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June  2021, 15(3): 445-474. doi: 10.3934/ipi.2020075

The interior transmission eigenvalue problem for elastic waves in media with obstacles

Fioralba Cakoni 1, , Pu-Zhao Kow 2,, and  Jenn-Nan Wang 3,

1. 

Department of Mathematics, Rutgers University, New Brunswick, USA
2. 

Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
3. 

Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan

* Corresponding author: Pu-Zhao Kow

Received  June 2020 Revised  September 2020 Published  November 2020

In this paper, we investigate the interior transmission eigenvalue problem for elastic waves propagating outside a sound-soft or a sound-hard obstacle surrounded by an anisotropic layer. This study is motivated by the inverse problem of identifying an object embedded in an inhomogeneous media in the presence of elastic waves. Our analysis of this non-selfadjoint eigenvalue problem relies on the weak formulation of involved boundary value problems and some fundamental tools in functional analysis.

Keywords: Interior transmission eigenvalue problem, elastic waves, anisotropic inhomogeneous media, media with obstacles, non-selfadjoint eigenvalue problem.
Mathematics Subject Classification: Primary: 35P99, 35Q74; Secondary: 47A53, 47A75.
Citation: Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, 2021, 15 (3) : 445-474. doi: 10.3934/ipi.2020075
References:
[1]

C. Bellis, F. Cakoni and B. B. Guzina, Nature of the transmission eigenvalue spectrum for elastic bodies, IMA J. Appl. Math., 78 (2013), 895-923.  doi: 10.1093/imamat/hxr070.  Google Scholar

[2]

C. Bellis and B. B. Guzina, On the existence and uniqueness of a solution to the interior transmission problem for piecewise-homogeneous solids, J. Elasticity, 101 (2010), 29-57.  doi: 10.1007/s10659-010-9242-0.  Google Scholar

[3]

E. Blåsten, L. Päivärinta and J. Sylvester, Corners always scatter, Comm. Math. Phys., 331 (2014), 725-753.  doi: 10.1007/s00220-014-2030-0.  Google Scholar

[4]

A. S. Bonnet-Ben Dhia, L. Chesnel and H. Haddar, On the use of T -coercivity to study the interior transmission eigenvalue problem, C. R. Math. Acad. Sci. Paris, 349 (2011), 647-651.  doi: 10.1016/j.crma.2011.05.008.  Google Scholar

[5]

F. Cakoni, D. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, C.R. Acad. Sci. Paris., 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[6]

F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Probl. Imaging, 6 (2012), 373-398.  doi: 10.3934/ipi.2012.6.373.  Google Scholar

[7]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[8]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar

[9]

A. Charalambopoulos, On the interior transmission problem in nondissipative, inhomogeneous, anisotropic elasticity, J. Elasticity, 67 (2002), 149-170.  doi: 10.1023/A:1023958030304.  Google Scholar

[10]

A. Charalambopoulos and K. A. Anagnostopoulos, On the spectrum of the interior transmission problem in isotropic elasticity, J Elasticity, 90 (2008), 295-313.  doi: 10.1007/s10659-007-9146-9.  Google Scholar

[11]

A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in three-dimensional linear elasticity, Inverse Probl., 18 (2002), 547-558.  doi: 10.1088/0266-5611/18/3/303.  Google Scholar

[12]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.  doi: 10.1080/00036810802713966.  Google Scholar

[13]

B. Davey, C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with less regular coefficients, Math. Ann., (2020). doi: 10.1007/s00208-020-02026-0.  Google Scholar

[14]

H. Diao, H. Liu and L. Wang, On generalized Holmgren's principle to the Lamé operator with applications to inverse elastic problems, Calc. Var., 59 (2020), Paper No. 179, 50 pp. doi: 10.1007/s00526-020-01830-5.  Google Scholar

[15]

J. Elschner and G. Hu, Acoustic scattering from corners, edges and circular cones, Arch. Ration. Mech. Anal., 228 (2018), 653-690.  doi: 10.1007/s00205-017-1202-4.  Google Scholar

