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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation


Authors: Tristan Milne and Abdol-Reza Mansouri
Journal: Trans. Amer. Math. Soc. 371 (2019), 8781-8810
MSC (2010): Primary 35R30; Secondary 35R01
DOI: https://doi.org/10.1090/tran/7630
Published electronically: March 28, 2019
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Abstract: We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold, where sources are applied and fields observed on disjoint sets. This problem was recently studied by Lassas and Oksanen, who provided a unique reconstruction result. As an extension of their research, we prove that the Dirichlet to Neumann operator for disjoint source and observation areas 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. 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, and we provide a constructive procedure for doing so.


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Additional Information

Tristan Milne
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada

Abdol-Reza Mansouri
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada

DOI: https://doi.org/10.1090/tran/7630
Received by editor(s): November 18, 2016
Received by editor(s) in revised form: May 9, 2018
Published electronically: March 28, 2019
Additional Notes: This work was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada, as well as through the Ontario Graduate Scholarship.
Article copyright: © Copyright 2019 American Mathematical Society