American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results
  • IPI Home
  • This Issue
  • Next Article
    Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials
August  2015, 9(3): 725-748. doi: 10.3934/ipi.2015.9.725

The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions

Fioralba Cakoni 1, , Shari Moskow 2, and  Scott Rome 2,

1. 

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, United States
2. 

Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, United States, United States

Received  August 2014 Revised  January 2015 Published  July 2015

This paper concerns the transmission eigenvalue problem for an inhomogeneous media of compact support containing small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the real transmission eigenvalues. Note that for practical applications the real transmission eigenvalues are important since they can be measured from the scattering data. In particular, in addition to proving the convergence rate for the eigenvalues corresponding to the perturbed media as inclusions' volume goes to zero, we also provide the explicit first correction term in the asymptotic expansion for simple eigenvalues. The correction terms involves the eigenvalues and eigenvectors of the unperturbed known background as well as information about the location, size and refractive index of small inhomogeneities. Thus, our asymptotic formula has the potential to be used to recover information about small inclusions from a knowledge of real transmission eigenvalues.
Keywords: transmission eigenvalues, homogenization., inverse scattering problem, Interior transmission problem, periodic inhomogeneous medium.
Mathematics Subject Classification: Primary: 35R30, 35Q60, 35J40, 78A2.
Citation: Fioralba Cakoni, Shari Moskow, Scott Rome. The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions. Inverse Problems & Imaging, 2015, 9 (3) : 725-748. doi: 10.3934/ipi.2015.9.725
References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems,, Van Nostrand-Reinhold, (1965).   Google Scholar

[2]

C. Amrouche and M. Fontes, Biharmonic problem in exterior domains of $\mathbbR^n$: An approach with weighted Sobolev spaces,, J. Math. Anal. Appl., 304 (2005), 552.  doi: 10.1016/j.jmaa.2004.09.039.  Google Scholar

[3]

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.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[4]

F. Cakoni, D. Colton and P. Monk, On the use of transmission eigenvalues to estimate the index of refraction from far field data,, Inverse Problems, 23 (2007), 507.  doi: 10.1088/0266-5611/23/2/004.  Google Scholar

[5]

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

[6]

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

[7]

_________, Transmission eigenvalues in inverse scattering theory,, in Inverse Problems and Applications: Inside Out. II, (2013), 529.   Google Scholar

[8]

F. Cakoni and S. Moskow, Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104014.  Google Scholar

[9]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13.  doi: 10.3934/ipi.2007.1.13.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, reprint of 1998 edition, (1998).   Google Scholar

[11]

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

[12]

I. Godberg and P. Lancaster, Indefinite Linear Algebra and Applications,, L. Rodman, (2005).   Google Scholar

[13]

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

[14]

S. Moskow, Nonlinear eigenvalue approximation for compact operators,, preprint, (2014).   Google Scholar

[15]

J. E. Osborn, Spectral approximation for compact operators,, Math. Comput., 29 (1975), 712.  doi: 10.1090/S0025-5718-1975-0383117-3.  Google Scholar

[16]

L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal, 40 (2008), 738.  doi: 10.1137/070697525.  Google Scholar

[17]

L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

show all references

References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems,, Van Nostrand-Reinhold, (1965).   Google Scholar

[2]

C. Amrouche and M. Fontes, Biharmonic problem in exterior domains of $\mathbbR^n$: An approach with weighted Sobolev spaces,, J. Math. Anal. Appl., 304 (2005), 552.  doi: 10.1016/j.jmaa.2004.09.039.  Google Scholar

[3]

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.  doi: 10.1016/j.crma.2010.02.003.  Google Scholar

[4]

F. Cakoni, D. Colton and P. Monk, On the use of transmission eigenvalues to estimate the index of refraction from far field data,, Inverse Problems, 23 (2007), 507.  doi: 10.1088/0266-5611/23/2/004.  Google Scholar

[5]

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

[6]

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

[7]

_________, Transmission eigenvalues in inverse scattering theory,, in Inverse Problems and Applications: Inside Out. II, (2013), 529.   Google Scholar

[8]

F. Cakoni and S. Moskow, Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104014.  Google Scholar

[9]

D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13.  doi: 10.3934/ipi.2007.1.13.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, reprint of 1998 edition, (1998).   Google Scholar

[11]

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

[12]

I. Godberg and P. Lancaster, Indefinite Linear Algebra and Applications,, L. Rodman, (2005).   Google Scholar

[13]

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

[14]

S. Moskow, Nonlinear eigenvalue approximation for compact operators,, preprint, (2014).   Google Scholar

[15]

J. E. Osborn, Spectral approximation for compact operators,, Math. Comput., 29 (1975), 712.  doi: 10.1090/S0025-5718-1975-0383117-3.  Google Scholar

[16]

L. Päivärinta and J. Sylvester, Transmission eigenvalues,, SIAM J. Math. Anal, 40 (2008), 738.  doi: 10.1137/070697525.  Google Scholar

[17]

L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

[1]

Yuebin Hao. Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1387-1397. doi: 10.3934/cpaa.2020068
[2]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025
[3]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[4]

David Colton, Lassi Päivärinta, John Sylvester. The interior transmission problem. Inverse Problems & Imaging, 2007, 1 (1) : 13-28. doi: 10.3934/ipi.2007.1.13
[5]

Vesselin Petkov, Georgi Vodev. Localization of the interior transmission eigenvalues for a ball. Inverse Problems & Imaging, 2017, 11 (2) : 355-372. doi: 10.3934/ipi.2017017
[6]

Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167
[7]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[8]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[9]

Fioralba Cakoni, Anne Cossonnière, Houssem Haddar. Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Problems & Imaging, 2012, 6 (3) : 373-398. doi: 10.3934/ipi.2012.6.373
[10]

Fioralba Cakoni, Houssem Haddar. A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Problems & Imaging, 2007, 1 (3) : 443-456. doi: 10.3934/ipi.2007.1.443
[11]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[12]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211
[13]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793
[14]

Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems & Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39
[15]

Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155
[16]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337
[17]

Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123
[18]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
[19]

Massimo Lanza de Cristoforis, aolo Musolino. A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2509-2542. doi: 10.3934/cpaa.2014.13.2509
[20]

Ha Pham, Plamen Stefanov. Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Problems & Imaging, 2014, 8 (3) : 795-810. doi: 10.3934/ipi.2014.8.795

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences