; TeX output 1998.09.02:1332z͍uF'0kN cmbx12kNenadAntonic%썑U{Nff cmbx12{H-measuresffappliedtosymmetricsystemsL͟P J̫]kAbstract W(CK`y cmr10CH-measures2wererecentlyintroGducedbyLucT*artarasatoGolwhichmightprovidebGetter understandingofpropagatingoscillations.IndepGendently*,4=PatrickGGerardintroGducedthesameob8jectsUUunderthenameofJp0J cmsl10JmicroloGcaldefectmeasuresC.(Partialdi erentialequationsofmathematicalphysicscanoftenbGewrittenintheformofasymmetricUUsystem:N;~0ercmmi7d rFu cmex10FX k+BٓRcmr7=1K"V cmbx10KAkD b> cmmi10D@kMm#R cmss10Mu8C+KBMuC=MfxD;Cwhere 0KA^kCandKBCarematrixfunctions,9whileMuCisanunknownvectorfunction,9andMfCaknownvectorUUfunction.(InEthisworkweproveageneralpropagationtheoremforH-measuresassoGciatedtosymmetricsystems.OThisresult,McombinedwiththeloGcalisationpropertyisthenusedtoobtainmorepreciseresults2onthebGehaviourofH-measuresassociatedtothewave2equation,*Maxwell'sandDirac'ssystems,UUandsecondorderequationsintwoUUvqariables.BJThis LworkissuppGortedinpartbyNationalScienceF*oundationgrant8803317andbyArmyResearchL͍OceUUcontractDAAL03-91-C-0023.}cXQ cmr12c10245" cmmi9th 3cJune,1997*z͍uF'Lk1.Intro`ductions卒 H-measures׍cInthestudyofconrtinuumphysicsequationsgoverningthebSehaviourofcontinuousL͍mediaNcanbSedividedinrtotwoclasses:33j}h! cmsl12jthebalancerelations_candjtheconstitutiveassump-tionsc.ҍWhiletheYVoungmeasureswrereagoSodtoolforthestudyofjoscillationce ectsofsolutions$9ofpartialdi erenrtialequations[1,2],2theyprovedinappropriateforthestudyofjconcenrtrationBce ects.5AsameasuredepSendingonthevXariablekxconlyV,$theYoungmeasurewrasnotwellsuitedtodescribSeanye ectthatdepSendsonaparticulardirectioninspace.AnwH-measureisaRadonmeasureonthecosphericalbundleorverwthedomain inconsideration (ingeneral,Rthebasespaceofthe brebundleisamanifold ,whilethe breaGistheunitspheredg cmmi12dSן2d5 cmsy93o cmr91φc).FVorasingleparametrisation(suppSose :e!", cmsy10ekR2d6cisaGanopendomain){itisameasureontheproSduct edSן2d1φc.In{ordertoapplytheFVouriertransform,functionsode nedonwholekR2d;cshouldbSeconsideredandthiscanbeacrhievedobyextendingtheUfunctionsbryzerooutsideofthedomain . Aftersuchadjustment,:thefollowingtheoremcanbSestated(fordetailssee[12]):܍kTheorem1. (existenceofH-measures)jIfc(mR6 cmss12munpc)s_jisasequenceincL22c(kR2d\c;kR2rc)=j,sucrhCthat@munDꍑ'L-=2pe i"d*#lm0{j(wreakly),*thenthereexistsasubsequencec(muVnjO!cmsy70c)jandacomplexmatrixRadonmeasuregDF cmmib10gjonkR2decS2d1Wjsucrhthatforalld'1d;'2uPe2URcC0c(kR2d\c)jandd Ëe2cC(S2d1c)j:;zc(1)!)limx䍑%!nj03ͦ甆fu cmex10fZ%0Т;t : cmbx9RjdBeF1 f[d'1muVnj0⩟ fe F1 fd'2muVnj0⩟ f Ud n9 fō !ng[z ! ΍ejg\ej \ f Kddge"c=URehgd;'1cd'2rd n9ei)Oe"c=Ef甆fZ%RjdSjd1-5yd'1c(kxc)d'2 tc(kxc)d n9c(g\c)ddgc(kxd;gc)URd:2zhޟ**f́KNotation. 7CTheUUF*ouriertransformusedabGoveUUisde nedinthefollowingway:?Oq^#9MuC(GDF cmmib10GMVC):=E!", cmsy10EF9MuC(GC):=cFZUR yf$cmbx7RO \cmmi5djDe2@Li*cmmib7x\MuC(KxC)DdKxD;OCwhileUUitsinverseUUis:Mx䍒ipEFMv#C(KxC):=cFZUR yRdjDe2@Lix[MvC(GMVC)DdGnD:OCTheUUderivqativeoftheF*ouriertransformandtheFouriertransformofthederivqativesatisfy:bYJD@8jC(EF9MuC)(GMVC)=EC2D[iEFC(Dxj6MuC(KxC))(GMVC) xEFC(D@jMuC)(GMVC)=2D[ij6EFMuC(GMVC)D:CWithMasOE MbCwedenotethecomplextensorproGductoftwovectors.xItisde nedasalinearopGerator, actingonavectorMvaCby:%(MaE MbC)MvC:=(Mv)nEMbC)MaC,ڈwhereMvEMbC=FPލ USd% USi=1tJDvi䍑C7TLDbri CisthecomplexscalarproGduct($DaCdenotescomplexconjugateofascalarDaC,orofeachentryforvectorsandmatrices.ADa^aC=\Da^>CdenotesHermitianconjugateUUforvectorsandmatrices). ;V*ariablesin aredenotedbyKxY C=(Dx^1|sD;:::;x^dC)(orKxY C=(Dx^0D;Kx^09C)=(Dx^0D;x^1D;:::;x^dC)whenthisismoreconvenient),‘andythederivqativesbyD@k C= UF@D&fe zG@n9xkC.YSimilarlyfortheJdualECvqariableGgC=(D1|sD;:::;dC),‘where ^ߍderivqativesUUaredenotedbyD@8^lgC= ?