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=ω2ρ and ∇ W retained. Three independent physical polarizations of the rank-three characteristic quotient yield a complete 3×3 matrix of polynomial moment identities. A Cauchy transform construction produces an exact relative transport matrix H, normalized to I3 in a common exterior region. Its invertibility follows from the determinant identity det H=√(μ2/μ1)>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.
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BibTeX
@misc{diao2026determination,
title = {{Determination of the Lamé parameters and density at a fixed frequency}},
author = {Huaian Diao and Yang Dong and Umidjon Durdiev and Weisheng Zhou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07642},
eprint = {2611.07642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07642},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Determination of the Lamé parameters and density at a fixed frequency
AU - Huaian Diao
AU - Yang Dong
AU - Umidjon Durdiev
AU - Weisheng Zhou
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07642
UR - https://arxiv.org/abs/2611.07642
AB - 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 ∇ W retained. Three independent physical polarizations of the rank-three characteristic quotient yield a complete 3×3 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=√(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.
ER -