Analysis on a class of carnot groups of heisenberg type

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Analysis on a class of carnot groups of heisenberg type

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Title:
Analysis on a class of carnot groups of heisenberg type
Creator:
McNamee, Meagan
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[Tampa, Fla.]
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University of South Florida
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English

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Subjects / Keywords:
Viscosity solutions
Partial differential equations
Sub-riemannian geometry
Taylor polynomial
Lie group
Dissertations, Academic -- Mathematics -- Masters -- USF ( lcsh )
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government publication (state, provincial, terriorial, dependent) ( marcgt )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Abstract:
ABSTRACT: In this thesis, we examine key geometric properties of a class of Carnot groups of Heisenberg type. After first computing the geodesics, we consider some partial differential equations in such groups and discuss viscosity solutions to these equations.
Thesis:
Thesis (M.A.)--University of South Florida, 2005.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
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Title from PDF of title page.
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Document formatted into pages; contains 40 pages.
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by Meagan McNamee.

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001670357 ( ALEPH )
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E14-SFE0001221 ( USFLDC DOI )
e14.1221 ( USFLDC Handle )

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Analysis on a class of carnot groups of heisenberg type
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ABSTRACT: In this thesis, we examine key geometric properties of a class of Carnot groups of Heisenberg type. After first computing the geodesics, we consider some partial differential equations in such groups and discuss viscosity solutions to these equations.
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AnalysisonaClassofCarnotGroupsofHeisenbergTypebyMeaganMcNameeAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofArtsDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:ThomasBieske,Ph.D.YanWu,Ph.D.ManougManougian,Ph.D.DateofApproval:July14,2005Keywords:ViscositySolutions,PartialDierentialEquations,Sub-RiemannianGeometry,TaylorPolynomial,LieGroupcCopyright2005,MeaganMcNamee

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TableofContentsAbstractii1BackgroundandMotivation12CarnotGroups62.1Geometry.................................62.2Calculus..................................83GroupsofHeisenbergType103.1Denition.................................103.2AClassofGroupsofHeisenbergType.................113.2.1Geodesics.............................123.2.2TaylorPolynomials........................204ViscositySolutions274.1AClassofPartialDierentialEquations................274.2Jets....................................28References37i

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AnalysisonaClassofCarnotGroupsofHeisenbergTypeMeaganMcNameeABSTRACTInthisthesis,weexaminekeygeometricpropertiesofaclassofCarnotgroupsofHeisenbergtype.Afterrstcomputingthegeodesics,weconsidersomepartialdierentialequationsinsuchgroupsanddiscussviscositysolutionstotheseequations.ii

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1BackgroundandMotivationWeconsiderRnendowedwiththeEuclideannormkkdenedforx=x1;x2;:::;xn2Rnbykxk=x21+x22++x2n1 2andwiththevectoreldsf@ @x1;@ @x2;:::;@ @xng:Notethatthesevectoreldsarealsothestandarddirectionalderivatives.IffisasucientlysmoothfunctiononRn,thenthegradientoffisgivenbyrf=@f @x1;@f @x2;:::;@f @xnandthesecondorderderivativematrix,denotedD2f,hasij-thentryD2fij=@2f @xi@xj:Notethatthismatrixissymmetric,sincemixedpartialsareequal.Sincefissucientlysmooth,itsTaylorPolynomialatapointp0=x01;x02;:::;x0nisgivenbyfp=fp0+hrfp0;p)]TJ/F27 11.955 Tf 11.955 0 Td[(p0i+1 2hD2fp0p)]TJ/F27 11.955 Tf 11.955 0 Td[(p0;p)]TJ/F27 11.955 Tf 11.955 0 Td[(p0i+okp)]TJ/F27 11.955 Tf 11.955 0 Td[(p0k2wherepisnearp0andh;iistheEuclideaninnerproductrelatedtothenormkk.Ifafunctionisnotsucientlysmooth,thentheTaylorpolynomialdoesnotexist.1

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Thispossiblelackofexistencemotivatesustodenesecondordersuperjets,denotedJ2;+,andsecondordersubjets,denotedJ2;)]TJ/F19 11.955 Tf 10.986 -4.338 Td[(byconsideringthefollowinginequality:fpfp0+h;p)]TJ/F27 11.955 Tf 11.955 0 Td[(p0i+1 2hXp)]TJ/F27 11.955 Tf 11.955 0 Td[(p0;p)]TJ/F27 11.955 Tf 11.955 0 Td[(p0i+okp)]TJ/F27 11.955 Tf 11.955 0 Td[(p0k2.0.1foravector2RnandX2Sn;whereSnisthesymmetricnnmatrices.Notethatifafunctionissucientlysmooth,thevectorcanbereplacedbyrfp0andthesymmetricmatrixXcanbereplacedbyD2fp0.Iffisnotsucientlysmooth,thevectorandthesymmetricmatrixXplaytheroleofageneralizedderivative,actingasasubstituteforthederivativesthatdonotexist.Itshouldbenotedthatthereisnoguaranteethatapair;Xdoesindeedexist.Thesecondordersuperjetandsubjetoffatapointp0aredenedby;X2J2;+fp0,.0.1holds.J2;)]TJ/F27 11.955 Tf 7.085 -4.936 Td[(fp0=)]TJ/F27 11.955 Tf 9.299 0 Td[(J2;+)]TJ/F27 11.955 Tf 9.299 0 Td[(fp0:Asmentionedabove,thesejetsmaybeempty.Wedenotetheset-theoreticclosureofthesuperjetby J2;+.Thatis,;X2 J2;+upifthereisasequencefpn;n;Xngsothatn;Xn2J2;+upnandfpn;n;Xngn!1)167(!p;;X:Usingthesejets,wecandeneviscositysolutionsforaclassofnon-linearpartialdierentialequations.WeletSnbethesetofnnsymmetricmatrices.GivenacontinuousfunctionF:RnRRnSn7!R;werequirethatFp;r;;XFp;s;;YwhenrsandYX.Thatis,Fisproperinthesenseof[3].Wethenformthe2

