Counterexamples to inverse problems for the wave equation

Tony Liimatainen; Lauri Oksanen

doi:10.3934/ipi.2021058

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2022, Volume 16, Issue 2: 467-479. Doi: 10.3934/ipi.2021058

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Counterexamples to inverse problems for the wave equation

Tony Liimatainen^1,2, and
Lauri Oksanen^2,

1.
Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland
2.
Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

Received: December 31, 2020

Early access: October 2021

Published: April 2022

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We construct counterexamples to inverse problems for the wave operator on domains in $ \mathbb{R}^{n+1} $, $ n \ge 2 $, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On $ \mathbb{R}^{n+1} $ the metrics are conformal to the Minkowski metric.

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Mathematics Subject Classification: 35R30, 35L05, 58J45.

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