Regularity and the behavior of eigenvalues for minimizers of a constrained Q-tensor energy for liquid crystals

We investigate minimizers defined on a bounded domain in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document} for the Maier–Saupe Q-tensor energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Majumdar’s modification of the Landau-de Gennes Q-tensor model, so as to constrain the competing states to have eigenvalues in the closure of a physically realistic range. We prove that minimizers are regular and in several model problems we are able to use this regularity to prove that minimizers have eigenvalues strictly within the physical range.


Introduction
In this paper we prove regularity properties and bounds on the eigenvalues for local minimizers of an energy derived from Maier-Saupe theory, a model that is used to describe order in nematic liquid crystal materials (see [9].) We examine the special case of a liquid crystal occupying a cylindrical region in R 3 with cross-section ⊂ R 2 , where is an open bounded domain with a smooth (C 2 ) boundary. The liquid crystal material is described at almost every (x 1 , x 2 ) in by a 3 × 3 matrix Q(x 1 , x 2 ) and is assumed to be uniform in the x 3 -direction. The function Q is a tensor-valued order parameter, by which we mean that for almost every where κ and b 0 are constants, κ ≥ 0, f is convex, f ∈ C ∞ (M), and lim Q→∂M f (Q) = +∞. Such a function is bounded below and hence we assume without loss of generality that Our assumptions on the bulk energy density f b are motivated by the work in the papers [2] by Ball and Majumdar and [7] by Katriel, Kventsel, Luckhurst, and Sluckin. In these papers a particular potential function f (Q) = f ms (Q) as described above was constructed consistent with Maier-Saupe theory. More precisely, it was proved in [2] and [7] that for each Q ∈ M there is a unique functionρ =ρ(Q) in the set A Q := ρ ∈ L 1 (S 2 ; R) : ρ ≥ 0, S 2 ρ(p) dp = 1, Q = S 2 (p ⊗ p − 1 3 I )ρ(p)dp satisfying f ms (Q) := inf ρ∈A Q S 2 ρ(p) ln(ρ(p))dp = S 2ρ (p) ln(ρ(p))dp. (1.3) Given Q ∈ M the set A Q is the family of (absolutely continuous) probability densities ρ with Q as their normalized second moments. The set A Q is nonempty for Q in S 0 if and only if Q ∈ M; if Q ∈ ∂M one can still represent it as the second moment of a density, however in this case the density μ will be a singular measure and this is not considered physical. (See [2].) The densities in A Q then provide all possible physically admissible statistics for the local orientation of the liquid crystal molecules with second moments given by Q. It is proved in [2] and [7] that f ms (Q) has the properties postulated above for f (Q) except for the fact that f ms ∈ C ∞ (M). This fact is proved in [5]. We next describe our assumptions on the elastic energy density f e . Let Note that D includes the tangent space for differentiable mappings Q : → S 0 where is an open subset of R 3 , i.e., [∂ x k Q i j (x)] ∈ D for each differentiable S 0 -valued Q and each x in . We assume that f e (Q, D) is continuous on M × D and that there are constants 0 < α 1 ≤ α 2 < ∞, 0 ≤ M 1 ≤ M 2 < ∞ so that  (3) where Here we used the summation convention for repeated indices among , m, h in the set {1, 2, 3}. We remark that for the case of a Landau-de Gennes elastic energy density which is a function of D only, denoted by f ld (Q, D) = f (1) ld (D), it is known that f (1) with the elasticity constants satisfying (See [8].) This is a classic example among Landau-de Gennes models. More general examples of Landau de Gennes energy densities are also given in [8] such as where ε k j is the Levi-Civita tensor. The L 4 term above allows the model to account for molecular chirality, and cubic expressions such as the L 5 term permit a more specific description of elastic contributions. Since M is a bounded set one can easily give conditions on the elasticity constants so that f ) and that Q(x) ∈ M for almost every x ∈ . Note that in this paper we apply the energy density f ld (Q, D) to functions Q in H 1 loc ( ; M) where is a subset of the x 1 x 2 plane, and as such for the functions considered here we have D i j3 = Q i j,x 3 = 0. We consider fixed boundary conditions, (1.5) One can show that there exists Q ∈ A 0 such that F (Q) < ∞.
