The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation

IF 1.2 4区 数学 Q2 MATHEMATICS, APPLIED Inverse Problems and Imaging Pub Date : 2021-01-01 DOI:10.3934/ipi.2021004
Jianlin Xiang, G. Yan
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Abstract

This paper considers the inverse elastic wave scattering by a bounded penetrable or impenetrable scatterer. We propose a novel technique to show that the elastic obstacle can be uniquely determined by its far-field pattern associated with all incident plane waves at a fixed frequency. In the first part of this paper, we establish the mixed reciprocity relation between the far-field pattern corresponding to special point sources and the scattered field corresponding to plane waves, and the mixed reciprocity relation is the key point to show the uniqueness results. In the second part, besides the mixed reciprocity relation, a priori estimates of solution to the transmission problem with boundary data in \begin{document}$ [L^{\mathrm{q}}(\partial\Omega)]^{3} $\end{document} ( \begin{document}$ 1 ) is deeply investigated by the integral equation method and also we have constructed a well-posed modified static interior transmission problem on a small domain to obtain the uniqueness result.
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基于混合互易关系的弹性波反散射问题的唯一性
本文研究了有界可穿透和不可穿透散射体对弹性波的反向散射。我们提出了一种新的技术,表明弹性障碍物可以唯一地由其与固定频率的所有入射平面波相关的远场模式确定。在本文的第一部分中,我们建立了特殊点源对应的远场图样与平面波对应的散射场之间的混合互易关系,混合互易关系是证明唯一性结果的关键。第二部分,除混合互反关系外,利用积分方程法深入研究了在\begin{document}$ [L^{\ mathm {q}}(\partial\Omega)]^{3} $\end{document} (\ begin{document}$ 1)中边界数据传输问题解的先验估计,并构造了一个小定域上的修正静态内部传输问题的唯一性结果。
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来源期刊
Inverse Problems and Imaging
Inverse Problems and Imaging 数学-物理:数学物理
CiteScore
2.50
自引率
0.00%
发文量
55
审稿时长
>12 weeks
期刊介绍: Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing. This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.
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