A lower bound for sums of eigenvalues of the Laplacian (2)

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1、PROCEEDINGSOFTHEAMERICANMATHEMATICALSOCIETYVolume131,Number2,Pages631{636S0002-9939(02)06834-XArticleelectronicallypublishedonSeptember25,2002ALOWERBOUNDFORSUMSOFEIGENVALUESOFTHELAPLACIANANTONIOSD.MELAS(CommunicatedbyBennettChow)Abstract.Letk()bethekthDirichleteige

2、nvalueofaboundeddomaininRn.AccordingtoWeyl'sasymptoticformulawehave()sC2=nkn(k=V()):TheoptimalinviewofthisasymptoticrelationlowerestimateforthesumsPk()hasbeenprovenbyP.LiandS.T.Yau(Comm.Math.Phys.88j=1j(1983),309-318).Herewewillimprovethisestimatebyaddingtoitsrigh

3、t-handsideatermoftheorderofkthatdependsontheratioofthevolumetothemomentofinertiaof.1.IntroductionLetRnbeaboundedopensetandlet0<()():::denotethe12eigenvalues(repeatedwithmultiplicity)oftheDirichletLaplacianon,thatis,oftheeigenvalueproblem(1.1)
u+u=0in;u=0on@:Th

4、easymptoticbehaviorofk()ask!1relatestogeometricpropertiesoftheopenset.InfactWeyl'sasymptoticformulaassertsthatk2=n(1.2)k()Cn()ask!1V()whereV()isthevolumeofandC=(2)2!2=nnnwith!nbeingthevolumeoftheunitballinRn.Polyaprovedin[4]thattheaboveasymptoticrelationisinfa

5、ctaone-sidedinequalityifisaplanedomainthattilesR2(andhisproofalsoworksinRn)andheconjectured,foranydomaininRn,theinequalityk2=n(1.3)k()Cn()V()forallk1.InthisdirectionLieb[3]provedaninequalitylike(1.3)foranydomaininRnbutwithaconstantC~nthatdiersfromtheconstantCnby

6、afactor.ReceivedbytheeditorsAugust28,2001.2000MathematicsSubjectClassication.Primary58G25;Secondary35P15,58G05.c2002AmericanMathematicalSociety631632ANTONIOSD.MELASThenP.LiandS.T.Yau[2]provedthatontheaverage(1.3)istrueforanydomaininRn,thatis,XknCnn+22(1.4)j()knV

7、()nn+2j=1whichissharpinviewofWeyl'sasymptoticformula.Thisalsogivesalowerboundforeachindividualeigenvalue,betterthanpreviouslyknownandtendingtobeoptimalasthedimensionn!1.ThisinequalitywascomplementedbyP.Kr•ogerin[1]whogaveanupperboundforthesumsoftheeigenvaluesdependi

8、ngongeometricpropertiesofthathavetodowiththebehaviorofthevolumeofthe"-neigbourhoodsoftheboundary@.Usingthisheobtainedclosetoopti-malestima