December  2019, 13(6): 1309-1325. doi: 10.3934/ipi.2019057

Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity

1. 

Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA

2. 

Institute for Advanced Study, The Hong Kong University of Science and Technology, Hong Kong, China

* Corresponding author

Received  January 2019 Revised  May 2019 Published  October 2019

Fund Project: The research of YY was partly supported by NSF Grant DMS-1715178, AMS-Simons travel grant, and start-up fund from Michigan State University

We consider a linearized inverse boundary value problem for the elasticity system. From the linearized Dirichlet-to-Neumann map at zero frequency, we show that a transversely isotropic perturbation of a homogeneous isotropic elastic tensor can be uniquely determined. From the linearized Dirichlet-to-Neumann map at two distinct positive frequencies, we show that a transversely isotropic perturbation of a homogeneous isotropic density can be identified at the same time.

Citation: Yang Yang, Jian Zhai. Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity. Inverse Problems & Imaging, 2019, 13 (6) : 1309-1325. doi: 10.3934/ipi.2019057
References:
[1]

T. Alkhalifah, Velocity analysis using nonhyperbolic moveout in transversely isotropic media, SEG Technical Program Expanded Abstracts 1996, (1996), 1499–1502. doi: 10.1190/1.1826401.  Google Scholar

[2]

T. AlkhalifahI. TsvankinK. Larner and J. Toldi, Velocity analysis and imaging in transversely isotropic media: Methodology and a case study, The Leading Edge, 15 (1996), 371-378.  doi: 10.1190/1.1437345.  Google Scholar

[3]

J. A. BarcelóM. Folch-GabayetS. Pérez-EstevaA. Ruiz and M. C. Vilela, Uniqueness for inverse elastic medium problems, SIAM Journal on Mathematical Analysis, 50 (2018), 3939-3962.  doi: 10.1137/17M1138315.  Google Scholar

[4]

O. A. Bauchau and J. I. Craig, Structural Analysis with Applications to Aerospace Structures, Springer-Verlag, New York, 2009. Google Scholar

[5]

E. Beretta, M. V. de Hoop, E. Francini, S. Vessella and J. Zhai, Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves, Inverse Problems, 33 (2017), 035013, 27 pp. doi: 10.1088/1361-6420/aa5bef.  Google Scholar

[6]

E. Beretta, E. Francini, A. Morassi, E. Rosset and S. Vessella, Lipschitz continuous dependence of piecewise constant Lamé coefficients from boundary data: The case of non-flat interfaces, Inverse Problems, 30 (2014), 125005, 18 pp. doi: 10.1088/0266-5611/30/12/125005.  Google Scholar

[7]

E. BerettaE. Francini and S. Vessella, Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements, Inverse Probl. Imag., 8 (2014), 611-644.  doi: 10.3934/ipi.2014.8.611.  Google Scholar

[8]

S. Bhattacharyya, Local uniqueness of the density from partial boundary data for isotropic elastodynamics, Inverse Problems, 34 (2018), 125001, 10 pp. doi: 10.1088/1361-6420/aade76.  Google Scholar

[9]

A. P. Calderón, On an inverse boundary value problem, Computational & Applied Mathematics, 25 (2006), 133-138.  doi: 10.1590/S0101-82052006000200002.  Google Scholar

[10]

C. I. CârsteaN. Honda and G. Nakamura, Uniqueness in the inverse boundary value problem for piecewise homogeneous anisotropic elasticity, SIAM J. Math. Anal., 50 (2018), 3291-3302.  doi: 10.1137/17M1125662.  Google Scholar

[11]

M. V. de Hoop, G. Nakamura and J. Zhai, Unique Recovery of Piecewise Analytic Density and Stiffness Tensor from the Elastic-Wave Dirichlet-to-Neumann Map, 2018, arXiv: 1803.01091. Google Scholar

[12]

M. V. de Hoop, G. Uhlmann and A. Vasy, Recovery of Material Parameters in Transversely Isotropic Media, 2019, arXiv: 1902.09394. Google Scholar

[13]

G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907-921.  doi: 10.1088/0266-5611/18/3/324.  Google Scholar

[14]

A. GreenleafM. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Mathematical Research Letters, 10 (2003), 685-693.  doi: 10.4310/MRL.2003.v10.n5.a11.  Google Scholar

[15]

M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, SIAM Journal on Applied Mathematics, 50 (1990), 1635-1644.  doi: 10.1137/0150097.  Google Scholar

[16]

M. Ikehata, The linearization of the Dirichlet to Neumann map in anisotropic plate theory, Inverse Problems, 11 (1995), 165-181.  doi: 10.1088/0266-5611/11/1/009.  Google Scholar

[17]

M. Ikehata, The linearization of the Dirichlet-to-Neumann map in the anisotropic Kirchhoff-Love plate theory, SIAM Journal on Applied Mathematics, 56 (1996), 1329-1352.  doi: 10.1137/S0036139994270437.  Google Scholar

[18]