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classofnon-linearpartialdierentialequationsFp;fp;rfp;D2fp=0:ExamplesofsuchequationsincludetheinniteLaplacian,denoted1,whichisdenedby1fp=hD2fprfp;rfpi:WemayalsoconsidertheP-Laplacian,whichisdenedfor2P<1andgivenbyPf=)]TJ/F32 11.955 Tf 9.298 16.856 Td[(krfpk2trD2fp)]TJ/F19 11.955 Tf 11.955 0 Td[(P)]TJ/F19 11.955 Tf 11.955 0 Td[(21fp:Wethendeneviscositysolutionsasfollows:Denition1.0.1Acontinuousfunctionfisaviscositysubsolution ofFp;fp;rfp;D2fp=0ifforall;X2 J2;+fpwehaveFp;fp;;X0Acontinuousfunctionfisaviscositysupersolution ofFp;fp;rfp;D2fp=0ifforall;Y2 J2;)]TJ/F27 11.955 Tf 7.085 -7.406 Td[(fpwehaveFp;fp;;Y0Thecontinuousfunctionfisaviscositysolution ifitisbothaviscositysubsolutionandaviscositysupersolution.WeshallfocusonaspecialsubclassofsuchfunctionsF.WecallthesubclassF3

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thosefunctionsFforwhichthereisauniversal>0sothatforeachrealrandswithr0isarealconstant.Inthiscase,=c.ConcerningoursubclassF,thefollowingtheoremisknowninRn.Theorem1.0.2CILLetF2FandbeaboundedopensetinRn.ThenifuisaviscositysubsolutiontoFp;up;rup;D2up=0andvisaviscositysupersolutiontoFp;vp;rvp;D2vp=0withuvon@.Thenuvin.Inthisthesis,welooktoextendthistheoremtoaclassofCarnotgroups,which4

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arenaturalgeneralizationsofRn.Asdiscussedbelow,RnisaspecicCarnotgroup,buttheclassweconsiderpossessesarichgeometrythatismuchdierentfromRn.5

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2CarnotGroups2.1GeometryALieGroupGisareal,connected,manifoldwithanon-abeliangroupstructuresuchthatthemapxy)]TJ/F25 7.97 Tf 6.586 0 Td[(1issmooth.TheLieAlgebragofaLieGroupGisthetangentspaceattheorigine.ItcanbeendowedwithaLiebracket,whichisanantisymmetricbilinearform[;]:gg7!gobeyingtheJacobiidentity[[X;Y];Z]+[[Y;Z];X]+[[Z;X];Y]=0:TheLieAlgebraisidentiedwiththeleftinvariantunderthegroupstructurevectoreldsonGand[X;Y]isthentheLiebracketofsuchvectoreldsXandY.Underanappropriatechangeofvariables,wemayidentifythepointsintheLieAlgebrawiththepointsintheLieGroup.Thatis,iffXigni=1isabasisoftheLieAlgebra,wehavenXi=1xiXi,x1;x2;x3;:::;xn:Thisidenticationhastwomainpurposes.First,thepracticaleectofidentifyingGwithgisthatitmimicstheimportantpropertythatRnisboththemanifoldandtangentspace.Secondly,itallowsustowritethegroupmultiplicationlawinaworkableform.Namely,pq=p+q+1 2[p;q]+Hp;q2.1.1whereHp;qconsistsofLiebracketsoforder2andhigher.NotethatHp;qneed6

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notbenon-zero.Infact,aLieAlgebragisnilpotentifH;consistsofonlyanitenumberofnon-zerobracketsandaLieGroupGisnilpotentifitscorrespondingLieAlgebraisnilpotent.WearenowreadytodeneaCarnotgroup.Denition2.1.1AnilpotentLieGroupGwithaLieAlgebragthatsatisesg=V1V2Vlwith[V1;Vj]=V1+jiscalledaCarnotgroup ,orhomogeneousgroup .NotethatthisrequiresV1togenerategasaLieAlgebra,sinceV2=[V1;V1];V3=[V1;V2]=[[V1;V1];V1]andsoon.Itshouldbenotedthatfromlinearalgebra,wecanndabasisofVi,denotedfXijgdij=1,wheredi=dimVi,sothattheidenticationofGwithgholds.TheCarnotgroupGhastwonaturaldimensionstomeasureitssize.Thetopo-logicaldimensionissimplythenumberofelementsinthebasisofg,namely,D=lXi=1dimVi=lXi=1diXj=1Xij:ThisdimensiondoesnotreectthestraticationinthedenitionofCarnotgroups.AmorenaturalchoicewouldbeonethattakesintoaccountthefactthatV1generatesgasaLieAlgebra.Thismotivatestheconceptofthehomogenousdimension,denotedQ,andgivenbyQ=lXi=1idi:NotethatthisisnosmallerthanthetopologicaldimensionofGand,ingeneral,ismuchlarger.EveryLieAlgebraofaCarnotgrouphasatleastonenon-isotropiclineardilation,denotedsfors>0;thathasthepropertysVi=siVi:7

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UsingtheaboveidenticationofGwithg,thatis,lXi=1diXj=1xijXij2g$x11;x12;:::;xldl2G;thedilationonginducesadilationonG,alsodenoteds,denedbysp=sx11;sx12;:::;sx1d1;s2x21;s2x22;:::;slxldl.1.2Additionally,wemaychooseaninvariantRiemannianinnerproductdenotedbyh;i,anditsassociatednormkkonthevectorspaceg.Acurve:R7!Giscalledhorizontalifthetangentvector0tisinV1.DenetheCarnot-Caratheodorydistancebythefollowing:dCp;q=inf)]TJ/F32 11.955 Tf 11.72 23.712 Td[(Z10k0tkdtwherepandqareinGand)-330(isthesetofallhorizontalcurvessuchthat=pand=q.Itisanon-trivialfactChow'stheorem,[1]thatanytwopointsinaCarnotgroupcanbeconnectedbyahorizontalcurve,whichmakesdCp;qaleft-invariantmetriconG.Inaddition,thereexistsatleastoneshortestcurveminimizinggeodesicconnectingpointspandq.ItshouldbenotedthatdCp;qisindependentofthechoiceofkk.DeneaCarnot-Caratheodoryballofradiusrcenteredatapointp0byB=Bp0;r=fp2G:dCp;p0
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ThisistheprojectionontoV1oftheusualgradientoff,whichisgivenbyrf=X11f;X12f;:::;X1d1f;X21f;X22f;:::;X2d2f;:::;Xl1f;Xl2f;:::;Xldlf:Thesecondisthesemi-horizontalgradient,denedbyr1f=X11f;X12f;:::;X1d1f;X21f;X22f;:::;X2d2f:ThisistheprojectionofthegradientontoV1V2andishasrstandsecondorderderivatives.Anotherimportantsecondorderderivativeisthesymmetrizedsecondordermatrix,denotedD2f?,whichisad1d1matrixwithentriesD2f?ij=1 2XiXj+XjXif:Wenotethatfortechnicalreasons,weneedasymmetricsecondorderderivativematrix.Becauseofthenon-abeliannatureoftheCarnotgroup,D2f,whichisthematrixgivenbyD2fij=XiXjfneednotbesymmetric.AfunctionfisC1ifXifiscontinuousforalli.AfunctionfisC2ifitisC1andXiXjfiscontinuousforalli;jwhichimpliesr1fhascontinuouscomponents.Lastly,wenoteallintegrationisdonewithrespecttoLebesguemeasure.9

