ONMINIMIZATIONOFSOMENON-SMOOTHCONVEXFUNCTIONALSARISING INMICROMAGNETICS By HongliGao ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof Mathematics|DoctorofPhilosophy 2015 ABSTRACT ONMINIMIZATIONOFSOMENON-SMOOTHCONVEXFUNCTIONALS ARISINGINMICROMAGNETICS By HongliGao Thisthesisismotivatedbystudyingthepropertiesofferromagneticmaterialsusing theLandau-Lifshitztheoryofmicromagnetics.Inthistheorythestateofaferromagnetic materialisdescribedbythemagnetizationvector m intermsofatotalmicromagneticenergy thatconsistsofseveralcompetingsub-energies:exchangeenergy,anisotropyenergy,external interactionenergyandmagnetostaticenergy.Forlargeferromagneticmaterialsandunder somelimitingregimesofthemodel,theexchangeenergycanbenegligibleandthetotal energybecomesareducedmodel.Ourinvestigationsfocusonthestudyofsuchareduced modelofLandau-Lifshitztheory. Theprimaryfocusofthethesisincludestwoparts:theminimization(static)studyand theevolution(dynamic)study.Weinvestigateanewmethodfortheexistenceofminimizers ofthereducedmicromagneticenergybasedonadualitymethod.Inthismethod,thereduced micromagneticenergyiscloselyrelatedtoaconvexfunctional(thedualfunctional)onthe curl-freevectorfunctions.Ourminimizationanddynamicsstudiesarebasedonthestudy oftheminimizationandgradientwofthisdualfunctional.Muchofthethesisisfocused ontheminimizationproblemoftwospecialcases:softcaseanduniaxialcaseontheannulus domain;inparticular,inthesoftcase,forsomerangeoftheparameter,theenergyminimizers oftheoriginalmicromagneticenergyareconstructedthroughtheEuler-Lagrangeequation ofthedualfunctionalusingthecharacteristicsmethodforareducedEikonaltypeequation. Theseconddirectionofourstudyofthisthesisisanattempttoobtaincertainreasonable dynamicprocessfortheevolutionof m ,wheretheasymptoticbehaviorofthegradientw ofthereducedenergyfunctionalisinvestigated. ACKNOWLEDGMENTS IwouldliketoexpressmydeepestgratitudetomyadvisorDr.BaishengYanforhisgenerous andcontinuoussupportthroughoutmyPh.Dstudy.Hisimmenseknowledge,patienceand supporthavehelpedmeovercomemanysituationsandcompletethisdissertation.I havebeenamazinglyfortunatetohavehimasmyadvisor.Hispersonalguidanceandsupport areofgreathelptomeduringthethetimeofbothmyPh.D.studyandjobapplication. IwouldalsoliketothankDr.ChichiaChiu,Dr.KeithPromislow,Dr.MoxunTang andDr.ZhengfangZhouforservingonmydissertationcommitteeandfortheirinsightful commentsandsuggestions. IwouldadditionallyliketoexpressmyspecialgratitudetoDr.PeiruWuandMSIM program.IhavebeenveryfortunatetojoinMSIMprogramduring2014-2015.Thisgreat opportunityhasopenedanotherdoortomebeyondtheacademia.WithoutDr.Wu's generoushelpandencouragement,itwouldnotbepossibletoachievemycareergoalsin industry. ManyfriendsatMSUhavehelpedmethroughtheseyears.Igreatlyvaluetheir friendship.IparticularlywouldliketothankWeiDeng,JiayinJin,SeonghakKim,JunLai andQiliangWufortheirhelpfuldiscussionsduringmyresearch. Lastbutnottheleast,IwouldliketothankmyfatherYanzhenGaoandmyboyfriend, YangZhoufortheirendlessloveandsupport. iv TABLEOFCONTENTS LISTOFFIGURES ................................... vii Chapter1Introduction ............................... 1 1.1Landau-Lifshitztheoryofmicromagnetics...................1 1.2ReducedmodelofLandau-Lifshitztheory....................3 1.3Mainresults....................................5 Chapter2Preliminaries ............................... 12 2.1Notationsand.............................12 2.2Thedualitymethodformicromagnetics.....................14 2.3Constantconstraintproblemin V ? .......................18 Chapter3GeneralResults ............................. 20 3.1Minimizationofthedualfunctional.......................20 3.1.1Theminimizerisharmonicoutside..................20 3.1.2Anecessaryandtconditionfortheminimizer.........21 3.1.3Adomaindependenceresultfortheminimizer.............27 3.1.4Theuniaxialanisotropyenergy.....................29 3.2Thegradientwofthedualfunctional....................31 3.2.1Thesubtial............................31 3.2.2Thegradientw.............................33 3.2.3Possibledynamicsformagnetization m .................37 3.2.4Studyofaspecialcase..........................38 Chapter4AMinimizationProblemonAnnulus ............... 44 4.1Thecharacteristicsmethod............................46 4.1.1Themaximalexistencetime ˝ M ( )...................50 4.1.2Invertingthecharacteristicsmap....................53 4.1.3Constructionofthelocalsolutions....................54 4.2Constructionneartheinnerquarter-circle...................54 4.2.1Thecharacteristicssolutions.......................55 4.2.2Invertingthecharacteristicsmap....................56 4.2.3Constructionofthesolutionon Z 1 ...................62 4.3Constructionneartheouterquarter-circle...................67 4.3.1Characteristicsstripconditions.....................67 4.3.2Invertingthecharacteristicmap.....................70 4.3.3Constructionofthesolutionon Z 2 ...................90 4.4Gluingthelocalsolutions:ProofofTheorem4.1.2...............93 4.4.1Thepreparations.............................94 4.4.2Constructionsfromthefunction 2 ...................95 v 4.4.3Theproofinthecase0 < 1 a n 1 ................105 4.4.4Theproofinthecase =1 a n 1 ..................107 BIBLIOGRAPHY .................................... 112 vi LISTOFFIGURES Figure3.1:Thenewdomainwhenasmallregion E isremoved.........28 Figure3.2:Thegraphsforthecaseof j H j 1 3 ...................41 Figure3.3:Thegraphsforthecaseof j H j > 1 3 ...................42 Figure4.1:Thefunction = ( t )by(4.27)isstrictlyincreasingand left-continuous..............................59 Figure4.2:Thedomain Z 1 andthesmoothincreasingfunction s = s 1 ( t )deter- minedinLemma4.2.7..........................60 Figure4.3:Thedomain S isbetweenthetwosmoothcurves 0 ( s )and ~ ( s ) ; while S 1 isthepartwith0 << (emptyif =0), S 2 isthepart boundedby ~ and ~ 1 with << ^ 1 ,and S 3 isthepartbetween ~ 1 and 0 with ^ 1 <<ˇ= 2 : Thenumber^ s isdeterminedinLemma 4.3.8....................................82 Figure4.4:Thefunction t ( s )= F ( s; ( s ))andthedomain Z 2 : Thenumber ^ t isdeterminedinLemma4.3.11andthesmoothfunction t 1 ( s )is determinedinLemma4.3.13.......................87 Figure4.5:Thedomains Z 1 ; Z 2 and Z A ,thecurve s = s 1 ( t )determinedin Lemma4.2.7,andthecurve s = r 1 ( t )thatistheinversefunctionof thefunction t = t 1 ( s )determinedinLemma4.3.13..........94 Figure4.6:Atypicalconstructionofsequence f a i g andfunctions r i ( t )on[ a i 1 ;a i ] inTheorem4.4.1.............................97 Figure4.7: (Thecase 0 < 1 a n 1 ). Thecurve s = r A ( t )determined inCorollary4.4.3,thecurve s = l A ( t )determinedLemma4.4.4,and thesub-domainsdividedby s = l A ( t )and s = r A ( t )inthedomain !: 104 Figure4.8: (Thecase =1 a n 1 ). Thecurve s = c ( t )on[ A 00 ; 1]determined inLemma4.4.7,thecurve s = l A ( t )determinedinLemma4.4.8,and thecurve s = r A ( t )intersectat t = a .................109 vii Chapter1 Introduction 1.1Landau-Lifshitztheoryofmicromagnetics Ourresearchisbasedonthewell-knownLandau-Lifshitztheoryofmicromagnetics;see Brailsford[7],Brown[11]andLandauetal[36].Underthistheory,observablemagnetic propertiesofaferromagneticmaterialaredescribedbyamagnetizationvector m througha formulationofatotalmicromagneticenergyincludingseveralcompetingenergies: E ( m )= 2 Z jr m ( x ) j 2 dx + Z ' ( m ( x )) dx Z H ( x ) m ( x ) dx + 1 2 Z R n j F m ( z ) j 2 dz; (1.1) whereisaboundeddomainin R n ( n =2 ; 3inpractice)occupiedbytheferromagnetic material, F m 2 L 2 ( R n ; R n )isamagneticinducedby m onthewhole R n thatis determinedbythedMaxwell'sequations: curl F m =0 ; div( F m + m ˜ )=0in R n ; (1.2) ' ( m )isagivenfunctionrepresentingtheanisotropyenergydensitythatisminimizedalong certainpreferredcrystallographicdirections,and H ( x )isagivenvectorfunctionrepresenting theexternalappliedHere > 0isamaterialconstant.Underthistheory,whenbelow 1 certaincriticaltemperature,themagnetization m shouldhaveconstantmagnitude: j m ( x ) j = M s ; (1.3) where M s > 0isasaturationconstant. Thetermintheenergy E ( m )iscalledthe exchangeenergy ,thesecondtermthe anisotropyenergy ,thethirdtermthe externalinteractionenergy ,andthelasttermisa non-localenergyandisusuallycalledthe magnetostaticenergy .Thenon-localityandnon- convexityofthetotalenergy E ( m )notonlypresentamajorandchallengingmathematical problembutalsoprovideaconcreteexampleforsomeotherphysicalproblemsofasimilar nature. TheLandau-Lifshitzmodelhasbeenatthecenterofmuchofcurrentactiveresearch; seethesurveybyKruzandProhl[35].Ononehand,thestaticLandau-Lifshitztheory ispostulatedbyminimizationofenergy E ( m )underthesaturationcondition(1.3).Onthe otherhand,thedynamictheoryfortimeevolutionofmagnetization m isgovernedbythe Landau-Lifshitzequation: @ t m = m F + j m j m ( m F )(1.4) on [0 ; 1 ) ; where < 0istheelectrongyromagneticratio, > 0istheLandau-Lifshitz phenomenologicaldampingparameter,and F isthetotal ctivemagnetic bythefunctionalderivativeof E ( m )as F = @ E @ m = m ' 0 ( m )+ H ( x )+ F m : 2 Thisequationisalsoequivalenttotheso-calledLandau-Lifshitz-Gilbertequation.Many results,suchasexistence([2,3,16,17,19,30,32,50]),stability([14])andasymptotic behaviorhavebeenwellestablishedfortheLandau-Lifshitzequationsthatincludetheso- calledexchangeenergy(when > 0).Suchexchangeenergyprovidesthemagnetization m with m 2 L 1 ((0 ; 1 ); H 1 whichallowsustohavesomecompactnessandstabilitythat areneededforusingthestandardmethods. 1.2ReducedmodelofLandau-Lifshitztheory Forlargeferromagneticmaterials,ithasbeenbyDeSimone[22](seealsoJamesand Kinderlehrer[33])thatthetotalmicromagneticenergycanbeapproximatedbythefollowing reducedform(ignoringtheexchangeenergy): I ( m )= Z ' ( m ( x )) dx Z H ( x ) m ( x ) dx + 1 2 Z R n j F m ( z ) j 2 dz: (1.5) Throughoutthisthesis,weassumethemagnetization m hasunitlength: j m ( x ) j =1 : Duetothesaturationconstraint j m j =1andtheanisotropyenergy,theexistenceofmin- imizersofthisenergy I ( m )isnotguaranteed;somorecarefulanalysisshouldbecarried out. Anewmethodforminimizationofthisfunctional I ( m )hasbeenintroducedbyPedregal andYan[43,44]basedontheideaofduality;seealso[33].Themainideaofthismethodis 3 motivatedbyrewritingthemagnetostaticenergyas 1 2 Z R n j F m j 2 =min div G =0 1 2 Z R n j m ˜ G j 2 by(1.2),wheretheminimumistakenoveralldivergence-free G in L 2 ( R n ; R n ) : In[44] ithasbeenprovedthat inf m 2 L 2 R n ) j m ( x ) j =1 I ( m )= min F 2 L 2 ( R n ; R n ) curl F =0 J ( F ) ; where J ( F )isaconvexfunctionalby J ( F )= Z F ( x )+ H ( x ) dx + 1 2 Z R n j F ( x ) j 2 dx (1.6) withaconvexfunctionby ˘ )=max h 2 R n ; j h j =1 ( ˘ h ' ( h ))( ˘ 2 R n ) : Itiseasilyseenthat J isstrictlyconvexon V = L 2 ( R n ; R n ) \f curl F =0 g .Theexistence oftheminimizer F 2 V isguaranteedbythegeneraltheory.Usingthisuniqueminimizer F of J ( F ),anecessaryandtconditionfortheexistenceofminimizersof I ( m )has beengivenin[44,52].In[52],anotionof generalizedminimizers of I ( m )hasbeenalso Asimilardualformulationtothefunctional J ( F )hasalsobeenusedbyMelcher [38]toapproachsomeregularityproblemsforthin Wefollowthislineofinvestigationstostudysomeconcreteproblemsregardingthemin- imizationoffunctional I ( m ) : Ourprimaryresultswillbetheconstructionofminimizersof 4 somespecialenergy I ( m )onanannulusdomain ; whichmayhavesomephysicalapplica- tionsinstudyingmagneticnanorings[15]. Theevolutionmodelbasedonthereducedenergy I ( m )leadstoacorrespondingreduced Landau-Lifshitzequation(1 : 4)with =0andhasbeenrecentlystudiedin[26,27,53,54]. Inthiscase,oneonlyhas m 2 L 1 ((0 ; 1 ); L 1 whichleadstothelackofcompactness andstability.Yan[54]discussedstabilityandasymptoticbehaviorsofsolutionsforadegen- erateLandau-Lifshitzequationinmicromagneticsinvolvingonlythenonlocalmagnetostatic energy.HeshowedthattheCauchyproblemsforsuchanequationarenotstableunderthe weak convergenceofinitialdata.Fortheasymptoticbehaviorsofweaksolutions,heestab- lishedanestimateontheweak ! -limitsetsthatisvalidforallinitialdatasatisfyingthe saturationcondition.DengandYan[27]havepresentedanewmethodfortheexistenceof globalweaksolutiontothereducedLandau-Lifshitzequation.Inaddition,theyalsoestab- lishedhighertimeregularitywhentheinitialvalue m 0 isconstant.Theystudiedtheweak ! -limitsetsforthesoftcaseandtheasymptoticbehaviorsinthecasewhenisellipsoid andinitialvalue m 0 isconstant. Inattempttoobtainotherreasonabledynamicprocessesfortheevolutionof m ,westudy thegradientwoftheconvexfunctional L : L 2 ( R n ; R n ) ! R :=( ; 1 ], L ( F )= 8 > > > < > > > : Z F ( x )+ H ( x ) dx + 1 2 Z R n j F ( x ) j 2 dx;F 2 V ; + 1 ;F 2 L 2 ( R n ; R n ) n V : (1.7) 1.3Mainresults Ourresultsconsistmainlyoftwoparts:minimizationofthefunctional L andtheasymptotic behaviorofthegradientw. 