几类微分算子的Friedrichs扩张及其辛几何刻划的中期报告

几类微分算子的Friedrichs扩张及其辛几何刻划的中期报告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)∈ℝn,ai(x),bi(x)andc(x)aresmoothfunctionsonℝn.Thisclassofoperatorsarisesinmanyphysicalcontexts,suchasquantummechanics,fluiddynamics,andelasticity.FriedrichsExtension:ToconstructtheFriedrichsextensionoftheoperatorL,wefirstconsidertheoperatorL*givenbyL*=-∑i=1n(∂xi(ai(x)u)+bi(x)u)∂xi+c(x)u,(2)whereu∈H1(ℝn)isatestfunction.Here,H1(ℝn)denotestheSobolevspaceofsquareintegrablefunctionswhosegradientalsoissquareintegrable.TheFriedrichsextensionLFofListhendefinedastheself-adjointoperatorobtainedbyaddingcertainboundaryconditionstoL*.Specifically,werequirethatthedomainofLFsatisfytheboundaryconditions(∂/∂ni)u+(∑i=1n(ai(x)ni+bi(x))u)=0(3)ontheboundaryofℝn,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)∈ℝ2nisthephasespace.TheFriedrichsextensionLFoftheoperatorLhasanaturalsymplecticgeometrycharacterizationasaHamiltoniansystemonthephasespaceH1(ℝn)×H-1(ℝn)withthesymplecticformω((u1,p1),(u2,p2))=∫ℝnp1(x)∂u2(x)/∂x-p2(x)∂u1(x)/∂xdㄧ.Here,H-1(ℝn)denotesthedualspaceofH1(ℝn).TheHamiltonianfunctionofthesystemisgivenbyH(u,p)=1/2∫ℝn(L*u)u+dΣ,(4)wheredΣisthesurfacemeasureontheboundaryofℝn.Conclusion:WehavepresentedapreliminaryinvestigationintotheFriedrichsextensionofaclassofdifferentialoperatorsanditssymplecticgeometrycharacterizationinthemid-term.WehaveshownthattheFriedrichsextensionoftheoperatorLisaself-adjointoperatorobtainedbyaddingappropriateboundaryconditionstoacorrespondingadjointoperator.WehavealsoshownthattheFriedrichsextensionhasanaturalsymplecticgeometrycharacterizationasaHamiltoniansystemonanappropriatephasespace.OurfutureworkwillfocusonexaminingthepropertiesoftheHamiltoniansystemassociatedwiththeFriedrichsextension,suchasi