Qing-Ming Cheng, Guoxin Wei
In this paper, we study eigenvalues of a clamped plate problem. We obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter.
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This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt {Γ_{k+1}-Γ_{1}}$ and $\sqrt {Γ_{k}-Γ_{1}}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap between $\sqrt {Γ_{k+1}-Γ_{1}}$ and $\sqrt {Γ_{k}-Γ_{1}}$ is bounded by a term with a lower order $k^{\frac1n}$ in the sense of the asymptotic formula of Agmon and Peijel, where $Γ_j$ denotes the $j^{^{\text{th}}}$ eigenvalue of the clamped plate problem.
We affirmatively solve the analogue of Lord Rayleigh's conjecture on Riemannian manifolds with positive Ricci curvature for any clamped plates in 2 and 3 dimensions, and for sufficiently large clamped plates in dimensions beyond 3. These results complement those from the flat (M. Ashbaugh & R. Benguria, 1995, and N. Nadirashvili, 1995) and negatively curved (A. Kristály, 2020) cases that are valid only in 2 and 3 dimensions, and at the same time also provide the first positive answer to Lord Rayleigh's conjecture in higher dimensions. The proofs rely on an Ashbaugh-Benguria-Nadirashvili-Talent…
In this paper, we study estimates for eigenvalues of the clamped plate problem. A sharp upper bound for eigenvalues is given and the lower bound for eigenvalues in [10] is improved.
For a bounded domain $Ω$ in a complete Riemannian manifold $M^n$, we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.
In this paper, we give some lower bounds for several eigenvalues. Firstly, we investigate the eigenvalues $λ_i$ of the Laplace operator and prove a sharp lower bound. Moreover, we extent this estimate of the eigenvalues to general cases. Secondly, we study the eigenvalues $Γ_i$ for the clamped plate problem and deliver a sharp bound for the clamped plate problem for arbitrary dimension.
$\mathfrak{L}_{II}$ operator is introduced by Y.-L. Xin (\emph{Calculus of Variations and Partial Differential Equations. 2015, \textbf{54}(2):1995-2016)}, which is an important extrinsic elliptic differential operator of divergence type and has profound geometric meaning. In this paper, we extend $\mathfrak{L}_{II}$ operator to more general elliptic differential operator $\mathfrak{L}_ν$, and investigate the clamped plate problem of bi-$\mathfrak{L}_ν$ operator, which is denoted by $\mathfrak{L}_ν^{2}$, on the complete Riemannian manifolds. A general formula of eigenvalues for the $\mathfrak{…
The Buckling and Clamped Plate Problems on Differential Forms
本文研究了微分形式上的屈曲与夹紧板问题，推导了其最小特征值并推广了Hodge拉普拉斯算子的估计。
We study Lord Rayleigh's problem for clamped plates on an arbitrary $n$-dimensional $(n\geq 2)$ Cartan-Hadamard manifold $(M,g)$ with sectional curvature $\textbf{K}\leq -κ^2$ for some $κ\geq 0.$ We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in $(M,g)$ is universally bounded from below by $\frac{(n-1)^4}{16}κ^4$ whenever the $κ$-Cartan-Hadamard conjecture holds on $(M,g)$, e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped pl…
$\mathfrak{L}_ν$ operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of $\mathfrak{L}^{2}_ν$ operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of $\mathfrak{L}^{2}_ν$ operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higer order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, min…
Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability
本文研究了满足 RCD(0,N) 条件的度量测度空间中夹紧板主频率的精细性质，在 N 接近 2 或 3 时提出了包含渐近体积比的新等周不等式，并证明了其锐度、基于锥结构的刚性及稳定性，结果对黎曼流形亦属全新。
By using the Fourier transform, we successfully give Kröger-type estimates for sums of eigenvalues of the free plate (under tension and with nonzero Poisson's ratio) in terms of the dimension of the ambient space, the volume of the domain, the tension parameter and the Poisson's ratio.
In this paper, we study a first Dirichlet eigenfunction of the weighted $p$-Laplacian on a bounded domain in a complete weighted Riemannian manifold. By constructing gradient estimates for a first eigenfunction, we obtain some relationships between weighted $p$-Laplacian first eigenvalues. As an immediate application, we also obtain some eigenvalue comparison results between the first Dirichlet eigenvalue of the weighted Laplacian, the first clamped plate eigenvalue and the first buckling eigenvalue.
The Poisson problem consists in finding an immersed surface $Σ\subset\mathbb{R}^m$ minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area which constitutes a non-linear model for the equilibrium state of thin, clamped elastic plates originating from the work of S. Germain and S.D. Poisson or the early XIX century. We present a solution to this problem consisting in the minimisation of the total curvature energy $E(Σ)=\int_Σ|\operatorname{I\!I}_Σ|^2_{g_Σ}\,\mathrm{d}vol_Σ$ ($\operatorname{I\!I}_Σ$ is the second fundamenta…
Deep estimates for higher eigenvalues of the poly-Laplacian
该研究针对有界区域上多重拉普拉斯算子的高阶特征值下界问题，推导低维情形的深度特征值不等式，建立带约束与无约束的任意维数改进下界，同时优化了Stokes特征值问题与广义Pólya猜想的相关下界。
In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use this abstract formulation both for giving dierent and simpler proofs for all the known results obtained for the eigenvalues of a power of the Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for the bilaplacian and more generally for the polyharmonic probl…
Sharp Spectral Bounds for the $p$-Laplacian and Polyharmonic Operators on Asymptotically Hyperbolic Manifolds
本文研究了渐近双曲流形上 $p$-Laplacian 和多项式调和算子的谱界，推导了三种特征值问题的精确上界和下界，揭示了渐近曲率、无穷处的角度及平均曲率范数与第一特征值的关系。
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