﻿ 微分流形上Laplace型算子的主特征值估计 The Principal Eigenvalue Estimations of La-place Type Operators on DifferentialManifold

Modern Physics
Vol. 09  No. 01 ( 2019 ), Article ID: 28285 , 11 pages
10.12677/MP.2019.91001

The Principal Eigenvalue Estimations of Laplace Type Operators on Differential Manifold

Jinnan Li, Xiang Gao*

School of Mathematical Sciences, Ocean University of China, Qingdao Shandong

Received: Dec. 7th, 2018; accepted: Dec. 21st, 2018; published: Dec. 28th, 2018

ABSTRACT

Based on the eigenvalue estimations of Li-Conjecture’s and Yang-Conjecture’s proposed and developed ideas, this paper summarized the variation of the upper and lower bounds of the estimated value when the Laplace operator principal eigenvalue estimation conditions were changed on a typical Riemannian manifold, and yielded some precise estimation results. The principal eigenvalue estimation of p-Laplacian on general Riemannian manifolds is studied. The estimation of principal eigenvalues of Finsler manifolds is studied. Since the potential function is introduced, the eigenfunction is changed, then the estimation of the principal eigenvalue of the new Laplace type operator-Schrödinger operator is studied. It shows the close connection between Riemannian manifold and general relativity. It also can simplify the solution of energy spectrum and other problems in quantum mechanics, and provide some new methods for quantum mechanics, quantum optics and solid physics.

Keywords:Differential Manifold, Laplace Operator, Schrödinger Operator, Principal Eigenvalue, P-Laplacian

1. 引言

1.1. 微分流形上特征值估计问题的研究背景与意义

20世纪，微分几何与物理学及数学中的分析数学、代数几何、拓扑学等相互影响相互促进，得到了迅猛发展。分析方法的引入更是对微分几何的发展产生了深远的影响，促进了许多著名问题的解决。一个微分流形的全体特征值能够反映出很多几何或拓扑信息？众所周知：一个等距的流形必然是等谱的。但是等谱的流形是否必然等距呢？1911年，Weyl证明了平面区域的面积可以由谱来决定；1964年，著名数学家Milnor构造出等谱而不等距的16维平坦环面；1966年，Kac提出一个问题“你能听出鼓的形状吗”化为数学问题即两个平面等谱区域是否等距同构？并证明答案是否定的，即等谱的流形不一定等距。Laplace算子的谱与流形的几何性质及拓扑性质有着密切的联系，谱理论对数学与物理具有重要的作用，但是已经求出谱的只有等腰直角三角形、标准单位球面、平坦球面、Klein瓶、复射影空间、酉群等很少的流形。鉴于大部分的流形的谱尚无法完全计算，而主特征值是谱的主项，故近几十年来数学家们主要研究主特征值(第一特征值)的尽可能精确的结果。主要有三类研究方法：一是Cheeger引入的等周常数的方法；二是Li-Yau发展的梯度估计方法；三是概率中的耦合方法。

1.2. 特征值估计问题的发展

Dirichlet边界问题：

$\left\{\begin{array}{l}\Delta u=-\lambda u\text{,}\text{ }\text{on}\text{\hspace{0.17em}}M\\ u=0,\text{on}\text{\hspace{0.17em}}\partial M\end{array}$

Neumann边界问题：

$\left\{\begin{array}{l}\Delta u=-\lambda u\text{,on}\text{\hspace{0.17em}}M\\ {\nabla }_{n}u=0,\text{}\text{ }\text{on}\text{\hspace{0.17em}}\partial M\end{array}$

