Logarithmic Stability for the Inverse Boundary Problem in Three-Dimensional Isotropic Elasticity
Abstract
We establish logarithmic stability for the simultaneous recovery of both Lamé parameters in three-dimensional static isotropic elasticity from the full Dirichlet-to-Neumann map. The L∞(Ω) estimate holds uniformly on every fixed bounded C22 class of uniformly strongly convex coefficients, without prescribed exterior coefficients, agreement of boundary values or derivatives, or smallness assumptions on material gradients. For every integer s≥2, we also obtain boundary stability estimates with exponents (s−q)/s for all Cartesian derivatives of orders 0≤ q<s, where the measurement error δ is the operator-norm discrepancy between the Dirichlet-to-Neumann maps. These exponents are sharp on admissible classes containing a constant material satisfying the a priori inequalities strictly. On such classes, no uniform modulus of continuity exists for boundary derivatives of order s. The proof combines an exact four-component reduction with finite-regularity complex geometrical optics constructions, while deriving complete matrix-valued integral identities from the Betti identity for elastic displacements. An amplitude annihilated by the leading physical adjoint lift produces, at the first nonvanishing order, the third polarization in a rank-three characteristic quotient. Holomorphically weighted matrix moments, Cauchy--Plemelj factorization, and a fixed holomorphic projection then yield quantitative control of the relative transport and recovery of both coefficients. Boundary stability estimates and paired extensions transfer the analysis from an auxiliary common-exterior setting to the original measurement domain. The exact-data argument also establishes global uniqueness for C22 Lamé parameters.
Related
- Tensor-Train Isogeometric Analysis for Large-Scale Three-Dimensional Linear Elasticity
- Global Solution, Decay, and Blow-Up of a Viscoelastic Kirchhoff System with Logarithmic Nonlinearity and Acoustic Fractional Delay Boundary Conditions
- One-dimensional second-gradient elasticity: pantograph-inspired energies and solitary waves
- Logarithmic Lefschetz fixed point formulae and resonant boundary indices
- Boundary layer analysis for the 2D Chemotaxis-Navier-Stokes system with logarithmic sensitivity, Part II. viscous vanishing limit
BibTeX
@misc{diao2026logarithmic,
title = {{Logarithmic Stability for the Inverse Boundary Problem in Three-Dimensional Isotropic Elasticity}},
author = {Huaian Diao and Umidjon Durdiev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24053},
eprint = {2610.24053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24053},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Logarithmic Stability for the Inverse Boundary Problem in Three-Dimensional Isotropic Elasticity
AU - Huaian Diao
AU - Umidjon Durdiev
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24053
UR - https://arxiv.org/abs/2610.24053
AB - We establish logarithmic stability for the simultaneous recovery of both Lamé parameters in three-dimensional static isotropic elasticity from the full Dirichlet-to-Neumann map. The L^∞(Ω) estimate holds uniformly on every fixed bounded C^22 class of uniformly strongly convex coefficients, without prescribed exterior coefficients, agreement of boundary values or derivatives, or smallness assumptions on material gradients. For every integer s≥2, we also obtain boundary stability estimates with exponents (s−q)/s for all Cartesian derivatives of orders 0≤ q<s, where the measurement error δ is the operator-norm discrepancy between the Dirichlet-to-Neumann maps. These exponents are sharp on admissible classes containing a constant material satisfying the a priori inequalities strictly. On such classes, no uniform modulus of continuity exists for boundary derivatives of order s. The proof combines an exact four-component reduction with finite-regularity complex geometrical optics constructions, while deriving complete matrix-valued integral identities from the Betti identity for elastic displacements. An amplitude annihilated by the leading physical adjoint lift produces, at the first nonvanishing order, the third polarization in a rank-three characteristic quotient. Holomorphically weighted matrix moments, Cauchy--Plemelj factorization, and a fixed holomorphic projection then yield quantitative control of the relative transport and recovery of both coefficients. Boundary stability estimates and paired extensions transfer the analysis from an auxiliary common-exterior setting to the original measurement domain. The exact-data argument also establishes global uniqueness for C^22 Lamé parameters.
ER -