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Math-2856 · mathematics · Pde

Full-boundary recovery of both smooth isotropic elastic moduli

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For any bounded connected smooth domain in $R^3$, do identical static isotropic-elasticity traction Dirichlet-to-Neumann maps on $H^{1/2}(boundary;R^3)\to H^{-1/2}(boundary;R^3)$ imply equality of both smooth real Lamé functions lambda and mu throughout the domain, assuming $\mu >0$ and $3\lambda +2\mu >0$?

Short English statement

For any bounded connected smooth domain in $R^3$, do identical static isotropic-elasticity traction Dirichlet-to-Neumann maps on $H^{1/2}(boundary;R^3)\to H^{-1/2}(boundary;R^3)$ imply equality of both smooth real Lamé functions lambda and mu throughout the domain, assuming $\mu >0$ and $3\lambda +2\mu >0$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2856-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2856

Smooth isotropic elasticity uniqueness: Both Lamé coefficients are smooth real functions, with the strict isotropic positivity conditions on the closure. Full traction-map equality implies equality of both functions throughout the domain; no reconstruction or partial-boundary conclusion is added.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Full-boundary recovery of both smooth isotropic elastic moduli

OpenAI's claim

Smooth isotropic elasticity uniqueness: The announced uniqueness statement recovers both smooth Lamé coefficients from full boundary displacement and traction data on a bounded smooth three-dimensional domain when $\mu>0$ and $3\lambda+2\mu>0$. It is a uniqueness assertion under these positivity hypotheses.

Mathematical statement of the claimed result

Mathematical claim 1.
For a bounded connected smooth domain $\Omega\subset\mathbb R^3$, define $e(u)=(\nabla u+\nabla u^{\mathsf T})/2$ and $L_{\lambda,\mu}u=\nabla\cdot[\lambda(\nabla\cdot u)I+2\mu e(u)]$. For two smooth real Lamé pairs on $\overline\Omega$ with $\mu_j>0$ and $3\lambda_j+2\mu_j>0$, equality of the full-boundary displacement-to-traction maps is claimed to imply \[\lambda_1=\lambda_2,\qquad\mu_1=\mu_2\quad\text{throughout }\Omega.\] The maps act $H^{1/2}(\partial\Omega;\mathbb R^3)\to H^{-1/2}(\partial\Omega;\mathbb R^3)$, for solutions of $L_{\lambda,\mu}u=0$. Both elastic moduli may vary, without an additional closeness or constant-modulus hypothesis.

Outline of the argument

Smooth isotropic elasticity uniqueness: Boundary determination supplies matching jets and common exterior coefficient extensions. Equality of displacement and traction permits physical solutions in the two media to be glued across the boundary, including their actual divergences. This exact exterior comparison is propagated through the interior system, identifying both moduli rather than only an auxiliary transformed coefficient.

Formal verification

Family 372 describes Lean coverage for uniqueness of the two smooth positive-isotropic Lamé moduli from the full static traction map.

Smooth isotropic elasticity uniqueness: The manuscript links ElasticityUniqueness.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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