Stable determination of an inclusion in an inhomogeneous elastic body by boundary measurements

TL;DR

This paper establishes logarithmic stability estimates for identifying an unknown inhomogeneous elastic inclusion within a body using boundary measurements, extending previous results to more general inhomogeneous materials.

Contribution

It extends the stability results for inverse elastic problems to inhomogeneous materials and simplifies the proof techniques involved.

Findings

Logarithmic stability estimate holds for inhomogeneous elastic bodies.

The proof is simplified with new arguments compared to previous work.

Results apply under mild smoothness and regularity assumptions.

Abstract

In this paper we consider the stability issue for the inverse problem of determining an unknown inclusion contained in an elastic body by all the pairs of measurements of displacement and traction taken at the boundary of the body. Both the body and the inclusion are made by inhomogeneous linearly elastic isotropic material. Under mild a priori assumptions about the smoothness of the inclusion and the regularity of the coefficients, we show that the logarithmic stability estimate proved in \cite{ADiCMR14} in the case of piecewise constant coefficients continues to hold in the inhomogeneous case. We introduce new arguments which allow to simplify some technical aspects of the proof given in \cite{ADiCMR14}.