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3GroupsofHeisenbergType3.1DenitionACarnotgroupisofHeisenbergtype[4]ifitsLieAlgebragsatisesg=V1V2andghasaninnerproducth;iwithassociatednormkksothatforeachz2V2,thelinearmapJz:V1!V1,denedbyhJzv;wi=hz;[v;w]isatisesthepropertyJ2z=kzk2Id:ThefunctionJzalsohasthefollowingusefulproperty[4].Proposition3.1.1kJzwk2=kzk2kwk2.hw;Jzvi=hJzv;wi=hz;[v;w]i=h)]TJ/F27 11.955 Tf 13.948 0 Td[(z;[w;v]i=hJzw;vi:ThisthenleadstokJzvk2=hJzv;Jzvi=hJ2zv;viandthepropositionfollowsfromthepropertyabove.WenotethatwhendimV2=1,thisguaranteestheexistenceoftheJz,andsointhiscase,wehaveagroupofHeisenbergtype.[4]10

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IngroupsofHeisenbergtype,thesemi-horizontalgradientandfullgradientareequal,andbotharedenotedbyrf.3.2AClassofGroupsofHeisenbergTypeLetn3andconsiderRn+1spannedbythelinearlyindependentvectoreldsfXi;Tg,wheretheindexirangesfrom1ton,denedbyX1=@ @x1)]TJ/F27 11.955 Tf 13.151 8.088 Td[(x2 2@ @tX2=@ @x2+x1 2@ @tX3=@ @x3...Xn=@ @xnT=@ @tThisLiealgebra,denotedhn,hasthepropertythatfori<>:Tifi=1;j=20otherwiseandforalli,[Xi;T]=0:Thus,hndecomposesasadirectsumhn=V1V2whereV1isspannedbytheXi'sandV2isspannedbyT.11

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Fromtheprevioussection,thecorrespondingLieGroup,denotedHn,isaCarnotgroupofHeisenbergtype.WenotethatalthoughRnisaCarnotgroupwithl=1,itisnotagroupofHeisenbergtype,becauseV2=f0g.Usingthegrouplawformula2.1.1,weseethatthegrouplawforourclassofgroupsofHeisenbergtypeispq=x1+y1;x2+y2;:::;xn+yn;t+s+1 2x1y2)]TJ/F27 11.955 Tf 11.955 0 Td[(x2y13.2.1wherep=x1;x2;:::;xn;tandq=y1;y2;:::;yn;s.3.2.1GeodesicsWerstwishtoexplorethegeodesicsfortheclassofgroupsHn.BecauseRnisnotagroupofHeisenbergtype,weexpectthegeodesicstobedierentfromthoseinRn.InordertocomputethegeodesicsofaCarnotgroup,weinvokethePontrjaginmaximumprinciple,asusedin[5]and[1].Thefollowinglemmaproducesaformulaforthesegeodesics.Lemma3.2.1Thegeodesicsstartingattheoriginhavethefollowingequations:x1=)]TJ/F27 11.955 Tf 10.494 8.087 Td[(A2 B)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+A1 BsinBx2=A1 B)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+A2 BsinBxj=Aj;j=3;4;:::;nt=A12+A22 2B2B)]TJ/F19 11.955 Tf 11.955 0 Td[(sinBwheretheAiandBareconstants.Wenotethatbyperiodicity,B2)]TJ/F27 11.955 Tf 9.298 0 Td[(;].Also,whenB=0,bycontinuity,wemaytakethelimitasB!0andobtainT=0andforalli,xi=Ai.Proof:Byleftmultiplicationunderthegrouplaw,wemayassumeallgeodesicsarebasedattheorigin,whichisdenoted0=;0;0;:::;0.Becauseapointp2Hnhas12

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coordinatesp=x1;x2;x3;:::;xn;t,weparameterizeageodesiccurveasx1;x2;:::;xn;tfor01.Thegeodesicsstartattheorigin=0andwillterminateatthepointcorrespondingto=1.WewishtosetupasystemofequationsinvolvingthevectorsXiandTasaboveandthecorrespondingcovectorsi.Notethatbythedenitionofhorizontalcurve,wedonotneedthevectorTinoursystemofequations.Wemayalsotreatthecovectorsascomponentsandwrite=1;2;3;:::;n;t:Beforecomputing,wenotethatbytheabovediscussion,ourinitialconditionsarexi=0andi=AiwheretheAiareconstants.Inordertobegincomputing,weneedtodenetheoperatorthatisthebasisofourcalculations.Thisoperatorusesbothvectorsandcovectorsandisdenedbyap;=nXi=1hXip;i2:Writingthisoutincoordinateform,weobtainap;=1)]TJ/F27 11.955 Tf 13.151 8.088 Td[(x2 2t2+2+x1 2t2+32+42+:::+n2:Wenormalizethisoperatorbymulitplyingap;by1 2,toobtain1 2ap;=1 21)]TJ/F27 11.955 Tf 13.151 8.087 Td[(x2 2t2+1 22+x1 2t2+1 232+1 242+:::+1 2n2:13

