Approximate continuity outside the jump set for bounded-deformation maps
Let $d \ge 2$ and let $u \in L^1_{\mathrm{loc}}(\mathbb{R}^d; \mathbb{R}^d)$ be of locally bounded deformation: its distributional symmetric gradient $Eu = \tfrac12 (Du + Du^T)$ is a locally finite Radon measure with values in the symmetric $d \times d$ matrices. Let $S_u$ be the set of points $x$ at which there is no $z \in \mathbb{R}^d$ 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 $x$ for which there are $a \ne b$ in $\mathbb{R}^d$ and $\nu \in \mathbb{S}^{d-1}$ 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 $B_r^\pm(x,\nu) = \{y \in B_r(x) : \pm (y - x) \cdot \nu > 0\}$.
For functions of bounded variation, $\mathcal{H}^{d-1}(S_u \setminus J_u) = 0$ (Federer, Vol'pert). For $u$ of locally bounded deformation, Ambrosio, Coscia and Dal Maso proved the weaker statement $|Ev|(S_u \setminus J_u) = 0$ for every $v$ of locally bounded deformation; Kohn proved that $S_u \setminus J_u$ has zero $(d-1)$-dimensional Riesz capacity, so that $\mathcal{H}^{d-1+\varepsilon}(S_u \setminus J_u) = 0$ for every $\varepsilon > 0$; and Arroyo-Rabasa and Skorobogatova proved that $S_u \setminus J_u$ is purely $\mathcal{H}^{d-1}$-unrectifiable. Conti, Focardi and Iurlano proved $\mathcal{H}^{d-1}(S_u \setminus J_u) = 0$ for $u \in SBD^p$, $p > 1$, that is, when $Eu$ has no Cantor part, $e(u) \in L^p$ and $\mathcal{H}^{d-1}(J_u) < \infty$.
Problem. Is $\mathcal{H}^{d-1}(S_u \setminus J_u) = 0$ for every $d \ge 2$ and every $u$ of locally bounded deformation on $\mathbb{R}^d$?
Source: De Philippis and Rindler (2020), Conjecture 1.3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, fine properties, functions of bounded deformation, jump sets
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{dephilippis2020,
author = {De Philippis, Guido and Rindler, Filip},
title = {Fine properties of functions of bounded deformation---an approach via linear {PDE}s},
journal = {Math. Eng.},
volume = {2},
number = {3},
year = {2020},
pages = {386--422},
doi = {10.3934/mine.2020018},
eprint = {1911.01356},
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},
}
@phdthesis{kohn1979,
author = {Kohn, Robert V.},
title = {New estimates for deformations in terms of their strains},
school = {Princeton University},
year = {1979},
}
@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{conti2018,
author = {Conti, Sergio and Focardi, Matteo and Iurlano, Flaviana},
title = {Which special functions of bounded deformation have bounded variation?},
journal = {Proc. Roy. Soc. Edinburgh Sect. A},
volume = {148},
number = {1},
year = {2018},
pages = {33--50},
doi = {10.1017/S030821051700004X},
eprint = {1502.07464},
archivePrefix = {arXiv},
}
@misc{aim-approximate-continuity-outside-the-jump-set-for-bounded-deformation-maps,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Approximate continuity outside the jump set for bounded-deformation maps}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/756-approximate-continuity-outside-the-jump-set-for-bounded-deformation-maps.md},
note = {Entry 756 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-01145, "Approximate continuity outside the jump set for bounded-deformation maps", clopen.solutions.