Minimizers of a Landau–de Gennes energy with a subquadratic elastic energy

Abstract

Ball in the occasion of his 70th birthday. Abstract We study a modified Landau-de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two-and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small compared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimizers on two-dimensional domains, when the temperature is sufficiently low.

Hölder regularity and convergence for a non-local model of nematic liquid crystals in the large-domain limit

“…Studying the convergence of minimisers of Landau-de Gennes towards the Oseen-Frank limit has attracted interest, with Majumdar and Zarnescu showing global W 1,2 convergence and uniform convergence away from singular sets in the one-constant case [35], Nguyen and Zarnescu proving convergence results in stronger topologies [37], Contreras, Lamy and Rodiac generalising the approach to other harmonic-map problems [10], and further extensions by Contreras and Lamy [9] and Canevari, Majumdar and Stroffolini [7] to more general elastic energies. In other settings, the W 1,2 -convergence does not hold globally but only locally, away from the singular sets, due to topological obstructions carried by the boundary data and/or the domain (see e.g.…”

Improved Partial Regularity for Manifold-Constrained Minimisers of Subquadratic Energies

“…In turns, the proof of Proposition 3.3 is based on "dimension reduction" argument, and uses the following lemma (whose proof may be found, e.g., in [13,Lemma 16]). Lemma 3.4.…”

“…Now configurations with line discontinuities, even non-orientable ones, do belong to the admissible class. As in the quadratic case, the model (5) can be obtained from a functional of the Landau-de Gennes type, in the limit of small nematic correlation length [13]. Let us consider a minimiser Q 0 of (5) on a bounded, smooth domain Ω ⊆ R 3 , subject to the boundary condition…”