|@K&fe =؟@n9lkVC. AA+summation+withrespGecttorepeatedindices(oneupper,4"anotherlower)+isalways+assumedover+thewholeUUrangeofindices,exceptwhereexplicitlystatedotherwise.C0|sC(KR^d IC)EdenotesthespaceofC^1 Cfunctionsvqanishingatin nity*,IwhichEistheL^1CclosureofthespaceofUUC^1 :CfunctionswithcompactsuppGort,denotedbyC^1cxC(KR^d IC)or,equivqalently*,EDGC(KR^d IC) d.W*eQshalldenotethedualityproGductwiththesamesymbGolEhD:;:EiC,regardlessQofthetypGe(scalarormatrix)$ofthefunctions.I Inanycasetheresultisascalar.IfthefunctionsappGearingarematrixfunctions,weUUassumethattheirscalarproGducthasbeentakenbeforeintegration. N8N8OK1dz͍CH-measuresUUappliedtosymmetricsystemsuF'cInacertainsensetheH-measurejmeasuresgxchorwfarthegivenweaklyconvergentse-L͍quencedBmuc(kxc)E:=URdbc(kxc)muc(kxc)d:*5kLemma1. c( rstcommutationlemma,U[12])jIfG~daWe2cC(dSן2d1φc)janddbe2cC0c(kR2d\c)jthenthewoabSorvede nedopSeratorsbelongtoeLc(L22c(kR2d\c)$)j(i.e.theyareboundedlinearoperatorson?cL22c(kR2d\c)*6j),and?theirnormscoincidewithsupremrumnormsofdajanddbj(respSectively).Moreorver,ithe*commutatordC;c:=_[dA;Bc]=dAB]e{WdBA*jisacompactopSeratoroncL22c(kR2d\c)j(denoteddC1e2URK,`c(L22c(kR2d\c)$)j).AB**fL͍cWVearereadytode nesymrbSolsandcorrespondingoperators.>Anjadmissiblesymrbol cisa-functiondPe2URcC(kR2dVedSן2d1φc)whicrhcanbSewrittenintheform: dPc(kxd;g\c)=fPkmdbkpc(kxc)dakc(g\c);with%{dak e2m0cC(dSן2d1φc)d;bke2cC0c(kR2d\c)%{andsucrhthatthefollowingbSoundednessconditionissatis ed:8fP㍟kwekdakpek17n1andaddingtheresultL͍totheequation(2)mrultipliedbyeAmun cfromtheleft,weobtain:^(4)3͍c=UReAmunekBmunekBeAmunemunL͍`c+eAmfnemunc+eAmunemfnehMmunemunehKmunemun id:^,cUsingthefactthat(|lkA2kpc) &4*c=URkA2kpc,thelefthandsidecanbSewrittenintheform:Wd@kpc(eAmunpc)ekA kmunc+eAmunekA kpd@kmun id:cWVewrouldliketowritethesetwotermsasaderivXativeofaproSduct; xclearlyV,thatproductcannotbSeascalar,butshouldbeamatrix(tensor),sothattheconrtractionwithkA2k )c(foreacrhddkgc)givesascalar.tAInaturalcandidateisthetensorproSductoftwovectors, theactionof<>whicrhonanarbitraryvectormvfcisgivenby(ma4e mbc)mv c:=0(mv embc)mad:<>cIfthescalarproSductoftrwo0matricesisde nedtobSe:$kA@ekAURc:=mtr c(MIkA2kAc)*(where0mtrcistheuniquelinearextensionofthemapmtr_Mc:URmae mbe7UY!maembc),thenthefollorwingidentityisvXalid:pmaekAmbURc=(mae mbc)ekAd:c(Thiscan]OeasilybSecrhecked]Ousingcomponenrts,yor,if]OanintrinsicproSofispreferred,ythegeneralcasecanbSereducedbrylinearitytothecasewherekAcisoftheformmce mdc.)Norwwecanrewritethelefthandsideof(4)as:TX[d@kpc(eAmunpc)e munc+eAmune d@kpmunpc]ekA kc=URd@kc(eAmune munpc)ekA kd:cWVeshoulddothesamewiththerighrthandside;itisthesumoftracesoftensorproSducts(herewreusesimpleidentityma&embURc=(ma&e mbc)ekIc,withkIcbSeingtheidentitymatrix).