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Stability in conductivity imaging from partial measurements of one interior current
Localization of the interior transmission eigenvalues for a ball
1. | Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France |
2. | Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2, rue de la Houssiniére, BP 92208,44322 Nantes Cedex, France |
We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball $\{x ∈ \mathbb{R}^d:\: |x| ≤ 1\}, \: d≥ 2,$ and the coefficients $c_j(x), \: j =1,2, $ and the indices of refraction $n_j(x), \: j =1,2,$ are constants near the boundary $|x| = 1$. We prove that in this case the eigenvalue-free region obtained in [
References:
[1] |
F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse
Problems and Applications, Inside Out Ⅱ, G. Uhlmann, editor, MSRI Publications, Cambridge University Press, Cambridge, 60 (2013), 529-580. |
[2] |
D. Colton and P. Monk,
The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium, Quart. J. Mech. Appl. Math., 41 (1988), 97-125.
doi: 10.1093/qjmam/41.1.97. |
[3] |
D. Colton and Y. -J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues
Inverse Problems, 29 (2013), 104008, 6pp.
doi: 10.1088/0266-5611/29/10/104008. |
[4] |
D. Colton, Y. -J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media Inverse Problems, 31 (2015), 035006, 19pp.
doi: 10.1088/0266-5611/31/3/035006. |
[5] |
M. Faierman,
The interior transmission problem: Spectral theory, SIAM J. Math. Anal., 46 (2014), 803-819.
doi: 10.1137/130922215. |
[6] |
M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta,
The interior transmission problem and bounds of transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293.
doi: 10.4310/MRL.2011.v18.n2.a7. |
[7] |
A. Kirsch,
The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37 (1986), 213-225.
doi: 10.1093/imamat/37.3.213. |
[8] |
E. Lakshtanov and B. Vainberg, Application of elliptic theory to the isotropic interior transmission eigenvalue problem Inverse Problems, 29 (2013), 104003, 19pp.
doi: 10.1088/0266-5611/29/10/104003. |
[9] |
Y. -J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media Inverse Problems, 28 (2012), 075005, 9pp.
doi: 10.1088/0266-5611/28/7/075005. |
[10] |
F. Olver,
Asymptotics and Special Functions, Academic Press, New York, London, 1974. |
[11] |
H. Pham and P. Stefanov,
Weyl asymptotics of the transmission eigenvalues for a constant index of refraction, Inverse Problems and Imaging, 8 (2014), 795-810.
doi: 10.3934/ipi.2014.8.795. |
[12] |
L. Robbiano, Spectral analysis of interior transmission eigenvalues Inverse Problems, 29 (2013), 104001, 28pp.
doi: 10.1088/0266-5611/29/10/104001. |
[13] |
L. Robbiano,
Counting function for interior transmission eigenvalues, Mathematical Control and Related Fields, 6 (2016), 167-183.
doi: 10.3934/mcrf.2016.6.167. |
[14] |
J. Sylvester, Transmission eigenvalues in one dimension Inverse Problems, 29 (2013), 104009, 11pp.
doi: 10.1088/0266-5611/29/10/104009. |
[15] |
V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, J. Spectral Theory, to appear. |
[16] |
G. Vodev,
Transmission eigenvalue-free regions, Comm. Math. Phys., 336 (2015), 1141-1166.
doi: 10.1007/s00220-015-2311-2. |
[17] |
G. Vodev,
Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336.
doi: 10.1007/s00208-015-1329-2. |
show all references
References:
[1] |
F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse
Problems and Applications, Inside Out Ⅱ, G. Uhlmann, editor, MSRI Publications, Cambridge University Press, Cambridge, 60 (2013), 529-580. |
[2] |
D. Colton and P. Monk,
The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium, Quart. J. Mech. Appl. Math., 41 (1988), 97-125.
doi: 10.1093/qjmam/41.1.97. |
[3] |
D. Colton and Y. -J. Leung, Complex eigenvalues and the inverse spectral problem for transmission eigenvalues
Inverse Problems, 29 (2013), 104008, 6pp.
doi: 10.1088/0266-5611/29/10/104008. |
[4] |
D. Colton, Y. -J. Leung and S. Meng, Distribution of complex transmission eigenvalues for spherically stratified media Inverse Problems, 31 (2015), 035006, 19pp.
doi: 10.1088/0266-5611/31/3/035006. |
[5] |
M. Faierman,
The interior transmission problem: Spectral theory, SIAM J. Math. Anal., 46 (2014), 803-819.
doi: 10.1137/130922215. |
[6] |
M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta,
The interior transmission problem and bounds of transmission eigenvalues, Math. Res. Lett., 18 (2011), 279-293.
doi: 10.4310/MRL.2011.v18.n2.a7. |
[7] |
A. Kirsch,
The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37 (1986), 213-225.
doi: 10.1093/imamat/37.3.213. |
[8] |
E. Lakshtanov and B. Vainberg, Application of elliptic theory to the isotropic interior transmission eigenvalue problem Inverse Problems, 29 (2013), 104003, 19pp.
doi: 10.1088/0266-5611/29/10/104003. |
[9] |
Y. -J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media Inverse Problems, 28 (2012), 075005, 9pp.
doi: 10.1088/0266-5611/28/7/075005. |
[10] |
F. Olver,
Asymptotics and Special Functions, Academic Press, New York, London, 1974. |
[11] |
H. Pham and P. Stefanov,
Weyl asymptotics of the transmission eigenvalues for a constant index of refraction, Inverse Problems and Imaging, 8 (2014), 795-810.
doi: 10.3934/ipi.2014.8.795. |
[12] |
L. Robbiano, Spectral analysis of interior transmission eigenvalues Inverse Problems, 29 (2013), 104001, 28pp.
doi: 10.1088/0266-5611/29/10/104001. |
[13] |
L. Robbiano,
Counting function for interior transmission eigenvalues, Mathematical Control and Related Fields, 6 (2016), 167-183.
doi: 10.3934/mcrf.2016.6.167. |
[14] |
J. Sylvester, Transmission eigenvalues in one dimension Inverse Problems, 29 (2013), 104009, 11pp.
doi: 10.1088/0266-5611/29/10/104009. |
[15] |
V. Petkov and G. Vodev, Asymptotics of the number of the interior transmission eigenvalues, J. Spectral Theory, to appear. |
[16] |
G. Vodev,
Transmission eigenvalue-free regions, Comm. Math. Phys., 336 (2015), 1141-1166.
doi: 10.1007/s00220-015-2311-2. |
[17] |
G. Vodev,
Transmission eigenvalues for strictly concave domains, Math. Ann., 366 (2016), 301-336.
doi: 10.1007/s00208-015-1329-2. |
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