Full interior regularity for the nondegenerate symmetric p-Laplace system
Let $n \ge 3$, $1 < p < \infty$ with $p \ne 2$, and let $B_1 \subset \mathbb{R}^n$ be the open unit ball. For $u \in W^{1,p}(B_1; \mathbb{R}^n)$ let $e(u) = \tfrac12 (\nabla u + \nabla u^T)$, and for matrices $P$ let $A(P) = (1 + |P|^2)^{(p-2)/2} P$, with $|P|$ the Frobenius norm. Consider weak solutions $u \in W^{1,p}(B_1; \mathbb{R}^n)$ of the system $\operatorname{div} A(e(u)) = 0$: $$\int_{B_1} A(e(u)) : e(\varphi) \, dx = 0 \qquad \text{for every } \varphi \in C_c^\infty(B_1; \mathbb{R}^n).$$ This is the Euler–Lagrange system of the convex energy $\int_{B_1} \frac1p \bigl((1 + |e(u)|^2)^{p/2} - 1\bigr)\, dx$.
In dimension two, local minimizers of such energies are of class $C^{1,\alpha}$, a known result for which Conti, Focardi and Iurlano give a proof; for $n \ge 3$, Conti, Focardi and Iurlano proved that they are of class $C^{1,\alpha}$ for every $\alpha \in (0,1)$ outside a closed set of Hausdorff dimension at most $\max\{n - q, 0\}$, for some $q > 2$. Behn, Cianchi, Diening and Peng proved that the stress $A(e(u))$ lies in $W^{1,2}_{\mathrm{loc}}$ for $p$ in a dimension-dependent range $p_-(n) < p < p_+(n)$, where $p_+(n) = \infty$ for $n \le 7$.
Problem. Is every such weak solution of class $C^{1,\alpha}_{\mathrm{loc}}(B_1; \mathbb{R}^n)$ for some $\alpha \in (0,1)$, for every $n \ge 3$ and every $p \in (1, \infty) \setminus \{2\}$?
Source: Conti, Focardi and Iurlano (2019), §1 and §4, who call it a major open problem to prove or disprove full regularity for n ≥ 3 and p ≠ 2, even in the nondegenerate symmetrized p-Laplacian case; stated for this system by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, elliptic regularity, p-Laplace systems, symmetric gradients
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{conti2019,
author = {Conti, Sergio and Focardi, Matteo and Iurlano, Flaviana},
title = {A note on the {H}ausdorff dimension of the singular set of solutions to elasticity type systems},
journal = {Commun. Contemp. Math.},
volume = {21},
number = {6},
year = {2019},
pages = {1950026},
doi = {10.1142/S0219199719500263},
eprint = {1804.09945},
archivePrefix = {arXiv},
}
@misc{behn2026,
author = {Behn, Linus and Cianchi, Andrea and Diening, Lars and Peng, Fa},
title = {Maximal {S}obolev regularity of the stress tensor for the symmetric gradient $p$-{L}aplace system},
year = {2026},
eprint = {2603.17036},
archivePrefix = {arXiv},
}
@misc{aim-full-interior-regularity-for-the-nondegenerate-symmetric-p-laplace-system,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Full interior regularity for the nondegenerate symmetric p-Laplace system}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/759-full-interior-regularity-for-the-nondegenerate-symmetric-p-laplace-system.md},
note = {Entry 759 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01151, "Full interior regularity for the nondegenerate symmetric p-Laplace system", clopen.solutions.