流形上的特征值问题

流形上的特征值问题（英文版）
	曾用价	78.00
	出版社	科学出版社
	版次	1
	出版时间	2017年05月
	开本	16
	著编译者	毛井，杜锋，吴传喜   .
	页数	188
	ISBN编码	9787030525680

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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
　　Let Ω be a bounded connected domain on an n-dimensional complete Riemannian manifold (M; g). The so-called Dirichlet eigenvalue problem is to find all possible real numbers . such that there exists a nontrivial solution u to the boundary value problem .
　　(1.1)
　　where △u = div(▽u) is the Laplacian of u given by
　　in a system of local coordinates on M, with (gij) = (gij).1 the inverse of the metric matrix. The desired real numbers . are called the eigenvalues of △. For a given ., the space of solutions of (1.1) is a vector space, since thefirst equation of (1.1) is linear in u. This vector space is called the eigenspace of .. The non-zero elements of each eigenspace are called eigenfunctions. Denote by L2(Ω) the space of all measurable functions f on Ω satisfying
　　with d Ω the volume element of Ω. We can define the usual inner product and induced norm on L2(Ω) by
　　for f; g 2 L2(Ω). Under this inner product, L2(Ω) is a Hilbert space. By the boundary condition in (1.1) and the Green's formula, it follows that △ is a self Ω adjoint operator on the Hilbert space L2(Ω). Furthermore, by the spectral theory of a self-adjoint compact operator, we know that the Laplacian △ in (1.1) has eigenvalues listed by
　　and each associated eigenspace hasfinite dimension (see, e.g., [30], p.169 or equiv Ω alently, Theorem 1.6 below). .i (i > 1) is called the ith Dirichlet eigenvalue of △. If . = 0 in (1.1), then together with the boundary condition u = 0 we know that u vanishes identically. This contradicts the fact that u is nontrivial on Ω. So, the low Ω est Dirichlet eigenvalue .1 is strictly positive. By Rayleigh's theorem and Max-min principle (see, e.g., [30], p.16-17), we know that (1.1) has positive weak solutions u in the space W1;2 0 (Ω), the completion of the set C10 (Ω) of smooth functions compactly supported on Ω under the Sobolev norm , and the first Dirichlet eigenvalue (Ω) can be characterized by
　　(1.2)
　　Remark 1.1 (1) In fact, according to the di.erent situations of boundary , one can consider di.erent eigenvalue problems of the Laplacian. If then one can consider the following closed eigenvalue problem
　　As in the case of the Dirichlet eigenvalue problem, it is not di±cult to get that .△ only has the discrete spectrum and all the eigenvalues can be listed increasingly also. However, in this case, thefirst closed eigenvalue .c1(Ω) satisfies .c1(Ω) = 0 and the corresponding eigenfunction should be nonzero constant function. Moreover,by Rayleigh's theorem and Max-min principle, thefirst nonzero closed eigenvalue .c1(Ω) can be characterized as follows
　　where W1;2(Ω) is the completion of the set C10 (Ω) of smooth functions under the Sobolev norm , and the constraint should be assured because of Courant's minimum principle. except the Dirichlet eigenvalue problem, one can consider the Neumann eigenvalue problem
　　(1.3)
　　where v is the outward unit normal vectorfield on , and the mixed eigenvalue problem
　　(1.4)
　　where N is an open submanifold of . Clearly, in (1.3) and (1.4),△ only has the discrete spectrum and all the eigenvalues can be listed increasingly. But by the boundary conditions and the maximum principle of second-order partial di.erential equations (PDEs for short), we know that thefirst Neumann eigenvalue .N1(Ω) satisfies .N1(Ω) = 0 with nonzero constant function as its eigenfunction, and the first mixed eigenvalue .M1(Ω) must be positive, i.e., .M1(Ω) > 0. Moreover, thefirst nonzero Neumann and mixed eigenvalues are the infimums of the Rayleigh's quotient in different functional spaces.
　　(2) One can consider the following nonlinear Dirichlet eigenvalue problem
　　(1.5)
　　where is the p-Laplacian with 1 < p < 1, and Ω is a bounded domain on an n-dimensional Riemannian manifold (M; g). In local coordinates on M, we have
　　(1.6)
　　where is the inverse of the metric matrix. Clearly, the p-Laplacian is a generalization of the linear Laplacian. Although many results about the linear Laplacian (p = 2) have been obtained, many rather basic questions about the spectrum of the nonlinear p-Laplacian remain to be solved. A well-known result about the above nonlinear eigenvalue problem states that it has a positive weak solution, which is unique modulo the scaling, in the space W1;p 0 (Ω), the completion of the set C10 (Ω) under the Sobolev norm . For a bounded simply connected domain with su±ciently smooth boundary in Euclidean space, one can get a simple proof of this fact in [15]. Moreover, thefirst Dirichlet eigenvalue (Ω) of the p-Laplacian can be characterized by
　　(1.7)
　　(3) In fact, except the Laplace and the p-Laplace operators, one can also consider eigenvalue problems for other elliptic operators (such as, the drifting Laplacian, the bi-harmonic operator, the bi-drifting Laplacian, the Paneitz operator, the poly-harmonic operator, and so on) on bounded connected domains, which will be investigated in the following chapters.
