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Inverse boundary spectral problem for Riemannian polyhedra

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thesis
posted on 14.11.2018, 16:55 authored by Anna Kirpichnikova
The 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 Kirpichnikova

Publisher 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

2006

Notes

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

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