5 Sinceevery F ( x ) 2 V canbewrittenas F ( x )= r u ( x ),where u 2 H 1 loc ( R n ),weintroduce thefollowingvariationalfunctional: L ( u )= L ( u ) Z r u ( x )+ H ( x )) dx + 1 2 Z R n jr u ( x ) j 2 dx (1.8) forall u 2 H 1 loc ( R n )with r u 2 L 2 ( R n ; R n ) : Forsimplicityandwhenthereisnoconfusion, wesimplyuse L ( u )todenote L ( u ). Notethat L ( u + c )= L ( u )forallconstants c 2 R : Totheidea,wethelinear space X by X = ˆ u 2 H 1 loc ( R n ) r u 2 L 2 ( R n ; R n ) ; Z @ udS =0 ˙ ; (1.9) where u = u j @ isthewdtracein H 1 = 2 ( @ (see[1]).Itiseasilyseenthat L is strictlyconvexon X : Hence L hasa uniqueminimizer on X ;wedenotethisuniqueminimizer by u = v˜ + w˜ c : Certainly,thisfunction u dependsonthedomaintheanisotropy function ' (intermsoffunctionandtheapplied H ( x ) : PedregalandYan[44]have shownthat u isuniquelydeterminedbyitsboundarydata g = u j @ and,inparticular,that w isharmonicon c : Theyhavealsoestablishedanecessaryandtconditionforthe existenceofminimizersofenergy I ( m )intermsoftheuniqueminimizer u = v˜ + w˜ c offunctional L ( u ) : Forexample,theyestablishedthefollowingtheorem. Theorem1.3.1. Let u = v˜ + w˜ c 2X betheuniqueminimizeroffunctional L d above.Then,theenergy I ( m ) hasaminimizerifandonlyifthereexistsafunction G 2 6 L 2 R n ) that 8 > > > < > > > : div( r u + G˜ )=0 in R n , G ( x ) 2 r v ( x )+ H ( x )) a:e:x 2 : (1.10) Here, )= f h 2 S n 1 j )= h ' ( h ) g : Inadditionany m 2 L 1 S n 1 ) satisfying m ( x )= G ( x ) a.e.on isaminimizerofenergy I: Wefocusonhowtotheminimizer u of L ( u ) : Thefollowingresulthasbeenproved in[44].WeprovideinChapter3aerentproofforit. Theorem1.3.2. (Chapter3,section3.1.2) Afunction u = v˜ + w˜ c 2X isaminimizer of L ( u ) ifandonlyifthereexistsavectorfunction G 2 L 2 R n ) suchthat 8 > > > < > > > : div( r u + G˜ )=0 in R n , G 2 @ r v + H ( x )) a.e. , (1.11) where @ ˘ ) denotestheentialof at ˘: Anysuchfunction G iscalleda generalized minimizer ofthefunctional I ( m ) . Dependingonthetanisotropydensityfunctions ' ,thefunctional L ( u )takesa tformintermsoftheconvexfunctionabove. Wesaythatthematerialisinthe softcase if ' 0;inthiscase ˘ )= j ˘ j on ˘ 2 R n . Wesaythematerialisinthe uniaxialcase if ' ( h )= (1 j h e j ),where > 0isaconstant and e 2 R n isagivenunitvector;inthiscase,thefunctioncanbeexplicitlycomputedand thetialset @ ˘ )hasaspecialstructure(seebelow),whichtheexistence ofthesolutiontotheproblem(1.11). 7 Wearealsointerestedinthedependenceof u onthedomain : Whenasmallregion E is removedfromthedomainwewanttostudyhowtheminimizerof L n E ( u )(orultimately theminimizersof I ( m ))shouldchange.Inparticular,cantheminimizersof L ( u )and L n E ( u )bethesameon n E ? Wehavethefollowingresult. Theorem1.3.3. (Chapter3,Section3.1.3) Let u = v˜ + w˜ c betheminimizerofthe functional L ( u ) and E ˆˆ . ~ w by 8 > > > < > > > : ~ w =0 in E; ~ w = u on @E . (1.12) Supposethatthereexists ~ G 2 L 2 n E ; R n ) satisfying 8 > > > > > > > > > > > > < > > > > > > > > > > > > : div( r v + ~ G )=0 in n E , ( ~ G + r v ) = @ w @ on @ ; ( ~ G + r v ) = @ ~ w @ on @E; ~ G 2 @ r v + H ( x )) a.e. n E . (1.13) Then ~ u = u˜ R n n E +~ w˜ E istheminimizerof L n E ( u ) : InChapter4,weapplythisresulttotheminimizationof I ( m )and L ( u )inthesoftcase (when ˘ )= j ˘ j )withconstantapplied H = e 1 foranannulus= f x 2 R n j a< j x j < 1 g ; where0 > > > > > > > > > > > < > > > > > > > > > > > > : jr ( s;t ) j = t n 2 in a 2 0, ( s;t )=0on s 2 + t 2 = a 2 ;t 0, ( s;t )= n 1 t n 1 on s 2 + t 2 =1 ;t 0, ( s;t )=0on t =0 ;a j s j 1 : (1.14) Weprovethattheboundaryvalueproblem(1 : 14)hasaLipschitzsolutionifandonlyif 0 1 n (1 a n 1 ) ; (1.15) andthat,inthiscase,wecanconstructmanyLipschitzsolutions.Thisconstruction istheprimarygoalofChapter4ofthethesis.Wesummarizetheresultasfollows: Theorem1.3.4. (Chapter4,Section4.1) If 0 1 n (1 a n 1 ) ,thentheproblem (1 : 14) hasmanyLipschitzsolutions ( s;t ) ,constructedinTheorems4.4.6and4.4.9in Chapter4.Inthiscase,theminimizers m of I ( m ) obtainedfromtheconstructedsolution willbetheconstant e 1 incertainsubdomains 0 = f ( x 1 ;x 0 ) 2 j ( j x 1 j ; j x 0 j ) 2Z 0 g away fromtheboundary @ : Theseconddirectionofourstudyisanattempttoobtaincertainreasonabledynamic processfortheevolutionof m .Westudythegradientwoftheconvexfunctional L on L 2 ( R n ; R n ): _ F ( t ) 2 @ L ( F ( t ))( t> 0) ;F (0)= F 0 ; (1.16) 9 wherethesubtial @ L ( F )isgivenby @ L ( F )= F + P V @ F ( x )+ H ( x )) ˜ + V ? 8 F 2 V ; with P V beingtheorthogonalprojectionon V : Theexistenceofthegradientwisstandard;see[10,28].Fortheasymptoticbehavioras t !1 ,Bruck[8]hasshownthatingeneralthegradientwofastrictlyconvexfunctional convergesweaklytoaminimizer,whileBaillon[9]hasgivenacounterexampleshowingthat ingeneralthegradientwdoesnotstronglyconvergetoaminimizer. Duetothefactthattheminimizerof L isharmonicoutsideweobtainthestrong convergenceofthegradientwforthefunction L outsideof Theorem1.3.5. (Chapter3,section3.2) Foreach F 0 2 V thereexistsauniquesolutionto thegradient (1 : 16) : Furthermore, F ( t ) * F as t !1 and F ( t ) ! F in L 2 ( ~ c ; R n ) for allcompactsets ~ containing : Notethatthegradientw(1.16)determinesa(nonunique)time-dependentvectorfunc- tion m ( t )= m ( x;t )withtheproperty 8 > > > < > > > : div( _ F ( t )+ F ( t )+ m ( t ) ˜ )=0( t> 0) ; m ( x;t ) 2 @ F ( x;t )+ H ( x )) a:e:x 2 ;t> 0 : This,alongasubsequence t k !1 ,determinesavectorfunction m ( x )satisfyingdiv( F + m ˜ )=0 : If,inaddition,onehasthat F ( t ) ! F stronglyin L 2 ( R n ; R n )thenitwould followthat m ( x ) 2 @ F ( x )+ H ( x ))fora.e. x 2 andthus m ( x )wouldbeageneralized minimizerforenergy I ( m ) : 10 AttheendofChapter3,wewilldiscussaparticularexampleforthesoftcasewith aconstantapplied H ontheunitballin R 3 .Inthiscase,thegradientwcanbe expressedasanordinarytialequation,wherewehavethestrongconvergencefor gradientwbothinsideandoutsideof 11 Chapter2 Preliminaries Inthischapter,wereviewsomepreliminaryandresultsinordertopresentour resultsinChapter3andChapter4. 2.1Notationsand Throughoutthisthesis,weuse H todenotetherealHilbertspace L 2 ( R n ; R n )withusual L 2 -innerproduct F G andnorm jjjj .Whenusingtheconvergencenotationinthisthesis, \ ! "denotesthestrongconvergencein L 2 ( U ; R n );\ * "denotestheweakconvergencein L 2 ( U ; R n )provided Z U u k vdx ! Z U uvdx as k !1 : foreach v 2 L 2 ( U ; R n ) : Here U R n isanysetin R n . Let V bethesubspaceof H by V = f F 2 H j curl F =0inthesenseofdistributionson R n g : Theneachelement F 2 V canberepresentedas F = r u forsomefunction u 2 H 1 loc ( R n ). Moreovertheorthogonalcomplementof V isexactlygivenby V ? = f G 2 H j div G =0inthesenseofdistributionson R n g ; 12 thatis, G 2 V ifandonlyif Z R n G ( x ) r ( x ) dx =0 8 2 C 1 c ( R n ) : Werefertothebooks[47,49]fortheproofoftheseresults. Wenextreviewsomenotationsandinconvexanalysis(see,e.g.,[6,45]).Let p : H ! R beagivenfunctionalon H : 2.1.1. The (convex)conjugate orthe Legendretransform p of p andthe conve p # of p are,respectively,dby p ( G )=sup F 2 H f F G p ( F ) g ;p # ( G )=sup F 2 H f F G p ( F ) g ; thatis, p # =( p ) : Bothareconvexfunctionalson H anditalsofollowsthat p =( p # ) : 2.1.2. The tial of p at G 2 H isdtobetheset @p ( G )= f F 2 H j p ( A ) p ( G )+ F ( A G ) 8 A 2 H g : (2.1) Notethat @p ( G ) 6 = ; onlyif p ( G ) < 1 ; andthatif @p ( G ) 6 = ; thenitisaconvexsubset of H .Moreover,0 2 @p ( G )ifandonlyif p ( G )istheabsoluteminimumof p on H : Wealsohavethefollowingproperty: F 2 @p ( G )ifandonlyif p ( F )= F G p ( G ) : (2.2) 13 Moreover,if q isaconvexfunctionalon H ,then k F k sup k A 1 f q ( G + A ) q ( G ) g8 F 2 @q ( G ) : (2.3) 2.2Thedualitymethodformicromagnetics Assumethat ' : S n 1 ! R isagivenfunctionrepresentingtheanisotropyenergydensity, isagivenboundeddomainwithpiece-wisesmoothboundaryoccupiedbytheferromagnetic material,and H 2 L 2 R n )isagivenappliedmagnetic.Considerthe(reduced) micromagneticenergyintroducedabove I ( m )= Z [ ' ( m ( x )) H ( x ) m ( x )] dx + 1 2 Z R n j F m j 2 dx; where F m 2 V isbyMaxwell'sequation(1.2)above. Notethat,by(1.2),themagnetostaticenergycanbeexpressedasavariationalproblem 1 2 Z R n j F m j 2 =min G 2 V ? 1 2 Z R n j m ˜ G j 2 : Introduceanauxiliaryfunctional A ( m ;G )for m 2 L 1 S n 1 ) ;G 2 H = L 2 ( R n ; R n )by A ( m ;G )= Z ' ( m ) Z H ( x ) m + 1 2 Z R n j m ˜ G j 2 ; (2.4) whichleadsto I ( m )=min G 2 V ? A ( m ;G ).Therefore inf j m j =1 I ( m )=inf j m j =1 " inf G 2 V ? A ( m ;G ) # =inf G 2 V ? " inf j m j =1 A ( m ;G ) # : 14 Now,for G 2 H , J ( G )=inf j m j =1 A ( m ;G ) ; wheretheum(infactaminimum)istakenoverall m 2 L 1 S n 1 ) : Thenoneeasily has inf j m j =1 I ( m )=inf G 2 V ? J ( G ) : Anelementarycomputationshowsthat J ( G )= Z ( x;G ( x )) dx + 1 2 Z c j G j 2 dx; (2.5) where ( x;˘ )= 1 2 ( j ˘ j 2 +1) ˘ + H ( x )) withdenotingtheconvexfunctionaboveby )=max h 2 S n 1 [ h ' ( h )] ; ( 2 R n ) : (2.6) )= f h 2 S n 1 j h ' ( h )= ) g : Then )= @ ) \ S n 1 ( 2 R n ) : Remark2.2.1. (1)Ifanisotropyenergydensity ' isgivenby ' =0 ,whichisthesoftcase, then ) j j : 15 (2)Ifanisotropyenergydensity ' isgivenby ' ( h )= (1 j h e j ) ; (2.7) where > 0 and e 2 S n 1 aregivenconstants.Then ' ( h ) 0 andequals 0 ifandonly if h 2f e ; e g ; thesearetheso-called easyaxes .Thisisthe uniaxialcase. Inthiscasethe function dabovecanbeeasilyfoundasfollows: )=max j h j =1 ( h + j h e j ) =max j h j =1 max t = 1 f h + e h g =max t = 1 max j h j =1 ( + e ) h =max t = 1 j + e j =( j j 2 +2 j e j + 2 ) 1 = 2 : Therefore, @ )= 8 > > > < > > > : + sgn( e ) e j + sgn( e ) e j ; if e 6 =0 ; ˆ + e ( j j 2 + 2 ) 1 = 2 : 1 t 1 ˙ ; if e =0 : Westudythesetwospecialcasesforminimizationorgradient 16 Ingeneral,theLegendretransformof ( x; )canbecomputedasfollows: ( x; )=sup ˘ 2 R n f ˘ ( x;˘ ) g =sup ˘ 2 R n ˆ ˘ 1 2 ( j ˘ j 2 +1)+( ˘ + H ( x )) ˙ =sup ˘ 2 R n ˆ ˘ 1 2 j ˘ j 2 1 2 +max h 2 S n 1 f ( ˘ + H ( x )) h ' ( h ) g ˙ =sup ˘ 2 R n max h 2 S n 1 ˆ ( + h ) ˘ 1 2 j ˘ j 2 1 2 + H ( x ) h ' ( h ) =sup h 2 S n 1 " sup ˘ 2 R n ˆ ( + h ) ˘ 1 2 j ˘ j 2 1 2 + H ( x ) h ' ( h ) ˙ # =sup h 2 S n 1 1 2 j + h j 2 1 2 + H ( x ) h ' ( h ) =sup h 2 S n 1 1 2 j j 2 +( + H ( x )) h ' ( h ) = 1 2 j j 2 + + H ( x )) : Therefore,theLegendretransformof J canbewrittenas J ( F )= Z F ( x )+ H ( x ) dx + 1 2 Z R n j F ( x ) j 2 dx (2.8) forall F ( x ) 2 H .Let L ( F )beby(1.7)on H : Theorem2.2.2. [44,Theorem1.2] Let F beaminimizerof L ( F ) .Avector m isa minimizerof I ( m ) ifandonlyif 8 > > > < > > > : div( F + m ˜ )=0 in R n , m ( x ) 2 F ( x )+ H ( x )) a.e. : (2.9) 17 Sincethereexists u ,suchthat F = r u .Accordingly,div( r u + m ˜ )=0 ; whichyields that u = div( m ˜ ) : Hence, u ( x )canbesolvedbyNewton'spotential: u ( x )= Z m ( y ) r x y ) dy = c n Z m ( y ) ( y x ) j y x j n dy; where z )isthefundamentalsolutionofLaplace'sequationand c n isaconstant. Remark2.2.3. If istheunitballin R n and m K 2 R n isaconstant,then u ( x ) can beexpressedexplicitlyby(see[34]or[44,Lemma4.2]): u ( x )= 8 > > > < > > > : K x n x 2 ; K x n j x j n x 2 c ; whichwillbelaterappliedtothecalculationofgradientinthesoftcaseinChapter3. 2.3Constantconstraintproblemin V ? Inthissection,wereviewsomeexistingresultsthatarehelpfultounderstandthemain resultsinthisthesis. In[44]ithasbeenshownthatthecondition(1.10)isequivalenttothefollowingcon- 18 strainedproblemforfunction ~ G 2 L 2 R n ): 8 > > > < > > > : div( ~ G˜ )=0on R n ; ~ G ( x ) 2 S ( x ) a:e:x 2 ; (2.10) where S ( x )issomeset-valuedfunction.Theconstrainedproblem(2.10)fordivergence-free withconstantset S ( x )= S hasbeenrecentlystudiedbymanyauthors;see,e.g., [5,12,18,20,33,44].Forexample,thefollowingresulthasbeenprovedin[5,18]. Theorem2.3.1. (cf,[5,Theorem4.15];[18,Theorem6.2]) Let n =3 andlet beany boundedopensetin R 3 ,andassume S ( x )= S isanyconstantboundedsetin R 3 : Then problem (2 : 10) hasasolutionifandonlyifeither 0 2 S orthereexistsasubset F S suchthat dim(span F ) 2 and 0 2 ri(con F ) : Moreover,inthiscase,asolution ~ G canbe obtainedby ~ G = r ! withsome ! 2 W 1 ; 1 0 R 3 ) : Thistheoremhasitsownlimitations:(1) S ( x )hastobeconstant;(2) ~ G hastobe 0ontheboundarybutsometimeswedonothavesuchcondition.Forexample,inthecase whenthedomainisannulus,suchconditionfailsandwecannotusethistheorem. 19 Chapter3 GeneralResults Inthischapter,wepresentourgeneralresultsintheminimizationandtheasymptotic behaviorofgradientwofthefunctional L .Somespecialcaseswillbealsodiscussed. 3.1Minimizationofthedualfunctional 3.1.1Theminimizerisharmonicoutside Theorem3.1.1. Supposethatthefunctional L isdby(1.7).Iffollowsthat F is harmonicon c . Proof. Let 2 C 1 c c )beatestfunctionwithcompactsupportin c : Since L ( F + " r ) L ( F )and 0on Z x; F dx + 1 2 Z c j F + " r j 2 dx Z x; F dx + 1 2 Z c j F j 2 dx: Let R c j F + " r j 2 dx = h ( " ),thentheaboveinequalityimpliesthat h ( " ) h (0) : Therefore h 0 (0)=0,i.e. Z c F r =0forany 2 C 1 c c ) : 20 Combingtheof F ,wehavethefollowingresults div F =0on c ; curl F =0on c ; inthesenseofdistribution.Weapplythe distributional identity curlcurl( N )+ 4 ( N )= r (div( N )) tohavethat F =0on c : indistributionandthusinclassicalsense. 3.1.2Anecessaryandtconditionfortheminimizer Ithasbeenestablishedin[44]anecessaryandsutconditionfortheexistenceofmini- mizersofenergy I ( m )intermsoftheuniqueminimizer u = v˜ + w˜ c offunctional L ( u ) onwhichisaboundeddomainin R n withpiecewisesmoothboundary. L ( u )is previously, L ( u )= Z r u + H ( x )) dx + 1 2 Z R n jr u j 2 dx Theorem3.1.2. Afunction u = v˜ + w˜ c 2X isaminimizerof L ( u ) ifandonlyif 21 thereexistsafunction G 2 L 2 R n ) suchthat 8 > > > < > > > : div( r u + G˜ )=0 in R n , G 2 @ r v + H ( x )) a.e. . (3.1) Anysuchfunction G iscalleda generalizedminimizer ofthefunctional I ( m ) . Remark3.1.3. Let betheoutwardunitnormaltotheboundary @ ofdomain .Then both G and @ w @ aredaselementsin H 1 = 2 ( @ (see,e.g.,[49,Page9]).Theabove necessaryandcondition(3.1)canbealsoreformulatedas: 8 > > > > > > > > > > > > < > > > > > > > > > > > > : div( r v + G )=0 in H 1 , ( G + r v ) = @ w @ on @ , w =0 in c ; G 2 @ r v ( x )+ H ( x )) a:e:x 2 : (3.2) HerewepresentatmethodtoproveTheorem3.1.2. Proof. Supposethatthereexist G and u satisfying(3.1)thenforany v 2X ,wehave L ( v ) L ( u ) = Z r v + H ( x )) r u + H ( x )) dx + 1 2 Z R n jr v j 2 jr u j 2 dx Z G ( r v r u )+ Z R n r u ( r v r u )+ 1 2 Z R n jr v r u j 2 dx = Z R n [ G˜ + r u ] ( r v r u )+ 1 2 Z R n jr v r u j 2 dx = 1 2 Z R n jr v r u j 2 dx 0 : 22 Therefore, u istheminimizerof(1.8). Next,assumethat u isaminimizerof L ( u ).Introduceafunction " ( ),for "> 0,which isas " ( )=min ˘ 2 R n ˆ 1 2 " j ˘ j 2 + ˘ ) ˙ : " ( )followsthefollowingproperties(refertoBrezis[10]): (1) " ( )isconvex; (2) " ( )istiable; (3) " ( ) ! )as " ! 0; (4) 0 " ( ) 1; (5) " ( ) ). Considerthefunctionalbelow L " ( u )= Z " ( r u + H ( x )) dx + 1 2 Z R n jr u j 2 dx = Z 1 2 " jr u + H ( x ) B j 2 + B ) dx + 1 2 Z R n jr u j 2 dx: Then,functional L " ( u )istiable,strictlyconvexon X andthushasauniqueminimizer u " in X : Let L " ( u " )=min L " ( u ).Thenweapplytheof " tohave L " ( u " )= Z " ( r u " + H ( x )) dx + 1 2 Z R n jr u " j 2 dx (3.3) = Z 1 2 " jr u " + H ( x ) B " j 2 + B " ) dx + 1 2 Z R n jr u " j 2 dx (3.4) Therefore,byEuler-Lagrangeequation, div 0 " ( r u " + H ( x )) ˜ + r u " =0 : 23 Let G " ( x )= 0 " ( r u " + H ( x )).Thenthereexist G ( x ),suchthat G " ( x ) *G ( x )weakly in L 1 : Consequently, div( G˜ + r u )=0 : Since L " ( u " ) L " (0)= Z " ( H ( x )) dx = Z H ( x )) dx< 1 .Inaddition, )is Lipschitzso j ) j c ( j j +1).Therefore, j " ( ) jj ) j c ( j j +1) ; whichyieldsthat jr u " j isuniformlyboundedfrom(3.4).Hence r u " isweaklyconvergentandthereexists ~ u 2X ,suchthat r u " * ~ u weaklyas " ! 0.Notethat L islowersemicontinuous, L (~ u ) lim " ! 0 L ( u " ) : (3.5) Notethat L " ( u " )and r u " areboundedandisLipschitz,thereexist M 1 ;M 2 > 0such that M 1 L " ( u " )= Z " ( r u " + H ( x )) dx + 1 2 Z R n jr u " j 2 dx = Z 1 2 " jr u " + H ( x ) B " j 2 + B " ) dx + 1 2 Z R n jr u " j 2 dx Z 1 2 " jr u " + H ( x ) B " j 2 j B " j dx M 2 : Notethat jr u " + H j is L 2 boundedandweapplythetriangleinequalityandCauchyin- 24 equalitytohavethat Z jr u " + H ( x ) B " j 2 dx 2 " Z j B " j dx +2 "M 1 2 " Z jr u " + H ( x ) B " j dx +2 " Z jr u " + H ( x ) j dx +2( M 1 M 2 ) " 2 " 1 2 Z jr u " + H ( x ) B " j 2 dx + 1 2 j j + "M 3 forsomeconstant M 3 ,whichyieldsthat Z jr u " + H ( x ) B " j 2 dx " 1 " M; forsome M> 0 : Thus Z jr u " + H ( x ) B " j 2 dx ! 0 ; as " ! 0 : (3.6) NotethatisLipschitz, L " ( u " )= Z 1 2 " jr u " + H ( x ) B " j 2 + B " ) dx + 1 2 Z R n jr u " j 2 dx Z B " ) dx + 1 2 Z R n jr u " j 2 dx Z r u " + H ) jr u " + H B " j ] dx + 1 2 Z R n jr u " j 2 dx whichyieldsthat L " ( u " ) L ( u " ) Z jr u " + H B " j dx: (3.7) Combining(3.5),(3.7)and(3.6),wehavethat L (~ u ) lim " ! 0 L ( u " ) lim " ! 0 L " ( u " )+ Z jr u " + H B " j dx lim " ! 0 L " ( v )+lim " ! 0 Z jr u " + H B " j dx = L ( v ) ; forany v 2X : 25 Hence~ u istheminimizerof L andthus~ u = u: Inaddition, L " ( u ) L " ( u " ) = Z " ( r u + H ( x )) dx Z " ( r u " + H ( x )) dx + 1 2 Z R n jr u j 2 dx 1 2 Z R n jr u " j 2 dx = Z " ( r u + H ( x )) " ( r u " + H ( x )) dx + 1 2 Z R n jr u r u " j 2 dx + Z R n r u " ( r u r u " ) dx Z 0 " ( r u " + H ( x )) ( r u r u " ) dx + Z R n r u " ( r u r u " ) dx + 1 2 Z R n jr u r u " j 2 dx = Z R n 0 " ( r u " + H ( x )) ˜ + r u " ( r u r u " ) dx + 1 2 Z R n jr u r u " j 2 dx = 1 2 Z R n jr u r u " j 2 dx yields 1 2 Z R n jr u r u " j 2 dx L " ( u ) L " ( u " ) L ( u ) L " ( u " ) L ( u " ) L " ( u " ) Z jr u " + H B " j dx ! 0 : Hence r u !r u a.e.in L 2 ( R n ) : (3.8) Bytheof G " ( x ), " ( ) G " ( x ) ( r u " H )+ " ( r u " + H ) 26 Let beatestfunctionwith 2 C 1 ( R n ),multiplybothsidesby andintegratebothsides, Z R n " ( ) Z R n G " ( x ) ( r u " H ) dx + Z R n " ( r u " + H ) dx: Let " ! 0whichyieldsthat Z R n ) Z R n G ( x ) ( r u H ) dx + Z R n r u + H ) dx: i.e. G 2 @ r v + H ( x )) : Thelasttermisobtainedbythefactthat j " ( r u " + H ) r u + H ) j j " ( r u " + H ) " ( r u + H ) j + j " ( r u + H ) r u + H ) j jr u " r u j ++ j " ( r u + H ) r u + H ) j =0as " ! 0 : Thiscompletesthetpart. 3.1.3Adomaindependenceresultfortheminimizer Supposethatthedomainhasasmallregion E removedfrominside. 27 E Figure3.1:Thenewdomainwhenasmallregion E isremoved Letusdenote L ( u )= Z r u ( x )+ H ( x )) dx + 1 2 Z R n jr u ( x ) j 2 dx; L n E ( u )= Z n E r u ( x )+ H ( x )) dx + 1 2 Z R n jr u ( x ) j 2 dx: Theorem3.1.4. Let u = v˜ + w˜ c betheminimizerofthefunctional L ( u ) and E ˆˆ . ~ w by 8 > > > < > > > : ~ w =0 in E; ~ w = u on @E . (3.9) Supposethatthereexists ~ G 2 L 2 n E ; R n ) satisfying 8 > > > > > > > > > > > > < > > > > > > > > > > > > : div( r v + ~ G )=0 in n E , ~ G + r v = @ w @ on @ ; ~ G + r v = @ ~ w @ on @E; ~ G 2 @ r v + H ( x )) a.e. n E . (3.10) 28 Then ~ u = u˜ R n n E +~ w˜ E istheminimizerof L n E ( u ) : Proof. ThisfollowsdirectlyfromRemark3.1.3. 3.1.4Theuniaxialanisotropyenergy Weconsidertheeasycasewheretheanisotropyenergydensity ' ( h )= (1 j h e j ),where > 0and e 2 S n 1 aregivenconstants.Inthiscase,letusrecallfromRemark2.2.1, )=( j j 2 +2 j e j + 2 ) 1 = 2 ; (3.11) and @ )= 8 > > > < > > > : + sgn( e ) e j + sgn( e ) e j ; if e 6 =0 ; ˆ + e ( j j 2 + 2 ) 1 = 2 : 1 t 1 ˙ ; if e =0 : Let= f x 2 R n j a< j x j < 1 g beanannulusdomainin R n .Consider L ( u )= Z r u + H ) dx + 1 2 Z R n jr u j 2 dx where H 2 R n isaconstantand H 6 =0. Theorem3.1.5. Supposethat isdas(3.11).Thentheminimizerof L cannotbe linearon : Proof. Noticethat =0istheonlyminimizerofin(3.11)on R n : Supposethat theminimizer u of L ( u )islinearon: u ( x )= x onThenrecallfromRemark2.2.3, 29 u = x˜ j x j < 1 + x j x j n ˜ j x j > 1 ; andthereexistsa G 2 L 2 R n )suchthat 8 > > > > > > > > > > > > < > > > > > > > > > > > > : div G =0in H 1 ; G = n on j x j =1 ; G =0on j x j = a; G ( x ) 2 @ + H )a.e. : Weproceedwith2cases. Case1: + H e 6 =0 : Then G ( x ) 0 ( + H ).Since G = n on j x j =1 ; then 0 ( + H )= n =0 : Hence =0and 0 ( H )=0.Accordingly, H =0and + H =0 whichisacontradiction. Case2: + H e =0 : Then G ( x )= + t ( x ) e q j j 2 + 2 forsomefunction t ( x ) 2 L 1 ; 1 t ( x ) 1,where = + H: Thenapplyingtheconditiondiv G =0in H 1 wehavethat @t @ e ( x )=0in H 1 whichimpliesthat t ( x )= t ( x 0 )if x = x 1 e + x 0 e 0 = x 1 e + x 2 e 2 + ::: + x n e n with e i ? e , e i e j =0for i 6 = j; and j e i j =1 : Sinceforany j x j = a , G ( x ) x =0.Wehavethat x + t ( x 0 ) e x =0 ; 8j x j = a: Ifassumethat 6 =0andlet x = j j a .Thenwehave =0 : Therefore, =0and = H: Accordingly, G ( x )= t ( x ) e : Let x 1 6 =0,then0= G ( x ) x = t ( x 0 ) x 1 yieldsthat t ( x 0 )=0. Thus G ( x ) 0andtherefore n = H =0,whichyieldsacontradiction. 