Global conductivity uniqueness for the p-Laplacian in higher dimensions
Let $n \ge 3$, let $1 < p < \infty$ with $p \ne 2$, and let $\Omega \subset \mathbb{R}^n$ be a bounded connected domain with smooth boundary. For a positive $\gamma \in C^\infty(\overline\Omega)$ and boundary data $f$, let $u$ solve $$\nabla \cdot \bigl(\gamma |\nabla u|^{p-2} \nabla u\bigr) = 0 \ \text{ in } \Omega, \qquad u|_{\partial\Omega} = f,$$ and let $\Lambda_{\gamma,p} f = \gamma |\nabla u|^{p-2} \partial_\nu u|_{\partial\Omega}$ be the nonlinear Dirichlet-to-Neumann map, interpreted weakly on the trace space of $W^{1,p}(\Omega)$. Salo and Zhong proved that $\Lambda_{\gamma,p}$ determines $\gamma$ on $\partial\Omega$, and Brander that it determines the gradient of $\gamma$ there. Guo, Kar and Salo proved that $\Lambda_{\gamma_1,p} = \Lambda_{\gamma_2,p}$ implies $\gamma_1 = \gamma_2$ when $\gamma_1 \ge \gamma_2$, in the plane, and in every dimension when moreover $\gamma_2$ is close to a constant. Cârstea and Feizmohammadi proved uniqueness for all smooth positive weights in the plane and, for $n \ge 3$, for two weights both constant along a common direction, and for real-analytic weights with a small derivative in one direction when $\partial\Omega$ is flat near some point.
Problem. Let $\gamma_1, \gamma_2 \in C^\infty(\overline\Omega)$ be positive. Does $\Lambda_{\gamma_1,p} f = \Lambda_{\gamma_2,p} f$ for all boundary data $f$ imply $\gamma_1 = \gamma_2$, for every such $n$, $p$ and $\Omega$? No ordering, smallness, analyticity or invariance in one direction is assumed.
Source: Salo and Zhong (2012), who introduced the inverse problem for the weighted p-Laplace equation (the p-Calderón problem) and proved boundary determination; recorded as open without monotonicity assumptions by Kar, Railo and Zimmermann (2023), §2.3.1. Stated in this form, for smooth weights in dimensions n ≥ 3, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, Calderón problem, Dirichlet-to-Neumann maps, p-Laplace equations
References
@article{salo2012,
author = {Salo, Mikko and Zhong, Xiao},
title = {An inverse problem for the $p$-{L}aplacian: boundary determination},
journal = {SIAM J. Math. Anal.},
volume = {44},
number = {4},
year = {2012},
pages = {2474--2495},
doi = {10.1137/110838224},
eprint = {1106.4229},
archivePrefix = {arXiv},
}
@article{kar2023,
author = {Kar, Manas and Railo, Jesse and Zimmermann, Philipp},
title = {The fractional $p$-biharmonic systems: optimal {P}oincar{\'e} constants, unique continuation and inverse problems},
journal = {Calc. Var. Partial Differential Equations},
volume = {62},
number = {4},
year = {2023},
pages = {130},
doi = {10.1007/s00526-023-02468-9},
eprint = {2208.09528},
archivePrefix = {arXiv},
}
@article{brander2016,
author = {Brander, Tommi},
title = {Calder{\'o}n problem for the $p$-{L}aplacian: first order derivative of conductivity on the boundary},
journal = {Proc. Amer. Math. Soc.},
volume = {144},
number = {1},
year = {2016},
pages = {177--189},
doi = {10.1090/proc/12681},
eprint = {1403.0428},
archivePrefix = {arXiv},
}
@article{guo2016,
author = {Guo, Chang-Yu and Kar, Manas and Salo, Mikko},
title = {Inverse problems for $p$-{L}aplace type equations under monotonicity assumptions},
journal = {Rend. Istit. Mat. Univ. Trieste},
volume = {48},
year = {2016},
pages = {79--99},
doi = {10.13137/2464-8728/13152},
eprint = {1602.02591},
archivePrefix = {arXiv},
}
@article{carstea2025,
author = {C{\^a}rstea, C{\u{a}}t{\u{a}}lin I. and Feizmohammadi, Ali},
title = {Two uniqueness results in the inverse boundary value problem for the weighted $p$-{L}aplace equation},
journal = {Forum Math. Sigma},
volume = {13},
year = {2025},
pages = {e147},
doi = {10.1017/fms.2025.10095},
eprint = {2405.04123},
archivePrefix = {arXiv},
}
@misc{aim-nonlinear-p-calderon,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global conductivity uniqueness for the p-Laplacian in higher dimensions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/049-nonlinear-p-calderon.md},
note = {Entry 049 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00539, "Global conductivity uniqueness for the p-Laplacian in higher dimensions", clopen.solutions.