Stability analysis of an inverse coefficients problem in a system of partial differential equations

Abstract

In this study, we address the inverse problem of recovering the Lamé parameters ($λ, μ$) and the density $ρ$ of a medium from the Neumann-to-Dirichlet map for any dimension $d\geq 2$. This inverse problem finds its motivation in the reconstruction of mechanical properties of tissues in medical diagnostics. We first assume that the Lamé parameters ($λ, μ$) are know and we look for the inverse problem of recovering the density $ρ$. In this context, we derive a constrcutive Lipschitz stability estimate in terms of the Neumann to Dirichlet map in the case of piecewise constant parameters. Then, we look for the inverse problem of recovering $λ$, $μ$ and $ρ$ simultameousely. We establish Lipschitz stability estimate, provided that the parameters $λ$, $μ$ and $ρ$ have upper and lower bounds and belong to a known finite-dimensional subspace. The proofs hinge on monotonicity relations between the parameters and the Neumann-to-Dirichlet operator, coupled with the techniques of localized potentials.