On generalized Holmgren’s principle to the Lamé operator with applications to inverse elastic problems

• DOI: 10.1007/s00526-020-01830-5
• Publication Date: 2020-09-22T15:04:20Z
• Authors: Huaian Diao; Hongyu Liu; Li Wang
• Keywords: Nabla symbol; Eigenfunction; Operator (biology)

Consider the Lam�� operator $\mathcal{L}(\mathbf{ u} ) :=����\mathbf{u}+(��+��) \nabla(\nabla \cdot \mathbf{ u} )$ that arises in the theory of linear elasticity. This paper studies the geometric properties of the (generalized) Lam�� eigenfunction $\mathbf{u}$, namely $-\mathcal{L}(\mathbf{ u} )=��\mathbf{ u}$ with $��\in\mathbb{R}_+$ and $\mathbf{ u}\in L^2(��)^2$, $��\subset\mathbb{R}^2$. We introduce the so-called homogeneous line segments of $\mathbf{u}$ in $��$, 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 Lam�� 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.