M. Ikehata, A relationship between two Dirichlet to Neumann maps in anisotrpoic elastic plate theory, Journal of Inverse and Ill-Posed Problems, 4 (1996), 233-243.  doi: 10.1515/jiip.1996.4.3.233.  Google Scholar

[19]

O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, On uniqueness of Lamé coefficients from partial Cauchy data in three dimensions, Inverse Problems, 28 (2012), 125002, 5 pp. doi: 10.1088/0266-5611/28/12/125002.  Google Scholar

[20]

O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004.  Google Scholar

[21]

D. Kumar, M. K. Sen and R. J. Ferguson, Traveltime calculation and prestack depth migration in tilted transversely isotropic media, SEG Technical Program Expanded Abstracts 2003, (2003). doi: 10.1190/1.1818099.  Google Scholar

[22]

Y.-H. Lin and G. Nakamura, Boundary determination of the Lamé moduli for the isotropic elasticity system, Inverse Problems, 33 (2017), 125004, 23 pp. doi: 10.1088/1361-6420/aa942d.  Google Scholar

[23]

J.-P. Montagner and H.-C. Nataf, A simple method for inverting the azimuthal anisotropy of surface waves, Journal of Geophysical Research: Solid Earth, 91 (1986), 511-520.  doi: 10.1029/JB091iB01p00511.  Google Scholar

[24]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar

[25]

G. Nakamura and G. Uhlmann, Inverse problems at the boundary for an elastic medium, SIAM J. Math. Anal., 26 (1995), 263-279.  doi: 10.1137/S0036141093247494.  Google Scholar

[26]

G. Nakamura and G. Uhlmann, Layer stripping for a transversely isotropic elastic medium, SIAM J. Appl. Math., 59 (1999), 1879-1891.  doi: 10.1137/S0036139998337164.  Google Scholar

[27]

G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity, Invent. Math., 152 (2003), 205-207.  doi: 10.1007/s00222-002-0276-1.  Google Scholar

[28]

L. V. Rachele, An inverse problem in elastodynamics: Uniqueness of the wave speeds in the interior, Journal of Differential Equations, 162 (2000), 300-325.  doi: 10.1006/jdeq.1999.3657.  Google Scholar

[29]

L. V. Rachele, Uniqueness of the density in an inverse problem for isotropic elastodynamics, Transactions of the American Mathematical Society, 355 (2003), 4781-4806.  doi: 10.1090/S0002-9947-03-03268-9.  Google Scholar

[30]

J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, New York, second edition, 2002. Google Scholar

[31]

P. Stefanov, G. Uhlmann and A. Vasy, Local recovery of the compressional and shear speeds from the hyperbolic DN map, Inverse Problems, 34 (2018), 014003, 13 pp. doi: 10.1088/1361-6420/aa9833.  Google Scholar

[32]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.  Google Scholar

show all references

References:
[1]

T. Alkhalifah, Velocity analysis using nonhyperbolic moveout in transversely isotropic media, SEG Technical Program Expanded Abstracts 1996, (1996), 1499–1502. doi: 10.1190/1.1826401.  Google Scholar

[2]

T. AlkhalifahI. TsvankinK. Larner and J. Toldi, Velocity analysis and imaging in transversely isotropic media: Methodology and a case study, The Leading Edge, 15 (1996), 371-378.  doi: 10.1190/1.1437345.  Google Scholar

[3]

J. A. BarcelóM. Folch-GabayetS. Pérez-EstevaA. Ruiz and M. C. Vilela, Uniqueness for inverse elastic medium problems, SIAM Journal on Mathematical Analysis, 50 (2018), 3939-3962.  doi: 10.1137/17M1138315.  Google Scholar

[4]

O. A. Bauchau and J. I. Craig, Structural Analysis with Applications to Aerospace Structures, Springer-Verlag, New York, 2009. Google Scholar

[5]

E. Beretta, M. V. de Hoop, E. Francini, S. Vessella and J. Zhai, Uniqueness and Lipschitz stability of an inverse boundary value problem for time-harmonic elastic waves, Inverse Problems, 33 (2017), 035013, 27 pp. doi: 10.1088/1361-6420/aa5bef.  Google Scholar

[6]

E. Beretta, E. Francini, A. Morassi, E. Rosset and S. Vessella, Lipschitz continuous dependence of piecewise constant Lamé coefficients from boundary data: The case of non-flat interfaces, Inverse Problems, 30 (2014), 125005, 18 pp. doi: 10.1088/0266-5611/30/12/125005.  Google Scholar

[7]

E. BerettaE. Francini and S. Vessella, Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements, Inverse Probl. Imag., 8 (2014), 611-644.  doi: 10.3934/ipi.2014.8.611.  Google Scholar

[8]

S. Bhattacharyya, Local uniqueness of the density from partial boundary data for isotropic elastodynamics, Inverse Problems, 34 (2018), 125001, 10 pp. doi: 10.1088/1361-6420/aade76.  Google Scholar

[9]

