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

Abstract

We investigate the Dirichlet weighted eigenvalue problem of the elliptic operator in divergence form on compact Riemannian manifolds . 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.

1. Introduction

Let be an -dimensional bounded compact Riemannian manifold, , and , where is the Riemannian volume measure on . Let 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 [1–3]). 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 , is the mean curvature vector, and .

In this paper, we consider the following eigenvalue problem: 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 . As we know (see [5]), this problem has a real and discrete spectrum: here each eigenvalue is repeated from its multiplicity.

In Section 2, we get a general inequality for the eigenvalue of the operator in divergence form 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.

2. 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 and any integer , we have where is any positive constant.

Proof. We define a trial function where , and then we have If we set , 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 Multiplying (19) by , we get 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 , we can refer to the fact that Substituting (23) into (22), we can finish the proof of Theorem 1.

3. 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 be a bounded domain with smooth boundary and let be a smooth function on 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 , be the standard coordinate functions of . Taking in (8), summing over from 1 to , we have Since is isometrically immersed in , we have and then, Also, we have
Let 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 , , , , , , and then one has

Proof. Obviously, we have Multiplying the equation by and integrating on , we have Considering 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 , , and , 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].

Remark 5. For some of the recent developments about universal inequalities for eigenvalues on Riemannian manifolds, we refer to [6–12] and the references therein.

4. 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 be a bounded domain with smooth boundary and let be a smooth function on in ; is a symmetry and positive definite matrix; let , , , , , ; suppose be the th eigenvalue of the problem: if there exists a function and a constant such that then

Proof. Taking in (8) and considering (38), (42), and , we have Taking we can complete the proof of Theorem 6.

Remark 7. Let be an -dimensional connected complete Riemannian manifold; suppose its Ricci curvature satisfies , . If there exists a smooth function satisfying , then . So the Bussemann functions on Cartan-Hadamard manifolds with Ricci curvature bounded below satisfy the condition in Theorem 6.

Theorem 8. Let be an -dimensional complete Riemannian manifold and let be a bounded domain with smooth boundary; let be a smooth function on in M; A is a symmetry and positive definite matrix; let , , , , , ; suppose be the th eigenvalue of the problem: if admits an eigenmap corresponding to an eigenvalue , that is then

Proof. Because of (47), we obtain Taking in (8) and summing over from 1 to , we get Taking 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].

5. 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their gratitude to the referee for his valuable comments and suggestions. The paper is supported by the National Natural Science Foundation of China (Grant no. 11401531) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 14KJD110004).