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

In this study, we address the inverse problem of recovering the Lam\'e parameters ($\lambda, \mu$) and the density $\rho$ 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 Lam\'e parameters ($\lambda, \mu$) are know and we look for the inverse problem of recovering the density $\rho$. 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 $\lambda$, $\mu$ and $\rho$ simultameousely. We establish Lipschitz stability estimate, provided that the parameters $\lambda$, $\mu$ and $\rho$ 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.