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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, Princeton, NJ, 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), 552-571. 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), 379-383. 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), 507-522. 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), 237-255. 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), 475-493. doi: 10.1080/00036810802713966.  Google Scholar

[7]

_________, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications: Inside Out. II, Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013, 529-580.  Google Scholar

[8]

F. Cakoni and S. Moskow, Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities, Inverse Problems, 29 (2013), 104014, 18pp. 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), 13-28. 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, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[11]

G. Giorgi and H. Haddar, Computing estimates of material properties from transmission eigenvalues, Inverse Problems, 28 (2012), 055009, 23pp. 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), 104011, 21pp. 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), 712-725. doi: 10.1090/S0025-5718-1975-0383117-3.  Google Scholar

[16]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal, 40 (2008), 738-753. doi: 10.1137/070697525.  Google Scholar

[17]

L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104014, 28pp. 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, Princeton, NJ, 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), 552-571. 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), 379-383. 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), 507-522. 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), 237-255. 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), 475-493. doi: 10.1080/00036810802713966.  Google Scholar

[7]

_________, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications: Inside Out. II, Math. Sci. Res. Inst. Publ., 60, Cambridge Univ. Press, Cambridge, 2013, 529-580.  Google Scholar

[8]

F. Cakoni and S. Moskow, Asymptotic expansions for transmission eigenvalues for media with small inhomogeneities, Inverse Problems, 29 (2013), 104014, 18pp. 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), 13-28. 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, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[11]

G. Giorgi and H. Haddar, Computing estimates of material properties from transmission eigenvalues, Inverse Problems, 28 (2012), 055009, 23pp. 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), 104011, 21pp. 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), 712-725. doi: 10.1090/S0025-5718-1975-0383117-3.  Google Scholar

[16]

L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal, 40 (2008), 738-753. doi: 10.1137/070697525.  Google Scholar

[17]

L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104014, 28pp. doi: 10.1088/0266-5611/29/10/104001.  Google Scholar

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