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.

中文翻译：

---

基于混合互易关系的弹性波逆散射问题的唯一性

本文考虑有界穿透或不可穿透散射体的逆弹性波散射。我们提出了一种新颖的技术来表明，弹性障碍物可以通过与固定频率下所有入射平面波相关联的远场模式来唯一地确定。在本文的第一部分中，我们建立了对应于特殊点源的远场模式与对应于平面波的散射场之间的混合互易关系，而混合互易关系是显示唯一性结果的关键。在第二部分中，除了混合互易关系之外，还对$ [L ^ {\ mathrm {q}}（\ partial \ Omega）] ^ {3} $（$ 1 <\ mathrm {q} <