30 3.2Thegradientwofthedualfunctional Inthissection,forsimplicity,wewrite x;F ) F ( x )+ H ( x )).Recallthat L ispreviously as: L ( F )= 8 > > > < > > > : Z x;F ) dx + 1 2 Z R n j F ( x ) j 2 dx;F 2 V ; + 1 ;F 2 H n V : 3.2.1Thesubtial Tostudytheasymptoticbehaviorofgradientwof L ( F ),wecalculatethesub tial @ L ( F ).Notethatthisfunctionalisconvexon H withalue)domain D ( L )= V : Foreach F 2 D ( L )= V ,thesubtialof L isas: @ L ( F )= f K 2 H jL ( X ) L ( F )+ K ( X F ) 8 X 2 H g : Theorem3.2.1. Foreach F 2 D ( L )= V ,wehave @ L ( F )= F + P V @ x;F ) ˜ + V ? ; where P V : H ! V istheprojectionoperatorand V ? istheorthogonalcomplementof V in H : Proof. Clearly, F + P V @ x;F ) ˜ + V ? @ L ( F ) : Nowlet K 2 @ L ( F ) : Withanabuse ofnotation,we L ( F )= Z x;F ) dx + 1 2 Z R n j F ( x ) j 2 dx; 8 F 2 H ; 31 namely, L ( F )= J ( F )on H ,in(2.8)before.Wemayassume K 2 V : Then L ( X ) L ( F )+ K ( X F )forall X 2 V : Thismeansthat F 2 V istheminimizerofthe functional ~ L ( X )= L ( X ) L ( F ) K ( X F ) over V : Foreach > 0considerthefunctional ~ L ( X )= ~ L ( X )+ 1 2 k X F k 2 + 1 2 k P V ? ( X ) k 2 on X 2 H ,where P V ? : H ! V ? istheprojectionoperator.Thefunctional ~ L isconvex andthusthestandarddirectmethodofthecalculusofvariationsshowsthatithasaunique minimizer F overwhole H : Forthisminimizer F ,since ~ L ( F ) ~ L ( F )= ~ L ( F )=0,we have ~ L ( F )+ 1 2 k F F k 2 + 1 2 k P V ? ( F ) k 2 0 : (3.12) Fromthisandthelineargrowthof L ( X ),wehavethat f F g 0 isbounded;therefore,bya subsequence,weassume F * ~ F as ! 0 + ,weaklyin H ,forsome ~ F 2 H : From(3.12)and thelowersemicontinuityof ~ L ,itfollowsthat ~ L ( ~ F )+ 1 2 k ~ F F k 2 0 ; k P V ? ( ~ F ) k lim ! 0 + k P V ? ( F ) k =0 : Therefore, ~ F 2 V andhence ~ L ( ~ F ) ~ L ( F )=0;thisimpliesthat ~ F = F and F ! F as ! 0 + : Finally,from0 2 @ ~ L ( F )theelementarycomputationsyieldthat K 2 2 F F + @ x;F ) ˜ + 1 P V ? ( F ) : 32 Let,viasubsubsequencesifnecessary, 1 P V ? ( F ) *G in H as ! 0 + : Then G 2 V ? and K 2 F + @ x;F ) ˜ + G; whichproves K 2 F + P V @ x;F ) ˜ + V ? : 3.2.2Thegradientw Since L isconvex,properandlowersemicontinuous,wehavethatforanyinitialdatum F 0 2 V ,thereexistsauniquefunction F :[0 ; 1 ) ! V suchthat 8 > > > < > > > : 0 2 _ F + @ L ( F ) ;t> 0 ; F (0)= F 0 : (3.13) Bruck[8]demonstratedthat,forgeneralgradientwofastrictlyconvexfunctional F ( t ) * F weaklyas t !1 ; where F istheuniqueminimizerofthefunctional.Next,wewillshowthatforthefunctional L above,wealsohavestrongconvergenceoutsideofLetussummarizeour discussionaboveinthetheorembelow: Theorem3.2.2. Foreach F 0 2 V thereexistsauniquesolutiontothegradientgiven by(3.13).Furthermore, F ( t ) * F as t !1 and F ( t ) ! F in L 2 ( ~ c ; R n ) forallcompact sets ~ containing : 33 Proof. Since F ( t ) 2 V ,thegradientwshouldbereducedto 8 > > > < > > > : 0 2 _ F + F + P V @ x;F ) ˜ ;t> 0 ; F (0)= F 0 : (3.14) Let N 2 P V @ x;F ) ˜ .thenwehavecurl( N )=0byanddiv( N )=0in c in thesenseofdistribution.Bytheidentitycurlcurl( N )+ 4 ( N )= r (div( N )) ; wehavethat 4 N =0in c inthesenseofdistribution.Hence 4 ( _ F + F )=0in c : Solvingthisevolutionequation,weobtainthat F ( x;t )= F 0 ( x ) e t + U ( x;t )for x 2 c and t> 0 ; where 4 U ( x;t )=0in c (0 ; 1 ) : Let V ( x;t )= U ( x;t ) F ( x ).Then V ( x;t )is harmonicin c and V ( x;t ) * 0as t !1 .Forany x 0 2 c , V ( x 0 ;t )= 1 j B r ( x 0 ) j Z B r ( x 0 ) V ( x;t ) dx ! 0as t !1 : Foranyball R n ˙ B R (0) B R ˙˙ c = B c R + B R n Let C ˆ B R n ~ beany compactsetwith d ( C;@ ~ = d 0 > 0. V ( x;t )= 1 B d 0 2 ( x ) Z B d 0 2 ( x ) V ( y;t ) dy; 8 x 2 C: = ) j V ( x;t ) j 1 B d 0 2 ( x ) jj V ( y;t ) jj L 2 B d 0 2 ( x ) 1 2 M: 34 Therefore, U ( x;t ) ! F ( x )stronglyin C; when t !1 ; bytheBoundedConvergenceTheorem. Let's t Forany x 2 B c R , j V ( x;t ) j = 1 j B j x R ( x ) j Z B j x R ( x ) V ( y;t ) dy C ( j x j R ) n= 2 ; (3.15) Nowlet x !1 ,wehavethat V ( x;t ) ! 0,forall t .Applyingthe KelvinTransform , (assume R = 1 2 below)wehavethat ~ V ( y;t )= 1 j y j n 2 V y j y j 2 ;t ; 0 j y j 1 : (3.16) Weproceedwiththreecaseswhen x 2 B c R : (i)If n =3,let x = y j y j 2 ,so j x j 1andthereexists c> 0,notdependingon t ,suchthat j ~ V ( y;t ) jj y j = j V ( x;t ) j c ( j x j R ) 3 = 2 c j y j 3 = 2 ; (3.17) i.e. j V ( y;t ) j c j y j 1 2 : So y =0isaremovablesingularpoint.Therefore,we W ( y;t )= 8 > > > < > > > : ~ V ( y;t ) ;y 6 =0 ; 0 ;y =0 : Thus j W ( y;t ) j c 1 j y j : andcombinedwith(3.17),itfollowsthat j V ( x;t ) j 2 C 1 j y j 4 = C 1 j x j 4 2 L 1 ( B c R ) : 35 BytheDominantConvergenceTheorem, V ( x;t ) ! 0stronglyin L 2 ( B c R )as t !1 . (ii)If n =4,theKelvinTransformis ~ V ( y;t )= 1 j y j 2 V y j y j 2 ;t andthereexistssome constant c 2 suchthat j ~ V ( y;t ) j c 2 .Supposethatthereexists a> 0,suchthat W ( y;t )= 8 > > > < > > > : ~ V ( y;t ) ;y 6 =0 ; a;y =0 : Hence 9 > 0,s.t.when j y j < , 1 j y j 2 j V ( x;t ) j a> 0 : Thus j V ( x;t ) j a j y j 2 = a j x j 2 62 L 2 ( B c R ) : Thiscontradictionimpliesthat a =0,i.e. W 0when y =0.Thusthereexistsa constant c 3 ,notdependingon t ,suchthat j W ( y;t ) j = j P c j y j j c 3 j y j : Therefore, j V ( x;t ) j c 3 j y j 3 = c 3 j x j 3 2 L 2 ( B c R ) : BytheDominantConvergenceTheorem, V ( x;t ) ! 0stronglyin L 2 ( B c R )as t !1 . (iii)If n 5,wealreadyhave j V ( x;t ) j C 1 j x j n 2 forany t> 0by(3.15). Y ( x )= C 2 j x j n 2 .Let Z ( x;t )= Y ( x ) V ( x;t ),fora R 1 > 0,wecanaconstant C 2 > 0 suchthat Z ( x;t ) j x j = R 1 > 0andweclaimthat Z ( x;t ) 0in n x 2 R n ; j x j R 1 o : Ifnot,thereexists x 0 2 R n , j x 0 j >R 1 and > 0suchthat Z ( x 0 ;t )= < 0.Therefore, thereexists R 2 >R 1 ,s.t. Z ( x;t )+ 2 j x j = R 2 > 0 : Applyingthemaximumprincipleto Z ( x;t )+ 2 on n x 2 R n ; R 1 j x j R 2 o yieldsacontradiction!Therefore, Z ( x;t ) 0in n x 2 R n ; j x j R 1 o : 36 i.e. V ( x;t ) Y ( x )in n x 2 R n ; j x j R 1 o and Y ( x ) 2 L 2 ( R n n B 1 ).Thereforewhen n 5, V ( x;t ) ! 0strongly ; as t !1 ; when x>R 1 bytheDominantConvergenceTheorem. Combiningalltheresults,foranycompact ~ ˙˙ F ( x;t ) ! F ( x )stronglyin L 2 ( ~ c ) ; as t !1 : 3.2.3Possibledynamicsformagnetizationm Notethatthegradientw(3.13)on H = L 2 ( R n ; R n )determinesa(nonunique)time- dependentvectorfunction m ( t )= m ( x;t )withtheproperty 8 > > > < > > > : div( _ F ( t )+ F ( t )+ m ( t ) ˜ )=0( t> 0) ; m ( x;t ) 2 @ F ( x;t )+ H ( x )) a:e:x 2 ;t> 0 : (3.18) Theconditionassertsthat F m ( t ) = _ F ( t ) F ( t ). Fromthegeneraltheoryofgradientw[10],wehave Z 1 0 k _ F ( t ) k 2 H dt< 1 ; whichimpliesthat,alongasubsequence t k !1 ,onehas _ F ( t k ) ! 0in H : Hence,alonga 37 furthersubsequenceof f t k g ,wehave m ( t k ) * m ;forthis m wehave div( F + m ˜ )=0 : If,inaddition,onehasthat F ( t ) ! F stronglyin L 2 ( R n ; R n )thenitwouldfollowthat m ( x ) 2 @ F ( x )+ H ( x ))fora.e. x 2 andthus m ( x )wouldbeageneralizedminimizer forenergy I ( m ) : Therefore,insomesense,thesystem(3.18)areasonableevolutionprocessforthe functional I ( m ) : 3.2.4Studyofaspecialcase Weinvestigateaspecialcaseofthegradientwinthesoftcasein R 3 : Assume ' 0 andthus ˘ )= j ˘ j : Let H beaconstantandbeaballin R 3 .Westudythegradient w(3.13)withinitialdatum F 0 2 V thatequalsaconstantvectorparallelto H ;thatis, F 0 = H inwhere 2 R isaconstant. Wearetryingtothesolution F ( t )suchthat F ( x;t )= ( t )onandacorresponding vectorfunction m ( t )= m ( x;t )of(3.18)isalsoconstantinthatis, m ( x;t )= k ( t ) 2 R 3 for x 2 Therefore P V ( k ( t ) ˜ )( x;t )= _ F ( x;t ) F ( x;t )for x 2 R 3 : LetusrecallfromSection2.2thatforunitball= B 1 (0) 2 R n andanyconstant k 2 R n , wehave P V ( k˜ )= 8 > > > < > > > : 1 n k in ; r 1 n k x j x j n in c : 38 Therefore,inourcase,since @ ˘ )= 8 > > > > < > > > > : ˘= j ˘ j ( ˘ 6 =0) ; B 1 (0)( ˘ =0) ; thegradientwimplies _ ( t )+ ( t )+ 1 3 K ( ( t )) 3 0( t> 0) ; (3.19) where K ( )istheset-valuedfunctionby K ( )= 8 > > > < > > > : + H j + H j ; 6 = H; B 1 (0) ; = H: Oncewesolve ( t )from(3.19),let k ( t )= K ( ( t ))andsolve f ( x;t )forall x 2 c by 8 > > > < > > > : _ f ( x;t )+ f ( x;t )+ r k ( t ) x 3 j x j 3 =0( t> 0) ; f ( x; 0)= F 0 ( x ) : Thenthefunction F ( x;t )= ( t ) ˜ ( x )+ f ( x;t ) ˜ c ( x )willbethesolutiontothegradient w(3.13).Consequently,thegradientw(3.13)becomesequivalenttothe3-Dgradient w(3.19)on R 3 : E ( )= 1 2 j j 2 + 1 3 j + H j ( 2 R 3 ) : 39 Thentheproblem(3.19)becomesthegradientwof E ( )on 2 R 3 : Wenowstudythesolution ( t )to(3.19)incases. Case1. j H j 1 = 3 . (Inthiscase,noticethattheminimizerof E ( )is = H .) I. If = 1,i.e. (0)= H ,then ( t ) H: Next,considerthat 6 = 1.The gradientwisgivenby _ + + 1 3 + H j + H j =0 ; (0)= H: Let P beavectorsuchthat P ? H andwedenote ( t ) P = h ( t )andwedotproductboth sidesby P ,then h 0 ( t )+ h ( t )+ 1 3 h ( t ) j + H j =0 ; h (0)=0 : Wecanconcludethat h ( t ) 0,whichimpliesthatthereexists g ( t ),suchthat ( t )= g ( t ) H: Consequently, 8 > > > < > > > : g 0 ( t )+ g ( t )+ 1 3 j H j g ( t )+1 j g ( t )+1 j =0 ; g (0)= : (3.20) Let g ( t )+1= r ( t ),thentheequationabovereads r 0 ( t )+ r ( t )+ 1 3 j H j r ( t ) j r ( t ) j =1 ; r (0)= +1 : 40 II. If > 1,then r (0) > 0.Thus r ( t ) > 0in0 t< t ,forsome t> 0.Hencethe equationbecomes r 0 ( t )+ r ( t )+ 1 3 j H j =1 ; then r ( t )=1 1 3 j H j + + 1 3 j H j e t ,andlet r ( t )=0 ; thenwehave t =ln h 3 j H j +1 3 j H j +1 i : III. < 1,i.e. r (0) < 0.Thus r ( t ) < 0in0 t< t ,forsome t> 0.Thusthe equationbecomes r 0 ( t )+ r ( t ) 1 3 j H j =1 ; then r ( t )=1+ 1 3 j H j + 1 3 j H j e t ,andlet r ( t )=0,thenwehave t =ln h 3 j H j 1 3 j H j 1 i ; accordingly. Combingallthecaseswhen j H j 1 3 ,thereexist t< 1 suchthat r ( t )=0,whichis equivalenttosay ( t )= H ; showninFigure3.2. r ( t ) O t t r ( t ) O t t > 1 ; j H j 1 = 3 < 1 ; j H j 1 = 3 Figure3.2:Thegraphsforthecaseof j H j 1 3 . Case2. j H j 1 = 3 . (Inthiscase,noticethattheminimizerof E ( )is = H 3 j H j : ) Nowwestillconsiderproblem(3.20). I. > 1 3 j H j ,i.e. g (0) > 1 3 j H j .Therefore g (0)+1 > 1 3 j H j +1 > 0,whichimplies that g ( t )+1 > 0in0 t< t ,forsome t> 0.Nowtheequationbecomes: g 0 ( t )+ g ( t )+ 1 3 j H j = 0.Hence g ( t )= 1 3 j H j + + 1 3 j H j e t : Itiseasytoseethat g ( t )isdecreasingand 41 1 3 j H j t 1 j H j > 1 = 3 t Figure3.3:Thegraphsforthecaseof j H j > 1 3 . g ( t ) ! 1 3 j A j as t !1 : II. 1 << 1 3 j H j ,i.e. 1 1+1=0, whichimpliesthat g ( t )+1 > 0in0 t< t ,forsome t> 0.Nowtheequationbecomes: g 0 ( t )+ g ( t )+ 1 3 j H j =0andthesolutionis g ( t )= 1 3 j H j + + 1 3 j H j e t : Itiseasytosee that g ( t )isincreasingand g ( t ) ! 1 3 j H j as t !1 . III. < 1,i.e. g (0) < 1.Inthiscase, g (0)+1 < 1+1=0,whichimpliesthat g ( t )+1 < 0in0 t 0.Nowtheequationbecomes g 0 ( t )+ g ( t ) 1 3 j H j =0, when0 t > > < > > > : g 0 ( t )+ g ( t ) 1 3 j H j =0 ; with g (0)= ; 0 t > > > < > > > > : 1 3 j H j + 1 3 j H j e t ; 0 t 1 3 ; Wecanalsoseethat g ( t ) ! 1 3 j H j as t !1 butneverequals 1 3 j H j ,showninFigure3.3. 