# Sharp Lower Bounds for the First Eigenvalues of the Bi-Laplace Operator

@article{Wang2018SharpLB, title={Sharp Lower Bounds for the First Eigenvalues of the Bi-Laplace Operator}, author={Qiaoling Wang and Changyu Xia}, journal={arXiv: Analysis of PDEs}, year={2018} }

## 4 Citations

Sharp Estimates for the First Eigenvalues of the Bi-drifting Laplacian

- Mathematics
- 2019

In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact…

SHARP ESTIMATES FOR EIGENVALUES OF BI-DRIFTING LAPLACIAN

- 2019

In this paper, we study the four types of eigenvalue problems for the bi-drifting Laplacian. By using the weighted Reilly formula, we get some sharp lower bounds for the first nonzero eigenvalue for…

On the spectra of three Steklov eigenvalue problems on warped product manifolds

- Mathematics
- 2019

Let $M^n=[0,R)\times \mathbb{S}^{n-1}$ be an $n$-dimensional ($n\geq 2$) smooth Riemannian manifold equipped with the warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a…

Bi-eigenfunctions and biharmonic submanifolds in a sphere

- Mathematics
- 2021

In this note, we classify biharmonic submanifolds in a sphere defined by bieigenmaps (∆φ = λφ) or buckling eigenmaps (∆φ = −μ∆φ). The results can be viewed as generalizations of Takahashi’s…

## References

SHOWING 1-10 OF 42 REFERENCES

Sharp Bounds for the First Eigenvalue of a Fourth-Order Steklov Problem

- Mathematics
- 2012

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold with smooth boundary. We give a sharp lower bound of the first eigenvalue of this problem, which depends only on…

Eigenvalues of the Wentzell-Laplace Operator and of the Fourth Order Steklov Problems

- Mathematics
- 2015

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same…

Extremum Problems for Eigenvalues of Elliptic Operators

- Mathematics
- 2006

Eigenvalues of elliptic operators.- Tools.- The first eigenvalue of the Laplacian-Dirichlet.- The second eigenvalue of the Laplacian-Dirichlet.- The other Dirichlet eigenvalues.- Functions of…

ON THE ESTIMATE OF THE FIRST EIGENVALUE OF A COMPACT RIEMANNIAN MANIFOLD

- Mathematics
- 1984

The main theorem proved in this work is: Let M be a compact Riemannian manifold withnon-negative Recci curvature, then the first eigenvalue -λ1, of the Laplace operator of Msatisfies λ1≥π2, where d…

Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

- Mathematics
- 2007

In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this…

On asymptotic properties of biharmonic Steklov eigenvalues

- Mathematics
- 2016

Abstract In this paper, by explicitly calculating the principal symbols of pseudodifferential operators, we establish two Weyl-type asymptotic formulas with sharp remainder estimates for the counting…

Universal Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains

- Mathematics
- 2007

In this paper we study the eigenvalues of the buckling problem on domains in a unit sphere. We obtain universal bounds on the (k + 1)th eigenvalue in terms of the first k eigenvalues independent of…

Universal Bounds for Eigenvalues of the Polyharmonic Operators

- Mathematics
- 2009

We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic…

Eigenvalues in Riemannian geometry

- Mathematics
- 1984

Preface. The Laplacian. The Basic Examples. Curvature. Isoperimetric Inequalities. Eigenvalues and Kinematic Measure. The Heat Kernel for Compact Manifolds. The Dirichlet Heat Kernel for Regular…

An extremal eigenvalue problem for the Wentzell-Laplace operator

- Mathematics
- 2014

We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Ventcel-Laplace operator of a domain $\Om$, involving only geometrical informations. We provide such an…