　　Optimal domains in isoperimetric inequalities relating eigenvalues to geometrical quantities such as volume and surface area quite often display some degree of symmetry. In many instances, this symmetry is actually the maximal possible, such as in the Rayleigh-Faber-Krahn and the Szeg.o-Weinberger inequalities, corresponding to Dirichlet and Neumann boundary conditions for Euclidean domains, respectively. It is thus quite natural that symmetrization plays a fundamental role in this aspect of spectral theory and is at the heart of many isoperimetric inequalities of this type. The Rayleigh-Faber-Krahn inequality, for instance, is a consequence of the fact that Schwarz symmetrization does not increase the Dirichlet integral while leaving the L2 norm unchanged. Even in some cases where the minimiser is not one but two balls, this symmetrization plays a role, as happens not only in the case of the second Dirichlet eigenvalue, but also when other restrictions are enforced--see, for instance, [21] and [71].
　　However, Schwarz and other similar symmetrization procedures are mostly Euclidean techniques, and do not extend to manifolds in general. But, as stated above, this is not to say that symmetry does not play a similar fundamental role in isoperimetrical eigenvalue inequalities on manifolds. This can be seen, for instance, from Hersch's result for two-dimensional spheres [86], which states that among all surfaces with the same area which are homeomorphic to S2, the round sphere (canonical metric) maximises thefirst nontrivial eigenvalue.
　　The purpose of Chapter 2 is to develop the usage of symmetrization techniques in the case of manifolds, allowing us to derive comparison isoperimetric inequalities there. To this end, we shall consider a symmerization procedure based on curvature. More precisely, given a complete n-manifold M and a point p in M such that we have lower and upper bounds for the radial Ricci and sectional curvatures within a geodesic disk of radius r0, which depend only on the distance t to the point p, we build two spherically symmetric manifolds centered at a point p and whose curvatures are determined by the respective bounds. In this way, we are then able to obtain that thefirst eigenvalue with Dirichlet boundary conditions is bounded from above and below by thefirst Dirichlet eigenvalue on geodesic disks centered at p*on these two manifolds|see Theorems 2.8 and 2.14 for the precise statements of these results.
　　Now, we would like to recall the history of radial curvature briefly and also mention some comparison theorems for radial curvature partially. It was thefirst time that Klingenberg introduced the notion of radial curvature in [101] to study compact Riemannian manifolds with radial curvatures pinched between 1/4 and 1. After that, mathematicians have been paying attention to the radial curvatures. In general, the reference manifolds for comparison theorems are space forms. However, Elerath [69] employed a Von Mangoldt surface of revolution (i.e., a complete surface of revolution homeomorphic to Euclidean plane whose Gaussian curvature is nonincreasing along each meridian) Z*R3 with nonnegative Gaussian curvature as the reference surface to prove the generalized Toponogov comparison theorem (we write GTCT for short) successfully for complete open Riemannian manifolds with radial curvatures bounded from below by that of Z. For complete open Riemannian manifolds whose radial Ricci curvatures are bounded from below by a nonnegative smooth function (t) of the distance parameter w.r.t. some point (as described in Definition 2.2), together with other constraints for (t), Abresch proved the GTCT in [1] (these special manifolds were called "asymptotically non-negatively curved" manifolds therein). Of course, there are other types of GTCT, which we do not need to also mention here. From these facts, we know that mathematicians have investigated manifolds with radial curvatures bounded by some continuous function of the distance parameter (of the original manifolds), and generalized some classical comparison theorems.
　　Theorems 2.8 and 2.14 may be seen as extensions of Cheng's bounds for the first eigenvalue, where the comparison is made between a geodesic disk on M and those on spaces of constant curvature which are obtained by taking lower and upper bounds of the curvature [50, 51]. More precisely, Cheng has proved the following conclusions.
　　Theorem 1.2 ([50]) Suppose M is a complete Riemannian manifold and Ricci curvature of with dimM = n: Then; for we have
　　and equality holds if and only if B(x0; r0) is isometric to Vn(k; r0); where B(x0; r0) denotes the open geodesic ball with center x0 and radius r0 on M; and Vn(k; r0) is a geodesic ball with radius r0 in the n-dimensional simply connected space form with constant curvature k; moreover; denotes thefirst Dirichlet eigenvalue of the Laplacian on the corresponding geodesic ball.
　　Theorem 1.3 ([51]) Let M be a complete Riemannian manifold all of whose sectional curvatures are less than or equal to a given constant k; and dimM = n: Then; for which B is within the cut-locus of p; we have
　　where symbols have the same meanings as those in Theorem 1.2.
　　The starting point behind Theorems 2.8 and 2.14 is twofold. On the one hand, it should be possible to replace the constant curvature spaces in Cheng's results by spherically symmetric spaces, in such a way that these still yield curvature bounds which imply the desired eigenvalue bounds. On the other hand, spherically symmetric manifolds posses a relatively simple characterization and thefirst Dirichlet eigenvalue on a geodesic disk is given by the zero of a solution to a second order ordinary di.erential equation (see (2.16)). Thus, there are many bounds for these eigenvalues, some of which providing quite accurate bounds--see [11], [12], [20], [77], for instance.
　　The heat equation, which can be used to describe the conduction of heat through a given medium, and related deformations of the heat equation, like the di.usion equation, the Fokker-Planck equation, and so on, are of basic importance in variable scientificfields. Given an n-dimensional Riemannian manifold M with the Laplace-Beltrami operator △. Then we are able to define a di.erential operator L, which is known as the heat operator, by
　　acting on functions in , which are C2 w.r.t. the variable x, varying on M, and C1 w.r.t. the variable t, varying on (0;1). Correspondingly, the heat equation is given by
　　with If we want to get the existence, or even give an explicit expression, of the solution for this heat equation with a prescribed initial condition or (Dirichlet or Neumann) boundary condition, we need to use a tool named heat kernel.
　　Definition 1.4 A fundamental solution; which is called the heat kernel; of the heat equation on a prescribed Riemannian manifold M is a continuous function H(x; y; t); defined on ; which is C2 with respect to x; C1 with respect to t; and