Logarithmic Stability for Three-Dimensional Isotropic Elasticity

This manuscript is a preliminary research paper intended for public preprint circulation and has not undergone peer review. The authors are still checking the mathematical arguments and do not guarantee the correctness or completeness of any statement, proof, or estimate in this version. Errors, omissions, or gaps may remain; results and arguments may be corrected, substantially revised, or withdrawn in later versions. Readers citing this work should identify the specific version and date.  We study simultaneous determination of the Lam\'e parameters in three-dimensional static isotropic elasticity from the physical displacement-to-traction Dirichlet-to-Neumann map. Our principal result is a single-logarithmic stability estimate for pairs of strongly convex interior coefficients in a uniformly bounded $C^{2m+14}$ class, $m\ge4$, without prescribed exterior coefficients or boundary matching; in particular, $C^{22}$ is sufficient. For an exponent parameter $\gamma_{*,\mathrm{ext}}\in(0,1)$ determined by fixed extension geometry, the estimate holds for every $0<\alpha<\gamma_{*,\mathrm{ext}}(m-3)/(28m)$. The proof combines an auxiliary common-exterior stability theorem obtained from CGO-weighted matrix integral identities and complex parallel transport with an independent quantitative stability theorem for boundary derivatives, $\mathcal J_q\le C\delta^{(s-q)/s}$ for $0\le q<s$, and a uniformly bounded scale-dependent exterior extension with uniform regularity.