Boundary recovery of smooth isotropic elastic moduli

Source and verification

This article explains family 372 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Static elasticity asks how a material deforms under boundary displacement. Family 372 claims that complete displacement-to-traction measurements determine both smooth isotropic elastic moduli in three dimensions. The proof must recover information about physical displacement fields, rather than assume access to an enlarged auxiliary system.

🏷️ Preliminaries: strain, stress, and traction

For a displacement $u:\Omega \to {\mathbb{R}}^3$, the linearized strain is

$$e(u)=\frac{\nabla u+(\nabla u{)}^T}{2}.$$

The antisymmetric part represents an infinitesimal rotation. An isotropic material has stress

$$\sigma (u)=\lambda (\mathrm{div}u)I+2\mu e(u),$$

where the Lamé coefficients $\lambda ,\mu$ may vary with position. Static equilibrium is $\mathrm{div}\sigma (u)=0$. Prescribed boundary displacement $f$ produces a solution $u_f$, and its boundary traction is $\sigma (u_f)\nu$.

The boundary map can also be defined by the energy pairing

$$\begin{array}{c}⟨{\Lambda }_{\lambda ,\mu }f,h⟩={\int }_{\Omega }\lambda \mathrm{div}{u}_f\mathrm{div}{u}_h\\ +2{\int }_{\Omega }\mu e(u_f):e(u_h).\end{array}$$

The matrix product $A:B$ sums corresponding entries. This weak definition is meaningful for Sobolev boundary traces.

🏷️ A calculation explaining positivity

Decompose a symmetric strain into its trace-free part and its scalar part:

$$e=e_0+\frac{\mathrm{tr}e}{3}I.$$

These parts are orthogonal, so

$$\begin{array}{c}\lambda (\mathrm{tr}e{)}^2+2\mu |e{|}^2\\ =2\mu |e_0{|}^2+\left(\lambda +\frac{2\mu }{3}\right)(\mathrm{tr}e{)}^2.\end{array}$$

Thus $\mu >0$ controls shear, while $3\lambda +2\mu >0$ controls volume change. The assumptions do not require $\lambda >0$ separately. With zero boundary displacement, Korn’s inequality rules out the rigid-motion kernel and gives the coercivity needed for the forward problem.

🏷️ The claimed inverse theorem

Family 372, as stated in the manuscript

Let $\Omega \subset {\mathbb{R}}^3$ be bounded, connected, and smooth. For two pairs of smooth real Lamé coefficients satisfying ${\mu }_j>0$ and $3{\lambda }_j+2{\mu }_j>0$ on the closure, equality of the full static displacement-to-traction maps implies ${\lambda }_1={\lambda }_2$ and ${\mu }_1={\mu }_2$ throughout $\Omega$. (OpenAI, 2026)

The coefficients need not be analytic or close to constants, and agreement near the boundary is not an extra hypothesis. The data are full-boundary, zero-frequency measurements in fixed Euclidean coordinates. This is a uniqueness assertion, not a reconstruction algorithm or a quantitative noise-stability estimate.

🏷️ Physical solutions and complex exponential tests

Boundary determination first recovers the coefficient jets. The manuscript uses them to choose common smooth extensions outside the domain. Equality of displacement and traction then permits an exact solution in one medium to be transferred to the other with agreement in the exterior.

The proof studies rapidly varying complex exponential solutions with phase $e^{\tau \theta \cdot x}$, where $\theta \cdot \theta =0$. This bilinear null condition cancels the leading Laplacian term. Real and imaginary parts of $\theta$ define a plane on which $\theta \cdot \nabla$ behaves like a Cauchy–Riemann operator.

The relevant leading data are a displacement amplitude together with its actual divergence. An auxiliary vector–scalar system is convenient, but not every auxiliary solution necessarily corresponds to a physical displacement. The manuscript therefore tracks the compatibility defect $\mathrm{div}u-d$ and constructs exact physical solutions realizing the required amplitudes. A Carleman estimate—a weighted estimate for the differential operator—controls the correction errors and their derivatives.

🏷️ Comparing transport in all complex directions

After normalization, the leading amplitudes satisfy first-order matrix transport equations. Transferring exact physical solutions supplies a comparison matrix equal to the identity in the exterior. Uniqueness for supported transport equations assembles these local comparisons and controls their dependence on the complex direction.

The source then uses holomorphic dependence in that direction and the shared upper transport equation to force the comparison to be the identity everywhere. Equality of the resulting transport coefficients recovers the elastic moduli.

The important proof obligation is the transfer from measured physical solutions to this family of amplitude equations. Assuming unrestricted auxiliary boundary data would bypass that obligation. The strain decomposition above justifies the forward positivity assumptions; it does not supply the inverse recovery argument.

