#### Year of Publication

2009

#### Degree Name

Doctor of Philosophy (PhD)

#### Document Type

Dissertation

#### College

Arts and Sciences

#### Department

Mathematics

#### First Advisor

Dr. Peter Hislop

#### Abstract

This thesis considers two isoperimetric inequalities for the eigenvalues of the Laplacian on a family of spherically symmetric Riemannian manifolds. The Payne-Pólya-Weinberger Conjecture (PPW) states that for a bounded domain Ω in Euclidean space R^{n}, the ratio λ1(Ω)/λ0(Ω) of the first two eigenvalues of the Dirichlet Laplacian is bounded by the corresponding eigenvalue ratio for the Dirichlet Laplacian on the ball B_{Ω}of equal volume. The Szegö-Weinberger inequality states that for a bounded domain Ω in Euclidean space R^{n}, the first nonzero eigenvalue of the Neumann Laplacian μ1(Ω) is maximized on the ball B_{Ω} of the same volume. In the first three chapters we will look at the known work for the manifolds R^{n} and H^{n}. Then we will take a family a spherically symmetric manifolds given by R^{n} with a spherically symmetric metric determined by a radially symmetric function *f*. We will then give a PPW-type upper bound for the eigenvalue gap, λ1(Ω) − λ0(Ω), and the ratio λ1(Ω)/λ0(Ω) on a family of symmetric bounded domains in this space. Finally, we prove the Szegö-Weinberger inequality for this same class of domains.

#### Recommended Citation

Miker, Julie, "Eigenvalue Inequalities for a Family of Spherically Symmetric Riemannian Manifolds" (2009). *University of Kentucky Doctoral Dissertations*. 783.

http://uknowledge.uky.edu/gradschool_diss/783