Determining Lamé coefficients by elastic Dirichlet-to-Neumann map on a Riemannian manifold

Abstract

For the Lamé operator $\mathcal{L}_{λ,μ}$ with variable coefficients $λ$ and $μ$ on a smooth compact Riemannian manifold $(M,g)$ with smooth boundary $\partial M$, we give an explicit expression for full symbol of the elastic Dirichlet-to-Neumann map $Λ_{λ,μ}$. We show that $Λ_{λ,μ}$ uniquely determines partial derivatives of all orders of the Lamé coefficients $λ$ and $μ$ on $\partial M$. Moreover, for a nonempty open subset $Γ\subset\partial M$, suppose that the manifold and the Lamé coefficients are real analytic up to $Γ$, we prove that $Λ_{λ,μ}$ uniquely determines the Lamé coefficients on the whole manifold $\bar{M}$.