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Wedierentiatethefunctionap;toobtain@a @p0=x10;x20;x30;:::;xn0)]TJ/F27 11.955 Tf 10.494 8.088 Td[(@a @p0=10;20;30;:::;n0;t0:UsingthePontrjaginmaximumprinciple[5]and[1],wehavetherelationsx10=1)]TJ/F27 11.955 Tf 13.151 8.088 Td[(x2 2tx20=2+x1 2tx30=3x40=4...xn0=nt0=)]TJ/F27 11.955 Tf 10.494 8.087 Td[(x2 21)]TJ/F27 11.955 Tf 13.151 8.087 Td[(x2 2t+x1 22+x1 2t10=)]TJ/F19 11.955 Tf 10.494 8.087 Td[(1 2t2+x1 2t20=1 2t1)]TJ/F27 11.955 Tf 13.151 8.088 Td[(x2 2t30=040=0...n0=0t0=0:Wenowneedtosolvethissystemofdierentialequations,usingtheaboveinitialconditions.Weeasilyconcludethatt=Bandi=Aifori=3ton.Thus,14

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Thusxj=Ajforj=3ton.Wearethenlefttoconsider10=)]TJ/F19 11.955 Tf 10.494 8.088 Td[(1 2t2+x1 2t=)]TJ/F27 11.955 Tf 10.494 8.087 Td[(t 2x2020=1 2t1)]TJ/F27 11.955 Tf 13.151 8.088 Td[(x2 2t=t 2x10:Simplifying,wehave1=)]TJ/F27 11.955 Tf 10.494 8.088 Td[(t 2x2+A12=t 2x1+A2:Weneedtosolveforx1andx2,x10=1)]TJ/F27 11.955 Tf 13.151 8.088 Td[(x2 2t:Pluggingin1=)]TJ/F28 7.97 Tf 10.494 5.698 Td[(t 2x2+A1andt=B,weobtainx10=)]TJ/F27 11.955 Tf 9.299 0 Td[(Bx2+A1x20=2+x1 2t:Similarly,pluggingin2=t 2x1+A2andt=B,wehavex20=Bx1+A2:Weconcludethatsincex10=)]TJ/F27 11.955 Tf 9.299 0 Td[(Bx2+A1;x001=)]TJ/F27 11.955 Tf 9.299 0 Td[(Bx2015

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andsubstitutingx20=Bx1+A2,weconcludex001=)]TJ/F27 11.955 Tf 9.298 0 Td[(B2x1)]TJ/F27 11.955 Tf 11.955 0 Td[(BA2:Bythemethodofundeterminedcoecientsx1=cosB+sinB+:Butx1=0so=)]TJ/F27 11.955 Tf 9.298 0 Td[(andsox1=)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+sinB:Tosolveforwetakethesecondderivativeofx1andcompareittox001obtainedearlierandseethatx001=)]TJ/F27 11.955 Tf 9.298 0 Td[(B2x1)]TJ/F27 11.955 Tf 12.008 0 Td[(BA2andx1=)]TJ/F19 11.955 Tf 12.008 0 Td[(cosB+sinB.Thusx001=)]TJ/F27 11.955 Tf 9.298 0 Td[(B2)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+sinB)]TJ/F27 11.955 Tf 11.955 0 Td[(BA2andsox001=)]TJ/F27 11.955 Tf 9.298 0 Td[(B2+B2cosB)]TJ/F27 11.955 Tf 11.955 0 Td[(B2sinB)]TJ/F27 11.955 Tf 11.955 0 Td[(BA2:Takingthesecondderivativeofthex1weobtained,weconcludex001=B2cosB)]TJ/F27 11.955 Tf 11.955 0 Td[(B2sinB:Thuswehave)]TJ/F27 11.955 Tf 9.298 0 Td[(B2+B2cosB)]TJ/F27 11.955 Tf 11.955 0 Td[(B2sinB)]TJ/F27 11.955 Tf 11.956 0 Td[(BA2=B2cosB)]TJ/F27 11.955 Tf 11.955 0 Td[(B2sinB16

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whichimplies)]TJ/F27 11.955 Tf 9.299 0 Td[(B2)]TJ/F27 11.955 Tf 11.955 0 Td[(BA2=0:Thatis,=)]TJ/F27 11.955 Tf 10.494 8.088 Td[(A2 B:Thusx1=)]TJ/F27 11.955 Tf 10.494 8.088 Td[(A2 B)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+sinB:Wenowsolveforx2.Fromabove,wehavex20=Bx1+A2:Thisimpliesx2=ZBx1+A2d:Thisimpliesx2=ZB)]TJ/F27 11.955 Tf 10.494 8.087 Td[(A2 B)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+sinB+A2d=BZsinBd+A2ZcosBdandsox2=)]TJ/F27 11.955 Tf 9.298 0 Td[(cosB+A2 BsinB+:Butx2=0,so=,givingusx2=)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+A2 BsinB:17

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Tosolveforwecomparethederivativesofx1.Weknowx10=)]TJ/F27 11.955 Tf 9.298 0 Td[(Bx2+A1andbysubstitutingx2=)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+A2 BsinBwegetx10=)]TJ/F27 11.955 Tf 9.299 0 Td[(B+BcosB)]TJ/F27 11.955 Tf 11.955 0 Td[(A2sinB+A1:Wefoundthatx1=)]TJ/F27 11.955 Tf 10.494 8.087 Td[(A2 B)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+sinB:Thusx10=BsinB)]TJ/F27 11.955 Tf 11.955 0 Td[(A2sinB:Weconclude)]TJ/F27 11.955 Tf 9.298 0 Td[(B+A1=0andso=A1 B:Pluggingintox1andx2,wehavex1=)]TJ/F27 11.955 Tf 10.494 8.088 Td[(A2 B)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+A1 BsinBx2=A1 B)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+A2 BsinB:Wenowneedtosolvefort.t0=)]TJ/F27 11.955 Tf 10.494 8.088 Td[(x2 21)]TJ/F27 11.955 Tf 13.15 8.088 Td[(x2 2t+x1 22+x1 2t18

