Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime

We study global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional domains, under a Dirichlet boundary condition. In a relevant range of parameters (which we call the Lyuksyutov regime), the main result establishes the nontrivial topology of the biaxiality sets of minimizers for a large class of boundary conditions including the homeotropic boundary data. To achieve this result, we first study minimizers subject to a physically relevant norm constraint (the Lyuksyutov constraint), and show their regularity up to the boundary. From this regularity, we rigorously derive the norm constraint from the asymptotic Lyuksyutov regime. As a consequence, isotropic melting is avoided by unconstrained minimizers in this regime, which then allows us to analyse their biaxiality sets. In the case of a nematic droplet, this also implies that the radial hedgehog is an unstable equilibrium in the same regime of parameters. Technical results of this paper will be largely employed in Dipasquale et al. (Torus-like solutions for the Landau- de Gennes model. Part II: topology of S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^1$$\end{document}-equivariant minimizers. https://arxiv.org/pdf/2008.13676.pdf; Torus-like solutions for the Landau- de Gennes model. Part III: torus solutions vs split solutions (In preparation)), where we prove that biaxiality level sets are generically finite unions of tori for smooth configurations minimizing the energy in restricted classes of axially symmetric maps satisfying a topologically nontrivial boundary condition.


Introduction
Nematic liquid crystals are mesophases of matter between the liquid and the solid phases. Nematic molecules typically have elongated shape, approximately rod-like, and can flow freely, like in a liquid, which forces their long axes to align locally along some common direction. This feature is the key for the extreme responsivity of nematics to external stimuli, which in turn is the reason why they are so useful in technological applications. Macroscopic configurations of nematics are usually described by continuum theories, the most successful being the phenomenological Landaude Gennes (LdG) theory ( [60,13,2,44]) which accounts for the most convincing description of the experimentally observed optical defects [30,34]. In the present paper, we study minimizers of the Landau-de Gennes energy functional in three dimensional domains under topologically nontrivial boundary conditions (e.g. radial anchoring). The goal here is to shed some light on the emergence of topological structure in the so-called biaxial surfaces associated to energy minimizing configurations, proving some mathematically rigorous results on the nature of the defects which, at least in model geometries, is expected to be of torus type [48,31,18,32].
According to the (LdG) theory, we let M 3×3 (R) be the real vector space made of 3 × 3-matrices, and we consider its 5-dimensional subspace where Q t denotes the transpose of Q, and tr(Q) the trace of Q. The space S 0 is endowed with the Hilbertian structure given by the usual (Frobenius) inner product. Since the matrices under consideration are symmetric, the inner product and the induced norm are given by P : Q := 3 i,j=1 P ij Q ij = tr(P Q) and |Q| 2 = tr(Q 2 ) .
Upon the choice of an orthonormal basis, S 0 can be identified with the Euclidean space R 5 . In particular, Q ∈ S 0 : |Q| = 1 = S 4 .
Let Ω ⊆ R 3 a bounded domain with (at least) C 1 -smooth boundary, and Q : Ω → S 0 a configuration in the Sobolev space W 1,2 (Ω; S 0 ). We consider the Landau-de Gennes energy functional of the form i.e., with the one-constant approximation for the elastic energy density with parameter L > 0 and quartic polynomial bulk potential where a, b and c are material-dependent strictly positive constants. It is convenient to subtract-off an additive constant and introduce so that the potential becomes nonnegative. In order to discuss the qualitative properties of energy minimizing configurations (such as isotropic/nematic phase transition, biaxial escape), we find convenient to modify the usual definition of biaxiality parameter as follows. Observe that if the matrix Q has a spectrum σ(Q) = {λ 1 , λ 2 , λ 3 } ⊆ R, and we order the eigenvalues increasingly, then β(Q) = ±1 iff the minimal/maximal eigenvalue is double (purely positive/negative uniaxial phase), β(Q) = 0 iff λ 2 = 0 and λ 1 = −λ 3 (maximal biaxial phase), and Q = 0 iff λ 1 = λ 2 = λ 3 (isotropic phase). It turns out that the potential is minimal when the signed biaxiality is maximal, and F B (Q) = 0 iff Q ∈ Q min , i.e., if Q is in the vacuum-manifold of positive uniaxial matrices Q min := Q ∈ S 0 : Q = s + n ⊗ n − 1 3 I , n ∈ S 2 , (1.5) where is the positive root of the characteristic equation Notice that, up to a multiplicative constant, Q min ∼ RP 2 , therefore it has nontrivial topology.
In particular, there are nontrivial homotopy groups π 2 (Q min ) = Z and π 1 (Q min ) = Z 2 , which are relevant for the presence of topological defects. The corresponding energy functional is (1.8) which is the sum of two nonnegative term, penalizing respectively spatial variations and deviations from the vacuum manifold Q min . We rescale the tensor by setting In this way, the vacuum manifold becomes exactly the real projective plane RP 2 = S 2 /{±1}, where RP 2 ⊆ S 4 is embedded as in (1.5), i.e., through the so-called Veronese immersion. We rewrite the energy functional as with F λ,µ (Q) := (1.10) The reduced parameters λ and µ are given by and the reduced smooth potential W : S 0 → R is nonnegative and vanishes exactly on RP 2 . More precisely, in view of (1.6)-(1.7), the potential W is explicitely given by 11) or equivalently, (1.12) The structure relations (1.10) and (1.11) suggests that, in a regime where µ is large, the energy F λ,µ favours rescaled configurations of approximatively unit norm.
In this paper, we first make the fundamental assumption that the norm of any admissible configuration is given by the constant value proper of the vacuum manifold [40], i.e., (1.14) The restriction of the potential W : S 0 → R to S 4 is given by (1. 15) In particular, W is nonnegative on S 4 , {W = 0} ∩ S 4 = RP 2 and ∇ tan W (Q) = 0 for any Q ∈ RP 2 . As a consequence, when further restricted to the subspace of uniaxial configurations W 1,2 (Ω; RP 2 ), the energy functional (1.14) reduces to the Dirichlet integral, i.e., the Frank-Oseen energy in the one-constant approximation. For an account on the qualitative properties of defects in the Frank-Oseen model, we refer the interested reader to e.g. [1,8].
A critical point Q λ ∈ W 1,2 (Ω; S 4 ) of E λ among S 4 -valued maps satisfies in the sense of distributions in Ω the Euler-Lagrange equation 16) with the tangential gradient of W along S 4 ⊆ S 0 given by Notice that the left hand side of (1.16) is the so-called tension field of Q, a tangent field along Q in S 4 , and equation (1.16) is nothing but the harmonic map equation for S 4 -valued map with the extra term λ∇ tan W (Q) as a source term. Any tensor field Q which is weakly harmonic among S 4 -valued maps and lying in the subspace W 1,2 (Ω; RP 2 ) is also a weakly harmonic among map in W 1,2 (Ω; RP 2 ) 1 , and provides a solution to (1.16). Since everywhere discontinuous weakly harmonic maps among maps in W 1,2 (Ω; RP 2 ) do exist (see [50]), we expect smoothness of solutions to (1.16) to fail in general, and to prove regularity we shall rely in an essential way on energy minimality.
We consider the minimization for the energy functional E λ among maps in W 1,2 (Ω; S 4 ) satisfying a Dirichlet boundary condition (strong anchoring) in the sense of traces. We fix a Lipschitz (at least) boundary trace Q b ∈ Lip(∂Ω; S 4 ), and we consider the set of admissible configurations A Q b (Ω) := Q ∈ W 1,2 (Ω; S 0 ) : Q |∂Ω = Q b , |Q| = 1 a.e. in Ω ⊆ W 1,2 (Ω; S 4 ) , (1.17) which is nonempty by [27]. Hence, one can fix a reference extensionQ b ∈ A Q b (Ω), which, as a matter of fact, can be chosen in C 0 (Ω; S 4 ), or even smooth in the interior since π 2 (S 4 ) = 0 (so that density of smooth maps in A Q b (Ω) holds). By the direct method in the Calculus of Variations, it is routine to show that there exist minimizers Q λ ∈ A Q b (Ω) of E λ . Requiring the boundary condition to be at least C 1,1 -regular allow some smoothness up to the boundary of the corresponding minimizers. Indeed, the first result of our paper is the following regularity theorem.
The proof of this theorem is based on ideas and techniques from the regularity theory of harmonic maps, starting from the pioneering papers [54,55,56], as summarized in the books [57,46,37]. The crucial point is to obtain Lipschitz continuity, while higher order regularity follows from standard Calderon-Zygmund and Schauder theories and the analyticity results in [45]. Both for the interior and for the boundary regularity, the main steps are : 1) monotonicity formulae; 2) strong compactness of blow-ups; 3) constancy of blow-up limits (Liouville property); 4) continuity under smallness of the scaled energy (ε-regularity); 5) Lipschitz continuity. Within this general scheme, our proof presents some differences and simplifications we want to comment on. The monotonicity formula here is not obtained by inner variations but instead by a penalty approximation which is, to our knowledge, new. More precisely, we relax the norm constraint, and passing to the limit in monotonicity formulae for approximated problems, we obtain interior and boundary monotonicity formulae. The validity of the latter requires the Lipschitz assumption on Q b as in the harmonic map case (see e.g. [11]). Strong compactness of blow-ups is obtained by construction of comparison maps arguing as in [57] for harmonic maps again. Indeed, the perturbed Dirichlet energy E λ is treated as in [47] with techniques based on the Luckhaus interpolation lemma [39], both in the interior and near the boundary. As strong limits of blow-ups are degree-zero homogeneous and minimizing harmonic maps into S 4 , i.e, Q * (x) = ω (x/|x|) for some harmonic sphere ω : S 2 → S 4 , their constancy in the interior case directly follows from [56]. At a boundary point, limits of blow-ups are of the form Q * (x) = ω + (x/|x|) with x 3 > 0 (up to rotations in the domain), for some harmonic half-sphere ω + : S 2 + → S 4 with constant trace on ∂S 2 + . The constancy of those ω + is derived in a way similar to [55]. We emphasize that boundary regularity is not deduced from minimality, but instead by a reflection argument and the 2d-interior regularity result for weakly harmonic maps from [25], and constancy of ω + follows from [35] as in [55]. Since our argument for boundary regularity does not rely on energy minimality, it will be possible to apply it also in the symmetric case we consider in our companion papers [15,16]. Our approach to ε-reguarity treats in a unified way the interior and the boundary case, adapting for the latter the clever reflection argument devised in [52] for harmonic maps (under the same regularity assumptions on ∂Ω and Q b ). Moreover, Hölder-continuity under smallness of the scaled energy is not deduced as usual from Hardy-BMO duality as in [17], or using the integrability by compensation due to the hidden 1 Observe that the converse implication is not true in general, because the Veronese immersion is minimal but it is not totally geodesic, and the tension field of Q in S 4 could be purely orthogonal to RP 2 but nonzero. Thus, if Q is weakly harmonic among map in the space W 1,2 (Ω; RP 2 ), i.e., it is a critical point of the Frank-Oseen energy, then it does not solve (1.16) in general. Hence it is not a critical point of the Landau-de Gennes energy under norm constraint.
antisymmetric structure of the quadratic gradient term in (1.16) as in [51] for harmonic maps. In the present paper, we adapt to our context the elementary iteration approach introduced in [10], based on the divergence structure of the quadratic gradient term and the decay properties of the BMO seminorm together with the integral characterization of Hölder-continuous functions due to Campanato. Finally, Lipschitz continuity is obtained using the harmonic replacement argument of [53] for harmonic maps.
With Theorem 1.2 at hand, we now remove the norm constraint (1.13), and we consider the full energy functional (1.10). We minimize F λ,µ over maps in W 1,2 (Ω; S 0 ) still satisfying a Dirichlet boundary condition (strong anchoring) in the sense of traces. Given a Lipschitz boundary trace Q b ∈ Lip(∂Ω; S 0 ), existence of minimizers Q µ λ ∈ W 1,2 (Ω; S 0 ) follows again from the direct method in the Calculus of Variations. In addition, applying usual interior and boundary regularity for semilinear elliptic equations to the Euler-Lagrange equation satisfied by critical points of F λ,µ (see (4.1)), we can infer that Q µ λ ∈ C α (Ω; S 0 ) ∩ C ω (Ω; S 0 ) for some α ∈ (0, 1). At this stage, we are interested in the asymptotic behaviour of minimizers Q µ λ in the range of parameters (that we call Lyuksyutov regime) These regimes resemble the low-temperature limit and the small elastic constant limit, respectively. Under these restrictions on the parameters, the last term in F λ,µ acts as a penalty approximation of the norm constraint (1.13), and it is natural to expect convergence of the family {F λ,µ } µ to the functional E λ (in the sense of Γ-convergence, see e.g. [7]), and that minimizers of F λ,µ converge to minimizers of E λ .