In this paper, under the assumptions described above on , f e , f b , and Q 0 , we analyze finite energy local minimizers of (1.1) and global minimizers in A 0 . Note that a Landau-de Gennes elastic energy density f e = f ld (Q, D) satisfies (1.4) if and only if it is of degree two in D for each Q and the mapping D ∈ D → f ld (Q, D) is strictly convex, uniformly in Q for Q ∈ M. In light of this, if f e = f ld the existence of global minimizers in A 0 follows in a standard way by using direct methods, just as in [2].
Our main results can be described as follows. In Sect. 2 we first prove that local minimizers of F in H 1 ( ; M) are uniformly Hölder continuous in subdomains whose closure is a compact subset of . This follows immediately from: is an open disk contained in with center o and radius 4r.
. Then there exists a constant σ in (0, 1) and c > 0 depending only on r 0 and on the constants in (1.2) and (1.4 We then consider local minimizers near ∂ . Let r 1 be a positive constant such that for each point o on ∂ , B 4r 1 (o) ∩ is a coordinate neighborhood for the C 2 structure of ∂ and is diffeomorphic to a half-disk.
In Maier-Saupe theory the set We next assume that f e = f ld (Q, D) [(and hence from our assumptions f ld satisfies (1.4)].
The condition Q ∈ S 0 allows us to express the nine components of Q in terms of the five independent variables as Q =Q(z), so that Q(x) =Q(z(x)). The energy takes the form where the coefficients are polynomials and there is a constant λ > 0 so that From elliptic regularity theory (see [6,10]) we obtain the following: Corollary 2 Assume f e = f ld (Q, D) and f ld satisfies our assumptions. Then: In Sect. 3 we analyze (Q) for local minimizers, in the case

If B is an open disk contained in , a finite energy local minimizer Q in H
Thus the natural ellipticity condition for (1.9) is It follows that if L 1 and L 5 satisfy (1.10) and L 4 is fixed then (1.4) holds for the elastic density (1.9) for some positive constants α 1 , α 2 , M 1 , M 2 . The uniform convexity of the elastic density f e (Q, D) in D is due to having Q constrained to values in M. This structure leads to the existence of minimizers for F over A 0 with f e as in (1.9) using direct methods. In contrast to this setting, in [2] Ball and Majumdar examined the classical unconstrained (Landau-de Gennes) energy that included the case where f e is as in (1.9) and f b is a polynomial in Q. They studied the problem of minimizing the energy over H 1 ( ; S 0 ) with Q = Q 0 on ∂ . They proved that if L 5 = 0 then the energy is not bounded below and the unconstrained minimum problem is ill-posed. Thus the unconstrained energy is ill suited for investigating certain relevant elastic effects. These observations motivate investigating the minimum problem for the Maier-Saupe energy. Minimizers for the constrained problem, however, may a priori take on un-physical states. We show that if f e is as in (1.9) then this is impossible. We prove the following physicality result.
Theorem 3 Let Q be a finite energy local minimizer for F (·; B) for a ball B ⊂ where f e is as in (1.9) satisfying (1.10). Then (Q) ∩ B = ∅ and we have Q ∈ C ∞ (B).
Theorem 3 generalizes a result of Ball and Majumdar [2] proved for the case in which is replaced by a domain D in R 3 , the elasticity density f e (∇ Q) = L 1 |∇ Q| 2 , and Q 0 is valued in a compact subset of M. They proved that minimizers Q in A 0 are in C ∞ (D) and have (Q) = ∅. In this paper we extend their result to minimizers and local minimizers defined on a domain ⊂ R n based on our approach. We work with a slightly more general setting than used above.

