Asymptotic Behavior of Nematic States in the Theory of Liquid Crystals under a Specific Applied External Fields

We consider the asymptotic behavior of nematic states of liquid crystals under a specific applied external fields. We use the Landau-de Gennes energy functional with the Dirichlet boundary condition for director fields. We show that in the case of equal elastic coefficients the pure nematic states are not global minimizers when we apply a strong field.


Introduction
In this paper, we consider nematic state of liquid crystals under non-constant applied external fields.Under the applied external (magnetic or electric) field H, we have to add a density −χ(H • n) 2 to the classical Oseen-Frank density of nematic liquid crystals F N (n, ∇n), and consider an energy functional: Here n : Ω → S 2 is a director field of the nematic liquid crystals and χ is a positive parameter.See de Gennes and Prost [4, p.287].
In the Landau-de Gennes theory, phase transitions from nematic states to smectic states can be described by minimizer (ψ, n) of the Landau-de Gennes energy functional: where κ, K 1 , K 2 , K 3 and χ are positive constants.K 4 is a constant, and q is a real number.Operator ∇ qn is denoted by ∇ qn = ∇ − iqn where i = √ −1.Without loss of generality, we may assume that q is nonnegative.
We consider the functional E under the strong anchoring condition which is the Dirichlet boundary condition on the director field n = u 0 on ∂Ω where u 0 is a given smooth boundary data.Then we can drop the divergence term Tr(∇n) 2 − |div n| 2 in (1.1) (see Hardt, et al. [5]).Let W 1,2 (Ω, C) and W 1,2 (Ω, R 3 ) be the usual Sobolev spaces of complex-valued functions and vector-valued functions, respectively, and Then we consider the functional E on the space W 1,2 (Ω, C) × W 1,2 (Ω, S 2 , u 0 ).
Throughout this paper, we only consider the case where its boundary ∂Ω is smooth, and assume that the applied external field H is non-constant and the boundary data u 0 is a constant unit vector with angle θ.Thus we assume that where h is a smooth unit vector field on Ω, u 0 = e is a constant unit vector and σ is nonnegative and denote the intensity of the applied external field H.In Lin and Pan [8], they treat the case where h and u 0 are constant unit vectors and θ = π/2.Also, in the case where h is a non-constant unit vector, u 0 is a constant unit vector and θ = π/2, we can see a result in Aramaki and Chinen [2].Moreover, it is treated in Aramaki [1] when an angle between h and u 0 is θ = π/2, that is, h • e = cos θ, and h and u 0 are constant unit vectors.We may assume that 0 < θ ≤ π/2 by replacing h and −h.For (1.2), when θ = π/2, since the result appears in [2], we assume that We write where is the simplified Oseen-Frank energy for nematics and is the Ginzburg-Landau energy for smectics.Moreover, we write The minimizers (ψ, n) of E describe the states of liquid crystals.The minimizers correspond to the nematic states, if ψ = 0, and the minimizers correspond to the smectic states, if ψ = 0. Then the energy functional E has trivial critical point.It is given by where n σ is a global minimizer of F σh : and we call (ψ, n) = (0, n σ ) the pure nematic state.
Here, we will give the main theorem.
Theorem 1.1.Let q, κ, K, h and e be given as above.Assume that (1.2) and (1.3) hold and curl h = 0 in Ω.Then if σ is large, the pure nematic states are not global minimizers.Moreover, we obtain

Preliminaries
In this section, we are setting ready to show the Theorem 1.1.This section is constructed based to [1] (cf. [8]). where We can show by a simple calculation, where and Hence where Thus we have the following condition: where ν is the outer unit normal vector field to ∂Ω.Define Then we have

Lemma 2.2. ([1]
) For any σ, K > 0, and h, e satisfy Here we consider the Euler-Lagrange equation for minimizer n of F σh .For any v ∈ W 1,2 0 (Ω, R 3 ), we compute the following equation: where λ is the Lagrange multiplier which depends on x.We use the identity curl 2 n = −Δn + ∇div n and n • n = 1, we have −Δn • n = |∇n| 2 .Hence we show easily that the Euler-Lagrange equation for n: Since h • e = cos θ > 0, we see that e is not a critical point of F σh for any σ > 0.

This implies (i).
Proof of (ii).From (i), we have Thus we see that Proof of (iii).First, since the boundary data e is smooth, from Schoen and Uhlenbeck [10, Theorem II] and [11, Proposition 3.1], we see that the singular set of n σ is a finite set and n σ is smooth near ∂Ω.
then the pure nematic states are not global minimizers of E. Assume that Ω is simply connected and σ > 0. If n is a minimizer of F σh , then F [n] > 0 and