散射与逆散射中的分裂角
Split Corners in Scattering and Inverse Scattering
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中文总结 AI 辅助
研究具有角奇点的源和可穿透介质的散射与逆散射，引入分裂角概念，建立相关结果并推广经典理论，得到源和介质逆散射的唯一性结果，如从单个远场测量恢复多边形凸包。
中文摘要 AI 辅助
我们研究由具有角奇点的源和可穿透介质产生的散射和逆散射。引入分裂角概念，它是统一源和介质散射中几何奇点与系数不连续性的局部模型。在此框架下，建立散射和逆散射结果，将经典理论扩展到分裂角，使源或介质对比度在角尖处不同扇区趋近不同极限值。对于源散射，证明每个可允许的双分裂角在一般有界非均匀背景中必然辐射，推广了经典角辐射原理。对于可穿透介质，建立类似结果及任意消失阶入射波的显式相容性条件。由此在逆源和逆介质散射中得到几个唯一性结果，包括从单个远场测量恢复多边形凸包。
英文摘要
We study scattering and inverse scattering generated by sources and penetrable media with corner singularities. We introduce the notion of split corners, a local model that unifies geometric singularities and coefficient discontinuities arising in source and medium scattering. Within this framework, we establish scattering and inverse scattering results that extend the classical theory to split corners, allowing the source or medium contrast to approach different limiting values in distinct sectors meeting at the corner tip. For source scattering, we prove that every admissible two-split corner necessarily radiates in a general bounded inhomogeneous background, thereby generalizing the classical corner radiation principle to configurations involving both geometric corners and jump discontinuities. For penetrable media, we establish analogous results together with explicit compatibility conditions for incident waves of arbitrary vanishing order. As consequences, we obtain several uniqueness results in inverse source and inverse medium scattering, including recovery of polygonal convex hulls from a single far-field measurement.
