Minimizers of a Landau-de Gennes Energy with a Subquadratic Elastic Energy

We study a modified Landau-de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two- and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small compared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimizers on two-dimensional domains, when the temperature is sufficiently low.


Introduction
Liquid crystals (LCs) are classical examples of mesophases that combine the fluidity of liquids with the orientational and positional order of solids [8]. Nematic liquid crystals (NLCs) are the simplest type of LCs for which the constituent asymmetric molecules have no translational order but exhibit a degree of long-range orientational order i.e. certain distinguished directions of averaged molecular alignment in space and time. The mathematics of NLCs is very rich and there are at least three continuum theories for NLCs in the literature -the Oseen-Frank, the Ericksen and the Landau-de Gennes theories. These theories typically have two key ingredients -the concept of a macroscopic order parameter and a free energy whose minimizers model the physically observable stable nematic equilibria. The Oseen-Frank theory is the simplest continuum theory restricted to purely uniaxial nematics with a single preferred direction of molecular alignment and a constant degree of orientational order. The Oseen-Frank order parameter is just a unit-vector field that models this single special direction, with two degrees of freedom, referred to as the director field. The Oseen-Frank energy density is a quadratic function of the director and its spatial derivatives; in the so-called one-constant approximation, the Oseen-Frank energy density reduces to the Dirichlet energy density. The Oseen-Frank theory has been remarkably successful but is limited to purely uniaxial materials and can only describe low-dimensional defects. For example, minimizers of the Dirichlet energy density can only support isolated point defects and these point defects have the celebrated radial-hedgehog profile with the molecules pointing radially outwards everywhere from the point defect [4]. Minimizers of the Oseen-Frank free energy with multiple elastic constants (subject to certain constraints) have a defect set of Hausdorff dimension less than one [18]. However, confined NLC systems frequently exhibit line defects and even surface defects or wall defects.
The Ericksen theory is also restricted to uniaxial nematics but can account for a variable degree of orientational order, labelled by an order parameter which vanishes at defect locations. This order parameter regularises higherdimensional defects. The Landau-de Gennes (LdG) theory is the most powerful continuum theory for nematic liquid crystals and the LdG order parameter is the LdG Q-tensor order parameter, which is mathematically speaking, a symmetric traceless 3 × 3 matrix with five degrees of freedom. The LdG Q-tensor can describe both uniaxial and biaxial nematic states, which have a primary and secondary direction of molecular alignment. The LdG free energy density usually comprises an elastic energy density (which is quadratic in the derivatives of the Q-tensor) and a bulk potential which drives the isotropic-nematic phase transition induced by lowering the temperature and the exact relation between Oseen-Frank and LdG minimizers has received a lot of mathematical interest in recent years.
We do not give an exhaustive review here; one of the first rigorous results in this direction is an asymptotic result in the limit of vanishing elastic constant studied by Majumdar & Zarnescu [24] and subsequently refined by Nguyen & Zarnescu [27]. The authors study qualitative properties of LdG minimizers with a one-constant elastic energy density and show that the LdG minimizers for appropriately defined Dirichlet boundary-value problems on three-dimensional bounded simply-connected domains, converge strongly in W 1,2 to a limiting minimizing harmonic map, which is the minimizer of the one-constant Oseen-Frank energy. The limiting map has a discrete set of point defects and the LdG minimizers converge uniformly to the limiting map, ev-erywhere away from the defects of the limiting map i.e. the limiting map is an excellent approximation of the LdG minimizers in this asymptotic limit, away from defects. In Contreras & Lamy [7] and Henao, Majumdar & Pisante [21], the authors study a different asymptotic limit, namely, the low temperature limit of minimizers of the LdG energy (with a one-constant elastic energy density) and prove that minimizers cannot have purely isotropic points with Q = 0 in this limit. Henao, Majumdar & Pisante demonstrate the uniform convergence of LdG minimizers to a minimizing harmonic map, away from the singularities of the limiting map, in this asymptotic limit. Using topological arguments, Canevari [5] shows that the non-existence of isotropic points for suitably prescribed Dirichlet data implies the existence of points with maximal biaxiality and negative uniaxiality (uniaxial with negative order parameter) in global LdG minimizers in this limit. These results clearly illustrate two features: using the one-constant Dirichlet elastic energy densities, the Oseen-Frank minimizers provide excellent approximations to the LdG minimizers in certain asymptotic limits; the differences are primarily contained near the defect sets of the limiting maps and the defects of the limiting map and the LdG minimizers can have different structures. However, we would expect the LdG defects (for minimizers) to shrink to the defects of the limiting minimizing harmonic map in the limit of vanishing nematic correlation length. For example, it is well known from numerical simulations that LdG minimizers have biaxial tori as defect structures with a negatively ordered uniaxial defect loop and as the nematic correlation length shrinks (in the limit of vanishing elastic constant), the biaxial torus shrinks to the radial-hedgehog defect, which is the corresponding defect for the limiting Oseen-Frank minimizer. In this respect, defects of the limiting map do give some insight into the defects of the LdG minimizers and vice-versa.
These continuum theories are variational theories with a quadratic elastic energy density or an energy density that is quadratic in the derivatives of the order parameter. However, there is little experimental evidence to support the quadratic behaviour in regions of large gradient i.e. near defects. Hence, it is reasonable to conjecture that the elastic energy density may be subquadratic near defects matched by a quadratic growth away from defects. For example, if the Oseen-Frank energy density was subquadratic for large values of the gradient, then line defects would be captured by the Oseen-Frank theory. This would be a significant development since one of the most popular reasons for choosing the LdG theory over the Oseen-Frank theory are the limitations of the Oseen-Frank approach with respect to defects.
Building on this idea, we propose a variant of the LdG energy, with a modified elastic energy density and the LdG bulk potential, for Dirichlet boundaryvalue problems on three-dimensional domains. The modified elastic energy exhibits a subquadratic growth in |∇Q| p with 1 < p < 2, for |∇Q| sufficiently large and interpolates to the usual Dirichlet energy density, |∇Q| 2 for bounded values of the gradient. This elastic energy density is necessarily not homogeneous, introducing various technical difficulties. A suquadratic variant of the Oseen-Frank theory was proposed by Ball & Bedford, [3].
We study minimizers of this modified LdG free energy, in the limit of vanishing elastic constant, by analogy with the work in Majumdar & Zarnescu [24]. The limiting map in our case, is a φ-minimizing map with a defect set of zero d− p Hausdorff measure, where d = 2 or d = 3 according to the dimension of the domain. The limiting map is C 1,α for α ∈ (0, 1) away from the defect set and we prove that the modified LdG minimizers converge uniformly to the φ-minimizing map away from the defect set of the φ-minimizing map. The essential difference is that the φ-minimizing map can support higherdimensional defects, in contrast to the minimizing harmonic map which can only support point defects. As noted above, we would expect the LdG defects to converge to the defects of the φ-minimizing map as the correlation length shrinks to zero and hence, a comprehensive study of the defects of the φminimizing map can yield new possibilities for the modified LdG defects too.