Theorem 4 Let be a domain in
Furthermore if ∂ is of class C 2,σ for some 0 < σ < 1 and u takes on boundary values u = u 0 ∈ C σ (∂ ; K), then u( ) ⊂⊂ K and it follows that u ∈ C σ ( ; K).
Applications of Theorem 4 to the Maier-Saupe energy functional are given in Corollaries 3 and 4. Further prior work concerning regularity results for the constrained minimum problem has been done by Evans et al. [4] where they investigated minimizers for energies of the form (1.1) assuming that ⊂ R n for n ≥ 2 and proved partial regularity results.

Hölder continuity and higher regularity
Our principal technique is to use elliptic replacements as a way of constructing comparison functions valued in M. where Proof Let tr A denote the trace of the square matrix A. We have that tr Q B (x) = 0 and where the C i are independent of s, ρ, and Q for ρ 2 ≤ s ≤ ρ. We next take s so that where ds denotes arc-length. This holds for almost every ρ (1.6). It follows that f (Q(z)) is a convex function of z and that it is a smooth function for all z such thatQ(z) is in this neighborhood.
It follows from this and the mean value theorem that Note that since f is convex we have As f is bounded below on M we can apply the dominated converges theorem to the right side and Fatou's lemma to the left as τ ↑ 1. Thus (2.4) holds for all s such that (2.3) is true. Next choose a constant C 6 = C 6 (b 0 , κ) and a value ρ Set s = s in (2.2) and (2.4); using these statements with (2.1) and (2.5) we arrive at where μ 1 , μ 2 , and C 7 are positive constants depending only on the constants in (1.2) and (1.4).
The above inequality allows us to do a "hole filling" argument. Adding We obtain, just as in the proof of Lemma 2.1 from [6], Ch. III the Morrey-type estimate for all x 0 ∈ B r (o) and ρ ≤ r 2 ≤ r 0 2 where σ > 0 depends on θ , and C depends on θ, r 0 , and C 8 . Since Q is bounded, the theorem follows from this and Morrey's theorem. (See [6], Ch. III, Theorems 1.1 and 1.2.) Proof of Theorem 2 Since ∂ is of class C 2 there exists 0 < ρ 0 ≤ r 1 so that satisfies the exterior sphere condition at each y ∈ ∂ with a ball B ρ 0 of radius ρ 0 . It follows that for each x ∈ and s > 0, B s (x) ∩ also satisfies the exterior sphere condition at each Just as in the Proof of Theorem 1, we have We next solve and note that since B s (x 0 ) ∩ satisfies the exterior sphere condition at each boundary point as described above we can construct a barrier function relative to each point in ∂(B s (x 0 ) ∩ ) so as to ensure that the normal derivative |∂ ν w(y)| ≤ C 2 s for each y ∈ ∂(B s (x 0 ) ∩ ) for which ∂ ν w(y) exists.
To see this let B ρ 0 (z) be the exterior sphere associated with y (so that B ρ 0 (z) (B s (x 0 ) ∩ ) = {y}) and set η = η(x) = |x − z|. Here we can take ρ 0 = ρ 0 (∂ ) > 0 fixed and assume without loss of generality that s < ρ 0 . Then B s (x 0 ) ∩ ⊂ B (2s+ρ 0 ) (z)\B ρ 0 (z) and the function This can then be used as a barrier so that It follows that We remark that one needs H (x) and f (H (x)) in H 2 (B s (x 0 ) ∩ ) to directly apply Green's identity as above, however these functions are only in H 1 (B s (x 0 )∩ ). We can circumvent this problem by using an approximation. We assume without loss of generality that s is sufficiently small so that B s (x 0 ) ∩ is strictly star-like with respect to a point p ∈ B s (x 0 ) ∩ and set We can then apply Green's identity for λ < 1 and obtain (2.7) in the limit as λ ↑ 1. We next replace the condition that H is separated from ∂M on ∂(B s (x 0 ) ∩ ) with the condition that ∂(B s (x 0 )∩ ) f b (Q)ds < ∞. This is done just as before by considering τ H for 0 < τ < 1 for which (2.7) is valid and letting τ ↑ 1.
