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內容簡介

本專著擬介紹黎曼流形有界連通區域上幾類自伴隨橢圓算子的特徵值問題。通過進一步地拓展黎曼幾何裏經典的體積比較定理，導齣瞭綫性Laplace算子與非綫性p-Laplace算子的Dirichlet特徵值問題下**非零特徵值的比較定理，對於Steklov特徵值問題，同樣可以得到**非零特徵值的比較定理，這些結論極大地推廣瞭已有的經典結果；通過更好地構造指定流形上的測試函數

目錄

Chapter 1 Introduction

Chapter 2 Eigenvalue comparison theorems
2.1 Basic notions and the geometry of spherically symmetric manifolds
2.2 A generalized eigenvalue comparison theorem for manifolds with radial Ricci curvature bounded from below
2.3 A generalized eigenvalue comparison theorem for manifolds with radial sectional curvature bounded from above
2.4 Properties of the model manifolds
2.5 Examples
2.6 A criterion for the existence of the model spaces
2.7 Estimates for the heat kernel
2.8 A Cheng-type isoperimetric inequality for the p-Laplace operator and a spectral result for the model spaces
2.9 Proof of Theorem 2.3 4
2.1 0Estimates for the first eigenvalue of the p-Laplacian

Chapter 3 Estimates for lower eigenvalues
3.1 Reilly-type inequalities for the first nonzero closed eigenvalue of the p-Laplacian
3.2 Isoperimetric bounds for the first eigenvalue of the p-Laplacian
3.3 Isoperimetric bounds for the first eigenvalue of the Paneitz operators
3.4 Lower bounds for the first eigenvalue of four kinds of eigenvalue problems of the bi-drifting Laplacian

Chapter 4 Universal Inequalities
4.1 Universal inequalities of the clamped plate problem of the drifting Laplacian
4.1.1 Recent developments for universal inequalities of the clamped plate problem
4.1.2 The clamped plate problem of the drifting Laplacian
4.1.3 Universal inequalities for the clamped plate problem of the drifting Laplacian
4.1.4 Universal inequalities on the Gaussian shrinking soliton
4.2 Universal inequalities for the buckling problem of the drifting Laplacian
4.2.1 Recent developments for universal inequalities of the buckling problem
4.2.2 The buckling problem of the drifting Laplacian
4.2.3 A general inequality on the Ricci solitons
4.2.4 Universal inequalities in a bounded domain on Ricci solitons
4.3 Further study
4.3.1 Some interesting conjectures
4.3.2 Some Problems

Chapter 5 Open problems
5.1 Spectral problem concerning quantum strips and quantum layers
5.2 Connection between curvature flows and eigenvalue problem
5.3 Eigenvalue problem in Finsler geometry
Appendix A Warped products
Bibliography
流形上的特徵值問題（英文版） 下載 mobi epub pdf txt 電子書

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