On a Ginzburg–Landau type equationwith applications to liquid crystals

Giacomo CANEVARIUPMC – Paris 6, Laboratoire J.L. Lions, 75005 Paris, France

Under the supervision of Fabrice BETHUEL

We consider a generalization of the classical Ginzburg–Landau equation, and perform theasymptotic analysis in the vanishing elastic constant limit. Our discussion applies, in par-ticular, to the Landau–de Gennes model for nematic liquid crystals.

The general problemFormulation and assumptions

Let Ω ⊆ R2 be a smooth, bounded domain, let g : ∂Ω −→ Rd be a boundary datum, anddefine the Sobolev space H1

g(Ω; Rd) as the set of maps in H1(Ω; Rd) which agree with g onthe boundary. We are interested in the problem

minu∈H1

g (Ω;Rd)Eε(u) (1)

whereEε(u) := Eε(u, Ω) =

∫Ω

1

2|∇u|2 +

1

ε2f (u)

and f : Rd −→ R is a non negative, smooth function, satisfying these assumptions.

(H1) The set N := f−1(0) is a non empty, smooth, compact and connected submani-fold of Rd without boundary, called the vacuum manifold.

(H2) There exists k > 0 so that, for all p ∈ N and all normal vector ν ∈ Rd to N at thepoint p,

ν ·D2f (p)ν ≥ k |ν|2 .

(H3) For all v ∈ Rd with |p| > 1, we have

f (p) > f

(p

|p|

).

(H4) Every conjugacy class in the fundamental group π1(N ) is finite.(H5) g : ∂Ω −→ Rd is smooth, and g(x) ∈ N for all x ∈ ∂Ω.

The existence of a minimizer uε for Problem (1) follows from the Direct Method in Calculusof Variations. Every minimizer is smooth and solves the Euler–Lagrange equation

−∆uε +1

ε2Df (uε) = 0 in Ω .

Question. What happens in the limit as ε 0?

Main results

Theorem 1. Assume that (H1)–(H5) hold. There exist λ0, δ0 > 0 (independent of ε) and, for eachδ ∈ (0, δ0), a finite set Xε = Xε(δ) ⊂ Ω, whose cardinality is bounded independently of ε, so that

dist(uε(x), N ) ≤ δ if dist(x, Xε) ≥ λ0ε .

Theorem 2. Under the assumptions (H1)–(H5), there exist a subsequence of εn 0, a finite setX ⊂ Ω and a function u0 ∈ C∞(Ω \X ; N ) so that

uεn −→ u0 strongly in H1loc ∩ C

0(Ω \X ; Rd) .

On every ball B ⊂⊂ Ω \X , the function u0 is minimizing harmonic, which means

1

2

∫B|∇u0|2 = min

1

2

∫B|∇v|2 : v ∈ H1(B; Rd), v = u0 on ∂B

.

In particular, u0 solves the harmonic map equation

∆u0(x) ⊥ Tu0(x)N for all x ∈ Ω \X .

Remarks on the proof of Theorem 1

• The structure of the proof is classical, and based on the analysis of the Ginzburg–Landau model (see [1]).•One needs to find a constant κ∗ = κ∗(g), so that the estimate

Eε(uε) ≤ κ∗ |log ε| + C (2)

holds. (For the Ginzburg–Landau model, κ∗ = π |d|, where d is the degree of g).•We proceed as in [2]. Assume, for simplicity, that Ω is simply connected and the

fundamental group π1(N ) is abelian. To each α ∈ π1(N ) we associate its length

λ(α) := inf

(2π

∫S1

∣∣c′(θ)∣∣2 dθ)1/2

: c ∈ α ∩H1(S1, N )

.

•Denote by γ the homotopy class of the boundary datum g : ∂Ω ' S1 −→ N . Thenγ belongs to the fundamental group π1(N ), and it may be decomposed as a productγ = γ1γ2 . . . γk, where γi 6= 1. Then, we define

κ∗ := inf

1

4π

k∑i=1

λ(γi)2 : k ∈ N, γi ∈ π1(N ), γ =

k∏i=1

γi

.

