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

Pisante, Adriano

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 2, pp. 301-326.

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. Enfin, nous discutons ces mêmes proprietés pour des minimiseurs sous contrainte de symétrie axiale et sous contrainte de norme. Dans ce dernier cas, nous montrons que les minimiseurs ne satisfont qu’une régularité partielle, à savoir la régularité en dehors d’un ensemble fini situé sur l’axe de symétrie. De plus, nous démontrons que ces singularités ponctuelles peuvent en effet exister pour des raisons énergétiques, et nous décrivons en détails le comportement asymptotique des minimiseurs près de ces points singuliers. Pour terminer, nous donnons quelques propriétés qualitatives des surfaces de biaxialité pour une classe de domaines et de données au bord montrant que les minimiseurs réguliers présentent une structure en tore biaxial comme celle prédite dans [16, 24, 25, 39].

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. Then, under suitable assumptions on the topology of the domain and on the Dirichlet boundary condition, we show that smoothness of energy minimizing configurations yields the emergence of nontrivial topological structure in their biaxiality level sets. Then, we discuss the previous properties under both the norm constraint and an axial symmetry constraint, showing that in this case only partial regularity is available, away from a finite set located on the symmetry axis. In addition, we show that singularities may appear due to energy efficiency and we describe precisely the asymptotic profile around singular points. 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].

Mots clés : Liquid crystals; axisymmetric torus solutions; harmonic maps

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