43 Chapter4 AMinimizationProblemonAnnulus Inthischapter,weassume n 3isanintegerand= f x 2 R n j a< j x j < 1 g isaspherical shellin R n : Westudythereducedmicromagneticenergy: I ( m )= Z 1 ( x ) dx + 1 2 Z R n j F m ( x ) j 2 dx forcertainconstants 0 : Relatedtotheminimizationofthisfunctional I ( m ),westudytheboundaryvalueprob- lemoftheEikonalequation: 8 > > > > > > > > > > > > < > > > > > > > > > > > > : jr ( s;t ) j = t n 2 in a 2 0, ( s;t )=0on s 2 + t 2 = a 2 ;t 0, ( s;t )= n 1 t n 1 on s 2 + t 2 =1 ;t 0, ( s;t )=0on t =0 ;a j s j 1 ; (4.1) where0 1 g ;F = w : Then,allsuchunit-lengthvectorfunctions G canbedescribedby div( F + G˜ )=0(4.4) inthesenseofdistributionson R n : Suppose isaLipschitzsolutionto(4.1)andlet m = G ( x )beby(4.3);so (4.4)holds.Since F = e 1 inforall F 2 L 2 ( R n ; R n )withcurl F =0,wehave j F + e 1 j = j F F j G ( F F )andhence J ( F ) J ( F ) Z R n F ( F F ) dx + Z G ( F F ) dx = Z R n ( F + G˜ ) ( F F ) dx =0 : Therefore F istheuniqueminimizerof J : For m = G ( x ),by(4.4),wehavethat F m = F andthat R R n F F dx = R F Gdx ,andhence I ( m )= Z F ( x ) G ( x ) dx + 1 2 Z R n j F ( x ) j 2 dx = 1 2 Z R n j F j 2 dx = J ( F ) : 45 Consequently, m = G ( x )isaminimizerof I . SomeexistenceresultsontheLipschitzsolutionstotheboundaryvalueproblemforgen- eralentialequationshavebeengivenin[20],buttoapplysuchanexistence theoremtoourproblem,onewouldneedtoaLipschitzfunction 0 satisfying,inaddi- tiontotherequiredboundaryconditions,theinequality jr 0 ( s;t ) j t n 2 ;theconstruction ofsuchafunction 0 isequallyasthatofasolution of(4.1).Evensuchafunc- tion 0 isknowntoexist,thegeneralexistencetheoremwouldonlyasserttheexistence ofitelymanyLipschitzsolutionsto(4.1)withoutspecifyingthestructuresofanysuch solutions. Ourmainideaistousethecharacteristicsmethodtoconstructthelocalsolutionsnear theboundariesandthengluethemtogetherwithcertaintrivialsolutionsawayfromthe boundaries.AsweshallseeinTheorems4.4.6and4.4.9below,thesolutionsconstructed thiswaydonothavemanyoscillations,whichwouldbeotherwiseexpectedbythe generalexistencetheoremsof[20]. 4.1Thecharacteristicsmethod TherestofthechapterisdevotedtotheconstructionoftheLipschitzsolutions ( s;t )of (4.1)thatareevenin s .Forthispurpose,let ! = f z =( s;t ) j s> 0 ;t> 0 ;a 2 > > > > > > > > > > > < > > > > > > > > > > > > : jr ( s;t ) j = t n 2 in ! , ( s;t )=0 on j z j = a with s> 0 ;t> 0 ; ( s;t )=0 on t =0 ;s 2 [ a; 1] ; ( s;t )= n 1 t n 1 on j z j =1 with s> 0 ;t> 0 . (4.6) Proofofthenecessityof(4.5). Suppose isaLipschitzsolutionon !: Then n n 1 = Z 1 a t (0 ;t ) dt Z 1 a jr (0 ;t ) j dt Z 1 a t n 2 dt = 1 n 1 (1 a n 1 ) : Thisprovesthat 1 n (1 a n 1 ) : Theproofoftheof(4.5)isthemainpurposeofthisthesis;westatethis partinthefollowingtheorem,includingsomeapplicationtocertainminimizers ofthefunctional I ( m ) : Theorem4.1.2. If 0 1 n (1 a n 1 ) ,thentheproblem (4 : 6) hasmany Lipschitzsolutions ( s;t ) ,constructedinTheorems4.4.6and4.4.9below.Inthiscase, when n 3 isintegerand isdasabove,theminimizers m = G ( x ) of I ( m ) given by (4 : 3) withtheconstructedsolution willbetheconstant e 1 incertainsubdomains 0 = f ( x 1 ;x 0 ) 2 j ( j x 1 j ; j x 0 j ) 2Z 0 g awayfromtheboundary @ : SomeexistenceresultsontheLipschitzsolutionstotheboundaryvalueproblemforgen- 47 eralentialequationshavebeengivenin[20],buttoapplysuchanexistence theoremtoourproblem,onewouldneedtoaLipschitzfunction 0 on ! satisfying, inadditiontotherequiredboundaryconditions,theinequality jr 0 ( s;t ) j t n 2 in ! . Constructionofsuchafunction 0 isequallyasthatofasolution of(4.6). Ourmainideaistousethecharacteristicsmethodtoconstructthelocalsolutionsnear thetwoquarter-circlesoftheboundaryof ! andthentogluethemtogetherwithcertain trivialsolutionsawayfromtheboundaries.Wewritetheequation jr j = t n 2 as F ( s;t; ; s ; t )=0 ; where F ( s;t;z;p;q )= 1 2 ( p 2 + q 2 t 2 n 4 ) : ThecharacteristicsODEsforthisPDE aregivenby(see[28]) ds d˝ = p; dt d˝ = q; dz d˝ = t 2 n 4 ; dp d˝ =0 ; dq d˝ =( n 2) t 2 n 5 : (4.7) Wesolvethesystem(4.7)on ˝ 0withgiveninitialdata ( s;t;z;p;q ) j ˝ =0 =( ( ) ; ( ) ; ( ) ;f ( ) ;g ( )) ; (4.8) where ( ) ; ( ) ; ( ) ;f ( )and g ( )dependonaparameter inaninterval I .Assumethe functions ( ) ; ( ) ; ( ) ;f ( )and g ( )aresmoothin I andsatisfythe characteristicsstrip conditions : 8 > > > < > > > : f ( ) 2 + g ( ) 2 = ( ) 2 n 4 ; f ( ) 0 ( )+ g ( ) 0 ( )= 0 ( ) 8 2 I: (4.9) 48 Foreach 2 I; thesmoothsolutionsto(4.7)-(4.8)willbedenotedby s = S ( ˝; ) ;t = T ( ˝; ) ;z = Z ( ˝; ) ;p = P ( ˝; ) ;q = Q ( ˝; ) : Weeasilysolve s = S ( ˝; )and p = P ( ˝; )tohave P ( ˝; )= f ( ) ;S ( ˝; )= ( )+ f ( ) ˝ 8 ˝ 0 ; 2 I: (4.10) Solving( t;q )in(4.7)wehavethat,foreach 2 I ,theuniquesmoothsolution t = T ( ˝; ) existsonamaximalinterval[0 ;˝ M ( ))and 8 > > > < > > > : t 0 2 = t 2 n 4 f ( ) 2 ; 0 <˝<˝ M ( ) ; t (0)= ( ) ;t 0 (0)= g ( ) ; (4.11) andhencethesolution Q ( ˝; )= T ˝ ( ˝; ) f ( ) 2 + Q ( ˝; ) 2 = T ( ˝; ) 2 n 4 8 2 I;˝ 2 [0 ;˝ M ( )) : (4.12) Afterwesolve T ( ˝; ),weeasilyobtain z = Z ( ˝; )byintegration: Z ( ˝; )= ( )+ Z ˝ 0 T ( ; ) 2 n 4 8 2 I;˝ 2 [0 ;˝ M ( )) : (4.13) Using T 2 n 4 = f 2 + Q 2 and T ˝ = Q ,anintegrationbypartsyieldsthat Z ( ˝; )= ( )+ f ( ) 2 ˝ + Z ˝ 0 T ( ; ) Q ( ; ) 49 = ( )+ f ( ) 2 ˝ +[ T ( ; ) Q ( ; )] ˝ 0 Z ˝ 0 T ( ; ) Q ( ; ) : Pluggingin Q ( ; )=( n 2) T ( ; ) 2 n 5 andrearrangingterms,wehave Z ( ˝; )= ( )+ 1 n 1 [ f ( ) 2 ˝ + T ( ˝; ) Q ( ˝; ) ( ) g ( )] = f + g n 1 + S ( ˝; ) f ( )+ T ( ˝; ) Q ( ˝; ) n 1 : (4.14) Weremarkthatwhen n =3theequationsfor( t;q )become linear andthesystem(4.7) canbesolvableinanexplicitform;however,thesubsequentcalculationsarecomplicated andtoospAsweseelater,thecase n> 3presentssometfeaturesfromthe case n =3 : 4.1.1Themaximalexistencetime ˝ M ( ) Inviewofthetwosubsequentcasestobeconsidered,wemakethefollowingassumption: ( ) > 0 ; ( ) > 0 ;f ( ) 6 =0 8 2 I: (4.15) Foragiven 2 I wethenumber ˝ M = ˝ M ( )accordingtothesignof g ( ) : Case (i):Assume g ( ) 0.Inthiscase,theequationin(4.11)becomes dt d˝ = q t 2 n 4 f ( ) 2 on ˝ 2 (0 ;˝ M ) : 50 Thesolution t = T ( ˝; )isincreasingin ˝ and T ( ˝; ) > ( )on(0 ;˝ M );moreover, since(0 ;˝ M )ismaximalintervalofexistencefor T ( ˝; ),wemusthave lim ˝ ! ˝ M T ( ˝; )= 1 : Givenany ˝ 2 [0 ;˝ M )and t ( ),itfollowsthat t = T ( ˝; )ifandonlyif ˝ = A ( t; ):= Z t ( ) dy p y 2 n 4 f ( ) 2 : (4.16) Therefore,weobtain ˝ M = ˝ M ( )= Z + 1 ( ) dy p y 2 n 4 f ( ) 2 if g ( ) 0 : Notethat ˝ M =+ 1 if n =3and ˝ M < + 1 if n> 3since f ( ) 6 =0 : Inthiscase, t = T ( ˝; ) istheinversefunctionof ˝ = A ( t; ),and Q ( ˝; ) > 0forall0 <˝<˝ M ( ) : Case (ii):Assume g ( ) < 0 : Inthiscase,wehavethat T ˝ ( ˝; )= Q ( ˝; ) < 0andthus T ( ˝; )isdecreasingin ˝ onsomeinterval ˝ 2 [0 ;˝ m ),where ˝ m = ˝ m ( ) > 0isanumber suchthat Q ( ˝ m ; )= T ˝ ( ˝ m ; )=0 : By(4.12),weobtain lim ˝ ! ˝ m T ( ˝; )= j f ( ) j 1 n 2 : Clearly,thefunctions t = T ( ˝; )and q = Q ( ˝; )satisfy dt d˝ = q = q t 2 n 4 f ( ) 2 8 ˝ 2 (0 ;˝ m ) : (4.17) 51 Itfollowsthat j f ( ) j 1 n 2 ˝ m : Asin thecase,using T ( ˝ m ; )= j f ( ) j 1 n 2 ,solutions t = T ( ˝; )and q = Q ( ˝; )satisfy dt d˝ = q = q t 2 n 4 f ( ) 2 8 ˝ 2 ( ˝ m ;˝ M ) : Hence t = T ( ˝; )isincreasingin ˝ and T ( ˝; ) > j f ( ) j 1 n 2 on( ˝ m ;˝ M ) : The maximalexistencetime ˝ M of T ( ˝; )mustsatisfy lim ˝ ! ˝ M T ( ˝; )= 1 : Moreover,given ˝ 2 [ ˝ m ;˝ M )and t j f ( ) j 1 n 2 ,itfollowsthat t = T ( ˝; )ifandonlyif ˝ ˝ m = Z t j f ( ) j 1 n 2 dy p y 2 n 4 f ( ) 2 ; thatis, ˝ =2 ˝ m ( )+ A ( ˝; )=2 ˝ m ( )+ Z t ( ) dy p y 2 n 4 f ( ) 2 : (4.20) 52 Letting ˝ ! ˝ M ,weobtain ˝ M ( )=2 ˝ m ( )+ Z + 1 ( ) dy p y 2 n 4 f ( ) 2 if g ( ) < 0 : Again, ˝ M =+ 1 if n =3and ˝ M < 1 if n> 3 : Inthiscase,itiseasilyshownthatthe solution T ( ˝; )soconstructedissmoothon[0 ;˝ M ( ))byverifying T ˝ ( ˝ m ; )= T ˝ ( ˝ + m ; ) : (Somerelatedcomputationisgivenbelow.)Furthermore, Q ( ˝; ) < 0for0 <˝<˝ m ( )and Q ( ˝; ) > 0for ˝ m ( ) <˝<˝ M ( ) : 4.1.2Invertingthecharacteristicsmap Let ˝ M ( )beasabove,and D = f ( ˝; ) j 2 I; 0 ˝<˝ M ( ) g : (4.21) the characteristicmap ( S ( ˝; ) ;T ( ˝; )): D! R 2 ; andconsiderthecurve= f ( ( ) ; ( )) j 2 I g : Wewouldliketoasubdomain Z of ! with ˆ @ Z andasubdomain Y of D suchthatforeach( s;t ) 2Z thereexistsaunique ( ˝; )=( ( s;t ) ;˘ ( s;t ))in Y satisfying ( s;t )=( S ( ˝; ) ;T ( ˝; )); 53 thatis,themap( S;T ): Y!Z is bijective. Let( ˝; )=( ( s;t ) ;˘ ( s;t )): Z!Y beits inversemap.Ofcourse,astandardmethodwouldbetostudytheJacobianofthemap ( s;t )=( S ( ˝; ) ;T ( ˝; )).However,weuset(but,eventually,equivalent)methods dependingonthespparametrizationofcurve : 4.1.3Constructionofthelocalsolutions Onceweobtaintheinversemap( ;˘ )ofthemap( S;T ),wedalocalsolution by ( s;t )= Z ( ( s;t ) ;˘ ( s;t )) 8 ( s;t ) 2Z ; where Z ( ˝; )isby(4.13)above.Bycontinuitywemayalsoextend tosomeofthe boundarypointsof Z : Notethatby(4.14)thesolution on Z canbecomputedas ( s;t )= " ( ) f ( ) ( )+ g ( ) ( ) n 1 + sf ( ) t p t 2 n 4 f ( ) 2 n 1 # = ˘ ( s;t ) (4.22) withthechoiceof\ "thesameasthesignof Q ( ( s;t ) ;˘ ( s;t )). Inthenexttwosections,wecarryouttheseconstructionsneartheinnercircle j z j = a andneartheoutercircle j z j =1separately. 4.2Constructionneartheinnerquarter-circle Inthiscase,wechoosetheinterval I =(0 ;ˇ= 2)and ( )= a cos ; ( )= a sin ; ( )=0 8 2 I: 54 Wethestripconditions(4.9)byselecting f ( )=( a sin ) n 2 cos ;g ( )=( a sin ) n 2 sin : Hencethecondition(4.15)holdsand g ( ) > 0forall 2 I andthedomain D in(4.21) becomes D = ( ( ˝; ) 2 (0 ;ˇ= 2) ; 0 ˝< Z + 1 ( ) dy p y 2 n 4 f ( ) 2 ) : 4.2.1Thecharacteristicssolutions Thefunction t = T ( ˝; )isdetermineduniquelyby ˝ = Z T ( ˝ ) ( ) dy p y 2 n 4 f ( ) 2 8 ( ˝; ) 2D : Also Q ( ˝; )= T ˝ ( ˝; )= p T ( ˝; ) 2 n 4 f ( ) 2 0andachangeofvariablesyieldsthat Z ( ˝; )= Z T ( ˝ ) ( ) y 2 n 4 dy p y 2 n 4 f ( ) 2 8 ( ˝; ) 2D : Let R := f ( t; ) j 2 I;t ( ) g and U ( t; )= Z t ( ) y 2 n 4 dy p y 2 n 4 f ( ) 2 8 ( t; ) 2R : (4.23) Then Z ( ˝; )= U ( T ( ˝; ) ; ) 8 ( ˝; ) 2D : 55 Asabove,let A ( t; )= Z t ( ) dy p y 2 n 4 f ( ) 2 8 ( t; ) 2R : Then,for( ˝; ) 2D and( t; ) 2R ,itfollowsthat t = T ( ˝; )ifandonlyif ˝ = A ( t; ) : 4.2.2Invertingthecharacteristicsmap B ( t; )= S ( A ( t; ) ; ),thatis, B ( t; )= ( )+ f ( ) Z t ( ) dy p y 2 n 4 f ( ) 2 8 ( t; ) 2R : Adirectcomputationyieldsthat B ( t; )= a sin + f 0 ( ) Z t ( ) y 2 n 4 dy ( y 2 n 4 f ( ) 2 ) 3 = 2 8 2 I: (4.24) Lemma4.2.1. Let U ( t; ) and B ( t; ) bedasabove.Then U ( t; )= B ( t; ) f ( ) 8 ( t; ) 2R : (4.25) Proof. By(4.23), U ( t; )= 0 2 n 4 g + ff 0 Z t y 2 n 4 dy ( y 2 n 4 f 2 ) 3 = 2 : Since ff 0 Z t y 2 n 4 dy ( y 2 n 4 f 2 ) 3 = 2 = f ( ) B ( t; )+ af ( ) sin ; 56 consequently,itfollowsthat U ( t; )= 0 2 n 4 g + af ( ) sin + B ( t; ) f ( )= B ( t; ) f ( ) ; resultingfromtheidentity 0 2 n 4 g = af ( ) sin : Notethat f 0 ( )= a ( a sin ) n 3 [( n 2) ( n 1)sin 2 ] : So,for ^ =arcsin( p ( n 2) = ( n 1)) ; itfollowsthat f 0 ( ) > 0 8 2 (0 ; ^ ) ;f 0 ( ) < 0 8 2 ( ^ ;ˇ= 2) : Hence,by(4.24), B ( t; ) < 0 8 2 [ ^ ;ˇ= 2) ;t ( ) : Lemma4.2.2. Forall t> 0 , lim ! 0 + B ( t; )= 8 > > > > < > > > > : a R 1 1 q 2 n 4 1 ( n> 3) ; + 1 ( n =3) : (4.26) Proof. Byachangeofvariablesandintegrationbyparts,wehave B ( t; )= a ( n 1) n 2 sin + 1 n 2 0 @ Z t=k =k f 0 f 3 n n 2 p 2 n 4 1 tf 0 p t 2 n 4 f 2 1 A ; 57 where k = f 1 n 2 : Since =k =1 = (cos ) 1 n 2 ! 1and t=k !1 as ! 0 + ; itfollowsthat lim ! 0 + Z t=k =k p 2 n 4 1 = 8 > > > > < > > > > : R 1 1 q 2 n 4 1 ( n> 3) ; + 1 ( n =3) : Notethat lim ! 