The second part of our paper concerns a qualitative study of minimizers of the modified LdG free energy in the low-temperature limit, for two-dimensional domains. Our qualitative conclusions are the same as Contreras and Lamy [7], who use the Dirichlet energy density i.e. the exclusion of purely isotropic points in global energy minimizers.
There are substantial technical differences between our work and previous work with the usual Dirichlet elastic energy density. The Euler-Lagrange equations in the modified case are only quasi-linear and not uniformly elliptic, we do not have exact monotonicity results for the normalized modified LdG energy on balls, we need different arguments for the regularity of the φ-minimizing limiting map and in the low-temperature limit, we need more technical details since the limiting map is a p-minimizing harmonic map for 1 < p < 2 as opposed to Contreras and Lamy who dealt with the p = 2 case for the low-temperature limit of LdG minimizers on two dimensional domains.
Our strategy for regularity is the following: first we get Morrey and C 1.α estimates for the minimizers Q L of the modified LdG functional, I mod , which possibly depend on L. These estimates are needed in order to prove L ∞ − L 1 estimates for the gradients using the Bernstein-Uhlenbeck method of passing through an uniformly elliptic equation, see [11]. The final goal of uniform estimates is reached by a combined use of monotonicity of the energy and a scaling procedure that does not affect the characteristics of φ, see Proposition 4.
From a physical standpoint, the overall story for a modified LdG elastic energy density that is subquadratic in |∇Q| in regions of large gradient, seems to be similar to the story for a Dirichlet elastic energy density with the difference being captured by the limiting φ-minimizing map as compared to the limiting minimizing harmonic map. The φ-minimizing map is expected to have a more complicated and higher-dimensional defect set and this will have consequences for the LdG minimizers too. However, it remains a difficult task to test these theoretical predictions for defect structures since the experimental resolution of defect structures or the determination of the elastic energy density near defects are open issues.

Setting of the problem and statement of the main result
Let Ω ⊂ R d be a bounded, smooth domain of dimension d ∈ {2, 3}. Let S 0 denote the space of symmetric, traceless 3 × 3 matrices given by where M 3×3 is the set of 3 × 3 matrices, Q = (Q ij ) and we have used Einstein summation convention. The matrix Q is the Landau-de Gennes tensor parameter. In particular, (i) Q is biaxial if it has three distinct eigenvalues; (ii) uniaxial if it has two non-zero degenerate eigenvalues such that the eigenvector associated with the non-degenerate eigenvalue is the distinguished director and (iii) isotropic if Q = 0. We study minimizers of the modified Landau-de Gennes energy functional where ψ(t 2 ) := φ(t). We will assume the following on φ: for all t > 0 and s ∈ R with |s| < t/2.
An example of admissible φ is where k > 0 is a fixed parameter (compare with [3]). Notice that the assumptions (H 1 )-(H 3 ), (H 5 ) guarantee the excess decay estimate for local minimizers of functional of Uhlenbeck type with φ-growth, see [11]. Further, f B is the usual quartic thermotropic potential that dictates the isotropic-nematic phase transition as a function of the temperature [23,2]: where trQ n = 3 i=1 λ n i for n ≥ 1, 3 i=1 λ i = 0, A is the re-scaled temperature and B, C are positive material-dependent constants whilst L > 0 in (2) is a fixed material-dependent elastic constant. The bulk potential f B is bounded from below and we add the constant M (A, B, C) to ensure that min S0 f B = 0. We work with temperatures below the critical nematic supercooling temperature or roughly speaking, we work with low temperatures so that A > 0 in this paper and f B attains its minimum on the set of uniaxial Q-tensors given below: and n ∈ S 2 an arbitrary unit-vector [26,23,2]. We take our admissible space to be where W 1,φ (Ω; S 0 ) is the Orlicz-Sobolev space of L φ -integrable Q-tensors with ∇Q ∈ L φ (Ω), see Section 2.1 . The Dirichlet boundary condition Q b ∈ W 1,φ (Ω; Q max ) by assumption, since this is a physically relevant choice that simplifies the subsequent analysis. In other words, we assume that where I is the 3×3 identity matrix, n b : Ω → S 2 and n b ⊗n b ∈ W 1,φ (Ω; M 3×3 ).

Remark 1
We have assumed that the boundary condition Q b is actually defined on the whole of the domain Ω, and belongs to the Sobolev-Orlicz space W 1,φ . However, in practical applications the behaviour of Q may only be assigned on the boundary of Ω, and one might ask whether there exists a map Q b ∈ W 1,φ (Ω; Q max ) that matches the prescribed behaviour at the boundary. A sufficient condition for the existence of such Q b is the following: let p ∈ (1, 2) be given by Assumption (H 4 ), and let P ∈ W 1−1/p,p (∂Ω; Q max ) be given; then, there exists a map Q b ∈ W 1,p (Ω; Q max ) such that Q b = P on ∂Ω, in the sense of traces [19, Theorem 6.2]. The assumption (H 4 ) implies that φ(t) t p + 1 and hence, we also have Q b ∈ W 1,φ (Ω; Q max ). Such an extension Q b ∈ W 1,p (Ω; Q max ) might not exist in case p = 2, due to topological obstructions associated with the manifold Q max (see e.g. [6,Proposition 6]).
In what follows, we re-scale the energy (2); letx = x D where D is a characteristic length scale of the domain Ω. It is a straightforward exercise to show that the re-scaled energy is In what follows, we will work with the re-scaled energy (8) and drop the bars for brevity. In particular, we will study qualitative properties of minimizers of (8) in the limitL → 0 which is the macroscopic limit that describes D 2 ≫ L A0 , for a typical correlation length ξ ∝ L A0 . To this purpose, we define a φ-minimizing uniaxial tensor-valued harmonic map, by analogy with the "minimizing harmonic map" employed for the Dirichlet elastic energy density i.e. |∇Q| 2 in [24]. All subsequent results and statements are to be interpreted in terms of the re-scaled energy (8).
Equivalently, a φ-minimizing uniaxial harmonic map is given by for a unit-vector field n 0 : Ω → S 2 such that the symmetric matrix n 0 ⊗ n 0 is a global minimizer of the functional in the admissible space and Q b and n b are related as in (7).
We can now state our main result.