Cheng [5] 给出了仅依赖于流形的直径及Ricci曲率的主特征值的经典的上界估计：

${\lambda }_{1}\ge \frac{{\left(m-1\right)}^{2}}{4}K.$

Li-Yau [6] 得到了只依赖于直径及Ricci曲率的主特征值的经典的下界估计：

${\lambda }_{1}\ge \frac{{\pi }^{2}}{4{d}^{2}}.$

${\lambda }_{1}\ge \frac{{\pi }^{2}}{{d}^{2}}.$

2. 主特征值估计

2.1. Li-猜想的发展

1958年，Lichnerowicz [9] 证明了特殊曲率条件下的第一非零特征值的一个下界估计。

${\lambda }_{1}\ge mK.$

${\lambda }_{1}\ge \frac{mK}{2}.$

Li-Yau [6] 得到了只依赖于流形M的直径d及Ricci曲率的主特征值的下界的估计(即定理1.2)，随后钟家庆与杨洪苍利用极大值原理 [11] 优化选择更恰当的试验函数后给出这类问题的最优估计(即定理1.3)同时，Li-Yau [6] 给出了依赖于直径d并满足一定Ricci曲率条件的一个下界估计。

${\lambda }_{1}\ge \frac{{C}_{1}}{{d}^{2}}\mathrm{exp}\left(-{C}_{2}d\sqrt{K}\right).$

${\lambda }_{1}\ge \frac{{\pi }^{2}}{{d}^{2}}\mathrm{exp}\left(-Cd\sqrt{K}\right).$

${\lambda }_{1}\ge \frac{{\pi }^{2}}{{d}^{2}}+\frac{1}{4}\left(m-1\right)K.$

${\lambda }_{1}\ge \frac{{\pi }^{2}}{{\stackrel{¯}{d}}^{2}}+\frac{1}{2}\left(m-1\right)K.$

${\lambda }_{1}\ge \frac{{\pi }^{2}}{{\stackrel{¯}{d}}^{2}}+\frac{34}{100}\left(m-1\right)K.$

${\lambda }_{1}\ge \frac{{\pi }^{2}}{{d}^{2}}+\frac{1}{2}\left(m-1\right)K.$

${\lambda }_{1}\ge 4s\left(1-s\right)\frac{{\pi }^{2}}{{d}^{2}}+s\left(m-1\right)K,$

2.2. P-Laplacian的特征值估计

Dirichlet边界问题：

$\left\{\begin{array}{l}{\Delta }_{p}u=-\lambda {|u|}^{p-2}\text{,on}\text{\hspace{0.17em}}M\\ u=0,\text{on}\text{\hspace{0.17em}}\partial M\end{array}$

Neumann边界问题：

$\left\{\begin{array}{l}{\Delta }_{p}u=-\lambda {|u|}^{p-2}\text{,}\text{ }\text{on}\text{\hspace{0.17em}}M\\ {\nabla }_{n}u=0,\text{on}\text{\hspace{0.17em}}\partial M\end{array}$

${\lambda }_{1,p}\left(M\right)=\mathrm{inf}\left\{\frac{{\int }_{M}{|\nabla u|}^{p}}{{\int }_{M}{u}^{p}}|u\in {W}^{1,p}\left(\Omega \right),u\ne 0,{\int }_{M}{|u|}^{p-1}u=0\right\}.$

p-Laplacian作为一般Laplace算子的自然推广有以下三种特征值估计 [21] ：

${\lambda }_{1,p}\left(B\left({x}_{0},r\right)\right)\le {\stackrel{¯}{\lambda }}_{1,p}\left({B}_{K}\left(r\right)\right)+C{\left({‖Ri{c}_{-}^{K}‖}_{\stackrel{¯}{q},B\left({x}_{0},r\right)}\right)}^{1/2}.$

${\lambda }_{1,p}^{\frac{2}{p}}\left(M\right)\ge \frac{\sqrt{m}\left(p-2\right)+m}{\left(p-1\right)\left(\sqrt{m}\left(p-2\right)+m-1\right)}\left[\left(m-1\right)K-2{‖Ri{c}_{-}^{K}‖}_{q}\right].$

$Ric\ge \left(m-1\right)K$ 时， ${\lambda }_{1,p}^{\frac{2}{p}}\left(M\right)\ge \frac{\sqrt{m}\left(p-2\right)+m}{\sqrt{m}\left(p-2\right)+m-1}\cdot \frac{\left(m-1\right)K}{p-1}\ge \frac{\left(m-1\right)K}{p-1}$