#\Therighthandsideof(4)thrusbSecomes:ec(eAmune munpc)ec(kBc+kB c)+(eAmfne munc+eAmune mfnpc)ekIec(hMmune munpc)ekIec(hKmune munpc)ekIURd:cMultiplyingtheequation(4)transformedinthiswraybyascalartestfunctiondwe2URcC21RAcc( ),andinrtegrating,weget:3͍!ehd@kpc(eAmune munpc)d;kA kdwReiEc=URehAmune munpd;wRc(kBc+kB c)eic+ehAmfne munc+eAmune mfnpd;wRkIeiL͍F{hhMmune munpd;wRkIeihhKmune munpd;wRkIeiURd:cAfterinrtegratingthelefthandsidebyparts,wecanpasstothelimitdnURe !UR1c,bSecauseof8thez-assumptionthatmfnDꍑ'L-=2pe i"d*#lm0c.bWVeshalldenotetheH-measure(itisa(2ec2)z-bloScrkmatrixmeasure,with(dr6edrSc)blocrks)associatedtothesequence(munpd;mfnc)bry:)3;gURc=qfٙ g11)-g12 g21)-g22=繟qfF=d:)4cClearlyV,g11 Lcisthealreadyde nedH-measureassoSciatedtothesequence(munpc).Inthelimit(dueatothecompactnessofhKc,,thelasttermontherighrthandsideconvergesto0)weobtain:(5).oehg11 d;a@kpc(kA kdwRc)edawc(kBc+kB c)edkpd@ lda@lhkA kdweic+ehg12 jc+g21 d;akIdweiURc=0d;cwheresymrbSolsofoperatorseAcandhMcappear.Letustakred n9c(kxd;g\c)UR:=dac(gc)dwRc(kxc).8ThentheProissonbracketofkPcandd Xcis:%_%̍efkPd; n9eg\c=URd@ lkPd@lhd ed@ ld n7@lkPg\c=URds2 l ekkA kpda@lhdwRed@ ldawRkd@lhkA k\c=URdakA kpd@kdwRedkd@ lda@lhkA kdw:%^cWVecannorwwrite(5)intheform:(6)Pehg11 d;efkPd; n9egc+d @kpkA k?ed c(kBc+kB c)eic+ehc2mRetr/ g12 d; eiURc=0d:cIfzdsIT>c1 C+Fu?vd?vzꍐx2 =c,thenH2sc(kR2d\c)edX21c(kR2dc).Thrusztheequation(6)hasameaningforanyd Ëe2URcCߍ[P vd+3vaHI0&2c] xL03c(kR2de{(dSן2d1φc).1ByfdensitryV, theformulacanbSeunderstoodevrenford AcofclassC21L0conkR2dedSן2d1.c(andhomogeneouslyextendedoutsidedSן2d1φc).VakQ.E.D.K6qПz͍CH-measuresUUappliedtosymmetricsystemsuF'KRemark.~KCAs TG11 :Cisameasureon 4EDS^d1yC,Sthe TtranspGortoperatorEfKPD;UP:Eg4C+D@kKA^k8D:jEC2KSD: TCshould bGe tangential.ThiscanbecorrectedbyreplacingthetermD@8^lCwhichisnottangentialingeneralbyitstangentialUUcompGonentD@^8ltC. ConsiderXatestfunctionD CextendedtoKR^dݴEn;3fK0EgCinG(Cbyhomogeneity:xD [C(KxD;GMVC)=D C(KxD;GMVC),YforXD>C0.W*eUUhave:D@8lD [lC=<$DdKwfe ɟ (֍d7G C(KxD;GMVC)=<$DdKwfe ɟ (֍d C(KxD;GMVC)=0D;CsoUUD@8^lD .CispGerpendicularUUtoGMVC.qThisimpliesthatD@lKPD@8^lD "C=bF\oD@lKPbFWD@^8ltD .CforUUhomogeneousD [C.IN8N8KRemark.CInRasimilarway*,startingRfromtheequationD@kC(KA^kMuC)i=Mf<CinsteadRof(2),onecouldobtainaformulaUUsimilarto(6): O䍒;EhG11xD;EfKPD; [Egi8C+EhC2MRe trqG12D; [EiC=00D:O䍒BN8N8kCorollary51. jUndertheassumptionsoftheorem3,3qtheH-measuregjsatis es,intheL͍senseofdistributionsonc edSן2d1φj,thefollorwing rstorderpartialdi erentialequation:c(7)d@lhc(d@ lkPeg11 c)ed@ lڍtc(d@lhkPeg11 c)+(ddec1)(d@lkPeg11 c)ds lc+(2kSed@lkA lc)eg11 Nc=UR2mRetr/ g12d;jwhered@2ltteQc:=URd@2leds2lVdkpd@2k &4jisthetangenrtialgradientontheunitsphere.mDem.cTVangenrtial59gradientonthespheredSן2d1cwithouterunitnormalmncisde nedbyertdgËc:=URerdgec(er=ow cmss9nGdgn9c)mnc.8Lemma16.1in[8]reads: B甆fZ? ^Sjd1*ertdgn7dËc=UR(ddec1)甆fZ ^Sjd1 :ddgn9mndd:"L܍cApplyingthisformrulatocompSonentsofproSductd n9mac,WwheremacisadCܞ21 cvector eldnotnecessarilytangenrtialtothesphere,andsummingthemup,onegets:/􍍟*G甆fZ y ^Sjd18 mtr@oertc(d n9mac)ddwyc=UR甆fZ ^Sjd1c(erd e mac)ec(kIemne mnc)ddc+甆fZ UT ^Sjd1 d n9ermaec(kIemne mnc)dd*wyc=UR(ddec1)甆fZ ^Sjd1 :dmtr(c(mae d n9mnc)ddË:cAsIallthefunctionsmrultiplyingmacaresmoSothenoughtheaborveIformulacanbSeappliedtovrector measuremacaswell,-ninterpretedinthesenseofdistributions.!FVord Lchomogeneousofdegreezero,ernGd Ëc=UR0(seetheremarkaftertheorem3),andtheformrulareads(8)nehmad;erd n9eic+ehrmad;c(kIemne mnc)d n9eiURc=(ddec1)ehmad; n9mneiURd:cReturningtoformrula(3)wehave:*鋍3, fD g11 d;@ lkPd@lhd ed@ ld n9@lkPc+d n9@kpkA k?ec2d kS fE?c=UR fD d@ lkPeg11 d;@lhd n9 fE n7e fD d@lkPeg11 d;@ ld n9 fEc+ fD c(d@kpkA k?ec2kSc)eg11 d; n9 fE d:cThe8 rsttermontherighrtcaneasilybSeintegratedbyparts(derivXativesareinkxc),givingehd@lhc(d@2lkPheg11 c)d; n9eic.