, and any such Q µ λ belongs to C ω (Ω) ∩ C 1,α (Ω) for every α ∈ (0, 1). In addition, as µ → ∞ with λ constant (Lyuksyutov regime), the following holds: In particular, for each λ > 0, there exists a value µ λ = µ λ (λ, Ω, Q b ) > 0 such that for µ > µ λ , any minimizer Q µ λ of F λ,µ satisfies |Q µ λ | > 0 in Ω, i.e., minimizers do not exibit the isotropic phase. In this theorem, existence and regularity properties of minimizers Q µ λ have already been recalled in the discussion above. Claim (1) and (2) can be seen as a standard consequence of the Γconvergence of the family {F λ,µ } µ to E λ , although for the reader's convenience such notion is not explicitly used in the proof (but just mentioned for readers familiar with it). As a matter of fact, the claims rely on a sharp two-sided bound on the energies {F λ,µ (Q µ λ )} µ , the lower semicontinuity property of the energy functionals (i.e., the Γ-liminf inequality), the construction of trial sequences (i.e., the recovery sequences for the Γ-limsup inequality), and the standard weak compactness in W 1,2 coming from the equicoercivity of the energies (i.e., the compactness in Γ-convergence). As a consequence, minimum points strongly converge to minimum points in W 1,2 , and the two claims follow as the upper and the lower bound mentioned above coincide. The second claim tells us in particular that the limiting map must be S 4 -valued as a direct consequence of Fatou's Lemma. Finally, claim (3) is by far the most interesting as it guarantees that the isotropic phase is avoided by energy minimizing configurations Q µ λ in the Lyuksyutov regime (as already proved in [9] and [12,26] in the low-temperature limit in 2D and 3D, respectively). Our proof of this property is based in a crucial way on Theorem 1.2. Indeed, smoothness of the limiting minimizer Q λ and the strong W 1,2 -convergence yield smallness of the scaled energy of Q µ λ at a sufficiently small scale. Combining monotonicity formulae with elliptic regularity in a way similar to [43], we are then able to show that |Q µ λ | has to converge to one uniformly as µ → ∞. To illustrate our results so far, let us now consider the model case of a nematic droplet, i.e., Ω = {|x| < 1} is the unit ball. The outer unit normal to the boundary is → n(x) = x/|x|, and a natural boundary datum is the so called radial anchoring, namely Since → n : ∂Ω → S 2 is harmonic, the homogeneous extensionH(x) = Q b (x/|x|) (the constantnorm hedgehog) is a weakly harmonic map from Ω into RP 2 , and it is energy minimizing for its own boundary value in W 1,2 (Ω; RP 2 ) by the lifting property of W 1,2 -maps in RP 2 in [3] and the celebrated result in [8]. Moreover, a direct computation shows thatH is also a weak solution to (1.16), i.e., a critical point of E λ . AsH is singular at the origin, Theorem 1.2 tells us thatH is not minimizing E λ in the class A Q b (Ω). We shall prove in Proposition 4.7 thatH is in fact strictly unstable, employing an argument similar to [56], an explicit computation of the second variation of energy, and a perturbation localized near the origin.
Still in the case of a nematic droplet subject to radial anchoring, the energy functional F λ,µ has an O(3)-equivariant (radial) critical point commonly known as radial hedgehog This solution is obtained from a unique function s λ µ (|x|) increasing from 0 to 3/2 solving an ODE with the prescribed values at 0 and ∞, see e.g. [42,29] and the references therein. It turns out to be the unique uniaxial critical point of F λ,µ among all critical points (not necessarily uniaxial), see [33]. As the origin is an isotropic point, Theorem 1.3 shows that H µ λ does not minimize F λ,µ in the class W 1,2 Q b (Ω; S 0 ), at least for µ large enough. Thus, biaxial escape must occur for minimizers. Using the strong convergence of H µ λ toH as µ → ∞, we are able to pass to the limit in the second variations of F λ,µ at H µ λ , and we infer in Theorem 4.8 the instability of H µ λ through biaxial perturbations for µ large enough. Both properties are the counterpart in the Lyuksyutov regime of the instability of the radial hedgehog in the low-temperature limit (essentially a 2 → ∞) already proved in [28] (see also [19,42]) together with the (infinitesimal) biaxial escape phenomenon obtained there.
Once the smoothness of Q λ and the absence of isotropic phase for Q µ λ are established, we can discuss for both cases the topological properties related to the presence of biaxial phase, and the way they are connected with the topology of the vacuum manifold. The starting point is that, under the assumption Q b ∈ C 1,1 (∂Ω; S 4 ), Q λ and Q µ λ are configurations satisfying the hypothesis The first assumption on the boundary condition that we impose is the following The caseβ = 1 occurs for the main and most natural example of such boundary condition, which is (in view of the embedding RP 2 ⊆ S 4 ) In particular, if ∂Ω is of class C 2 , the choice v(x) = → n(x) (the outer unit normal to the boundary ∂Ω) corresponds to the so-called homeotropic boundary condition (or radial anchoring).
If Ω ⊆ R 3 is a bounded open set with boundary of class at least C 1 , we know that ∂Ω is a finite union of embedded C 1 -surfaces. More precisely, ∂Ω = ∪ N i=1 S i where the surfaces S i are of class (at least) C 1 , disjoint, embedded, connected, orientable, and boundaryless. The second assumption we make on (the boundary of) Ω is Ω is connected and simply connected. Under this assumption, each surface S i has zero genus, so it is an embedded sphere (see Lemma 5.1). The domain Ω is thus a topological ball with finitely many disjoint closed balls removed from its interior. By assumption (HP 1 ), the maximal eigenvalue λ max (x) of Q b (x) is simple for every x ∈ ∂Ω. Hence there exists a corresponding well defined C 1 -smooth eigenspace map V max ∈ C 1 (∂Ω; RP 2 ), and this map as a (nonunique) lifting v max ∈ C 1 (∂Ω; S 2 ) since each surface S i has zero genus. To enforce the emergence of topology in the minimizers, we finally make a third assumption Notice that this property depends only on the map V max , it and does not depend on the choice of the lifting v max . In case of radial anchoring (i.e., is satisfied whenever N is odd, that is whenever ∂Ω has an odd number of connected components (or, equivalently, if the domain Ω is a topological ball with an even number of disjoint closed ball removed from its interior).
In order to emphasize the consequence of assumptions (HP 0 )-(HP 3 ) on a configuration Q satisfying Q = Q b on ∂Ω, let us assume for a moment that Q b takes values into the vacuum manifold, i.e., Q b ∈ C 1 (∂Ω; RP 2 ). First, observe that Q b has a lifting by (HP 2 ), i.e., Q b is of the form (1.21). Moreover, any lifting v ∈ C 1 (∂Ω; S 2 ) of Q b admits a finite energy extensionv ∈ W 1,2 (Ω; S 2 ) (see e.g. [27]), but no continuous extension because of assumption (HP 3 ). As a consequence, Q b admits an extensionQ b ∈ W 1,2 (Ω; RP 2 ) of the form In view of [3] and (HP 3 ), any extensionQ b ∈ W 1,2 (Ω; RP 2 ) of Q b is in fact of the form (1.22) for a suitable (necessarily) discontinuous mapv ∈ W 1,2 (Ω; S 2 ). The configuration Q being smooth and without isotropic phase by assumption (HP 0 ), it cannot be purely uniaxial (i.e., RP 2 -valued) and biaxial escape must occur for purely topological reasons.
To describe the way a configuration Q encodes some topological information, we shall make use of the biaxiality function as follows.
Definition 1.4. For a configuration Q ∈ C 0 (Ω; S 0 \ {0}) and t ∈ [−1, 1], we define the associated biaxiality regions as the closed subsets of Ω given by where β is the signed biaxiality parameter (1.4). The corresponding biaxial surfaces are Observe that if t ∈ (−1, 1) is a regular value of β • Q, then biaxial surfaces are smooth surfaces inside Ω, possibly with boundary which is anyway smooth and contained in ∂Ω. Moreover, the regions in (1.23) are homotopically equivalent to their interior {β < t} and {β > t}, since the biaxial surfaces are actually smooth and serve as their common boundary.
We now introduce a notion of "mutual linking", a property that will (partially) encode the topological nontriviality of the biaxiality regions. To illustrate this definition, let us discuss again the case of a nematic droplet. If Ω is the unit ball and Q b is the hedgehog boundary data (1.19), we expect the minimizers Q λ or Q µ λ to be axially symmetric around a fixed axis (in a sense made precise below). In particular, we expect their biaxiality regions (1.23) to be axially symmetric as well. More precisely, {β < t} with t ∈ (−1, 1) should be an increasing family of axially symmetric solid tori, and the complementary regions {β > t} should be kind of distance neighborhoods from the boundary ∂Ω with cylindrical neighborhoods of the symmetry axis added. In the extreme case t = ±1, we expect {β = −1} to be a circle with axial symmetry, and {β = 1} to be the sphere ∂Ω with the segment connecting the two antipodal points lying on the symmetry axis added. Clearly sub and superlevel of the biaxiality function should be mutually linked in the sense of Definition 1.5 above. This conjectural picture is supported by numerical simulations as already detailed in [48,31,18,32], where authors refer to it as the "torus solution" of the Landau-de Gennes model. For the nematic droplet with radial anchoring, the situation clearly reminds the one corresponding to the Hopf fibration where the subsets {|z 1 | 2 − |z 2 | 2 > t} and {|z 1 | 2 − |z 2 | 2 < t} with t ∈ (−1, 1) form a decomposition of S 3 into two disjoint mutually linked solid tori (a so-called Heegaard splitting).
The weak counterpart of the conjectural picture described in the example above is the main topological result of the paper. 2 As an example, if Ω is the unit ball, A is an unknotted embedded copy of S 1 into Ω, and B = Ω \ A δ with A δ a sufficiently small tubular neighborhood of A, then A and B are mutually linked. Theorem 1.6. Let Ω ⊆ R 3 be a bounded connected open set with C 3 -smooth boundary, and assume that either Q = Q λ is a minimizer of E λ as in Theorem 1.2 or Q = Q µ λ is a minimizer of F λ,µ as in Theorem 1.3, so that (HP 0 ) holds. Suppose that assumptions (HP 1 )-(HP 3 ) also hold (e.g., suppose that Ω is connected and simply connected, ∂Ω has an odd number of connected components, and that Q b (x) = 3/2( Claim 1) on discreteness of the set of singular values is a consequence of the analytic Morse-Sard theorem from [58]. The rest of the claim together with claim 2) is proved by contradiction using a degree-counting argument. The key observation is that on each spherical component of a biaxial surface {β = t}, the pull back bundle E = v max * F of the tangent bundle F = T S 2 → S 2 under the lifting v max of the eigenspace map V max must be trivial (hence its Euler number vanishes). Then the contradiction coming essentially from (HP 3 ) ensures that some S i has positive genus. The argument for 2) and 3) above holds for regular values t ∈ (−1,β), and the extension to arbitrary values is based on the analytic regularity of Q and the Lojasiewicz retraction theorem [38] (it is the only instance where this property is used). Finally, the linking property in 4) follows easily by contradiction using a deformation of the biaxial regions along the positive/negative gradient flow of β. We expect analogous properties to hold also for t ∈ (β, 1), but this case seems to be more subtle since the biaxial surfaces touch the boundary ∂Ω, and we do not have rigorous result in this direction at present.
As the conclusions of the theorem are weak counterparts of the properties conjectured for the torus solution on a nematic droplet, we refer to such solutions on a general domain as "torus-like solutions". It is a very challenging open problem to obtain a precise estimate on the genus of the surfaces S i , if any. Any control on it should depend on a subtle role of the genus in giving a possible lower order correction term in the energy expansion of the minimizing configurations.
In our subsequent papers [15] and [16] of the series, we continue this analysis focusing on axially symmetric configurations. Letting S 1 act by rotation around the vertical axis on an S 1 -invariant domain Ω ⊆ R 3 , and on S 0 by the induced action S 0 ∋ A → R t A R ∈ S 0 , R ∈ S 1 , we consider Sobolev maps Q ∈ W 1,2 (Ω; S 0 ) satisfying the equivariance property Minimizing the energy functional (1.14) or (1.10) in the appropriate class of equivariant configurations will provide minimizers which are either smooth and nowhere vanishing or with singularities/isotropic points, depending on the geometry of the domain and on the chosen boundary data.
In case such defects are not present, we will be able to show that the level sets of the signed biaxiality parameter are generically finite union of axially symmetric tori. On the other hand, when singularieties/isotropic points occur, the regularity/absence of isotropic phase results of the present paper will show that axial symmetry of minimizers is not inherited from the boundary condition, and axial symmetry breaking and nonuniqueness phenomena must occur. Such phenomena were already proved in [1] for minimizers of the Frank-Oseen energy, and our results are the natural counterpart for the Landau-de Gennes model, in agreement with the numerical simulations in [14].

