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June  2022, 16(3): 595-611. doi: 10.3934/ipi.2021063

## 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

Received  March 2021 Revised  August 2021 Published  June 2022 Early access  October 2021

Consider the transmission eigenvalue problem
 $(\Delta+k^2\mathbf{n}^2) w = 0, \ \ (\Delta+k^2)v = 0\ \ \mbox{in}\ \ \Omega;\quad w = v, \ \ \partial_\nu w = \partial_\nu v\ \ \mbox{on} \ \partial\Omega.$
It is shown in [16] that there exists a sequence of eigenfunctions
 $(w_m, v_m)_{m\in\mathbb{N}}$
associated with
 $k_m\rightarrow \infty$
such that either
 $\{w_m\}_{m\in\mathbb{N}}$
or
 $\{v_m\}_{m\in\mathbb{N}}$
are surface-localized, depending on
 $\mathbf{n}>1$
or
 $0<\mathbf{n}<1$
. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions
 $(w_m, v_m)_{m\in\mathbb{N}}$
associated with
 $k_m\rightarrow \infty$
such that both
 $\{w_m\}_{m\in\mathbb{N}}$
and
 $\{v_m\}_{m\in\mathbb{N}}$
are surface-localized, no matter
 $\mathbf{n}>1$
or
 $0<\mathbf{n}<1$
. Though our study is confined within the radial geometry, the construction is subtle and technical.
Citation: Youjun Deng, Yan Jiang, Hongyu Liu, Kai Zhang. On new surface-localized transmission eigenmodes. Inverse Problems and Imaging, 2022, 16 (3) : 595-611. doi: 10.3934/ipi.2021063
##### 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.
Schematic illustration of $\int_{0}^{1}f(r)dr$ which is bigger than the area of the triangle under the tangent of $f(1)$
Graphical illustration of the classification of transmission eigenvalues
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