Recover both Lamé moduli from static boundary measurements
Let $\Omega \subset \mathbb{R}^3$ be a bounded connected domain with smooth boundary and outward unit normal $\nu$. For real $\lambda, \mu \in C^\infty(\overline{\Omega})$ with $\mu > 0$ and $3\lambda + 2\mu > 0$ on $\overline{\Omega}$, set $$\varepsilon(u) = \tfrac12\bigl(\nabla u + \nabla u^{\mathsf T}\bigr), \qquad \sigma_{\lambda,\mu}(u) = \lambda (\nabla \cdot u) I + 2\mu\, \varepsilon(u).$$ For each boundary displacement $f \in H^{1/2}(\partial\Omega; \mathbb{R}^3)$ let $u$ solve $\nabla \cdot \sigma_{\lambda,\mu}(u) = 0$ in $\Omega$ with trace $f$, and define the traction map $\Lambda_{\lambda,\mu} f = \sigma_{\lambda,\mu}(u)\nu$. Nakamura and Uhlmann proved that $\Lambda_{\lambda,\mu}$ determines the full Taylor series of $\lambda$ and $\mu$ at the boundary. Nakamura and Uhlmann, and Eskin and Ralston, proved that it determines both moduli when $\|\nabla \mu_j\|_{C^{k-1}}$ is sufficiently small for $k$ large, relative to bounds on $\lambda_j$, $\mu_j$ and $\mu_j^{-1}$ in $C^k$. In two dimensions, Imanuvilov and Yamamoto proved global uniqueness for the isotropic Lamé system without smallness assumptions.
Problem. For two such coefficient pairs $(\lambda_1, \mu_1)$ and $(\lambda_2, \mu_2)$, does $\Lambda_{\lambda_1,\mu_1} = \Lambda_{\lambda_2,\mu_2}$ imply $\lambda_1 = \lambda_2$ and $\mu_1 = \mu_2$ in $\Omega$? Both coefficients are arbitrary smooth functions.
Source: The inverse boundary problem for isotropic elasticity of Nakamura and Uhlmann (1994); Eskin and Ralston (2002, Introduction) describe uniqueness as widely conjectured. Stated in this form, in three dimensions for smooth moduli, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, inverse problems, Lamé parameters, linear elasticity
References
@article{nakamura1994,
author = {Nakamura, Gen and Uhlmann, Gunther},
title = {Global uniqueness for an inverse boundary problem arising in elasticity},
journal = {Invent. Math.},
volume = {118},
number = {1},
year = {1994},
pages = {457--474},
doi = {10.1007/BF01231541},
}
@article{nakamura2003,
author = {Nakamura, Gen and Uhlmann, Gunther},
title = {Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity},
journal = {Invent. Math.},
volume = {152},
number = {1},
year = {2003},
pages = {205--207},
doi = {10.1007/s00222-002-0276-1},
}
@article{nakamura1995,
author = {Nakamura, Gen and Uhlmann, Gunther},
title = {Inverse problems at the boundary for an elastic medium},
journal = {SIAM J. Math. Anal.},
volume = {26},
number = {2},
year = {1995},
pages = {263--279},
doi = {10.1137/S0036141093247494},
}
@article{eskin2002,
author = {Eskin, Gregory and Ralston, James},
title = {On the inverse boundary value problem for linear isotropic elasticity},
journal = {Inverse Problems},
volume = {18},
number = {3},
year = {2002},
pages = {907--921},
doi = {10.1088/0266-5611/18/3/324},
}
@article{imanuvilov2015,
author = {Imanuvilov, Oleg Yu. and Yamamoto, Masahiro},
title = {Global uniqueness in inverse boundary value problems for the {N}avier--{S}tokes equations and {L}am{\'e} system in two dimensions},
journal = {Inverse Problems},
volume = {31},
number = {3},
year = {2015},
pages = {035004},
doi = {10.1088/0266-5611/31/3/035004},
eprint = {1309.1694},
archivePrefix = {arXiv},
}
@misc{aim-static-lame-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Recover both Lamé moduli from static boundary measurements}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/050-static-lame-uniqueness.md},
note = {Entry 050 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove that the full static displacement-to-traction map uniquely determines both real smooth Lamé moduli on every bounded connected smooth domain in $\R^3$, provided $\mu>0$ and $3\lambda+2\mu>0$ on the closure. This resolves the smooth three-dimensional isotropic elastic Calderón uniqueness problem under these positivity assumptions.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00721, "Recover both Lamé moduli from static boundary measurements", clopen.solutions.