Small energy regularity theory: a tool box
The aim of this section is to provide several regularity estimates, both in the interior and at the boundary, for weak solutions of (1.16) under certain general conditions. We emphasize that the material developed here is not restricted to minimizers of the energy functional E λ , but it applies to rather general critical points satisfying suitable energy monotonicity formulae. With this respect, we shall make a crucial use of the results of this section in our companion papers [15,16] where we considered solutions obtained by minimization of E λ in restricted (symmetric) classes.
Before going further, let us precise for completeness the (usual) notion of critical point of E λ over the nonlinear space W 1,2 (Ω; S 4 ), and show that critical points are exactly the distributional solutions of (1.16) belonging to W 1,2 (Ω; S 4 ).
for every Φ ∈ C 1 c (Ω; S 0 ). The Euler-Lagrange equation for critical points of E λ reads as follows.
2.1. Monotonicity formulae. In this subsection, our goal is (essentially) to derive the afore mentioned monotonicity formulae for certain critical points of E λ . Concerning minimizers, such formulae can be classically obtained by inner variations of the energy. However this argument can not be used when considering energy minimizers over symmetric classes as we do in [15,16]. To circumvent this difficulty, we consider critical points of E λ which can be (strongly) approximated by critical points of a suitable Ginzburg-Landau functional in which the constraint to be S 4 -valued is relaxed. In this way, the approximate solution is smooth enough to derive the monotonicity formulae from the Euler-Lagrange equation, and we conclude by taking the limit in the approximation parameter. This procedure applies of course to minimizers (as we shall see in Section 3), but also to the symmetric solutions of (1.16) considered in [15,16]. Let us now describe it in details.
Given a bounded open set Ω ⊆ R 3 , a reference map Q ref ∈ A Q b (Ω) and a small parameter ε ∈ (0, λ −1/2 ), we consider the energy functional GL ε (Q ref ; ·) defined over W 1,2 (Ω; S 0 ) by If Q ref can be achieved as a (strong) limit of critical points of GL ε (Q ref ; ·) when ε → 0, then Q ref satisfies the monotonicity formulae stated in the following proposition.
Step 2: Interior Monotonicity Formula. Without loss of generality, we may assume that x 0 = 0. Let us take the inner product of (2.10) with (x · ∇)Q ε , and integrate by parts over the ball B t of radius t ∈ (ρ, r). It yields Dividing both sides by t 2 , we obtain d dt Integrating this identity between ρ and r yields In view of the convergences established in Step 1, letting ε → 0 in this last identity leads to (2.8).
Step 3: Boundary Monotonicity Inequality. We first claim that there exists a constant C Ω > 0 depending only on Ω such that To prove this estimate, let us introduce Φ Ω ∈ C 2,α (Ω) the unique solution of see e.g. [21, Theorem 6.14]. We consider V : Ω → R 3 the C 1,α -vector field given by V : Taking the inner product of (2.10) with (V · ∇)Q ε , and integrating by parts over Ω leads to for some universal constant C > 0. On the other hand, by the Hopf lemma, there is a constant c 0 Ω > 0 depending only on Ω such that V · ν c 0 Ω on ∂Ω, and (2.11) follows. We now fix x 0 ∈ ∂Ω. By the smoothness assumption on ∂Ω, there are two constants r Ω > 0 and c 1 Ω > 0 (depending only Ω) such that for every t ∈ (0, r Ω ), In what follows, we assume without loss of generality that x 0 = 0. Let us fix 0 < ρ < r < r Ω . Taking once again the inner product of (2.10) with (x · ∇)Q ε , we integrate the result by parts in B t ∩ Ω with t ∈ (ρ, r). Similarly to Step 2, it yields (after dividing by t 2 ) d dt an orthonormal basis of the tangent space of ∂Ω at x, we have Then we infer from (2.12) that for a constant C Ω > 0 depending only on the constants r Ω and c 1 Ω . Still by (2.12), we have Inserting (2.14), (2.15), and (2.11) in (2.13), and integrating the resulting inequality between ρ and r yields and C Ω > 0 is a constant depending only on r Ω , c 1 Ω , (c 0 Ω ) −1 ∇Φ Ω C 1 (Ω) , and the (2-dimensional) measure of ∂Ω. In view of the convergences established in Step 1, letting ε → 0 in this last inequality leads to (2.9).
Remark 2.5 (Specific geometry [16]). In our companion paper [16], we consider a domain Ω and a boundary condition Q b for which the following situation occurs: 0 ∈ ∂Ω, In this situation, the boundary monotonicity inequality (2.9) for points on B 1 ∩ ∂Ω becomes an equality of the following form: for every point x 0 ∈ B 1 ∩ ∂Ω and every 0 < ρ < r < 1 − |x 0 |, Indeed, it suffices to notice that (x − x 0 ) · ν = 0 and ∇ tan Q b = 0 on B 1 ∩ ∂Ω, and then use this facts in identity (2.13).
One of the main consequences of the monotonicity formulae in Proposition 2.4 is a uniform control of the energy in small balls. Recalling thatQ b ∈ A Q b (Ω) is a given S 4 -valued extension to the domain Ω of the boundary condition Q b , we have (2) there exist two constants r Proof.
Step 1: proof of (1). We assume without loss of generality that x 0 = 0, and we consider an arbitrary ball B ρ (x) ⊆ B r/2 . By the interior monotonicity formula (2.8), we have and the claim is proved.
Step 2: proof of (2). We choose r Ω ∈ (0, r Ω ) (where r Ω is given by Proposition 2.4) in such a way that the nearest point projection π Ω on ∂Ω is well defined in the r (1) Ω -tubular neighborhood of ∂Ω. Once again, we may assume that x 0 = 0, and we consider B ρ (x) ⊆ B r/6 . We now distinguish different cases.

