Approximate continuity of the Cantor part for complex-elliptic systems
Let $n \ge 2$, let $V, W$ be nonzero finite-dimensional real inner-product spaces, and let $A_1, \dots, A_n \colon V \to W$ be linear maps. Put $$\mathcal{A}u = \sum_{j=1}^n A_j \partial_j u, \qquad \mathbb{A}(\xi) = \sum_{j=1}^n \xi_j A_j,$$ and assume that $\mathcal{A}$ is complex-elliptic: the symbol $\mathbb{A}(\xi) \colon \mathbb{C} \otimes V \to \mathbb{C} \otimes W$ is injective for every $\xi \in \mathbb{C}^n \setminus \{0\}$. Let $B_1 \subset \mathbb{R}^n$ be the open unit ball and let $BV^{\mathcal{A}}(B_1)$ be the space of $u \in L^1(B_1; V)$ for which $\mathcal{A}u$ is a finite $W$-valued Radon measure on $B_1$. For such $u$, let $S_u$ be the set of $x \in B_1$ at which there is no $z \in V$ with $$\lim_{r \downarrow 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} |u(y) - z| \, dy = 0,$$ and let $J_u$ be the set of approximate jump points: the $x \in B_1$ for which there are $a \ne b$ in $V$ and $\nu \in \mathbb{S}^{n-1}$ with $$\lim_{r \downarrow 0} \frac{1}{|B_r(x)|} \int_{B_r^+(x,\nu)} |u(y) - a| \, dy = 0, \qquad \lim_{r \downarrow 0} \frac{1}{|B_r(x)|} \int_{B_r^-(x,\nu)} |u(y) - b| \, dy = 0,$$ where $B_r^\pm(x,\nu) = \{y \in B_r(x) : \pm (y - x) \cdot \nu > 0\}$.
The gradient and the symmetric gradient are complex-elliptic. For the gradient, $|Du|(S_u \setminus J_u) = 0$ by the Federer–Vol'pert theorem; Ambrosio, Coscia and Dal Maso proved $|Eu|(S_u \setminus J_u) = 0$ for the symmetric gradient $E$, and the same holds for operators admitting one-dimensional slicing representations (Arroyo-Rabasa), a strict subclass of the complex-elliptic operators. For every first-order complex-elliptic $\mathcal{A}$, Arroyo-Rabasa and Skorobogatova proved that $S_u \setminus J_u$ is purely $\mathcal{H}^{n-1}$-unrectifiable and that $S_u$ has zero $(n-1)$-dimensional Riesz capacity.
Problem. Is $|\mathcal{A}u|(S_u \setminus J_u) = 0$ for every $n \ge 2$, every such $V$, $W$ and complex-elliptic $\mathcal{A}$, and every $u \in BV^{\mathcal{A}}(B_1)$?
Source: Arroyo-Rabasa and Skorobogatova (2025), Conjecture 4 of arXiv:1911.08474v4 (Conjecture 2.12 of the published version), for open sets Ω ⊂ ℝⁿ; stated on the unit ball by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, complex elliptic operators, functions of bounded A-variation, jump sets
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{arroyorabasa2025,
author = {Arroyo-Rabasa, Adolfo and Skorobogatova, Anna},
title = {A look into some of the fine properties of functions with bounded {$\mathscr{A}$}-variation},
journal = {ESAIM Control Optim. Calc. Var.},
volume = {31},
year = {2025},
pages = {50},
doi = {10.1051/cocv/2025003},
eprint = {1911.08474},
archivePrefix = {arXiv},
}
@article{ambrosio1997,
author = {Ambrosio, Luigi and Coscia, Alessandra and Dal Maso, Gianni},
title = {Fine properties of functions with bounded deformation},
journal = {Arch. Ration. Mech. Anal.},
volume = {139},
number = {3},
year = {1997},
pages = {201--238},
doi = {10.1007/s002050050051},
}
@misc{arroyorabasa2020,
author = {Arroyo-Rabasa, Adolfo},
title = {Slicing and fine properties for functions with bounded {$\mathcal{A}$}-variation},
year = {2020},
eprint = {2009.13513},
archivePrefix = {arXiv},
}
@misc{aim-approximate-continuity-of-the-cantor-part-for-complex-elliptic-systems,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Approximate continuity of the Cantor part for complex-elliptic systems}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/753-approximate-continuity-of-the-cantor-part-for-complex-elliptic-systems.md},
note = {Entry 753 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-01144, "Approximate continuity of the Cantor part for complex-elliptic systems", clopen.solutions.