~FVorthesecondterm,Gwreuseformula(8)withdalc:=0d@lhkPheg11candobtain::t fDAd@ lc(d@lhkPeg11 c)eds lVdkpd@ k;c(d@lkPeg11 c)d; n9 fE n7ec(ddec1) fDUVc(d@lkPeg11 c)ds lVd; n9 fE d;r cwherewrehaveusedthefactthatmnURc=gGucontheunitsphere.gnComrbiningtheabSovewecanwrite(7)inthesenseofdistributions.VakQ.E.D.OK7z͍CH-measuresUUappliedtosymmetricsystemsuF'KRemark.CTheequationinthecorollarycanbGerewrittenas(thetermswithD@lD@8^lKPEGCandD@8^lD@lKPEG CcancelUUoutbGecauseofoppositesigns): <_(9)lD@8lKP8ED@lG11 ED@lKPED@8lG11 C+D@lKPED@8kyG11xDkDuǟlHC+8(DdEC1)(D@lKPEG11xC)Duǟl}rHC+8D@8kyD@lKPEG11xDkDuǟloC+(2KSED@lKAlC)EG11 ?C=2MRe trqG12D:퍒BN8N8Y :^k3.Examples) ThewaveequationEcLetusconsiderthewraveequationinddc-dimensionalspace: (du 0c) 06emdivcc(kAerduc)UR=dgË:cWVe%oassumethatdmc:kR2dJekR+0e"!kR2+Vcand%okAc:kR2dekR+0e"!cPsym'1(vXalues%oofkAcareL͍symmetricpSositivrede nitematrices).0oWVewouldliketorewritethewaveequationasasymmetric1hrypSerbolicsystem.|Denotingthetimebrydtc=dx20 /candthetimederivXativrebyd@0uPc:=Fu4@z\@t{c,thewraveequationcanbSewritteninthefollorwingform:'U~(10)Kd@0c(d@0duc)e 2#dfXi;j=1d@ iJc(da ijZd@ jduc)UR=dgË:&TcInOPordertoreducethesecondorderequationtoa rstordersystemwremustintroSducenewvXariables:8dv j!bc:=URd@ jduc,fordj%c=0d;:::ʜ;dc.Theprevioustransformationgivresusonlyoneequation.VInordertomakethesystemwithddxpc+2unknorwnsformallydeterministic,dvwehavetoprovideddxpc+1moreequations.ClearlyV,Paddingthede nitionequationsfordvn92i}cwrouldleadtoaformallydeterministicsystem,whicrh,unfortunatelyV,isnotsymmetric.Besidesthese,wehave,bytheSchwarz'stheorem,them follorwing(ddnc+1)(ddc+2)d=c2m symmetryrelationsd@ iJdv jPc=3@d@ jdv ic,form di;jc=0d;:::ʜ;dm caswrell.OneOcrhoiceof(dd:c+2)Oequations,whichleadstoasymmetrichypSerbolicOsystem,requirestakingderivXativresoftheproSductin(10)(summationoverdi;j%c=UR1d;:::ʜ;dc): d@0dv0ʦeda ijZd@ iJdv jvc+d@0dv0ʦec(d@ iJda ijZc)dv j!bc=URdgË:cThismwillbSethesecondequationofthesystem.FVorthe rst,wreshalljusttakethede nitionL͍of dv0c.TheremainingddcequationswillbSethesymmetryrelations,withoneindexbeing0,but]1mrultipliedbythematrixkA2> c=FkAc.|So,ythesystemweshallstudyreads(summationorverdi;j%c=UR1d;:::ʜ;dc):+*(11)Hd@0duedv0#c=UR0dtd@0dv0ʦeda ijZd@ iJdv jvc+db 0dv0ʦc+db jdv j#c=URdgrȍ [a ijZd@0dv iTeda ijd@ iJdv0#c=UR0d;,scwhereEdb20 c:=d@0d;b2jc:=ed@ iJda2ij c=[emdivIkA2>c]2jc,\fordjc=1d;:::ʜ;dc.JThissystemcanbSewrittenintherequiredform.Beforewritingitdorwn,letusnotethatthe rstequationin(11)istheonlyonewheredu cappSearsexplicitlyV.Thrus,we cansolvrethesystemfordv ic rst,andlaterusethesolutionOK8 z͍CH-measuresUUappliedtosymmetricsystemsuF'cinZordertoobtainduc.Thisreducesthesystemtoddӑc+1Zunknorwnsmvc=UR(dv0d;:::ʜ;vdJc)andddӑc+1L͍equations:LV(12)3!5(f2Ǎ5(6385(65(4ލ?(dPc0be|Wc0?@>0*@v.@v.@v.d~kA?@>c07f3Ǎ773877757d@0mvc+dfXNi=1f2Ǎ63864ލ) c0Ceda2i1g@eda2id ueda2i1**Xc.*X.*X.jk0eda2idf3Ǎ73875d@ iJmv=c+f2Ǎ63864ލ db20"db218hgeR(]db2dc0*U.U.U.<fk0c0^r2f3Ǎ^r2738^r27^r25hr1mvc=URf2ǍUR638UR6UR4ލ UQdg lc0*... l0cf3Ǎc738c7c5&d:NcItisclearthatkA2i=fcareallsymmetric,9kA20 cisevrenpSositivede nite(bSecausedt >c0andkAcisRpSositivrede nite).jThus,|wehavewrittenthewaveequationintheformofasymmetrichrypSerbolicsystem.rKRemark.