On new surface-localized transmission eigenmodes

Youjun Deng; Yan Jiang; Hongyu Liu; Kai Zhang

doi:10.3934/ipi.2021063

Article Contents

2022, Volume 16, Issue 3: 595-611. Doi: 10.3934/ipi.2021063

This issue Previous Article Uniqueness and numerical reconstruction for inverse problems dealing with interval size search Next Article An inverse problem for a fractional diffusion equation with fractional power type nonlinearities

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35P25, 78A46; Secondary: 35Q60, 78A05.

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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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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.