We study global minimizers of the Landau–de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, t and some constraints on the material parameters, and we work in the t→∞ limit that captures features of the widely used Lyuksyutov constraint (Kralj and Virga in J Phys A 34:829–838, 2001). We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map; (iii) estimates for the size of “strongly biaxial” regions in terms of the parameter t. We further show that global LdG minimizers in the restricted class of uniaxial Q-tensors cannot be stable critical points of the LdG energy in this limit.
|Number of pages||22|
|Journal||Calculus of Variations and Partial Differential Equations|
|Early online date||4 Apr 2017|
|Publication status||Published - 30 Apr 2017|
Henao, D., Majumdar, A., & Pisante, A. (2017). Uniaxial versus biaxial character of nematic equilibria in three dimensions. Calculus of Variations and Partial Differential Equations, 56(2), . https://doi.org/10.1007/s00526-017-1142-8