=CSuchJasystemisJsymmetrichypGerbolicyC(seeJ[4])ifthereisavector\q~G >Csuchthat\q~Dk )KA^kCisapGositiveUUde nitematrix.qClearly*,\q~G C:=(1D;C0D;:::;C0)UUgives^F GDKM0 \0rKA#^F)5C,whichispGositivede nite.qύInuparticular,ithesystemtowhichwereducedthewaveequationisJhypGerbolicuinthesenseofPetrovski: CforeveryvectorG4Cthereisavector\q~G 1Csuchthatmatrices::KAC(GMVD;C):=DkKA^kHE]hD\qXC~Dk KKA^k(Chavesimpleelementarydivisors,UUandtheequationdetKAC(GMVD;C)=0UUhasrealsolutionsDC.N8N8?cIfwreassumethattheinitialdataweregivenforthewaveequationbyduc(0d;:c)UR=du0candL͍du20c(0d;:c)UR=du1c,wrecantake: hdv0c(0d;:c);=URdu1L͍mdv iJc(0d;:c);=URd@ iJdu0 ʢd;cfordic=1d;:::ʜ;d gcastheinitialdataforthesystem(12).Therelationduc(0d;:c)6=du0 %cdeterminestheinitialconditionforthetimederivXativreofduc.Duetothefactthatdu0cisde nedonkR2dc,ĀwrecancomputeitsderivXativesinthespatialdirections. JWVevshouldstillcrheckvwhethertheidenrtitiesde ningdvd cc(andthereforethesymmetryrelations)arevXalid.FVoranrydiURc=1d;:::ʜ;dcwehave:΍d@0dv ic=URd@ iJdv0uPc=d@ id@0duc=d@0d@ iJdu:c(TheV rstequalitryfollowsfromtheregularityofkA2>c,tbSecausekA2>c(d@0mve}rVxj0$dv0c)UR=0VimpliesL͍d@0dv iWc=Rd@ iJdv0c.)-WVeknorwd@0c(dv iދe4Ad@ iduc)R=0,Handthatdv iދe4Ad@ iduRc=0atdtRc=0,HthrusthelastidenrtityholdsforanrydtUR>c0.'LetusnorwapplythegeneralresultforH-measurestothesystem(12)(notethatforhnotationalconrveniencehwedenotekx*c=(dx20d;kx20c)=(dx20d;x21d;:::ʜ;x2dJc)handgzc=*(d0d;g\͟20`[c)=(d0d;1d;:::ʜ;dJc),thrusworkinginddc+1dimensions).ThesymrbSolofthedi erentialopSeratoris:gčvkPc(kxd;g\c)UR=dkpkA kc(kxc)UR=qfٙd0d4kec(kAg20`[c)2> ekAg20Ad0kAdB qfod:!?cWVe2assumethatthesequencemvn *ed*m0cwreaklyinthespaceL22c(kR+0n4ekR2d]c)J,eachmvncsatisfyingthesystem(12),andthatthesequencede nestheH-measure: pdgURc=qfٙ ad00)-g01 g10)-g11=繟qf qc(whered00 cisa1ec1bloScrk,whileg11cisaddeddcbloScrk).OK9 z͍CH-measuresUUappliedtosymmetricsystemsuF'cTheloScalisationpropertrygivesus:卑Lk0URc=kPgc=qfVȆ Bd0d00 jekAg\͟20 eVc*g10d0g01 jec(L11 kAg\͟20`[c)*>L͍ ed00 kAg\͟20 c+d0kAg10zh^ekAg\͟20 e g2>L01 jc+d0kAg11}qf(8d:cThisgivresusthefollowingrelationsbSetweencompSonentsoftheH-measuregc:3nXEd0d00Ќc=URkAg\͟ 0 eVc*g10Svd0g01Ќc=UR(g >ڍ11 kAg\͟ 0`[c)*>L͍d00 kAg\͟ 0Ќc=URd0kAg10gKkAg\͟ 0 e g >ڍ01Ќc=URd0kAg113cThe4 rstidenrtity4isbSetrween4scalars,×thelastbetrween4matrices,×whileremainingtrwo4areL͍bSetrweenvectors.The#secondequalitrygivesus(aftertakingthehermitianconjugateofmatrices,'andusing$thehermitianpropSertryofH-measures)d0g10c=g11 kAg\͟20`[c.%UIfwemultiplythelastequalitrybyd0dkA21L01c=d0g\͟20e 6g\͟20`[dyc(wreusethealgebraicidenrtitykAg\͟20$e Dg10 Nc=URkAc(g\͟20e g10 c)).(@ThisgivresusasimpleexpressiongURc=ge g\dzc(atleastwhered0uPe6c=UR0).FVromUthe rstequalitrywe nallyget:F;d0c(d2s2L0ce4kAg\͟20eg\͟20`[c)d"wc=`0,whichUinthecasewhend0uPe6c=UR0givresusthatthesuppSortofdociscontainedinthejlightconecinthedualspace.KRemark.XCAtgthispGointwehavelostsomeinformationcontainedinthewaveequation,=bGecausewe discardedL0&d@ lkPed@lhgžc=URertdqerxc(d0dc)edqn9@0dF&@lhkPed@ lgžc=URd@0dqn9loc+erxdqertc(d0dc)~%d@lhkPed@ k;gdkpds lžc=URerxdqeg\c(d0dloc+ertc(d0dc)eg\c)Kd@lhkPegds lžc=URd@ k;d@lhkPegdkpds lL͍žc=URerxdqeg\c(d0dc)gtڸ(yl|2kSed@lhkA lc) Oegžc=URd@0dqn9;Gcwhicrhgivestheresult.VakQ.E.D.