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On new surface-localized transmission eigenmodes
1. | School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan, China |
2. | Department of Mathematics, Jilin University, Changchun, Jilin, China |
3. | Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China |
$ (\Delta+k^2\mathbf{n}^2) w = 0, \ \ (\Delta+k^2)v = 0\ \ \mbox{in}\ \ \Omega;\quad w = v, \ \ \partial_\nu w = \partial_\nu v\ \ \mbox{on} \ \partial\Omega. $ |
$ (w_m, v_m)_{m\in\mathbb{N}} $ |
$ k_m\rightarrow \infty $ |
$ \{w_m\}_{m\in\mathbb{N}} $ |
$ \{v_m\}_{m\in\mathbb{N}} $ |
$ \mathbf{n}>1 $ |
$ 0<\mathbf{n}<1 $ |
$ (w_m, v_m)_{m\in\mathbb{N}} $ |
$ k_m\rightarrow \infty $ |
$ \{w_m\}_{m\in\mathbb{N}} $ |
$ \{v_m\}_{m\in\mathbb{N}} $ |
$ \mathbf{n}>1 $ |
$ 0<\mathbf{n}<1 $ |
References:
[1] |
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Edited by Milton Abramowitz and Irene A. Stegun Dover Publications, Inc., New York 1966. |
[2] |
E. Blåsten,
Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270.
doi: 10.1137/18M1182048. |
[3] |
E. Blåsten, X. Li, H. Liu and Y. Wang, On vanishing and localization of transmission eigenfunctions near singular points: A numerical study, Inverse Problems, 33 (2017), 24pp.
doi: 10.1088/1361-6420/aa8826. |
[4] |
E. Blåsten and Y.-H. Lin, Radiating and non-radiating sources in elasticity, Inverse Problems, 35 (2019), 16pp.
doi: 10.1088/1361-6420/aae99e. |
[5] |
E. Blåsten and H. Liu,
On corners scattering stably and stable shape determination by a single far-field pattern, Indiana Univ. Math. J., 70 (2021), 907-947.
doi: 10.1512/iumj.2021.70.8411. |
[6] |
E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, Inverse Problems, 36 (2020), 16pp.
doi: 10.1088/1361-6420/ab958f. |
[7] |
E. Blåsten and H. Liu,
Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, SIAM J. Math. Anal., 53 (2021), 3801-3837.
doi: 10.1137/20M1384002. |
[8] |
E. Blåsten and H. Liu,
On vanishing near corners of transmission eigenfunctions, J. Funct. Anal., 273 (2017), 3616-3632.
doi: 10.1016/j.jfa.2017.08.023. |
[9] |
E. Blåsten, L. Päivärinta and J. Sylvester,
Corners always scatter, Comm. Math. Phys., 331 (2014), 725-753.
doi: 10.1007/s00220-014-2030-0. |
[10] |
E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Analysis & PDE, in press, 2021.
![]() |
[11] |
F. Cakoni, D. Colton, and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
doi: 10.1137/1.9781611974461.ch1. |
[12] |
F. Cakoni and M. Vogelius, Singularities almost always scatter: Regularity results for non-scattering inhomogeneities, preprint, arXiv: math/2104.05058. |
[13] |
F. Cakoni and J. Xiao,
On corner scattering for operators of divergence form and applications to inverse scattering, Comm. Partial Differential Equations, 46 (2021), 413-441.
doi: 10.1080/03605302.2020.1843489. |
[14] |
X. Cao, H. Diao and H. Liu,
Determining a piecewise conductive medium body by a single far-field measurement, CSIAM Trans. Appl. Math., 1 (2020), 740-765.
|
[15] |
Y. T. Chow, Y. Deng, H. Liu and M. Sunkula, Surface concentration of transmission eigenfunctions, preprint, arXiv: math/2109.14361. |
[16] |
Y. T. Chow, Y. Deng, Y. He, H. Liu and X. Wang,
Surface-localized transmission eigenstates, super-resolution imaging and pseudo surface plasmon modes, SIAM J. Imaging Sci., 14 (2021), 946-975.
doi: 10.1137/20M1388498. |
[17] |
S. Cogar, D. Colton and Y. L. Leung, The inverse spectral problem for transmission eigenvalues, Inverse Problems, 33 (2017), 15pp.
doi: 10.1088/1361-6420/aa66d2. |
[18] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^nd$ edition, Appl. Math. Sci. 93, Springer, Cham, 2019.
doi: 10.1007/978-3-030-30351-8. |
[19] |
Y. Deng, C. Duan and H. Liu, On vanishing near corners of conductive transmission eigenfunctions, preprint, arXiv: math/2011.14226. |
[20] |
H. Diao, X. Cao and H. Liu,
On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, Comm. Partial Differential Equations, 46 (2021), 630-679.
