doi: 10.3934/ipi.2021063
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & 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. BlåstenL. 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. BlåstenH. 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. CaoH. 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. ChowY. DengY. HeH. 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. DiaoX. 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. HuM. 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. LiuZ. J. ShangH. 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. SaloL. 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. BlåstenL. 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. BlåstenH. 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. CaoH. 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. ChowY. DengY. HeH. 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. DiaoX. 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. HuM. 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. LiuZ. J. ShangH. 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. SaloL. 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) $
Figure 2.  Graphical illustration of the classification of transmission eigenvalues
[1]

Vesselin Petkov, Georgi Vodev. Localization of the interior transmission eigenvalues for a ball. Inverse Problems & Imaging, 2017, 11 (2) : 355-372. doi: 10.3934/ipi.2017017

[2]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5135-5148. doi: 10.3934/dcdsb.2020336

[3]

Josselin Garnier. Optimal transmission through a randomly perturbed waveguide in the localization regime. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 597-621. doi: 10.3934/dcdsb.2011.15.597

[4]

Delfina Gómez, Sergey A. Nazarov, Eugenia Pérez. Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions. Networks & Heterogeneous Media, 2011, 6 (1) : 1-35. doi: 10.3934/nhm.2011.6.1

[5]

Q-Heung Choi, Changbum Chun, Tacksun Jung. The multiplicity of solutions and geometry in a wave equation. Communications on Pure & Applied Analysis, 2003, 2 (2) : 159-170. doi: 10.3934/cpaa.2003.2.159

[6]

Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257

[7]

Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031

[8]

Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153

[9]

Alan Compelli, Rossen Ivanov. Benjamin-Ono model of an internal wave under a flat surface. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4519-4532. doi: 10.3934/dcds.2019185

[10]

Bei Gong, Zhen-Hu Ning, Fengyan Yang. Stabilization of the transmission wave/plate equation with variable coefficients on $ {\mathbb{R}}^n $. Evolution Equations & Control Theory, 2021, 10 (2) : 321-331. doi: 10.3934/eect.2020068

[11]

Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure & Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97

[12]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1987-2020. doi: 10.3934/cpaa.2021055

[13]

Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255

[14]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399

[15]

Weinan E, Weiguo Gao. Orbital minimization with localization. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 249-264. doi: 10.3934/dcds.2009.23.249

[16]

Radu Balan, Peter G. Casazza, Christopher Heil and Zeph Landau. Density, overcompleteness, and localization of frames. Electronic Research Announcements, 2006, 12: 71-86.

[17]

Luisa Berchialla, Luigi Galgani, Antonio Giorgilli. Localization of energy in FPU chains. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 855-866. doi: 10.3934/dcds.2004.11.855

[18]

Gary Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 671-689. doi: 10.3934/dcds.2007.17.671

[19]

Bernd Kawohl, Friedemann Schuricht. First eigenfunctions of the 1-Laplacian are viscosity solutions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 329-339. doi: 10.3934/cpaa.2015.14.329

[20]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

2020 Impact Factor: 1.639

Article outline

Figures and Tables

[Back to Top]