EXTRINSIC ESTIMATES FOR EIGENVALUES OF THE LAPLACE OPERATOR. 1. introduction

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1 EXTRINSIC ESTIMATES FOR EIGENVALUES OF THE LAPLACE OPERATOR DAGUANG CHEN AND QING-MING CHENG* Abstract. For a bouded domai i a complete Riemaia maifold M isometrically immersed i a Euclidea space, we derive extrisic estimates for eigevalues of the Dirichlet eigevalue problem of the Laplace operator, which deped o the mea curvature of the immersio. Further, we also obtai a upper boud for the (k + 1) th eigevalue, which is best possible i the meaig of order o k. 1. itroductio Let R be a bouded domai i a -dimesioal Euclidea space R. The Dirichlet eigevalue problem of the Laplacia is give by { u λu, i (1.1) u 0, o. It is well kow that the spectrum of this problem is real ad purely discrete 0 < λ 1 < λ 2 λ 3, where each λ i has fiite multiplicity which is repeated accordig to its multiplicity. The ivestigatio of uiversal iequalities for eigevalues of (1.1) was iitiated by Paye, Pólya ad Weiberger [13] ad [14]. They proved λ k+1 λ k 4 k λ i. (1.2) Although the result of Paye, Pólya ad Weiberger has bee exteded by may mathematicias i several way, there are two mai cotributios due to Hile ad Protter [9] ad Yag [15]. I 1980, Hile ad Protter improved the result of Paye, Pólya ad Weiberger to λ i k λ k+1 λ i 4. (1.3) 2001 Mathematics Subject Classificatio: 35P15. 58C40 * Research partially Supported by a Grat-i-Aid for Scietific Research from the Japa Society for the Promotio of Sciece. Key words ad phrases. uiversal iequality for eigevalues, Yag-type iequality, trial fuctio. 1

2 Further, Yag [15] (cf. [6]) has obtaied very sharp iequality, that is, he has derived From (1.4), oe ca ifer (λ k+1 λ i ) ( λ k+1 (1 + 4 )λ i) 0. (1.4) λ k+1 1 k (1 + 4 ) λ i. (1.5) The iequalities (1.4) ad (1.5) are called Yag s first iequality ad secod iequality, respectively (cf. [1], [2]). By makig use of the Chebyshev s iequality, it is ot difficult to prove the followig relatio (1.4) (1.5) (1.3) (1.2). I [1] ad [2], Ashbaugh has also give a differet proof. O the other had, from the Weyl s asymptotic formula, oe has λ k 4π 2 k 2, k, (ω vol) 2 where ω is the volume of the uit ball i R. Further, Pólya cojectured eigevalue λ k should satisfy 4π 2 λ k k 2 (ω vol) 2, for k 1, 2,. O the cojecture of Pólya, Li ad Yau [12] attacked it ad obtaied λ k 4π 2 k (ω vol) 2, for k 1, 2,. Recetly, Cheg ad Yag [6] have obtaied a very sharp upper boud of λ k+1, that is, they have proved λ k+1 C 0 (, k)k 2/ λ 1, where j/2,1 2, for k 1 C 0 (, k) j/2 1,1 2 a(mi{, k 1}) 1 +, for k 2 ad a(1) 2.64 ad a(m) log(1 + m 3 ) for m 2 is a costat depedig 50 oly o m, ad j p,k deotes the k th positive zero of the stadard Bessel fuctio J p (x) of the first kid of order p. From the Weyl s asymptotic formula, we kow that the upper boud of Cheg ad Yag is best possible i the meaig of order o k. It is atural ad importat to obtai uiversal iequalities for eigevalues of the Dirichlet eigevalue problem o a bouded domai i a complete Riemaia maifold. Sice the Weyl s asymptotic formula also holds i this case, it is also importat to obtai the lower boud ad upper boud of λ k. 2

