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

Jianli Xiang; Guozheng Yan

doi:10.3934/ipi.2021004

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2021, Volume 15, Issue 3: 539-554. Doi: 10.3934/ipi.2021004

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

Jianli Xiang^1, and
Guozheng Yan^1, ,

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

^* Corresponding author: Guozheng Yan
^* Corresponding author: Guozheng Yan

Received: May 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: June 2021

The first author is supported by the Innovative Funding Project from Central China Normal University (Grant No. 2019CXZZ078). The second author is supported by the National Natural Science Foundation of China (Grant No. 11571132)

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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 $ [L^{\mathrm{q}}(\partial\Omega)]^{3} $ ($ 1<\mathrm{q}<2 $) 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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Mathematics Subject Classification: Primary: 31B10, 35J25, 74B05, 74J25; Secondary: 45Q05.

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Figure 1. Possible choice of $ x^{*} $

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