Theorem 1 Suppose that the elastic energy density φ satisfies the Assumptions (H 1 )-(H 5 ) above. Let Q L be a minimizer of the functional (8) in the admissible class A defined by (6). Then, there exists a subsequence L k → 0 as k → +∞ and a φ-minimizing uniaxial harmonic map Q 0 such that the following properties hold: where p ∈ (1, 2) is given by Assumption (H 4 ), is closed and there holds

Notation, Orlicz spaces
In what follows, we use the notations f ∼ g and f g as short-hand for c 0 f ≤ g ≤ c 1 f and f ≤ c 2 g respectively, c 0 , c 1 , c 2 being some positive constants. We recall here some standard facts about N-functions (see e.g. [28] for more details). A real function φ : [0, +∞) → [0, +∞) is said to be an N-function if φ(0) = 0, φ is differentiable, the derivative φ ′ is right continuous, non-decreasing and satisfies φ ′ (0) = 0 and φ ′ (t) > 0 for t > 0. In particular, an N-function is convex. We say that φ satisfies the ∆ 2 -condition if there exists c > 0 such that φ(2t) ≤ cφ(t) for any t ≥ 0. We denote by ∆ 2 (φ) the smallest constant c such that the previous inequality holds. Given two N-functions φ 1 , φ 2 , we define ∆ 2 (φ 1 , φ 2 ) := max i=1,2 ∆ 2 (φ i ). If φ is an N-function that satisfies the ∆ 2 -condition, then If φ ′ is strictly increasing, then we denote by (φ ′ ) −1 : [0, +∞) → [0, +∞) the inverse function of φ, and we define The function φ * is called the Young-Fenchel-Yosida dual function of φ. The functions φ and φ * satisfy the so-called Young inequality, namely, for any ǫ > 0 there is C ǫ > 0 such that If ǫ = 1, then we can take C ǫ = 1. We can restate (H 3 ) in this way: uniformly in t > 0. The constants in (14) are called the characteristics of φ. We remark that under these assumptions ∆ 2 (φ, φ * ) < ∞ will be automatically satisfied, where ∆ 2 (φ, φ * ) depends only on the characteristics of φ and φ * . Next, we define the Orlicz space L φ (Ω) as the space of measurable function u such that´Ω φ(|u(x)|)dx < ∞. The Orlicz space is a Banach space, also it is reflexive if the function φ verifies the ∆ 2 condition and its dual is the space L φ * . The Orlicz-Sobolev space W 1,φ (Ω) is defined accordingly by requiring that both u and the distributional gradient ∇u belong to L φ .
For a given N-function φ, we define the N-function ω by We remark that if φ satisfies the condition (14), then also φ * , ω, and ω * satisfy this condition.
Define A, V : R d ⊗ S 0 → R d ⊗ S 0 in the following way: The function A represents the leading term of the φ-Laplacian system, while the function V , called the "excess" function, is the nonlinear expression for the excess decay, see Theorem 2.
Another important set of tools are the shifted N-functions {φ a } a≥0 . We The families {φ a } a≥0 and {(φ a ) * } a≥0 satisfy the ∆ 2 -condition uniformly in a ≥ 0. The connection between A, V (see [10]) is the following: uniformly in D.
is an N -function and that φ, φ * both satisfy the ∆ 2 -condition. Then we have, uniformly in λ ∈ [0, 1] and a ≥ 0, The proof of this lemma is a starightforward computation, based on the definition of the shifted function (17). In the paper [11], the authors proven that the analogue of Uhlenbeck result holds true for functionals with general growth.
There exists a constant c ≥ 1 and an exponent γ ∈ (0, 1) depending only on n, N and the characteristics of φ such that the following statement holds true: where Φ(h; x 0 , r, (∇h) x0,r ) is defined through the function V: 3 Asymptotic analysis of minimizers

Preliminaries
Proof The proof follows immediately from the direct methods in the calculus of variations. The admissible space A is non-empty since the Dirichlet boundary condition Q b ∈ A for each A > 0. The energy density in (8) is bounded from below and the energy density is a convex function of ∇Q for each A > 0, hence the functional I mod in (8) is bounded from below, W 1,φ -coercive and lower semi-continuous for each A > 0 [14]. Furthermore, the set A is weakly closed. This is enough to guarantee the existence of a global energy minimizer The critical points of I mod in (8) are C 1,α -solutions (for some 0 < α < 1) of the following system of Euler-Lagrange equations: for i, j, k = 1, 2, 3. The C 1,α -regularity of the weak solutions of the system (21) was first proven in the p-Laplacian case for p > 2 [30], for 1 < p < 2 see [1], for convex functions of general growth, see [25], and [11] where there is an excess decay. See Section 3.2 below for problems with a right-hand side. In [31], the author uses the C 1,α -regularity of solutions of a p-Ginzburg Landau-type system to deduce qualitative properties of minimizers of a p-Ginzburg Landau functional.
Our first result is a maximum principle argument for all solutions of the system (21).
Proof The Euler-Lagrange equations (21) can be written as We assume that |Q| 2 attains a maximum at an interior point x 0 ∈ Ω. Note that if |Q| 2 attains a maximum at an interior point x 0 ∈ Ω, then Q ij Q ij,k = 0 at x 0 . We multiply both sides of the above equation with Q ij using Einstein summation convention, and obtain the following equality at the interior maximum point x 0 ∈ Ω: where g (Q) = 1 L −A|Q| 2 − BtrQ 3 + C|Q| 4 . In [23], it is explicitly shown that g (Q) > 0 for |Q| 2 > 2 3 s 2 + . Using the definition of ψ(t) and an interior maximum point, we have that the left-hand side of (23) is non-positive whereas the right-hand side is strictly positive if |Q| (x 0 ) 2 > 2 3 s 2 + . Therefore, we must have at an interior maximum point x 0 ∈ Ω. Since the boundary datum Q b defined in (7) satisfies |Q b | 2 ≤ 2 3 s 2 + , the conclusion of the lemma follows.
Next, we consider the rescaled energy on balls B(x, r) ⊂ Ω defined to be where the number p ∈ (1, 2), depending on φ, is given by Assumption (H 4 ).
In [24], the authors consider the usual Dirichlet energy density, φ(|∇Q|) = 1 2 |∇Q| 2 , and show that the rescaled energy is an increasing function of the ball radius r. In our setting, we can show instead an "almost-monotonicity" formula, i.e. we show that the rescaled energy as defined above, is an increasing function of r "up to an error" that we can bound.
Lemma 4 Let Q L ∈ A be a minimizer of the functional I mod , and let Then, for any x 0 ∈ Ω and any 0 < r ≤ R such that B(x 0 , R) ⊂ Ω, there holds for some constant M > 0 that only depends on d, φ and p.
Since φ(t) ∼ t 2 for small t, we have φ(t) > 0 for small t > 0, whence the extra term. However, that term is bounded and small if R is small, so it is not a problem (see proof of Prop. 4).
