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流形上的特征值問題-英文版 版權(quán)信息
- ISBN：9787030525680
- 條形碼：9787030525680 ; 978-7-03-052568-0
- 裝幀：一般膠版紙
- 冊(cè)數(shù)：暫無
- 重量：暫無
- 所屬分類：>>
流形上的特征值問題-英文版 本書特色
本專著擬介紹黎曼流形有界連通區(qū)域上幾類自伴隨橢圓算子的特征值問題。通過進(jìn)一步地拓展黎曼幾何里經(jīng)典的體積比較定理，導(dǎo)出了線性Laplace算子與非線性p-Laplace算子的Dirichlet特征值問題下*非零特征值的比較定理，對(duì)于Steklov特征值問題，同樣可以得到*非零特征值的比較定理，這些結(jié)論極大地推廣了已有的經(jīng)典結(jié)果；通過更好地構(gòu)造指定流形上的測試函數(shù)
流形上的特征值問題-英文版 內(nèi)容簡介
本專著擬介紹黎曼流形有界連通區(qū)域上幾類自伴隨橢圓算子的特征值問題。通過進(jìn)一步地拓展黎曼幾何里經(jīng)典的體積比較定理，導(dǎo)出了線性Laplace算子與非線性p-Laplace算子的Dirichlet特征值問題下非零特征值的比較定理，對(duì)于Steklov特征值問題，同樣可以得到非零特征值的比較定理，這些結(jié)論極大地推廣了已有的經(jīng)典結(jié)果；通過更好地構(gòu)造指定流形上的測試函數(shù)，對(duì)于緊致完備流形上p-Laplace算子的閉特征值以及乘積流形上平面夾板（特征值）問題的非零特征值的下界給出了較好的估計(jì)，另外，對(duì)于滿足特定條件的乘積Ricci孤立子上的Buckling（特征值）問題，我們可以給出涉及低階特征值的萬有不等式，這些新穎的結(jié)果對(duì)已有的經(jīng)典結(jié)論做了很好的拓展。
流形上的特征值問題-英文版 目錄
Contents
Chapter 1 Introduction 1
Chapter 2 Eigenvalue comparison theorems 10
2.1 Basic notions and the geometry of spherically symmetric manifolds 11
2.2 A generalized eigenvalue comparison theorem for manifolds with radial Ricci curvature bounded from below 17
2.3 A generalized eigenvalue comparison theorem for manifolds with radial sectional curvature bounded from above 22
2.4 Properties of the model manifolds 27
2.5 Examples 30
2.6 A criterion for the existence of the model spaces 40
2.7 Estimates for the heat kernel 43
2.8 A Cheng-type isoperimetric inequality for the p-Laplace operator and a spectral result for the model spaces 51
2.9 Proof of Theorem 2.34 54
2.10 Estimates for the first eigenvalue of the p-Laplacian 59
Chapter 3 Estimates for lower eigenvalues 65
3.1 Reilly-type inequalities for the first nonzero closed eigenvalue of the p-Laplacian 65
3.2 Isoperimetric bounds for the first eigenvalue of the p-Laplacian 74
3.3 Isoperimetric bounds for the first eigenvalue of the Paneitz operators 77
3.4 Lower bounds for the first eigenvalue of four kinds of eigenvalue problems
of the bi-drifting Laplacian 81
Chapter 4 Universal Inequalities 91
4.1 Universal inequalities of the clamped plate problem of the drifting
Laplacian 93
4.1.1 Recent developments for universal inequalities of the clamped plate problem 93
4.1.2 The clamped plate problem of the drifting Laplacian 95
4.1.3 Universal inequalities for the clamped plate problem of the drifting Laplacian 104
4.1.4 Universal inequalities on the Gaussian shrinking soliton 115
4.2 Universal inequalities for the buckling problem of the drifting Laplacian 118
4.2.1 Recent developments for universal inequalities of the buckling problem 119
4.2.2 The buckling problem of the drifting Laplacian 120
4.2.3 A general inequality on the Ricci solitons 125
4.2.4 Universal inequalities in a bounded domain on Ricci solitons 138
4.3 Further study 147
4.3.1 Some interesting conjectures 147
4.3.2 Some Problems 151
Chapter 5 Open problems 154
5.1 Spectral problem concerning quantum strips and quantum layers 154
5.2 Connection between curvature flows and eigenvalue problem 160
5.3 Eigenvalue problem in Finsler geometry 163
Appendix A Warped products 166
Bibliography 172
Chapter 1 Introduction 1
Chapter 2 Eigenvalue comparison theorems 10
2.1 Basic notions and the geometry of spherically symmetric manifolds 11
2.2 A generalized eigenvalue comparison theorem for manifolds with radial Ricci curvature bounded from below 17
2.3 A generalized eigenvalue comparison theorem for manifolds with radial sectional curvature bounded from above 22
2.4 Properties of the model manifolds 27
2.5 Examples 30
2.6 A criterion for the existence of the model spaces 40
2.7 Estimates for the heat kernel 43
2.8 A Cheng-type isoperimetric inequality for the p-Laplace operator and a spectral result for the model spaces 51
2.9 Proof of Theorem 2.34 54
2.10 Estimates for the first eigenvalue of the p-Laplacian 59
Chapter 3 Estimates for lower eigenvalues 65
3.1 Reilly-type inequalities for the first nonzero closed eigenvalue of the p-Laplacian 65
3.2 Isoperimetric bounds for the first eigenvalue of the p-Laplacian 74
3.3 Isoperimetric bounds for the first eigenvalue of the Paneitz operators 77
3.4 Lower bounds for the first eigenvalue of four kinds of eigenvalue problems
of the bi-drifting Laplacian 81
Chapter 4 Universal Inequalities 91
4.1 Universal inequalities of the clamped plate problem of the drifting
Laplacian 93
4.1.1 Recent developments for universal inequalities of the clamped plate problem 93
4.1.2 The clamped plate problem of the drifting Laplacian 95
4.1.3 Universal inequalities for the clamped plate problem of the drifting Laplacian 104
4.1.4 Universal inequalities on the Gaussian shrinking soliton 115
4.2 Universal inequalities for the buckling problem of the drifting Laplacian 118
4.2.1 Recent developments for universal inequalities of the buckling problem 119
4.2.2 The buckling problem of the drifting Laplacian 120
4.2.3 A general inequality on the Ricci solitons 125
4.2.4 Universal inequalities in a bounded domain on Ricci solitons 138
4.3 Further study 147
4.3.1 Some interesting conjectures 147
4.3.2 Some Problems 151
Chapter 5 Open problems 154
5.1 Spectral problem concerning quantum strips and quantum layers 154
5.2 Connection between curvature flows and eigenvalue problem 160
5.3 Eigenvalue problem in Finsler geometry 163
Appendix A Warped products 166
Bibliography 172
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