Stable determination of cavities in elastic bodies

Abstract

We prove a conditional stability estimate for the inverse problem of determining either cavities inside an elastic body or unknown boundary portions, from a single measurement of traction and displacement taken on the accessible part of the exterior boundary of .

Three spheres inequalities for a two-dimensional elliptic system and its application

“…We will follow the approach used in [9] where a log-log type stability estimate was proved for the star-shaped cavity in the Lamé system. For more general cavities in the Lamé system, a log-log type estimate has been derived in [14]. The result in [14] is also applied to the case of determining unknown boundary portions of the exterior boundary from a single pair of traction-displacement measurement taken on the accessible part of the boundary.…”

“…For more general cavities in the Lamé system, a log-log type estimate has been derived in [14]. The result in [14] is also applied to the case of determining unknown boundary portions of the exterior boundary from a single pair of traction-displacement measurement taken on the accessible part of the boundary. For the similar problem in the scalar elliptic equation, a log type stability estimate was obtained in [1].…”

Smoothness dependent stability in corrosion detection

“…In Theorem 4.4 we provide a lower bound on the local vanishing rate of the solution u. The main ingredient of the proof is the so called Lipschitz Propagation of Smallness, see also [3,16]. Finally in Proposition 4.5 we state a weighted interpolation inequality which was previously introduced in [8] and we conclude by giving the proof of Theorem 2.1.…”

Cavity identification in linear elasticity and thermoelasticity

“…The uniqueness in the identification of two-dimensional cavities in a heterogeneous isotropic elastic medium has been proved by Ang et al [6,7], who studied electric detection of cavities from Cauchy data measured on the boundary and proved a uniqueness result. Stability results for crack and interface identification via Laplace equation using boundary measurements have been derived in [8][9][10][11] and more recently Morassi and Rosset [12] proved stability estimate of log-log type for cavities in a homogeneous isotropic elastic body under regularity assumptions on the geometry. Most of the existing stability results for these problems are focused on global estimates for simply connected shapes (usually of logarithmic type), we take a different approach deriving directional Lipschitz stability for multiply connected shapes.…”