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]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075
[2]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004
[3]

Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020050
[4]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391
[5]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
[6]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[7]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[8]

Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021017
[9]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005
[10]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006
[11]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339
[12]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227
[13]

Kuo-Chih Hung, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020281
[14]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351
[15]

Weihong Guo, Yifei Lou, Jing Qin, Ming Yan. IPI special issue on "mathematical/statistical approaches in data science" in the Inverse Problem and Imaging. Inverse Problems & Imaging, 2021, 15 (1) : I-I. doi: 10.3934/ipi.2021007
[16]

Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230
[17]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398
[18]

Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29 (1) : 1753-1782. doi: 10.3934/era.2020090
[19]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258
[20]

Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021002

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences