Theoretical athematics & Applications, vol.3, no., 03, 39-48 ISSN: 79-9687 print, 79-9709 online Scienpress Ltd, 03 Universal inequalities for eigenvalues of elliptic operators in divergence form on domains in complete noncompact Riemannian manifolds Yanli Li and Feng Du Abstract In this paper, we study the eigenvalue problem of elliptic operators in divergence form, and obtain some universal inequalities for eigenvalues of elliptic operators in divergence form on domains in complete simple connected noncompact Riemannian manifolds admitting special functions which include hyperbolic space. Especially, by using our universal inequalities, we can get the different universal inequalities including Yang inequality. athematics Subject Classification: 35P5, 53C4, 58G5 Keywords: Elliptic operators in divergence form, eigenvalues, universal inequalities, complete noncompact Riemannian manifolds, hyperbolic space School of Electronic and Information Science, Jingchu University of Technology, Hubei Jingmen 448000, P.R. China. School of athematics and Physics Science, Jingchu University of Technology, Hubei Jingmen 448000, P.R. China. Article Info: Received : April 0, 03. Revised : ay 30, 03 Published online : June 5, 03
40 Universal inequalities for eigenvalues Introduction Let be a bounded domain in an n-dimensional complete Riemannian manifold. Let be the Laplacian operator acting on functions on and consider the following eigenvalues problem for the Laplacian operator u = λu, in, u = 0, on, it is known that this eigenvalue problem has a discrete spectrum, 0 < λ < λ λ k,. where each eigenvalue is repeated with its multiplicity. When = R n, = n, Payne-Pólya-Weinberger [] in 956 proved x i λ k+ λ k 4 kn In 980, Hile-Protter [9] strengthened., and proved λ i.. kn 4 λ i..3 λ k+ λ i In 99, Yang [3] gave the following much stronger inequality λ k+ λ i 4 n λ k+ λ i λ i..4 From inequality.3, we can get a weaker but explicit form λ k+ + 4 λ i..5 n k These inequalities are called universal inequalities because they do not involve domain dependence. In the following, we introduce an elliptic operator in divergence form in Riemannian manifold. Let,, be an n-dimensional compact Riemannian manifold with boundary possibly empty. Let A : EndT be a
Yanli Li and Feng Du 4 smooth symmetric and positive definite section of the bundle of all endomorphisms of T. Let V be a nonnegative continuous function on. Denote by and the Laplacian and the gradient operator of respectively. Define Lu = diva u + V u,.6 where for a vector field X on, divx denotes the divergence of X. The operator L defined in.5 is an elliptic operator in divergence form. It is easy to see that the Laplacian operator and Schrödinger operator are it s special cases. In 00, do Carmo-Wang-Xia [6] considered the eigenvalue problem of the elliptic operator in divergence form with weight such that Lu = λρu, in, u = 0, on,.7 where ρ is a weight function which is positive and continuous on. They got a Yang type inequality λ k+ λ i 4ξ ρ nρ λ k+ λ i ξ λ i V 0 + n H0,.8 ρ 4ρ where ξ I A and tra nξ throughout, ρ ρx ρ, x, I is the identity map, ξ, ξ, ρ, ρ are positive constants, H 0 = max H, V 0 = min V, and H is the mean curvature of immersed into some Euclidean space R N. Recently, Du-Wu-Li[7] considered the problem.7 on complete Riemannian manifolds, they proved the inequality fλ i ρ nρ nξ ξ gλ i fλ i λ k+ λ i gλ i n where f, g I λk+,s = sup ξ 4 H V vector field. λ i + S,.9 ρ, and H is the mean curvature In this paper, we will consider the eigenvalue problem.7 on bounded domains in the complete simply connected noncompact Riemannian manifolds. Then we obtain.
