Stable determination of an inclusion in an elastic body by boundary measurements (unabridged)

TL;DR

This paper establishes a logarithmic stability estimate for identifying an unknown elastic inclusion within a body using boundary measurements, extending techniques from electrical and thermal inverse problems.

Contribution

It introduces a novel approach for the elastic Lamé system, combining propagation of smallness and local approximation methods for inverse boundary value problems.

Findings

Logarithmic stability estimate for inclusion identification

Extension of three-spheres inequality to elastic systems

Refined approximation of fundamental solutions in elastic media

Abstract

We consider the inverse problem of identifying an unknown inclusion contained in an elastic body by the Dirichlet-to-Neumann map. The body is made by linearly elastic, homogeneous and isotropic material. The Lam\'e moduli of the inclusion are constant and different from those of the surrounding material. Under mild a-priori regularity assumptions on the unknown defect, we establish a logarithmic stability estimate. For the proof, we extend the approach used for electrical and thermal conductors in a novel way. Main tools are propagation of smallness arguments based on three-spheres inequality for solutions to the Lam\'e system and refined local approximation of the fundamental solution of the Lam\'e system in presence of an inclusion.