2.2.
Reflection across the boundary. To obtain regularity estimates at the boundary for critical points of E λ in the class A Q b (Ω), we rely on arguments developed by C. Scheven in [52]. The main idea is to construct a suitable reflection across the boundary taking into account the prescribed boundary condition Q b in such a way that the reflected critical point satisfies an equation similar in nature to (2.2) in a larger domain. Boundary regularity can then be treated as an interior regularity problem. The aim of this subsection is to construct such reflection and to derive the resulting equation in the extended domain. We proceed as follows.
We still assume that the boundary of the bounded open set Ω ⊆ R 3 is of class C 3 . In this way, we can find a small number δ Ω > 0 such that the nearest point projection π Ω on ∂Ω is well defined and of class C 2 in the (2δ Ω )-tubular neighborhood of ∂Ω (see e.g. [57, Chapter 2, Section 2.12.3]). We set for δ ∈ (0, 2δ Ω ),

It satisfies
σ Being involutive, its (matrix) differential satisfies Moreover, for every x ∈ ∂Ω we have where p x denotes the orthogonal projection of R 3 onto the tangent plane T x (∂Ω), i.e., in this case Dσ Ω (x) is the (linear) reflection across the tangent plane T x (∂Ω). In particular, where I is the identity matrix. We now extend the domain Ω to the domain and we simplify the notation by setting On the extended domain Ω, we consider the Lipschitz continuous field of symmetric 3 × 3-matrices where J(σ Ω ) denotes the Jacobian determinant of σ Ω . Note that the continuity of A across ∂Ω follows from (2.19). In addition, (2.18) implies that A is uniformly elliptic, i.e., in the sense of quadratic forms for some constants m Ω > 0 and M Ω > 0 depending only on Ω.
Let us now consider for any given (Q 1 , Q 2 ) ∈ S 0 × S 0 their tensor product Q 1 ⊗ Q 2 as the linear mapping Q 1 ⊗ Q 2 : S 0 → S 0 defined by (Q 1 ⊗ Q 2 )P := (P : Q 2 )Q 1 for any P ∈ S 0 . The geodesic reflection on S 4 ⊆ S 0 with respect to a point N ∈ S 4 is given by the linear mapping (2N ⊗ N − id), where id denotes the identity map on S 0 . Note that (2N ⊗ N − id) is simply the orthogonal symmetry with respect to N which is the identity along N and minus the identity along any orthogonal direction to N . In particular, it is involutive, isometric, and symmetric. Given a boundary data Q b ∈ C 1,1 (∂Ω; S 4 ), we consider the mapping Σ : U → GL(S 0 ) of class C 1,1 given by With the help of Σ, we define the extension procedure of maps in A Q b (Ω) to the domain Ω as follows: to a map Q ∈ A Q b (Ω) we associate Q ∈ W 1,2 ( Ω; S 4 ) given by If no confusion arises, we shall simply write Q instead of Q the extension of a map Q.

∇P, ∇Q
where A is the matrix field defined in (2.21).
We are now in position to present the equation satisfied by the extension to Ω of a critical point of E λ in the class A Q b (Ω).
The proof of Proposition 2.8 essentially rests on the following lemma.
for some constants C Ω > 0 (depending only on Ω) and C Q b > 0 (depending only on Ω and Q b ).
Proof. If Φ ∈ W 1,2 ( Ω; Q λ * T S 4 ) is compactly supported in Ω, then (2.25) reduces to (2.1). Therefore, it suffices to consider the case where Φ is compactly supported in U . Following the argument in [52], we decompose Φ into its equivariant and anti-equivariant parts with respect to the involution Here equivariance is understood in terms of the joint reflections across the boundary and on S 4 . Thus, one simply obtains We shall prove (2.25) for Φ e and Φ a separately, starting with Φ a . To this purpose, we consider Q λ as extended to the whole U as in (2.22) and we also introduce for x ∈ U , We start from the identity To compute the II-term, we integrate by parts. Since A is the identity matrix on ∂Ω and ∂ ν Σ = 0 on ∂Ω, the boundary term vanishes, and we are left with Concerning the I-term, we use the anti-equivariance of Φ a to derive (2.28) Next we change variables in the first term of the last identity, and by (2.18) we obtain Since Σ 2 = id, we have the identities everywhere (resp. a.e.) in U , Consequently, Clearly, F : U ex ×S 4 ×(S 0 ) 3 → S 0 is Carathéodory and it is sublinear in its third argument because Σ ∈ C 1,1 and |Q λ | 1 in U . It now remains to perform the computations with the equivariant part Φ e . First, we observe that Φ e = 0 on ∂Ω. Indeed, since the function (Q λ : Φ) belongs to W 1,1 (U ), it has a trace on ∂Ω, and this trace is equal to the inner product of the traces on ∂Ω. Since (Q λ : Φ) = 0 in U , and Thanks to the regularity of ∂Ω, (2.1) holds for every test Next, from the definition of Q * λ we have an identity analogous to (2.26), namely Summing up the contributions for III and IV , in view of the identities for Σ and its derivatives we infer Changing variables once again, we derive with f := |J(σ Ω )|. Combining (2.37) and (2.38), the conclusion follows.
Proof of Proposition 2.8. Starting from Lemma 2.9, we proceed as in the proof of Proposition 2.2.
Before closing the subsection, we provide a counterpart to Lemma 2.6 for reflected maps.
There exist two constants r Ω > 0 and κ = κ Ω ∈ (0, 1) depending only on Ω such that for every x 0 ∈ ∂Ω and r ∈ (0, r Ω > 0 is given by Lemma 2.6. Given a point x 0 ∈ ∂Ω and a radius r ∈ (0, r Using the facts that Σ(x) is isometric for every x ∈ U and |Q ref | = 1, we estimate where the last inequality follows from a change of variables. Setting y := σ Ω (x), we observe that instead of (2.41) (with x 0 = 0 and r = 1).