[16]

G. Giorgi and H. Haddar, Computing estimates of material properties from transmission eigenvalues, Inverse Problems, 28 (2012), 055009, 23 pp. doi: 10.1088/0266-5611/28/5/055009.  Google Scholar

[17]

I. Harris, F. Cakoni and J. Sun, Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids, Inverse Problems, 30 (2014), 035016, 21pp. doi: 10.1088/0266-5611/30/3/035016.  Google Scholar

[18]

G. Hu, M. Salo and E. V. Vesalainen, Shape identification in inverse medium scattering problems with a single far-field pattern, SIAM J. Math. Anal., 48 (2016), 152-165.  doi: 10.1137/15M1032958.  Google Scholar

[19]

X. Ji and H. Liu, On isotropic cloaking and interior transmission eigenvalue problems, European J. Appl. Math., 29 (2018), 253-280.  doi: 10.1017/S0956792517000110.  Google Scholar

[20]

A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci., 21 (1998), 619-651.  doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P.  Google Scholar

[21]

C. Kenig and J.-N. Wang, Unique continuation for the elasticity system and a counterexample for second order elliptic systems, Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory, 4 (2016), 159-178.  doi: 10.1007/978-3-319-30961-3_10.  Google Scholar

[22]

A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Problems, 29 (2013), 104011, 21pp. doi: 10.1088/0266-5611/29/10/104011.  Google Scholar

[23]

J. Li, X. Li, H. Liu and Y. Wang, Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking, IMA Journal of Applied Mathematics, 82 (2017), 1013-1042.  doi: 10.1093/imamat/hxx022.  Google Scholar

[24]

C.-L. Lin, G. Nakamura, G. Uhlmann and J.-N. Wang, Quantitative strong unique continuation for the Lamé system with less regular coefficients, Methods Appl. Anal., 18 (2011), 85-92.  doi: 10.4310/MAA.2011.v18.n1.a5.  Google Scholar

[25]

C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients, Math. Ann., 331 (2005), 611-629.  doi: 10.1007/s00208-004-0597-z.  Google Scholar

show all references

References:
[1]

C. Bellis, F. Cakoni and B. B. Guzina, Nature of the transmission eigenvalue spectrum for elastic bodies, IMA J. Appl. Math., 78 (2013), 895-923.  doi: 10.1093/imamat/hxr070.  Google Scholar

[2]

C. Bellis and B. B. Guzina, On the existence and uniqueness of a solution to the interior transmission problem for piecewise-homogeneous solids, J. Elasticity, 101 (2010), 29-57.  doi: 10.1007/s10659-010-9242-0.  Google Scholar

[3]

E. Blåsten, L. Päivärinta and J. Sylvester, Corners always scatter, Comm. Math. Phys., 331 (2014), 725-753.  doi: 10.1007/s00220-014-2030-0.  Google Scholar

[4]

A. S. Bonnet-Ben Dhia, L. Chesnel and H. Haddar, On the use of T -coercivity to study the interior transmission eigenvalue problem, C. R. Math. Acad. Sci. Paris, 349 (2011), 647-651.  doi: 10.1016/j.crma.2011.05.008.  Google Scholar

[5]

F. Cakoni, D. Colton and H. Haddar, On the determination of Dirichlet or transmission eigenvalues from far field data, C.R. Acad. Sci. Paris., 348 (2010), 379-383.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[6]

F. Cakoni, A. Cossonnière and H. Haddar, Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Probl. Imaging, 6 (2012), 373-398.  doi: 10.3934/ipi.2012.6.373.  Google Scholar

[7]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[8]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.  Google Scholar

[9]

A. Charalambopoulos, On the interior transmission problem in nondissipative, inhomogeneous, anisotropic elasticity, J. Elasticity, 67 (2002), 149-170.  doi: 10.1023/A:1023958030304.  Google Scholar

