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:

show all references

References:
 [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, 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 [8] 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 [9] 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 [10] 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 [11] 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 [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] 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 [14] 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 [15] Fioralba Cakoni, Shixu Meng, Jingni Xiao. A note on transmission eigenvalues in electromagnetic scattering theory. Inverse Problems & Imaging, 2021, 15 (5) : 999-1014. doi: 10.3934/ipi.2021025 [16] 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 [17] 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 [18] 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 [19] 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 [20] Alexandre Thorel. A biharmonic transmission problem in Lp-spaces. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3193-3213. doi: 10.3934/cpaa.2021102

2020 Impact Factor: 1.639