3 For the Dirichlet eigevalue problem of the Laplacia o a compact homogeeous Riemaia maifold or o a compact miimal submaifold i a sphere, may mathematicias have studied uiversal iequalities for eigevalues (for examples [3], [5], [7], [8], [10], [11], [16] ad so o). More recetly, Cheg ad Yag [3], [5] have derived uiversal iequalities for eigevalues of the Dirichlet eigevalue problem of the Laplacia o a domai i a sphere or i a complex projective space. The upper boud for λ k+1 ca also be obtaied by the same proof as i [6]. Ufortuately, for a geeral complete Riemaia maifold, it is very hard to fid a appropriate trial fuctio with ice properties such that oe ca ifer uiversal iequalities for eigevalues. Fortuately, we have the Nash s theorem: each complete Riemaia maifold ca be isometrically immersed i a Euclidea space. I this paper, we shall make use of this theorem to costruct appropriate trial fuctios with ice properties. By makig use of these trial fuctios, we derive uiversal iequalities for eigevalues ad the upper bouds for eigevalues, which is best possible i the meaig of order o k. Theorem 1.1. Let be a bouded domai i a -dimesioal complete Riemaia maifold M isometrically immersed i R N. For the Dirichlet eigevalue problem of the Laplacia: { u λu i, (1.6) u 0, we have (µ k+1 µ i ) 2 4 (µ k+1 µ i )µ i, (1.7) where µ i λ i H 2, λ i deotes the i th eigevalue of (1.6) ad H is the mea curvature vector field of M with H 2 sup H 2. Sice the formula (1.7) is a quadratic iequality of µ k+1, it is easy to ifer the followig: Corollary 1.1. Uder the same assumptios as i the theorem 1.1, we have ( µ k ) 1 µ i. (1.8) k Remark 1.1. Our uiversal iequality (1.7) is the Yag-type first iequality ad (1.8) is the Yag-type secod iequality. I particular, whe M is isometrically miimally immersed i R N, we have Corollary 1.2. Let be a bouded domai i a -dimesioal complete Riemaia maifold M isometrically miimally immersed i R N. The, we have (λ k+1 λ i ) (λ k+1 λ i )λ i. (1.9)

4 Remark 1.2. Sice R ca be see as a totally geodesic miimal hypersurface i R +1, we kow that the results of Yag is icluded i the corollary 1.2. Further, sice there exist may complete miimal submaifolds i R N, we kow that the Yag s iequalities for eigevalues also hold for ay bouded domai i ay complete miimal submaifold i R N. Sice the -dimesioal uit sphere S (1) ca be see as a totally umbilical hypersurface with the mea curvature 1 i R +1, from our theorem 1.1, we have (λ k+1 λ i ) 2 4 (λ k+1 λ i )(λ i ). which is the Yag-type iequality for eigevalues of the Dirichlet eigevalue problem of Laplacia o a domai i a uit sphere obtaied by Cheg ad Yag [3]. I order to obtai the upper boud for λ k+1, the uiversal iequality for lower order eigevalues of the eigevalue problem (1.6) is ecessary. Theorem 1.2. Uder the same assumptios as i the theorem 1.1, we have µ 2 + µ µ +1 µ (1.10) Corollary 1.3. Let be a bouded domai i a -dimesioal complete Riemaia maifold M isometrically miimally immersed i R N. The, we have λ 2 + λ λ +1 λ (1.11) Remark 1.3. Accordig to the same argumets as i the remark 1.2, we kow that the result for lower order eigevalues of Paye, Pólya ad Weiberger [14] does ot oly hold for a bouded domai i R, but also for a bouded domai i ay complete miimal submaifod i R N. Sice the upper boud for λ k+1 of Cheg ad Yag [6] does hold ot oly for eigevalues, but also for ay positive real umbers which satisfy Yag s first iequality ad the iequality of Paye, Pólya ad Weiberger which is same as (1.11), we ifer, from the theorem 1.1 ad the therorem 1.2, Theorem 1.3. Uder the same assumptios as i theorem 1.1, we have a(mi{, k 1}) µ k+1 (1 + )k 2/ µ 1, where the boud of a(m) ca be formulated as: a(0) 4, a(1) 2.64, a(m) log(1 + 1 (m 3)), for m I particular, for 41 ad k 41, we have µ k+1 k 2/ µ 1. 4

