Sharp gradient regularity for higher-dimensional p-Poisson equations
Let $d \ge 3$, $p > 2$, and let $B_r$ be the centred ball of radius $r$ in $\mathbb{R}^d$. Consider weak solutions $u \in W^{1,p}(B_1)$ of $$-\operatorname{div}\bigl(|\nabla u|^{p-2} \nabla u\bigr) = f, \qquad f \in L^\infty(B_1),$$ that is, $\int |\nabla u|^{p-2} \nabla u \cdot \nabla \varphi = \int f \varphi$ for all compactly supported smooth $\varphi$. With $p' = p/(p-1)$, the function $|x|^{p'}$ has constant $p$-Laplacian, so the gradient of a solution cannot in general be better than Hölder continuous with exponent $1/(p-1)$; that solutions are locally of class $C^{1,1/(p-1)}$ is the $C^{p'}$-regularity conjecture. Araújo, Teixeira and Urbano proved the estimate below in the plane (2017), and the regularity for restricted classes of solutions in higher dimensions, such as radially symmetric ones and those without saddle critical points (2018). Lee and Lee proved the estimate in every dimension for $2 < p < 2 + \varepsilon_d$ (2026).
Problem. For every integer $d \ge 3$ and every $p > 2$, does there exist $C = C(d,p)$ such that every such weak solution satisfies $$\|u\|_{C^{1,1/(p-1)}(\overline{B}_{1/2})} \le C \left( \|u\|_{L^p(B_1)} + \|f\|_{L^\infty(B_1)}^{1/(p-1)} \right)?$$ The endpoint exponent $1/(p-1)$ is required, not merely every smaller exponent.
Source: The C^{p'}-regularity conjecture, so named by Araújo, Teixeira and Urbano (2017); in this quantitative form, Lee and Lee (2026), Conjecture 1.1 and Theorem 1.2. Stated by AIM — Open Applied Problems for dimensions d ≥ 3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, gradient regularity, p-Laplace equations
References
@article{araujo2017,
author = {Ara{\'u}jo, Dami{\~a}o J. and Teixeira, Eduardo V. and Urbano, Jos{\'e} Miguel},
title = {A proof of the {$C^{p'}$}-regularity conjecture in the plane},
journal = {Adv. Math.},
volume = {316},
year = {2017},
pages = {541--553},
doi = {10.1016/j.aim.2017.06.027},
eprint = {1901.03827},
archivePrefix = {arXiv},
}
@article{araujo2018,
author = {Ara{\'u}jo, Dami{\~a}o J. and Teixeira, Eduardo V. and Urbano, Jos{\'e} Miguel},
title = {Towards the {$C^{p'}$}-regularity conjecture in higher dimensions},
journal = {Int. Math. Res. Not. IMRN},
volume = {2018},
number = {20},
year = {2018},
pages = {6481--6495},
doi = {10.1093/imrn/rnx068},
}
@misc{lee2026,
author = {Lee, Se-Chan and Lee, Taehun},
title = {The {$C^{p'}$}-regularity conjecture near $p=2$},
year = {2026},
eprint = {2609.09966},
archivePrefix = {arXiv},
}
@misc{aim-sharp-p-poisson-gradient-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp gradient regularity for higher-dimensional p-Poisson equations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/420-sharp-p-poisson-gradient-regularity.md},
note = {Entry 420 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-00642, "Sharp gradient regularity for higher-dimensional p-Poisson equations", clopen.solutions.