PAGE 22

whichthenimpliest0=)]TJ/F27 11.955 Tf 10.494 8.087 Td[(x2x01 2+x1x20 2=1 2x1x20)]TJ/F27 11.955 Tf 11.955 0 Td[(x2x10=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[()]TJ/F27 11.955 Tf 10.494 8.088 Td[(A2 B)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+A1 BsinBA1sinB+A2cosB)]TJ/F19 11.955 Tf 11.955 0 Td[(A1 B)]TJ/F19 11.955 Tf 11.955 0 Td[(cosB+A2 BsinBA1cosB)]TJ/F27 11.955 Tf 11.955 0 Td[(A2sinB=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(A1 BsinB)]TJ/F27 11.955 Tf 13.15 8.088 Td[(A2 B+A2 BcosBA1sinB+A2cosB)]TJ/F19 11.955 Tf 11.955 0 Td[(A1 B)]TJ/F27 11.955 Tf 13.151 8.088 Td[(A1 BcosB+A2 BsinBA1cosB)]TJ/F27 11.955 Tf 11.955 0 Td[(A2sinB=1 2)]TJ/F27 11.955 Tf 6.675 -1.596 Td[(A12 Bsin2B)]TJ/F27 11.955 Tf 13.15 8.088 Td[(A1A2 BsinB+A1A2 BsinBcosB+A1A2 BsinBcosB)]TJ/F27 11.955 Tf 13.15 8.087 Td[(A22 BcosB+A22 Bcos2B)]TJ/F27 11.955 Tf 13.151 8.088 Td[(A12 BcosB+A12 Bcos2B)]TJ/F27 11.955 Tf 13.151 8.088 Td[(A1A2 BsinBcosB+A1A2 BsinB)]TJ/F27 11.955 Tf 13.151 8.088 Td[(A1A2 BsinBcosB+A22 Bsin2B=1 2A12 B+A22 B)]TJ/F27 11.955 Tf 13.15 8.088 Td[(A22 BcosB)]TJ/F27 11.955 Tf 13.15 8.088 Td[(A12 BcosB=1 2A12+A22 B)]TJ/F19 11.955 Tf 13.151 8.088 Td[(1 2A12+A22 BcosBsot0=1 2A12+A22 B)]TJ/F19 11.955 Tf 13.15 8.087 Td[(1 2A12+A22 BcosB:Integrating,wearriveatt=1 2A12+A22 BZ1d)]TJ/F19 11.955 Tf 13.15 8.088 Td[(1 2A12+A22 BZcosBd=1 2A12+A22 B)]TJ/F19 11.955 Tf 13.151 8.088 Td[(1 2A12+A22 B2sinB+:19

PAGE 23

Butt=0,so=0andsot=1 2A12+A22 B)]TJ/F19 11.955 Tf 13.15 8.088 Td[(1 2A12+A22 B2sinB=A12+A22 2B2B)]TJ/F19 11.955 Tf 11.955 0 Td[(sinB: 3.2.2TaylorPolynomialsTaylorpolynomialsinRnareusefultoapproximatefunctions.Thisrolewillbeex-tendedtogroupsofHeisenbergtype.However,inthissetting,theTaylorpolynomialsmustbemodiedtocompensateforthegroupoperation.1.1andthestructureoftheHeisenbergtypegroups.ThefollowingpropositiongivestheformulafortheTaylorpolynomialbasedatanarbitrarypoint.Proposition3.2.2Letf:Hn7!RbeaC2function.Letthebasepointbedenotedbyp0.Then,fp=fp0+nXi=1Xifp0xi)]TJ/F27 11.955 Tf 11.955 0 Td[(x0i+Tfp0t)]TJ/F27 11.955 Tf 11.955 0 Td[(t0+1 2x1x02)]TJ/F27 11.955 Tf 11.955 0 Td[(x01x2+1 2nXi;j=11 2XiXj+XjXifp0xi)]TJ/F27 11.955 Tf 11.955 0 Td[(x0ixj)]TJ/F27 11.955 Tf 11.955 0 Td[(x0j+odCp0;p2=Pp+odCp0;p2:BythedenitionoftheTaylorPolynomial,fpshouldsatisfyeachofthefollow-ing:XjPp0=Xjfp0XiXjPp0=XiXjfp0TPp0=Tfp0:20

PAGE 24

So,wewillcheckeachofthese.Clearly,TPp0=Tfp0andwehaveagreement.WenowturntoX1.X1Pp=X1fp0+x02 2Tfp0+1 2nXj=11 2X1Xj+XjX1fp0xj)]TJ/F27 11.955 Tf 11.955 0 Td[(x0j+1 2nXi=11 2XiX1+X1Xifp0xi)]TJ/F27 11.955 Tf 11.955 0 Td[(x0i)]TJ/F27 11.955 Tf 13.15 8.088 Td[(x2 2Tfp0:ThisgivesusX1Pp0=X1fp0+x02 2Tfp0)]TJ/F27 11.955 Tf 13.15 8.088 Td[(x02 2Tfp0andsowehaveagreement.LookingatX2,wehaveX2Pp=X2fp0+)]TJ/F27 11.955 Tf 9.298 0 Td[(x01 2Tfp0+1 2nXj=11 2X2Xj+XjX2fp0xj)]TJ/F27 11.955 Tf 11.955 0 Td[(x0j+1 2nXi=11 2XiX2+X2Xifp0xi)]TJ/F27 11.955 Tf 11.955 0 Td[(x0i)]TJ/F27 11.955 Tf 13.15 8.088 Td[(x1 2Tfp0:ThisgivesusX2Pp0=X2fp0)]TJ/F27 11.955 Tf 13.151 8.088 Td[(x01 2Tfp0+x01 2Tfp0andwehaveagreement.21