Proof The monotonicity formula has been shown in the quadratic case [24, Lemma 2] by multiplying both sides of the Euler-Lagrange equation by x k ∂ k Q ij and integrating by parts. This argument could be adapted to our case but, in order to make the computation rigourous, some control on the second derivatives of Q L is needed. In order to avoid this technicality, we adopt here another approach. For simplicity, we assume, x 0 = 0, we write Q instead of Q L and B ρ instead of B(0, ρ). We first note that Now, we consider the so-called "inner variation" of Q, that is, we take a regular vector field X ∈ C ∞ c (Ω, R d ) and, for t ∈ R, we define the maps X t := Id +tX. For |t| small enough, X t maps Ω diffeomorphically into itself and so it makes sense to define Q t := Q • X t (here the • denotes the composition of maps). We can now compute that On the other hand, by the implicit function theorem we can express the inverse of X t as (X t ) −1 = Id −tX + o(t). Therefore, by using the identity det(Id +tA) = 1 + t tr A + o(t), we obtain Using the previous information, we can evaluate the energy of Q t by making the change of variable y = X t (x) in the expression for I mod : Now, we differentiate both sides with respect to t and evaluate for t = 0. Due to the minimality of Q, the left-hand side vanish, and hence, by rearranging, we obtainˆΩ In particular for a radial vector field X(x) = γ(|x|)x where γ is a smooth, compactly supported scalar function, we compute where ν := x |x| . Thus, (28) becomeŝ Rearranging the terms, we obtain Using this equality and the formula for the derivative of the rescaled energy (27), we get: Integrating the above formula with respect to ρ, and keeping in mind the definition of ξ (25), we get: so the first inequality in (26) follows. To conclude the proof we observe that, by Assumption (H 4 ), the function ξ is bounded from above. In particular, we have´B ρ ξ(|∇Q|) ≤ Cρ d for some constant C that only depends on φ, d, so the second line of (26) follows.
Proof The proof closely follows Lemma 3 in [24]. Let Q * be a φ-minimizing uniaxial tensor-valued map, in the sense of Definition 1. (The existence of such a map follows by routine arguments, based on the direct method of the calculus of variations). We note that Q * ∈ A and since Q * (x) ∈ Q max for a.e. x (recall Q max is the set of minimizers of the bulk potential f B in (8)), we have f B (Q * ) = 0 a.e. in Ω. We get the following chain of inequalitieŝ (30) This shows that the L φ -norms of ∇Q L are uniformly bounded in the parameter L and hence, we can extract a weakly convergent subsequence Q L k such that converges to zero pointwise almost everywhere in Ω, up to extraction of a non-relabelled subsequence. The bulk potential f B (Q) = 0 if and only if Q ∈ Q max ( [23,2]). Hence, the weak limit Q 0 is of the form Using the convexity of φ, which implies the weak lower semicontinuity of the corresponding integral functional, from (30) we get and, because Q * is φ-minimizing harmonic, These inequalities, together, imply that Q 0 is φ-minimizing harmonic and that Then, passing to the limit into (30), we deduce that 1 Finally, if there existed a smooth subdomain ω ⊂⊂ Ω and a further subsequence which contradicts the lower semi-continuity. Therefore, we must havê for any smooth subdomain ω ⊂⊂ Ω, whence the lemma follows.

Splitting type functionals
The aim of this section is to prove the following result: Proposition 1 Let L > 0 be fixed, and letQ L be a minimizer of the functional I LdG defined by (8). Suppose that the assumptions (H 1 )-(H 5 ) are satisfied. Then, there exists α ∈ (0, 1) (only depending on the characteristics of φ) such thatQ L ∈ C 1,α loc (Ω).
Throughout this section, L is fixed, so we will writeQ instead ofQ L . We deduce the proposition from the regularity results in [11] (see Theorem 2 above). However, these results apply to a problem without right-hand side (i.e., the case f B = 0). In order to reduce to this case, we consider a ball B(x 0 , R) ⊆ Ω. We compareQ with the solution P of the φ-harmonic system with boundary dataQ on the boundary of the ball B(x 0 , R): in the classQ+W 1,φ 0 (B(x 0 , R), S 0 ). The existence and uniqueness of P follows by the strict convexity of φ, by a routine application of the direct method of the calculus of variations. Recall that, by the maximum principle (Lemma 3), Q is bounded independently of L.
Proof Let us set A(∇Q) := φ ′ (|∇Q|) |∇Q| . We observe thatQ is a solution of the Euler-Lagrange system: We consider the difference between the above system and (35), testing with Q − P. Using (18), we can rewrite the right-hand side as follows: The right-hand side can be estimated taking into account that tr(Q − P) = 0, f B is a polynomial in Q andQ is bounded by the maximum principle (Lemma 3); this yieldŝ On the other hand, using Young (13) and Poincaré inequalities and (11), we obtain The minimality of P implies that the average of φ(|∇P|) is bounded from above by the average of φ(|∇Q|), and hence, the proposition follows.
Proof (of Proposition 1) We first show thatQ ∈ C α loc (Ω). Let B(x 0 , R) ⊆ Ω be a fixed ball, and let P be the solution of (35) on B(x 0 , R) such that P =Q on ∂B(x 0 , R). Let B ρ be a ball of radius ρ with B ρ ⊆ B(x 0 , R). We apply the previous estimate (Proposition 2), so to deducê Since P is φ-harmonic, we can use the L ∞ − L 1 estimate proven in [11] (Theorem 2, Eq. (20)): The last term can be bounded from above by− BR φ(|∇Q|)dx using the minimality of P. The term φ * (R) is also bounded from above, because R ≤ diameter(Ω) < +∞. Let us consider the quantity h(r) :=´B r φ(|∇Q|)dx. We have proven that for any 0 < ρ < R/2 it holds: for some positive constants C 1 that only depends on φ, d. By modifying the value of C 1 , if necessary, we can make sure that the same inequality holds for any 0 < ρ < R. We apply Giaquinta for all 0 < ρ < R and all 0 < σ < d, where c σ > 0 depends only on φ, d, σ. Thanks to Morrey's characterisation of Hölder continuous functions, we conclude thatQ ∈ C α (B(x 0 , R/2)) for any α ∈ (0, 1]. One we know thatQ is locally Hölder continuous, we can prove thatQ ∈ C 1,α loc (Ω) by adapting the arguments in [12, Proposition 5.1, third step]. (In the power case the proof was done in [13,Lemma 5].) Let B ρ ⊆ B(x 0 , R) be balls contained in Ω. Let P be the solution of the system (35) on the smaller ball B ρ , in the classQ + W 1,φ (B ρ , S 0 ). In the proof of Proposition 2 (see Equation (36)), we have shown that SinceQ is locally α-Hölder continuous, we have: for an arbitrary α ∈ (0, 1) and some constant C α,L depending on α, L but not on ρ. We can apply the convex-hull property for P: the image of P(B ρ ) is contained in the convex hull of P(∂B ρ ) =Q(∂B ρ ). Therefore, keeping in mind that P =Q on ∂B ρ , Let us consider the excess functional Φ, defined by Then, for any κ ∈ (0, 1/2), we have It follows from Theorem 2 that there exists γ > 0 and c > 0 only depending on d and the characteristics of φ, such that Φ(P, B κρ ) ≤ c κ 2γ Φ(P, B ρ ). Thus (the excess of P can be controlled by the excess ofQ using the minimality of P). Now, choose κ ∈ (0, 1/2) such that cκ 2γ ≤ 1/2. Then By iterating the previous inequality, for each j ∈ N we obtain Thus, there exists β > 0, c β > 0 (depending on κ) such that for all r ∈ (0, ρ) there holds Recall that B ρ is an arbitrary ball with B ρ ⊆ B(x 0 , R). Let B(x, r) ⊆ B(x 0 , R/2) be an arbitrary ball of radius r < R/4. We choose ρ := R/2 and s := (rρ) 1/2 = (rR/2) 1/2 , so that B(x, r) ⊆ B(x, s) ⊆ B(x, ρ) ⊆ B(x 0 , R). We apply (39) first on the balls B(x, r) ⊆ B(x, s), then on B(x, s) ⊆ B(x, ρ). We obtain Since Φ(Q, B(x, R/2)) Φ(Q, B(x 0 , R)) and α ∈ (0, 1) is arbitrary, for fixed x 0 , R, L the right-hand side grows as r β/2 , when r is small. This estimate, combined with Campanato's characterisation of Hölder functions, proves that V(∇Q) ∈ C β/4 (B(x 0 , R/4)). Since the function V is invertible and the inverse V −1 is µ-Hölder continuous, for some µ > 0 which only depends on the characteristics of φ [11, Lemma 2.10], we conclude that ∇Q ∈ C µβ/2 loc (Ω).

