A nonlinear Weyl lemma at the minimal p-harmonic integrability threshold
Let $d \ge 3$, $p > 2$ and let $\Omega \subset \mathbb{R}^d$ be open. Consider $u \in W^{1,p-1}_{\mathrm{loc}}(\Omega)$ satisfying $$\int_\Omega |\nabla u|^{p-2} \nabla u \cdot \nabla\varphi \, dx = 0 \qquad (\varphi \in C_c^\infty(\Omega)).$$ The assumption makes the flux $|\nabla u|^{p-2}\nabla u$ locally integrable, so the identity is meaningful, but it does not give finite $p$-energy, and test functions such as $\zeta^p u$ are not available. For $p = 2$, Weyl's lemma makes every such solution harmonic. Iwaniec and Martin proved, and Lewis gave a simpler proof, that the assumption $u \in W^{1,p-\varepsilon}$ for some small $\varepsilon > 0$ suffices for $u$ to be $p$-harmonic.
Problem. Must every such $u$ belong to $W^{1,p}_{\mathrm{loc}}(\Omega)$? Equivalently, must this distributional solution have a representative that is an ordinary weak $p$-harmonic function?
Source: Lindqvist (2017), Notes on the p-Laplace Equation, Section 11, "Very Weak Solutions", for u in W^{1,p-1}(Ω). Stated by AIM — Open Applied Problems for d ≥ 3 and p > 2, with local Sobolev classes.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, p-Laplace equation, very weak solutions
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},
year = {2017},
isbn = {978-951-39-7120-5},
url = {https://jyx.jyu.fi/bitstreams/2a8cbc94-58e9-4e75-94d6-0f1f5bc22ac1/download},
}
@article{iwaniec1993,
author = {Iwaniec, Tadeusz and Martin, Gaven},
title = {Quasiregular mappings in even dimensions},
journal = {Acta Math.},
volume = {170},
number = {1},
year = {1993},
pages = {29--81},
doi = {10.1007/BF02392454},
}
@article{lewis1993,
author = {Lewis, John L.},
title = {On very weak solutions of certain elliptic systems},
journal = {Comm. Partial Differential Equations},
volume = {18},
number = {9-10},
year = {1993},
pages = {1515--1537},
doi = {10.1080/03605309308820984},
}
@misc{aim-very-weak-p-harmonic-energy-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A nonlinear Weyl lemma at the minimal p-harmonic integrability threshold}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/433-very-weak-p-harmonic-energy-regularity.md},
note = {Entry 433 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-00842, "A nonlinear Weyl lemma at the minimal p-harmonic integrability threshold", clopen.solutions.