Strict physicality for anisotropic singular Q-tensor minimizers
Let $S_0$ be the space of symmetric trace-free real $3 \times 3$ matrices, and let $f$ be the Ball–Majumdar singular potential, $$f(Q) = \inf_\rho \Bigl\{\int_{\mathbb{S}^2} \rho \log \rho \, dS : \rho \ge 0,\ \int_{\mathbb{S}^2} \rho \, dS = 1,\ \int_{\mathbb{S}^2} \bigl(p \otimes p - I/3\bigr) \rho(p)\, dS = Q\Bigr\},$$ with $f(Q) = +\infty$ when no admissible density exists. Let $\Omega \subset \mathbb{R}^3$ be bounded and smooth, and let $Q_b$ be smooth boundary values whose eigenvalues lie in $[-1/3 + \delta_0, 2/3 - \delta_0]$ for some $\delta_0 > 0$. For $L_1, L_2, \theta, \kappa > 0$ consider minimizing $$\int_\Omega \Bigl[L_1 |\nabla Q|^2 + L_2 |\operatorname{div} Q|^2 + \theta f(Q) - \kappa |Q|^2\Bigr]\, dx$$ over $Q \in H^1(\Omega; S_0)$ with trace $Q_b$, where $(\operatorname{div} Q)_i = \sum_j \partial_j Q_{ij}$. Ball and Majumdar showed that $f$ is smooth and strictly convex where $\lambda_{\min}(Q) > -1/3$ and tends to $+\infty$ as $\lambda_{\min}(Q) \to -1/3$, and proved for the one-constant energy ($L_2 = 0$) that minimizers have $\lambda_{\min}$ bounded away from $-1/3$. Ball noted that such a bound would make a minimizer a weak solution of the Euler–Lagrange equations, smooth by elliptic regularity.
Problem. For arbitrary $L_1, L_2, \theta, \kappa > 0$, must every global minimizer admit some $\delta > 0$ with $\lambda_{\min}(Q(x)) \ge -1/3 + \delta$ for almost every $x \in \Omega$? (Since $Q$ is trace-free, this also keeps the largest eigenvalue away from $2/3$.)
Source: Ball (2017), §6.2, who asks whether minimizers with the Ball–Majumdar singular potential are strictly physical for elastic energies beyond the one-constant one. Stated by AIM — Open Applied Problems for the energy L₁|∇Q|² + L₂|div Q|².
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Landau–de Gennes theory, Q-tensors, singular potentials
References
@article{ball2017,
author = {Ball, J. M.},
title = {Mathematics and liquid crystals},
journal = {Mol. Cryst. Liq. Cryst.},
volume = {647},
number = {1},
year = {2017},
pages = {1--27},
doi = {10.1080/15421406.2017.1289425},
eprint = {1612.03792},
archivePrefix = {arXiv},
}
@article{ballmajumdar2010,
author = {Ball, John M. and Majumdar, Apala},
title = {Nematic liquid crystals: from {M}aier--{S}aupe to a continuum theory},
journal = {Mol. Cryst. Liq. Cryst.},
volume = {525},
number = {1},
year = {2010},
pages = {1--11},
doi = {10.1080/15421401003795555},
}
@misc{gimperlein2025,
author = {Gimperlein, Heiko and Maity, Ruma R.},
title = {Numerical analysis for constrained and unconstrained {$Q$}-tensor energies for liquid crystals},
year = {2025},
eprint = {2506.04880},
archivePrefix = {arXiv},
}
@misc{aim-anisotropic-q-physicality,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Strict physicality for anisotropic singular Q-tensor minimizers}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/221-anisotropic-q-physicality.md},
note = {Entry 221 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-00574, "Strict physicality for anisotropic singular Q-tensor minimizers", clopen.solutions.