# American Institute of Mathematical Sciences

• Previous Article
Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems
• IPI Home
• This Issue
• Next Article
On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives
May  2016, 10(2): 369-378. doi: 10.3934/ipi.2016004

## On a transmission eigenvalue problem for a spherically stratified coated dielectric

 1 Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553, United States, United States

Received  July 2015 Published  May 2016

Suppose that the boundary of the unit ball in $R^3$ is coated with a very thin layer of a highly conductive material and the refractive index $n(x)$ inside the ball is spherically stratified. We show that in this case the set of transmission eigenvalues behave quite differently than in the previous studied case of an uncoated ball. In particular, if the index of refraction varies smoothly across the boundary of the unit ball we show that complex eigenvalues always exist and accumulate on the real axis and that the real and complex eigenvalues uniquely determine the index of refraction without any restriction on its magnitude.
Citation: 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
##### References:
 [1] T. Aktosun, D. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/11/115004. Google Scholar [2] T. Aktosun and V. G. Papanicolaou, Reconstruction of the wave speed from transmission eigenvalues for spherically symmetric variable-speed wave equation,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/6/065007. Google Scholar [3] A. Baker, Transcendental Number Theory,, Cambridge University Press, (1975). Google Scholar [4] F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory,, Applied Mathematical Sciences Series Volume 188, (2014). doi: 10.1007/978-1-4614-8827-9. Google Scholar [5] F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem,, SIAM J. Math Anal., 42 (2010), 2912. doi: 10.1137/100793542. Google Scholar [6] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume I,, Interscience Publishing, (1953). Google Scholar [7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, 3rd ed., (2013). doi: 10.1007/978-1-4614-4942-3. Google Scholar [8] D. Colton and Y. J. Leung, Complex Eigenvalues and the inverse spectral problem for transmission eigenvalues,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104008. Google Scholar [9] D. Colton, Y. J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media,, Inverse Problems, 31 (2015). doi: 10.1088/0266-5611/31/3/035006. Google Scholar [10] R. Duffin and A. C. Schaeffer, Some properties of functions of exponential type,, Bull. Amer. Math. Soc., 44 (1938), 236. doi: 10.1090/S0002-9904-1938-06725-0. Google Scholar [11] B. Levin, Distribution of Zeros of Entire Functions,, American Mathematical Society Translation, (1980). Google Scholar [12] J. McLaughlin and P. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues,, J. Differential Equations, 107 (1994), 351. doi: 10.1006/jdeq.1994.1017. Google Scholar [13] H. Pham and P. Stefanov, Weyl asymptotics for the transmission eigenvalues for a constant index of refraction,, Inverse Prob. Imaging, 8 (2014), 795. doi: 10.3934/ipi.2014.8.795. Google Scholar [14] W. Rundell and P. Sacks, Reconstruction techniques for classical inverse Sturm-Liouville problems,, Math. Comp., 58 (1992), 161. doi: 10.1090/S0025-5718-1992-1106979-0. Google Scholar [15] J. Sylvester, Transmission eigenvalues in one dimension,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104009. Google Scholar [16] G. Wei and H. Xu, Inverse spectral analysis for the transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/11/115012. Google Scholar [17] W. Young, Introduction to Nonharmonic Fourier Series,, Academic Press, (2001). Google Scholar

show all references

##### References:
 [1] T. Aktosun, D. Gintides and V. G. Papanicolaou, The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/11/115004. Google Scholar [2] T. Aktosun and V. G. Papanicolaou, Reconstruction of the wave speed from transmission eigenvalues for spherically symmetric variable-speed wave equation,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/6/065007. Google Scholar [3] A. Baker, Transcendental Number Theory,, Cambridge University Press, (1975). Google Scholar [4] F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory,, Applied Mathematical Sciences Series Volume 188, (2014). doi: 10.1007/978-1-4614-8827-9. Google Scholar [5] F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem,, SIAM J. Math Anal., 42 (2010), 2912. doi: 10.1137/100793542. Google Scholar [6] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume I,, Interscience Publishing, (1953). Google Scholar [7] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, 3rd ed., (2013). doi: 10.1007/978-1-4614-4942-3. Google Scholar [8] D. Colton and Y. J. Leung, Complex Eigenvalues and the inverse spectral problem for transmission eigenvalues,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104008. Google Scholar [9] D. Colton, Y. J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media,, Inverse Problems, 31 (2015). doi: 10.1088/0266-5611/31/3/035006. Google Scholar [10] R. Duffin and A. C. Schaeffer, Some properties of functions of exponential type,, Bull. Amer. Math. Soc., 44 (1938), 236. doi: 10.1090/S0002-9904-1938-06725-0. Google Scholar [11] B. Levin, Distribution of Zeros of Entire Functions,, American Mathematical Society Translation, (1980). Google Scholar [12] J. McLaughlin and P. Polyakov, On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues,, J. Differential Equations, 107 (1994), 351. doi: 10.1006/jdeq.1994.1017. Google Scholar [13] H. Pham and P. Stefanov, Weyl asymptotics for the transmission eigenvalues for a constant index of refraction,, Inverse Prob. Imaging, 8 (2014), 795. doi: 10.3934/ipi.2014.8.795. Google Scholar [14] W. Rundell and P. Sacks, Reconstruction techniques for classical inverse Sturm-Liouville problems,, Math. Comp., 58 (1992), 161. doi: 10.1090/S0025-5718-1992-1106979-0. Google Scholar [15] J. Sylvester, Transmission eigenvalues in one dimension,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/10/104009. Google Scholar [16] G. Wei and H. Xu, Inverse spectral analysis for the transmission eigenvalue problem,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/11/115012. Google Scholar [17] W. Young, Introduction to Nonharmonic Fourier Series,, Academic Press, (2001). Google Scholar
 [1] 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 [2] Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167 [3] 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 [4] 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 [5] Fioralba Cakoni, Drossos Gintides. New results on transmission eigenvalues. Inverse Problems & Imaging, 2010, 4 (1) : 39-48. doi: 10.3934/ipi.2010.4.39 [6] Andreas Kirsch. On the existence of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (2) : 155-172. doi: 10.3934/ipi.2009.3.155 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] Yalin Zhang, Guoliang Shi. Continuous dependence of the transmission eigenvalues in one dimension. Inverse Problems & Imaging, 2015, 9 (1) : 273-287. doi: 10.3934/ipi.2015.9.273 [13] 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 [14] Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems & Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041 [15] Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031 [16] 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 [17] 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 [18] Huicong Li, Jingyu Li. Asymptotic behavior of Dirichlet eigenvalues on a body coated by functionally graded material. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1493-1516. doi: 10.3934/cpaa.2017071 [19] Hui Wan, Jing-An Cui. A model for the transmission of malaria. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 479-496. doi: 10.3934/dcdsb.2009.11.479 [20] 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

2018 Impact Factor: 1.469