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

On new surface-localized transmission eigenmodes

Youjun Deng 1, , Yan Jiang 2, , Hongyu Liu 3,, and  Kai Zhang 2,,

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.
Keywords: Transmission eigenfunctions, spectral geometry, surface localization, wave concentration.
Mathematics Subject Classification: Primary: 35P25, 78A46; Secondary: 35Q60, 78A05.
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.

Figure 1.  Schematic illustration of $ \int_{0}^{1}f(r)dr $ which is bigger than the area of the triangle under the tangent of $ f(1) $
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Figure 2.  Graphical illustration of the classification of transmission eigenvalues
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