全涂层非均匀介质外部反散射的传输特征值问题

丁慧; 刘立汉

doi:10.21656/1000-0887.450207

全涂层非均匀介质外部反散射的传输特征值问题

doi: 10.21656/1000-0887.450207

丁慧,
刘立汉^,

重庆师范大学数学科学学院，重庆 401331

基金项目:

重庆市留学人员回国创业创新支持计划（cx2021061；cx2019022）；重庆市巴渝学者计划（BYQNCS2020002）

KJZD-K2021000503）

国家自然科学基金青年科学基金（12001075）；重庆市自然科学基金面上项目（cstc2020jcyj-msxmX0167）

重庆市高校创新研究群体项目（CXQT20014）

重庆市教育委员会科学技术研究计划项目重点项目（KJZD-K202300506

详细信息

作者简介:
丁慧（1999—），女，硕士生(E-mail: dinghuidddh@163.com);刘立汉（1987—），男，教授，博士，硕士生导师（通讯作者. E-mail: 20132130@cqnu.edu.cn）.

通讯作者:
刘立汉（1987—），男，教授，博士，硕士生导师（通讯作者. E-mail: 20132130@cqnu.edu.cn）.

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出版历程
- 收稿日期: 2024-07-12
- 修回日期: 2024-09-03
- 网络出版日期: 2025-06-30

The Transmission Eigenvalue Problem of Exterior Inverse Scattering in Fully Coated Inhomogeneous Media

DING Hui,
LIU Lihan^,

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P.R.China

Funds:

The National Science Foundation of China(12001075)

摘要

摘要: 研究了具有全涂层边界条件的非均匀介质外部反散射的传输特征值问题.首先，根据经典过程建立了四阶非线性公式，利用Lax-Milgram定理及Fredholm理论证明了全涂层非均匀介质外部反散射传输特征值的存在性和离散性.其次，通过一个带有辅助变量的等价混合公式，将问题转换为线性特征值问题，利用Riesz表示定理、 Rellich紧性定理等构造了恰当的算子，再利用Cauchy收敛准则、 Brezzi理论、 Poincaré不等式证明了算子的紧性和强制性.
- 全涂层边界 /
- 外部反散射 /
- 传输特征值
Abstract: The transmission eigenvalue problem of exterior inverse scattering in fully coated inhomogeneous media was studied. Firstly, a nonlinear 4th-order formulation was established based on the classical process, and the existence and discreteness were proved by the Lax-Milgram theorem and the Fredholm theory. Next, by means of an equivalent mixed formulation with an auxiliary variable, the problem is transformed into a linear eigenvalue problem, and an appropriate operator was constructed by the Riesz representation theorem and the Rellich-Kondrachov compactness theorem, etc. The compactness and coerciveness of the operators were proved with the Cauchy convergence criterion, the Brezzi theory and the Poincaré inequality.
- fully coated boundary /
- exterior inverse scattering /
- transmission eigenvalue

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参考文献(16)

[2]CAKONI F, GINTIDES D, HADDAR H. The existence of an infinite discrete set of transmission eigenvalues[J]. SIAM Journal on Mathematical Analysis,2010,42(1): 237-255.

CAKONI F, CAYOREN M, COLTON D. Transmission eigenvalues and the nondestructive testing of dielectrics[J]. Inverse Problems,2008,24(6): 065016.

[3]COLTON D, KRESS R. Inverse Acoustic and Electromagnetic Scattering Theory[M]. Berlin: Springer, 2019.

[4]CAKONI F, COLTON D, HADDAR H. Inverse Scattering Theory and Transmission Eigenvalues[M]. Philadelphia: Society for Industrial and Applied Mathematics, 2016.

[5]SYLVESTER J. Discreteness of transmission eigenvaluesvia upper triangular compact operators[J]. SIAM Journal on Mathematical Analysis,2012,44(1): 341-354.

[6]张亚林. 逆散射理论中传播特征值问题的若干结果[D]. 天津: 天津大学, 2015.(ZHANG Yalin. Some results on the transmission eigenvalue problem in inverse scattering theory[D]. Tianjin: Tianjin University, 2015. (in Chinese))

[7]COLTON D, LEUNG Y J. Complex eigenvalues and the inverse spectral problem for transmission eigenvalues[J]. Inverse Problems,2013,29(10): 104008.

[8]ROBBIANO L. Spectral analysis of the interior transmission eigenvalue problem[J]. Inverse Problems,2013,29(10): 104001.

[9]CAKONI F, COLTON D, MONK P. The determination of the surface conductivity of a partially coated dielectric[J]. SIAM Journal on Applied Mathematics,2005,65(3): 767-789.

[10]陈林冲, 李小林. 二维Helmholtz方程的插值型边界无单元法[J]. 应用数学和力学, 2018,39(4): 470-484.(CHEN Linchong, LI Xiaolin. An interpolating boundary element-free method for 2D Helmholtz equations[J]. Applied Mathematics and Mechanics,2018,39(4): 470-484. (in Chinese))

[11]LIU Q, LI T X, ZHANG S. A mixed element scheme for the Helmholtz transmission eigenvalue problem for anisotropic media[J]. Inverse Problems,2023,39(5): 055005.

[12]戴海, 潘文峰. 谱元法求解Helmholtz方程透射特征值问题[J]. 应用数学和力学, 2018,39(7): 833-840.(DAI Hai, PAN Wenfeng. A spectral element method for transmission eigenvalue problems of the Helmholtz equation[J]. Applied Mathematics and Mechanics,2018,39(7): 833-840. (in Chinese))

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全涂层非均匀介质外部反散射的传输特征值问题

doi: 10.21656/1000-0887.450207

作者简介:
丁慧（1999—），女，硕士生(E-mail: dinghuidddh@163.com);刘立汉（1987—），男，教授，博士，硕士生导师（通讯作者. E-mail: 20132130@cqnu.edu.cn）.

通讯作者:
刘立汉（1987—），男，教授，博士，硕士生导师（通讯作者. E-mail: 20132130@cqnu.edu.cn）.

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The Transmission Eigenvalue Problem of Exterior Inverse Scattering in Fully Coated Inhomogeneous Media

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留言板

全涂层非均匀介质外部反散射的传输特征值问题

doi: 10.21656/1000-0887.450207

作者简介: 丁慧（1999—），女，硕士生(E-mail: dinghuidddh@163.com);刘立汉（1987—），男，教授，博士，硕士生导师（通讯作者. E-mail: 20132130@cqnu.edu.cn）.

通讯作者: 刘立汉（1987—），男，教授，博士，硕士生导师（通讯作者. E-mail: 20132130@cqnu.edu.cn）.

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出版历程

The Transmission Eigenvalue Problem of Exterior Inverse Scattering in Fully Coated Inhomogeneous Media

计量

出版历程

目录

作者简介:
丁慧（1999—），女，硕士生(E-mail: dinghuidddh@163.com);刘立汉（1987—），男，教授，博士，硕士生导师（通讯作者. E-mail: 20132130@cqnu.edu.cn）.

通讯作者:
刘立汉（1987—），男，教授，博士，硕士生导师（通讯作者. E-mail: 20132130@cqnu.edu.cn）.