PAGE 25

Wenowletk3.XkPp=Xkfp0+1 2nXj=11 2XkXj+XjXkfp0xj)]TJ/F27 11.955 Tf 11.955 0 Td[(x0j+1 2nXi=11 2XiXk+XkXifp0xi)]TJ/F27 11.955 Tf 11.955 0 Td[(x0iandsowehaveagreement.Thus,alltheXiandTderivativescoincide.WenowfocusonthederivativesoftheformXiXj.Lettingi=j=1,wehave,usingtheaboverstordercalculations,X1X1Pp=1 2)]TJ/F19 11.955 Tf 6.675 -1.596 Td[(1 2X1X1+X1X1fp0+1 2X1X1+X1X1fp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.683 Td[(2X1X1fp0=X1X1fp0andsoX1X1Pp0=X1X1fp0:Now,lettingi=j=2,wehave,usingtheaboverstordercalculations,X2X2Pp=1 2)]TJ/F19 11.955 Tf 6.675 -1.596 Td[(1 2X2X2+X2X2fp0+1 2X2X2+X2X2fp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(2X2X2fp0=X2X2fp0andsoX2X2Pp0=X2X2fp0:Now,for3mnand3kn,wehaveXmXkPp=1 2)]TJ/F19 11.955 Tf 6.675 -1.596 Td[(1 2XkXm+XmXkfp0+1 2XmXk+XkXmfp0=1 2)]TJ/F19 11.955 Tf 5.48 -9.684 Td[(XkXm+XmXkfp0:22

PAGE 26

Since[Xk;Xm]=0,wehaveXkXm=XmXk,soXmXkPp=1 2)]TJ/F19 11.955 Tf 5.479 -9.683 Td[(2XmXkfp0=XmXkfp0thusXmXkPp0=XmXkfp0:LookingatX1X2,wehaveX1X2Pp=1 2)]TJ/F19 11.955 Tf 6.675 -1.596 Td[(1 2X2X1+X1X2fp0+1 2X1X2+X2X1fp0+1 2Tfp0:Now,T=X1X2)]TJ/F27 11.955 Tf 11.955 0 Td[(X2X1,sowehaveX1X2Pp=1 2)]TJ/F19 11.955 Tf 5.48 -9.684 Td[(X2X1+X1X2fp0+1 2)]TJ/F19 11.955 Tf 5.48 -9.684 Td[(X1X2)]TJ/F27 11.955 Tf 11.955 0 Td[(X2X1fp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(X2X1+X1X2+X1X2)]TJ/F27 11.955 Tf 11.955 0 Td[(X2X1fp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(2X1X2fp0=X1X2fp0thereforeX1X2Pp0=X1X2fp0:LookingatX2X1,wehaveX2X1Pp=1 2)]TJ/F19 11.955 Tf 6.675 -1.596 Td[(1 2X1X2+X2X1fp0+1 2X2X1+X1X2fp0)]TJ/F19 11.955 Tf 13.151 8.088 Td[(1 2Tfp0:23

PAGE 27

NowT=X1X2)]TJ/F27 11.955 Tf 11.955 0 Td[(X2X1,sowehaveX2X1Pp=1 2)]TJ/F19 11.955 Tf 5.48 -9.683 Td[(X1X2+X2X1fp0)]TJ/F19 11.955 Tf 13.151 8.087 Td[(1 2)]TJ/F19 11.955 Tf 5.48 -9.683 Td[(X1X2)]TJ/F27 11.955 Tf 11.955 0 Td[(X2X1fp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.683 Td[(X1X2+X2X1)]TJ/F27 11.955 Tf 11.955 0 Td[(X1X2+X2X1fp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.683 Td[(2X2X1fp0=X2X1fp0andsoX2X1Pp0=X2X1fp0:LookingatX1Xkfor3kn,wehaveX1XkPp=1 2)]TJ/F19 11.955 Tf 6.675 -1.596 Td[(1 2XkX1+X1Xkfp0+1 2X1Xk+XkX1fp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(XkX1+X1Xkfp0:Since[X1;Xk]=0,wehaveX1Xk=XkX1,thusX1XkPp=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(2X1Xkfp0=X1Xkfp0henceX1XkPp0=X1Xkfp0:LookingatXkX1for3kn,wehaveXkX1Pp=1 2)]TJ/F19 11.955 Tf 6.675 -1.596 Td[(1 2X1Xk+XkX1fp0+1 2XkX1+X1Xkfp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(X1Xk+XkX1fp0:24

PAGE 28

Since[X1;Xk]=0,wehaveX1Xk=XkX1,soXkX1Pp=1 2)]TJ/F19 11.955 Tf 5.479 -9.683 Td[(2XkX1fp0=XkX1fp0thereforeXkX1Pp0=XkX1fp0:LookingatX2Xkfor3kn,wehaveX2XkPp=1 2)]TJ/F19 11.955 Tf 6.675 -1.596 Td[(1 2XkX2+X2Xkfp0+1 2X2Xk+XkX2fp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(XkX2+X2Xkfp0:Since[X2;Xk]=0,wehaveX2Xk=XkX2,sowehaveX2XkPp=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(2X2Xkfp0=X2Xkfp0andsoX2XkPp0=X2Xkfp0:LookingatXkX2for3kn,wehaveXkX2Pp=1 2)]TJ/F19 11.955 Tf 6.675 -1.596 Td[(1 2X2Xk+XkX2fp0+1 2XkX2+X2Xkfp0=1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(X2Xk+XkX2fp0:Since[X2;Xk]=0,wehaveX2Xk=XkX2,sowehaveXkX2Pp=1 2)]TJ/F19 11.955 Tf 5.479 -9.683 Td[(2XkX2fp0=XkX2fp025

PAGE 29

thusXkX2Pp0=XkX2fp0: 26

PAGE 30

4ViscositySolutions4.1AClassofPartialDierentialEquationsWelooktoextendtheresultsinRnfromChapter1toourclassofgroupsofHeisenbergtype.WerecallthecalculusonCarnotgroupsfromChapter2.Thehorizontalgradientisgivenbyr0fp=X1fp;X2fp;X3fp;:::;Xnfpandthegradientisgivenbyrfp=X1fp;X2fp;X3fp;:::;Xnfp;Tfp:Wealsorecallthatthisisalsothesemi-horizontalgradient.Inaddition,weconsiderthesymmetrizedsecondorderderivativematrixD2f?pwithentriesD2f?ij=1 2XiXj+XjXif:NotethatneitherthehorizontalgradientnorthesecondorderderivativematrixusethevectorTdirectly.FollowingChapter1andrecallingthatSnisthesetofnnsymmetricmatrices,weconsideraclassofpartialdierentialequationsoftheformFp;up;rup;D2up?=027

