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doi: 10.3934/ipi.2021063
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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 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 & Imaging, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10] E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Analysis & PDE, in press, 2021.   Google Scholar
[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.  Google Scholar

[12]

F. Cakoni and M. Vogelius, Singularities almost always scatter: Regularity results for non-scattering inhomogeneities, preprint, arXiv: math/2104.05058. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

Y. T. Chow, Y. Deng, H. Liu and M. Sunkula, Surface concentration of transmission eigenfunctions, preprint, arXiv: math/2109.14361. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Y. Deng, C. Duan and H. Liu, On vanishing near corners of conductive transmission eigenfunctions, preprint, arXiv: math/2011.14226. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

B. G. Korenev, Bessel functions and their applications, Integral Transforms and Special Functions, 25 (2002), 272-282.   Google Scholar

[24]

I. Krasikov, Uniform bounds for Bessel functions, J. Appl. Anal., 12 (2006), 83-91.  doi: 10.1515/JAA.2006.83.  Google Scholar

[25]

H. Liu, On local and global structures of transmission eigenfunctions and beyond, preprint, arXiv: math/2008.03120. doi: 10.1515/jiip-2020-0099.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Salo, L. Päivärinta and E. Vesalainen, Strictly convex corners scatter, Rev. Mat. Iberoam, 33 (2017), 1369-1396.  doi: 10.4171/RMI/975.  Google Scholar

[31]

M. Salo and H. Shahgholian, Free boundary methods and non-scattering phenomena, preprint, arXiv: math/2106.15154. Google Scholar

[32]

M. Vogelius and J. Xiao, Finiteness results concerning non-scattering wave numbers for incident plane- and Herglotz waves, preprint, arXiv: math/2104.05058. Google Scholar

[33]

G. Vodev, Parabolic transmission eigenvalue-free regions in the degenerate isotropic case, Asymptot. Anal., 106 (2018), 147-168.  doi: 10.3233/ASY-171443.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10] E. Blåsten, H. Liu and J. Xiao, On an electromagnetic problem in a corner and its applications, Analysis & PDE, in press, 2021.   Google Scholar
[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.  Google Scholar

[12]

F. Cakoni and M. Vogelius, Singularities almost always scatter: Regularity results for non-scattering inhomogeneities, preprint, arXiv: math/2104.05058. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

Y. T. Chow, Y. Deng, H. Liu and M. Sunkula, Surface concentration of transmission eigenfunctions, preprint, arXiv: math/2109.14361. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Y. Deng, C. Duan and H. Liu, On vanishing near corners of conductive transmission eigenfunctions, preprint, arXiv: math/2011.14226. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

B. G. Korenev, Bessel functions and their applications, Integral Transforms and Special Functions, 25 (2002), 272-282.   Google Scholar

[24]

I. Krasikov, Uniform bounds for Bessel functions, J. Appl. Anal., 12 (2006), 83-91.  doi: 10.1515/JAA.2006.83.  Google Scholar

[25]

H. Liu, On local and global structures of transmission eigenfunctions and beyond, preprint, arXiv: math/2008.03120. doi: 10.1515/jiip-2020-0099.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Salo, L. Päivärinta and E. Vesalainen, Strictly convex corners scatter, Rev. Mat. Iberoam, 33 (2017), 1369-1396.  doi: 10.4171/RMI/975.  Google Scholar

[31]

M. Salo and H. Shahgholian, Free boundary methods and non-scattering phenomena, preprint, arXiv: math/2106.15154. Google Scholar

[32]

M. Vogelius and J. Xiao, Finiteness results concerning non-scattering wave numbers for incident plane- and Herglotz waves, preprint, arXiv: math/2104.05058. Google Scholar

[33]

G. Vodev, Parabolic transmission eigenvalue-free regions in the degenerate isotropic case, Asymptot. Anal., 106 (2018), 147-168.  doi: 10.3233/ASY-171443.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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