Sharp strain integrability in planar Griffith fracture
Let $\Omega \subset \mathbb{R}^2$ be a bounded Lipschitz domain, and fix Lamé coefficients $\mu > 0$ and $\lambda + \mu > 0$; put $\mathbb{C}\xi = \lambda \operatorname{tr}(\xi) I + 2\mu \xi$ for symmetric $2 \times 2$ matrices $\xi$. For a relatively closed set $K \subset \Omega$, a displacement $u \in W^{1,2}_{\mathrm{loc}}(\Omega \setminus K; \mathbb{R}^2)$ with symmetric gradient $e(u) = \tfrac12 (Du + Du^T)$, and a ball $B$, let $$G(u, K; B) = \int_{B \setminus K} \mathbb{C}e(u) : e(u) \, dx + \mathcal{H}^1(K \cap B).$$ Such a pair $(u, K)$ is a local minimizer if $K$ is coral, that is, $\mathcal{H}^1(K \cap B(x,r)) > 0$ for every $x \in K$ and $r > 0$, if $G(u, K; B) < \infty$ for every ball $B \Subset \Omega$, and if $G(u, K; B) \le G(v, L; B)$ for every ball $B \Subset \Omega$ and every relatively closed $L \subset \Omega$ and $v \in W^{1,2}_{\mathrm{loc}}(\Omega \setminus L; \mathbb{R}^2)$ with $L \setminus B = K \setminus B$ and $v = u$ almost everywhere on $\Omega \setminus (K \cup B)$.
Friedrich, Labourie and Stinson proved, for general elasticity tensors, that the singular set of the crack of a local minimizer has Hausdorff dimension less than one and that $e(u) \in L^p_{\mathrm{loc}}$ for some $p > 2$ depending on the tensor; Labourie and Lemenant had proved both for minimizers with connected cracks. Labourie, Lamberti and Lemenant proved that $u$ is locally bounded and that its full gradient $\nabla u$ is locally square integrable off the crack.
Problem. Is $e(u) \in L^{4,\infty}_{\mathrm{loc}}(\Omega)$ for every such $\Omega$, $\lambda$, $\mu$ and every local minimizer $(u, K)$, that is, $$\sup_{s > 0} s^4 \, \mathcal{L}^2\bigl(\{x \in U \setminus K : |e(u)(x)| > s\}\bigr) < \infty$$ for every open $U \Subset \Omega$, with $|\cdot|$ the Frobenius norm?
Source: Labourie and Lemenant (arXiv 2021; 2023), §1, last paragraph, who conjecture integrability of e(u) for every p < 4; the weak-L⁴ endpoint for general elasticity tensors is expected by Friedrich, Labourie and Stinson (2026), §1. Stated for isotropic tensors and local minimizers by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, free-discontinuity problems, Griffith energy, linearized elasticity
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{labourie2023,
author = {Labourie, Camille and Lemenant, Antoine},
title = {Regularity improvement for the minimizers of the two-dimensional {G}riffith energy},
journal = {Rend. Lincei Mat. Appl.},
volume = {34},
number = {2},
year = {2023},
pages = {337--357},
doi = {10.4171/rlm/1009},
eprint = {2111.13081},
archivePrefix = {arXiv},
}
@article{friedrich2026,
author = {Friedrich, Manuel and Labourie, Camille and Stinson, Kerrek},
title = {Hausdorff dimension of the singular set for {G}riffith almost-minimizers in the plane},
journal = {Trans. Amer. Math. Soc.},
volume = {379},
number = {11},
year = {2026},
pages = {7789--7818},
doi = {10.1090/tran/9776},
eprint = {2510.08670},
archivePrefix = {arXiv},
}
@misc{labourie2026,
author = {Labourie, Camille and Lamberti, Lorenzo and Lemenant, Antoine},
title = {Good access to the crack and integrability of the full gradient for {G}riffith almost-minimizers in the plane},
year = {2026},
eprint = {2607.19229},
archivePrefix = {arXiv},
}
@misc{aim-sharp-strain-integrability-in-planar-griffith-fracture,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp strain integrability in planar Griffith fracture}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/757-sharp-strain-integrability-in-planar-griffith-fracture.md},
note = {Entry 757 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-01162, "Sharp strain integrability in planar Griffith fracture", clopen.solutions.