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Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries

  • *Corresponding author: Hongyu Liu

    *Corresponding author: Hongyu Liu 

Dedicated to the memory of Professor Victor Isakov (1947–2021)

The first author was supported by the Austrian Science Fund (FWF): P 32660. The second author was supported in part by NSFC/RGC Joint Research Grant No. 12161160314 and the startup fund from Jilin University. The third author was supported by Hong Kong RGC General Research Funds (project numbers, 11300821, 12301218 and 12302919) and the NSFC/RGC Joint Research Grant (project number, N_CityU101/21). The fourth author was supported by Hong Kong RGC General Research Fund (projects 14306718 and 14306719)

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  • We consider the unique determinations of impenetrable obstacles or diffraction grating profiles in $ \mathbb{R}^3 $ by a single far-field measurement within polyhedral geometries. We are particularly interested in the case that the scattering objects are of impedance type. We derive two new unique identifiability results by a single measurement for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [12,13]. In this paper, we derive novel geometric properties that generalize and extend the related results in [13], which further enable us to establish the new unique identifiability results. It is pointed out that in addition to the shape of the obstacle or the grating profile, we can simultaneously recover the boundary impedance parameters.

    Mathematics Subject Classification: Primary: 35P05, 35P25, 35R30; Secondary: 35Q60.


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  • Figure 1.  A schematic illustration for an edge corner with the dihedral angle $ \phi_0 $

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