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

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