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 $\mathbb{R}^2$ for the Maier--Saupe Q--tensor energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Mujamdar's modification of the Ginzburg--Landau 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 [8]. ) We examine the special case of a liquid crystal occupying a cylindrical region in R 3 with cross-section Ω ⊂ R 2 , where Ω 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 Ω 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 x = (x 1 , x 2 ) in Ω, Q(x) ∈ S 0 := {Q ∈ R 3×3 : Q = Q t and tr Q = 0}.
where R 3×3 denotes the space of 3 × 3 real-valued matrices. The energy considered here is: To describe our assumptions on the energy density of F , let Q ∈ S 0 and let λ 1 (Q) ≤ λ 2 (Q) ≤ λ 3 (Q) denote its eigenvalues. Define the set M ⊂ S 0 as the set of matrices Q ∈ S 0 for which λ i (Q) ∈ (− 1 3 , 2 3 ) for all 1 ≤ i ≤ 3. Then M is an open, bounded, and convex subset of S 0 . Note that ∂M is the set of all Q in S 0 such that λ i (Q) ∈ [− 1 3 , 2 3 ] for all 1 ≤ i ≤ 3 and λ j (Q) ∈ {− 1 3 , 2 3 } for at least one j. We assume throughout the paper that f b satisfies 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 [1] by Ball and Mujamdar and [6] 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 [1] and [6] that for each Q ∈ M there is a unique functionρ =ρ(Q) in the set ρ(p) ln(ρ(p))dp = S 2ρ (p) ln(ρ(p))dp.
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 [1].) 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 [1] and [6] 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 [4].
We next describe our assumptions on the elastic energy density f e . Let Note that D includes the tangent space for differentiable mappings Q : We assume that f e (Q, D) is continuous on M × D and that there are constants 0 for all (Q, D) ∈ M × D. A specific class of elastic energy density functions that we have in mind are the Landau-de Gennes elastic energy densities f ld (Q, D) defined as polynomials in terms of the components of Q and D that are Here we used the summation convention for repeated indices among ℓ, 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, with the elasticity constants satisfying (See [7].) This is a classic example among Landau-de Gennes models. More general examples of Landau de Gennes energy densities are also given in [7] such as where ε ℓkj 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 (2) ld satisfies (1.4). (See (1.10).) Note that in this paper we apply the energy density f ld (Q, D) to functions Q in H 1 loc (Ω, M) where Ω is a subset of the x 1 x 2 plane, and as such for the functions considered here we have We consider fixed boundary conditions, One can show that there exists Q ∈ A 0 such that F (Q) < ∞.
In this paper, under the assumptions described above on Ω, 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 [1].
Our main results can be described as follows. In Section 2 we first prove that local minimizers of F in H 1 (Ω; M) are uniformly Hölder continuous in subdomains whose closure is a compact subset of Ω. This follows immediately from: Theorem 1. Assume r 0 > 0, 0 < r < r 0 , and B 4r (o) is an open disk contained in Ω 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) such that Q ∈ C σ (B r ) and We then consider local minimizers near ∂Ω. Let r 1 be a positive constant such that for each point o on ∂Ω, B r 1 (o) ∩ Ω is a coordinate neighborhood for the C 2 structure of ∂Ω and is diffeomorphic to a half-disk.
In Maier-Saupe theory the set corresponds to locations where perfect nematic order occurs and this is interpreted as not physical. (See [1].) Note that by (1.2), if F (Q; E) < ∞ then Λ(Q) ∩ E has measure zero. Thus we have: Then Q is Hölder continuous in Ω and hence Λ(Q) is compact.
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 (1.6) z = (z 1 , . . . , z 5 ) = (Q 11 , Q 12 , Q 13 , Q 22 , Q 23 ), 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 a lk ij (z)ζ i l ζ j k ≥ λ|ζ| 2 for all (1.8) ζ ∈ R 5×2 and z such thatQ(z) ∈ M.
From elliptic regularity theory ( see [5] and [9] ) we obtain the following: Assume f e = f ld (Q, D) and f ld satisfies our assumptions. Then:

If
In Section 3 we analyze Λ(Q) for local minimizers, in the case 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 [1] 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 (Ω; S 0 ) with Q = Q 0 on ∂Ω. They proved that if L 5 = 0 then the energy is not bounded below and the unconstrained minimum problem is ill-posed. Thus the unconstained 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 ⊂ Ω where f e is as in (1.9) satisfying (1.10). Then Λ(Q) ∩ B = ∅ and we have Q ∈ C ∞ (B).
Theorem 3 generalizes a result of Ball and Majumdar [1] proved for the case in which Ω 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 M. They prove that minimizers Q in A 0 are in C ∞ (D) and have Λ(Q) = ∅. In this paper we extend their result to local minimizers defined on a domain Ω ⊂ R n based on our approach. (See Theorem 4 and Corollary 3.) Further prior work concerning regularity results for the constrained minimum problem has been done by L. Evans, O. Kneuss, and H. Tran in [3] where they investigated minimizers for energies of the form (1.1) assuming that Ω ⊂ 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.
Proof. Let trA denote the trace of the square matrix A. We have that Proof of Theorem 1.
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 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 2 ≤ s ≤ ρ so that 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 µ 2 Iterating this inequality and setting we obtain 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 [5], Ch. III, Theorem 1.1, 1.2, and Lemma 2.1.) Proof of Theorem 2. Since ∂Ω is of class C 2 there exists 0 < ρ 0 ≤ r 1 so that Ω satisfies the exterior sphere condition at each y ∈ ∂Ω with a ball B ρ 0 of radius ρ 0 . It follows that for each x ∈ Ω and s > 0, B s (x) ∩ Ω also satisfies the exterior sphere condition at each y ∈ ∂(B s (x) ∩ Ω) with a ball B ρ 0 .
Assume the hypotheses of Theorem 2. Let o ∈ ∂Ω, x 0 ∈ B r (o) ∩ Ω, 0 < ρ ≤ r, ρ 2 ≤ s ≤ ρ, and using Definition 2.1 set Just as in the proof of Theorem 1, we have Assume for now that H is separated from ∂M on ∂(B s (x 0 ) ∩ Ω). Then f is C ∞ on a neighborhood of H| Bs(x 0 )∩Ω and We next solve and note that since B s (x 0 ) ∩ Ω 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 ) ∩ Ω) so as to ensure that the normal derivative To see this let B ρ 0 (z) be the exterior sphere associated with y (so that Here we can take ρ 0 = ρ 0 (∂Ω) > 0 fixed and assume without loss of generality that s < ρ 0 . Then B s (x 0 ) ∩ Ω ⊂ B (2s+ρ 0 ) (z) \ B ρ 0 (z) and the function satisfies g(ρ 0 ) = 0, g(η) ≤ 0 and ∆g ≥ 1 for ρ 0 ≤ η ≤ ρ 0 + 2s.
This can then be used as a barrier so that |∂ ν w(y)| ≤ |g ′ (ρ 0 )| = 6s. The set B s (x 0 ) ∩ Ω is such that w ∈ H 2 (B s (x 0 ) ∩ Ω) and we have It follows that (2.7) We remark that one needs H(x) and f (H(x)) in H 2 (B s (x 0 ) ∩ Ω) to directly apply Green's identity as above, however these functions are only in H 1 (B s (x 0 ) ∩ Ω). 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 ) ∩ Ω is strictly star-like with respect to a point p ∈ B s (x 0 ) ∩ Ω and set H λ (x) = H(p + λ(x − x 0 )). Then for 0 < λ < 1 we have ∩ Ω) as λ ↑ 1. 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 )∩ Ω) with the condition that We write ∂(B s (x 0 ) ∩ Ω) = (∂B s (x 0 ) ∩ Ω) ∪ (B s (x 0 ) ∩ ∂Ω). Then using (1.5) 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 Ω ′ of Ω so that and extend f b ≡ 0 in Ω ′ \Ω. We then have for all x 0 in a neighborhood of ∂Ω and all ρ sufficiently small. Our assertion follows just as in the proof for Theorem 1.

Proof of Corollary 2.
To prove the interior regularity result, let B be an open disk in Ω and Q a local minimizer for F in H 1 (B; M). Assume B 4r (o) ⊂ B\Λ(Q). As outlined in Section 1 it is enough to consider solutions to the quasilinear Euler-Lagrange equation for F ′ from (1.7) satisfying the strong ellipticity condition (1.8) in B 4r (o). It suffices to show that our solutions have Höldercontinuous first derivatives in B r (o); higher regularity then follows using techniques from linear theory [5,9]. 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 [5], Ch. VI, Proposition 3.1 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 Ω ∩ B 4r (o), where o ∈ ∂Ω and r < r 1 , B 4r (o) ∩ ∂Ω is of class C k,α for some k ≥ 2 and α ∈ (0, 1), Q 0 ∈ C k,α (B 4r (o) ∩ ∂Ω), and B 4r (o) ∩ Λ(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) ∩ Ω. Here it is enough to prove that Q has Hölder-continuous first derivatives in B r 2 (o) ∩ Ω. One can derive a reverse-Hölder type inequality on sets B 2r (x 0 ) ∩ Ω where x 0 ∈ B 2r (o) ∩ ∂Ω in the spirit of that given in [5], Ch. V, Sec. 2, and then apply the same argument to prove the estimate near the boundary asserting that the solution is in H 2,p (B r 2 (o) ∩ Ω). The details for this inequality are given in the Appendix. As before, higher regularities follows from linear theory.