PAGE 31

wherecontinuousF:HnRhnSn7!RhasthepropertythatFp;r;;XFp;s;;YwhenrsandYX.Again,thisclassofequationsisproperinthesenseof[3].WemayconsiderthesameexamplesinChapter1,substitutingtheCarnotgroupderivativesinplaceoftheRnderivatives.4.2JetsLet=1;2;:::;n;t2hnandX2Sn.Givenafunctionf:Hn7!R,considerthefollowinginequalitybasedontheTaylorPolynomialfromtheprevioussection:fpfp0+nXi=1ixi)]TJ/F27 11.955 Tf 11.955 0 Td[(x0i+tt)]TJ/F27 11.955 Tf 11.955 0 Td[(t0+1 2x1x02)]TJ/F27 11.955 Tf 11.955 0 Td[(x01x2+1 2nXi;j=1Xijxi)]TJ/F27 11.955 Tf 11.955 0 Td[(x0ixj)]TJ/F27 11.955 Tf 11.955 0 Td[(x0j+odCp0;p2.2.1FollowingChapter1,wedenethesecondordersuperjetoffatp0,denotedJ2;+fp0,andthesecondordersubjetoffatp0,denotedJ2;)]TJ/F27 11.955 Tf 7.084 -4.338 Td[(fp0,bythefollowing:;X2J2;+fp0$.2.1holds.J2;)]TJ/F27 11.955 Tf 7.085 -4.936 Td[(fp0=)]TJ/F27 11.955 Tf 9.299 0 Td[(J2;+)]TJ/F27 11.955 Tf 9.299 0 Td[(fp0:WenoteasinChapter1,thatjetsmaybeemptyatapoint,andsothereisnoguaranteethatsuchapair;Xexists.Additionally,weagaindene J2;+upastheset-theoreticclosureofJ2;+up.Wethendeneviscositysolutionsasfollows:Denition4.2.1Acontinuousfunctionfisaviscositysubsolution ofFp;fp;rfp;D2f?p=028

PAGE 32

ifforall;X2 J2;+fpwehaveFp;fp;;X0Acontinuousfunctionfisaviscositysupersolution ofFp;fp;rfp;D2f?p=0ifforall;Y2 J2;)]TJ/F27 11.955 Tf 7.085 -7.405 Td[(fpwehaveFp;fp;;Y0Thecontinuousfunctionfisaviscositysolution ifitisbothaviscositysubsolutionandaviscositysupersolution.AsinChapter1,wefocusonaspecialsubclassofsuchfunctionsF.WecallthesubclassFthosefunctionsFforwhichthereisauniversal>0sothatforeachrealrandswithr
PAGE 33

Theorem4.2.2LetF2FandbeaboundedopensetinHn.ThenifuisaviscositysubsolutiontoFp;up;rup;D2u?p=0andvisaviscositysupersolutiontoFp;vp;rvp;D2v?p=0withuvon@.Thenuvin.OurproofwillnotbeaseasyastheRncase,becausethetechnicallemmasareunproveningroupsofHeisenbergtype.Instead,weneedtheBieske-Manfredi[2]twistinglemmathatconnectsjetsonourgroupsofHeisenbergtypetojetsinRn.Westatethefullversionofthetwistinglemma.Lemma4.2.3LetDLp0bethedierentialoftheleftmultiplicationmapatthepointp0,letJ2;+euclup0bethetraditionalEuclideansuperjetofuatthepointp0asdenedinChapter1andlet;X2Rn+1Sn+1.Then,;X2 J2;+euclup0givestheelementDLp0;DLp0XDLp0Tn2 J2;+up0withtheconventionthatforanymatrixM,Mnisthennprincipalminor.Foranypointp=x1;x2;x3;:::;t,wecancomputeDLpdirectlyforourclassofgroupsofHeisenbergtypebecausethemultiplicationlawforourclassisgivenexplic-30

PAGE 34

itlybyEquation.2.1.Ifweletvbethen1vectorgivenbyv=0BBBBBBBBBBB@)]TJ/F28 7.97 Tf 10.494 4.813 Td[(x2 2x1 200...01CCCCCCCCCCCAandObethe1nvectorgivenbyO=;0;0;:::;0,thenwehaveDLp=24InvO135:Ifisthevector1;2;3;:::;n+1,wecanthencomputeDLpexplicitlytoobtainDLp=0BBBBBBBBBBBBBB@1)]TJ/F28 7.97 Tf 13.151 4.813 Td[(x2 2n+12+x1 2n+134...nn+11CCCCCCCCCCCCCCA:Inaddition,ifthematrixXhasij-thentryXij,thenwecancomputethennprincipalminorofDLpXDLTp,denotedDLpXDLTpn,directlyalso.Thematrixis31

