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Counterexamples to inverse problems for the wave equation

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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.

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


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