ƍKRemark. CTheUUequationinthestatementofthetheoremcanalsobGewrittenintheformO|Z^EfDq[;0|sDEg8C+(ErxGwDqEGMVC)bF*G6ErC(D0DC)+(DdC+2)(D0DC)bFcC+D@0Dq[jC=2MRe D UP:CThisresultisageneralisationoftheresultinT*artar[12,)3.3]andFrancfort&Murat[3,(3.28)].o#Under afstrongerassumptionthatDCandKACdonotdepGendonDtC=Dx^0|sC,thesefresultscoincide(forexample,withD@0|sDq"C=0UUandzerorighthandside,theabGovecoincideswith(3.28)of[3])._N8N8+v6kTheMaxwellsystem90cWVeXshallnorwpresentamorecomplicatedexample|thesystemofMaxwell'sequationsL͍inamaterialwithelectricpSermeabilitrygc,conductivityg%icandmagneticsusceptibilitygc(wreassumethatgc,g candgcdepSendonlyonspacevXariableskx20Xc=UR(dx21d;x22d;x23c)).>Thesystemreads:(13)͍2mD 0Йc=URmrotgHemJc+mFL͍NB 0Йc=URemrotEc+mGd;!jctogetherwithmdiv4cDURc=dc,mdivBc=0,andwithconstitutivrelaws:)1s3mDc(d:;tc)u=URgmEc(d:;tc)L͍mJc(d:;tc)u=URgqmEc(d:;tc)k#mBc(d:;tc)u=URgmHc(d:;tc)d:2BcChoSosing(mEcandmHcasunknorwnfunctionsandintroSducingmuc:=qfٙ ~mE DHPqfc,87the(system(13)cančbSewritteninaformofasymmetricsystem:$Ӎ 3x_fXi=0͵kA iJd@ imuc+kBmuURc=mf&5d;$acwhere:덑kA 0uPc=URqfٙ $Bg k0 0*g(/qf4Sd;kA 1c:=URqfٙmk0&*Q2>L1 kQ1,8k0:yqfBd;kA 2c:=URqfٙmk0&*Q2>L2 kQ2,8k0:yqfd;kA 3c:=URqfٙmk0&*Q2>L3 kQ3,8k0:yqfd: @cTheconstanrtantisymmetricmatriceskQk caregivenby:'>w?bkQ1uPc:=9URf2UR43 UQc0E05?0 UQ0E009ec1 UQ0E15?09Aʊf3Aʊ5Kʉd;kQ2c:=9URf2UR43c0(J0910(J090 UQec1(J0909Aʊf3Aʊ5d;kQ3c:=9URf2UR43 UQc0Eec190 UQ1#090 UQ0#0909Aʊf3Aʊ5d:PK11 :z͍CH-measuresUUappliedtosymmetricsystemsuF.)cThematrixkBcisoftheform:lkBc=qfٙ g Ck0L͍ G0 C0)qf/Xc, #whiletherighrthandsideismfRpc=qfٙ kUmFL͍ G0qfc.InthefpabSorvewehaveusedthefactthattherotator(curl)ofavector eldmEccanbSewrittenas:(ߍmrotEURc=9f243 UQd@2dE23~ed@3dE22L͍ UQd@3dE21~ed@1dE23 UQd@1dE22~ed@2dE219Q.f3Q.5] c=9f243 UQc0E05?0L͍ UQ0E009ec1 UQ0E15?09Aʊf3Aʊ5Kʉd@1mEc+9f243USc0'09?1L͍US0'09?0 ec1'09?09Af3A5Kd@2mEc+9f243 c0Jec19?0L͍ 1"G09?0 0"G09?09Af3A5Kd@3mEURd:(pcIf!wreassumeduniformbSoundednessandsymmetryofpermeabilitryandsusceptibilityL͍tensors,theabSorvesystemwrouldevenbSesymmetrichypSerbolic.m5InK@ordertoapplytheH-measuretheorywreshouldconsiderasequencemun ied*URm0cweaklyin]othespace(L22c( )e)26}mc(i.e.mEnpd;mHn ied*URm0c). Therighrthandsidetermmf.Rcisallowedtooscillateaswrell;sotakemfn ied*URm0cweaklyinthespace(L22c( )e)26c.TheMH-measurecorrespSondingto(asubsequenceof8)thesequence(munpc)willbedenotedbry:Zg11 Nc=URqfٙge+Igem gme-p c=URekc),sokPcisasymmetricmatrix.TheloScalisationpropertrystatesthatkPg11 Nc=URk0c.8Inourcasethistakestheform:!.Xqfٙ6d0gRek:5 R|^d0gg<qfoqٙ{uge~gemwZgmegmeqfQc=URqfٙ d0gewc+k2>gmef;d0gemc+k2>gm*kgewc+d0gmehkgemc+d0gm懟qfđ0c=URk0d:!=dcInrtordertosimplifytheabSorvertequations,hletus rststudythecasewhered0 \uc=sod@0kA20uPc=URk0c.AskQkTcareconstanrtmatrices,wrehaved@kpkA2kc=URk0c.ThepropagationpropSertrythusreads:ZzZehBg11 d;efkPd; n9egc2d kSeiBc+ehUSc2mRetr/ g12 d; n9eiK/c=UR0d:cNextwrecomputethePoissonbracket:rsefkPd; n9eg'c=URd@ lkPd@lhd ed@ ld n9@lkP5O'c=GURf2ǍUR638UR6UR6UR6UR6UR6UR4ۙLڍAd@0d n9g%e9f243m|c04zed@3d a@2d L͍@3d ?nc0\ed@1d ed@2d 9%@1d hc`c09{f3{5,3N9 UQ2 UQ43"c0?Bed@3d l@2d L͍@3d K=c0hK4ed@1d UPed@2d Dz@1d sc09,f3,5W#d@0d n9gG)Tf3Ǎ)T738)T7)T7)T7)T7)T7)T54ved0qfٙuQd@lhgd@2ld Kk0L͍0<d@lhgd@2ld cezqfZf'c=GURf2ǍUR638UR6UR6UR6UR6UR6UR4ۙLڍ$>d@0d n9ged0d@2ld n9@lhg%e9f243m|c04zed@3d a@2d L͍@3d ?nc0\ed@1d ed@2d 9%@1d hc`c09{f3{5,3N9 UQ2 UQ43"c0?Bed@3d l@2d L͍@3d K=c0hK4ed@1d UPed@2d Dz@1d sc09,f3,5pd@0d n9ged0d@2ld n9@lhgG)Tf3Ǎ)T738)T7)T7)T7)T7)T7)T5:vd:{͍cTVaking&inrtoaccountthatkSc=Fu'1'z2 