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Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem
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Abstract
We prove that the full static displacement-to-traction map uniquely determines both real smooth Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and $3\lambda+2\mu\gt 0$ on the closure. This resolves the smooth three-dimensional isotropic elastic Calderón uniqueness problem under these positivity assumptions.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Huaian Diao, Umidjon Durdiev, Unique Determination of the Lamé Parameters in Three Dimensions (arXiv:2610.14741, 10 Oct 2026). Claims the same full-data global uniqueness theorem (smooth λ, μ, μ > 0 and 3λ+2μ > 0, bounded smooth domain in ℝ³) by a four-component Laplace-type reduction, CGO solutions and Cauchy–Plemelj factorization; its Section 1.2 says the authors later became aware of "the related OpenAI preprint" and describe their work as independent.
- Huaian Diao, Umidjon Durdiev, Logarithmic Stability for the Inverse Boundary Problem in Three-Dimensional Isotropic Elasticity (arXiv:2610.24053, 16 Oct 2026). Proves logarithmic interior stability and boundary derivative estimates for both Lamé parameters, and states global uniqueness for C^22 parameters, which it says "relaxes the C∞ regularity assumptions in the independent uniqueness results" including the OpenAI preprint.
Formalization
- OpenAI math release, Lean scope note for family 372: Global uniqueness in smooth isotropic elasticity (lean/docs/372.md, 2026-10-06). The release's own Lean scope note: the formalization covers the uniqueness statement (Theorem 1.1) for both smooth real Lamé moduli under the paper's positivity and full displacement-to-traction data assumptions, via the comparator challenge
ElasticityUniqueness.lean; reconstruction is outside its scope. [family-level]
Related prior literature mentioned by others
- Lu Chen, Yan Jiang, Hongyu Liu, Longyue Tao, On an inverse boundary problem in elastodynamics (arXiv:2609.20134v1, 17 Sep 2026). Proves uniqueness for smooth coefficients when μ is axisymmetric, separable on a product domain, or quasianalytic in one direction; both the paper and the Diao–Durdiev follow-ups cite it as the most recent structured-case result before the general theorem.
Possibly relevant literature (found by the agent, not checked)
- Oleg Imanuvilov, Gunther Uhlmann, Masahiro Yamamoto, On reconstruction of Lamé coefficients from partial Cauchy data in three dimensions (arXiv:1205.4436, 2012; related DOI 10.1088/0266-5611/28/12/125002). Recovers both Lamé coefficients in dimension ≥ 3 from partial Cauchy data when μ is close to a constant; relevant to the paper's history paragraph on near-constant three-dimensional results, and not in its bibliography.
- Elena Beretta, Elisa Francini, Sergio Vessella, Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements (arXiv:1303.2443, 2013). Uniqueness and Lipschitz stability for piecewise constant Lamé parameters in 3D from the Dirichlet-to-Neumann map; a non-smooth special class relevant to the paper's claim of treating "the full smooth class", and not in its bibliography.
- Elena Beretta, Elisa Francini, Antonino Morassi, Edi Rosset and Sergio Vessella, Lipschitz continuous dependence of piecewise constant Lamé coefficients from boundary data: the case of non flat interfaces (arXiv:1406.1899, 2014-06-07). Establishes Lipschitz stability for piecewise constant Lamé coefficients on a known finite partition with C¹,α interfaces, a comparison point for the paper's recovery of both moduli in the general smooth class.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 372
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026,
author = {{OpenAI}},
title = {{Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf}{OAI:Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/372.md. The Lean statements have not been checked by people against the paper's claims.
Family 372: Global uniqueness in smooth isotropic elasticity
Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ3, provided μ > 0 and $3\lambda+2\mu\gt 0$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.
This is the only paper in this family.