🏷️ Key steps and proof outlines

Realizing the divergence, as well as the displacement

Write $m=\mu$, $k=\lambda +\mu$, and $\ell =\lambda +2\mu$. The positivity assumptions imply all three are positive. For $E_{\tau }=e^{\tau \theta \cdot x}$, consider a displacement expansion $E_{\tau }{\sum }_{j=0}^N{\tau }^{-j}{a}_j$. Its divergence starts as

$$E_{\tau }^{-1}\mathrm{div}u=\tau \theta \cdot a_0+\mathrm{div}{a}_0+\theta \cdot a_1+O({\tau }^{-1}).$$

Thus the leading scalar amplitude $b_0$ cannot simply be declared to equal $\mathrm{div}{a}_0$. It must satisfy $\theta \cdot a_0=0$ and $\theta \cdot a_1=b_0-\mathrm{div}{a}_0$. More generally the source imposes

$$\theta \cdot a_j=b_{j-1}-\mathrm{div}{a}_{j-1}.$$

Proof outline. At each order, first choose the component of $a_j$ transverse to ${\theta }^{\perp }$ to enforce this identity. The remaining unknown belongs to the three-dimensional space ${\theta }^{\perp }\oplus \mathbb{C}$. The augmented equations give a square first-order transport system on this space; an operator compatibility identity ensures its vector source belongs to ${\theta }^{\perp }$. A planar Cauchy transform solves the inhomogeneous equation after conjugation by a local invertible transport frame. This recursion simultaneously cancels the equation residual and the unwanted divergence terms.

Truncation leaves a physical elasticity residual $O({\tau }^{1-N})$. The source’s Carleman solvability estimate corrects the displacement equation with an error of that size. Taking its conjugated divergence costs one power of $\tau$, leaving $O({\tau }^{2-N})$. Choosing $N\ge 4$ therefore gives strong convergence of both physical amplitudes, with derivative bounds sufficient for transferring them between the two media. The high-order expansion pays for this derivative loss. (OpenAI, 2026)

From exterior agreement to a rigid comparison

Set $g=\nabla \log m$ and change amplitude variables to

$$p=\sqrt{m}a,s=-\frac{k}{2\sqrt{m}}b-\frac{1}{2}g\cdot p.$$

The normalized transport has the particularly restrictive upper equation $D_{\theta }p=\theta s$, where $D_{\theta }=\theta \cdot \nabla$. Transferred physical solutions produce a comparison matrix $G$ between the two transport systems, equal to the identity outside a compact set.

Proof outline. Uniqueness of a compactly supported solution to the homogeneous planar transport equation makes comparisons constructed from different local frames agree. A supported estimate then upgrades their parameter dependence to smoothness. Differentiating in the conjugate complex direction parameter, at fixed physical $x$, gives another homogeneous supported equation, so that derivative vanishes. Thus $G$ depends holomorphically on the null direction.

The source uses a geometric lemma about these direction-dependent polarization spaces: a global holomorphic vector lying in the relevant ${\theta }^{\perp }$ at each direction must vanish, and a suitable quotient is a trivial line. Treat that lemma as an input here. Together with the upper transport equation, it forces the comparison into the form

$$G=\left(\begin{array}{cc}fI& 0\\ A& f\end{array}\right).$$

The upper-left transport equation now gives

$$(D_{\theta }f)I_{{\theta }^{\perp }}=\theta A.$$

The left side has rank two unless $D_{\theta }f=0$, while the right side has rank at most one. Hence $D_{\theta }f=0$ and $A=0$. Null directions span ${\mathbb{C}}^3$, so $f$ is spatially constant; its exterior value is one. This is the final elementary rigidity mechanism after the parameter and polarization lemmas. (OpenAI, 2026)

Closing coefficient recovery by an ordinary differential equation

Equality of the scalar transport entries first yields equality of $k/\ell$, hence of $m/\ell$ and $k/m$. For the remaining difference $\eta =\log m_1-\log m_2$, equality of the restricted vector entries produces

$${\partial }_i{\partial }_j\eta =\sum _a{B}_{ij}^a{\partial }_a\eta +{\delta }_{ij}q,$$

with smooth coefficients $B$ and an initially unknown scalar $q$. One must eliminate that scalar ambiguity too.

Fix $s$ and choose $i\ne s$. Differentiate the diagonal $ii$ equation in direction $s$ and the off-diagonal $si$ equation in direction $i$, and commute third derivatives. This expresses ${\partial }_{s}q$ linearly in $\nabla \eta$ and $q$. Together with the displayed Hessian equation, it gives a homogeneous first-order system for $(\nabla \eta ,q)$. Its restriction to any path is a linear ODE. Since the common exterior extension makes both quantities initially zero, uniqueness propagates zero throughout the connected domain. Then $m_1=m_2$ and the already recovered ratio gives ${\lambda }_1={\lambda }_2$. This last argument requires neither analyticity of the coefficients nor smallness of their variation.

See Also

• Family 365 discusses coordinate and frame ambiguities in related elliptic inverse problems.
• Family 368 addresses smooth approximation of globally injective deformations, a nonlinear elasticity motivation.

🏷️ Manuscripts and verification scope

• Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope page describes uniqueness of both smooth Lamé moduli from the full static boundary map under the stated energy-positivity conditions. It does not provide a reconstruction algorithm. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem.