Calderón uniqueness for bounded measurable conductivities
Let $\Omega \subset \mathbb{R}^3$ be a bounded connected domain with smooth boundary. For a real scalar conductivity $\gamma \in L^\infty(\Omega)$ with $0 < c \le \gamma \le C < \infty$ almost everywhere, for some constants $c$ and $C$, and for $f \in H^{1/2}(\partial\Omega)$, let $u_f \in H^1(\Omega)$ be the weak solution of $$\nabla \cdot (\gamma \nabla u_f) = 0 \ \text{ in } \Omega, \qquad u_f|_{\partial\Omega} = f.$$ The Dirichlet-to-Neumann map $\Lambda_\gamma : H^{1/2}(\partial\Omega) \to H^{-1/2}(\partial\Omega)$ is defined weakly by $\langle \Lambda_\gamma f, h \rangle = \int_\Omega \gamma \nabla u_f \cdot \nabla v_h \, dx$, where $v_h \in H^1(\Omega)$ is any extension of $h \in H^{1/2}(\partial\Omega)$. Sylvester and Uhlmann proved that $\Lambda_\gamma$ determines $\gamma$ for smooth conductivities in dimension $n \ge 3$. Haberman extended this to uniformly elliptic conductivities in $W^{1,n}$ for $n = 3, 4$, and Caro and Rogers to Lipschitz conductivities in every dimension $n \ge 3$; Caro, García-Ferrero and Rogers gave a reconstruction for Lipschitz conductivities. In the plane, Astala and Päivärinta proved that $\Lambda_\gamma$ determines every such $\gamma \in L^\infty$ on bounded simply connected domains.
Problem. Does $\Lambda_{\gamma_1} = \Lambda_{\gamma_2}$ imply $\gamma_1 = \gamma_2$ almost everywhere in $\Omega$, for every pair of such conductivities? No continuity, differentiability or known partition of $\Omega$ is assumed.
Source: Calderón (1980), who asked whether boundary measurements determine a bounded positive conductivity; see Uhlmann (2013). Stated in this form, in three dimensions with no regularity, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Calderón problem, Dirichlet-to-Neumann map, electrical impedance tomography
References
@incollection{calderon1980,
author = {Calder{\'o}n, Alberto P.},
title = {On an inverse boundary value problem},
booktitle = {Seminar on Numerical Analysis and its Applications to Continuum Physics},
publisher = {Sociedade Brasileira de Matem{\'a}tica},
address = {Rio de Janeiro},
year = {1980},
pages = {65--73},
note = {Reprinted in Comput. Appl. Math. 25 (2006), no. 2--3, 133--138, doi:10.1590/S0101-82052006000200002},
}
@article{uhlmann2013,
author = {Uhlmann, Gunther},
title = {30 years of {C}alder{\'o}n's problem},
journal = {S{\'e}minaire Laurent Schwartz --- EDP et applications (2012--2013)},
year = {2013},
pages = {Expos{\'e} no. XIII, 1--25},
doi = {10.5802/slsedp.40},
}
@article{sylvester1987,
author = {Sylvester, John and Uhlmann, Gunther},
title = {A global uniqueness theorem for an inverse boundary value problem},
journal = {Ann. of Math. (2)},
volume = {125},
number = {1},
year = {1987},
pages = {153--169},
doi = {10.2307/1971291},
}
@article{haberman2015,
author = {Haberman, Boaz},
title = {Uniqueness in {C}alder{\'o}n's problem for conductivities with unbounded gradient},
journal = {Comm. Math. Phys.},
volume = {340},
number = {2},
year = {2015},
pages = {639--659},
doi = {10.1007/s00220-015-2460-3},
eprint = {1410.2201},
archivePrefix = {arXiv},
}
@article{caro2016,
author = {Caro, Pedro and Rogers, Keith M.},
title = {Global uniqueness for the {C}alder{\'o}n problem with {L}ipschitz conductivities},
journal = {Forum Math. Pi},
volume = {4},
year = {2016},
pages = {e2},
doi = {10.1017/fmp.2015.9},
eprint = {1411.8001},
archivePrefix = {arXiv},
}
@article{caro2025,
author = {Caro, Pedro and Garc{\'i}a-Ferrero, Mar{\'i}a {\'A}ngeles and Rogers, Keith M.},
title = {Reconstruction for the {C}alder{\'o}n problem with {L}ipschitz conductivities},
journal = {Anal. PDE},
volume = {18},
number = {8},
year = {2025},
pages = {2033--2060},
doi = {10.2140/apde.2025.18.2033},
eprint = {2401.06120},
archivePrefix = {arXiv},
}
@article{astala2006,
author = {Astala, Kari and P{\"a}iv{\"a}rinta, Lassi},
title = {Calder{\'o}n's inverse conductivity problem in the plane},
journal = {Ann. of Math. (2)},
volume = {163},
number = {1},
year = {2006},
pages = {265--299},
doi = {10.4007/annals.2006.163.265},
}
@misc{aim-rough-conductivity-calderon,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Calderón uniqueness for bounded measurable conductivities}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/046-rough-conductivity-calderon.md},
note = {Entry 046 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We construct two distinct uniformly positive bounded measurable scalar conductivities on a ball in $\R^3$ with the same full Dirichlet-to-Neumann operator. Both conductivities equal one near the boundary. This gives nonuniqueness in the scalar Calderón problem at bounded measurable regularity.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00712, "Calderón uniqueness for bounded measurable conductivities", clopen.solutions.