0 + f 0 ( )= 8 > > > < > > > : a ( n =3) ; 0( n> 3) andlim ! 0 + f 0 ( ) f ( ) 3 n n 2 = a ( n 2) : Combiningtheselimits,wehave(4.26). Foreach t> 0,let 0 ( t )= 8 > > > < > > > : arcsin t a 0 0and,by(4.26), ( t ) > 0forall t> 0 : Furthermore, B ( t; ( t ))=0 ;B ( t; ) 0 8 ( t ) < 0 ( t ) : (4.28) Lemma4.2.3. Foreach t> 0 ,thefunction B ( t; ) isone-to-oneontheinterval 2 58 ˇ 2 a t 0 = ( t ) = 0 ( t ) ^ Figure4.1:Thefunction = ( t )by(4.27)isstrictlyincreasingandleft-continuous. [ ( t ) ; 0 ( t )) : Moreover,thefunction ( t ) isstrictlyincreasingandleft-continuouson t> 0 and (0 + )=0 : Proof. Let a;b 2 [ ( t ) ; 0 ( t ))besuchthat B ( t;a )= B ( t;b ) : Weshow a = b: If a 0 8 0 < ^ ;t ( ) : Wehavethat B ( t 0 ; ( t )) >B ( t; ( t ))=0 ; whichgivesacontradiction.Toshowtheleft- continuityof ,given t> 0 ; let l = ( t );then0 0 : Furthermore,from ( ( t )) < ( 0 ( t )) t , itfollowsthat (0 + )=0 : s t 0 Z 1 s + ( t ) s ( t ) s 1 ( t ) a a Figure4.2:Thedomain Z 1 andthesmoothincreasingfunction s = s 1 ( t )determinedin Lemma4.2.7. (seeFigure4.2) s + ( t )= B ( t; ( t )) ;s ( t )= q ( a 2 t 2 ) + 8 t> 0 ; and Z 1 = f ( s;t ): t> 0 ;s ( t ) 0 : (4.29) 60 Lemma4.2.4. Thereexistsauniquefunction = ˘ ( s;t ) on Z 1 suchthat s = B ( t;˘ ( s;t )) 8 ( s;t ) 2Z 1 : Moreover,thefunction = ˘ ( s;t ) iscontinuouson Z 1 andisentiableateverypoint ( s 0 ;t 0 ) of Z 1 where B ( t 0 ;˘ ( s 0 ;t 0 )) 6 =0 and,atanysuchpoint ( s;t ) ,wehave ˘ t = B t ( t;˘ ) =B ( t;˘ ) ;˘ s =1 =B ( t;˘ ) : Proof. Let( s;t ) 2Z 1 ;then t> 0and s ( t ) 0 andtheEikonalequation jr 1 ( s;t ) j = t n 2 in Z 1 .Furthermore, 1 canbeextendedcontinuouslytothecurve s = s ( t ) forall t> 0 suchthat 1 ( s ( t ) ;t )= 1 n 1 ( t n 1 a n 1 ) + 8 t> 0 : (4.30) Proof. Thefunction 1 isclearlycontinuousin Z 1 : Ateachpoint( s;t ) 2Z 1 where B ( t;˘ ( s;t )) 6 = 0, 1 istiable,andby(4.25), 1 s ( s;t )= U ( t;˘ ) ˘ s ( s;t )= U ( t;˘ ) B ( t;˘ ) = f ( ˘ ( s;t )) > 0 : Ontheotherhand,with = ˘ ( s;t ), 1 t ( s;t )= U t ( t; )+ U ( t; ) ˘ t ( s;t )= U t ( t; )+ f ( ) B ( t; ) ˘ t ( s;t ) = U t ( t; ) f ( ) B t ( t; )= U t ( t; ) f ( ) 2 A t ( t; ) = t 2 n 4 p t 2 n 4 f ( ) 2 f ( ) 2 p t 2 n 4 f ( ) 2 62 andhence 1 t ( s;t )= q t 2 n 4 f ( ˘ ( s;t )) 2 : Theformulasof 1 s and 1 t alsoshowthat 1 istiableateverypointof Z 1 ,with 1 s ( s;t )= f ( ˘ ( s;t )) > 0,andtheequation jr 1 j = t n 2 : Finally,weextend 1 to thecurve s = s ( t )byletting 1 ( s ( t ) ;t )=lim s ! ( s ( t )) + 1 ( s;t ) 8 t> 0 : Weshow 1 ( s ( t ) ;t )= 1 n 1 ( t n 1 a n 1 ) + 8 t> 0 : Fix t> 0 : By(4.29),wehave lim s ! ( s ( t )) + ˘ ( s;t )= 0 ( t ) : Therefore lim s ! ( s ( t )) + 1 ( s;t )=lim s ! ( s ( t )) + U ( t;˘ ( s;t )) = U ( t; 0 ( t ))= Z t ( 0 ( t )) y 2 n 4 dy p y 2 n 4 f ( 0 ( t )) 2 = 1 n 1 ( t n 1 a n 1 ) + : Lemma4.2.6. Forall ( s;t ) 2Z 1 \f s> 0 ;s 2 + t 2 =1 g ,wehave 1 ( s;t ) > 1 a n 1 n 1 t n 1 : (4.31) 63 Proof. Let( s;t ) 2Z 1 besuchthat s> 0 ;s 2 + t 2 =1 : Then 1 ( s;t )= U ( t;h ( t )),where h ( t )= ˘ ( p 1 t 2 ;t ) : Weshowthat h ( t ) < arcsin t: Tothisend,let ~ =arcsin t ;so ( ~ )= a p 1 t 2 and ( ~ )= at .Wethenhave B ( t; ~ )= ( ~ )+ f ( ~ ) Z t ( ~ ) ( y 2 n 4 f ( ~ ) 2 ) 1 = 2 dy h ( t ) : Finally,bytheformulaof U ( t; ),wehave 1 ( s;t )= U ( t;h ( t )) Z t ( h ( t )) y n 2 dy = t n 1 ( h ( t )) n 1 n 1 > t n 1 ( ~ ) n 1 n 1 = 1 a n 1 n 1 t n 1 : Lemma4.2.7. Thereexistsasmooth increasing function s 1 ( t ) 2 ( s ( t ) ;s + ( t )) suchthat, forall t> 0 , 1 ( s;t ) < t n 1 n 1 8 s ( t ) t n 1 n 1 8 s 1 ( t ) 0andso,for each t> 0, K ( s;t )isstrictlyincreasingin s 2 [ s ( t ) ;s + ( t )] : Wewillshowthat K ( s ( t ) ;t ) < 0 ;K ( s + ( t ) ;t ) > 0 8 t> 0 : (4.32) Thiswillprovetheexistenceof s 1 ( t ) 2 ( s ( t ) ;s + ( t ))suchthat K ( s 1 ( t ) ;t )=0 : Moreover,since K s ( s 1 ( t ) ;t ) > 0,bytheimplicitfunctiontheorem,thefunction s 1 ( t )isalso tiablein t> 0 ; with s 0 1 ( t )= t n 2 1 t 1 s > 0 8 t> 0 ; whichcompletestheproof.Toprove(4.32),ofall,notethat K ( t )= K ( s ( t ) ;t )= U ( t; 0 ( t )) t n 1 n 1 : If0 0.Wenowshowthat K + ( t )= K ( s + ( t ) ;t )= U ( t; ( t )) t n 1 n 1 > 0 8 t> 0 : 65 Given t> 0,wehave B ( t; ( t ))=0andhence ( t ) 2 (0 ; ^ ) : Wewillactuallyshowthat U ( t; ) > t n 1 n 1 whenever B ( t; )=0 : (4.33) Notethat,by(4.14), ( n 1) U ( t; )= f ( ) 2 A ( t; )+ tQ ( A ( t; ) ; ) ( ) g ( ) = f ( ) 2 A ( t; )+ t q t 2 n 4 f ( ) 2 ( ) g ( ) : Fromtheof A ( t; ),integrationbypartsyieldsthat A ( t; )= t p t 2 n 4 f 2 g +( n 2) Z t y 2 n 4 dy ( y 2 n 4 f 2 ) 3 = 2 : Assume,atpoint( t; ), B ( t; )=0;so0 << ^ : By(4.24),wehave Z t y 2 n 4 dy ( y 2 n 4 f 2 ) 3 = 2 = a 2 f 0 : Hence A ( t; )= t p t 2 n 4 f 2 g + ( n 2) a 2 f 0 : Therefore, ( n 1) U ( t; )= tf 2 p t 2 n 4 f 2 f 2 g + ( n 2) a 2 f 2 f 0 + t q t 2 n 4 f 2 g = t 2 n 3 p t 2 n 4 f 2 2 n 3 g + ( n 2) a 2 f 2 f 0 >t n 1 2 n 3 g + ( n 2) a 2 f 2 f 0 : 66 Clearly 0 < f 0 f =( n 2) cos sin sin cos < ( n 2) cos sin : Hence ( n 2) a 2 f 2 f 0 > ( n 2) a 2 f sin ( n 2)cos = 2 n 3 g ; fromwhichitfollowsthat( n 1) U ( t; ) >t n 1 : 4.3Constructionneartheouterquarter-circle Inthiscase,againlet I =(0 ;ˇ= 2)but ( )=cos ; ( )=sin ; ( )= n 1 (sin ) n 1 8 2 I: 4.3.1Characteristicsstripconditions Toselectthefunctions f ( )and g ( )tothestripcondition f 2 ( )+ g 2 ( )=(sin ) 2 n 4 ; (4.34) f ( )sin + g ( )cos = 0 ( )= (sin ) n 2 cos ; (4.35) we f ( )=(sin ) n 2 cos ';g ( )=(sin ) n 2 sin '; where ' = ' ( )isafunctionof tobeselectedbelow.Inviewof(4.35),wehave sin( ' )= cos : 67 Thisconditionalonedoesnotdeterminesin ' andcos ' uniquely.Werequirethecharac- teristiccurvegoinsidethedisc s 2 + t 2 < 1forsmall ˝> 0 : Tothisend,let ˆ ( ˝; )= S 2 ( ˝; )+ T 2 ( ˝; ) : Werequirethat dˆ d˝ (0 + ; )=2 ( ) f ( )+2 ( ) g ( )=2(sin ) n 2 cos( ' ) < 0 andsothat cos( ' )= q 1 ( cos ) 2 : Inthisway, f and g areuniquelydeterminedifweset,forall 2 I; sin ' = cos 2 sin p 1 ( cos ) 2 ; cos ' = cos sin cos p 1 ( cos ) 2 : Lemma4.3.1. Itfollowsthat 1 ' 0 ( ) < 2 ;' 00 ( ) 0 8 2 I: Proof. tiatingsin( ' )= cos twice,wehave cos( ' )( ' 0 1)= sin ; sin( ' )( ' 0 1) 2 +cos( ' ) ' 00 = cos = sin( ' ) : Hence ' 0 =1 sin cos( ' ) =1+ sin p 1 ( cos ) 2 ; 68 cos( ' ) ' 00 =sin( ' ) ' 0 ( ' 0 2) : Theequationimplies1 ' 0 < 2since0 ( ) 2 < 1;so,fromthesecondequation,it followsthat ' 00 0,duetotheinequalitiessin( ' ) 0andcos( ' ) < 0 : Notethat f ( ) < 0but g ( )changessignson I .Wesolve g ( )=0toobtain = arctan( ) 2 [0 ;ˇ= 4)suchthat g ( ) > 0 8 2 (0 ; ) ;g ( ) < 0 8 2 ( ;ˇ= 2) : Let t =sin = p 1+( ) 2 ;s =cos = 1 p 1+( ) 2 : Lemma4.3.2. Thereexistsa(unique)number 2 ( ;ˇ= 2) suchthat f 0 ( ) < 0 8 2 (0 ; ); f 0 ( ) > 0 ;f 00 ( ) > 0 8 2 ( ;ˇ= 2) : Proof. Weeasilyhave f 0 ( )=(sin ) n 3 h ( ) ; where h ( )=( n 2)cos cos ' (sin sin ' ) ' 0 : So f 0 ( )=(2 n )( t ) n 2 s < 0and f 0 ( ˇ= 2)=1+ > 0 : Itiseasytocheck,byLemma 4.3.1,that h 0 ( )= ( n 2)(sin cos ' + ' 0 cos sin ' ) ' 0 cos sin ' (sin cos ' )( ' 0 ) 2 (sin sin ' ) ' 00 > 0 8 2 ( ;ˇ= 2); hence h isstrictlyincreasingon( ;ˇ= 2) : Therefore, h and f 0 haveauniquezero 2 69 ( ;ˇ= 2) ; thatis, f 0 ( ) < 0 8 2 [ ; ) ;f 0 ( ) > 0 8 2 ( ;ˇ= 2) : Fromthiswealsoobtainthat f 00 ( ) > 0forall 2 ( ;ˇ= 2).Finally,itiseasytoseethat f 0 ( ) < 0forall 2 (0 ; ] : Thiscompletestheproof. Asabove,let S ( ˝; ) ;T ( ˝; ) ;P ( ˝; ) ;Q ( ˝; )and Z ( ˝; )bethecharacteristicsolutions onthedomain D = f ( ˝; ) j 2 (0 ;ˇ= 2) ; 0 ˝<˝ M ( ) g : Here ˝ M ( )=+ 1 if n =3,andif n> 3, ˝ M ( )isby ˝ M ( )= 8 > > > > > < > > > > > : R + 1 ( ) dy q y 2 n 4 f ( ) 2 8 2 (0 ; ] ; 2 ˝ m ( )+ R + 1 ( ) dy q y 2 n 4 f ( ) 2 8 2 ( ;ˇ= 2) ; where ˝ m ( )= Z ( ) j f ( ) j 1 n 2 dy p y 2 n 4 f ( ) 2 8 2 ( ;ˇ= 2) : 4.3.2Invertingthecharacteristicmap Unlikethecaseoftheinnercircle,wesolve ˝ from s = S ( ˝; )tohave ˝ = C ( s; )= s ( ) f ( ) 8 0 <<ˇ= 2 : 70 Considerthefunction F ( s; )= T ( C ( s; ) ; )(4.36) forall( s; )with s> 0and0 > > < > > > : if n =3, ( )+ f ( ) ˝ M ( )if n> 3. Let l 1 ( )= ( )+ f ( ) ˝ m ( ) 8 2 ( ;ˇ= 2) : Notethat0 3 ,then l (0 + )=1 andthereexistsa(unique)number ^ 2 (0 ;ˇ= 2) suchthat l ( ) > 0 ;l 0 ( ) < 0 8 2 (0 ; ^ ); l ( ) 0 8 2 ( ^ ;ˇ= 2) : 71 (b)Forall n 3 ,itfollowsthat l 1 (( ) + )= s andthereexistsa(unique)number ^ 1 2 ( ;ˇ= 2) suchthat l 1 ( ) > 0 ;l 0 1 ( ) < 0 8 2 ( ; ^ 1 ); l 1 ( ) 0 8 2 ( ^ 1 ;ˇ= 2) : Proof. Let k = j f ( ) j 1 n 2 : Wehave Z + 1 dy p y 2 n 4 f ( ) 2 = k 3 n Z 1 k p 2 n 4 1 8 2 (0 ;ˇ= 2) ; ˝ m ( )= k 3 n Z k 1 p 2 n 4 1 8 2 ( ;ˇ= 2) : 1.Weprovepart(a).Inthiscase, n> 3and l = + f˝ M canbewrittenas l ( )= 8 > > > > > < > > > > > : k R 1 k q 2 n 4 1 8 2 (0 ; ] ; k R 1 k q 2 n 4 1 2 k R k 1 q 2 n 4 1 8 2 ( ;ˇ= 2) : Itthenfollowsthat l (0 + )=1and l (( ˇ= 2) )=0 : Inorderto l 0 ( ),weusetheelementary identities k 0 k = f 0 ( n 2) f ; k 0 = k ' 0 sin ' ( n 2)cos ' (4.37) toobtain l 0 ( )= 8 > > > > > < > > > > > : 0 ' 0 n 2 k 0 R 1 k q 2 n 4 1 ! 8 2 (0 ; ) ; 0 ' 0 n 2 k 0 R 1 k q 2 n 4 1 +2 R k 1 q 2 n 4 1 ! 8 2 ( ;ˇ= 2) : (4.38) 72 Fromthis,weseethat l 0 existsat andalso l 0 (( ˇ= 2) )=+ 1 : Ineithercaseoftheformula (4.38),thetermintheparenthesisequals l k andsowesimplify(4.38)toobtainthat l 0 ( )= 0 ' 0 n 2 f 0 ( n 2) f + f 0 l ( n 2) f = 1 sin 0 ( n 2[ sin + p 1 ( cos ) 2 ] + f 0 l ( n 2) f : (4.39) Since l (( ˇ= 2) )=0,thereexistsa 0 <ˇ= 2closedto ˇ= 2suchthat l ( 0 ) < 0 : Let be determinedinLemma4.3.2.Then,by(4.38)and(4.37),itfollowsthat l 0 ( ) < 0forall 2 (0 ; ],andby(4.39),itfollowsthat l 0 ( ) < 0whenever l ( ) 0and 2 [ ;ˇ= 2) : 2.Weproceedintwocases. Case 1: l ( ) 0 : Inthiscase,since l (0 + )=1 > 0,thereexistsanumber ^ 2 (0 ; ] suchthat l ( ^ )=0 : Weshowthatthis ^ theconclusionofthelemma.Clearly l ( ) > 0 ;l 0 ( ) < 0forall 2 (0 ; ^ ) : Toshow l ( ) 0forall 2 ( ^ ;ˇ= 2) ; supposeotherwise, forsome d 2 ( ^ ;ˇ= 2), l ( d ) > 0 : Thenthemaximumof l on[ ^ ;d ]mustattainatsome c 2 ( ^ ;d ],where l ( c ) > 0and l 0 ( c ) 0 ; andso c 2 ( ;ˇ= 2);thisisacontradictionto(4.39). Case 2: l ( ) > 0 : Inthiscase,thereexistsanumber ^ 2 ( ; 0 )suchthat l ( ^ )=0 : Weshowthatthis ^ theconclusionofthelemma.Weshow l ( ) 0on 2 ( ^ ;ˇ= 2) : Supposeotherwise,forsome d 2 ( ^ ;ˇ= 2), l ( d ) > 0 : Thenthemaximumof l on [ ^ ;d ]ispositiveandattainsatsomepoint c 2 ( ^ ;d )with l ( c ) > 0and l 0 1 ( c )=0;thisisa contradictionto(4.39).Wenowshowthat l ( ) > 0 ;l 0 ( ) < 0forall 2 (0 ; ^ ) : It toshow l 0 ( ) < 0forall 2 (0 ; ^ ) : Supposeotherwise l 0 ( e ) 0forsome e 2 (0 ; ^ ) : Then e 2 ( ; ^ ) : Themaximumof l on[ e; ^ ]mustattainatsome f 2 ( e; ^ ] : Atthispoint f we musthave l ( f ) 0and l 0 ( f ) 0;thisisagainacontradictionto(4.39). 