Lower Order Eigenvalues Research Articles

Universal inequalities for eigenvalues of a system of elliptic equations of the drifting Laplacian

Let \(\Omega \) be a bounded domain in a n-dimensional Euclidean space \(\mathbb {R}^{n}\). We study eigenvalues of an eigenvalue problem of a system of elliptic equations of the drifting Laplacian $$\begin{aligned} \left\{ \begin{array}{ll} \mathbb {L_{\phi }}\mathbf{{u}} + \alpha (\nabla (\mathrm {div}{} \mathbf{{u}}) - \nabla \phi \mathrm {div}{} \mathbf{{u}})= -\bar{\sigma }\mathbf{{u}}, &{} \hbox {in} \,\Omega ; \\ \mathbf{{u}}|_{\,\partial \Omega }=0. \end{array} \right. \end{aligned}$$ Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, a universal inequality for lower order eigenvalues of the problem is also derived. Finally, we prove an universal inequality type Ashbaugh and Benguria for the drifting Laplacian on Riemannian manifold immersed in an unit sphere or a projective space.

Estimates for eigenvalues of fourth-order elliptic operators in divergence form

In this paper, we study the eigenvalue problem of fourth-order elliptic operators in divergence form with weight on compact Riemannian manifolds with boundary (possibly empty) and get two types of general inequalities for them. By using these general inequalities, we can obtain some universal inequalities for eigenvalues of fourth-order elliptic operators in divergence form with weight on bounded domains in complete Riemannian manifolds. Furthermore, we also obtain some estimates for lower order eigenvalues of fourth-order elliptic operators in divergence form with weight on compact submanifolds in Euclidean spaces.

Reduced-order modeling for rotating rotor-bearing systems with cracked impellers using three-dimensional finite element models

A novel reduced-order modeling method is presented in this paper for dynamics analysis of rotating impeller–shaft-bearing assembly with cracked impellers. Based on three-dimensional finite element model, the complex component mode synthesis (CMS) method is employed to generate an efficient reduced-order model (ROM) for studying the effects of crack on the global vibration of the rotating assembly. First, a modeling framework for impeller–shaft-bearing systems in rotating frame is presented. Rotational effects, including Coriolis matrix and centrifugal softening, have been taken into account. Then, the governing equation of motion of the damped gyroscopic system is reduced by the complex CMS method. Finally, the obtained ROM is employed to study the effects of crack on assembly׳s vibration. During the steady-state response analysis, external excitations on the impeller due to rotor–stator interactions have been taken into account, which was however neglected in previous investigations on rotordynamics. Numerical results show that the lower-order eigenvalues and the unbalance response of the assembly are not sensitive to the local crack on impeller. Nevertheless, the flexible coupling between impeller and shaft becomes more complex when the air flow-induced excitations are considered. Under EO1 traveling wave excitations, a crack leads to slight changes in the assembly׳s response. In contrast, the effect of crack becomes significant when the assembly is excited by EO2 and higher EO excitations. Moreover, the nonlinear crack breathing effects affect the assembly׳s response obviously. Finally, a potential technique for detecting the crack on impeller during operation is discussed.

Inequalities for lower order eigenvalues of second order elliptic operators in divergence form on Riemannian manifolds

In this paper, we investigate the Dirichlet eigenvalue problems of second order elliptic operators in divergence form on bounded domains of complete Riemannian manifolds. We discuss the cases of submanifolds immersed in a Euclidean space, Riemannian manifolds admitting spherical eigenmaps, and Riemannian manifolds which admit l functions \({f_\alpha : M \longrightarrow \mathbb{R}}\) such that \({\langle \nabla f_\alpha, \nabla f_\beta \rangle = \delta_{\alpha \beta}}\) and Δfα = 0, where ∇ is the gradient operator. Some inequalities for lower order eigenvalues of these problems are established. As applications of these results, we obtain some universal inequalities for lower order eigenvalues of the Dirichlet Laplacian problem. In particular, the universal inequality for eigenvalues of the Laplacian on a unit sphere is optimal.

Inequalities for lower-order eigenvalues of a fourth-order elliptic operator

In this paper, we investigate the Dirichlet weighted eigenvalues problem of a fourth-order elliptic operator with variable coefficients on a bounded domain with smooth boundary in ℝ n . We establish some inequalities for lower-order eigenvalues of this problem. In particular, our results contain an inequality for eigenvalues of the biharmonic operator derived by Cheng, Huang, and Wei.

EIGENVALUES OF THE LAPLACIAN ON RIEMANNIAN MANIFOLDS

For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L2(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

Extrinsic estimates for the eigenvalues of Schrödinger operator

We consider the eigenvalues of Schrodinger operator with weight on compact Riemannian manifolds with boundary (possibly empty) and prove two types of general inequalities for them. In particular, we study the eigenvalues of Schrodinger operator with weight on compact submanifolds of Euclidean spaces, of spheres, and of projective spaces. We also study the same problem on Riemannian manifolds admitting spherical eigenmaps. We give explicit bound for the (k + 1)-th eigenvalue of Schrodinger operator with weight on such objects in terms of its first k eigenvalues. Furthermore, we also obtain some estimates for lower order eigenvalues of Schrodinger operator with weight on such objects. Our results extend recent estimates on eigenvalues of the Laplacian obtained by Sun et al. [28, Manuscripta Math 125:139–156] and Wang and Xia [30, Ann Acad Sci Fenn Math 33:319–336].

Novel Modal Method for Efficient Calculation of Complex Eigenvector Derivatives

A novel modal superposition method is presented for calculating eigenvector derivatives in self-adjoint or non-self-adjoint systems. The method is especially appropriate for determination of derivatives of many eigenvectors and requires only one eigenvalue and its corresponding right and left eigenvectors. Two parameters for the eigenvalue under consideration are introduced to obtain a generic derivative formula, which contains most of the available modal superposition methods as a branch. Matrices of the modified system are decomposed with available lower modes to avoid repeatedly solving equations for different eigenvalues. The contribution of truncated higher modes is expanded in a continued product of a few matrices, polynomials, and one generalized power series, whereby rapid convergence can be achieved easily, even in the vicinity of the convergent boundary, without additional computation cost. All other modal superposition coefficients can be neglected intrinsically with the determination of introduced parameters, but a simple condition for coefficients associated with lower-order eigenvalues should be satisfied in a special case. Numerical examples show that the method is efficient and can give results comparable to the exact solutions. The method is applicable to various damped systems and closed-mode cases.