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Stable determination of an elastic medium scatterer by a single far-field measurement and beyond

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Abstract

We are concerned with the time-harmonic elastic scattering due to an inhomogeneous elastic material inclusion located inside a uniformly homogeneous isotropic medium. We establish a sharp stability estimate of logarithmic type in determining the support of the elastic scatterer, independent of its material content, by a single far-field measurement when the support is a convex polyhedral domain in \({\mathbb {R}}^n\), \(n=2,3\). Our argument in establishing the stability result is localized around a corner of the medium scatterer. This enables us to further establish a byproduct result by proving that if a generic medium scatterer, not necessary to be a polyhedral shape, possesses a corner, then there exists a positive lower bound of the scattered far-field patterns. The latter result indicates that if an elastic material object possesses a corner on its support, then it scatters every incident wave stably and invisibility phenomenon does not occur.

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Acknowledgements

The authors would like to thank three anonymous referees for their constructive comments and suggestions, which have led to significant improvement on the earlier version of this paper. Particular, anoymous referees reminded us to pay attentions to the related stability study [46] on the rigid obstacle in linear elasticity. The research of Z Bai was partially supported by the National Natural Science Foundation of China under grant 11671337 and the Natural Science Foundation of Fujian Province of China under grant 2021J01033. The work of H Diao was supported in part by the NSFC/RGC Joint Research Fund (project 12161160314) and the startup fund from Jilin University. The work of H Liu was supported by the startup fund from City University of Hong Kong and the Hong Kong RGC General Research Fund (projects 12301420, 11300821 and 12301218), and the NSFC/RGC Joint Research Fund (project N_CityU101/21).

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Zhengjian Bai & Qingle Meng

  2. School of Mathematics, Jilin University, Changchun, 130012, China

    Huaian Diao

  3. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China

    Hongyu Liu

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  1. Zhengjian Bai
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  2. Huaian Diao
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  3. Hongyu Liu
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  4. Qingle Meng
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Correspondence to Huaian Diao.

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Communicated by Y. Giga.

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Bai, Z., Diao, H., Liu, H. et al. Stable determination of an elastic medium scatterer by a single far-field measurement and beyond. Calc. Var. 61, 170 (2022). https://doi.org/10.1007/s00526-022-02278-5

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