2.3.
The ε-regularity theorem. In this subsection, we present the main regularity estimate which provides local Hölder regularity for weak solutions of (2.2) under a smallness assumption on the energy. To treat interior and boundary estimates in a unified way, we consider the case of a general system with diagonal principal part, corresponding to the scalar operator Lv = −div(A∇v), as it appears in Proposition 2.8.
Theorem 2.12. Let r 0 ∈ (0, 1] and A : B r0 → M sym 3×3 (R) be a Lipschitz field of symmetric matrices, and assume that A is uniformly elliptic (i.e., mI A M I for some constants m > 0 and M > 0).
There exist two constants ε A > 0 and C A > 0, and an exponent α = α(A) ∈ (0, 1) depending only on the Lipschitz norm of A in B r0 and the ellipticity bounds m and M such that the condition We postpone the proof of this theorem as we require some preliminary lemmas. To this purpose, let us first recall the notion of function of bounded mean oscillation. Given an open ball B ⊆ R d , a function u ∈ L 1 (B) belongs to the space BMO(B) if where the supremum is taken over closed balls B ρ (y) as above. Analogously, for p > 1 a function where as above the supremum is taken over closed balls B ρ (y). It is well known that taking closed cubes inside B or closed balls B ρ (y) such that B 2ρ (y) ⊆ B gives equivalent definitions where the previous quantities are equivalent norms (see [59]). A first ingredient coming into play is the classical John-Nirenberg inequality, see e.g. [24,Chapter 19].
Lemma 2.13 (John-Nirenberg inequality). For every 1 < p < ∞, there exists a constant C p > 1 depending only on p and the dimension such that for every u ∈ BMO(B).
The second result is a standard scaling-invariant local regularity estimate for solutions of linear elliptic PDE's. Since the result is standard but we were not able to find a reference in the literature we sketch the proof for the reader's convenience.
Lemma 2.14. For d 3, let A : Ω ⊆ R d → M d×d (R) be a Lipschitz field of symmetric matrices, and assume that A is uniformly elliptic (i.e., mI A M I in Ω for some constants m > 0 and M > 0). Let f ∈ L 2 ( Ω; R d ), g ∈ L 2 ( Ω) and for each B r ⊆ Ω, 0 < r 1, consider u ∈ W 1,2 0 (B r ) the (unique) weak solution of −div(A∇u) = div f + g in B r , u = 0 on ∂B r .
For every q ∈ ( d d−1 , 2), there exists a constant C A = C A (q) depending only on q, d and the Lipschitz norm of A in Ω (i.e., not on the radius r) such that Proof. (Sketch) Since all the norms in the inequality have the same scaling properties and the Lipschitz norm of A is decreasing under scaling with factor r 1 we may assume r = 1. Then the estimate for q = 2 just follows testing with u, integrating by parts and using Sobolev inequality. The case q ∈ (2, d) follows from the case q = 2 and the combination of [21,Theorem 9.15] for the case f ≡ 0 with [22,Theorem 10.17] for the case g ≡ 0. Finally, standard duality arguments give the desired conclusion in the dual range of exponents q ∈ ( d d−1 , 2). The final ingredient is the following local gradient estimate for A−harmonic functions.
be a Lipschitz field of symmetric matrices, and assume that A is uniformly elliptic (i.e., mI A M I in Ω for some constants m > 0 and M > 0). If B r ⊆ Ω, 0 < r 1, and u ∈ W 1,2 (B r ) satisfies in the weak sense

46)
then u ∈ C 1 (B r ) and for some constant C A > 0 depending only on d and the Lipschitz norm of A in Ω (i.e., not on the radius r).
Proof. Since u − ξ also solves (2.46), we may assume that ξ = 0. By standard elliptic regularity theory, u is of class C 1,α locally inside B r , and the following estimate holds (see e.g. [ Next we observe that |u| 2 ∈ W 1,1 (B r ) satisfies (in the W −1,1 -sense) − div(A∇|u| 2 ) = −2(A∇u) · ∇u 0 in B r . (2.48) According to [21,Theorem 9.15], there exists a unique strong solution ϕ of −div(A∇ϕ) = 1 in B r , ϕ = 0 on ∂B r , which belongs to W 2,p (B r ) for every p < ∞. In particular, ϕ ∈ C 1 (B r ) by Sobolev embedding whenever p > d, and an elementary scaling argument (using r 1) leads to for some constant C A > 0 depending only on d and the Lipschitz norm of A in Ω (and independent of r). Moreover, ϕ 0 in B r by the maximum principle. Next we write |u| 2 = −|u| 2 div(A∇ϕ), and we integrate by parts over B r to obtain Proof of Theorem 2.12. We start with some useful pointwise identities which hold a.e. in the domain and which allow to perform the so-called Helein's trick and rewrite the quadratic term in the r.h.side of (2.44) in divergence form. From the identity |Q| 2 = 1, we first infer that Q : ∂ k Q = 0 for each k ∈ {1, 2, 3}. As a consequence, which in turn implies that with the vector fields We now claim that in view of the previous pointwise identities for every i, j, k, l ∈ {1, 2, 3}, Indeed, given a test function ϕ ∈ D(B r0 ), we integrate by parts using equation (2.44) to obtain and the claim follows.
(One can choose for instance q = 7/4.) Using the identity |Q| = 1 and Hölder's inequality, we estimate with the help of (2.54), , as well as .
According to Lemma 2.14, we thus have .
Applying Theorem 2.12 to our main equation (2.2) yields the following interior regularity estimate.
Proof. Since Q λ is a weak solution of(1.16), it solves (2.44) in B r (x 0 ) with the matrix A = I, and for some universal constant C > 0. Hence, we can choose ε in and r in small enough in such a way that (2.45) holds (with ε A = ε I ), and the conclusion follows from Theorem 2.12.
Concerning boundary regularity estimates under a Dirichlet boundary condition, we apply the refection procedure of the previous subsection, and then Theorem 2.12 to equation (2.23).
Corollary 2.17. Assume that ∂Ω is of class C 3 and Q b ∈ C 1,1 (∂Ω; S 4 ). Let Q λ ∈ A Q b (Ω) be a critical point of E λ , and Q λ its extension to Ω given by (2.22). There exist two constants ε bd > 0 and r bd > 0 depending only on Ω and Q b such that for every ball B r (x 0 ) ⊆ Ω with x 0 ∈ ∂Ω and 0 < r < r bd (1 + λ) −1/2 , the condition for some constants α ∈ (0, 1) and C Q b > 0 depending only on Ω and Q b (and not on λ).
Proof. By Proposition 2.8, Q λ solves (2.44) in B r (x 0 ) with the matrix field A given by (2.21), and the map G given by G := G λ (·, Q λ , ∇ Q λ ) where G λ satisfies the growth condition (2.24). In particular, for a constant C Q b > 0 depending only on Ω and Q b . Hence, we can choose ε bd and r bd small enough in such a way that (2.45) holds, and the conclusion follows from Theorem 2.12.

2.4.
Higher order regularity. In this subsection, we improve Hölder continuity estimates from the previous one into Lipschitz estimates and finally we deduce analytic regularity both in the interior and at the boundary, whenever boudary data permit.
Proposition 2.18. Let r ∈ (0, 1] and let A : B r → M sym 3×3 (R) be a Lipschitz field of symmetric matrices. Assume that A is uniformly elliptic, i.e., mI A M I for some constants m > 0 and M > 1. Let G : B r × S 4 × (S 0 ) 3 → S 0 be a Carathéodory map satisfying
Proof. Let us fix an arbitrary point x 0 ∈ B r/2 , and set A 0 := A(x 0 ), r 1 := r/(2 √ M ) < 1. We change variables by setting for x ∈ B r1 (so that A We observe thatĀ is Lipschitz continuous in B r1 , and m M I Ā M m I andĀ(0) = I .