Combining (2.6) and (2.7) we have and it follows that At this point we argue just as in the Proof of Theorem 1 arriving at for fixed positive constants β 1 , β 2 , and C 9 , where here they depend on Q 0 as well. We next extend Q in a neighborhood of of so that and extend f b ≡ 0 in \ . We then have for all x 0 in a neighborhood of ∂ and all ρ sufficiently small. Our assertion follows just as in the proof for Theorem 1.  [6,10]. Note that Q =Q(z(x)), f b (Q(z)) is a C ∞ function if z in a neighborhood of the range of z(x) for x in B 4r (o), and the coefficients of f ld are polynomials in z. By Theorem 1, z(x) ∈ C σ (B 4r (o)) and we can apply the result from [6], Ch. VI, Proposition1 asserting that in two space dimensions a continuous weak solution is in H 2, p for some p > 2, and thus, first order derivations are Hölder continuous. The argument given there proves an interior estimate. To prove regularity near the boundary, we assume Q is a local minimizer in ∩ B 4r (o), where o ∈ ∂ and r < r 1 , B 4r (o) ∩ ∂ is of class C k,α for some k ≥ 2 and α ∈ (0, 1), Q 0 ∈ C k,α (B 4r (o) ∩ ∂ ), and B 4r (o) ∩ (Q) = ∅. As before, Q =Q(z(x)) and f p (Q(z)) is a C ∞ function if z on a neighborhood of the range of z(x) for x ∈ B 4r (o) ∩ . Here it is enough to prove that Q has Hölder-continuous first derivatives in B r 2 (o)∩ . One can derive a reverse-Hölder type inequality on sets B 2r (x 0 )∩ where x 0 ∈ B 2r (o) ∩ ∂ in the spirit of that given in [6], Ch. V, Sec. 2, and then apply the same argument to prove the estimate near the boundary asserting that the solution is in

Proof of Corollary 2 To prove the interior regularity result, let B be an open disk in and
The details for this inequality are given in the Appendix. As before, higher regularities follows from linear theory.

Analysis of ( Q)
In Sect. 2 we proved that a local minimizer on a ball B 4r (o) ⊂ satisfies for a fixed 0 < δ < 1 by establishing the Morrey-type estimate for all y ∈ B r (o) and 0 < ρ ≤ r 2 . This was done by constructing comparison functions using harmonic replacements for each component of Q. In this section we consider f e satisfying (1.9) and (1.10), and prove that if Q is a finite energy local minimizer for F on an open set E then (Q) ∩ E = ∅. We do this by refining the notion of replacement in this case so as to prove that (3.2) holds for any 0 < δ < 1. We are then able to carry this one step further to get for all 0 < ρ sufficiently small. If o ∈ (Q) we know that lim y→o f b (Q(y)) = ∞ and if (Q) ∩ E is nonempty, this leads to a contradiction.

Proof of Theorem 3
Assume f e satisfies (1.9) and (1.10) and let Q be a local minimizer for F on B 4r (o) ⊂ . Assume without loss of generality that o ∈ (Q). We associate with f e and any V ∈ M the quadratic form corresponding to Using (1.10) we have that there exist constants μ 1 , μ 2 > 0 so that Let y = Ax be a linear nonsingular change of variables in R 2 . Given V (x) ∈ M set V (y) = V (A −1 (y)) and define the quadratic form Here whereŴ (y) is the 2×2 matrix whose lk entry is the lk entry ofV (y). Thus c i j (V (y)) are affine functions ofV lk (y) for 1 ≤ l, k ≤ 2. Choose and fix A so that By the chain rule, the energy F transforms tô Let 0 < s ≤ r . Then B s (y) ⊂ E and we can construct the L-replacement comparison function Q s (y) :=Q B s (y) (y) such that for each 1 ≤ , k ≤ 3 In the following estimates the constants M i depend on A 2 and δ from (3.2), however they are independent of s and r for 0 < s ≤ r ≤ r 0 . Since homogeneous solutions to L(y) satisfy the strong maximum principleQ s has the same properties as the harmonic replacements used earlier. We use two other features ofQ s . SinceQ s lk minimizes B s (y) c i j (Q(y))g y i g y j dy and let w(y) solve where C is constant on B s (y) such that Here we have used (3.1) and the formulas for the c i j . It follows that |∇w − ∇u| ≤ M 5 sr δ on B s (y).
