Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation
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- by Tristan Milne and Abdol-Reza Mansouri PDF
- Trans. Amer. Math. Soc. 371 (2019), 8781-8810 Request permission
Corrigendum: Trans. Amer. Math. Soc. 374 (2021), 8305-8306.
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.References
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Additional Information
- Tristan Milne
- Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
- MR Author ID: 1324108
- Abdol-Reza Mansouri
- Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
- MR Author ID: 235698
- 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.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 8781-8810
- MSC (2010): Primary 35R30; Secondary 35R01
- DOI: https://doi.org/10.1090/tran/7630
- MathSciNet review: 3955564