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 $L^\infty(Ω)$ 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 $s\ge2$, we also obtain boundary stability estimates with exponents $(s-q)/s$ for all Cartesian derivatives of orders $0\le q<s$, 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.