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doi: 10.3934/ipi.2021058
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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

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 & Imaging, 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. Google Scholar

[2]

A. L. Besse, Einstein Manifolds, Classics in Mathematics. Springer-Verlag, Berlin, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737–1758. doi: 10.1080/03605300701382340.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Lassas and T. Liimatainen, Conformal harmonic coordinates, To appear in Communications in Analysis and Geometry, Preprint arXiv: 1912.08030, 2019. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.  doi: 10.1126/science.1125907.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9.  Google Scholar

show all references

References:
[1]

S. Alexakis, A. Feizmohammadi and L. Oksanen, Lorentzian Calderón problem under curvature bounds, Preprint arXiv: 2008.07508, 2020. Google Scholar

[2]

A. L. Besse, Einstein Manifolds, Classics in Mathematics. Springer-Verlag, Berlin, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737–1758. doi: 10.1080/03605300701382340.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Lassas and T. Liimatainen, Conformal harmonic coordinates, To appear in Communications in Analysis and Geometry, Preprint arXiv: 1912.08030, 2019. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.  doi: 10.1126/science.1125907.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. Uhlmann, Inverse problems: Seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9.  Google Scholar

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