4 Universal inequalities for eigenvalues Theorem.. Let,, be an nn 3-dimensional complete noncompact simply connected Riemannian manifold with Sectional curvature Sec satisfying K Sec k. For a bounded domain, Let A : EndT be a smooth symmetric and positive definite section of the bundle of all endomorphisms of T, and assume that ξ I A ξ I throughout, x, ρ ρx ρ, here ξ, ξ, ρ, ρ are positive constants. Let λ i be the i th eigenvalue of the eigenvalue problem.7, then for any f, g I λk+, we have fλ i { } { gλ i fλ i λ k+ λ i gλ i } ρ ξ ρ λ i V 0 n k + n K ξ.0 where V 0 = min x V x. From Theorem., we can get the following. Corollary.. Under the assumption of Theorem., if is a hyperbolic space H n, we have fλ i { ρ ξ { where V 0 = min x V x. gλ i } fλ i λ k+ λ i gλ i } ρ λ i V 0 n., ξ Remark. Taking fλ i, gλ i = λ k+ λ i, λ k+ λ i in., we can get a Yang inequality that λ k+ λ i ρ ξ λ k+ λ i ρ λ i V 0 n ξ So, by choosing different fλ i, gλ i, we can get different universal inequalities.
Yanli Li and Feng Du 43 Preliminaries [8]. In this section, firstly, we shall introduce a family of couples of functions Definition.. Let λ > 0, a couple f, g of functions defined on [0, λ] belongs to I λ as that i f and g are positive, ii f and g satisfy the following condition, for any x, y [0, λ], such that x y, fx fy fx + x y gxλ x + fy gx gy 0. gyλ y x y. We can easily find that gx is a nonincreasing function. Li[7]. In the following, we will introduce a lemma which is obtained by Du-Wu- Lemma.. Let,, be an n-dimensional compact Riemannian manifold with boundary possibly empty. Let λ i be the i th eigenvalue of the eigenvalue problem of elliptic operators in divergence form with weight ρ such that Lu = λρu in, u = 0 on, and u i be the orthonormal eigenfunction corresponding to λ i, that is, Lu i = λ i ρu i in, and u i = 0 on, ρu i u j = δ ij, i, j =,, Then for any h C and f, g I λk+, we have fλ i u i h δ gλ i u i h, A h + u i, h + ρ u i h fλ i δλ k+ λ i gλ i,.
44 Universal inequalities for eigenvalues where δ is any positive constant and f = f. From this Lemma, if we can find a nice function on bounded domains in complete Riemannian manifolds and take it in. and.3, we can get universal inequalities in complete Riemannian manifolds. In the following, we will introduce a function on a bounded domain in complete noncompact simple connected Riemannian manifoldsfor the more details about this function, we refer [5]. Let,, be an n-dimensional complete noncompact Riemannian manifold, and be a bounded connected domain in. Sec is the section curvature satisfying K Sec k, where k, K is constants and 0 k K. When p is not in, define the distance function rx = dx, p, then n k coshkr sinhkr because of r r = Hess r Ric r, r, so r n K coshkr sinhkr,.3 r r n K coshkr sinhkr n k,.4 where Hess, Ric are the Hessian operator and Ricci curvature operator, respectively. 3 Proofs of the main results In this section, we will give the proofs of Theorem. and Corollary.. Proof of Theorem.. Taking h = r in the., we can get fλ i u i δ gλ i u i r, A r + u i, r + ρ u i r fλ i δλ k+ λ i gλ i, 3.
