Strong comparison for ordered p-harmonic fields
Let $d \ge 3$, $1 < p < \infty$, $p \ne 2$, and let $\Omega \subset \mathbb{R}^d$ be a connected open set. Let $u, v \in W^{1,p}_{\mathrm{loc}}(\Omega) \cap C(\Omega)$ be weak solutions of $\operatorname{div}(|\nabla u|^{p-2} \nabla u) = \operatorname{div}(|\nabla v|^{p-2} \nabla v) = 0$. For $p = 2$ the question below is answered by the strong maximum principle applied to $v - u$; for $p \ne 2$ the difference is not $p$-harmonic. When one of the two functions is constant, it is the strong maximum principle, which holds for every $p$. Manfredi proved the strong comparison principle in the plane (1988). Tolksdorf (1983), and Guedda and Véron (1989), proved it when the gradient of one of the functions has no zeros, and Damascelli proved it on each connected component of the complement of the set where both gradients vanish (1998). Björn, Björn, Eriksson-Bique and Zhou showed that it fails for $p$-harmonic functions on $\mathbb{R}^2$ with a suitable weight and a bi-Lipschitz equivalent metric (2026).
Problem. If $u \le v$ on $\Omega$ and $u(x_0) = v(x_0)$ at one point $x_0 \in \Omega$, must $u \equiv v$? The common contact point may be a critical point of both functions.
Source: Listed as an open problem by Lindqvist (2017), §11, who credits the planar case to Manfredi (1988). Stated by AIM — Open Applied Problems for d ≥ 3 and p ≠ 2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, comparison principles, p-harmonic functions
References
@techreport{lindqvist2017,
author = {Lindqvist, Peter},
title = {Notes on the $p$-{L}aplace equation},
edition = {Second},
institution = {University of Jyv{\"a}skyl{\"a}, Department of Mathematics and Statistics},
type = {Report},
number = {161},
address = {Jyv{\"a}skyl{\"a}},
year = {2017},
isbn = {978-951-39-7120-5},
}
@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{tolksdorf1983,
author = {Tolksdorf, Peter},
title = {On the {D}irichlet problem for quasilinear equations in domains with conical boundary points},
journal = {Comm. Partial Differential Equations},
volume = {8},
number = {7},
year = {1983},
pages = {773--817},
doi = {10.1080/03605308308820285},
}
@article{guedda1989,
author = {Guedda, Mohammed and V{\'e}ron, Laurent},
title = {Quasilinear elliptic equations involving critical {S}obolev exponents},
journal = {Nonlinear Anal.},
volume = {13},
number = {8},
year = {1989},
pages = {879--902},
doi = {10.1016/0362-546X(89)90020-5},
}
@article{damascelli1998,
author = {Damascelli, Lucio},
title = {Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {15},
number = {4},
year = {1998},
pages = {493--516},
doi = {10.1016/S0294-1449(98)80032-2},
}
@article{bjorn2026,
author = {Bj{\"o}rn, Anders and Bj{\"o}rn, Jana and Eriksson-Bique, Sylvester and Zhou, Xiaodan},
title = {Uniqueness and nonuniqueness of $p$-harmonic {G}reen functions on weighted {$\mathbf{R}^n$} and metric spaces},
journal = {J. Differential Equations},
volume = {453},
year = {2026},
pages = {113932},
doi = {10.1016/j.jde.2025.113932},
eprint = {2505.19074},
archivePrefix = {arXiv},
}
@misc{ye2026,
author = {Ye, Dong and Zhang, Weimin},
title = {Comparison principles and symmetry for subquadratic fractional $p$-{L}aplacian equations},
year = {2026},
eprint = {2609.02626},
archivePrefix = {arXiv},
}
@misc{aim-p-harmonic-strong-comparison,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Strong comparison for ordered p-harmonic fields}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/427-p-harmonic-strong-comparison.md},
note = {Entry 427 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00630, "Strong comparison for ordered p-harmonic fields", clopen.solutions.