Thesis-2006-Kirpichnikova.pdf (5.5 MB)
Inverse boundary spectral problem for Riemannian polyhedra
thesis
posted on 2018-11-14, 16:55 authored by Anna KirpichnikovaThe object under consideration is an admissible Riemannian polyhedron M with a
piece-wise smooth boundary δM. This is a finite n-dimensional simplicial complex
equipped with a family of Riemannian metrics smooth inside each simplex. We introduce
an anisotropic Dirichlet Laplace operator in a weak sense for the admissible
Riemannian polyhedron and define a set of boundary spectral data Γ, {λκ, δυφκ|Γ}∞κ=1
on an open part Γ ∩ δM, where λκ are the eigenvalues on Γ and δυφκ|Γ are the traces
of normal derivatives of eigenfunctions of the Laplacian. The main result of the work
is: if two admissible Riemannian polyhedra M and M have open diffeomorphic parts
of the boundaries Γ ∩ δM and Γ ∩ δM such that the set of boundary spectral data
on Γ coincides with the set of boundary spectral data on Γ, then there is one-to-one
correspondence between M and M as simplicial complexes and they are also isometric
as metric spaces. A new technique was developed to tackle the problem. That
technique incorporated two methods: BC-method generalized and adjusted for the
admissible Riemannian polyhedra and the technique of Gaussian beams extended for
anisotropic piecewise smooth media.
Funding
Loughborough University. ORS Award Scheme.
History
School
- Science
Department
- Mathematical Sciences
Publisher
© Anna KirpichnikovaPublisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Publication date
2006Notes
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy at Loughborough University.Language
- en