doi: 10.1080/03605302.2020.1857397. |
[21] |
J. Elschner and G. Hu,
Acoustic scattering from corners, edges and circular cones, Arch. Ration. Mech. Anal., 228 (2018), 653-690.
doi: 10.1007/s00205-017-1202-4. |
[22] |
G. Hu, M. Salo and E. V. Vesalainen,
Shape identification in inverse medium scattering problems with a single far-field pattern, SIAM J. Math. Anal., 48 (2016), 152-165.
doi: 10.1137/15M1032958. |
[23] |
B. G. Korenev,
Bessel functions and their applications, Integral Transforms and Special Functions, 25 (2002), 272-282.
|
[24] |
I. Krasikov,
Uniform bounds for Bessel functions, J. Appl. Anal., 12 (2006), 83-91.
doi: 10.1515/JAA.2006.83. |
[25] |
H. Liu, On local and global structures of transmission eigenfunctions and beyond, preprint, arXiv: math/2008.03120.
doi: 10.1515/jiip-2020-0099. |
[26] |
H. Liu and J. Zou,
Zeros of the Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering, IMA J. Appl. Math., 72 (2007), 817-831.
doi: 10.1093/imamat/hxm013. |
[27] |
H. Liu, Z. J. Shang, H. Sun and J. Zou,
Singular perturbation of the reduced wave equation and scattering from an embedded obstacle, J. Dynam. Differential Equations, 24 (2012), 803-821.
doi: 10.1007/s10884-012-9270-5. |
[28] |
Y. J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media, Inverse Problems, 28 (2012), 9pp.
doi: 10.1088/0266-5611/28/7/075005. |
[29] |
J. R. McLaughlin and P. L. Polyakov,
On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, J. Differential Equations, 107 (1994), 351-382.
doi: 10.1006/jdeq.1994.1017. |
[30] |
M. Salo, L. Päivärinta and E. Vesalainen,
Strictly convex corners scatter, Rev. Mat. Iberoam, 33 (2017), 1369-1396.
doi: 10.4171/RMI/975. |
[31] |
M. Salo and H. Shahgholian, Free boundary methods and non-scattering phenomena, preprint, arXiv: math/2106.15154. |
[32] |
M. Vogelius and J. Xiao, Finiteness results concerning non-scattering wave numbers for incident plane- and Herglotz waves, preprint, arXiv: math/2104.05058. |
[33] |
G. Vodev,
Parabolic transmission eigenvalue-free regions in the degenerate isotropic case, Asymptot. Anal., 106 (2018), 147-168.
doi: 10.3233/ASY-171443. |
[34] |
G. Vodev,
High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues, Anal. PDE, 11 (2018), 213-236.
doi: 10.2140/apde.2018.11.213. |
[35] |
R. Wong and C. K. Qu,
Best possible upper and lower bounds for the zeros of the Bessel function $J_{\nu}$(x), Trans. Amer. Math. Soc., 351 (1999), 2833-2859.
doi: 10.1090/S0002-9947-99-02165-0. |
show all references
References:
[1] |
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Edited by Milton Abramowitz and Irene A. Stegun Dover Publications, Inc., New York 1966. |
[2] |
E. Blåsten,
Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), 6255-6270.
doi: 10.1137/18M1182048. |
[3] |
E. Blåsten, X. Li, H. Liu and Y. Wang, On vanishing and localization of transmission eigenfunctions near singular points: A numerical study, Inverse Problems, 33 (2017), 24pp.
doi: 10.1088/1361-6420/aa8826. |
[4] |
E. Blåsten and Y.-H. Lin, Radiating and non-radiating sources in elasticity, Inverse Problems, 35 (2019), 16pp.
doi: 10.1088/1361-6420/aae99e. |
[5] |
E. Blåsten and H. Liu,
On corners scattering stably and stable shape determination by a single far-field pattern, Indiana Univ. Math. J., 70 (2021), 907-947.
doi: 10.1512/iumj.2021.70.8411. |
[6] |
E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, Inverse Problems, 36 (2020), 16pp.
doi: 10.1088/1361-6420/ab958f. |
[7] |
E. Blåsten and H. Liu,
Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, SIAM J. Math. Anal., 53 (2021), 3801-3837.
doi: 10.1137/20M1384002. |
[8] |
E. Blåsten and H. Liu,
On vanishing near corners of transmission eigenfunctions, J. Funct. Anal., 273 (2017), 3616-3632.
doi: 10.1016/j.jfa.2017.08.023. |
[9] |
E. Blåsten, L. Päivärinta and J. Sylvester,
Corners always scatter, Comm. Math. Phys., 331 (2014), 725-753.
doi: 10.1007/s00220-014-2030-0. |
[10] |
E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Analysis & PDE, in press, 2021.