•With this choice of κ∗, Equation (2) follows easily from a comparison argument.Moreover, κ∗ is optimal, since minimizers uε satisfy∫

Ω|∇uε|2 ≥ κ∗ |log ε| − C .

• Thanks to (H2), we argue a clearing–out property: For δ > 0 small enough, thereexist λ0, µ0 > 0 so that∫

B(x0, 2λ0ε)f (uε) ≤ µ0 implies dist(uε(x), N ) ≤ δ for x ∈ B(x0, λ0ε) .

This property, combined with local Pohozeav identities and covering arguments,leads to Theorem 1.

The Landau–de Gennes modelBiaxiality and uniaxiality

We consider now a special case: The target space Rd is replaced by the matrix set

S0 :=Q ∈M3(R) : QT = Q , trQ = 0

and the potential is given by

f (Q) =a

2trQ2 − b

3trQ3 +

c

4

(trQ2

)2. (3)

The potential (3) is proved to satisfy (H1)–(H4), hence the previous discussion applies.Moreover, in this case N ' P2 (R) (the real projective plane).

Define the biaxiality parameter by

β(Q) := 1− 6

(trQ3

)2(trQ2

)3for Q ∈ S0 \ 0 .

It can be checked that 0 ≤ β(Q) ≤ 1, and that β(Q) ∈ 0, 1 iff two eigenvalues of Qcoincide. In this case Q is said to be uniaxial, and biaxial otherwise. It holds that

Q ∈ N implies β(Q) = 0 .

Question. Assume that the boundary datum satisfies β(g(x)) = 0 for all x ∈ ∂Ω. Do mini-mizers Qε of (1)–(3) are uniaxial?

Theorem 3. Assume that the boundary datum satisfy (H5) (in particular, β(g) ≡ 0) and is nothomotopically trivial. There exist t0 > 0 with the following property: If

ac

b2≥ t0 ,

then there is a constant ε0 = ε0(a, b, c) > 0 so that for ε ≤ ε0 any minimizer Qε of (1)–(3) satisfies

maxΩ

β(Qε) = 1 .

Under the assumptions of the theorem, minimizers are not uniaxial. Biaxial configurationshave been investigated numerically (and there are experimental evidences for them).

A biaxial configuration, known as “biaxial torus”, as it appears in numerical simulations of the core of defects (thesefigures are taken from [3]).

• On the left, a schematic representation of a section with a plane through the symmetry axis of the core. The shape ofthe ellipses represents the degree of biaxiality.

• On the right, logarithmic plot of the biaxiality parameter β.

Idea of the proof of Theorem 3

• The potential given by (3), up to rescaling, behaves approximately as

f (Q) ' µ (1− |Q|)2 + σβ(Q) |Q|3 ,

where |Q|2 =∑ij Q

2ij and, setting t := ac/b2,

µ

a(t) = O(1) ,

σ

a(t) = O(t−1/2) as t −→ +∞ . (4)

•With the help of the coarea formula, we show that any functionQwith max β(Q) < 1fulfills

Eε(Q) ≥ κ∗ |log ε| + κ∗2

log µ− C1 . (5)

•We construct explicitly a comparison function Pε so that

Eε(Pε) ≤ κ∗ |log ε| + κ∗2

log σ + C2 . (6)

• The conclusion follows from (5) and (6), with the help of (4).

References[1] F. Bethuel, H. Brezis and F. Hélein, Ginzburg–Landau vortexes, Birkhäuser, Basel and Boston, 1994.

[2] D. Chiron, Étude mathématique de modèles issus de la physique de la matière condensée, PhD thesis, UniversitéPierre et Marie Curie, Paris, 2004.

[3] S. Kralj and E. G. Virga, Universal fine structure of nematic hedgehogs, J. Phys. A 34 (2001), no. 4, pp. 829–838.