Vc(TkBc+kB2c)@lp=qfٙFu D|1 D|z2c(ogoc+gqǟ2c)Nk0L͍$?s0N0Wqf^c,6and&thebloSckformofthemeasureg11 c,wreget3J]fX fD ged;@0d n9ged0d@ ld n9@lhg fE•c+ fD gm6d;@0d n9geds@ ld @lhg fE}ehUSged; n7c(goc+gqǟ c)0)eiWc+ehUSc2mRetr/ g12 d; n9ei$f{•c+zf* UVgmeegem d;9f243m|c04zed@3d a@2d L͍@3d ?nc0\ed@1d ed@2d 9%@1d hc`c09{f3{5ڟz+ֲc=UR0d:3J[cAfter&placingallthederivXativresfromthetestfunctiond ,_conthemeasures,3theabSoveL͍equalitryreads *ged@0gewed0d@lhged@ lڍtgewc+(ed@x8dwRc+d@y`dvc=UR0d:cTheunknorwnducappSearsonlyinthe rstequation;sowecantakethisequationasL͍itsTde nition(assumingthattheinitialconditionforducisgivren),o/andsolvethesystemoftrworemainingequations, withunknorwsdvA9canddwRc.WVeintroSducethevectornotationforunknorwns:qadu1 Ec:=_Gdvn9;u2c:=dwRc. (FVorvXariables,wreoSccasionallywritekxcfor(dx;yn9c)andgcfor(ds;n9c).)WVeharvearrivedatanequivXalentsystem:(15)Rfqfٙda=`c0>0ec1qfZd@x8muc+qfٙ c01 10$qf,@d@y`muURc=m0d;cwhicrhissymmetric,andfordaURedOqf[?Md::kLemma*3. jTheH-measuregjcorrespSondingtoasubsequenceofacL22jwreaklyconvergentsequence|ofsolutionsofthesymmetricsystem(15),assoSciatedtotheequation(12),canbSewrittenas:fogURc=qfٙ 7ds22#["ds n9#?n9221Kqf9d;jwhere[dbjisanonnegativre(scalar)Radonmeasureonc edSן21j,suppSorted[insidethesetdN6c=URef USc(dx;yn9;s;c)URe2c edSן21 ('c:URdas22_c+d22c=UR0egj.!mDem.<cFVromatheloScalisationpropertrykPgc=k0c,9OsoagcissupportedinsidethesetwheredetQkPic=0,candKthisreducestothesetdN@c.[WVritingallthetermsoftheproSductexplicitlywreget:ÍCqfٙL2ndahc0)d2c=0.iInthejparabSolicregionc(dac=0)L͍thesuppSortofdocisconrtainedinthesetwheredËc=UR0(andthusdc=URec1).InLthehrypSerbolicLregion(daφC0,[paraboliconthelineDy+_C=φ0andhyperbolicinthelower halfUUplaneDy"<C0.TheNcharacteristicsofT*ricomi'sequation(intheclosedlowerhalfplaneonly)aresolutionsoftheordinaryUUdi erentialequation:qDy[dy^2,C+8Ddx^2CC=0;UUorequivqalently(EDy"EC0):Z<$Ddxԟwfe 뎟 (֍:dyC=ErpUWrfe {Dy&:r׍CF*orWDy+>C0(uppGerhalfplane,ellipticregion),DsC=0.NAtthecoGordinatelineDxC(parabolicJregionC),DCis suppGortedontwo oppositepointsonthecircleDS^1C,8/namelyforDQiC=0(andthusDkWC=EC1).:F*orDyQiEfC(EDy[C)Duǟ^2"Cor, forgivenDy[C, ontheintersectionUUofthecircleDS^1eUCwiththelinesDE8mp 7mfe DyMDdx%zwfe  (֍d#>MC=2<$lsin*2/D#33wfet (֍Ccos cr4D#<$y dy%zwfe  (֍d#>MC=EC2<$sinkD#33wfet (֍Ccos cr3D#>:& mCWithUUappropriateinitialconditionsthesolutionsareLqˋDxC=<$K2Kwfe (֍3 -tg43D#Cand8yDy"C=ECtg 2 D#:/؍CLetusconstructameasureD;Cwhichisasolutionoftheequation(18).;Usingtheparametrisationof theIsuppGort,wetakethefollowingJansatz/SCforDC: DwDC(D#C)D`C( 33233&fes3bCtgiRH3iD#;ECtg RH2 D#;#C).XT*akingID'GXE2CCcbF8KR^2El-h 3333&fe -p2 D; ۟&feox~2}EibFWC,theUUactionofDKConD'CcanbGewrittenas:ՍuEhD;'EiC=cFZqDwDC(D#C)D'^F<$ :OC2 :Owfe (֍3mtgt3D#;ECtg 2 D#;#^FVDd#:VCAssumingUUthat\qa}~D E2CCcbF8KR^2SE8h 3333&fe -p2 D; ۟&feox~2}EibFWC,wehave:6h#1T<$*[Dd'ewfe  (֍d#\q5C~3D bF"C2sin Ɵ*2D#@x\q C~D !-E8C2cosD#CsinnD#@y\qگC~·D C+coso4ÊD#@#\q``C~T8D 3bF%eD: K17az͍CH-measuresUUappliedtosymmetricsystemsuF'Thus,UUtheequation(18)reducesto:7cFZ'BDwDC(D#C)cos745RD#<$(>d33wfe  (֍d#\qC~ D ^F<$C2wfe (֍3$tg-32UjD#;ECtg 2 D#;#^FkIDd#8C+cFZ DwC(D#C)4cos735RD#CsinnD#\q (C~D ^F<$C2wfe (֍3Ltg T 3$|D#;ECtg 2 D#;#^F]Dd#C=0D:CThealastequationiscertainlytruefor:#DwDC(D#C)[=DCC,ĤwhereaDCe}Cisanarbitraryconstant.|So,themeasureDC, givenUUby:q΍EhD;'EiC=cFZqDC'^F<$ :OC2 :Owfe (֍3mtgt3D#;ECtg 2 D#;#^FVDd#;CisyWasolutionoftheequationinthecorollary*.