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doi: 10.3934/ipi.2021058
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## Counterexamples to inverse problems for the wave equation

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

Received  January 2021 Early access October 2021

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.

Citation: Tony Liimatainen, Lauri Oksanen. Counterexamples to inverse problems for the wave equation. Inverse Problems & Imaging, doi: 10.3934/ipi.2021058
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