular 76 2.1 SavinRegularityTheoryinC2,α-Spaces 76 2.2 ThePartialRegularityResult 94 BibliographicalNotes 99 3 TheRecessionStrategy 100 3.1 TheRecessionFunction 100 3.2 ApplicationstoRegularityTheoryinSobolevSpaces 104 3.3 ApplicationstoRegularityTheoryinHölderSpaces 114 3.4 WeakRegularityTheory:DensityResults 120 3.5 LimitationsoftheRecessionStrategy 122 BibliographicalNotes 125 4 ARegularityTheoryfortheIsaacsEquation 127 4.1 SomeContext 127 4.2 TheBellmanEquation 130 vii Published online by Cambridge University Press viii Contents 4.3 RegularityfortheIsaacsEquationinSobolevSpaces 132 4.4 RegularityfortheIsaacsEquationinHölderSpaces 145 BibliographicalNotes 150 5 RegularityTheoryforDegenerateModels 151 5.1 ARegularityTheoryforthep-LaplaceOperator 151 5.2 FullyNonlinearDegenerateProblems 162 5.3 FurtherRemarksonDegenerateDiffusions 178 BibliographicalNotes 180 References 181 Index 189 Published online by Cambridge University Press Preface This set of notes focuses on regularity theory for elliptic partial differential equations (PDEs). In particular, it details regularity results obtained through approximation(perturbative)methods,inlinewiththetechniqueslaunchedin theworksofCaffarelli(1988,1989). Our goal is to tell a story; it starts with the fundamental breakthroughs of the 1980s, namely, the Krylov–Safonov and the Evans–Krylov results for nonvariational PDEs, jointly with Caffarelli’s regularity theory for fully nonlinear elliptic equations. At this point we emphasize the importance of convexity.WhileuniformellipticityimpliesHöldercontinuityofthegradient, a convexity condition leads to C2,α-regularity. Our perspective is that such a fact entails two fundamental directions. First, one asks whether or not C1,α- regularityisoptimalformerelyellipticequations.Then,oneseeksconditions, weakerthanconvexity,thatarecapableofunlockinggeneralregularityresults inbetweentheKrylov–SafonovandtheEvans–Krylovtheories. The first direction is well understood and documented in the corpus of results due to Nadirashvili and Vla˘du¸t (2007, 2008, 2011). As far as the second direction is concerned, an important bifurcation arises. On one hand, differentiabilityoftheoperatorstandsoutasaconditionaffectingtheregularity ofthesolutions. Inthisregard,wediscussthedevelopmentsduetoSavin(2007),namely,the factthatflatsolutionstoequationsdrivenbyoperatorsofclassC2 arelocally ofclassC2,α.ItisworthnoticingthecontributioninSavin(2007)transcends thisdiscussionasitrelatesalsotothecaseofdegenerateellipticequations.A further development under differentiability conditions is the so-called partial regularityresult,duetoArmstrongetal.(2012). The alternative route concerns the analysis of particular models through a set of techniques capable of producing new information on the regularity ix https://doi.org/10.1017/9781009099899.001 Published online by Cambridge University Press

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