Unique continuation for the higher-dimensional p-Laplace equation
Let $n \ge 3$, let $1 < p < \infty$ with $p \ne 2$, and let $\Omega \subset \mathbb{R}^n$ be a connected open set. Consider $u \in W^{1,p}_{\mathrm{loc}}(\Omega)$ with $$\int_\Omega |\nabla u|^{p-2} \nabla u \cdot \nabla \phi \, dx = 0 \qquad \text{for every } \phi \in C_c^\infty(\Omega),$$ a weak solution of the $p$-Laplace equation. For $p = 2$ the solutions are harmonic, hence real analytic. In the plane, $p$-harmonic functions have the strong unique continuation property for every $p > 1$, as proved by Bojarski and Iwaniec, with a gap filled by Manfredi, and by Alessandrini. Granlund and Marola proved that a solution vanishing on a ball vanishes identically, provided a $p$-version of Almgren's frequency function is bounded on every interval of radii on which it is defined.
Problem. If $u = 0$ almost everywhere on some nonempty open subset of $\Omega$, must $u = 0$ almost everywhere in $\Omega$?
Source: Alessandrini (2020), Problem 2.1 (weak unique continuation), who records that Gene Fabes first proposed the problem to him in 1981, in a stronger form; see also Granlund and Marola (2014).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, p-harmonic functions, unique continuation
References
@article{alessandrini2020,
author = {Alessandrini, Giovanni},
title = {A small collection of open problems},
journal = {Rend. Istit. Mat. Univ. Trieste},
volume = {52},
year = {2020},
pages = {591--600},
doi = {10.13137/2464-8728/30921},
eprint = {2003.11441},
archivePrefix = {arXiv},
}
@article{granlund2014,
author = {Granlund, Seppo and Marola, Niko},
title = {On the problem of unique continuation for the {$p$}-{L}aplace equation},
journal = {Nonlinear Anal.},
volume = {101},
year = {2014},
pages = {89--97},
doi = {10.1016/j.na.2014.01.020},
eprint = {1105.1241},
archivePrefix = {arXiv},
}
@incollection{bojarski1987,
author = {Bojarski, Bogdan and Iwaniec, Tadeusz},
title = {{$p$}-harmonic equation and quasiregular mappings},
booktitle = {Partial Differential Equations (Warsaw, 1984)},
series = {Banach Center Publ.},
volume = {19},
publisher = {PWN},
address = {Warsaw},
year = {1987},
pages = {25--38},
doi = {10.4064/-19-1-25-38},
}
@article{manfredi1988,
author = {Manfredi, Juan J.},
title = {{$p$}-harmonic functions in the plane},
journal = {Proc. Amer. Math. Soc.},
volume = {103},
number = {2},
year = {1988},
pages = {473--479},
doi = {10.1090/S0002-9939-1988-0943069-2},
}
@article{alessandrini1987,
author = {Alessandrini, Giovanni},
title = {Critical points of solutions to the {$p$}-{L}aplace equation in dimension two},
journal = {Boll. Un. Mat. Ital. A (7)},
volume = {1},
year = {1987},
pages = {239--246},
}
@misc{aim-p-harmonic-unique-continuation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Unique continuation for the higher-dimensional p-Laplace equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/068-p-harmonic-unique-continuation.md},
note = {Entry 068 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-00544, "Unique continuation for the higher-dimensional p-Laplace equation", clopen.solutions.