Consider the Lamé operator \({\mathcal {L}}({\mathbf {u}}) :=\mu \Delta {\mathbf {u}}+(\lambda +\mu ) \nabla (\nabla \cdot {\mathbf {u}} )\) that arises in the theory of linear elasticity. This paper studies the geometric properties of the (generalized) Lamé eigenfunction \({\mathbf {u}}\), namely \(-{\mathcal {L}}({\mathbf {u}})=\kappa {\mathbf {u}}\) with \(\kappa \in {\mathbb {R}}_+\) and \({\mathbf {u}}\in L^2(\Omega )^2\), \(\Omega \subset {\mathbb {R}}^2\). We introduce the so-called homogeneous line segments of \({\mathbf {u}}\) in \(\Omega \), on which \({\mathbf {u}}\), its traction or their combination via an impedance parameter is vanishing. We give a comprehensive study on characterizing the presence of one or two such line segments and its implication to the uniqueness of \({\mathbf {u}}\). The results can be regarded as generalizing the classical Holmgren’s uniqueness principle for the Lamé operator in two aspects. We establish the results by analyzing the development of analytic microlocal singularities of \({\mathbf {u}}\) with the presence of the aforesaid line segments. Finally, we apply the results to the inverse elastic problems in establishing two novel unique identifiability results. It is shown that a generalized impedance obstacle as well as its boundary impedance can be determined by using at most four far-field patterns. Unique determination by a minimal number of far-field patterns is a longstanding problem in inverse elastic scattering theory.

考虑Lamé运算符\（{\ mathcal {L}}（{\ mathbf {u}}）：= \ mu \ Delta {\ mathbf {u}} +（\ lambda + \ mu）\ nabla（\ nabla \ cdot {\ mathbf {u}}）\）出现在线性弹性理论中。本文研究（广义）Lamé特征函数\（{\ mathbf {u}} \）的几何性质，即\（-{\ mathcal {L}}（{\ mathbf {u}}）= \ kappa {\ mathbf【U}} \）与\（\卡帕\在{\ mathbb {R}} _ + \）和\（{\ mathbf【U}} \在L ^ 2（\欧米茄）^ 2 \），\ （\ Omega \ subset {\ mathbb {R}} ^ 2 \）。我们在\（\ Omega \）中引入\（{\ mathbf {u}} \）的所谓齐次线段，在其上\（{\ mathbf {u}} \，其牵引力或通过阻抗参数的组合将消失。我们对表征一个或两个这样的线段的存在及其对\（{\ mathbf {u}} \）的唯一性进行了全面的研究。结果可以看作是从两个方面推广了Lamé算子的经典Holmgren唯一性原理。我们通过分析\（{\ mathbf {u}} \）的解析微局部奇异性的发展来建立结果存在上述线段。最后，我们将结果应用于反弹性问题，以建立两个新颖的独特可识别性结果。结果表明，通过使用最多四个远场图形可以确定广义阻抗障碍及其边界阻抗。在逆弹性散射理论中，由最少数量的远场图案进行唯一确定是一个长期存在的问题。