5 Especially, whe M is a complete miimal submaifold i R N, we have Corollary 1.4. Uder the same assumptios as i the corollary 1.2, we have ad whe 41 ad k 41, λ k+1 (1 + a(mi{, k 1}) )k 2/ λ 1 λ k+1 k 2/ λ Proof of Theorem 1.1 Throughout this paper we will agree the followig covetio o rages of idices: 1 i, j,, ; 1 α, β,, N; + 1 A, B,, N. Let M be a -dimesioal complete Riemaia maifold isometrically immersed i R N. Let M be a bouded domai of M ad P be a arbitrary poit of. Let (x 1,, x ) be a arbitrary coordiate system i a eighborhood U of P i M. Assume that y with compoets y α defied by y α y α (x 1,, x ), 1 α N, is the positio vector of P i R N. Sice M is isometrically immersed i R N, the ( g ij g x, ) i x j y α x i y, N α β1 y β x j y N β y α x i y α x j, (2.1) where g deotes the iduced metric of M from R N,, is the stadard ier product i R N. At the poit P, g( y α, y α ) i,j1 where is the gradiet operator o M. y α x i y α x j gij Lemma 2.1. For ay fuctio u C (M ), we have g ij g ij, (2.2) i,j1 ( g( y α, u)) 2 u 2, (2.3) ( y α ) 2 2 H 2, (2.4) y α y α 0, (2.5) where H is the mea curvature of M. 5

6 Proof. Let ad h deote the coectio of R N ad the secod fudametal form of M, respectively. We choose a ew coordiate system ȳ (ȳ 1,, ȳ N )( of R N give ) by y y(p ) ȳa such that ( ) ȳ 1 P,, ( ) ȳ P spa T P M ad at P, g, δ ȳ i ȳ j ij, where A (a α β ) O(N) is a N N orthogoal matrix. The we have x i x N j Therefore, from the formula of Gauss, we ifer 2 ȳ α x i x j ȳ α. (2.6) h A ij h A ( x, i x ) 2 ȳ A j x i x, (2.7) j where h A ij, deotes the compoet of the secod fudametal form h of x i x j ȳ A M. For u C (M ), at P, we have ( g( y α, u) ) 2 N [ ( g (y α (P ) + [ g ( N ( N β1 β1 ( N a α i a α i u 2. β1 )] 2 a α β ȳ β, u a α ȳ β u ) 2 β ȳ i ȳ i ) u ȳ i u ȳ i )] 2 a α βȳ β ), u where u 2 g( u, u). Sice P is ay poit, it fiishes the proof of (2.3). Let H be the mea curvature vector of M. The, from the stadard calculatio, we have y H. (2.8) Therefore, we derive ( y α ) 2 2 H 2. (2.9) Sice i y is taget to M, we have y α i y α y, i y 0. Thus, we have y α y α 0. (2.10) 6

7 The followig theorem of Cheg ad Yag [5] will play a importat role to prove our theorem 1.1 Theorem CY. Let λ i be the i th eigevalue of the Dirichlet eigevalue problem o a -dimesioal compact Riemaia maifold with boudary ad u i be the orthoormal eigefuctio correspodig to λ i. The, for ay fuctio f C 3 () C 2 ( ) ad ay iteger k, we have (λ k+1 λ i ) 2 u i f 2 (λ k+1 λ i ) 2 f u i + u i f 2, where f 2 M f 2 ad f u i g( f, u i ). Proof of Theorem 1.1 Let u i be the eigefuctio correspodig to the eigevalue λ i such that {u i } i N becomes a orthoormal basis of L 2 (). Put f α y α, 1 α N. Sice M is complete ad is a bouded domai, we kow that is a compact Riemaia maifold with boudary. From the theorem CY of Cheg ad Yag, we ifer (λ k+1 λ i ) 2 u i f α 2 (λ k+1 λ i ) 2 f α u i + u i f α 2. (2.11) Takig sum o α from 1 to N, we have N (λ k+1 λ i ) 2 u i f α 2 (λ k+1 λ i ) From (2.2) ad the lemma 2.1, we ifer N (λ k+1 λ i ) 2 u i f α (λ k+1 λ i ) 2 (λ k+1 λ i ) 2 y α u i + u i y α 2 (λ k+1 λ i ) (λ k+1 λ i ) { 4 y α u i 2 + u 2 i u i H 2 2 f α u i + u i f α 2. u 2 i y α 2 ( y α ) (λ k+1 λ i ) (λ k+1 λ i )λ i + 2 H 2 (λ k+1 λ i ), (λ k+1 λ i ) 2, } ( y α y α ) u 2 i where H 2 sup H 2. Therefore, we derive (λ k+1 λ i ) 2 4 (λ k+1 λ i )λ i + 2 H 2 (λ k+1 λ i ). 7

