Recover a potential on a general known Riemannian manifold
Let $(M,g)$ be a known compact connected smooth Riemannian manifold of dimension $n \ge 3$ with smooth nonempty boundary. For a real $q \in C^\infty(M)$ such that zero is not a Dirichlet eigenvalue of $-\Delta_g + q$, the Dirichlet-to-Neumann map is $$\Lambda_{g,q} f = \partial_{\nu_g} u|_{\partial M}, \qquad (-\Delta_g + q)u = 0 \ \text{ in } M, \quad u|_{\partial M} = f.$$ Dos Santos Ferreira, Kenig, Salo and Uhlmann proved that $\Lambda_{g,q}$ determines $q$ when $M \subset \mathbb{R} \times M_0$ and $g = dt^2 + g_0$ with $(M_0, g_0)$ a simple manifold. In dimension two a smooth potential is determined by the Dirichlet-to-Neumann map at any fixed frequency (Guillarmou and Tzou; see Ma, Sahoo and Salo). Ma, Sahoo and Salo proved uniqueness at a large fixed frequency $\lambda$, for the operator $-\Delta_g + q - \lambda^2$, on nontrapping manifolds whose geodesic ray transform is stably invertible, for perturbations of $q$ with bounded ratio of Sobolev to $L^2$ norm.
Problem. Must $\Lambda_{g,q_1} = \Lambda_{g,q_2}$ imply $q_1 = q_2$, for every such manifold and every such pair of potentials? The metric is fixed and known, and the map is measured at the single frequency zero.
Source: Report of the BIRS workshop Dirichlet-to-Neumann Maps: Spectral Theory, Inverse Problems and Applications (16w5083, 2016), "Variants of the anisotropic Calderón's problem", for manifolds not of product form; also Ma, Sahoo and Salo (2024), introduction. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, Calderón problem, Dirichlet-to-Neumann map, inverse problems
References
@misc{birs2016dtn,
author = {Levitin, Michael and Nachman, Adrian and Oksanen, Lauri and Polterovich, Iosif},
title = {Dirichlet-to-{N}eumann maps: spectral theory, inverse problems and applications (16w5083)},
howpublished = {Workshop report, Banff International Research Station, Casa Matem{\'a}tica Oaxaca},
year = {2016},
url = {https://www.birs.ca/cmo-workshops/2016/16w5083/report16w5083.pdf},
note = {Section ``Variants of the anisotropic Calder{\'o}n's problem''},
}
@article{ma2024,
author = {Ma, Shiqi and Sahoo, Suman Kumar and Salo, Mikko},
title = {The anisotropic {C}alder{\'o}n problem at large fixed frequency on manifolds with invertible ray transform},
journal = {J. Lond. Math. Soc. (2)},
volume = {110},
number = {4},
year = {2024},
pages = {e13006},
doi = {10.1112/jlms.13006},
eprint = {2207.02623},
archivePrefix = {arXiv},
}
@article{dossantosferreira2009,
author = {Dos Santos Ferreira, David and Kenig, Carlos E. and Salo, Mikko and Uhlmann, Gunther},
title = {Limiting {C}arleman weights and anisotropic inverse problems},
journal = {Invent. Math.},
volume = {178},
number = {1},
year = {2009},
pages = {119--171},
doi = {10.1007/s00222-009-0196-4},
}
@article{guillarmou2011,
author = {Guillarmou, Colin and Tzou, Leo},
title = {Calder{\'o}n inverse problem with partial data on {R}iemann surfaces},
journal = {Duke Math. J.},
volume = {158},
number = {1},
year = {2011},
pages = {83--120},
doi = {10.1215/00127094-1276310},
}
@misc{aim-potential-on-general-manifold,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Recover a potential on a general known Riemannian manifold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/051-potential-on-general-manifold.md},
note = {Entry 051 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove uniqueness of a smooth real scalar potential from its full Dirichlet-to-Neumann map at one energy on an arbitrary known compact, connected, smooth Riemannian manifold of dimension at least three with smooth nonempty boundary. The only spectral assumption is that this energy is not a Dirichlet eigenvalue; negative eigenvalues and sign-changing potentials are allowed. Equal boundary maps first give …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00765, "Recover a potential on a general known Riemannian manifold", clopen.solutions.