# American Institute of Mathematical Sciences

April  2022, 16(2): 467-479. doi: 10.3934/ipi.2021058

## 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 Published  April 2022 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, 2022, 16 (2) : 467-479. doi: 10.3934/ipi.2021058
##### References:

show all references