Counterexamples to inverse problems for the wave equation

2021/01/26 by Tony Liimatainen, Lauri Oksanen, Liimatainen, Tony +1
Mathematics · #35L05 #35R30 #58J45 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:35L05 #msc:35R30 #msc:58J45

paper · pdf · doi:10.48550/arxiv.2101.10740

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arxiv created 2021/01/26 · arxiv updated 2021/01/27

Abstract

We construct counterexamples to inverse problems for the wave operator on domains in ℝⁿ⁺¹, n ≥ 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 ℝⁿ⁺¹ the metrics are conformal to the Minkowski metric.

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