Result 372, Partial differential equations

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 3λ+2μ>03\lambda+2\mu\gt 0 on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.

Lean formalization Proof

The bigger picture

Why it matters

If a material's elastic response is the same in every direction, can surface measurements reveal its interior stiffness? The manuscript claims that complete static boundary data uniquely identify the two elastic parameters, even when they vary throughout the material.

What changes?

The manuscript reports uniqueness on every bounded, connected three-dimensional domain with smooth boundary. The unknowns are two smooth, real-valued Lamé moduli, lambda and mu, describing an isotropic material's elastic response. It assumes mu is positive and three times lambda plus twice mu is positive throughout the closure, including the boundary. The data give boundary force per unit area for every imposed static boundary displacement. No analyticity, closeness to constant coefficients, or prior knowledge near the boundary is required.

What does that help mathematicians do?

The claimed uniqueness rules out an exact ambiguity: two different admissible pairs of elastic parameters cannot produce identical responses to every static boundary displacement. Researchers could therefore separate the question of whether the parameters are identifiable from the harder operational questions of reconstructing them and controlling errors. Crucially, the conclusion would not depend on already knowing a layer of material near the surface.

Are there practical applications?

The immediate value is foundational for inverse elasticity: it establishes, under the stated assumptions, what ideal boundary measurements can determine about an interior. This is relevant to attempts to infer material properties without accessing the interior directly. The supplied abstract does not provide a reconstruction algorithm, noise guarantees, or a result for finitely many measurements.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

September 24, 2026 24 pages

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 3λ+2μ>03\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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Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/372.md.

Global uniqueness in smooth isotropic elasticity

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves global uniqueness in the three-dimensional static isotropic elasticity inverse problem. On a bounded connected smooth domain, let two pairs of smooth real Lamé moduli satisfy μ>0\mu>0 and 3λ+2μ>03\lambda+2\mu>0 on the closure. If their full displacement-to-traction maps agree, then both Lamé moduli agree throughout the domain. The selected statement is uniqueness; it does not supply a reconstruction algorithm.

Comparator links

Result Comparator statement
Global uniqueness of smooth isotropic Lamé moduli ElasticityUniqueness.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.