Cracks with impedance, stable determination from boundary data

TL;DR

This paper develops a stability estimate for detecting cracks inside a body using boundary measurements, advancing inverse problem techniques with quantitative unique continuation and singular solutions.

Contribution

It introduces an optimal single-logarithm stability estimate for crack detection from boundary data, applicable to various boundary measurement types.

Findings

Proven stability estimate for inverse crack detection problem.

Applicable to Dirichlet-to-Neumann, Neumann-to-Dirichlet, and local boundary maps.

Uses singular solutions and unique continuation techniques.

Abstract

We discuss the inverse problem of determining the possible presence of an (n-1)-dimensional crack \Sigma in an n-dimensional body \Omega with n > 2 when the so-called Dirichlet-to-Neumann map is given on the boundary of \Omega. In combination with quantitative unique continuation techniques, an optimal single-logarithm stability estimate is proven by using the singular solutions method. Our arguments also apply when the Neumann-to-Dirichlet map or the local versions of the D-N and the N-D map are available.