A. P. Calderón, On an inverse boundary value problem, Computational & Applied Mathematics, 25 (2006), 133-138.  doi: 10.1590/S0101-82052006000200002.  Google Scholar

[10]

C. I. CârsteaN. Honda and G. Nakamura, Uniqueness in the inverse boundary value problem for piecewise homogeneous anisotropic elasticity, SIAM J. Math. Anal., 50 (2018), 3291-3302.  doi: 10.1137/17M1125662.  Google Scholar

[11]

M. V. de Hoop, G. Nakamura and J. Zhai, Unique Recovery of Piecewise Analytic Density and Stiffness Tensor from the Elastic-Wave Dirichlet-to-Neumann Map, 2018, arXiv: 1803.01091. Google Scholar

[12]

M. V. de Hoop, G. Uhlmann and A. Vasy, Recovery of Material Parameters in Transversely Isotropic Media, 2019, arXiv: 1902.09394. Google Scholar

[13]

G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907-921.  doi: 10.1088/0266-5611/18/3/324.  Google Scholar

[14]

A. GreenleafM. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Mathematical Research Letters, 10 (2003), 685-693.  doi: 10.4310/MRL.2003.v10.n5.a11.  Google Scholar

[15]

M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, SIAM Journal on Applied Mathematics, 50 (1990), 1635-1644.  doi: 10.1137/0150097.  Google Scholar

[16]

M. Ikehata, The linearization of the Dirichlet to Neumann map in anisotropic plate theory, Inverse Problems, 11 (1995), 165-181.  doi: 10.1088/0266-5611/11/1/009.  Google Scholar

[17]

M. Ikehata, The linearization of the Dirichlet-to-Neumann map in the anisotropic Kirchhoff-Love plate theory, SIAM Journal on Applied Mathematics, 56 (1996), 1329-1352.  doi: 10.1137/S0036139994270437.  Google Scholar

[18]

M. Ikehata, A relationship between two Dirichlet to Neumann maps in anisotrpoic elastic plate theory, Journal of Inverse and Ill-Posed Problems, 4 (1996), 233-243.  doi: 10.1515/jiip.1996.4.3.233.  Google Scholar

[19]

O. Y. Imanuvilov, G. Uhlmann and M. Yamamoto, On uniqueness of Lamé coefficients from partial Cauchy data in three dimensions, Inverse Problems, 28 (2012), 125002, 5 pp. doi: 10.1088/0266-5611/28/12/125002.  Google Scholar

[20]

O. Y. Imanuvilov and M. Yamamoto, Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions, Inverse Problems, 31 (2015), 035004, 46 pp. doi: 10.1088/0266-5611/31/3/035004.  Google Scholar

[21]

D. Kumar, M. K. Sen and R. J. Ferguson, Traveltime calculation and prestack depth migration in tilted transversely isotropic media, SEG Technical Program Expanded Abstracts 2003, (2003). doi: 10.1190/1.1818099.  Google Scholar

[22]

Y.-H. Lin and G. Nakamura, Boundary determination of the Lamé moduli for the isotropic elasticity system, Inverse Problems, 33 (2017), 125004, 23 pp. doi: 10.1088/1361-6420/aa942d.  Google Scholar

[23]

J.-P. Montagner and H.-C. Nataf, A simple method for inverting the azimuthal anisotropy of surface waves, Journal of Geophysical Research: Solid Earth, 91 (1986), 511-520.  doi: 10.1029/JB091iB01p00511.  Google Scholar

[24]

A. I. Nachman, Reconstructions from boundary measurements, Annals of Mathematics, 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar

[25]

G. Nakamura and G. Uhlmann, Inverse problems at the boundary for an elastic medium, SIAM J. Math. Anal., 26 (1995), 263-279.  doi: 10.1137/S0036141093247494.  Google Scholar

[26]

G. Nakamura and G. Uhlmann, Layer stripping for a transversely isotropic elastic medium, SIAM J. Appl. Math., 59 (1999), 1879-1891.  doi: 10.1137/S0036139998337164.  Google Scholar

[27]

G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity, Invent. Math., 152 (2003), 205-207.  doi: 10.1007/s00222-002-0276-1.  Google Scholar

[28]

L. V. Rachele, An inverse problem in elastodynamics: Uniqueness of the wave speeds in the interior, Journal of Differential Equations, 162 (2000), 300-325.  doi: 10.1006/jdeq.1999.3657.  Google Scholar

[29]

L. V. Rachele, Uniqueness of the density in an inverse problem for isotropic elastodynamics, Transactions of the American Mathematical Society, 355 (2003), 4781-4806.  doi: 10.1090/S0002-9947-03-03268-9.  Google Scholar

[30]

J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, New York, second edition, 2002. Google Scholar

[31]

P. Stefanov, G. Uhlmann and A. Vasy, Local recovery of the compressional and shear speeds from the hyperbolic DN map, Inverse Problems, 34 (2018), 014003, 13 pp. doi: 10.1088/1361-6420/aa9833.  Google Scholar

[32]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.  Google Scholar

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