Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime

Abstract

We study global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional domains, under a Dirichlet boundary condition. In a relevant range of parameters (which we call the Lyuksyutov regime), the main result establishes the nontrivial topology of the biaxiality sets of minimizers for a large class of boundary conditions including the homeotropic boundary data. To achieve this result, we first study minimizers subject to a physically relevant norm constraint (the Lyuksyutov constraint), and show their regularity up to the boundary. From this regularity, we rigorously derive the norm constraint from the asymptotic Lyuksyutov regime. As a consequence, isotropic melting is avoided by unconstrained minimizers in this regime, which then allows us to analyse their biaxiality sets. In the case of a nematic droplet, this also implies that the radial hedgehog is an unstable equilibrium in the same regime of parameters. Technical results of this paper will be largely employed in Dipasquale et al. (Torus-like solutions for the Landau- de Gennes model. Part II: topology of $$\mathbb {S}^1$$ S 1 -equivariant minimizers. https://arxiv.org/pdf/2008.13676.pdf; Torus-like solutions for the Landau- de Gennes model. Part III: torus solutions vs split solutions (In preparation)), where we prove that biaxiality level sets are generically finite unions of tori for smooth configurations minimizing the energy in restricted classes of axially symmetric maps satisfying a topologically nontrivial boundary condition.

Torus-like solutions for the Landau–De Gennes model.

“…The regularity properties claimed in the theorem could be obtained combining the results in [42,43,44]. However, as commented below, the proof in [12] is somewhat different and it is organized so that the argument also covers the case of minimization in a symmetric class of maps, as considered in [10,11], with only minor modifications.…”

“…Note that the monotonicity formulae are not obtained by inner variations, as in [45] for the interior case, but instead by a penalty approximation, passing to -309 -the limit in monotonicity formulae for smooth solutions of approximated problems. This approach is more flexible, as it applies also on the boundary and it is of use even in the symmetric case considered in [10,11], where inner variations no longer give admissible deformations.…”

“…Thus, the main feature of -303 -the Landau-De Gennes model will be the presence in the energy minimizing configuration of biaxial phase { β • Q( • ) = ±1} = ∅ (biaxial escape). The goal here is to account on some results from [10,11,12] on the regularity of minimizers and the emergence of topological structure in the corresponding biaxial surfaces { β • Q( • ) = t}, t ∈ (−1, 1), which in the model case of a nematic droplet are expected to be of torus type [16,24,25,39].…”

“…Adriano Pisante (1)ABSTRACT. -In this note we report on some recent progress [10,11,12] about the study of global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals in three-dimensional domains. First, we discuss absence of singularities for minimizing configurations under norm constraint, as well as absence of the isotropic phase for the unconstrained minimizers, together with the related biaxial escape phenomenon.…”

“…Finally, in an appropriate class of domains and boundary data we obtain qualitative properties of the biaxial surfaces, showing that smooth minimizers exibit torus structure, as predicted in [16,24,25,39].RÉSUMÉ. -Dans cette note, nous présentons des avancées récentes [10,11,12] sur l'etude des minimiseurs globaux d'une énergie continue de Landau-De Gennes dans des domaines 3D utilisée dans la modélisation des cristaux liquides nématiques.Dans un premier temps, nous décrivons l'absence de singularités des configurations minimisantes sous contrainte de norme, ainsi que l'absence de phase isotrope pour les minimiseurs non contraints, et le phénomène de fuite biaxiale en résultant. Sous certaines hypothèses sur la topologie du domaine et la condition de Dirichlet au bord, nous montrons ensuite comment la régularité / absence de phase isotrope des configurations minimisantes permet de déduire une structure topologique non triviale des ensembles de niveau de la biaxialité.…”