Subharmonicity
The aim of this subsection is to prove that, given a minimizerQ L of the functional (8), φ(|∇Q L |) is a subsolution of a scalar elliptic problem. This was the so-called Bernstein-Uhlenbeck trick used in the power case [30] for the case p ≥ 2, adapted for the general growth in [11]. Then there exists G : R d ⊗ S 0 → R d×d that is uniformly elliptic and satisfieŝ for all η ∈ C 1 0 (2B) such that η ≥ 0. Moreover, for all D ∈ R d ⊗ S 0 and all ξ ∈ R n there holds where α 0 , α 1 are positive constants that only depend on the characteristics of φ ′ .
Before proving the theorem, we need to prove the existence of second derivatives ofQ L because in the following computations we will encounter terms of the form´η |∇V(∇Q)| 2 dx. We can apply the results in [10] to deduce higher integrability and existence of second derivatives; in particular, we have V(∇Q L ) ∈ W 1,2 loc (Ω). To prove Theorem 3, it is convenient to work on an approximated system. For λ > 0 and t ≥ 0 we define and It follows from assumption (H 3 ) that there exist positive constants c 0 , c 1 (the characteristics of φ ′ ) such that for all t ≥ 0 and all λ > 0. Given L > 0 and a critical pointQ of the functional (8), we consider the approximated system in Q λ : subject to the boundary conditions Q λ =Q = Q b on ∂Ω. (Note that the righthand side is a function ofQ, not of Q λ , and hence can be trated as a given source term.) Since φ λ is strictly convex, this system has a unique solution Q λ for any given L,Q. Moreover, Q λ converges weakly toQ in W 1,φ as λ → 0 (see e.g. [11,Theorem 4.6]). The next results shows that φ λ (|∇Q λ |) is a subsolution to a uniformly elliptic problem, where the constants of ellipticity do no depend on λ > 0; we can then recover Theorem 3 by passing to the limit λ → 0.
Proof The proof parallels the one presented in [11](Lemma 5.4) , with an additional lower order term. Let η ∈ C 1 0 (2B). Let B R be a ball of radius R and let h ∈ R d \{0} with |h| ≤ min{dist(spt(η), ∂(2B)), 1}. Let τ h be the finite difference operator, defined by for an arbitrary function F : , so ξ is an admissible test function. By multiplying both sides of the system by ξ, and using that tr ξ = 0, we obtain We choose h := re l with l ∈ {1, . . . , d} and 0 < r ≤ dist(spt(η), ∂(2B)). Then, as r → 0, the left hand side converges to We now have to deal with the terms in the right-hand side, following the strategy in [11]. We only list the principal steps: and for r → 0 and the integral is well defined in L 1 . After summation over l = 1, . . . , n l,j,k Note that the constant does not depend on η ∈ C 1 0 (2B).
Then, for any index k we have This together with (47) implieŝ For all Q ∈ R N ×n and all ξ ∈ R n holds This implies where c 0 and c 1 are the constants from (42).

An
The aim is to prove that global minimizers of I mod , where I mod is defined in (8), converge uniformly to Q 0 everywhere away from the singularities of Q 0 . This is parallel to the work in [24] where the authors prove that global minimizers of a Landau-de Gennes energy with φ (|∇Q|) = 1 2 |∇Q| 2 converge uniformly to a limiting minimizing harmonic map in the limit L → 0, everywhere away from the singularities of the minimizing harmonic map and the limit L → 0 in [24] is equivalent to the asymptotic limit in this paper, modulo some scaling.
The key step in this section is the following result, which in inspired by [31, Lemma 2.3].
Proposition 4 There exist positive numbers r 0 , ε, Λ with the following property. Let B(x * , r) be a ball of radius r ≤ r 0 , let L > 0, and let Q L ∈ W 1,p (B(x * , r), S 0 ) be a minimizer of the functional I mod on B(x * , r). If there holds Before giving the proof of the proposition, we present some auxiliary material. Let δ 0 > 0 be a small parameter, to be specified later. For Q ∈ S 0 such that dist(Q, Q max ) ≤ δ 0 , we can write in a unique way where we assume that λ 1 ≤ λ 2 < λ 3 , 3 i=1 λ i = 0 and n, m, p are unit vectors. As in [24,Lemma 6], we define the projection of Q on Q max as Note that Π(Q) can be written as Π(Q) = s + (u ⊗ u − I/3) for some unit vector u, so Π(Q) ∈ Q max indeed. In fact, it can be shown that Π(Q) is the nearest-point projection of Q onto Q max , that is |Q − P| ≥ |Q − Π(Q)| for any P ∈ Q max (see e.g. [6,Lemma 12]). Then, the matrix defined by is normal to the manifold Q max at the point Π(Q). We note that Lemma 6 If δ 0 > 0 is small enough, then any matrix Q ∈ S 0 such that dist (Q, Q max ) < δ 0 has the following properties: for positive constants α 1 , α 2 , α 3 , α 4 that only depend on A, B, C.