Analysis of Λ(Q)
In Section 2 we proved that a local minimizer on a ball B 4r (o) ⊂ Ω 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 Λ(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 ∈ Λ(Q) we know that lim y→o f b (Q(y)) = ∞ and if Λ(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) ⊂ Ω. Assume without loss of generality that o ∈ Λ(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 and V ∈ M.
By the chain rule, the energy F transforms tô c ij (Q(y))ĝ y i y j .
Let 0 < s ≤ r. Then B s (y) ⊂ E and we can construct the L-replacement comparison functionQ s (y) :=Q Bs(y) (y) such that for each 1 ≤ ℓ, 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 Bs(y) c ij (Q(y))g y i g y j dy among g ∈ H 1 (B s (y)) satisfying g =Q lk on ∂B s (y) it follows that Then where C is constant on B s (y) such that Here we have used (3.1) and the formulas for the c ij . 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 (3.9) ∂Bs(y) Also from (3.8) we get Next, using the fact that Thus adding I to the left side of (3.7) together with (3.9) gives Just as before we can drop the assumption thatQ| ∂Bs(y) is separated from ∂M and replace it with the condition that f b (Q) ∈ L 1 (∂B s (y)). Also as in Section 2, usingQ s as a comparison function and the above inequality we get We rewrite this as The left side of (3.10) equals

Using (3.3) and (3.4) the first integral in (3.11) equals and satisfies
Bs(y) c ij (Q(y))(Q lk,y i −Q slk,y i )(Q lk,y j +Q slk,y j )dy = Bs(y) c ij (Q(y))(Q lk,y i −Q slk,y i )(Q lk,y j −Q slk,y j )dy 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 Thus we haveμ 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) s η ≤ C 1 for 0 < s ≤ 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 We can use the harmonic replacement method to give a short proof of a local version of the Ball-Majumdar result. We do this here in a slightly more general setting. Let K ⊂ R m be an open, bounded, convex set where m ≥ 1.
Proof. Let B = B r (x 0 ) ⊂ Ω 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 , 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 Ω δ = {x ∈ Ω : dist(x, ∂Ω) ≥ δ}. Then just as in the proof of Theorem 3 we find that for x ∈ Ω δ and r < δ/2 that where C 4 = C 4 (δ, Ωf (u) dx). It follows thatf (u(x)) ≤ C 4 almost everywhere on Ω δ . From the structure off we see that there is a β > 0, β = β(C 4 ) so that u(x) ∈ K β = {p ∈ K : dist(p, ∂K) ≥ β} for almost every x ∈ Ω 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. Since f (·) ∈ C ∞ (K) it follows that ∆u ∈ L ∞ loc (Ω) and we have that u ∈ C 1,α (Ω) for 0 < α < 1. Higher regularity follows from elliptic estimates.
We can apply Theorem 4 to tensor-valued functions as in Section 1. Proof. Let {E 1 , . . . , E 5 } be an orthonormal basis for S 0 and parameterize W ∈ S 0 by W (u) = 5 i=1 u i E i for u ∈ R 5 . Then K := {u ∈ R 5 : W (u) ∈ M} is a bounded, open, and convex. For V ∈ H 1 (D; M) we can write and the assertions follow from Theorem 4.

A Appendix
In this section we derive a reverse Hölder inequality that when combined with the proof from [5]; Ch. VI, Proposition 3.1 demonstrates that local minimizers are regular near ∂Ω ∩ Λ(Q). Let x 0 ∈ ∂Ω\Λ(Q). Fix r > 0 so that B 4r (x 0 ) ∩ Λ(Q) = ∅ and take r sufficiently small so that B 4r (x 0 ) ∩ Ω is diffeomorphic to a half-disk. From what has been proved we can assume with boundary condition here the coefficients are C 1 functions on a neighborhood K of the range of z| B 2r (x 0 )∩Ω , such that for λ > 0 we assume that A lk ij (z)ζ i l ζ j k ≥ λ|ζ| 2 for ζ ∈ R 5×2 and z ∈ K, and z 0 ∈ C 2 (∂Ω). Set We prove Proposition A.1. There exist constants δ 0 , C > 0 depending on λ, M, z 0 , and Ω so that if δ ≤ δ 0 then D 2 z, |Dz| 2 ∈ L 2 (B r (x 0 ) ∩ Ω) and for 0 < ρ ≤ r The general problem reduces to this case by flattening the boundary. This introduces an explicit x-dependence for the coefficients in (A.1) that does not affect the estimates, however the constants C i appearing below will also depend on Ω.