8 Puttig µ i λ i H, we obtai the iequality (1.7). 3. Proof of Theorem 1.2 I this sectio we shall give a proof of the theorem 1.2 Proof. Let u i be the eigefuctio correspodig to the eigevalue λ i such that {u i } i N becomes a orthoormal basis of L 2 (). Hece, u iu j δ ij for i, j N. We cosider the N N-matrix B (b αβ ) defied by b αβ yα u 1 u β+1, where y (y α ) is the positio vector of the immersio i R N. From the orthogoalizatio of Gram ad Schmidt, there exist a upper triagle matrix R (R αβ ) ad a orthogoal matrix Q (q αβ ) such that R QB. Thus, R αβ q αγ b γβ q αγ y γ u 1 u β+1 0, for 1 β < α N. (3.1) γ1 γ1 Defiig g α N γ1 q αγy γ, we have g α u 1 u β+1 q αγ y γ u 1 u β+1 0, for 1 β < α N. (3.2) γ1 Therefore, the fuctios defied by Ψ α (g α a α )u 1, a α satisfy Ψ α u β+1 0, for 0 β < α N. From the Rayleigh-Ritz iequality, we have, for 1 α N, From the defiitio of Ψ α, we derive g α u 2 1, for 1 α N (3.3) λ α+1 Ψ α 2 Ψ α 2. (3.4) Ψ α g α u g α u 1 λ 1 u 1 g α + λ 1 a α u 1. (3.5) Therefore, (3.4) ca be writte as (λ α+1 λ 1 ) Ψ α 2 ( g α u 1 2 g α u 1 )Ψ α. (3.6) From the Cauchy-Schwarz iequality, we obtai ( ) 2 ( g α u 1 2 g α u 1 )Ψ α Ψ α 2 ( g α u g α u 1 ) 2. (3.7) 8

9 Multiplyig (3.7) by (λ α+1 λ 1 ), we ifer, from (3.6), ( ) 2 (λ α+1 λ 1 ) ( g α u 1 2 g α u 1 )Ψ α (λ α+1 λ 1 ) Ψ α 2 ( g α u g α u 1 ) 2 ( ( g α u 1 2 g α u 1 )Ψ α ) ( g α u g α u 1 ) 2. Hece, we derive (λ α+1 λ 1 ) ( g α u 1 2 g α u 1 )Ψ α ( g α u g α u 1 ) 2. (3.8) From the lemma 2.1 ad the defiitio of g α, takig sum o α from 1 to N, we have ( g α u g α u 1 ) 2 2 { ( g α ) 2 u ( g α u 1 ) 2 + 2( g α g α ) u 2 1} H 2 u λ H 2. u 1 2 (3.9) From the Stokes theorem ad the defiitio of Ψ α, we coclude ( g α u 1 2 g α u 1 )Ψ α ( g α u 1 2 g α u 1 )(g α u 1 a α u 1 ) g α g α u (gα ) 2 u 2 1 g α 2 u 2 1. (3.10) By (3.8), (3.9), (3.10) ad (2.2), we deduce λ α+1 g α 2 u 2 1 (4 + )λ H 2. (3.11) For ay poit P, we use the same trasformatio of coordiates as i the proof of the lemma 2.1 y y(p ) ȳa. Sice A ad Q are orthogoal matrices, QA is also a 9