Also, since f (Q s ) is a subsolution for L(y), we have We need to express this in terms of f b . In order to do this set Since c lk (Q) is a fixed affine function such that c lk (Q(o)) = δ lk it follows that Adding I to the right side of (3.7) then gives Also from (3.8) we get Next, using the fact that B s c dy = s 2 ∂ B s c ds if c is a constant and using (3.6) we have the estimate Thus adding I to the left side of (3.7) together with (3.9) gives Just as before we can drop the assumption thatQ| ∂ B s (y) is separated from ∂M and replace it with the condition that f b (Q) ∈ L 1 (∂ B s (y)). Also as in Sect. 2, usingQ s as a comparison function and the above inequality we get The left side of (3.10) equals For the second integral in (3.11), note that b(Q(y)) is constant andQ =Q s on ∂ B s (y). Thus this integral is zero. The first integral on the right side of (3.10) can be estimated using (3.1), (3.2), (3.5), (3.6) and the formula forf e . We find that it is bounded by We arrive at (3.12) Thus we havẽ Here m = m(r 0 ) such that 0 < m < 2 and can be taken as close to 2 as desired provided r 0 is taken sufficiently small to begin with. Now set for almost every 0 < s ≤ r . In what follows the constants C i will depend on r, δ, and η. Let 0 < η < min(3δ, m). Dividing by s 1+η and using the fact that γ (s) ≥ 0 we find Since min(3δ, m) > η we can integrate this to obtain γ (s) for η < min(3δ, m). We next get a corresponding, improved estimate for |∇Q| 2 . Since γ (s) is absolutely continuous on [0, r ], given 0 < t ≤ r we can select s 0 so that t 2 ≤ s 0 ≤ t and tγ (s 0 ) ≤ 2γ (t). Then from (3.13) we get (3.14) SinceQ s 0 solves (3.3) and (3.4) we have that (See [6]; Ch. III.) Thus using (3.5) and (3.14) B ρ (y) and it follows that for 0 < ρ ≤ t ≤ r , |y| ≤ r , and r ≤ r 0 . Then from [6]; Ch. III, Lemma 2.1 we find that B t (y) |∇Q| 2 dy ≤ C 9 t η for |y| ≤ r, t ≤ r.
Given 0 < θ < 1 set δ 1 = min( 5δ 4 , θ). We have strengthened (3.2) by showing that the inequality holds with δ replaced by δ 1 . The argument can be repeated a finite number of times to obtain the estimate with δ replaced by θ . We return to (3.12) setting y = o, s = r , and δ = θ . Using the definition of γ (r ) we find that there exists r 1 > 0 so that for almost every 0 < r ≤ r 1 . We need θ > 2 3 and set θ = 3 4 . Multiplying (3.15) by the integrating factor 2r −3 (1 + 2M 8 r 3/4 ) 5/3 we obtain Integrating from r to r 1 we get for 0 < r ≤ r 1 . We assumed thatQ(o) ∈ ∂M implying that lim y→o f b (Q(y)) = ∞ which is impossible. We next apply the harmonic replacement method to a problem with its domain in R n .