ConcerningḠ, it satisfies
for some constant C * > 0 depending only on C * and A.
We now fix an arbitrary radius ρ ∈ (0, r 1 ], and we consider H ∈ W 1,2 (B ρ ; S 0 ) ∩ C 0 (B ρ ) the (unique) solution of Representing H through the Poisson integral formula, one easily obtains for some constant C > 0 depending only A and κ (and osc is meant for oscillation). Since H −Q = 0 on ∂B ρ , we deduce that with C > 0 depending only A and κ.
On the other hand, for the harmonic function H, we have H ∈ C ∞ (B ρ ) and also ∆|∇H| 2 = 2|D 2 H| 2 0, hence ρ → ρ −2 |x|=ρ |∇H| 2 dH 2 is nondecreasing and in turn ρ → ρ −3 Bρ |∇H| 2 dx is nondecreasing. As a consequence, since H is equal toQ on ∂B ρ it satisfies We are now ready to estimate 67) and we shall treat separately the two terms I and II. Since A is Lipschitz andĀ(0) = I, we have |Ā − I| C A ρ in B ρ , and we infer from (2.66) that where we have used that 0 < ρ r 1 1. Using again this property together with the ellipticity bounds on A and |Ā − I| C A ρ in B ρ we conclude, Next we write for a constant C A > 0 depending only on A, C * , and κ and for all 0 < ρ r 1 1.
In view of the arbitrariness of ρ, we can apply (2.73) with ρ k := 2 −k r 1 and k ∈ N. It leads to and {y k } ⊆ [0, ∞) satisfy y k+1 θ k y k + σ k for each k 0, then a simple induction argument gives y k+1 θ(y 0 + σ) for each k 0. As a consequence, if we let for some constant C > 0 depending only on A, C * , κ, and α. Finally, if x 0 was chosen to be a Lebesgue point of |∇Q| 2 (which holds for a.e. x 0 ∈ B r0/2 by the Lebesgue differentiation theorem), then 0 is a Lebesgue point for |∇Q| 2Ā , and letting k → ∞ in (2.74) yields (recall thatĀ(0) = I) Changing variables again and using uniform ellipticity of A, by the definition of r 1 we deduce that for some constants C > 0 and Λ > 0 depending only on A, C * , κ, and α and the conclusion follows.
Once Lipschitz continuity is obtained it is easy to obtain higher regularity.
where r in and ε in are given by Corollary 2.16, then Q λ ∈ C ω (B r/4 (x 0 )). In addition, Q λ satisfies for each k ∈ N,

75)
for a constant C k > 0 depending only on k. Proof.
Step 2. In this second step, our aim is to prove the remaining estimate (2.75) for k 2. Let us fix a point y ∈ B r/8 (x 0 ), and rescale variables setting Q(x) := Q λ (y + rx). Then, with λ := r 2 λ ∈ (0, r in ). Let us fix j ∈ {1, 2, 3}, and set v := ∂ j Q. Differentiating (2.76) with respect to the j-th variable, we obtain that v satisfies a linear system of the form where the coefficients b, c, and d satisfy From the arbitrariness of j, we conclude that ∇ 2 Q L ∞ (B 1/16 ) C. Now we can proceed by induction on k following the same strategy (differentiating (k − 1)-times equation (2.76)) to prove that ∇ k Q L ∞ (B 2 −(k+2) ) C k for a constant C k depending only on k. Scaling variables back, we obtain that |∇ k Q λ (y)| C k r −k , and (2.75) follows from the arbitrariness of y.
A similar argument then yields higher regularity near the boundary when the boundary data are sufficiently regular.
Finally, under the assumption that ∂Ω is of class C k,β and Q b ∈ C k,β (∂Ω; S 4 ) with k 2, the fact that Q λ ∈ C k,β loc (B r/4 (x 0 ) ∩ Ω) now follows from equation (2.2) and standard elliptic regularity at the boundary, see e.g. [21,Chapter 6]. The corresponding conclusion within the analytic class follows again from the results in e.g. [45, Chapter 6].
2.5. Bochner inequality and uniform regularity estimates. In this subsection we refine the previous analysis and clarify the dependence of the regularity estimates for the smooth solutions Q λ of (1.16) on the parameter λ. The results of this subsection are not used in the present paper but they will be a fundamental tool in the subsequent papers [15], [16], of our series where we will study (axially symmetric) minimizers in the asymptotic limit λ → ∞.
Proposition 2.21. Let Q λ ∈ W 1,2 (B r ; S 4 ) be a smooth solution of (1.16) in B r . There exists a universal constant ε reg > 0 such that the condition for a further universal constant C > 0.

Lemma 2.23 (Bochner inequality).
Let Q λ be a smooth solution of (1.16) in B r . Setting e λ := From (1.16), we derive that It then follows from Lemma 2.22 (applied to Q = Q λ and T = ∂ k Q λ ) that and it follows from (1.16) that Noticing that tr Q 4 = 1/2, we obtain from Lemma 2.22, Combining (2.79) and (2.80), we are led to exactly as in Lemma 2.6.
Proof of Proposition 2.21. We argue as in [11], where the scaling argument first presented in [53] for harmonic maps is adapted to the harmonic heat flow. Since Q λ is smooth, we can find σ λ ∈ (0, r/2) such that In addition, by continuity we can find x λ ∈ B σ λ such that sup Bσ λ e λ = e λ (x λ ) := e λ .
for a universal constant C > 0. Here we have used that B 1/ √ e λ (x λ ) ⊆ B r/2 since 1/ √ e λ < ρ λ . Therefore, 1 2Cε reg which is clearly a contradiction if ε reg is small enough.
Choosing σ = r/4 now yields e λ 128r −2 in B r/4 , and the proof is complete.

Regularity of minimizers under norm constraint
The aim of this section is to prove Theorem 1.2, and the proof is divided according to the following subsections. Recall that in the statement of Theorem 1.2, we assume that the boundary ∂Ω is of class C 3 and Q b ∈ C 1,1 (∂Ω; S 4 ).
3.1. Monotonicity formulae. We start establishing the monotonicity formulae for minimizers of E λ over A Q b (Ω) applying the general principle in Proposition 2.4. First, let us recall that Q b ∈ A Q b (Ω) is a given S 4 -valued extension to the domain Ω of the boundary condition Q b .