Proof The lemma follows by [5, Remark 2.5 and Lemma 3.6]; see also [24,Lemma 6]. The crucial point is to show that for some α > 0 that only depends on A, B, C; then, the lemma follows by Taylor-expanding the function t ∈ R → f B (Π(Q) + tν(Q)) about t = 0.
Lemma 7 Let δ 0 > 0 be as in Lemma 6. Let B R be a ball of radius R. Let L > 0 be fixed, and let Q L ∈ W 1,φ (B R , S 0 ) be a solution of the Euler-Lagrange system in B R such that Then, there holds for a constant M > 0 that only depends on the characteristics of φ and on A, B, C.
Proof As in [31, Lemma 2.2], we take a cut-off η ∈ C ∞ c (B R ) such that η = 1 on B R/2 , 0 ≤ η ≤ 1 on B R , |∇η| 1/R and multiply the system by ν(Q L )η 2 . After integration by parts, and dropping the subscript L, we get Due to Lemma 6, the left-hand side of this formula bounds the left-hand side of (54) from above; therefore, it suffices to bound I. By expanding ∇(ν(Q)η 2 ) with the chain rule and using the fact that For the first term in the right-hand side, we use that φ ′ (t)t ∼ φ(t). For the second term, we apply the Young inequality (13) and Lemma 1: where φ * is defined in Section 2.1. Hence, the lemma follows.
Lemma 8 Let Q L be a critical point of the modified LdG energy I mod in (8), on the unit ball B 1 , for a fixed L > 0. We assume that where δ 0 is sufficiently small for (52) to hold, and that Then, for any integer q ≥ 1 and any 0 < θ < 1, there exist δ q ∈ [θ q , 1] and a positive constant C q > 0 (depending on Λ, θ, q, f B and the characteristics of φ but not on L) such that Proof From Lemma 7 and the bound (57), this is true for q = 1 and δ 1 := 1. By induction, we can assume that (58) is true for an integer q ≥ 2. Then by Fubini's theorem, there exists δ q+1 ∈ (θδ q , δ q ) such that We multiply both sides of the Euler-Lagrange equations (21) by L −q |Q − Π(Q)| q ν ij (Q) to get at the right-hand side (dropping the subscript L for brevity) where we have used the inequality (52). Similarly, at the left-hand side we haveˆB We estimate each integral on the right-hand side separately. The boundary integral can be estimated easily as shown below using the inequality (59): for a positive constantc independent of L; C q+1 has been defined in (59). The right-hand side is bounded due to (57). We then consider It is relatively straightforward to see that where we have used the hypotheses (57) and (59). It remains to note that [31] and the conclusion of the lemma follows.
We can now turn to the proof of Proposition 4.

Proof (of Proposition 4)
The proof follows from Lemma 7 of [24] and Lemma 2.3 of [31]. There is an additional step towards the end of the proof which was not needed/considered in either [24] or [31]. We can take x * = 0 without loss of generality. Choose r L 1 ∈ (r/2, r) and and max Such r L 1 and x L exist, because Q L ∈ C 1,α loc (Ω) (Proposition 1) and hence e L (Q L ) is continuous. Since r 2 < r+r L 1 2 < r, by definition of r L 1 we have Next, we set r L 2 := r−r L 1 2 K L andL := LK p L , with the scaled map the scaled elastic modulus and the scaled energy densitȳ We compute that |∇Q L | = K L |∇vL| and e L (Q L ) = K p LēL (vL), so we have from (66) and (67) that Further, from the Euler-Lagrange equations (21), we have that vL is a solution of div φ KL (|∇vL|) (in fact, vL is a minimizer of the associated functional, which is obtained from I LdG by scaling). We now consider two possibilities. If r L 2 ≤ 1 then, by choosing s = r/2 in Eq. (65), we deduce that verifying the pointwise bound on the energy density in (49). The second case is r L 2 > 1. We claim that if r L 2 > 1, then there exists a positive constant C 0 independent of L such that Assuming that (72) holds, we will get the required contradiction with (48), by appealing to the monotonicity of the normalised energy in Lemma 5. Indeed, since r L 2 > 1 by assumption, we have K −1 L < r and from (72) we deduce, by scaling, ≤ ε + M r p 0 (we have used that r ≤ r 0 ). Thus, we obtain a contradiction if we choose ε, r 0 small enough. The rest of the proof is dedicated to proving the inequality (72). By Equation (70) and Theorem 3, we know that wL :=φ KL (|∇vL|) is a subsolution of an elliptic problem, namely where the tensor field GL = GL(∇vL) is bounded and elliptic: Although the functionφ KL does depend on K L , the constants α 0 , α 1 are independent of K L . Indeed, by Proposition 3 we know that α 0 , α 1 only depend on the ratioφ which is bounded form above and below, independently of K L , thanks to Assumption (H 3 ). Now, let q > d be fixed, and let us set x + := min{x, 0} for x ∈ R. We claim that for some C q independent on L, K L . Let δ 0 > 0 be given by Lemma 6, and let Since f B (Q) > 0 if Q / ∈ Q max and f B (Q) → +∞ as |Q| → +∞, we have that γ > 0. We now consider two cases separately. Proof of (75), (76) -Case I:L ≥ 2 −p γ. This is straightforward because in this case, the left-hand sides of (75), (76) are uniformly bounded in terms of γ, A, B, and C (since we have the maximum principle, Lemma 3). Proof of (75), (76) -Case II:L < 2 −p γ. From (69), the energy densities are uniformly bounded and in particular f B (vL) ≤ 2 pL < γ on B 1 ⊆ B r L 2 , so, because of our choice of γ in (77), we have Then, we can consider the projection Π(vL) onto Q max . Since the potential f B is smooth, and in particular the second derivatives of f B are Lipschitzcontinuous, we have for some constant M that only depends on the coefficients of f B . Since Π(vL) belongs to the minimizing manifold of f B , the Hessian matrix of f B at Π(vL) is positive semi-definite and hence the first term in the right-hand side is nonpositive. Recalling that |∇vL| is bounded by (69), we obtain |vL − Π(vL)| q and the right-hand side is bounded by Lemma 8, so (75) follows. The proof of (76) follows by a similar reasoning, using the fact that Proof of (72): Because of (69), we must either haveL −1 f B (vL(0)) ≥ 1/2 or wL(0) =φ KL (|∇vL(0)|) ≥ 1/2. In caseL −1 f B (vL(0)) ≥ 1/2, we compute using the chain rule Now,L −1 (∇ Q f B )(vL) is uniformly bounded in L q (B 1/2 ) due to (76), while ∇vL is uniformly bounded in L ∞ (B 1 ) due to (69). It follows thatL −1 f B (vL) is uniformly bounded in W 1,q (B 1/2 ) with q > d and hence, by Sobolev embedding,L −1 f B (vL) is (1 − d/q)-Hölder continuous on B 1/2 , with uniform bound on the Hölder norm. Therefore, (72) follows immediately if we havē Finally, it only remains to consider the case wL(0) ≥ 1/2. In this case, we use the elliptic inequality (73), together with the ellipticity bounds (74) and the bound on the right hand side (75). By applying the theory for elliptic equations (see e.g. [16,Theorem 8.3]), we deduce that for any R ∈ (0, 1/2) and for some constants C 1 , C 2 that are independent of L. We choose , so that C 2 R 2(1−d/q) ≤ 1/4 and the second term in the right-hand side of (79) can be absorbed into the left-hand side. Then, using the interpolation inequality, we obtain whence (72) follows.