10 orthogoal matrix. Hece, we have, for ay α, g α 2 g( g α, g α ) q αγ q αβ g( y γ, y β ) β,γ1 q αγ q αβ g( a γ µ ȳ µ, β,γ1 β,γ,µ,ν1 µ1 ν1 a β ν ȳ ν ) q αγ a γ µq αβ a β ν g( ȳ µ, ȳ ν ) ( N ) q αβ a β 2 j 1. j1 β1 (3.12) Therefore, from (3.12), we have λ α+1 g α 2 λ i+1 g i 2 + λ +1 N A+1 λ i+1 g i 2 + λ +1 ( λ i+1 g i 2 + λ +1 λ i+1 g i 2 + λ i+1. From (3.11) ad (3.13), we ifer µ 2 + µ µ +1 µ g A 2 g i 2 ) (1 g i 2 ) λ i+1 (1 g i 2 ) (3.13) with µ i λ i H 2. Ackowledgmets. The authors would like to express their gratitude to Professor Hogcag Yag for his valuable suggestios. Refereces [1] Mark S. Ashbaugh, Isoperimetric ad uiversal iequalities for eigevalues, i Spectral theory ad geometry (Ediburgh,1998), E. B. Davies ad Yu Safalov eds., Lodo Math.Soc. Lecture Notes, vol. 273 (1999), Cambridge Uiv. Press, Cambridge, [2] Mark S. Ashbaugh, Uiversal eigevalue bouds of Paye-Polya-Weiberger, Hile-Prottter, ad H.C. Yag, Proc. Idia Acad. Sci. Math. Sci. vol. 112 (2002),

11 [3] Q. -M. Cheg ad H. C. Yag, Estimates o eigevalues of Laplacia, Math. A., 331 (2005), [4] Q. -M. Cheg ad H. C. Yag, Iequalities for eigevalues of a clamped plate problem, Tras. Amer. Math. Soc., 358 (2006), [5] Q. -M. Cheg ad H. C. Yag, Iequalities for eigevalues of Laplacia o domais ad compact hypersurfaces i complex projective spaces, J. Math. Soc. Japa, 58 (2006), [6] Q. -M. Cheg ad H. C. Yag, Bouds o eigevalues of Dirichlet Laplacia, Math. A., 337 (2007), [7] E. M. Harrell II ad P. L. Michel, Commutator bouds for eigevalues with applicatios to spectral geometry, Comm. i Part. Diff. Eqs., 19 (1994), [8] E. M. Harrell II ad J. Stubbe, O trace idetities ad uiversal eigevalue estimates for some partial differetial operators, Tras. Amer. Math. Soc., 349 (1997), [9] G.N. Hile, M.H. Protter, Iequalities for eigevalues of the Laplacia, Idiaa Uiv. Math. J. 29 (1980), [10] P. -F. Leug, O the cosecutive eigevalues of the Laplacai of a compact miimal submaifold i a sphere, J. Austral. Math. Soc., 50 (1991), [11] P. Li, Eigevalue estimates o homogeeous maifolds, Commet. Math. Helve., 55 (1980), [12] P. Li ad S. T. Yau, O the Schrödiger equatio ad the eigevalue problem, Comm. Math. Phys., 88 (1983), [13] L. E. Paye, G. Polya ad H. F. Weiberger, Sur le quotiet de deux fréqueces propres cosécutives, Comptes Redus Acad. Sci. Paris, 241 (1955), [14] L. E. Paye, G. Polya ad H. F. Weiberger, O the ratio of cosecutive eigevalues, J. Math. ad Phis., 35 (1956), [15] H. C. Yag, A estimate of the differace betwee cosecutive eigevalues, preprit IC/91/60 of ICTP,Trieste, [16] P. C. Yag ad S. T. Yau, Eigevalues of the Laplacia of compact Riemaia surfaces ad miimal submaifolds, A. Scuola Norm. Sup. Pisa CI. Sci., 7 (1980), Daguag Che Istitute of Mathematics Academy of Mathematics ad Systematical Scieces CAS, Beijig , Chia chedg@amss.ac.c Preset address: Departmet of Mathematical Scieces Tsighua Uiversity Beijig , Chia Qig-Mig Cheg Departmet of Mathematics Faculty of Sciece ad Egieerig Saga Uiversity Saga , Japa cheg@ms.saga-u.ac.jp 11