Calculus of variations and nonlinear partial differential equations: lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27-July 2, 2005 PDF
Preview Calculus of variations and nonlinear partial differential equations: lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27-July 2, 2005
Lecture Notes in Mathematics 1927 Editors: J.-M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris C.I.M.E.meansCentroInternazionaleMatematicoEstivo,thatis,InternationalMathematicalSummer Center.Conceivedintheearlyfifties,itwasbornin1954andmadewelcomebytheworldmathematical communitywhereitremainsingoodhealthandspirit.Manymathematiciansfromallovertheworld havebeeninvolvedinawayoranotherinC.I.M.E.’sactivitiesduringthepastyears. SotheyalreadyknowwhattheC.I.M.E.isallabout.Forthebenefitoffuturepotentialusersandco- operatorsthemainpurposesandthefunctioningoftheCentremaybesummarizedasfollows:every year,duringthesummer,Sessions(threeorfourasarule)ondifferentthemesfrompureandapplied mathematicsareofferedbyapplicationtomathematiciansfromallcountries.Eachsessionisgenerally basedonthreeorfourmaincourses(24−30hoursoveraperiodof6-8workingdays)heldfromspe- cialistsofinternationalrenown,plusacertainnumberofseminars. AC.I.M.E.Session,therefore,isneitheraSymposium,norjustaSchool,butmaybeablendofboth. Theaimisthatofbringingtotheattentionofyoungerresearcherstheorigins,laterdevelopments,and perspectivesofsomebranchoflivemathematics. Thetopicsofthecoursesaregenerallyofinternationalresonanceandtheparticipationofthecourses covertheexpertiseofdifferentcountriesandcontinents.Suchcombination,gaveanexcellentopportu- nitytoyoungparticipantstobeacquaintedwiththemostadvanceresearchinthetopicsofthecourses andthepossibilityofaninterchangewiththeworldfamousspecialists.Thefullimmersionatmosphere ofthecoursesandthedailyexchangeamongparticipantsareafirstbuildingbrickintheedificeof internationalcollaborationinmathematicalresearch. C.I.M.E.Director C.I.M.E.Secretary PietroZECCA ElviraMASCOLO DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica UniversitàdiFirenze UniversitàdiFirenze ViaS.Marta,3 vialeG.B.Morgagni67/A 50139Florence 50134Florence Italy Italy e-mail:zecca@unifi.it e-mail:[email protected]fi.it FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it CIME’sactivityissupportedby: – IstitutoNationalediAltaMathematica“F.Severi” – Ministerodell’Istruzione,dell’UniversitàedelleRicerca – MinisterodegliAffariEsteri,DirezioneGeneraleperlaPromozioneela Cooperazione,UfficioV ThisCIMEcoursewaspartiallysupportedby:HyKEaResearchTrainingNetwork(RTN)financedby theEuropeanUnioninthe5thFrameworkProgramme“ImprovingtheHumanPotential”(1HP).Project Reference:ContractNumber:HPRN-CT-2002-00282 Luigi Ambrosio · Luis Caffarelli Michael G. Crandall · Lawrence C. Evans Nicola Fusco Calculus of Variations and Nonlinear Partial Differential Equations Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 27–July 2, 2005 WithahistoricaloverviewbyElviraMascolo Editors:BernardDacorogna,PaoloMarcellini ABC LuigiAmbrosio LawrenceC.Evans ScuolaNormaleSuperiore DepartmentofMathematics PiazzadeiCavalieri7 UniversityofCalifornia 56126Pisa,Italy Berkeley,CA94720-3840,USA [email protected] [email protected] LuisCaffarelli NicolaFusco DepartmentofMathematics DipartimentodiMatematica UniversityofTexasatAustin UniversitàdegliStudidiNapoli 1UniversityStationC1200 ComplessoUniversitarioMonteS.Angelo Austin,TX78712-0257,USA ViaCintia [email protected] 80126Napoli,Italy MichaelG.Crandall [email protected] DepartmentofMathematics PaoloMarcellini UniversityofCalifornia SantaBarbara,CA93106,USA ElviraMascolo [email protected] DipartimentodiMatematica UniversitàdiFirenze BernardDacorogna VialeMorgagni67/A SectiondeMathématiques 50134Firenze,Italy EcolePolytechniqueFédérale [email protected]fi.it deLausanne(EPFL) [email protected]fi.it Station8 1015Lausanne,Switzerland bernard.dacorogna@epfl.ch ISBN978-3-540-75913-3 e-ISBN978-3-540-75914-0 DOI10.1007/978-3-540-75914-0 LectureNotesinMathematicsISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 LibraryofCongressControlNumber:2007937407 MathematicsSubjectClassification(2000):35Dxx,35Fxx,35Jxx,35Lxx,49Jxx (cid:1)c 2008Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper 987654321 springer.com Preface WeorganizedthisCIMECoursewiththeaimtobringtogetheragroupoftop leadersonthefieldsofcalculusofvariations andnonlinearpartialdifferential equations.Thelistofspeakersandthetitlesoflectureshavebeenthefollowing: - LuigiAmbrosio,Transport equation and Cauchy problem for non-smooth vectorfields. - LuisA.Caffarelli,Homogenizationmethodsfornondivergenceequations. - Michael Crandall, The infinity-Laplace equation and elements of the cal- culusofvariationsinL-infinity. - GianniDalMaso,Rate-independentevolutionproblemsinelasto-plasticity: avariationalapproach. - LawrenceC.Evans,WeakKAMtheoryandpartialdifferentialequations. - NicolaFusco,Geometricalaspectsofsymmetrization. IntheoriginallistofinvitedspeakersthenameofPierreLouisLionswas alsoincluded,buthe,attheverylastmoment,couldnotparticipate. TheCourse,justlookingatthenumberofparticipants(morethan140,one ofthelargestinthehistoryoftheCIMEcourses),wasagreatsuccess;mostof themwereyoungresearchers,someotherswerewellknownmathematicians, experts in the field. The high level of the Course is clearly proved by the qualityofnotesthatthespeakerspresentedforthisSpringerLectureNotes. WealsoinvitedElviraMascolo,theCIMEscientificsecretary,towritein thepresentbookanoverviewofthehistoryofCIME(whichshepresentedat Cetraro)withspecialemphasisincalculusofvariationsandpartialdifferential equations. Most of the speakers are among the world leaders in the field of viscos- ity solutions ofpartialdifferentialequations,inparticularnonlinearpde’sof implicit type. Our choice has not been random; in fact we and other mathe- maticians have recently pointed out a theory of almost everywhere solutions of pde’s of implicit type, which is an approach to solve nonlinear systems of pde’s.ThusthisCoursehasbeenanopportunitytobringtogetherexpertsof viscositysolutionsandtoseesomerecentdevelopmentsinthefield. VI Preface WebrieflydescribeherethearticlespresentedinthisLectureNotes. Starting from the lecture by Luigi Ambrosio, where the author studies thewell-posednessoftheCauchyproblemforthehomogeneous conservative continuityequation d µ +D ·(bµ)=0, (t,x)∈I×Rd dt t x t andforthetransportequation d w +b·∇w =c , dt t t t whereb(t,x)=b(x)isagiventime-dependentvectorfieldinRd.Theinter- t esting case is when b(·) is not necessarily Lipschitz and has, for instance, a t SobolevorBV regularity.Vectorfieldswiththis“low”regularityshowup,for instance,inseveralPDE’sdescribingthemotionoffluids,andinthetheory ofconservationlaws. ThelectureofLuisCaffarelligaverisetoajointpaperwithLuisSilvestre; wequotefromtheirintroduction: “Whenwelookatadifferentialequationinaveryirregularmedia(com- posite material, mixed solutions, etc.) from very close, we may see a very complicatedproblem.However,ifwelookfromfarawaywemaynotseethe detailsandtheproblemmaylooksimpler.Thestudyofthiseffectinpartial differential equations isknown ashomogenization. Theeffectof theinhomo- geneities oscillating at small scales is often not a simple average and may be hard to predict: a geodesic in an irregular medium will try to avoid the badareas,theroughnessofasurfacemayaffectinnontrivialwaytheshapes of drops laying on it, etc... The purpose of these notes is to discuss three problemsinhomogenizationandtheirinterplay. In the first problem, we consider the homogenization of a free boundary problem.Westudytheshapeofadroplyingonaroughsurface.Wediscuss in what case the homogenization limit converges to a perfectly round drop. It is taken mostly from the joint work with Antoine Mellet (see the precise referencesinthearticlebyCaffarelliandSilvestreinthislecturenotes).The secondproblemconcernstheconstructionofplanelikesolutionstothemini- malsurfaceequationinperiodicmedia.Thisisrelatedtohomogenizationof minimalsurfaces.ThedetailscanbefoundinthejointpaperwithRafaelde la Llave. The third problem concerns existence of homogenization limits for solutions to fully nonlinear equations in ergodic random media. It is mainly basedonthejointpaperwithPanagiotisSouganidisandLiheWang. We will try to point out the main techniques and the common aspects. Thefocushasbeensettothebasicideas.Themainpurposeistomakethis advancedtopicsasreadableaspossible.” Michael Crandall presents in his lecture an outline of the theory of the archetypalL∞variationalprobleminthecalculusofvariations.Namely,given Preface VII an open U ⊂ Rn and b ∈ C(∂U), find u ∈ C(U) which agrees with the boundaryfunctionbon∂U andminimizes F∞(u,U):=(cid:4)|Du|(cid:4)L∞(U) amongallsuchfunctions.Here|Du|istheEuclideanlengthofthegradientDu ofu.Heisalsointerestedinthe“Lipschitzconstant”functionalaswell:ifK isanysubsetofRn andu:K→R,itsleastLipschitzconstantisdenotedby Lip(u,K):=inf{L∈R: |u(x)−u(y)|≤L|x−y|,∀x,y∈K}. OnehasF∞(u,U)=Lip(u,U) ifU