# The sharp estimates of eigenvalues of Polyharmonic operator and higher order Stokes operator

@article{Chen2014TheSE, title={The sharp estimates of eigenvalues of Polyharmonic operator and higher order Stokes operator}, author={Daguang Chen and He-Jun Sun}, journal={arXiv: Mathematical Physics}, year={2014} }

In this paper, we establish some lower bounds for the sums of eigenvalues of the polyharmonic operator and higher order Stokes operator, which are sharper than the recent results in \cite{CSWZ13, I13}. At the same time, we obtain some certain bounds for the sums of positive and negative powers of eigenvalues of the polyharmonic operator.

## References

SHOWING 1-10 OF 32 REFERENCES

### Lower bounds for sums of eigenvalues of elliptic operators and systems

- Mathematics
- 2013

Two-term lower bounds of Berzin-Li-Yau type are obtained for the sums of eigenvalues of elliptic operators and systems with constant coefficients and Dirichlet boundary conditions. The polyharmonic…

### LOWER BOUNDS FOR THE SPECTRUM OF THE LAPLACE AND STOKES OPERATORS

- Mathematics
- 2009

We prove Berezin-Li-Yau-type lower bounds with additional
term for the eigenvalues of the Stokes operator and
improve the previously known estimates for the Laplace
operator. Generalizations to…

### Sharp Semiclassical Bounds for the Moments of Eigenvalues for Some Schrödinger Type Operators with Unbounded Potentials

- Mathematics
- 2013

We establish sharp semiclassical upper bounds for the moments of some negative powers for the eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporated in the problem, we…

### On the spectrum of the stokes operator

- Mathematics
- 2008

We prove Li-Yau type lower bounds for the eigenvalues of the Stokes operator and give applications to the attractors of the Navier-Stokes equations.

### Multidimensional lower bounds for the eigenvalues of Stokes and Dirichlet Laplacian operators

- Mathematics
- 2012

This article is concerned with the improvements of certain eigenvalue inequalities of Stokes operator and Dirichlet Laplacian related to the Berezin-Li-Yau type inequalities. The formulas proved…

### Estimates on the eigenvalues of the clamped plate problem on domains in Euclidean spaces

- Mathematics
- 2013

The purpose of this article is two-fold. First, we obtain some certain bounds for the sums of (positive and negative) powers of the eigenvalues of the clamped plate problem of the Dirichlet…

### Improved Berezin-Li-Yau inequalities with a remainder term

- Mathematics
- 2007

We give an improvement of sharp Berezin type bounds on the Riesz means $\sum_k(\Lambda-\lambda_k)_+^\sigma$ of the eigenvalues $\lambda_k$ of the Dirichlet Laplacian in a domain if $\sigma\geq 3/2$.…

### A lower bound for sums of eigenvalues of the Laplacian

- Mathematics
- 2002

Let λ k (Ω) be the kth Dirichlet eigenvalue of a bounded domain Ω in R n . According to Weyl's asymptotic formula we have λ k (Ω) ∼ C n (k/V(Ω)) 2/n . The optimal in view of this asymptotic relation…

### Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity

- Mathematics
- 1985

Lower bounds for the eigenvalues of some elliptic equations and elliptic systems over bounded regions are obtained. The bounds are universal in that they depend only upon the volume of the region.…

### Two-Dimensional Berezin-Li-Yau Inequalities with a Correction Term

- Mathematics
- 2009

We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is…