3.2.
Compactness of blow-ups and smallness of the scaled energy. When proving regularity the main issue is to analyse the asymptotic behavior of minimizers at small scales, and the key property is the compactness of rescaled maps. When rescaling around an interior point, we have the following statement.
To prove Proposition 3.2, we need two auxiliary lemmata. Lemma 3.3. Let Q λ,rn be as in Proposition 3.2 and ρ > 0. For each n ∈ N such that ρr n < r 0 , let v n ∈ W 1,2 (B ρ ; S 4 ) be such that v n = Q λ,rn on ∂B ρ in the sense of traces. Then, Proof. By minimality of Q λ and a change of variables, Q λ,rn is minimizing E λr 2 n (·, B ρ ) among all maps in W 1,2 (B ρ ; S 4 ) having the same trace Q λ,rn on ∂B ρ . Since v n is an admissible competitor and the potential W is bounded on S 4 , we have for a constant C depending only on W . Then the claim follows letting n → ∞.
for every 0 < R 1 < R 2 r 0 /r n . As a consequence, for every 0 < R < r 0 /r n , we have Consequently, we can find a (not relabeled) subsequence such that Q λ,rn converges to a map Q * weakly in W 1,2 loc (R 3 ) and strongly in L 2 loc (R 3 ). Up to a further subsequence, Q λ,rn → Q * a.e. in R 3 , and thus Q * ∈ W 1,2 loc (R 3 ; S 4 ). By the monotonicity formula (2.8) satisfied by Q λ , we have for every R > 0. Consequently, letting n → ∞ in (3.5) yields by W 1,2 -weak convergence and lower semicontinuity, for every 0 < R 1 < R 2 , which shows that Q * is 0-homogeneous. Now we aim to prove that, for every radius R > 0, Q λ,rn → Q * strongly in W 1,2 (B R ), and that for every competitorQ ∈ W 1,2 (B R ; S 4 ) such thatQ − Q * is compactly supported in B R (i.e., Q * is a minimizing harmonic map into S 4 on the whole space R 3 w.r.to compactly supported perturbations). By homogeneity of Q * , the value of the radius R does not play a role, and it is enough to show strong W 1,2 -convergence and energy minimality in a ball B ρ for some radius ρ ∈ (0, 1).
We now aim to perform a similar blow-up analysis around a boundary point. To this purpose, let us recall that ∂Ω is assumed to be of class C 3 , and Q b ∈ C 1,1 (∂Ω; S 4 ). We consider the enlarged domain Ω defined in (2.20), and we extend Q b to Ω\Ω by setting Q b (x) := Q b (π Ω (x)) for x ∈ Ω\Ω, where π Ω is the nearest point projection on ∂Ω. By the regularity assumption on ∂Ω and Q b , we have Q b ∈ C 1,1 ( Ω \ Ω).
Proposition 3.5. Let Q λ be a minimizer of E λ over A Q b (Ω), and denote by Q λ the extension of Q λ to Ω given by Q λ = Q b in Ω \ Ω. Given x 0 ∈ ∂Ω and 0 < r r 0 such that B r0 (x 0 ) ⊆ Ω, consider the rescaled map Q λ,r ∈ W 1,2 (B r0/r ; S 4 ) defined by For every sequence r n → 0, there exist a (not relabeled) subsequence and Q * ∈ W 1,2 loc (R 3 ; S 4 ) such that Q λ,rn → Q * strongly in W 1,2 loc (R 3 ). In addition, Q * is homogeneous of degree zero, and up to a rotation of coordinates, Q * is a minimizing harmonic map in the upper half space {x 3 > 0} and Proof. Up to a translation and a rotation, we may assume that {x 3 = 0} is the tangent plane to ∂Ω at x 0 and the vector (0, 0, −1) is the outward unit normal. By Proposition 3.1, Q λ satisfies the Boundary Monotonicity Inequality (2.9), and by rescaling variables, for every 0 < R 1 < R 2 r 0 /r n , where we have set Ω n := r −1 n (Ω − x 0 ). As a consequence, for every 0 < R < r 0 /r n . Since Q b ∈ C 1,1 ( Ω \ Ω) and Q λ,rn (x) = Q b (x 0 + r n x) for x ∈ B R \ Ω n and 0 < R < r 0 /r n , in view of (3.9) the sequence { Q λ,rn } is bounded in W 1,2 loc (R 3 ). Consequently, there exists a (not relabeled) subsequence such that Q λ,rn converges to a map Q * weakly in W 1,2 loc (R 3 ; S 4 ) and strongly in L 2 loc (R 3 ). Up to a further subsequence, Q λ,rn → Q * a.e. in R 3 , and thus Q * ∈ W 1,2 loc (R 3 ; S 4 ). Now observe that Ω n → {x 3 > 0} locally in the Hausdorff metric.
, and it has constant trace on the plane {x 3 = 0}. Arguing essentially as in the proof of Proposition 3.2, we can let n → ∞ in (3.9) to infer that Since the map Q * is constant in {x 3 < 0}, it follows that Q * is 0homogeneous in the whole R 3 . Now it remains to show the strong convergence of Q λ,rn in W 1,2 loc (R 3 ), and the local energy minimality of Q * in {x 3 > 0}. As in the proof of Proposition 3.2, by homogeneity, it is enough to show strong W 1,2 -convergence in a ball B ρ ⊆ B 1 (perhaps up to a subsequence), and energy minimality of Q * in B ρ ∩ {x 3 > 0}. We first notice that, Q b being C 1,1 in Ω \ Ω, we have and we only need to show that to establish the strong convergence of Q λ,rn in W 1,2 (B ρ ). The rest of the proof is quite similar to the one used for the interior case discussed in Proposition 3.2. For this reason; we only sketch few differences in the construction of comparison maps when gluing different maps near the boundary. The starting point of the construction is to flatten the boundary ∂Ω near x 0 . Assuming {r n } suitably small (depending only on x 0 and the curvature of ∂Ω at x 0 ), there exists a sequence of diffeomorphisms {Φ n } ⊆ C 2 (B 1 ; R 3 ) satisfying the following properties: 10) where we set B + r := B r ∩ {x 3 > 0}, 0 < r 1. We fix 0 < δ < 1/4 and a competitorQ ∈ W 1,2 loc (R 3 ; S 4 ) such thatQ = Q * a.e. in R 3 \ B + 1−δ . Notice that Q λ,rn • Φ n ⇀ Q * weakly in W 1,2 (B + 1 ; S 4 ) as n → ∞. In addition, Q λ,rn (Φ n (x)) = Q b (x 0 + r n Φ n (x)) andQ(x) = Q b (x 0 ) for x ∈ B 1 ∩ {x 3 = 0} because of (3.10). Consequently, since Q b ∈ C 1,1 (∂Ω; S 4 ) we get lim n→∞ B1∩{x3=0} | Q λ,rn • Φ n −Q| 2 dH 2 = 0 and lim n→∞ B1∩{x3=0} Hence we can argue as in the interior case: by Fatou's lemma and Fubini's theorem, extracting a further subsequence if necessary, we can select ρ ∈ (1 − δ, 1) and a constant C > 0 such that We then choose the sequence σ n → 0 with 0 < σ n < δ as σ n := Q λ,rn • Φ n −Q Before going further, let us notice that we can argue as in Lemma 3.8 (using the weak convergence of Q λ,rn , its energy minimality on Ω n ∩ B ρ , and (3.10)) to prove the following: for any bounded sequence {v n } ⊆ W 1,2 (B + ρ ; S 4 ) such that v n = Q λ,rn • Φ n on ∂B + ρ , we have where the last equality follows from a change of variables and (3.10). Now, to construct an effective sequence of comparison maps, it is convenient to introduce a biLipschitz map Ψ : B 1 → B + 1 . By means of Ψ, the comparison maps can be constructed as in the interior case. More precisely, we apply Lemma 3.4 to the pair of maps from the two-sphere S 2 , namely u(·) = Q λ,rn • Φ n (ρΨ(·)) and v(·) =Q(ρΨ(·)). As in the interior case, the lemma produces a sequence {w n } ⊆ W 1,2 (B 1 ; S 0 ) satisfying and dist(w n , S 4 ) → 0 uniformly in B 1 \ B 1−σn as n → ∞.
All possible limiting maps Q * obtained by either Proposition 3.2 or Proposition 3.5 are often referred to as (minimizing) tangent maps to Q λ at the given point x 0 . By the monotonicity formulae and the strong compactness of rescaled maps, triviality (i.e., constancy) of all tangent maps implies smallness of the rescaled energy at sufficiently small scale. In our setting, triviality of tangent maps together with smallness of the scaled energy are established in the following propositions.
Proof. Let us fix an arbitrary point x 0 ∈ Ω and a sequence r n → 0. According to Proposition 3.2, up to a subsequence, the rescaled maps satisfy Q λ,rn → Q * strongly in W 1,2 loc (R 3 ) as n → ∞ for some Q * ∈ W 1,2 loc (R 3 ; S 4 ). Moreover, Q * is a degree-zero homogeneous energy minimizing harmonic map, so that there exists a smooth harmonic sphere ω : S 2 → S 4 such that Q * (x) = ω x |x| . On the other hand, according to [56,Theorem 2.7] the map Q * is smooth. In particular, Q * is smooth at the origin which implies that ω must be constant, and thus Q * itself is a constant map. Then the interior monotonicity formula (see Proposition 3.1) and the strong W 1,2 -convergence yield which completes the proof.
Proof. As in the previous proof, by the strong W 1,2 -compactness of rescaled maps, it is enough to prove that any limiting map Q * obtained from Proposition 3.5 applied at a point x 0 ∈ ∂Ω is a constant map, i.e., Q * ≡ Q b (x 0 ). Indeed, by the Boundary Monotonicity Inequality (see Proposition 3.1), we have where we have set Ω n := r −1 n (Ω − x 0 ). Let us now consider a degree zero homogeneous map Q * ∈ W 1,2 loc (R 3 ; S 4 ) which is an energy minimizing harmonic map in {x 3 > 0}, and such that Q * = Q b (x 0 ) =: e 0 in {x 3 < 0}. Setting S 2 + := S 2 ∩ {x 3 > 0}, the homogeneity of Q * implies that Q * (x) = ω x |x| in {x 3 > 0} where ω ∈ W 1,2 (S 2 + ; S 4 ) is a weakly harmonic map on S 2 + satisfying ω = e 0 on ∂S 2 + in the sense of traces. It now suffices to show that ω ∈ C ∞ (S 2 + ). Indeed, by Lemaire rigidity theorem [35, Theorem 3.2], a smooth harmonic map on the (closed) half 2-sphere which is constant on the boundary has to be constant. In other words ω ≡ e 0 , whence Q * ≡ e 0 .
The smoothness of ω in the interior S 2 + follows from Hélein's theorem [25]. Smoothness up to the boundary ∂S 2 + could be asserted directly from [49], but we prefer to give a short argument illustrating in this simple case the reflection principle in Subsection 2.2.
Consider the map Q * ∈ W 1,2 loc (R 3 ; S 4 ) defined by is the reflection of x = (x 1 , x 2 , x 3 ) across the plane {x 3 = 0}, and Σ := 2e 0 ⊗ e 0 − id is the geodesic reflection on S 4 with respect to the point e 0 . Following the proof of Proposition 2.8 with λ = 0 (see also Remark 2.11), we infer that the reflected matrix A(x) is the identity and Q * is weakly harmonic in R 3 . Since Q * clearly inherits homogeneity from Q * , we have Q * (x) = ω x |x| for a weakly harmonic map ω ∈ W 1,2 (S 2 ; S 4 ). By Hélein's theorem [25], ω is smooth on S 2 , and the conclusion follows since ω = ω in S 2 + .
3.3. Full regularity. Combining the results from the subsections above with the ε−regularity theorem and the higher regularity theorem from Section 2.1, we are finally in the position to prove the first regularity result of the paper.
Proof of Theorem 1.2. Let Q λ be a minimizer of E λ over A Q b (Ω). First, we prove interior regularity of Q λ by showing smoothness in a neighborhood of an arbitrary point x 0 ∈ Ω. In view of Proposition 3.6, we have 1 r E λ (Q λ , B r (x 0 )) → 0 as r → 0. Combining Proposition 3.1 and Lemma 2.6 (with Q ref = Q λ ) with Corollary 2.19, we infer that Q λ ∈ C ω (B ρ (x 0 )) for some radius ρ > 0 possibly depending on the point x 0 . Since x 0 ∈ Ω is arbitrary, we conclude that Q λ ∈ C ω (Ω).

