Counterexamples to inverse problems for the wave equation

Tony Liimatainen; Lauri Oksanen

doi:10.3934/ipi.2021058

Article Contents

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

This issue Previous Article Weighted area constraints-based breast lesion segmentation in ultrasound image analysis Next Article

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: January 2021

Early access: October 2021

Published: April 2022

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	S. Alexakis, A. Feizmohammadi and L. Oksanen, Lorentzian Calderón problem under curvature bounds, Preprint arXiv: 2008.07508, 2020.
[2]	A. L. Besse, Einstein Manifolds, Classics in Mathematics. Springer-Verlag, Berlin, 2008.
[3]	S. N. Curry and A. R. Gover, An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity, volume 443 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2018, 86-170.
[4]	T. Daudé, N. Kamran and F. Nicoleau, A survey of non-uniqueness results for the anisotropic Calder´on problem with disjoint data, In Nonlinear Analysis in Geometry and Applied Mathematics. Part 2, volume 2 of Harv. Univ. Cent. Math. Sci. Appl. Ser. Math., Int. Press, Somerville, MA, 2018, 77-101.
[5]	T. Daudé, N. Kamran and F. Nicoleau, Non-uniqueness results for the anisotropic Calderón problem with data measured on disjoint sets, Ann. Inst. Fourier (Grenoble), 69 (2019), 119-170. doi: 10.5802/aif.3240.
[6]	T. Daudé, N. Kamran and F. Nicoleau, On the hidden mechanism behind non-uniqueness for the anisotropic Calderón problem with data on disjoint sets, Ann. Henri Poincaré, 20 (2019), 859-887. doi: 10.1007/s00023-018-00755-2.
[7]	T. Daudé, N. Kamran and F. Nicoleau, The anisotropic Calderón problem for singular metrics of warped product type: The borderline between uniqueness and invisibility, J. Spectr. Theory, 10 (2020), 703-746. doi: 10.4171/JST/310.
[8]	T. Daudé, N. Kamran and F. Nicoleau, On nonuniqueness for the anisotropic Calderón problem with partial data, Forum Math. Sigma, 8 (2020), Paper No. e7, 17 pp. doi: 10.1017/fms.2020.1.
[9]	G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737–1758. doi: 10.1080/03605300701382340.
[10]	A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-693. doi: 10.4310/MRL.2003.v10.n5.a11.
[11]	L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Classics in Mathematics. Springer, Berlin, 2007. doi: 10.1007/978-3-540-49938-1.
[12]	A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, volume 123 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220.
[13]	C. Kenig and M. Salo, The Calderón problem with partial data on manifolds and applications, Analysis and PDE, 6 (2013), 2003-2048. doi: 10.2140/apde.2013.6.2003.
[14]	Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on {R}iemannian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, (2019), 5087–5126. doi: 10.1093/imrn/rnx263.
[15]	Y. Kian, Y. Kurylev, M. Lassas and L. Oksanen, Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets, J. Differential Equations, 267 (2019), 2210-2238. doi: 10.1016/j.jde.2019.03.008.
[16]	M. Lassas and T. Liimatainen, Conformal harmonic coordinates, To appear in Communications in Analysis and Geometry, Preprint arXiv: 1912.08030, 2019.
[17]	M. Lassas, T. Liimatainen and M. Salo, The Calderón problem for the conformal Laplacian, To appear in Communications in Analysis and Geometry, Preprint arXiv: 1612.07939, 2016.
[18]	M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems, 26 (2010), 085012, 19 pp. doi: 10.1088/0266-5611/26/8/085012.
[19]	M. Lassas and L. Oksanen, Inverse problem for the Riemannian wave equation with Dirichlet data and {N}eumann data on disjoint sets, Duke Mathematical Journal, 163 (2014), 1071-1103. doi: 10.1215/00127094-2649534.
[20]	M. Lassas, M. Taylor and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Geom. Anal., 11 (2003), 207-222. doi: 10.4310/CAG.2003.v11.n2.a2.
[21]	J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91. doi: 10.1090/S0273-0979-1987-15514-5.
[22]	W. R. B. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems, 13 (1997), 125-134. doi: 10.1088/0266-5611/13/1/010.
[23]	J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782. doi: 10.1126/science.1125907.
[24]	Rakesh, Characterization of transmission data for Webster's horn equation, Inverse Problems, 16 (2000), L9–L24. doi: 10.1088/0266-5611/16/2/102.
[25]	D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr and D. R. Smith, Metamaterial electromagnetic cloak at microwave frequencies, Science, 314 (2006), 977-980. doi: 10.1126/science.1133628.
[26]	G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279. doi: 10.1007/s13373-014-0051-9.