Book contents
- Frontmatter
- Contents
- 1 Preface
- 2 List of lecture courses
- 3 List of participants
- 4 Basic Riemannian geometry
- 5 The Laplacian on Riemannian manifolds
- 6 Computational Spectral Theory
- 7 Isoperimetric and universal inequalities for eigenvalues
- 8 Estimates of heat kernels on Riemannian manifolds
- 9 Spectral theory of the Schrödinger operators on noncompact manifolds: qualitative results
- 10 Lectures on wave invariants
7 - Isoperimetric and universal inequalities for eigenvalues
Published online by Cambridge University Press: 08 January 2010
- Frontmatter
- Contents
- 1 Preface
- 2 List of lecture courses
- 3 List of participants
- 4 Basic Riemannian geometry
- 5 The Laplacian on Riemannian manifolds
- 6 Computational Spectral Theory
- 7 Isoperimetric and universal inequalities for eigenvalues
- 8 Estimates of heat kernels on Riemannian manifolds
- 9 Spectral theory of the Schrödinger operators on noncompact manifolds: qualitative results
- 10 Lectures on wave invariants
Summary
Abstract
This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szegö-Weinberger, Payne-Pólya-Weinberger, Sperner, Hile-Protter, and H.C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, Sn or Hn.
Introduction
The Eigenvalue Problems
The first eigenvalue problem we shall introduce is that of the fixed membrane, or Dirichlet Laplacian. We consider the eigenvalues and eigenfunctions of –Δ on a bounded domain (=connected open set) Ω in Euclidean space Rn, i.e., the problem
It is well-known that this problem has a real and purely discrete spectrum where
Here each eigenvalue is repeated according to its multiplicity. An associated orthonormal basis of real eigenfunctions will be denoted u1, u2, u3, …. In fact, throughout this paper we will assume that all functions we consider are real. This entails no loss of generality in the present context.
The next problem we introduce is that of the free membrane, or Neumann Laplacian.
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- Spectral Theory and Geometry , pp. 95 - 139Publisher: Cambridge University PressPrint publication year: 1999
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