73 3.Toprovethepart(b),notethat,similarto l ( ),wehave l 1 ( )= k Z k 1 p 2 n 4 1 8 2 ( ;ˇ= 2) : Itfollowsthat l 1 (( ) + )= s and l 1 (( ˇ= 2) )=0;moreover, l 0 1 ( )= 0 ' 0 n 2 k 0 Z k 1 p 2 n 4 1 8 2 ( ;ˇ= 2) : (4.40) Wesimplify(4.40)toobtainthat l 0 1 ( )= 0 ' 0 n 2 f 0 ( n 2) f f 0 l 1 ( n 2) f = 1 sin 0 ( n 2)[ sin + p 1 ( cos ) 2 ] f 0 l 1 ( n 2) f : (4.41) So l 0 1 (( ˇ= 2) )=+ 1 : Furthermore,by(4.40)and(4.37),itfollowsthat l 0 1 ( ) < 0forall 2 ( ; ],andby(4.41),itfollowsthat l 0 1 ( ) < 0whenever l 1 ( ) 0and 2 [ ;ˇ= 2) : Therefore,inacompletelyanalogouswaytotheproofofpart(b),wecanprovepart(b). Let ˝ 1 ( )=min f ˝ 0 ( ) ;˝ m ( ) g =min ˆ ( ) f ( ) ;˝ m ( ) ˙ andconsiderthefollowingsubsetsof D 0 : D 1 = f ( ˝; ) j 0 << ; 0 <˝<˝ 0 ( ) g ; D 2 = f ( ˝; ) j <<ˇ= 2 ;˝ 1 ( ) <˝<˝ 0 ( ) g ; D 3 = f ( ˝; ) j <<ˇ= 2 ; 0 <˝<˝ 1 ( ) g : 74 R = f ( t; ) j 2 (0 ;ˇ= 2) ;t j f ( ) j 1 n 2 g and A ( t; )= Z t ( ) dy p y 2 n 4 f ( ) 2 8 ( t; ) 2R : By(4.16),(4.18)and(4.20),itfollowsthat ˝ = A ( T ( ˝; ) ; ) 8 ( ˝; ) 2D 1 ; ˝ = A ( T ( ˝; ) ; )+2 ˝ m ( ) 8 ( ˝; ) 2D 2 ; ˝ = A ( T ( ˝; ) ; ) 8 ( ˝; ) 2D 3 : (4.42) Let S k = f ( s; ) 2Sj ( C ( s; ) ; ) 2D k g , k =1 ; 2 ; 3 ; besubdomainsof S ;namely, S 1 = f ( s; ) j 0 << ; ~ l ( ) > > > > > > > < > > > > > > > > : p F 2 n 4 f 2 ( C A ( F; ))if( s; ) 2S 1 ; p F 2 n 4 f 2 ( C A ( F; ) 2 ˝ 0 m )if( s; ) 2S 2 ; p F 2 n 4 f 2 ( C + A ( F; ))if( s; ) 2S 3 : (4.43) Inorderto F ( s; ),weneedtoderivetheformulafor A ( t; ) : Assume( t; ) 2R with 6 = and t> j f ( ) j 1 n 2 : Let k ( )= j f ( ) j 1 n 2 .Thenfrom A ( t; )= k ( ) 3 n Z t=k ( ) ( ) =k ( ) p 2 n 4 1 8 ( t; ) 2R ; itfollowsthat A ( t; )=(3 n ) k 2 n k 0 Z t=k =k p 2 n 4 1 + k 3 n " ( t=k ) 0 p ( t=k ) 2 n 4 1 ( =k ) 0 p ( =k ) 2 n 4 1 # ; which,by(4.37),to A ( t; )= sgn( g ) ' 0 ( n 2) f ( n 3) f 0 ( n 2) f A tf 0 ( n 2) f p t 2 n 4 f 2 : (4.44) Alsowehave ˝ 0 m ( )= n 3 n 2 f 0 f ˝ m ' 0 ( n 2) f 8 2 ( ;ˇ= 2) : (4.45) 76 Proposition4.3.4. Itfollowsthat F ( s; )= 8 > > > < > > > : p F 2 n 4 f 2 L ( s; )+ f 0 F ( n 2) f if ( s; ) 2S 1 ; 2 ; p F 2 n 4 f 2 L ( s; )+ f 0 F ( n 2) f if ( s; ) 2S 3 ; (4.46) F s ( s; )=( n 2) F 2 n 5 L ( s; ) f ( ) 8 ( s; ) 2S ; (4.47) where L ( s; )= f 1+ ' 0 n 2 f 0 ( n 2) f C ( s; ) : (4.48) Moreover, L ( s; ) < 0 forall s 2 [0 ; 1] and 2 (0 ;ˇ= 2); therefore, F s ( s; ) > 0 forall ( s; ) 2S : Proof. If 6 = ,thentwoformulasin(4.46)followfrom(4.43),(4.44)and(4.45).For = ,theformulafollowsbycontinuity.Formula(4.47)followsfrom(4.46).Weonlyneed toprove L ( s; ) < 0 : Usingtheidentities f 0 =( n 2) f g' 0 and 2 + 2 =1,wecompute that ( n 2) f ( ) 2 L ( s; )=( n 2) f + f' 0 f 0 fC =( n 2) f + f' 0 + f 0 sf 0 =( n 2) f (1 s )+( f g + sg ) ' 0 < 0 forall s 2 [0 ; 1]and 2 (0 ;ˇ= 2),thankstothefactthat f< 0 ;' 0 > 0andtheelementary calculationusing(4.35), f g + sg =(sin ) n 2 ( s sin ' cos ) 77 =(sin ) n 2 [ cos 2 s sin q 1 ( cos ) 2 cos ] < (sin ) n 2 ( cos 2 cos ) < 0 : Let ~ Z ( s; )= Z ( C ( s; ) ; ).Then,afterachangeofvariables, ~ Z ( s; )= ( )+ 1 f ( ) Z s cos F ( y; ) 2 n 4 dy 8 ( s; ) 2S : (4.49) Thisfunctionservesinthesameroleasdoesthefunction U ( t; )usedabove.Forexample, wehavethefollowingresult. Lemma4.3.5. Let ~ Q ( s; )= Q ( C ( s; ) ; ) : Then ~ Z ( s; )= ~ Q ( s; ) F ( s; ) 8 ( s; ) 2S : Proof. Adirectproofbybrutalcalculationsseemstoocomplicatedandgettingnowhere; instead,for ,weconsiderfunction ˆ ( s )= ~ Z ( s; ) ~ Q ( s; ) F ( s; )oninterval s 2 ( l ( ) ; cos )andshowthat ˆ 0 ( s )=0and ˆ ((cos ) )=0 : Thisprovesthat ˆ ( s )=0and theproof.By(4.49),wehave ~ Z ( s; )= 0 f 0 f 2 Z s cos F ( y; ) 2 n 4 dy + 2 n 4 sin f + 1 f Z s cos (2 n 4) F ( y; ) 2 n 5 F ( y; ) dy: 78 From(4.51),intermsoffunction ~ Q ( s; )= Q ( C ( s; ) ; ) ; wehave F ( s; )= ~ Q ( s; ) L ( s; )+ f 0 F ( n 2) f 8 ( s; ) 2S : (4.50) Therefore lim s ! (cos ) ˆ ( s )= 0 + 2 n 4 sin f g g f (1+ ' 0 n 2 )+ f 0 ( n 2) f =0 : Wealsocomputethat ˆ 0 ( s )= F 2 n 4 f 0 f 2 + (2 n 4) F 2 n 5 F f ~ Q s F ~ QF s : Using(4.50)and ~ Q s = Q ˝ C s = ( n 2) F 2 n 5 f ;F s = ( n 2) F 2 n 4 L f ; wehave ˆ 0 ( s )=0 : Thiscompletestheproof. Let 0 ( s )=arccos( s ).ByProposition4.3.3,weseethatdomains S and S 3 canbewritten as S = f ( s; ):0 3)and ~ 1 ( s )aretheinversefunctionsof l ( )(if n> 3)on(0 ; ^ )and l 1 ( )on(0 ; ^ 1 ),respectively.Notethat S hasthepropertythatevery verticalorhorizontalline-segmentbelongsto S whenevertheendpointsbelongto S : 79 Inwhatfollows,foreach s 2 (0 ; 1) ; westudy F ( s; )asafunctionof on interval( ~ ( s ) ; 0 ( s )) : Wehavethefollowingresult. Lemma4.3.6. Itfollowsthat lim ! ( 0 ( s )) F ( s; )= p 1 ( ) 2 p 1 s 2 + s p 1 ( ) 2 8 s 2 (0 ; 1) ; (4.51) lim ! ( ~ ( s )) + F ( s; )= 8 s 2 (0 ; 1) ; (4.52) F ( s; ~ 1 ( s ))= 1 2 n f 0 ( ~ 1 ( s )) j f ( ~ 1 ( s )) j 3 n n 2 8 s 2 (0 ;s ) : (4.53) Proof. 1.Weprove(4.51).Notethat,as ! ( 0 ( s )) ,itfollowsthat C ( s; ) ! 0 ;F ( s; )= T ( C ( s; ) ; ) ! ( 0 ( s ))= p 1 s 2 : If < 0 ( s )andistlyclosedto 0 ( s ),wehave( s; ) 2S 1 ; 3 andhence,by(4.46), lim ! ( 0 ( s )) F ( s; )= g ( )sin f ( ) + g' 0 sin ( n 2) f + sin f 0 ( n 2) f = 0 ( s ) = g ( )sin f ( ) +cos = 0 ( s ) = p 1 ( ) 2 p 1 s 2 + s p 1 ( ) 2 : 2.Toprove(4.52),weassume n> 3.Inthiscase, l ( ~ ( s ))= s andso C ( s; ~ ( s ))= ˝ M ( ~ ( s ));hence, lim ! ( ~ ( s )) + F ( s; )=lim ! ( ~ ( s )) + T ( C ( s; ) ; )=+ 1 : 80 Thus,with L by(4.48),wehave lim ! ( ~ ( s )) + " L ( s; )+ f 0 ( n 2) f F p F 2 n 4 f 2 # = L ( s; ~ ( s )) < 0 : Fromthis,(4.52)followsby(4.46). 3.Nowassume n =3 : Then ~ ( s )=0 : First,assume > 0(so 6 =0).Then,forall 0 << ,wehavethat C ( s; )= A ( F ( s; ) ; )andhence s ( )= k Z F=k =k p 2 1 ; where k = j f ( ) j : Since k ( ) ! 0and ( ) =k ( ) ! = 1 p 1 9 2 > 1as ! 0 + ; fromthe aboveequation,weobtainthat F ( s; ) k ( ) (1 s ) M k ( ) 2 as ! 0 + ,where M> 0isaconstant.Hence F ( s; ) ! + 1 as ! 0 + : Notethat,by (4.46), lim ! 0 + F ( s; )=lim ! 0 + p F 2 f 2 f 2 " f 2 L ( s; )+ f 0 f F p F 2 f 2 # : (4.54) By(4.48),wehave f 2 L ( s; )= f (1+ ' 0 ) ( s ) f 0 : Since f 0 (0)= p 1 9 2 < 0,we have lim ! 0 + " f 2 L ( s; )+ f 0 f F p F 2 f 2 # =(1 s ) f 0 (0) < 0 : Consequently,(4.52)followsfrom(4.54).Nowassume =0(so =0).Inthiscase, 81 ˇ 2 s 0 ^ 1 ^ 0 ( s )=arccos s ~ 1 ( s ) ~ ( s ) ( s ) ^ 1 s ^ s Figure4.3:Thedomain S isbetweenthetwosmoothcurves 0 ( s )and ~ ( s ) ; while S 1 isthe partwith0 << (emptyif =0), S 2 isthepartboundedby ~ and ~ 1 with << ^ 1 , and S 3 isthepartbetween ~ 1 and 0 with ^ 1 <<ˇ= 2 : Thenumber^ s isdeterminedin Lemma4.3.8. C ( s; )= A ( F ( s; ) ; )+2 ˝ m ( ) ; whichgives s ( )= k Z F=k =k p 2 1 2 k Z =k 1 p 2 1 : where k = j f ( ) j : Asabove,westillhavethat F ( s; ) ! + 1 as ! 0 + and,again,that (4.52)followsfrom(4.54). 4.Finally(4.53)isimmediatefrom(4.46).Thiscompletestheproof. Foreach s 2 (0 ; 1),(seeFigure4.3) ( s )=inf f 2 ( ~ ( s ) ; 0 ( s )): F ( s; 0 ) 0 8 0 2 ( ; 0 ( s )) g : Thewof followsfromLemma4.3.6;moreover,forall s 2 (0 ; 1),italso 82 followsthat ~ ( s ) < ( s ) < 0 ( s )and F ( s; ( s ))=0 ;F ( s; ) 0 8 ( s ) < 0 ( s ) : Furthermore,thereexistsasequence 0 i ! ( s ) suchthat F ( s; 0 i ) < 0 ; whichshowsthat F ( s; ( s )) 0 : Lemma4.3.7. Foreach s 2 (0 ; 1) ,function F ( s; ) isone-to-oneontheclosedinterval 2 [ ( s ) ; 0 ( s )) : Moreover,thefunction ( s ) isstrictlydecreasing,right-continuouson (0 ; 1) ,andthat (1 )=0 : Proof. TheproofissimilartothatofLemma4.2.3.Let a;b 2 [ ( s ) ; 0 ( s ))besuchthat F ( s;a )= F ( s;b ) : Weshow a = b: If a 0on S ; wehave F ( s; ( s 0 )) ~ ( s ) : Giveneach 0 2 ( l; 0 ( s )),forall s 0 >s tlyclosed to s ,wehave ( s 0 ) 1 n 1 F ( s; ( s )) n 1 8 ^ s s< 1 : (4.55) 83 Proof. Fromtheof ~ Z ( s; ),by(4.13),wehave ( n 1) ~ Z ( s; )=( n 1) g + C ( s; ) f 2 + F ( s; ) ~ Q ( s; ) : Since F = ~ QL + f 0 F ( n 2) f and F ( s; ( s ))=0,itfollowsthat L ( s; ( s ))= f 0 ( ( s )) F ( s; ( s )) ( n 2) f ( ( s )) ~ Q ( s; ( s )) 8 s 2 (0 ; 1) : Substitutionintotheionof L ( s; )yields C ( s; ( s ))= F ( s; ) ~ Q ( s; ) + ( n 2) ' 0 ) f 0 = ( s ) : Simplifying,weobtainthat ( n 1) ~ Z ( s; ( s ))= ( n 1) g + ( n 2+ ' 0 ) f 2 f 0 = ( s ) + F ( s; ( s )) 2 n 3 ~ Q ( s; ( s )) : (4.56) First,let s 0 2 (0 ; 1)suchthat ( s ) 2 (0 ; )forall s 2 [ s 0 ; 1) ; where 2 ( ;ˇ= 2)is determinedinLemma4.3.2.Hence f 0 ( ( s )) < 0andthus0 < ~ Q ( s; ( s )) F ( s; ( s )) n 1 8 s 0 s< 1 : 84 Wenowshowthatthereexistsanumber 0 2 (0 ; )suchthat ( n 1) g + ( n 2+ ' 0 ) f 2 f 0 > 0 8 0 << 0 : Thisisprovedbycomputingthat ( n 1) g + ( n 2+ ' 0 ) f 2 f 0 =(sin ) n 1 2 4 1 sin ' + ( n 2+ ' 0 )sin cos ' n 2 ' 0 sin sin ' cos cos ' cos 3 5 ; andnoticingthat lim ! 0 + 2 4 1 sin ' + ( n 2+ ' 0 )sin cos ' n 2 ' 0 sin sin ' cos cos ' cos 3 5 =1 > 0 : Finally,let^ s 2 [ s 0 ; 1)besuchthat ( s ) 2 (0 ; 0 )forall s 2 [^ s; 1) : Then,forthis^ s ,(4.55) followsfrom(4.56). Lemma4.3.9. Let t ( s )= F ( s; ( s )) forall s 2 (0 ; 1) : Thenthefunction t ( s ) isright- continuouson (0 ; 1) with t (1 )=0 and 0 0tlysmalland <ˇ= 2tlycloseto ˇ= 2,wehave( s; ) 2S 3 andhence s cos f ( ) = Z sin F ( ) dy p y 2 n 4 f ( ) 2 : Let s ! 0 + andwehave cos f ( ) = Z sin 1 dy p y 2 n 4 f ( ) 2 : forall <ˇ= 2tlycloseto ˇ= 2;thisisimpossibleasseenbytakingthelimitsas ! ˇ= 2 : Therefore0 0.Hence, from F ( s; ( s ))=0,by(4.46),itfollowsthat ( n 2) f ( ( s )) L ( s; ( s ))= f 0 ( ( s )) t ( s ) p t ( s ) 2 n 4 f ( ( s )) 2 : (4.57) Notethat fL = (1+ ' 0 n 2 ) f 0 C n 2 ;C = Z F dy p y 2 n 4 f 2 : So(4.57)implies [ ( n 2+ ' 0 )] = ( s ) = " f 0 t ( s ) p t ( s ) 2 n 4 f 2 + Z t ( s ) f 0 dy p y 2 n 4 f 2 # = ( s ) Takinglimitas s ! 0 + ,since (0 + )= ˇ= 2and f 0 (( ˇ= 2) )= ' 0 (( ˇ= 2) )=1+ we 86 s t 0 Z 2 t ( s ) ^ t t (0 + ) t 1 ( s ) 1 ^ s 1 Figure4.4:Thefunction t ( s )= F ( s; ( s ))andthedomain Z 2 : Thenumber ^ t isdetermined inLemma4.3.11andthesmoothfunction t 1 ( s )isdeterminedinLemma4.3.13. havethat^ y = t (0 + ) n 2+(1+ )=(1+ ) ^ y 3 n + Z 1 ^ y y 2 n dy : Fromthis,wesolvefor^ y toobtain t (0 + )= 8 > > > < > > > : e 1 1+3 if n =3, 1+ n 2+ 1 n 3 if n> 3. (4.58) Therefore0 0and ^ t t< 1andhence,forallsuch( s;t ), ~ Q ( s;˘ ( s;t ))= q t 2 n 4 f ( ˘ ( s;t )) 2 ; fromwhichthesecondlimitfollows. 2.Wenowprovethatthereexistsanumber ^ t suchthatthelimitholds.Let ˘ ( s i ;t i ) ! 0 alongasequence( s i ;t i ) ! (0 ;t 0 )in Z 2 : Since ( s ) <˘ ( s;t ) < 0 ( s )forall ( s;t ) 2Z 2 ,itfollowsthat 0 < (0 + ) 0 ˇ= 2 : If ^ = (0 + )= ˇ= 2(the Case (b)intheproofofLemma4.3.9),then 0 = ˇ= 2and,inthis case,thenumber ^ t canbechosentobe ^ t = t (0 + ) : 3.Nowassume ^ <ˇ= 2 : First,let 1 2 ( ^ ;ˇ= 2)besuchthat j f ( ) j 2 < 1 = 2forall 2 [ 1 ;ˇ= 2)andsuchthat( s; ) 2S 3 forall0 0 isasmallnumber.Weclaimthatthereexistsanumber0 < 2 < 1suchthat 8 ^ <<ˇ= 2 ;F (0 + ; ) 2 ) 1 : 89 Ifnot,thenthereexistnumbers i 2 ( ^ ;ˇ= 2)suchthat F (0 + ; i ) 1 1 i ; i < 1 8 i =1 ; 2 ; : Assume i ! 3 1 as i !1 : Then F (0 + ; )=1forall 2 ( 3 ;ˇ= 2) : Thisisimpossible, asprovedinthe Case (a)oftheproofofLemma4.3.9.Let0 < 4 < 1beanumbersuch that 2 n 4 4 > 2 = 3 ; p 6(1 4 ) < inf 2 [ ^ ;ˇ= 2) cos f ( ) : (4.59) Finally,let ^ t =max f t (0 + ) ; 2 ; 4 g : Then t (0 + ) ^ t< 1.Weclaimthat 0 = ˇ= 2forall ^ t t 0 < 1 : Suppose,forthecontrary,that ^ 0 <ˇ= 2 : Then t 0 = F (0 + ; 0 ) 2 ;so 0 1 : Hence j f ( 0 ) j 2 < 1 = 2and( s; 0 ) 2S 3 foralltlysmall s> 0 : So s cos 0 f ( 0 ) = Z sin 0 F ( 0 ) dy p y 2 n 4 f ( 0 ) 2 : Let s ! 0 + and,by(4.59),wearriveatadesiredcontradiction: cos 0 f ( 0 ) = Z sin 0 t 0 dy p y 2 n 4 f ( 0 ) 2 < Z 1 t 0 dy p y 2 n 4 f ( 0 ) 2 p 6(1 4 ) < inf 2 [ ^ ;ˇ= 2) cos f ( ) : 4.3.3Constructionofthesolutionon Z 2 Let ˘ ( s;t )bethefunctionon Z 2 aboveand ~ Z ( s; )beby(4.49). 90 Theorem4.3.12. 