Stability of half-degree point defect profiles for 2-D nematic liquid crystal

In this paper, we prove the stability of half-degree point defect profiles in $\mathbb{R}^2$ for the nematic liquid crystal within Landau-de Gennes model.

1. Introduction. Defects in liquid crystal are known as the places where the degree of symmetry of the nematic order increases so that the molecular direction cannot be well defined. The most striking feature of liquid crystal is a variety of visual defect patterns. Predicting the profiles of defect as well as stability is thus of great practical importance and theoretical interest. We mention some works [2,16,23,27] on the defects based on the topological properties of the order parameter manifolds.
There exist three commonly used continuum theories describing the nematic liquid crystal: Oseen-Frank model, Ericksen model and Landau-de Gennes model. In the Oseen-Frank model, the state of nematic liquid crystals is described by a unitvector filed which represents the mean local orientation of molecules, and defects are interpreted as all singularities of this vector field [9,10,6,19]. However, the core structure of defects in nematic liquid crystals, such as the disclination lines observed in experiments, cannot be represented by the usual director field and requires description by Landau-de Gennes model [4]. In this model, the state of nematic liquid crystals is described by a 3 × 3 order tensor Q belonging to 6228 ZHIYUAN GENG, WEI WANG, PINGWEN ZHANG AND ZHIFEI ZHANG For Q ∈ Q, one can find s, b ∈ R, n, m ∈ S 2 with n · m = 0 such that where I is a 3 × 3 identity matrix. The local physical properties of nematic liquid crystals depend on the degree of symmetry of order tensor Q. Specifically, there are three different states: 1. s = b = 0, which describes the isotropic distribution; 2. s = 0, b = 0, which corresponds to the uniaxial distribution; 3. s = 0, b = 0, which describes the biaxial distribution.
Configuration of nematic liquid crystals corresponds to local minimizers of the Landau-de Gennes energy functional, whose simplest form is given by where L > 0 is a material-dependent elastic constant, and f B is the bulk energy density, which can be taken as follows where a 2 ,b 2 ,c 2 are material-dependent and non-zero constants, which may depend on temperature. A well-known fact is that f B (Q) attains its minimum on a manifold N given by . It is easy to see that N is a smooth submanifold of Q, homemorphic to the real projective plane RP 2 , and contained in the sphere Q ∈ Q : |Q| = 2 3 s + . Critical points of Landau-de Gennes functional satisfy the Euler-Lagrange equation The Landau-de Gennes energy (1) and Euler-Lagrange equation (2) are widely used to study the behavior of defects, see [1,3,7,22] and references therein. However, there still exist many challenging problems in understanding the mechanism which generates defects and predicting their profiles as well as stability, see [11] for many conjectures. The radial symmetric solution in a ball or in R 3 , named hedgehog solution, is regarded as a potential candidate profile for the isolated point defect in 3-D region. The property and stability of this solution are well studied and it is shown that the radial symmetric solution are not stable for large a 2 and stable for small a 2 [13]. We also refer [26,21,17,12] and references therein for related works.
In this paper, we are concerned with a class of point defects in R 2 , which correspond to "radial" solutions of the Euler-Lagrange equation (2). Here "radial" means that the eigenvectors of Q don't change along the radial direction. Precisely speaking, we study the solution with the form where (r, ϕ) is the polar coordinate in R 2 , and The boundary condition on these solutions is taken to be which has degree k 2 about origin as an RP 2 -valued map. Here k ∈ Z \ {0}. Note that if we assume the invariance of Q along the defect line, then disclination line in 3-D domain can be ideally treated as a point defect in 2-D domain.
In [14], Ignat, Nguyen, Slastikov and Zarnescu proved the existence of the radial solution for any non-zero integer k. Moreover, the solution is also a local minimizer of the 1-dimensional reduced functional (see (9)). An important question is whether the radical solution they constructed is a local minimizer of the energy F LG . This problem was also partially answered in [14], where the instability result is proved for |k| > 1. However, the question of whether the k-radially symmetric solutions (3) subject to (6) for k = ±1 are stable remains open.
The goal of this paper is to give a positive answer to this question. Precise result will be stated in next section. We remark that this problem is somewhat analogous to the stability of radial solutions of the Ginzburg-Landau equation(see [18,24,20,8,25] for example).
2. The stability of radially symmetric solution with k = ±1. We make the following rescaling and let t = a 2 c 2 b 4 . Then Landau-de Gennes energy functional (1) is rescaled into the form(drop the tildes): Therefore, without loss of generality, we may take L = b = c = 1 and a 2 = t > 0. In such case, substituting (3) into (2), (u, v) satisifes the following ODE system(see [5,11]): together with the boundary conditions where . In [14], it has been constructed a solution to (7)-(8) with u > 0, v < 0 and Define the 1-D reduced energy density A solution (u, v) of (7)-(8) is called a local minimizer of the 1-D reduced energy of (1) if is a local minimizer of the 1-D reduced energy of (1), then The definition in the last line can be extended to all function V ∈ H 1 (R 2 , Q). We say that a solution Q to the Euler-Lagrange equation is a local minimizer of the Landau-de Gennes energy (1) The main result of this paper is stated as follows.
The equality holds if and only if This result implies that the solution (3) for k = ±1 can be regarded as the profile of point defects in R 2 or the local profile of line defects in R 3 . The proof is based on the following properties of (u, v) on (0, ∞): We will prove these properties in Proposition 1.

