幾類微分算子的Friedrichs擴(kuò)張及其辛幾何刻劃的中期報(bào)告

幾類微分算子的Friedrichs擴(kuò)張及其辛幾何刻劃的中期報(bào)告Abstract:TheFriedrichsextensionofdifferentialoperatorsisawell-studiedoperatortheoreticalconceptthathasbeenextensivelyusedinthestudyofpartialdifferentialequations.ThisreportpresentsapreliminaryinvestigationintotheFriedrichsextensionofaclassofdifferentialoperatorsanditssymplecticgeometrycharacterizationinthemid-term.Introduction:TheconceptofaFriedrichsextensionofadifferentialoperatorwasfirstintroducedbytheGermanmathematicianKurtFriedrichsinthe1930s.Thisextensionisawayofaddingboundaryconditionstoadifferentialoperatorthatmaynothavewell-definedboundaryconditionsassociatedwithit.TheFriedrichsextensionensuresthattheoperatorisself-adjoint,whichisadesirablepropertyinmanycontexts,especiallyinthestudyofpartialdifferentialequations.Inthisreport,wefocusontheFriedrichsextensionofaclassofdifferentialoperatorsthatariseinavarietyofphysicalapplications.Inparticular,weconsideroperatorsoftheformL=-∑i=1n(ai(x)?xi+bi(x))?xi+c(x),(1)wherex=(x1,x2,...,xn)∈?n,ai(x),bi(x)andc(x)aresmoothfunctionson?n.Thisclassofoperatorsarisesinmanyphysicalcontexts,suchasquantummechanics,fluiddynamics,andelasticity.FriedrichsExtension:ToconstructtheFriedrichsextensionoftheoperatorL,wefirstconsidertheoperatorL*givenbyL*=-∑i=1n(?xi(ai(x)u)+bi(x)u)?xi+c(x)u,(2)whereu∈H1(?n)isatestfunction.Here,H1(?n)denotestheSobolevspaceofsquareintegrablefunctionswhosegradientalsoissquareintegrable.TheFriedrichsextensionLFofListhendefinedastheself-adjointoperatorobtainedbyaddingcertainboundaryconditionstoL*.Specifically,werequirethatthedomainofLFsatisfytheboundaryconditions(?/?ni)u+(∑i=1n(ai(x)ni+bi(x))u)=0(3)ontheboundaryof?n,whereni=(0,...,0,1)istheoutwardunitnormalvectortotheboundaryatx.TheoperatorLFisthendefinedastheself-adjointoperatorobtainedbyrestrictingL*tothesetoftestfunctionsthatsatisfytheboundaryconditions(3).SymplecticGeometryCharacterization:Insymplecticgeometry,theHamiltonianflowofaHamiltonianfunctiononaphasespaceisgovernedbyasetofHamilton'sequations,whicharepartialdifferentialequationsoftheform(?H/?qi)(x)=-?pi(x)/?xj,(?H/?pi)(x)=?qi(x)/?xj,whereH(q,p)istheHamiltonianfunction,and(q,p)∈?2nisthephasespace.TheFriedrichsextensionLFoftheoperatorLhasanaturalsymplecticgeometrycharacterizationasaHamiltoniansystemonthephasespaceH1(?n)×H-1(?n)withthesymplecticformω((u1,p1),(u2,p2))=∫?np1(x)?u2(x)/?x-p2(x)?u1(x)/?xdㄧ.Here,H-1(?n)denotesthedualspaceofH1(?n).TheHamiltonianfunctionofthesystemisgivenbyH(u,p)=1/2∫?n(L*u)u+dΣ,(4)wheredΣisthesurfacemeasureontheboundaryof?n.Conclusion:WehavepresentedapreliminaryinvestigationintotheFriedrichsextensionofaclassofdifferentialoperatorsanditssymplecticgeometrycharacterizationinthemid-term.WehaveshownthattheFriedrichsextensionoftheoperatorLisaself-adjointoperatorobtainedbyaddingappropriateboundaryconditionstoacorrespondingadjointoperator.WehavealsoshownthattheFriedrichsextensionhasanaturalsymplecticgeometrycharacterizationasaHamiltoniansystemonanappropriatephasespace.OurfutureworkwillfocusonexaminingthepropertiesoftheHamiltoniansystemassociatedwiththeFriedrichsextension,suchasi