PAGE 35

givenby)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(DLpXDLTpn1;1=x1;1)]TJ/F27 11.955 Tf 13.15 8.088 Td[(x2 2xn+1;1+x1;n+1)]TJ/F27 11.955 Tf 13.151 8.087 Td[(x2 2xn+1;n+1)]TJ/F27 11.955 Tf 10.494 8.087 Td[(x2 2)]TJ/F19 11.955 Tf 5.479 -9.683 Td[(DLpXDLTpn1;2=x1;2)]TJ/F27 11.955 Tf 13.15 8.087 Td[(x2 2xn+1;2+x1;n+1)]TJ/F27 11.955 Tf 13.151 8.087 Td[(x2 2xn+1;n+1x1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(DLpXDLTpn1;j=x1;j)]TJ/F27 11.955 Tf 13.151 8.088 Td[(x2 2xn+1;j;j=3;4;:::;n)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(DLpXDLTpn2;1=x2;1+x1 2xn+1;1+x2;n+1+x1 2xn+1;n+1)]TJ/F27 11.955 Tf 10.494 8.088 Td[(x2 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(DLpXDLTpn2;2=x2;2+x1 2xn+1;2+x2;n+1+x1 2xn+1;n+1x1 2)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(DLpXDLTpn2;j=x2;j+x1 2xn+1;jj=3;4;:::;n)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(DLpXDLTpni;1=xi;1+xi;n+1)]TJ/F27 11.955 Tf 10.494 8.088 Td[(x2 2i=3;4;:::;n)]TJ/F19 11.955 Tf 5.479 -9.683 Td[(DLpXDLTpni;2=xi;2+xi;n+1x1 2i=3;4;:::;n)]TJ/F19 11.955 Tf 5.479 -9.684 Td[(DLpXDLTpni;j=xi;ji;j=3;4;:::;nThistwistingwillallowustousethemaximumprinciplefromtheRnenvironment,butitwillnotproduceestimatesthatareaseasytocontrol.Themaximumprincipleworksbyusingapenaltyfunction"thatguaranteestheexistenceofjetelementsatcertainpointsclosetothedesiredpoint.WepresentaweakversionofthemaximumprincipleinHn.[2]Theorem4.2.4LetuandvbecontinuousfunctionsinaboundeddomainHn.Ifu)]TJ/F27 11.955 Tf 11.955 0 Td[(vhasapositiveinteriorlocalmaximumsupu)]TJ/F27 11.955 Tf 11.955 0 Td[(v>0thenwehave:For>0wecanndpointsp;q2Hnsuchthatilim!1p;q=0;wherep;q=kp)]TJ/F27 11.955 Tf 11.955 0 Td[(qk2eucl:32

PAGE 36

iiThereexistsapoint^p2suchthatp!^pandsodoesqbyiandsupu)]TJ/F27 11.955 Tf 11.955 0 Td[(v=u^p)]TJ/F27 11.955 Tf 11.955 0 Td[(v^p>0;iiithereexistsymmetricmatricesX;Yandvectors+,)]TJ/F28 7.97 Tf -0.429 -7.294 Td[(sothativ+;X2 J2;+u;p;v)]TJ/F28 7.97 Tf -0.429 -7.891 Td[(;Y2 J2;)]TJ/F19 11.955 Tf 7.084 -7.406 Td[(v;q;vi+)]TJ/F27 11.955 Tf 11.955 0 Td[()]TJ/F28 7.97 Tf -0.429 -7.891 Td[(=RwhereR!0as!1.andviiXY+ZwhereZ!0as!1.Proof:BytheRnversion[3],Dpp;q;X2 J2;+euclupand)]TJ/F27 11.955 Tf 9.298 0 Td[(Dqp;q;Y2 J2;)]TJ/F25 7.97 Tf -6.587 -10.361 Td[(euclvqwhereDpandDqareeuclideanderivativesandassymmetricmatrices,XY.By33

PAGE 37

theabovetwistinglemma,wehave+=DLpDpp;q)]TJ/F28 7.97 Tf -0.429 -7.891 Td[(=)]TJ/F27 11.955 Tf 9.298 0 Td[(DLqDqp;qX=DLpXDLTpnY=DLqYDLTqnLetbeaxedn1vectorandlet bethen+11vectorgivenby =;0.ThenhX;i)-222(hY;i=hDLpXDLTpn;i)-222(hDLqYDLTqn;i=hDLpXDLTp ; i)-222(hDLqYDLTq ; i=hXDLTp ;DLTp i)-222(hYDLTq ;DLTq i:BytheresultsfromRn[3],wehavehXDLTp ;DLTp i)-222(hYDLTq ;DLTq i3kDLTp )]TJ/F27 11.955 Tf 11.955 0 Td[(DLTq k2:WethereforeobtainhX;i)-222(hY;i3kDLTp)]TJ/F27 11.955 Tf 11.955 0 Td[(DLTq k2=3k k2kDLTp)]TJ/F27 11.955 Tf 11.955 0 Td[(DLTqk2p;qandwearriveattheresultbyletting!1.34

PAGE 38

Examiningthevectordierence,thefactthatDp=)]TJ/F27 11.955 Tf 9.298 0 Td[(Dqgivesusk+k)-222(k)]TJ/F28 7.97 Tf -0.429 -7.892 Td[(k=kDLpDpp;qk)-222(k)]TJ/F27 11.955 Tf 41.179 0 Td[(DLqDqp;qk=kDLpDpp;qk)-222(kDLqDpp;qkkDLp)]TJ/F27 11.955 Tf 11.955 0 Td[(DLqDpp;qkkDLp)]TJ/F27 11.955 Tf 11.955 0 Td[(DLqkkDpp;qkp;q1 2p;q1 2:Theresultfollowsbytaking!1. Wearenowreadytorestateandproveourtheorem.Theorem4.2.5LetHnbeaboundedopenset.LetF2F.IfuisaviscositysubsolutiontoFp;up;rup;D2up?=0andvisaviscositysupersolutiontoFp;vp;rvp;D2vp?=0suchthatuvon@thenuvin.Proof:Supposethereisa^pinsidesothatu^p>v^p.Bythemaximumprincipleandcontinuityofuandv,wehaveup>vqforlarge.Inaddition,pandqareinteriorpoints.35

PAGE 39

UsingthefactthatF2F,weobtain0
PAGE 40

References[1]Bellache,Andre.TheTangentSpaceinSub-RiemannianGeometry.InSub-RiemannianGeometry;Bellache,Andre.,Risler,Jean-Jacques.,Eds.;ProgressinMathematics;Birkhauser:Basel,Switzerland.1996;Vol.144,1{78.[2]Bieske,Thomas.;Manfredi,Juan.PropertiesofInniteHarmonicFunctionsonCarnotGroups.Preprint.[3]Crandall,Michael.;Ishii,Hitoshi.;Lions,Pierre-Louis.User'sGuidetoViscositySolutionsofSecondOrderPartialDierentialEquations.Bull.ofAmer.Math.Soc.1992,27,1{67.[4]Kaplan,Aroldo.LieGroupsofHeisenbergType.Rend.Sem.Mat.Univ.Politec.Torino1983SpecialIssue1984,117{130.[5]Monti,Roberto.SomePropertiesofCarnot-CaratheodoryballsintheHeisenberggroup.Rend.Mat.Acc.Lincei2000,91,155{167.37


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