Proof of Theorem 4 1. Let B = B r (x 0 ) ⊂ ⊂ R n and let u B (x) be defined to have harmonic components and satisfy u B = u on ∂ B. It follows from the maximum principle and the fact that K is convex that u B (x) ∈ K for x ∈ B. We can then compare u and u B , (3.16) Since u B is harmonic we have that Using the fact that K is bounded and the Poincaré and Hölder inequalities we have that Next, just as in the proof for Theorem 1 we have that Combining these facts with (3.16) we get Given δ > 0 set δ = {x ∈ : dist(x, ∂ ) ≥ δ}. Then just as in the Proof of Theorem 3 we find that for x ∈ δ and r < δ that where C 4 = C 4 (γ , δ) and C 5 = C 5 (γ , δ, f (u) dx). It follows that f (u(x)) ≤ C 5 almost everywhere on δ . From the structure off we see that there is a constant β > 0, β = β(C 5 ) so that u(x) ∈ K β = {p ∈ K : dist(p, ∂K) ≥ β} for almost every x ∈ δ . Due to this we can take smooth variations with compact support about u for F 1 and as a consequence u satisfies the energy's Euler-Lagrange equation. Sincef (·) ∈ C ∞ (K) it follows that u ∈ L ∞ loc ( ) and we have that u ∈ C 1,α ( ) for 0 < α < 1. Higher regularity follows from elliptic estimates. with u B = u on ∂(B ). Then just as above we have Note that since u 0 ∈ C σ (∂ ; K) we have u 0 (∂ ) ⊂ K β for some β > 0 and as such u 0 is separated from ∂K. It then follows that there is a constant C 2 such that Using barriers we have |∂ ν w| ≤ C 3 r on ∂(B ). Since 1 2n (|x − o| 2 − r 2 ) ≤ w we have the specific estimate ∂ ν w ≤ r n on ∂ B. Using that f o (·) ≥ 0 we get We Thus We arrive at Multiplying the inequality by n r n+1 and integrating we get It follows that We can now prove that f (u) is bounded on . Indeed if x ∈ r 1 2 then from (3.17) we have f (u(x)) ≤ C 5 . If x ∈ \ r 1 2 then |x − o| = dist(x, ∂ ) ≤ r 1 2 for some o ∈ ∂ . Set r 2 = |x − o|. From (3.17) and (3.19), for ρ < r 2 it holds that and we find that f (u(x)) ≤ C 11 . It follows as above that u( ) ⊂ K β for some β > 0. We have that ∂ is C 2,σ , u 0 ∈ C σ and u ∈ L ∞ ( ). It holds that u ∈ C σ ( ). Indeed, we write u = h + z where h has harmonic components and h = u 0 on ∂ . It follows from elliptic L p estimates that z ∈ C 1,α ( ) for each 0 < α < 1 and from Schauder theory that h ∈ C σ ( ). See and the assertions follow from Theorem 4.
In the same manner we have the following result for the Dirichlet problem.

Corollary 4 Assume D ⊂ R 3 is a bounded domain with a C 2,σ boundary and Q
The last result was first proved by Ball-Majumdar in [2]. Specifically they worked with Q 0 ∈ H 1/2 (∂ D; M) that is valued in a compact subset of M and proved that Q(D) ⊂⊂ M.
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Appendix 1
In this section we derive a reverse Hölder inequality that when combined with the proof from [6]; Ch. VI, Proposition 1 demonstrates that local minimizers are regular near ∂ ∩ (Q). Let x 0 ∈ ∂ \ (Q). Fix r > 0 so that B 4r (x 0 ) ∩ (Q) = ∅ and take r sufficiently small so that B 4r (x 0 ) ∩ is diffeomorphic to a half-disk. From what has been proved we can assume that Q = Q(z) as in (1.6) Here the coefficients are C 1 functions on a neighborhood K of the range of z| B 2r (x 0 )∩ , such that For λ > 0 we assume that A lk i j (z)ζ i l ζ j k ≥ λ|ζ | 2 for ζ ∈ R 5×2 and z ∈ K, and z 0 ∈ C 2 (∂ ). Set δ = sup z − z 0 C(B 2r (x 0 )∩ ) .
We prove The general problem reduces to this case by flattening the boundary. This introduces an explicit x-dependence for the coefficients in (4.1) that does not affect the estimates; however the constants C i appearing below will also depend on .