OP-00664 Solution claimed

Calderón uniqueness from an arbitrary local boundary patch

Let $n \ge 3$, let $\Omega \subset \mathbb{R}^n$ be a bounded connected domain with smooth boundary, and let $\Gamma$ be a nonempty relatively open subset of $\partial\Omega$. For a positive $\gamma \in C^\infty(\overline\Omega)$ and $f \in C_c^\infty(\Gamma)$, extended by zero to $\partial\Omega$, let $u_f$ solve $$\nabla \cdot (\gamma \nabla u_f) = 0 \ \text{ in } \Omega, \qquad u_f|_{\partial\Omega} = f,$$ and let $\Lambda^\Gamma_\gamma f = \gamma\, \partial_\nu u_f|_\Gamma$, with $\nu$ the outward unit normal: voltages are imposed and currents measured on $\Gamma$ only. Sylvester and Uhlmann proved that for $\Gamma = \partial\Omega$ the map $\Lambda^\Gamma_\gamma$ determines $\gamma$. For proper subsets $\Gamma$, uniqueness is known when $\partial\Omega \setminus \Gamma$ is part of a hyperplane or of a sphere (Isakov), and Kenig, Sjöstrand and Uhlmann proved it with voltages on a possibly very small set and currents measured on a slightly larger set than its complement.

Problem. Let $\gamma_1, \gamma_2 \in C^\infty(\overline\Omega)$ be positive. Does $\Lambda^\Gamma_{\gamma_1} = \Lambda^\Gamma_{\gamma_2}$ imply $\gamma_1 = \gamma_2$ throughout $\Omega$, for every such $\Omega$ and $\Gamma$? No condition is imposed on the shape of the inaccessible boundary, and the conductivities need not be known near it.

Source: Kenig and Salo (2014), Section 8, Question 1, for bounded potentials of the Schrödinger equation; stated for smooth conductivities by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: AP — Analysis of PDEs
Keywords: AIM Open Applied Problems, Calderón problem, inverse conductivity problem, partial data

References

@incollection{kenig2014,
  author        = {Kenig, Carlos and Salo, Mikko},
  title         = {Recent progress in the {C}alder{\'o}n problem with partial data},
  booktitle     = {Inverse Problems and Applications},
  series        = {Contemp. Math.},
  volume        = {615},
  publisher     = {Amer. Math. Soc.},
  address       = {Providence, RI},
  year          = {2014},
  pages         = {193--222},
  doi           = {10.1090/conm/615/12245},
  eprint        = {1302.4218},
  archivePrefix = {arXiv},
}
@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{kenig2007,
  author  = {Kenig, Carlos E. and Sj{\"o}strand, Johannes and Uhlmann, Gunther},
  title   = {The {C}alder{\'o}n problem with partial data},
  journal = {Ann. of Math. (2)},
  volume  = {165},
  number  = {2},
  year    = {2007},
  pages   = {567--591},
  doi     = {10.4007/annals.2007.165.567},
}
@article{isakov2007,
  author  = {Isakov, Victor},
  title   = {On uniqueness in the inverse conductivity problem with local data},
  journal = {Inverse Probl. Imaging},
  volume  = {1},
  number  = {1},
  year    = {2007},
  pages   = {95--105},
  doi     = {10.3934/ipi.2007.1.95},
}
@misc{aim-arbitrary-local-calderon-data,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Calderón uniqueness from an arbitrary local boundary patch}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/048-arbitrary-local-calderon-data.md},
  note         = {Entry 048 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Smooth anisotropic uniqueness in the Calderón problem from one boundary patch v1 GitHub ↗

OpenAI

We resolve the smooth anisotropic Calderón uniqueness problem with arbitrary same-patch measurements. A smooth Riemannian metric on a compact connected manifold of dimension at least three is determined, up to a diffeomorphism fixing the measured patch, by its zero-frequency Dirichlet-to-Neumann energy form with both input and observation on any nonempty open boundary patch.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

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-00664, "Calderón uniqueness from an arbitrary local boundary patch", clopen.solutions.