THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY*

doi:10.1007/s10473-024-0118-y

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (1): 339-354.doi: 10.1007/s10473-024-0118-y

THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY^*

Jianli XIANG¹, Guozheng YAN^2,†

1. Three Gorges Mathematical Research Center, College of Science, China Three Gorges University, Yichang 443002, China;
2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

收稿日期:2022-08-04 修回日期:2023-06-19 出版日期:2024-02-25 发布日期:2024-02-27
通讯作者: † Guozheng YAN, E-mail: yangz@ccnu.edu.cn
作者简介:Jianli XIANG, E-mail: xiangjianli@ctgu.edu.cn
基金资助:
National Natural Science Foundation of China (11571132, 12301542), the Natural Science Foundation of Hubei (2022CFB725) and the Natural Science Foundation of Yichang (A23-2-027).

THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY^*

Jianli XIANG¹, Guozheng YAN^2,†

1. Three Gorges Mathematical Research Center, College of Science, China Three Gorges University, Yichang 443002, China;
2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received:2022-08-04 Revised:2023-06-19 Online:2024-02-25 Published:2024-02-27
Contact: † Guozheng YAN, E-mail: yangz@ccnu.edu.cn
About author:Jianli XIANG, E-mail: xiangjianli@ctgu.edu.cn
Supported by:
National Natural Science Foundation of China (11571132, 12301542), the Natural Science Foundation of Hubei (2022CFB725) and the Natural Science Foundation of Yichang (A23-2-027).

摘要/Abstract

摘要： We consider the interior transmission eigenvalue problem corresponding to the scattering for an anisotropic medium of the scalar Helmholtz equation in the case where the boundary $\partial\Omega$ is split into two disjoint parts and possesses different transmission conditions. Using the variational method, we obtain the well posedness of the interior transmission problem, which plays an important role in the proof of the discreteness of eigenvalues. Then we achieve the existence of an infinite discrete set of transmission eigenvalues provided that $n\equiv1$, where a fourth order differential operator is applied. In the case of $n\not\equiv1$, we show the discreteness of the transmission eigenvalues under restrictive assumptions by the analytic Fredholm theory and the T-coercive method.

关键词: interior transmission eigenvalue, anisotropic medium, partially coated boundary, the analytic Fredholm theory, T-coercive method

Abstract: We consider the interior transmission eigenvalue problem corresponding to the scattering for an anisotropic medium of the scalar Helmholtz equation in the case where the boundary $\partial\Omega$ is split into two disjoint parts and possesses different transmission conditions. Using the variational method, we obtain the well posedness of the interior transmission problem, which plays an important role in the proof of the discreteness of eigenvalues. Then we achieve the existence of an infinite discrete set of transmission eigenvalues provided that $n\equiv1$, where a fourth order differential operator is applied. In the case of $n\not\equiv1$, we show the discreteness of the transmission eigenvalues under restrictive assumptions by the analytic Fredholm theory and the T-coercive method.

Key words: interior transmission eigenvalue, anisotropic medium, partially coated boundary, the analytic Fredholm theory, T-coercive method

中图分类号:

35P25

Jianli XIANG, Guozheng YAN. THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY^*[J]. 数学物理学报(英文版), 2024, 44(1): 339-354.

Jianli XIANG, Guozheng YAN. THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY^*[J]. Acta mathematica scientia,Series B, 2024, 44(1): 339-354.

