Stable determination of a rigid scatterer in elastodynamics

TL;DR

This work addresses the inverse elastic scattering problem of determining a rigid scatterer from limited elastic far-field data in time-harmonic regimes. The authors develop a geometric stability framework that yields a local log log stability bound for recovering the scatterer from a single incident wave, assuming a priori closeness encoded via Friedrichs-type constants and $C^{2,\alpha}$ regularity. The approach combines Helmholtz decompositions into longitudinal and transversal components, a far-field to near-field continuation using three-spheres inequalities, and Friedrichs/Korn-type estimates to control interior fields, culminating in a quantitative Hausdorff-distance bound between two admissible scatterers. This result advances the theoretical understanding of stability under minimal measurement data in elasticity and provides explicit a priori conditions that depend only on dimension and material constants, enabling potential applications in nondestructive testing and inverse problem design.

Abstract

We deal with an inverse elastic scattering problem for the shape determination of a rigid scatterer in the time-harmonic regime. We prove a local stability estimate of log log type for the identification of a scatterer by a single far-field measurement. The needed a priori condition on the closeness of the scatterers is estimated by the universal constant appearing in the Friedrichs inequality.

Figures (2)

• Figure 1: The geometric construction from $x=x_2$ to $\tilde{x}$.
• Figure 2: From $\tilde{x}$ to $x_3$.

Theorems & Definitions (16)

• Definition 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5
• Lemma 2.6
• Theorem 2.7
• Theorem 3.1
• Remark 3.2
• Lemma 4.1