3.5 Uniform C 1,α estimates and the proof of Theorem 1 Proposition 4 provides a uniform bound, independent of L, in the regions where the energy is small (i.e., away from the singularities of the limiting harmonic map). In this section, we deduce a uniform C 1,α bound on Q L from the uniform bound on e L (Q L ); this will allow us to complete the proof of Theorem 1. To this end, given a ball B(x 0 , R) ⊂ Ω and a minimizer Q L of our functional, we consider again the φ-harmonic replacement of Q L inside B(x 0 , R), i.e. the unique solution P of div Lemma 9 Let Q L be a minimizer of the modified LdG energy I mod in (8), on the ball B(x 0 , 2R), for a fixed (but arbitrary) L > 0. We assume that Let P be the solution of (80). Then, there exists a positive constant C Λ (depending on Λ, but not on L) such that Proof By Equation (36) the proof of Proposition 2, we know that Under the assumption (81), we can apply Lemma 7 to obtain an uniform (i.e. L-independent) L q -bound for L −1 ∇ Q f B (Q L ), where q ∈ [1, +∞) is arbitrary. Then, the Hölder inequality giveŝ The assumption (81) also implies that Q L is Lipschitz continuous, and its Lispchitz constant is bounded in terms of Λ. Then, we can use the convex hull property, exactly as in the proof of (38), to check that Q L −P L ∞ (BR) ≤ C Λ R. Therefore, the lemma follows.
The C 1,α -bound for Q L is now obtained by repeating verbatim the arguments of Proposition 1 in Section 3.2. Since Lemma 9 gives a L-independent bound, we are now able to deduce regularity estimates that do not depend on L. As a result, we obtain the Lemma 10 Let Q L be a Lipschitz minimizer of functional (8), on the ball B(x 0 , R). Let Λ be defined as in (81). Then, there exists α ∈ (0, 1) (only depending on the characteristics of φ) and a constant C Λ,R , depending on Λ, R but not on L, such that We are now ready to prove Theorem 1.
Proof (of Theorem 1) Let Q L be a minimizer of I mod , in the class defined by (6). By Lemma 5 we know that, up to a non relabelled subsequence, Q L ⇀ Q in W 1,φ as L → 0, where Q 0 is a minimizing harmonic map. The set is closed, due to the monotonicity formula (Lemma 4), and moreover we have H d−p (S[Q 0 ]) = 0 , see [17]. Let ε > 0 be given by Proposition 4. For a fixed x 0 ∈ Ω \ S[Q 0 ], we can find a radius R = R(x 0 ) > 0 such that R p−d´B (x0,R) φ(|∇Q 0 |) ≤ ε/2. Then, due to Lemma 5, we have for any L small enough. We can then apply Proposition 4 and deduce that e L (Q L ) ≤ C R on B(x 0 , R/2), for some constant C R that depends on R but not on L. Finally, we apply Lemma 10 to obtain the uniform bound Q L C 1,α (B(x0,R/8)) ≤ C R . Thanks to this uniform bound and to Ascoli-Arzelà theorem, we deduce at once that Q 0 ∈ C 1,α (B(x 0 , R/8)) and that Q L → Q 0 , ∇Q L → ∇Q 0 uniformly on B(x 0 , R/8). This completes the proof of the theorem.

Biaxiality in the low temperature limit
In this section, we show the biaxial character of bidimensional defect cores, if the temperature is low enough. Throughout the section, we assume that Ω is a bounded, smooth domain in R 2 . To simplify the analysis, we take φ(t) := t p /p so that the elastic energy density reduces to 1 p |∇Q| p . This is consistent with our assumptions (H 1 )-(H 5 ) and, in this section, we are only interested in regions of large gradients. Therefore, we do not expect that this simplification should affect the qualitative conclusions of the analysis.
We introduce the rescaled temperature and we are interested in the limit as t → +∞. After a suitable non-dimensionalisation, along the lines of [21], the modified Landau-de Gennes free energy functional (8) reduces to is the optimal value of the scalar order parameter. Finally, is a potential that penalizes biaxiality (it is straightforward to verify that g(Q) is minimized by uniaxial tensors of unit norm). Since s + grows as t 1/2 as t → +∞, we have the asymptotic estimates Therefore, both the parameters T (t) and H(t) diverge as t → +∞, but the penalization associated with deviation from unit norm is stronger than the one associated with biaxiality.
Our main result of this section is the following.
Theorem 4 Let Q t be a minimizer of the functional F t , in the admissible class A defined by (6). For any δ > 0 there exists a positive number t 0 (δ) such that, if t ≥ t 0 (δ) and x ∈ Ω satisfies dist(x, ∂Ω) ≥ δ, then Q t (x) = 0.
This result guarantees that, if the temperature (measured in dimensionless units) is low enough, then the minimizer Q t does not possess any isotropic point, except possibly in a neighbourhood of the boundary. For a quadratic energy, i.e. when p = 2, it is already known that in the low temperature regime, minimizers do not have isotropic points, and this holds both in twodimensional [5,9] and three-dimensional domains [7,21]. While, on the one hand, the Landau-de Gennes potential favours biaxial phases over the isotropic one when the temperature is low, on the other hand isotropic defect cores such as the radially symmetric, uniaxial hedgehog are less heavily penalized by a subquadratic elastic energy, compared to the quadratic case.
In contrast with the quadratic case, we are not able to exclude the presence of isotropic points in a neighbourhood of the boundary. This boundary layer is related to the fact that it is difficult to obtain boundary regularity for the p-Laplace equation.
The absence of isotropic points means that biaxial escape takes place in the defect cores and indeed, the presence of biaxiality can be deduced from Theorem 4 by topological arguments. Given a matrix Q ∈ S 0 , we denote its eigenvalues by λ max (Q) ≥ λ mid (Q) ≥ λ min (Q). We also introduce the biaxiality parameter We recall that the biaxiality parameter satisfies 0 ≤ β 2 (Q) ≤ 1 and that β 2 (Q) = 0 if and only if Q is uniaxial.