2 ( s;t )= ~ Z ( s;˘ ( s;t )) 8 ( s;t ) 2Z 2 : Then 2 isentiablein Z 2 ,with 2 s ( s;t ) < 0 ,andtheEikonalequation jr 2 ( s;t ) j = t n 2 ateverypoint ( s;t ) 2Z 2 : Moreover, 2 ( s;t ) canbecontinuouslyextendedtothecurve s = p 1 t 2 ; 0 > > < > > > : n 1 t n 1 on s = p 1 t 2 ; 0 t n 1 n 1 8 t ( s ) 0on s 2 (^ s; 1) : There- fore,thereexistsaunique t 1 ( s ) 2 ( t ( s ) ;t 0 ( s ))foreach s 2 (^ s; 1)suchthat K ( s;t 1 ( s ))=0 : Thisfunction t 1 ( s )therequirementofthelemma.Moreover,since K t ( s;t 1 ( s )) > 0, bytheimplicitfunctiontheorem,thefunction t 1 ( s )iserentiablein s 2 (^ s; 1) : ti- ating K ( s;t 1 ( s ))=0yieldsthat 2 s + 2 t t 0 1 = t n 2 1 t 0 1 andhence t 0 1 ( s )= 2 s t n 2 1 2 t = f ( ˘ ( s;t 1 )) t n 2 1 ~ Q ( s;t 1 ) < 0 8 ^ s > > < > > > : p A 2 s 2 (0 s B 1 ) ; t 1 ( s )( B 1 > > < > > > : p A 2 t 2 ( A 2 t A ) ; s 1 ( t )(0 a 1 and n 1 t n 1 < 2 ( t )forall a 1 0for a 1 0 forall a 1 0andhence r 2 ( t )isstrictlyincreasingin a 1 r 2 ( a 1 )= b 1 ; acontradiction.Therefore, a 2 < q 1 b 2 1 : Furthermore, bytheof a 2 ,wehave r 2 ( a 2 )= q 1 a 2 2 : 2.Weconstruct a m +1 inductivelyfor m 2 : Suppose,forsome m 2,wehave thenumbers a 2 < 0 and h ( q 1 b 2 2 q )= 2 ( b 2 q ; q 1 b 2 2 q ) 2 q +1 ( q 1 b 2 2 q ) < 0 : 99 Therefore,thereexistsauniquenumber a 2 q +1 with a 2 q 0in a 2 q a 2 q 1 and n 1 t n 1 < 2 q ( t ) 8 a 2 q 1 0andhence h ( t ) > 0for a 2 q 1 0and hence r 2 q ( t )isstrictlyincreasingintheinterval.If a 2 q q 1 b 2 2 q 1 ; then,letting t ! q 1 b 2 2 q 1 in(4.71),wewouldobtainthat b 2 q 1 = r 2 q ( q 1 b 2 2 q 1 ) >r 2 q ( a 2 q 1 )= b 2 q 1 ; acontradiction.Therefore, a 2 q < q 1 b 2 2 q 1 < 1 : 3.Finallywecompletedtheinductionprocessandthustheproof. Lemma4.4.2. Let f a i g bethesequenceconstructedinTheorem4.4.1.Thenthereexistsan integer k 2 suchthat a k 1 0suchthat 2 s ( s;t ) 0 8 ( s;t ) 2 : (4.72) Let h ( t )bethefunctionby(4.69).Then,forsome c 2 ( b 2 q ; q 1 a 2 2 q )andhence( c;a 2 q ) 2 wehavethat h ( a 2 q )= 2 ( b 2 q ;a 2 q ) 2 ( q 1 a 2 2 q ;a 2 q )= 2 s ( c;a 2 q )( b 2 q q 1 a 2 2 q ) =( 2 s ( c;a 2 q ))( q 1 a 2 2 q q A 2 0 a 2 2 q ) 0 (1 A 0 ) : Since h ( a 2 q +1 )=0,thereexistsanumber d 2 ( a 2 q ;a 2 q +1 )suchthat h ( a 2 q )= h 0 ( d )( a 2 q a 2 q +1 ) : Therefore a 2 q +1 = a 2 q h ( a 2 q ) h 0 ( d ) = a 2 q + h ( a 2 q ) h 0 ( d ) a 2 q + 0 (1 A 0 ) 2 ; since h 0 ( d )= j h 0 ( d ) jj 2 t ( b 2 q ;d ) j + d n 2 2 d n 2 < 2 : Hence a 2 q +1 a 2 q 0 (1 A 0 ) 2 for all q =1 ; 2 ; ; whichyields A>a 2 q +1 a 2 > q X j =1 ( a 2 j +1 a 2 j ) 0 (1 A 0 ) 2 q 8 q =1 ; 2 ; ; acontradiction. Let k betheintegerdeterminedinLemma4.4.2,with a k 1 > > > > > > > < > > > > > > > > : (i)Condition(4.63)holds, (ii) a 2 ( s;t ) 8 ( s;t ) 2Z A \Z 1 : (4.73) Forsuchanumber A ,let A and r A bethefunctionsdeterminedinCorollary4.4.3above. Notethat a< a 1 (0 ;A ) 2 (0 ;A ) > 0 : Hencethereexistsauniquenumber a 2 ( a;A )suchthat h ( a )=0 : Forthis a wehave h ( t ) 0forall a t a .Furthermore,if a A 2 then 1 (~ s 1 ( t ) ;t )= 1 n 1 t n 1 A ( t ) forall0 A 2 ; thenby(4.73)(iii), 1 (~ s 1 ( t ) ;t ) > 2 (~ s 1 ( t ) ;t ) 2 ( r A ( t ) ;t )= A ( t )forall A 2 t a : Thereforewehave provedthat 1 ( s ( t ) ;t ) A ( t ) 1 (~ s 1 ( t ) ;t ) 8 0 2 ( r A ( t 1 ) ;t 1 )= A ( t 1 ) ; a contradiction. 106 thefunctions ~ l A ( t )= 8 > > > < > > > : 0( a > > < > > > : 0(~ a K +1 > > > > > > > < > > > > > > > > : 1 ( s;t )( s;t ) 2Z l ; A ( t )( s;t ) 2Z 0 ; 2 ( s;t )( s;t ) 2Z r isaLipschitzsolutiontotheproblem (4 : 6) . 4.4.4Theproofinthecase =1 a n 1 Inthiscase,let A 0 beanumbersuchthat max f A 0 2 ; ^ t g 2 ( s;t ) 8 A 00 t< 1 ;c ( t ) 0 : Furthermore, h (0)= t n 1 a n 1 n 1 1+ t n 1 n 1 = 2 t n 1 2 n 1 < 0 8 A 0 t< 1 ; andbyLemma4.2.6, h ( p 1 t 2 ) > 1 a n 1 n 1 t n 1 n 1 t n 1 =0 : Therefore,thereexista function s = c ( t )on t 2 [ A 0 ; 1)with0 0 ; bytheimplicitfunctiontheorem, s = c ( t )isalsotiablein( A 0 ; 1) : tiating 1 ( c ( t ) ;t )= 2 ( c ( t ) ;t )yieldsthat c 0 ( t )= 2 t ( c ( t ) ;t ) 1 t ( c ( t ) ;t ) 1 s ( c ( t ) ;t ) 2 s ( c ( t ) ;t ) : Notethat 1 s 2 s > 0and 1 t > 0.Clearly c (1 )=0andsothereexists A 00 2 ( A 0 ; 1)such that 2 t < 0near s =0and t =1.Hence, c 0 ( t ) < 0forany t 2 ( A 00 ; 1) : 108 s t 0 c ( t ) r A ( t ) Z r Z l Z 0 s t a K 1 1 a t 1 a a 1 A A 000 A 00 ~ a K +1 l A ( t ) Figure4.8: (Thecase =1 a n 1 ). Thecurve s = c ( t )on[ A 00 ; 1]determinedinLemma 4.4.7,thecurve s = l A ( t )determinedinLemma4.4.8,andthecurve s = r A ( t )intersectat t = a . ByLemma4.2.6,wenowselect A 000 2 ( A 00 ; 1)suchthat 1 ( s;t ) > 2 ( s;t ) 8 ( s;t ) 2Z A 000 \Z 1 \f 0 > > < > > > : (i)Condition(4.63)holds, (ii) A 000 > > > > > > > < > > > > > > > > : c ( t )( A 00 t 2 ( r A ( t ) ;t )= A ( t ).Consequently,foreach t 2 (0 ;a ), thereexistsanumber s = l A ( t )with s ( t ) > > < > > > : c ( t )( a > > < > > > : c ( t )( a > > > > > > > < > > > > > > > > : 1 ( s;t )( s;t ) 2Z l ; A ( t )( s;t ) 2Z 0 ; 2 ( s;t )( s;t ) 2Z r isaLipschitzsolutiontotheproblem (4 : 6) . 111 BIBLIOGRAPHY 112 BIBLIOGRAPHY [1] R.A.AdamsandJ.Fournier,\SobolevSpaces,"SecondEdition.AcademicPress,New York,2003. [2] F.AlougesandA.Soyeur,OnglobalweaksolutionsforLandau-Lifshitzequations: ExistenceandNonuniqueness, NonlinearAnalysis,TMA , 18 (11),(1992),1071{1084. [3] M.Bertsch,P.Podio-GuidugliandV.Valente,Onthedynamicsofdeformableferro- magnets.I.Globalweaksolutionsforsoftferromagnetsatrest, Ann.Mat.PuraAppl., 179 (4)(2001),331{360. [4] J.M.Ball,A.TaheriandM.Winter,Localminimizersinmicromagneticsandrelated problems, Calc.Var., 14 (2002),1{27. [5] S.Bandyopadhyay,A.Barroso,B.DacorognaandJ.Matias,tialinclusions fortialforms, Calc.Var., 28 (2007),449{469. [6] V.BarbuandTh.Precupanu,\ConvexityandOptimizationinBanachSpaces,"Second Edition,D.ReidelPublishingCo.,1986. [7] F.Brailsford,\PhysicalPrinciplesofMagnetism."VanNostrand,London,1966. [8] R.Bruck,Jr.,AsymptoticconvergenceofnonlinearcontractionsemigroupsinHilbert space, J.Funct.Anal., 18 (1)(1975),15{26. [9] J.B.Baillon,Unexempleconcernantlecomportementasymptotiquedelasolutiondu probleme, JournalofFunctionalAnalysis, 28 (1978),369{376. [10] H.Brezis,\Operateursmaximauxmonotonesetsemi-groupesdecontractionsdansles espacesdeHilbert."North-HollandMathematicsStudies,No.5.NotasdeMatematica (50).North-HollandPublishingCo.,Amsterdam-London;AmericanElsevierPublish- ingCo.,Inc.,NewYork,1973. [11] W.F.Brown,Jr.,\Micromagnetics."Interscience,NewYork,1963. [12] P.Celada,G.CupiniandM.Guidorzi,Asharpattainmentresultfornonconvexvari- ationalproblems, Calc.Var., 20 (2004),301{328. 113 [13] R.ChoksiandR.V.Kohn,Boundsonthemicromagneticenergyofaunixialferro- magnet. Commun.PureAppl.Math. 51 (1998),259{289. [14] I.CimrakandR.V.Keer,Higherorderregularityresultsin3DfortheLandau-Lifshitz equationwithanexchange NonlinearAnalysis, 68 (2008),1316{1331. [15] G.D.Chaves-O'Flynn,A.D.KentandD.L.Stein,Micromagneticstudyofmagneti- zationreversalinferromagneticnanorings, PhysicalReview B79 (2009),184421-1{13. [16] G.CarbouandP.Fabrie,Timeaverageinmicromagnetics, Journalofential Equations, 147 (1998),383{409. [17] G.CarbouandP.Fabrie,RegularsolutionsforLandau-Lifshitzequationinabounded domain, entialIntegralEquations, 14 (2)(2001),213{229. [18] B.DacorognaandI.Fonseca,A-Bquasiconvexityandimplicitpartialtialequa- tions, Calc.Var., 14 (2)(2002),115{149. [19] S.DingandB.Guo,Initial-boundaryvalueproblemforhigher-dimensionalLandau- Lifshitzsystems, Appl.Anal., 83 (2003),673{697. [20] B.DacorognaandP.Marcellini,GeneralexistencetheoremsforHamilton-Jacobiequa- tionsinthescalarandvectorialcases, ActaMath., 178 (1)(1997),1{37. [21] B.DacorognaandP.Marcellini,\ImplicitPartialtialEquations."Birkhauser, Boston,1999. [22] A.DeSimone,Energyminimizersforlargeferromagneticbodies, ArchRationalMech. Anal., 125 (1993),99-143. [23] A.DeSimone,R.V.Kohn,S.MullerandF.Otto,Areducedtheoryfor micromagnetics, Comm.PureandAppl.Math., LV (2002),1408{1460. [24] A.DeSimone,R.V.Kohn,S.MullerandF.Otto,Recentanalyticdevelopmentsin micromagnetics,in ScienceofHysteresis, G.BertottiandI.Magyergyoz,Eds,(2005), Elsevier,269{381. [25] A.DeSimone,R.V.Kohn,S.Muller,F.OttoandR.Scafer,Two-dimensionalmod- ellingofsoftferromagnetic Proc.R.Soc.Lond.A, 457 (2001),2983{2991. 114 [26] W.DengandB.Yan,Quasi-stationarylimitandadegenerateLandau-Lifshitzequation offerromagnetism, Appl.Math.Res.Express.AMRX 2 (2013),277{296. [27] W.DengandB.Yan,OnLandau-Lifshitzequationsofno-exchangeenergymodelsin ferromagnetics, Evol.Equ.ControlTheory2 4 (2013)599{620. [28] L.C.Evans,\PartialtialEquations."Amer.Math.Soc.,Providence,1998. [29] I.FonsecaandS.Muller, A -quasiconvexity,lowersemicontinuityandYoungmeasures, SIAMJ.Math.Anal., 30 (1999),1355{1390. [30] B.GuoandM.Hong,TheLandau-Lifshitzequationoftheferromagneticspinchain andharmonicmaps, Calc.Var., 1 (1993),311{334. [31] G.GioiaandR.D.James,Micromagneticsofverythin Proc.R.Soc.Lond.A, 453 (1997),213{223. [32] S.Gustafson,K.NakanishiandT.P.Tsai,Asymptoticstability,concentration,and oscillationinharmonicmapw.Landau-Lifshitz,andSchrodingermapson R 2 , Commun.Math.Phys., 300 (2010),205{242. [33] R.D.JamesandD.Kinderlehrer,Frustrationinferromagneticmaterials, Cont.Mech. Thermodyn., 2 (1990),215{239. [34] O.Kellogg,\FoundationsofPotentialTheory."Ungar,NewYork,1970. [35] M.KruzandA.Prohl,Recentdevelopmentsinthemodeling,analysis,andnumerics offerromagnetism, SIAMReview, 48 (3)(2006),439{483. [36] L.Landau,E.LifshitzandL.Pitaevskii,\ElectrodynamicsofContinuousMedia," PergamonPress,NewYork,1984. [37] M.LuskinandL.Ma,Analysisoftheeelementapproximationofmicrostructure inmicromagnetics, SIAMJ.Number.Anal., 29 (2)(1992),320{331. [38] C.Melcher,Adualapproachtoregularityinthinmicromagnetics, Calc.Var. PDE, 29 (1)(2007),85{98. [39] S.MullerandM.Sychev,Optimalexistencetheoremsfornonhomogeneoustial inclusions, J.Funct.Anal., 181 (2)(2001),447{475. 115 [40] P.Pedregal,Relaxationinferromagnetism:therigidcase, J.NonlinearScience, 4 (1994),105{125. [41] P.Pedregal,\ParametrizedMeasureandVariationalPrinciples."Birkhauser,Basel, 1997. [42] P.Pedregal,Relaxationinmagnetostriction, Calc.Var., 10 (2000),1{19. [43] P.PedregalandB.Yan,Ontwo-dimensionalferromagnetism, Proc.R.Soc.Edinburgh, 139A (2009),575{594. [44] P.PedregalandB.Yan,ADualityMethodForMicromagnetics, SIAMJ.Math.Anal., 6 (2009),2431{2452. [45] R.T.Rockafellar,\ConvexAnalysis."PrincetonUniversityPress,Princeton,1972. [46] T.RoubandM.KruzMesoscopicmodelforferromagnetswithisotropichard- ening, Z.angewMath.Phys., 56 (2005),107{135. [47] E.Stein,\HarmonicAnalysis,"PrincetonUniversityPress,Princeton,1993. [48] L.Tartar,BeyondYoungmeasures, Meccanica, 30 (1995),505{526. [49] R.Temam,\Navier-StokesEquations:TheoryandNumericalAnalysis,"RevisedEdi- tion.ElsevierSciencePublishers,Amsterdam,1984. [50] A.Visintin,OnLandau-Lifshitzequationforferromagnetism, JapanJ.Appl.Math., 2 (1)(1985),69{84. [51] B.Yan,Minimizing L 1 -normfunctionalondivergence-free Ann.Inst.H. Poincare,AnalyseNonlineaire, 28 (2011),325{355. [52] B.Yan,Characterizationofenergyminimizersinmicromagnetics, J.Math.Anal.Appl., 374 (2011),230{243. [53] B.Yan,Ontheequilibriumsetofmagnetostaticenergybytialinclusion, Calc. Var.PartialentialEquations, 47 (2013),547{565. [54] B.Yan,OnstabilityandasymptoticbehaviorsforadegenerateLandau-Lifshitzequa- tion, Proc.R.Soc.Edinburgh, 145A (2015),657{668. 116