Determination of the Lamé parameters and density at a fixed frequency

Abstract

We establish global uniqueness for the inverse boundary value problem of three-dimensional time-harmonic isotropic elasticity with smooth real Lamé parameters satisfying strong convexity and a smooth positive mass density. At a fixed known nonzero frequency for which the corresponding Dirichlet problems are uniquely solvable, the full displacement-to-traction map uniquely determines all three coefficients. The proof first recovers the full boundary Taylor expansions of $λ$, $μ$, and $ρ$, and derives an exact Green--Betti identity for a four-component reduction, with the terms involving $W=ω^2ρ$ and $\nabla W$ retained. Three independent physical polarizations of the rank-three characteristic quotient yield a complete $3\times3$ matrix of polynomial moment identities. A Cauchy transform construction produces an exact relative transport matrix $H$, normalized to $I_3$ in a common exterior region. Its invertibility follows from the determinant identity $\det H=\sqrt{μ_2/μ_1}>0$. Holomorphic dependence on the projective null direction, established by differentiating the transport equation at a fixed physical point, leads to a finite-type rigidity system that determines the Lamé parameters. Finally, products of physical elastic solutions and Fourier uniqueness recover the mass density.