Corollary 1 Suppose that the boundary datum Q b is topologically non-trivial, i.e. there is no continuous map Q : Let Q t be a minimizer of F t in the admissible class A defined by (6). Then, for any δ > 0 there exists a positive number t 0 (δ) with the following property: if t ≥ t 0 (δ) and λ max (Q t (x)) > λ mid (Q t (x)) for any x ∈ Ω with dist(x, ∂Ω) ≤ δ, then max x∈Ω β 2 (Q t (x)) = 1.
We prove Theorem 4 by contradiction, following the strategy in [7]. Suppose that there exists a number δ > 0 a sequence t j ր +∞ and a sequence of points (x tj ) in Ω such that From now on, we omit the subscript j and write t, x t instead of t j , x tj . Let δ t := δt 2/p−1/2 → +∞ as t → +∞. We define the blown-up map The mapQ t minimizes the rescaled functional subject to its own boundary conditions. Here,T (t) andH(t) are defined bȳ Thanks to (84), we see thatH(t) ∼ t −1/2 → 0 as t → +∞, whileT (t) is a bounded function of t and, in fact, it converges to a finite limit as t → +∞: Moreover,Q t is a solution of the Euler-Lagranges equations: on the ball B δt . A straightforward modification the maximum principle arguments in Lemma 3 shows that the solutions of this system of equations satisfy |Q t | ≤ 1 pointwise a.e. on B δt . Then, the right-hand side of (88) is bounded in L ∞ , uniformly in the parameter t > 0. The regularity theory for p-Laplace systems implies that for any R > 0 and any t large enough, for some uniform θ ∈ (0, 1) and some constant C R that depends on R but not on t. As a consequence, up to extraction of a subsequence, the mapsQ t converge locally uniformly to a continuous mapQ ∞ : Lemma 11 For any R > 0, the mapQ ∞ minimizes the functional Recall that γ has been defined in (87). Moreover, |Q ∞ | ≤ 1 on R 2 and there holdsQ ∞ (0) = 0.
Proof By the locally uniform convergenceQ t →Q ∞ , we immediately see that |Q ∞ | ≤ 1 on R 2 ,Q ∞ (0) = 0. Let Q ∈ W 1,p (B R ; S 0 ) be an admissible competitor forQ ∞ , i.e. a map such that (1 − |Q| 2 ) 2 ∈ L 1 (B R ) and Q =Q ∞ on ∂B R . By a truncation argument, we can assume w.l.o.g. that Q ∈ L ∞ (B R ). If t is large enough, so that B R ⊆ B δt , then Q +Q t −Q ∞ is an admissible competitor forQ t and we haveF t (Q t ) ≤F t (Q+Q t −Q ∞ ). Moreover, thanks to the uniform bound (89) and to Ascoli-Arzelà theorem, we deduce that ∇Q t → ∇Q ∞ locally uniformly in R 2 . We can hence pass to the limit as t → +∞ in the inequalityF t (Q t ) ≤F t (Q +Q t −Q ∞ ) and deduce thatF ∞ (Q ∞ ; B R ) ≤ F ∞ (Q; B R ).

Lemma 12
For any R > 0, there holds for some constant C that does not depend on R.
Proof We can reproduce the arguments of [21, Lemma 3.6] using the monotonicity formula given by Lemma 4.
For any R > 0, we define the map u R : Due to Lemma 11, u R satisfies |u R | ≤ 1 on B 1 , u R (0) = 0 and is a minimizer of the functional subject to its own boundary condition. Moreover, thanks to Lemma 12, the quantity G R (u R ; B 1 ) ≤ C is uniformly bounded, with respect to R. Therefore, up to extraction of a (non.relabelled) subsequence, the maps u R converge W 1,p -weakly to a limit map u * ∈ W 1,p (B 1 ; S 0 ) that satisfies |u * | = 1 a.e. on B 1 . In other words, u * takes values in the unit sphere of the 5-dimensional Euclidean space S 0 . We denote this sphere by S 4 .
The proof of Lemma 13 builds upon classical compactness arguments for harmonic maps which are due to Luckhaus [22], and is based on the following result, which is an adaptation of Lemma 1 in [22]. In contrast with the case considered in [22], we are dealing here with 2-dimensional domains only; on the other hand, we have to include in our analysis the Ginzburg-Landau potential (1 − |u| 2 ) 2 , which is not present in [22]. Similar results, for quadratic energies on three-dimensional domains, have been proven in [6]. We denote Lemma 14 There exist positive constants δ and C with the following property. Let 1/2 < ρ 0 < 1, 0 < λ < 1/4, R > 4 be given numbers, and set µ := λ+R −1 . Let u, v ∈ W 1,p (∂B ρ0 ; S 0 ) be given maps that satisfy Then, there exists a map w ∈ W 1,p (B ρ0 \ B ρ0(1−µ) ; S 0 ) such that w(x) = u(x) for a.e. x ∈ ∂B ρ0 , w(x) = v((1 − µ) −1 x) for a.e. x ∈ ∂B ρ0(1−µ) , and We postpone the proof of Lemmas 13 and 14, and conclude the proof of Theorem 4 first. The last ingredient is the following regularity result for minimizing p-harmonic maps:
With the help the previous results, Theorem 4 and Corollary 1 follow easily.
Proof (Conclusion of the proof of Theorem 4) Thanks to Lemma 13 and Proposition 5, we have strong convergence u R → u * in W 1,p loc (B 1 ), and moreover u * ∈ C 1,α loc (B 1 ). Then, by adapting the uniform convergence arguments above, we see that u R → u * locally uniformly in the open ball B 1 . But then we must have u * (0) = 0, since u R (0) = 0 for any R. This yields the desired contradiction.
Thanks to (94)-(96) and to the arguments in [5, Lemma 3.11], we can construct a continuous extension P : Ω → Q max of the boundary datum Q b . This map has the form P(x) := s + (n(x) ⊗ n(x) − I/3), where n(x) is a unit eigenvector associated with the leading eigenvalue λ max (Q t (x)). The existence of such an extension contradicts the topological non-triviality of Q b and completes the proof.
We now come back to the proof of the auxiliary results we used.
Finally, we give the proof of Proposition 5. In case p = 2, the result is known by the work of Hélein [20]. In case p > 2, the proposition follows by the results of Hardt and Lin [19, Corollary 2.6 and Theorem 3.1]. In case 1 < p < 2, it suffices to prove the following lemma: Lemma 15 Let B 1 be the unit disk in R 2 , let k ≥ 2, 1 < p < 2, and let w : B 1 → S k be a p-minimizing harmonic map that is homogeneous of degree 0, i.e. it satisfies w(x) = w(x/|x|) for a.e. x ∈ B 1 . Then, w is constant.