Further results on generalized Holmgren's principle to the Lam\'e operator and applications

TL;DR

This paper extends the generalized Holmgren's principle for the Lamé operator to include all physical boundary conditions in linear elasticity, enabling new uniqueness results for inverse elastic obstacle and diffraction grating problems with variable impedance parameters.

Contribution

It provides a comprehensive characterization of the GHP for various physical boundary conditions, including variable impedance, and applies it to solve inverse scattering problems.

Findings

Unique identifiability of elastic obstacles with few measurements

Extension of GHP to variable impedance conditions

Application to inverse elastic diffraction grating problem

Abstract

In our earlier paper [9], it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lam\'e system must be identically zero, which is referred to as the generalized Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse scattering problem of determining a polygonal elastic obstacle of general impedance type by at most a few far-field measurements. In this paper, we include all the possible physical boundary conditions from linear elasticity into the GHP study with additionally the soft-clamped, simply-supported as well as the associated impedance-type conditions. We derive a comprehensive and complete characterisation of the GHP associated with all of the aforementioned physical conditions. 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. We follow the general strategy from [9] in establishing the results. However, we develop technically new ingredients to tackle the more general and challenging physical and mathematical setups. It is particularly worth noting that in [9], the impedance parameters were assumed to be constant whereas in this work they can be variable functions.