Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation

We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.

Department dc:contributor.department

Mathematics and Statistics

Year dc:date.issued

2016

Author dc:creator

• Milne, Tristan

Advisor dc:contributor.supervisor

• Mansouri, Abdol-Reza

• Partial Differential Equations
• Inverse Problems

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1974/14738

OAI identifier oai:identifier

oai:queensu.scholaris.ca:1974/14737