Determining Lam\'{e} coefficients by elastic Dirichlet-to-Neumann map on a Riemannian manifold

TL;DR

This paper derives an explicit formula for the elastic Dirichlet-to-Neumann map on a Riemannian manifold and proves it uniquely determines the Lamé coefficients throughout the manifold under certain analyticity conditions.

Contribution

It provides an explicit symbol expression for the elastic Dirichlet-to-Neumann map and establishes unique determination of Lamé coefficients from boundary measurements.

Findings

Explicit formula for the Dirichlet-to-Neumann map's symbol.

Unique determination of Lamé coefficients on the boundary.

Global uniqueness result under analyticity assumptions.

Abstract

For the Lam\'{e} operator $\mathcal{L}_{\lambda,\mu}$ with variable coefficients $\lambda$ and $\mu$ 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 $\Lambda_{\lambda,\mu}$. We show that $\Lambda_{\lambda,\mu}$ uniquely determines partial derivatives of all orders of the Lam\'{e} coefficients $\lambda$ and $\mu$ on $\partial M$. Moreover, for a nonempty open subset $\Gamma\subset\partial M$, suppose that the manifold and the Lam\'{e} coefficients are real analytic up to $\Gamma$, we prove that $\Lambda_{\lambda,\mu}$ uniquely determines the Lam\'{e} coefficients on the whole manifold $\bar{M}$.