Finiteness of interior defects in Oseen–Frank minimizers
Let $\Omega \subset \mathbb{R}^3$ be a bounded smooth domain, let $g \colon \partial\Omega \to \mathbb{S}^2$ be smooth with a nonempty $H^1$ extension class, and let $k_1, k_2, k_3 > 0$. For $n \in H^1(\Omega; \mathbb{S}^2)$ with trace $g$ set $$E(n) = \frac12 \int_\Omega \Bigl[k_1 (\operatorname{div} n)^2 + k_2 (n \cdot \operatorname{curl} n)^2 + k_3 |n \times \operatorname{curl} n|^2\Bigr]\, dx.$$ A point of $\Omega$ is singular for $n$ if no neighbourhood of it admits a smooth representative of $n$. The saddle-splay term of the Oseen–Frank energy is a null Lagrangian and does not change the minimizers with this trace. Hardt, Kinderlehrer and Lin proved that every minimizer is analytic outside a closed interior singular set of Hausdorff dimension less than one, and that this set is locally finite when $k_1, k_2, k_3$ are sufficiently close to $1$. For $k_1 = k_2 = k_3$ the minimizers are minimizing harmonic maps into $\mathbb{S}^2$, whose interior singularities are isolated points (Schoen and Uhlenbeck). Dai, Fu, Wang and Wang proved that for arbitrary $k_1, k_2, k_3 > 0$ every global minimizer is smooth in a neighbourhood of $\partial\Omega$ (2026).
Problem. Is the interior singular set of every global minimizer of $E$ locally finite, that is, does every compact subset of $\Omega$ contain only finitely many singular points?
Source: Lin and Wang (2014), Question 1.4; Dai, Fu, Wang and Wang (2026) trace the question whether singularities are isolated to Lin and Liu (2001). Stated by AIM — Open Applied Problems without the saddle-splay term, for locally finite interior singular sets.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, nematic liquid crystals, Oseen–Frank energy, point defects
References
@article{lin2014,
author = {Lin, Fanghua and Wang, Changyou},
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doi = {10.1098/rsta.2013.0361},
eprint = {1408.4138},
archivePrefix = {arXiv},
}
@article{linliu2001,
author = {Lin, Fang-Hua and Liu, Chun},
title = {Static and dynamic theories of liquid crystals},
journal = {J. Partial Differential Equations},
volume = {14},
number = {4},
year = {2001},
pages = {289--330},
}
@article{hardt1986,
author = {Hardt, Robert and Kinderlehrer, David and Lin, Fang-Hua},
title = {Existence and partial regularity of static liquid crystal configurations},
journal = {Comm. Math. Phys.},
volume = {105},
number = {4},
year = {1986},
pages = {547--570},
doi = {10.1007/BF01238933},
}
@article{hardt1988,
author = {Hardt, R. and Kinderlehrer, D. and Lin, Fang-Hua},
title = {Stable defects of minimizers of constrained variational principles},
journal = {Ann. Inst. H. Poincar{\'e} Anal. Non Lin{\'e}aire},
volume = {5},
number = {4},
year = {1988},
pages = {297--322},
doi = {10.1016/S0294-1449(16)30340-7},
}
@article{schoen1982,
author = {Schoen, Richard and Uhlenbeck, Karen},
title = {A regularity theory for harmonic maps},
journal = {J. Differential Geom.},
volume = {17},
number = {2},
year = {1982},
pages = {307--335},
doi = {10.4310/jdg/1214436923},
}
@misc{dai2026,
author = {Dai, Zhiyuan and Fu, Haotong and Wang, Huaijie and Wang, Wei},
title = {Boundary regularity for {O}seen--{F}rank minimizers with arbitrary positive splay, twist, and bend constants},
year = {2026},
eprint = {2608.22287},
archivePrefix = {arXiv},
}
@misc{aim-oseen-frank-interior-defects,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finiteness of interior defects in Oseen–Frank minimizers}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/220-oseen-frank-interior-defects.md},
note = {Entry 220 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00629, "Finiteness of interior defects in Oseen–Frank minimizers", clopen.solutions.