is convex,butequality does nothold in general. TheauthorshowsthatafunctionwhichisabsolutelyminimizingforLip is also absolutely minimizing for F∞ and conversely. It turns out that the absolutely minimizing functions for Lip and F∞ are precisely the viscosity solutionsofthefamouspartialdifferentialequation (cid:1)n ∆∞u = uxiuxjuxixj =0. i,j=1 The operator ∆∞ is called the “∞-Laplacian” and “viscosity solutions” of theaboveequationaresaidtobe∞−harmonic. InhislectureLawrenceC.EvansintroducessomenewPDEmethodsde- veloped over the past 6 years in so-called “weak KAM theory”, a subject pioneeredbyJ.MatherandA.Fathi.Succinctlyput,thegoalofthissubject istheemployingofdynamicalsystems,variationalandPDEmethodstofind “integrablestructures”withingeneralHamiltoniandynamics.Mainreferences (see the precise references in the article by Evans in this lecture notes) are Fathi’sforthcomingbookandanarticlebyEvansandGomes. Nicola Fusco in his lecture presented in this book considers two model functionals: the perimeter of a set E in Rn and the Dirichlet integral of a scalar function u. It is well known that on replacing E or u by its Steiner symmetral oritssphericalsymmetrization,respectively,boththesequantities decrease. This fact is classical when E is a smooth open set and u is a C1 function. On approximating a set of finite perimeter with smooth open sets oraSobolevfunctionbyC1 functions,theseinequalitiescanbeextendedby lowersemicontinuitytothegeneralsetting.However,anapproximationargu- mentgivesnoinformationabouttheequalitycase.Thus,ifoneisinterested in understanding when equality occurs, one has to carry on a deeper analy- sis,basedonfinepropertiesofsetsoffiniteperimeterandSobolevfunctions. Briefly,thisisthesubjectofFusco’slecture. Finally, as an appendix to this CIME Lecture Notes, as we said Elvira Mascolo, the CIME scientific secretary, wrote an interesting overview of the historyofCIMEhavinginmindinparticularcalculusofvariationsandPDES. VIII Preface Wearepleasedtoexpressourappreciationtothespeakersfortheirexcel- lentlecturesandtotheparticipantsforcontributingtothesuccessoftheSum- merSchool.WehadatCetraroaninteresting,rich,nice,friendlyatmosphere, created by the speakers, the participants and by the CIME organizers; also for this reason we like to thank the Scientific Committee of CIME, and in particular Pietro Zecca (CIME Director) and Elvira Mascolo (CIME Secre- tary). We also thank Carla Dionisi, Irene Benedetti and Francesco Mugelli, whotookcareofthedaytodayorganizationwithgreatefficiency. BernardDacorognaand PaoloMarcellini Contents Transport Equation and Cauchy Problem for Non-Smooth Vector Fields LuigiAmbrosio .................................................. 1 1 Introduction.................................................. 1 2 TransportEquationandContinuityEquation withintheCauchy-LipschitzFramework.......................... 4 3 ODEUniquenessversusPDEUniqueness......................... 8 4 VectorFieldswithaSobolevSpatialRegularity ................... 19 5 VectorFieldswithaBV SpatialRegularity....................... 27 6 Applications.................................................. 31 7 OpenProblems,BibliographicalNotes,andReferences ............ 34 References ...................................................... 37 Issues in Homogenization for Problems with Non Divergence Structure LuisCaffarelli,LuisSilvestre...................................... 43 1 Introduction.................................................. 43 2 HomogenizationofaFreeBoundaryProblem:CapillaryDrops ...... 44 2.1 ExistenceofaMinimizer ................................... 46 2.2 PositiveDensityLemmas................................... 47 2.3 MeasureoftheFreeBoundary .............................. 51 2.4 Limitasε→0............................................ 53 2.5 Hysteresis ................................................ 54 2.6 References................................................ 57 3 The Construction of Plane Like Solutions to Periodic Minimal SurfaceEquations ............................................. 57 3.1 References................................................ 64 4 ExistenceofHomogenizationLimitsforFullyNonlinearEquations... 65 4.1 MainIdeasoftheProof.................................... 67 4.2 References................................................ 73 References ...................................................... 74
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