Extrinsic upper bounds for the first eigenvalue of elliptic operators

Adaptive Eigenvalue Computation for Elliptic Operators

This article is concerned with recent developments of adaptive wavelet solvers for elliptic eigenvalue problems. We describe the underlying abstract iteration scheme of the preconditioned perturbed iteration. We apply the iteration to a simple model problem in order to identify the main ideas which a numerical realization of the abstract scheme is based upon. This indicates how these concepts c...

Estimates for the First Nonzero Eigenvalue of Elliptic Operators in Divergence Form

We consider elliptic operators in divergence form L = div (Φ · grad ) either on a closed Riemannian manifold or in a domain with compact closure and piecewise smooth boundary M where Φ : M → End(TM) is a positive definite symmetric smooth section of the bundle of all endomorphisms of TM . We show that the first nonzero L-eigenvalues in the closed or Dirichlet eigenvalue problems can be bounded ...

Upper Bounds for the First Eigenvalue of the Dirac Operator on Surfaces. *

In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on an isometrically immersed surface M2 →֒ R3 as well as intrinsic bounds for 2-dimensional compact manifolds of genus zero and genus one. Moreover, we compare the different estimates of the eigenvalue of the Dirac operator for special families of metrics. Subj. Class.: Differential geometry. 1991 MS...

Upper Bounds for the First Eigenvalue of the Laplacian on Compact Submanifolds

Let (M, g) be a compact Riemannian manifold isometrically immersed in a simply connected space form (euclidean space, sphere or hyperbolic space). The purpose of this paper is to give optimal upper bounds for the first nonzero eigenvalue of the Laplacian of (M, g) in terms of r-th mean curvatures and scalar curvature. As consequences, we obtain some rigidity results. In particular, we prove tha...

Bounds for Elliptic Operators in Weighted Spaces