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

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 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.

Keywords: Inverse problems, counterexamples, wave equation, conformal scaling, Lorentzian manifold, partial data, hidden conformal invariance.
Mathematics Subject Classification: 35R30, 35L05, 58J45.
Citation: Tony Liimatainen, Lauri Oksanen. Counterexamples to inverse problems for the wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 467-479. doi: 10.3934/ipi.2021058
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.

show all references

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.

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