Remark 1.
Since we have the invariance of the energy F LG under translation: as well as the invariance of the energy F LG under rotation: one can obtain the following functions in the "null space": Here E 0 , E 1 , · · · , E 4 are defined in (13). Although these functions do not belong to H 1 (R 2 ), they inspire us to construct suitable identities to prove the theorem.
Remark 2. The result (with slight difference) have been independently obtained in the work [15]. They also considered the uniqueness of solution of (7)-(8) for t large.
3. The second variation of Landau-de Gennes energy. To prove the stabillity, we need to compute the second variation of F LG at critical point Q = u(r)F 1 + v(r)F 2 . For any V ∈ H 1 (R 2 , Q), we have We define A straightforward calculation shows which implies that {E i } 0≤i≤4 is an orthonormal basis in Q. Thus, we can write V ∈ H 1 (R 2 , Q) as a linear combination of this basis in the polar coordinate Using (14), a direct calculation yields that thus we obtain Here f r and f ϕ denote ∂ r f and ∂ ϕ f respectively. In summary, we conclude that In order to prove I(V ) ≥ 0, it suffices to show that for |k| = 1, and The following lemma shows that , it suffices to show that any C ∞ c (R 2 ) function can be approximated by C ∞ c (R 2 \{0}). For this, we introduce a smooth cut-off function χ(r) defined by It is easy to see that the first two terms on the right hand side tend to zero as N → +∞. While, the third term is bounded by which tends to zero as N → +∞.

Some important integral identities.
In this section, let us derive some important integral identities, which will play crucial roles in our proof. In the sequel, we assume that η ∈ C ∞ c ((0, +∞), R). Using (7), we deduce that Taking derivative to (7) gives Therefore, we have and In addition, we have 5. Monotonicity of u and v. In this section, we will prove that u and v are monotonic if (u, v) is a local minimizer for the 1-D reduced energy functional. The proof is based on several contradiction arguments, which is similar to the work [17] on the monotonicity of the scalar order parameter of uniaxial radial hedgehog solution.
When k = 1, we can write Noticing that m 2 ≥ 1, (1 − m) 2 ≥ 0 for m ≥ 1, and using the following simple relations we conclude that Thus, we only need to show that for any q 0 , q 1 ∈ C ∞ c ((0, ∞), R),
6.2. Non-negativity of I A (w 0 , w 1 , w 2 ). First of all, we expand w i (r, ϕ) as Since Thus, we can decompose I A (w 0 , w 1 , w 2 ) as where and From the assumption of local stability of (u, v), we have In addition, it follows from (19) that It remains to prove I A n ≥ 0 for all n ≥ 1, which is a consequence of the following proposition.
If the left hand side is zero, then we must have (up to multiply a nonzero constant) q 1 = u, q 0 = −3v.