[10]

A. Charalambopoulos and K. A. Anagnostopoulos, On the spectrum of the interior transmission problem in isotropic elasticity, J Elasticity, 90 (2008), 295-313.  doi: 10.1007/s10659-007-9146-9.  Google Scholar

[11]

A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in three-dimensional linear elasticity, Inverse Probl., 18 (2002), 547-558.  doi: 10.1088/0266-5611/18/3/303.  Google Scholar

[12]

F. Cakoni and H. Haddar, On the existence of transmission eigenvalues in an inhomogeneous medium, Appl. Anal., 88 (2009), 475-493.  doi: 10.1080/00036810802713966.  Google Scholar

[13]

B. Davey, C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with less regular coefficients, Math. Ann., (2020). doi: 10.1007/s00208-020-02026-0.  Google Scholar

[14]

H. Diao, H. Liu and L. Wang, On generalized Holmgren's principle to the Lamé operator with applications to inverse elastic problems, Calc. Var., 59 (2020), Paper No. 179, 50 pp. doi: 10.1007/s00526-020-01830-5.  Google Scholar

[15]

J. Elschner and G. Hu, Acoustic scattering from corners, edges and circular cones, Arch. Ration. Mech. Anal., 228 (2018), 653-690.  doi: 10.1007/s00205-017-1202-4.  Google Scholar

[16]

G. Giorgi and H. Haddar, Computing estimates of material properties from transmission eigenvalues, Inverse Problems, 28 (2012), 055009, 23 pp. doi: 10.1088/0266-5611/28/5/055009.  Google Scholar

[17]

I. Harris, F. Cakoni and J. Sun, Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids, Inverse Problems, 30 (2014), 035016, 21pp. doi: 10.1088/0266-5611/30/3/035016.  Google Scholar

[18]

G. Hu, M. Salo and E. V. Vesalainen, Shape identification in inverse medium scattering problems with a single far-field pattern, SIAM J. Math. Anal., 48 (2016), 152-165.  doi: 10.1137/15M1032958.  Google Scholar

[19]

X. Ji and H. Liu, On isotropic cloaking and interior transmission eigenvalue problems, European J. Appl. Math., 29 (2018), 253-280.  doi: 10.1017/S0956792517000110.  Google Scholar

[20]

A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci., 21 (1998), 619-651.  doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P.  Google Scholar

[21]

C. Kenig and J.-N. Wang, Unique continuation for the elasticity system and a counterexample for second order elliptic systems, Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory, 4 (2016), 159-178.  doi: 10.1007/978-3-319-30961-3_10.  Google Scholar

[22]

A. Kirsch and A. Lechleiter, The inside-outside duality for scattering problems by inhomogeneous media, Inverse Problems, 29 (2013), 104011, 21pp. doi: 10.1088/0266-5611/29/10/104011.  Google Scholar

[23]

J. Li, X. Li, H. Liu and Y. Wang, Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking, IMA Journal of Applied Mathematics, 82 (2017), 1013-1042.  doi: 10.1093/imamat/hxx022.  Google Scholar

[24]

C.-L. Lin, G. Nakamura, G. Uhlmann and J.-N. Wang, Quantitative strong unique continuation for the Lamé system with less regular coefficients, Methods Appl. Anal., 18 (2011), 85-92.  doi: 10.4310/MAA.2011.v18.n1.a5.  Google Scholar

[25]

C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with Lipschitz coefficients, Math. Ann., 331 (2005), 611-629.  doi: 10.1007/s00208-004-0597-z.  Google Scholar

Figure 1.  Plot of $ F(k) $ in (21) with $ \mu = 1 $, $ \lambda = 1 $ and $ n = 1/2 $ (GNU Octave)
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Figure 2.  Plot of $ F(k) $ in (22) with $ \mu = 1 $, $ \lambda = 1 $ and $ n = 1/2 $ (GNU Octave)
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