$\alpha {\lambda }_{1,p}\left(M\right)\ge {\stackrel{¯}{\lambda }}_{1,p}\left(\stackrel{¯}{M}\right).$

${\lambda }_{1,p}\left(M\right)\ge \mathrm{sup}\left\{\underset{\Omega }{\mathrm{inf}}\left(\left(1-p\right){‖X‖}^{q}+div\left(X\right)\right)|X\in {W}^{1,1}\left(M\right)\right\}.$

${\lambda }_{1,p}\left(M\right)\ge \frac{{\left(m-1\right)}^{p}{c}^{p}}{{p}^{p}}.$

${\lambda }_{1,p}\ge \left(p-1\right)\frac{{\pi }_{p}^{p}}{{d}^{p}}.$

${\lambda }_{1,p}\ge \frac{1}{p-1}{\left(\frac{\pi }{4d}\right)}^{p}.$

${\lambda }_{1,p}\ge \left(p-1\right){\left(\frac{{\pi }_{p}}{2d}\right)}^{p}.$

2.3. Finsler流形上的特征值的估计

$F{\left({x}^{1},{x}^{2},\cdots ,{x}^{n},\text{d}{x}^{1},\text{d}{x}^{2},\cdots ,\text{d}{x}^{n}\right)}^{2}$ 是黎曼度量 $\underset{i,j=1}{\overset{n}{\sum }}{g}_{ij}\left({x}^{1},{x}^{2},\cdots ,{x}^{n}\right)\text{d}{x}^{i}\text{d}{x}^{j}$ 的推广。同黎曼流形一样，Finsler流形上两点之间的距离定义为连接这两点的曲线弧长的下确界。由于Finsler流形是度量空间，其度量拓扑和原来微分流形拓扑一致，显然黎曼流形上的许多性质可以推广到Finsler空间。芬斯勒(Finsler)于1918年在学位论文中系统地研究了这种度量，把经典的曲线和曲面论中的许多概念和定理进行推广，开展了整体Finsler几何的研究。Finsler流形几何理论在广义相对论和其他物理学领域中有许多应用，近年来无限维Finsler流形在非线性分析中也有越来越重要的作用。由于Finsler流形是比黎曼流形更广泛的流形，自然地可以研究Finsler流形上的Finsler-Laplace算子的特征值的估计。但由于Finsler流形上的Laplace算子是一个非线性微分算子，故很多黎曼流形上的估计方法不再适用，为了克服这些困难，需引进加权梯度、加权Ricci曲率及加权Laplace算子。比较黎曼流形上Laplace算子特征值估计的经典结果，也有很多关于Finsler流形上的Finsler-Laplace算子的特征值估计的重要结论 [26] [27] 。

${\lambda }_{1}\ge \frac{m-1}{N-1}NK.$

${\lambda }_{1}\ge mK.$

${\lambda }_{1}\ge \frac{{\pi }^{2}}{{d}^{2}}.$

2.4. Schrödinger算子的特征值估计

${\lambda }_{k}\le \frac{Ck+{\delta }^{-1}{\int }_{M}{V}^{+}\text{d}\mu -\delta {\int }_{M}{V}^{-}\text{d}\mu }{\mu }.$

${\lambda }_{1}\le \frac{1}{\mu }{\int }_{M}V\text{d}\mu .$

${\lambda }_{1}\ge {V}_{m}+{I}^{2},\text{}I\le \frac{\pi }{L},$

${\lambda }_{1}\le {V}_{m}+\frac{{\pi }^{2}}{L},\text{}I>\frac{\pi }{L}.$

3. 结论

The Principal Eigenvalue Estimations of La-place Type Operators on DifferentialManifold[J]. 现代物理, 2019, 09(01): 1-11. https://doi.org/10.12677/MP.2019.91001

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30. NOTES

*通讯作者。