No diffuse Cantor damage in cohesive fracture minimizers
Let $\Omega \subset \mathbb{R}^n$, $n \ge 2$, be a bounded Lipschitz domain. Let $F, G \colon [0,\infty) \to [0,\infty)$ be continuously differentiable and increasing, with $F$ strictly convex and superlinear, $F(0) = F'(0) = 0$, and $G$ strictly concave, $G(0) = 0$, $\sigma := G'(0) > 0$ and $$\lim_{s \downarrow 0} \frac{G(s) - \sigma s}{s^2} < 0,$$ the limit existing in $[-\infty, 0)$. Let $e_M > 0$ be the solution of $F'(e_M) = \sigma$, and let $\overline{F}(t) = F(t)$ for $0 \le t \le e_M$ and $\overline{F}(t) = F(e_M) + \sigma (t - e_M)$ for $t > e_M$. For $u \in BV(\Omega)$ write $Du = \nabla u \, \mathcal{L}^n + [u] \nu_u \, \mathcal{H}^{n-1} \llcorner J_u + D^c u$, with approximate gradient $\nabla u$, jump set $J_u$, jump $[u]$ and Cantor part $D^c u$, and let $$\mathcal{E}(u) = \int_\Omega \overline{F}(|\nabla u|) \, dx + \int_{J_u} G(|[u]|) \, d\mathcal{H}^{n-1} + \sigma |D^c u|(\Omega).$$ A local minimizer is a $u \in BV(\Omega)$ with $\mathcal{E}(u) < \infty$ for which there is $\eta > 0$ such that $\mathcal{E}(u) \le \mathcal{E}(v)$ for every $v \in BV(\Omega)$ with $\operatorname{supp}(v - u) \Subset \Omega$ and $\|v - u\|_{L^1} + |D(v - u)|(\Omega) < \eta$.
The functional $\mathcal{E}$ represents on $BV(\Omega)$ the relaxation in $L^1$ of the cohesive energy $\int_\Omega F(|\nabla u|)\,dx + \int_{J_u} G(|[u]|)\,d\mathcal{H}^{n-1}$ on $SBV(\Omega)$ (Bouchitté, Braides and Buttazzo). In dimension one, Braides, Dal Maso and Garroni proved that every local minimizer of the corresponding functional has no Cantor part and that the density $\dot u$ of the absolutely continuous part of $u'$ satisfies $|\dot u| \le e_M$ almost everywhere. Dal Maso and Garroni proved that a local minimizer satisfies $|\nabla u| \le e_M$ on every open set on which it is of class $C^1$.
Problem. Does every local minimizer satisfy $|D^c u|(\Omega) = 0$ and $|\nabla u| \le e_M$ almost everywhere in $\Omega$?
Source: Dal Maso and Garroni (2008), §1, equation (5), where it is expected for G concave; stated for strictly concave G on bounded Lipschitz domains by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, cohesive fracture, free-discontinuity problems, functions of bounded variation
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{dalmaso2008,
author = {Dal Maso, Gianni and Garroni, Adriana},
title = {Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics},
journal = {Calc. Var. Partial Differential Equations},
volume = {31},
number = {2},
year = {2008},
pages = {137--145},
doi = {10.1007/s00526-006-0084-3},
eprint = {math/0507088},
archivePrefix = {arXiv},
}
@article{bouchitte1995,
author = {Bouchitt{\'e}, Guy and Braides, Andrea and Buttazzo, Giuseppe},
title = {Relaxation results for some free discontinuity problems},
journal = {J. Reine Angew. Math.},
volume = {458},
year = {1995},
pages = {1--18},
doi = {10.1515/crll.1995.458.1},
}
@article{braides1999,
author = {Braides, Andrea and Dal Maso, Gianni and Garroni, Adriana},
title = {Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case},
journal = {Arch. Ration. Mech. Anal.},
volume = {146},
number = {1},
year = {1999},
pages = {23--58},
doi = {10.1007/s002050050135},
}
@misc{aim-no-diffuse-cantor-damage-in-cohesive-fracture-minimizers,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{No diffuse Cantor damage in cohesive fracture minimizers}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/754-no-diffuse-cantor-damage-in-cohesive-fracture-minimizers.md},
note = {Entry 754 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-01160, "No diffuse Cantor damage in cohesive fracture minimizers", clopen.solutions.