Yamabe流下Laplace算子的第一特征值

本文得到Yamabe流下拉普拉斯算子的第一特征值的发展方程.我们证明出,在光滑的齐性流形(M(t),g)上,若λ(t)表示拉普拉斯算子的特征值,那么沿着规范化后的Yamabe 流,λ(t)=d,而且沿着非规范化的Yambe流,λ(t)=ded,这里d是一个常数,c表示齐性流形的数量曲率.而且作为发展方程的应用,我们得到...     

Yamabe流上Laplace算子特征值的一个单调性应用

讨论了紧致无边流形上Laplace算子的特征值在Yamabe流上随时间的变化情况,结合极值原理得到了Laplace算子特征值的单调性.     

Laplace算子特征值和的精细下界

该文研究了Rⁿ中Laplace算子在有界域Ω上的Dirichlet特征值和的下界.众所周知:第k个Dirichlet特征值λ_k(Ω)服从Weyl渐近公式,即λ_k(Ω)～4π²/[w_nV(Ω)]^2/nk^2/n当k→∞时,其中w_n和V(Ω)分别为是Rⁿ中n维单位球的体积和Ω的体积.根据上述公式,Pólya猜测λ_k(Ω)≥4π²/[w_nV(Ω)]^2/nk^2/n,?k∈N.这就是著名的Pólya猜想.对这一问题的研究由来已久,已有很多的工作.特别是,近几十年来最显著的成就之一是由Berezin^[4],以及李伟光和丘成桐^[3]分别独立取得的.他们部分解决了Pólya猜想,只是多了一个因子n/(n+2).后来,Melas^[7]改进...     

平均曲率流下拉普拉斯算子的特征值

考虑紧的、一致凸的n维无边曲而光滑地进入到具有常曲率的黎曼流形中,得到了平均曲率流下拉普拉斯算子特征值的发展方程,并且构造了一些沿着平均曲率流单调的几何量.     

Yamabe 流下完备Riemannian 流形在无穷远的性质

本文研究完备的局部共形平坦的Riemannian 流形Mⁿ. 证明了在Yamabe 流下, 流形在无穷远处曲率趋向于零的性质是随时间保持的. 作为应用, 可以得到这个流形的渐近体积比是一个常数.     

Riemann流形上Laplace算子第一特征值的一点注记

本文研究了黎曼流形上Laplace算子的第一特征值,利用流形的测地球上的Sobolev常数进行讨论并进行Moser迭代,得到闭的黎曼流形上Laplace算子第一特征值的一个下界估计.     

多项式Laplace算子的特征值估计

讨论了多项式Laplace算子Dirichlet问题,首先通过选取适当的函数,根据RayLeigh-Ritz不等式,得到了该问题用前k个特征值来估计第k+1个特征值的不等式,然后通过选取适当的系数,发现不等式蕴含成庆明和杨洪苍的结论及吴发恩和曹林芬的结论,且根据Chebyshev不等式等,证明了该不等式优于陈祖墀和钱春林的结论.     

任意阶Laplace算子的特征值估计

设D为n维Euclid空间Rn的一个有界区域,且0＜λ1≤λ2≤…≤λk≤…是l阶Laplace算子的Dirichlet问题{(-△)lu=λu, 在D中,u=(e)u/(e)n=…=(e)l-1u/(e)nl-1=0,在(e)D上的特征值.得到了该问题用其前k个特征值来估计第(k+1)个特征值λk+1的不等式k∑i=1(λk+1-λi)≤1/n(4l(n+2l-2)]1/2{k∑i=1(λk+1-λi)1/2λil-1/lk∑i=1(λk+1-λi)1/2λi1/l}1/2,此不等式不依赖于区域D.对l≥3,上述不等式比所有已知的结果都要好.陈庆民与杨洪苍考虑了l=2的情形.我们的结果是他们结果的自然推广.当l=1时,我们的不等式蕴含杨洪苍不等式的弱形式.文中还给出了陈和杨的一个断言的直接证明.     

小负曲率流形上Laplace算子第一特征值的下界估计

本文首先对流形的测地球上的Sobolev常数进行讨论,并利用它进行Moser迭代,最终得到具有小负曲率的闭的黎曼流形上Laplace算子特征值的一个下界估计.     

小负曲率流形上Laplace算子第一特征值的下界估计

本文首先对流形的测地球上的Sobolev常数进行讨论,并利用它进行Moser迭代,最终得到具有小负曲率的闭的黎曼流形上Laplace算子特征值的一个下界估计.     

The First Eigenvalue of p-Laplace Operator Under Powers of the mth Mean Curvature Flow

In this paper, we derive the evolution equation for the eigenvalues of p-Laplace operator. Moreover, we show the following main results. Let ( ${M^{n}, g(t)), t\in [0,T),}$ be a solution of the unnormalized powers of the mth mean curvature flow on a closed manifold and λ_1,p (t) be the first eigenvalue of the p-Laplace operator (p ≥ n). At the initial time t = 0, if H > 0, and $$h_{ij}\geq \varepsilon Hg_{ij}\quad \left(\frac{1}{p}\leq\epsilon\leq \frac{1}{n}\right),$$ then λ_1,p (t) is nondecreasing and differentiable almost everywhere along the unnormalized powers of the mth mean curvature flow on [0,T). At last, we discuss some interesting monotonic quantities under unnormalized powers of the mth mean curvature flow.     

Lower Bound of the First Eigenvalue for the Laplace Operator on Compact Riemannian Manifold

Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant C_m=max{2~(1/m-1),2~(1/2)}, Such thatλ_1≥π~2/d~2·1/(2-(11)/(2π~2))+11/2π~2e~cm、(?)     

Eigenvalue Problem for the Laplace Operator with Nonlocal Boundary Conditions

The suggested approach to maximizing the difference between the first and second eigenvalues of the Laplace operator is based on the introduction of nonlocal boundary conditions of a special form. It is shown that the difference can be arbitrarily large.     

Monotonicity of First Eigenvalues along the Yamabe Flow

Czechoslovak Mathematical Journal - We construct some nondecreasing quantities associated to the first eigenvalue of $$ - {{rm{Delta }}_varphi } + cRleft( {c ge {1 over 2}left( {n - 2}...     

The Higher Spin Laplace Operator

This paper deals with a certain class of second-order conformally invariant operators acting on functions taking values in particular (finite-dimensional) irreducible representations of the orthogonal group. These operators can be seen as a generalisation of the Laplace operator to higher spin as well as a second-order analogue of the Rarita-Schwinger operator. To construct these operators, we will use the framework of Clifford analysis, a multivariate function theory in which arbitrary irreducible representations for the orthogonal group can be realised in terms of polynomials satisfying a system of differential equations. As a consequence, the functions on which this particular class of operators act are functions taking values in the space of harmonics homogeneous of degree k. We prove the ellipticity of these operators and use this to investigate their kernel, focusing on polynomial solutions. Finally, we will also construct the fundamental solution using the theory of Riesz potentials.     

Convergence of Inverse Power Method for First Eigenvalue of p-Laplace Operator

In this article, convergence of an iterative scheme to approximate the first eigenfunction and related eigenvalue for p-Laplace operator is shown. Moreover, numerical examples are presented that show the e?ciency and accuracy of the algorithm.     

$p$-双调和算子的第一特征值的等周上界和 Reilly 型不等式

在本文中,我们给出了嵌入到欧氏空间中的$n$维闭超曲面上$p$-双调和算子的第一特征值的一些等周上界.我们也给出了浸入到高维流形如欧氏空间,球面和射影空间中的闭子流形上$p$-双调和算子的第一特征值的一些Reilly-型不等式.