Yanli Li and Feng Du 45 Because of ρ ρx ρ, ξ I A ξ I, and., we have u i 3. ρ ρ ξ r, A r ξ. 3.3 Take 3. and 3.3 into 3., we have fλ i ξ fλ i δ gλ i + ρ ρ 4δλ k+ λ i gλ i u i, r + u i r ρ 3.4 It is known that λ i = which yields u i Lu i = = u i diva u i + V u i Aui, u i + V u i ξ u i + V u i, u i λ i V 0. 3.5 ξ ρ From.4 and.5, we can get u i, r + u i r ρ = ρ r, u i + u i r 4 r, u i u i r u i r, r ρ u i u i r u i r r ρ λ i V 0 n k u cosh kr i ρ ξ ρ ρ sinh kr n K + u cosh Kr n k i ρ sinh Kr ρ ρ λ i V 0 n k n K + ρ ξ ρ ρ ρ ρ ρ n ρ u i k n sinh + kr ρ u i K sinh Kr 3.6
46 Universal inequalities for eigenvalues Since K k 0, r > 0, we obtain Since n 3, we get K sinhkr k sinhkr, 3.7 n k sinh kr n K so, by 3.7-3.9, we can get ρ u i r + r, u i λ i V 0 ρ ξ ρ Taking 3.0 into 3.4, we obtain sinh Kr n n 3 k n k ρ ρ + sinh kr 0, 3.8 n K ρ ρ 3.9 ρ fλ i ξ ρ δ gλ i + ρ ξ λ i V 0 ρ fλ i 4δλ k+ λ i gλ i n k + ρ ρ n K ρ ρ 3.0, above inequality implies that fλ i ρ ξ δ ρ gλ i + fλ i 4δλ k+ λ i gλ i ρ λ i V 0 n k + n K ρ ξ 3. δ = 4ρ ξ k 3. becomes fλ i λ k+ λ i gλ i k gλ i ξ ρ λ i V 0 n k + n K fλ i { } { gλ i fλ i λ k+ λ i gλ i } ρ ξ ρ λ i V 0 n k + n K ξ 3. This completes the proof of Theorem..
Yanli Li and Feng Du 47 Proof of Corollary.. Taking K = k = in 3., we immediately obtain.. ACKNOWLEDGEENTS. The research work is supported by Key Laboratory of Applied athematics of Hubei Province and The research project of Jingchu University of Technology. References [].S. Ashbaugh, Isoperimetric and unversal inequalities for eigenvalues, In: Davies, E.B., Safalov, Y.eds. Spectral theory and geometry Edinburgh, 998, London ath. Soc. Lecture Notes, Cambridge University Press, Cambridge 999, pp. 95-39. [] D. Chen and Q.. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. ath. Soc. Jpn., 60, 008, 35-339. [3] Q.. Cheng, T. Ichikawa and S. ametsuka, Estimates for eigenvalues of the poly-laplacian with any order, Communication in Contemporary athematics, 4, 009, 639-655. [4] Q.. Cheng and H.C. Yang, Estimates on eigenvalues of Laplacian, ath. Ann., 33, 005, 445-660. [5] D. Chen, T. Zheng and. Lu, Eigenvalue estimates on domain in complete noncompact Riemannian manifolds, Pacific Journal of athematics, 55, 0, 4-54. [6].P. do Carmo, Q. Wang and C. Xia, Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds. Annali di athematica, 89, 00, 643-660. [7] F. Du, C. Wu and G. Li, Extrinsic estimates for eigenvalues of elliptic operators in divergence form, Submitted to Umzh. mathe.
48 Universal inequalities for eigenvalues [8] S. Ilias and O. akhoul, Universal inequalities for the eigenvalues of a power of the Laplace operator, anuscripta ath., 3, 00, 75-0. [9] G.N. Hile and.h. Protter, Inequalities for eigenvalues of the Laplacian, Indiana Univ. ath. J., 9, 980, 53-538. [0] E.. Harrel and J. Stubbe, On trace inequalities and the universal eigenvalue estimates for some partial differntial operators, Trans. Am. ath. Soc., 349, 997, 797-809. [] L.E. Payne, G. Pólya and H.F. Weinberger, On the ratio of consecutive eigenvalues, J. ath. Phys., 35, 956, 89-98. [] Q. Wang and C. Xia, Universal bounds for eigenvalues of Schrödinger operator on Riemannian manifolds, Ann. Acad. Sci. Fenn. ath., 33, 008, 39-336. [3] H.C. Yang, An estimate of the difference between consecutive eigenvalues, preprint IC/9/60 of ICTP, Trieste, 99.