We construct an order reconstruction (OR)-type Landau-de Gennes critical point on a square domain of edge length λ, motivated by the well order reconstruction solution numerically reported in [1]. The OR critical point is distinguished by an uniaxial cross with negative scalar order parameter along the square diagonals. The OR critical point is defined in terms of a saddle-type critical point of an associated scalar variational problem. The OR-type critical point is globally stable for small λ and undergoes a supercritical pitchfork bifurcation in the associated scalar variational setting. We consider generalizations of the OR-type critical point to a regular hexagon, accompanied by numerical estimates of stability criteria of such critical points on both a square and a hexagon in terms of material-dependent constants.

Abstract. We consider the Landau-de Gennes variational problem on a bounded, two dimensional domain, subject to Dirichlet smooth boundary conditions. We prove that minimizers are maximally biaxial near the singularities, that is, their biaxiality parameter reaches the maximum value 1. Moreover, we discuss the convergence of minimizers in the vanishing elastic constant limit. Our asymptotic analysis is performed in a general setting, which recovers the Landau-de Gennes problem as a specific case.

We prove the existence of a well order reconstruction solution (WORS) type Landaude Gennes critical point on a square domain with an isotropic concentric square inclusion, with tangent boundary conditions on the outer square edges. There are two geometrical parametersthe outer square edge length λ, and the aspect ratio ρ, which is the ratio of the inner and outer square edge lengths. The WORS exists for all geometrical parameters and for all temperatures; we prove that the WORS is globally stable for either λ small enough or for ρ sufficiently close to unity. We study three different types of critical points in this model setting: critical points with the minimal two degrees of freedom consistent with the imposed boundary conditions, critical points with three degrees of freedom and critical points with five degrees of freedom. In the two-dimensional case, we use Γ-convergence techniques to identify the energy-minimizing competitors. We decompose the second variation of the Landau-de Gennes energy into three separate components to study the effects of different types of perturbations on the WORS solution and find that it is most susceptible to inplane perturbations. In the three-dimensional setting, we numerically find up to 28 critical points for moderately large values of ρ and we find two critical points with the full five degrees of freedom for very small values of ρ, with an escaped profile around the isotropic square inclusion.

We introduce an operator S on vector-valued maps u which has the ability to capture the relevant topological information carried by u. In particular, this operator is defined on maps that take values in a closed submanifold N of the Euclidean space R m , and coincides with the distributional Jacobian in case N is a sphere. More precisely, the range of S is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use S to characterise strong limits of smooth, N -valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with N -well potentials.