流形上的特征值问题

本专著拟介绍黎曼流形有界连通区域上几类自伴随椭圆算子的特征值问题。通过进一步地拓展黎曼几何里经典的体积比较定理，导出了线性Laplace算子与非线性p-Laplace算子的Dirichlet特征值问题下第一非零特征值的比较定理，对于Steklov特征值问题，同样可以得到第一非零特征值的比较定理，这些结论极大地推广了已有的经典结果；通过更好地构造指定流形上的测试函数，对于紧致完备流形上p-Laplace算子的第一闭特征值以及乘积流形上平面夹板（特征值）问题的第一非零特征值的下界给出了较好的估计，另外，对于满足特定条件的乘积Ricci孤立子上的Buckling（特征值）问题，我们可以给出涉及低阶特征值的万有不等式，这些新颖的结果对已有的经典结论做了很好的拓展。

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 2.34
2.10 Estimates 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