Estimates for Eigenvalues of the Elliptic Operator in Divergence Form on Riemannian Manifolds

Shenyang Tan, Tiren Huang, and Wenbin Zhang 1Taizhou Institute of Sci. & Tech., NUST., Taizhou 225300, China 2Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China Correspondence should be addressed to Shenyang Tan; ystsy@163.com Received 5 December 2014; Accepted 19 January 2015 Academic Editor: Yao-Zhong Zhang Copyright © 2015 Shenyang Tan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the Dirichlet weighted eigenvalue problem of the elliptic operator in divergence form on compact Riemannian manifolds (M, g, edV). We establish a Yang-type inequality of this problem. We also get universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below and any complete manifolds admitting eigenmaps to a sphere.


Introduction
Let (, ⟨, ⟩) be an -dimensional bounded compact Riemannian manifold,  ∈  2 (), and  =  − V, where V is the Riemannian volume measure on (, ⟨, ⟩).Let Δ and ∇ be the Laplacian and the gradient operator on , respectively.The witten Laplacian (or the drifting Laplacian) with respect to the weighted volume measure  is given by In recent years, many mathematicians have paid their attention to the eigenvalue problem of the drifting Laplacian on Riemannian manifolds (see [1][2][3]).They have studied the following eigenvalue problem: In particular in [4], Xia and Xu got a Payne-Plya-Weinberger-Yang-type inequality of the eigenvalues of this problem: where  0 = sup Ω ||,  is the mean curvature vector, and  0 = max Ω |∇|.
In this paper, we consider the following eigenvalue problem: − div (∇) + ⟨∇, ∇⟩ +  = , in , where  is a nonnegative potential function,  is a positive function continuous on , and  is symmetric and positive definite matrices.Through integration by part, we can find where  and  are smooth functions on  with |  = |  = 0.As we know (see [5]), this problem has a real and discrete spectrum: here each eigenvalue is repeated from its multiplicity.

Advances in Mathematical Physics
In Section 2, we get a general inequality for the eigenvalue of the operator in divergence form − div(∇(⋅)) + ⟨∇, ∇(⋅)⟩ +  through the way of trial function.In Section 3, we obtain a Payne-Plya-Weinberger-Yang-type inequality through defining special trial function.In Section 4, we prove some universal inequalities for eigenvalues of the divergence operator on manifolds admitting special functions.

A General Inequality
Firstly, we give a useful inequality about the eigenvalues.
Theorem 1.Let   be the th eigenvalue of problem (4) and let   be the orthonormal eigenfunction corresponding to   ; that is, Then, for any ℎ ∈  3 () ∩  2 () and any integer , we have where  is any positive constant.
Proof.We define a trial function where   = ∫  ℎ     =   , and then we have If we set  = − div(∇(⋅)) + ⟨∇, ∇(⋅)⟩ + , then through direct calculation, we have Substituting (11) into the well known Rayleigh-Ritz inequality we can get We set Through direct calculation, we have Combining with (13), we get Setting then through direct calculation, we have where  is any positive constant.Summing over  from 1 to , we have Because of   =   ,   = −  , we infer Considering the property of the measure on this weighted manifold that  =  − V, we can refer to the fact that Substituting ( 23) into (22), we can finish the proof of Theorem 1.

The Main Theorem and the Proof
In this section, we give some estimates about the eigenvalues of the operator in divergence form.
Lemma 2. Let  be an -dimensional complete Riemannian manifold and let Ω be a bounded domain with smooth boundary and let  be a smooth function on Ω in ;  is a symmetry and positive definite matrix; suppose   be the th eigenvalue of the problem: If  is isometrically immersed in   with mean curvature vector , then Proof.Let   ,  = 1, 2, . . .,  be the standard coordinate functions of   .Taking ℎ =   in (8), summing over  from 1 to , we have Since  is isometrically immersed in   , we have and then, Also, we have Let  1 , . . .,   be orthonormal tangent vector fields locally defined on ; we have and then, Substituting (28), (29), and (31) into (26), we can finish the proof of Lemma 2.

Theorem 3. Under the same assumption of Lemma 2, let 𝜏 = (sup
, and then one has Proof.Obviously, we have then we can obtain Solving this inequality, we have Substituting ( 33) and ( 38) into (25) and taking where we can finish the proof of Theorem 3.
Remark 4. If we set  = constant,  = ,  = 1 and  = 0, the divergence operator becomes the usual laplace operator on Riemannian manifolds and we can find our result is sharper than the result in [4,6].

Eigenvalues on Manifolds Admitting Special Functions
In this section, we get some universal inequalities for eigenvalues of the divergence operator on manifolds admitting special functions.
Theorem 6.Let  be an -dimensional complete Riemannian manifold and let Ω be a bounded domain with smooth boundary and let  be a smooth function on Ω in ;  is a symmetry and positive definite matrix; let  = (sup suppose   be the th eigenvalue of the problem: if there exists a function  : Ω →  and a constant  0 such that
Remark 7. Let  be an -dimensional connected complete Riemannian manifold; suppose its Ricci curvature satisfies Ric  ≥ −( − 1) Proof.Because of (47), we obtain Taking ℎ =   in (8) and summing over  from 1 to  + 1, we get then the proof of Theorem 8 is finished.
Remark 9. Any compact homogeneous Riemannian manifold admits eigenmaps to some unit sphere for the first positive eigenvalues of the Laplacian which satisfy the condition in Theorem 8 [13].

Physical Interpretation
In quantum mechanics, eigenvalue is the dynamics of macro possible values.The wave function is superposition of a number of eigenstates.Different eigenstate is corresponding to the specific eigenvalue (of course there may be degenerate case; namely, the same eigenvalue corresponds to different intrinsic state).The experimental measurement of the mechanical quantity must be one of eigenvalues, and wave function in the measurement is the eigenstate of the corresponding eigenvalue.The gap between different eigenvalues means the difference between the energy levels.That is why many researchers pay much attention to this problem.In this paper, we find a relatively accurate upper bound between any two different eigenvalues.