On Generalized Holmgren’s Principle To The Lamé Operator With Applications To Inverse Elastic Problems

In this thesis,we consider the Lamé operator L(u):=μΔu+(λ+μ)▽(▽·u)that arises in the theory linear elasticity.This thesis studies the geometric properties of the(generalized)Lamé eigenfunction u,namely-L(u)=κu with κ∈R+ and u∈L2(Ω)2,Ω(?)R2.We introduce the so-called six homogeneous boundary lines of u in Ω,that is,rigid line,traction-free line,impedance line,soft-clamped line,simply-supported line and generalized-impedance line of u.We give a comprehensive study on characterizing the presence of one or two such line segments and its implication to the uniqueness of 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 u with the presence of the aforesaid line segments.The generalized Holmgren principle(GHP)enables us to uniquely determine the polygonal elastic obstacle of general impedance type by at most a few far-field measurements.The polygonal elastic obstacle is uniquely determined by a minimal number of far-field patterns is a longstanding problem in inverse elastic scattering theory.As significant applications,we establish novel unique identifiability results by at most a few scattering measurements not only for the inverse elastic obstacle problem but also for the inverse elastic diffraction grating problem within polygonal geometry in the most general physical scenario.