UPPER BOUNDS ON THE SPLITTING OF THE EIGENVALUES

Author

Phuoc L. Ho, University of KentuckyFollow

Date Available

5-13-2011

Year of Publication

2010

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department

Mathematics

Advisor

Dr. Peter Hislop

Abstract

We establish the upper bounds for the difference between the first two eigenvalues of the relative and absolute eigenvalue problems. Relative and absolute boundary conditions are generalization of Dirichlet and Neumann boundary conditions on functions to differential forms respectively. The domains are taken to be a family of symmetric regions in Rⁿ consisting of two cavities joined by a straight thin tube. Our operators are Hodge Laplacian operators acting on k-forms given by the formula Δ^(k) = dδ+δd, where d and δ are the exterior derivatives and the codifferentials respectively. A result has been established on Dirichlet case (0-forms) by Brown, Hislop, and Martinez [2]. We use the same techniques to generalize the results on exponential decay of eigenforms, singular perturbation on domains [1], and matrix representation of the Hodge Laplacian restricted to a suitable subspace [2]. From matrix representation, we are able to find exponentially small upper bounds for the difference between the first two eigenvalues.

Recommended Citation

Ho, Phuoc L., "UPPER BOUNDS ON THE SPLITTING OF THE EIGENVALUES" (2010). University of Kentucky Doctoral Dissertations. 119.
https://uknowledge.uky.edu/gradschool_diss/119