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

Abstract

We consider a non-local free energy functional, modelling a competition between entropy and pairwise interactions reminiscent of the second order virial expansion, with applications to nematic liquid crystals as a particular case. We build on previous work on understanding the behaviour of such models within the large-domain limit, where minimisers converge to minimisers of a quadratic elastic energy with manifold-valued constraint, analogous to harmonic maps. We extend this work to establish Hölder bounds for (almost-)minimisers on bounded domains, and demonstrate stronger convergence of (almost)-minimisers away from the singular set of the limit solution. The proof techniques bear analogy with recent work of singularly perturbed energy functionals, in particular in the context of the Ginzburg-Landau and Landau-de Gennes models.

Effective surface energies in nematic liquid crystals as homogenised rugosity effects

“…We recall that in a nematic described by the Landau-de Gennes model, the "bulk" energy F b is minimised at Q of the form Q = s0 n ⊗ n − 1 3 I , where s0 > 0 is fixed and n can be any unit vector. It is known that in domains much larger than the characteristic correlation length of the molecules the penalisation of the bulk energy dominates, and it is appropriate to impose the constraint that Q minimises the bulk energy pointwise [9,16,28,38,40,55]. In certain cases, at least, this becomes equivalent to the Oseen-Frank energy, in which we use only the director n : Ω → S 2 to describe the nematic [7].…”