Global uniqueness in smooth isotropic elasticity

Proves that boundary push-and-measure data fix both stiffness values everywhere inside a smooth 3D elastic body.

1 paper September 24, 2026 Explained by AI

Imagine a solid whose stiffness changes from place to place inside. You may push its surface in any way and measure the force it pushes back with. The question is whether all such surface data fix the stiffness everywhere inside. The paper shows yes for smooth materials that behave the same in every direction. Earlier proofs needed the stiffness to be nearly constant or very special.

First step

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Significance

Major

It settles the three-dimensional elastic version of Calderón's problem for smooth isotropic materials. An earlier global claim had needed correction and a smallness assumption.

Trust

Formal proof listed

Field

Partial differential equations

1 paper

The explanation

Start with two springs joined end to end. One has stiffness 2 and the other has stiffness 6. Pull the free ends and measure the force. The pair acts like one spring of stiffness 1/(1/2 + 1/6) = 1.5. Swap the order and you get 1.5 again. So measurements at the two ends cannot tell you where the stiff spring is. The question is whether a full three-dimensional body hides its inside in the same way.

An isotropic elastic material resists in the same way in every direction. Two numbers, the Lamé moduli λ and μ, describe it at each point. The shear modulus μ measures resistance to sliding layers. The bulk modulus λ + 2μ/3 measures resistance to squeezingSource 1. Take μ = 80 and λ = 120 in gigapascals, roughly steel. The paper asks for μ > 0 and 3λ + 2μ > 0. Here 3λ + 2μ = 520. A material that grows fatter when stretched can have λ < 0. With μ = 200 and λ = −100, the test still gives 100 > 0.

The data are the displacement-to-traction map. You choose how to move every point of the surface. The map returns the force per area, called traction, that the surface then pushes back with. Calderón posed the matching question for electrical conductivitySource 2. For elasticity, a first global claim by Nakamura and Uhlmann was later correctedSource 3. The corrected version needed μ to vary only slightlySource 4. In two dimensions, Imanuvilov and Yamamoto proved full uniquenessSource 5.

The paper claims that in three dimensions the full map fixes both λ and μ everywhereSource 6. The moduli need not be analytic, close to constant, or known near the boundarySource 7. Analytic means given everywhere by convergent power series. The obstacle was that a known reduction to simpler fields loses information. Its map to physical displacements is not one-to-oneSource 8. So the paper works with real displacements and keeps their actual divergence, the local rate of volume changeSource 9. Equal boundary data let it glue a solution in one body to a solution in the other across the surfaceSource 10. Special exponential test solutions then compare the two bodies point by point.

How sure is this? Lean, a computer proof checker, has a formal statement of the uniqueness theorem. So the paper proves it, as far as the Lean statement matches the paper. Lean covers uniqueness only, not any way to recover the moduli. No specialist review of the paper is published.

What it doesn't show. This does not give a method or algorithm to compute the stiffness from the data. It does not treat rough materials or materials with direction-dependent stiffness.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

A major result with a Lean statement of the main theorem. It gives uniqueness only, with no reconstruction method, and no specialist review is published.

Words used

Isotropic: Behaving the same in every direction at a given point.
Lamé moduli: The two numbers λ and μ that describe the stiffness of an isotropic elastic material at a point.
Shear modulus: The modulus μ, which measures resistance to layers sliding past each other.
Bulk modulus: The value λ + 2μ/3, which measures resistance to being squeezed in all directions.
Traction: The force per unit area that the surface of a body pushes back with.
Displacement-to-traction map: The rule that sends each chosen surface movement to the traction the body returns.
Divergence: The local rate at which a displacement field changes volume.
Analytic: Given everywhere by a convergent power series, a strong smoothness condition.

The original paper

Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem (PDF, opens in a new tab)
September 24, 2026

Lean notes

All quoted lines (10)

Here $\mu$ is the shear modulus and $\lambda+2\mu/3$ is the bulk modulus.
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text
The scalar conductivity problem posed by Calder\'on and the complex geometric optics uniqueness theorem of Sylvester and Uhlmann are important antecedents, but elasticity couples two moduli through vector displacements.
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text
Their original global claim was corrected in.
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text
Its corrected theorem proves uniqueness when both $\|\nabla\mu_j\|_{C^m}$ are sufficiently small.
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text
In two dimensions, Imanuvilov and Yamamoto proved global uniqueness for smooth coefficients satisfying $\mu>0$ and $\lambda+\mu>0$ , without a smallness assumption.
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text
We prove that the full static displacement-to-traction map uniquely determines both real smooth Lam\'e moduli on every bounded connected smooth domain in $\R^3$ , provided $\mu>0$ and $3\lambda+2\mu>0$ on the closure.
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text
The coefficients need not be analytic or close to constants, and their agreement near the boundary is not an additional hypothesis.
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text
A direct inspection of the differential map from their auxiliary vector–scalar fields to physical displacement shows that it is noninjective.
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text
The proof here transfers exact physical solutions while retaining their actual divergence.
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text
Given a solution in the first medium, equality of displacement and traction then lets us glue a solution in the second medium to it across $\partial\Omega$ .
Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem Back to text

Other sources

openai/math, Lean scope note for family 372 The formalization proves global uniqueness in the three-dimensional static isotropic elasticity inverse problem.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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