![]() |
[11] |
F. Cakoni, D. Colton, and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics, 88. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
doi: 10.1137/1.9781611974461.ch1. |
[12] |
F. Cakoni and M. Vogelius, Singularities almost always scatter: Regularity results for non-scattering inhomogeneities, preprint, arXiv: math/2104.05058. |
[13] |
F. Cakoni and J. Xiao,
On corner scattering for operators of divergence form and applications to inverse scattering, Comm. Partial Differential Equations, 46 (2021), 413-441.
doi: 10.1080/03605302.2020.1843489. |
[14] |
X. Cao, H. Diao and H. Liu,
Determining a piecewise conductive medium body by a single far-field measurement, CSIAM Trans. Appl. Math., 1 (2020), 740-765.
|
[15] |
Y. T. Chow, Y. Deng, H. Liu and M. Sunkula, Surface concentration of transmission eigenfunctions, preprint, arXiv: math/2109.14361. |
[16] |
Y. T. Chow, Y. Deng, Y. He, H. Liu and X. Wang,
Surface-localized transmission eigenstates, super-resolution imaging and pseudo surface plasmon modes, SIAM J. Imaging Sci., 14 (2021), 946-975.
doi: 10.1137/20M1388498. |
[17] |
S. Cogar, D. Colton and Y. L. Leung, The inverse spectral problem for transmission eigenvalues, Inverse Problems, 33 (2017), 15pp.
doi: 10.1088/1361-6420/aa66d2. |
[18] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4$^nd$ edition, Appl. Math. Sci. 93, Springer, Cham, 2019.
doi: 10.1007/978-3-030-30351-8. |
[19] |
Y. Deng, C. Duan and H. Liu, On vanishing near corners of conductive transmission eigenfunctions, preprint, arXiv: math/2011.14226. |
[20] |
H. Diao, X. Cao and H. Liu,
On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, Comm. Partial Differential Equations, 46 (2021), 630-679.
doi: 10.1080/03605302.2020.1857397. |
[21] |
J. Elschner and G. Hu,
Acoustic scattering from corners, edges and circular cones, Arch. Ration. Mech. Anal., 228 (2018), 653-690.
doi: 10.1007/s00205-017-1202-4. |
[22] |
G. Hu, M. Salo and E. V. Vesalainen,
Shape identification in inverse medium scattering problems with a single far-field pattern, SIAM J. Math. Anal., 48 (2016), 152-165.
doi: 10.1137/15M1032958. |
[23] |
B. G. Korenev,
Bessel functions and their applications, Integral Transforms and Special Functions, 25 (2002), 272-282.
|
[24] |
I. Krasikov,
Uniform bounds for Bessel functions, J. Appl. Anal., 12 (2006), 83-91.
doi: 10.1515/JAA.2006.83. |
[25] |
H. Liu, On local and global structures of transmission eigenfunctions and beyond, preprint, arXiv: math/2008.03120.
doi: 10.1515/jiip-2020-0099. |
[26] |
H. Liu and J. Zou,
Zeros of the Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering, IMA J. Appl. Math., 72 (2007), 817-831.
doi: 10.1093/imamat/hxm013. |
[27] |
H. Liu, Z. J. Shang, H. Sun and J. Zou,
Singular perturbation of the reduced wave equation and scattering from an embedded obstacle, J. Dynam. Differential Equations, 24 (2012), 803-821.
doi: 10.1007/s10884-012-9270-5. |
[28] |
Y. J. Leung and D. Colton, Complex transmission eigenvalues for spherically stratified media, Inverse Problems, 28 (2012), 9pp.
doi: 10.1088/0266-5611/28/7/075005. |
[29] |
J. R. McLaughlin and P. L. Polyakov,
On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues, J. Differential Equations, 107 (1994), 351-382.
doi: 10.1006/jdeq.1994.1017. |
[30] |
M. Salo, L. Päivärinta and E. Vesalainen,
Strictly convex corners scatter, Rev. Mat. Iberoam, 33 (2017), 1369-1396.
doi: 10.4171/RMI/975. |
[31] |
M. Salo and H. Shahgholian, Free boundary methods and non-scattering phenomena, preprint, arXiv: math/2106.15154. |
[32] |
M. Vogelius and J. Xiao, Finiteness results concerning non-scattering wave numbers for incident plane- and Herglotz waves, preprint, arXiv: math/2104.05058. |
[33] |
G. Vodev,
Parabolic transmission eigenvalue-free regions in the degenerate isotropic case, Asymptot. Anal., 106 (2018), 147-168.
doi: 10.3233/ASY-171443. |
[34] |
G. Vodev,
High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues, Anal. PDE, 11 (2018), 213-236.
doi: 10.2140/apde.2018.11.213. |
[35] |
R. Wong and C. K. Qu,
Best possible upper and lower bounds for the zeros of the Bessel function $J_{\nu}$(x), Trans. Amer. Math. Soc., 351 (1999), 2833-2859.
doi: 10.1090/S0002-9947-99-02165-0. |


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