参考文献

[1] Bellis C, Cakoni F, Guzina B B. Nature of the transmission eigenvalue spectrum for elastic bodies. IMA J Appl Math, 2013, 78: 895-923
[2] Bondarenko O, Harris I, Kleefeld A. The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary. Appl Anal, 2017, 96: 2-22
[3] Bonnet-Ben Dhia A S, Chesnel L, Haddar H. On the use of T-coercivity to study the interior transmission eigenvalue problem. C R Math Acad Sci Paris, 2011, 349: 647-651
[4] Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer, 2011
[5] Cakoni F, Colton D.A Qualitative Approach to Inverse Scattering Theory. Berlin: Springer, 2014
[6] Cakoni F, Colton D, Haddar H. The linear sampling method for anisotropic media. J Comput Appl Math, 2002, 146: 285-299
[7] Cakoni F, Colton D, Haddar H. The computation of lower bounds for the norm of the index of refraction in an anisotropic media from far field data. J Integral Equations Appl, 2009, 21: 203-227
[8] Cakoni F, Colton D, Haddar H.Inverse Scattering Theory and Transmission Eigenvalues. Philadelphia, PA: SIAM, 2016
[9] Cakoni F, Colton D, Haddar H. The interior transmission problem for regions with cavities. SIAM J Math Anal, 2010, 42: 145-162
[10] Cakoni F, Colton D, Haddar H. The interior transmission eigenvalue problem for absorbing media. Inverse Problems, 2012, 28: 045005
[11] Cakoni F, Colton D, Monk P. The determination of the surface conductivity of a partially coated dielectric. SIAM J Appl Math, 2005, 65: 767-789
[12] Cakoni F, Cossonniére A, Haddar H. Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Probl Imag, 2012, 6: 373-398
[13] Cakoni F, Gintides D, Haddar H. The existence of an infinite discrete set of transmission eigenvalues. SIAM J Math Anal, 2010, 42: 237-255
[14] Cakoni F, Haddar H. Interior transmission problem for anisotropic media//Cohen G C, Joly P, Heikkola E, Neittaanmaki P. Mathematical and Numerical Aspects of Wave Propagation. Berlin: Springer, 2003: 613-618
[15] Cakoni F, Haddar H. On the existence of transmission eigenvalues in an inhomogeneous medium. Appl Anal, 2009, 88: 475-493
[16] Cakoni F, Haddar H.Transmission eigenvalues in inverse scattering theory//Uhlmann G. Inverse Problems and Applications: Inside Out II. Cambridge: Cambridge Univ Press, 2012: 527-578
[17] Cakoni F, Kirsch A. On the interior transmission eigenvalue problem. Int J Comput Sci Math, 2010, 3: 142-167
[18] Cakoni F, Kow P Z, Wang J N. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Probl Imag, 2021, 15: 445-474
[19] Cakoni F, Kress R. A boundary integral equation method for the transmission eigenvalue problem. Appl Anal, 2016, 96: 23-38
[20] Charalambopoulos A, Anagnostopoulos K A. On the spectrum of the interior transmission problem in isotropic elasticity. J Elasticity, 2008, 90: 295-313
[21] Chesnel L. Interior transmission eigenvalue problem for Maxwell's equations: the T-coercivity as an alternative approach. Inverse Problems, 2012, 28: 065005
[22] Colton D, Kirsch A, Päivärinta L. Far-field patterns for acoustic waves in an inhomogeneous medium. SIAM J Math Anal, 1989, 20: 1472-1483
[23] Colton D, Kress R.Inverse Acoustic and Electromagnetic Scattering Theory. 3rd ed. Berlin: Springer, 2013
[24] Colton D, Monk P. The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium-Numerical experiments. Quar J Mech Appl Math, 1989, 26: 323-350
[25] Cossonniére A, Haddar H. Surface integral formulation of the interior transmission problem. J Integral Equations Appl, 2013, 25: 341-376
[26] Hähner P. On the uniqueness of the shape of a penetrable, anisotropic obstacle. J Comput Appl Math, 2000, 116: 167-180
[27] Harris I, Cakoni F, Sun J. Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids. Inverse Problems, 2014, 30: 035016
[28] Harris I, Kleefeld A. The inverse scattering problem for a conductive boundary condition and transmission eigenvalues. Appl Anal, 2020, 99(3): 508-529
[29] McLaughlin J R, Polyakov P L. On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues. J Differential Equations, 1994, 107: 351-382
[30] McLaughlin J R, Polyakov P L, Sacks P E. Reconstruction of a spherically symmetric speed of sound. SIAM J Appl Math, 1994, 54: 1203-1223
[31] Päivärinta L, Sylvester J. Transmission eigenvalues. SIAM J Math Anal, 2008, 40: 738-753
[32] Rynne B P, Sleeman B D. The interior transmission problem and inverse scattering from inhomogeneous media. SIAM J Math Anal, 1991, 22: 1755-1762
[33] Xiang J L, Yan G Z. Uniqueness of the inverse transmission scattering with a conductive boundary condition. Acta Math Sci, 2021, 41B(3): 925-940

THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY^*

THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY^*

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