Second-order Sobolev regularity of p-harmonic functions beyond the Cordes range
A weak solution of the $p$-Laplace equation $\operatorname{div}(|\nabla u|^{p-2}\nabla u) = 0$ on an open set $\Omega \subset \mathbb{R}^n$ is a $p$-harmonic function. For $p \ge 2$ the field $|\nabla u|^{(p-2)/2}\nabla u$ lies in $W^{1,2}_{\mathrm{loc}}$, and the gradient $\nabla u$ is locally Hölder continuous. Manfredi and Weitsman proved $u \in W^{2,2}_{\mathrm{loc}}(\Omega)$, for $n \ge 3$, in the range $1 < p < 3 + 2/(n-2)$ where the Cordes condition holds; for $n = 3$ this is $1 < p < 5$.
Problem. For every $p \ge 5$, every open $\Omega \subset \mathbb{R}^3$, and every weak solution $u \in W^{1,p}_{\mathrm{loc}}(\Omega)$ of $\operatorname{div}(|\nabla u|^{p-2}\nabla u) = 0$, must $u \in W^{2,2}_{\mathrm{loc}}(\Omega)$? That is, every distributional second derivative of $u$ is to lie locally in $L^2$, including near critical points where $\nabla u = 0$.
Source: Lindqvist (2017), Section 11, “Second Derivatives”, for p > 2 in every dimension. Stated for three dimensions beyond the Cordes range by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, degenerate elliptic equations, p-Laplace equation, Sobolev regularity
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},
}
@article{manfredi1988,
author = {Manfredi, Juan J. and Weitsman, Allen},
title = {On the {F}atou theorem for $p$-harmonic functions},
journal = {Comm. Partial Differential Equations},
volume = {13},
number = {6},
year = {1988},
pages = {651--668},
doi = {10.1080/03605308808820556},
}
@misc{aim-p-harmonic-second-sobolev-derivatives,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Second-order Sobolev regularity of p-harmonic functions beyond the Cordes range}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/434-p-harmonic-second-sobolev-derivatives.md},
note = {Entry 434 at commit a153f46, 2026-10-13},
}
Resolution claims
For every finite $p\geq5$, a weak $p$-harmonic function on an arbitrary open subset of $\R^3$ belongs to $W^{2,2}_{\loc}$. The conclusion concerns the full unweighted distributional Hessian and includes neighborhoods of critical points without assumptions on the critical set. We prove an estimate for smooth regularized solutions in which the Hessian energy is divided by the actual gradient norm. Gaussian curvature …
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-00763, "Second-order Sobolev regularity of p-harmonic functions beyond the Cordes range", clopen.solutions.