(sItmovesyWalongacharacteristics,Wstartingfrom(E1D;E1D;E 3333&feox~2SC),bGouncesUUattheDxCaxis,andcontinuesUUtowards(E1D;E1D; ۟&feox~2}C).N8N8KRemark.BfCAszitwaskindlynotedbythereferee,6thetranspGortequationsforsecondorderpartialdi erentiallequationswithrealprincipalsymbGolcanbeobtainedbyadi erentmethoGd,qindependentlylofaUUpGossiblereductiontosymmetricsystems(cf.[3],Remark3.15).6<N8N8 L͍JI7thank[professorLucT*artarforsuggestingtheresearchtopicandforconstantencouragementandL͍advice>EwhilewritingthispapGer.,ItwasaninterestingexpGeriencetowitnessthedevelopmentofanewtheory*,UUH-measures.&L͍kReferences# C[1]3\DNenad8AntoniGc:MemoryE ectsinHomogenisation:LinearSecond-OrderEquations,qJArch.Ra- 3\DtionalUUMech.Anal.C,K125C,1{24(1993)# [2]3\DLawrence)CraigEvqans:JW*eakconvergencemethoGdsforNonlinearPartialDi erentialEqua-3\Dtions;CCBMSUUvol.74,AMS,Providence,1990.# [3]3\DGillesI:F*rancfort,3Fran9coisMurat:YOscillationsandEnergyDensitiesintheWaveEquation,3\DJComm.V\IPartialUUDi erentialEquationsC,K17C,1785{1865(1992)# [4]3\DKurt8O.F*riedrichs:SymmetricHypGerbolic8LinearDi erentialEquations,qJComm.PureAppl.Math.C,3\DVIGI,UU345{392(1954)# [5]3\DKurt8O.F*riedrichs:SymmetricPositiveLinearDi erentialEquations,qJComm.PureAppl.Math.C,3\DXI,UU333{418(1958)# [6]3\DPatrick`GGerard:-LCompacite`parcompGensationetregularite`2-microloGcale,JSeminaire`Equations3\DauxUUDGeriveesUUPartielles1988{89(EcolePolytechnique,Palaiseau,exp.qVI)# C[7]3\DPatrickGGerard:IMicroloGcalDefectMeasures,JComm.PartialDi erentialEquationsC,K16C,1761{3\D1794UU(1991)# [8]3\DDavid*Gilbarg,NeilS.T*rudinger:?qJEllipticPartialDi erentialEquationsofSecondOrderC,3\DSpringer-V*erlag,UUBerlin,1983.# [9]3\DLars,Hormander:]oJTheAnalysisofLinearPartialDi erentialOpGeratorsI{IV,CSpringer-V*erlag,3\DBerlin,UU1983{1985.[10]3\DCathleengS.Morawetz: AgUniquenessgTheoremforF*rankl'sproblem,`JComm.PureAppl.Math.C,3\DXI,UU697{703(1958)[11]3\DLucCT*artar:CRemarksonhomogenization, inJHomogenizationande ectivemoGduliofmaterials3\DandUUmediaC,(ed.J.L.Eriksenetal.),IMAvol.1(228{246),Springer-V*erlag,NewYork,1986.[12]3\DLucxT*artar: H-measures,anewapproachforstudyinghomogenisation,oscillations,andcon-3\DcentrationTe ectsinpartialdi erentialequations,JProGc.RoyalSoGc.EdinburghC,K115AC,193{2303\D(1990)[13]3\DLucT*artar:H-measuresandapplications,JProGc.Int.CongressofMathematicians,Kyoto,1990,3\Dvol.UU2,1215{1223,CMathematicalsoGcietyofJapan,1991.[14]3\DF*ran9cois=TrGeves:JIntroGductiontopseudodi erentialandF*ourierintegralopGeratorsC,6Plenum3\DPress,UUNewY*ork,1980.[15]3\DF*rancescoTricomi:GSulleequazioniallederivqateparzialidi2^0Cordine,ditipGomisto,JMemorie3\DdellaUUR.AccademiaNazionaledeiLinzeiC,ser.V,vol.XIV,fasc.VIGI(1923)[16]3\DAlexanderW*einstein:TheSingularSolutionsandtheCauchyProblemforGeneralizedTricomi3\DEquations,UUJComm.PureAppl.Math.C,VIGI,105{116(1954)PK18|z͍CH-measuresUUappliedtosymmetricsystemsuF'3J-cDepartmenrtofMathematicsL͍3J-UnivrersityofZagreb3J-BijenirsckXacesta303J-Zagreb,Croatia%03J-l߆T cmtt12lnenad@anak.math.hrPK19;D{Nff cmbx12mR6 cmss12l߆T cmtt12kN cmbx12j}h! cmsl12hɌ cmbsy10gDF cmmib10fu cmex10e!", cmsy10dg cmmi12cXQ cmr12Mm#R cmss10K"V cmbx10Jp0J cmsl10GDF cmmib10Fu cmex10E!", cmsy10D b> cmmi10CK`y cmr10=ow cmss9;t : cmbx97W6 cmmib95 cmsy945" cmmi93o cmr9f$cmbx7*cmmib7O!cmsy70ercmmi7ٓRcmr7O \cmmi5