LdG-minimizers in the Lyuksyutov regime
The main objective of this section is to prove Theorem 1.3, and in particular to prove that isotropic melting (i.e., presence of the zero phase) is avoided by minimizers of the energy functional F λ,µ in (1.10) for values of the parameters in the Lyuksyutov regime µ → ∞. More precisely, our main goal is to prove that the pointwise norm of any minimizer Q µ λ of F λ,µ subject to an S 4 -valued boundary condition is uniformly bounded from below by a positive constant whenever µ is large enough (and λ of order one). As a consequence we deduce that the radial hedgehog (1.20) is not energy minimizing and in Theorem 4.8 below we will show that it is not even a stable critical point of the energy functional F λ,µ .
Throughout this section, we assume again that the boundary ∂Ω is of class C 3 , and that the boundary condition Q b belongs to C 1,1 (∂Ω; S 4 ). Given λ > 0 and µ > 0, we shall consider critical point of F λ,µ over the class W 1,2 (Ω; S 0 ), including as a particular case solutions of the variational problem min F λ,µ (Q) : Q ∈ W 1,2 Q b (Ω; S 0 ) whose resolution follows from the direct method of calculus of variations. We may denote by Q µ λ a critical point of F λ,µ , or simply by Q µ (if no confusion arises) hiding the dependence on the fixed parameter λ to simplify the notation. We start with elementary/classical considerations and a priori estimates on Q µ .

4.1.
A priori estimates. In view of the explicit expression (1.12) of the potential W , the Euler-Lagrange equation characterizing a critical point Q µ ∈ W 1,2 Q b (Ω; S 0 ) reads as follows with the term 1 3 |Q µ | 2 I due to the traceless constraint. Let us start the analysis by establishing the regularity of critical points.
We now prove an a priori estimate on the modulus and on the gradient of a critical point reminiscent from the Ginzburg-Landau theories.
for a constant C depending only on Ω and Q b .
The last ingredients we need are the following monotonicity formulae.
The proof of this lemma follows word by word the one in Proposition 2.4 (Step 2 & Step 3), and we shall omit it. We just observe that the constant C λ Q b in (4.5) is independent of µ because Q b has always unit norm on ∂Ω.

4.2.
Lyuksyutov regime and absence of isotropic melting. We now complete the proof of Theorem 1.3 analyzing the asymptotic behavior as µ → +∞ of minimizers of F λ,µ over the class W 1,2 Q b (Ω; S 0 ). The heart of the matter is Proposition 4.5 below. We emphasize that Proposition 4.5 does not rely on energy minimality but on the a priori strong convergence towards a smooth limiting map. This allows for more flexibility in its application, see our companion paper [16].
Step 2. It now remains to prove that |Q µn λ | → 1 uniformly in Ω. Given δ ∈ (0, 1) arbitrary, we thus have to prove that |Q µn λ | > δ on Ω for n large enough. We argue by contradiction assuming that, along a (not relabeled) subsequence, there exists x n ∈ Ω such that |Q µn λ (x n )| δ. Extracting a further subsequence if necessary, we can assume that x n → x 0 as n → ∞ for some x 0 ∈ Ω. In view of Lemma 4.3 (and the fact that |Q µ | = 1 on ∂Ω), we can find a constant κ ∈ (0, 1) independent of n such that for r n := κµ −1/2 n → 0 and for all n we have We now distinguish two cases: Case 1: x 0 ∈ Ω. The limiting map Q λ being of class C 1 , we can find a radius r 0 ∈ (0, dist(x 0 , ∂Ω)) such that 1

From
Step 1, we deduce that for n large enough, On the other hand, still for n large enough, we have |x n − x 0 | < r 0 /2 and r n < r 0 /2. Then we infer from (4.6) and (4.4) that which contradicts (4.7).
Case 2: x 0 ∈ ∂Ω. Once again, since Q λ ∈ C 1 (Ω) and ∂Ω is of class C 3 , we can find a small radius r 0 ∈ (0, r Ω ) where r Ω is given by Lemma 4.4 such that the nearest point projection on ∂Ω is well defined in the r 0 -tubular neighborhood of ∂Ω, and where the constant C λ Q b is also given by Lemma 4.4 (notice that ∇Q µn λ L 2 (Ω) is bounded by Step 1). From Step 1, we deduce that for n large enough, If we denote y n ∈ ∂Ω the projection of x n on ∂Ω, we have for n large enough (by (4.6)), r n |y n − x n | = dist(x n , ∂Ω) |x n − x 0 | < r 0 4 , so that |y n − x 0 | < r 0 /2. Arguing as in Case 1 and setting d n := |y n − x n |, we infer from (4.6) and (4.4)-(4.5) that which contradicts (4.8).
Proof of Theorem 1.3. Let us consider an arbitrary sequence µ n → +∞ and corresponding Q µn λ minimizing F λ,µn over W 1,2 Q b (Ω; S 0 ). Since the mapQ b ∈ A Q b (Ω) is an admissible competitor to the minimality of Q µn λ , we have Therefore, the sequence {Q µn λ } is bounded in W 1,2 (Ω; S 0 ), and we can extract a (not relabeled) subsequence such that Q µn λ ⇀ Q λ weakly in W 1,2 (Ω) for some Q λ ∈ W 1,2 Q b (Ω; S 0 ). By the compact embedding W 1,2 (Ω) ֒→ L 4 (Ω), we have Ω (1 − |Q µn λ | 2 ) 2 dx → Ω (1 − |Q λ | 2 ) 2 dx, and it follows from (4.9) that Hence |Q λ | = 1 a.e. in Ω, so that Q λ ∈ A Q b (Ω). Since any Q ∈ A Q b (Ω) is in fact admissible to test the minimality of Q µn λ , we can proceed as in (4.9) and use the lower semicontinuity of E λ to infer that for every Q ∈ A Q b (Ω). Hence Q λ is a minimizer of E λ over A Q b (Ω), and we deduce from Theorem 1.2 that Q λ ∈ C 1,α (Ω). In addition, using Q = Q λ as competitor in (4.10) we obtain that F λ,µn (Q µn λ ) → E λ (Q λ ). The conclusion now follows from Proposition 4.5. 4.3. Instability of the melting hedgehog. In this subsection we discuss instability of the melting hedgehog H µ λ given in (1.20) in the Lyuksyutov regime µ → ∞. The conclusion here is similar to the one in [28], where the low-temperature regime a 2 → ∞ is considered. However, instead of the careful spectral decomposition considered there to analyse the linearized operator, here we will use different and somewhat simpler perturbation arguments; more precisely, here the instability property of H µ λ will follow essentially from the corresponding one for the constant norm hedgehoḡ H seen as a degree-zero homogeneous harmonic map into S 4 .
First we recall that the constant norm hedgehoḡ , it is a critical point of E λ both for λ = 0, i.e., a weakly harmonic map into S 4 , and a critical point also for λ > 0, because ∇ tan W (H) ≡ 0. In order to discuss its stability properties we first set for any Φ ∈ C ∞ 0 (B Proof. Let i = (1, 0, 0) t , j = (0, 1, 0) t , k = (0, 0, 1) t be the canonical basis of R 3 . From these vectors, we construct a distinguished orthonormal basis of S 0 by setting In terms of the latitude θ ∈ [0, π] and of the colatitude φ ∈ [0, 2π) on S 2 , the components ofH w.r.to this basis are easily seen to bē for any i, j = 0, . . . , 4. As a consequence, if we writev = iv i e i with |v| 2 = iv Next, we notice thatH is a degree-zero homogeneous harmonic map and |∇H| 2 = |∇ tanH | 2 = 6 |x| 2 , hence ∆ S 2h = − ∇ tanH 2h = −6h , and in view of (4.15) we obtain Finally, evaluating E ′′ 0 in (4.11) for Φ = ηv and integrating by parts, since η is radial and (4.16) holds we conclude that and the proof is complete.
The instability property ofH for the Dirichlet energy E 0 along some vector field can be derived from the general instability result for harmonic tangent maps from R 3 in to S 4 proved in [56] and [36]. Here, exploiting O(3)-equivariance ofH and using Lemma 4.6 we obtain a stronger and more explicit instability result forH as critical point of all the functionals E λ .
Proof. As alredy proved in Lemma 4.6 above, we have for any radial function η ∈ C ∞ 0 (B 1 \ {0}). In view of the standard Hardy inequality in R 3 the quadratic form is not bounded from below and there exists a radial function whence E ′′ 0 (η nv ;H) → −∞ as n → ∞ and in particular E ′′ 0 (η nv ;H) < 0 for n large enough. Finally, as η n ≡ 0 for |x| < 1/n and |x| > 1/4, taking ξ = η n * ρ ε a regularization by convolution with ε < 1/n and ρ ε a family of radial mollifiers, we have a family of radial functions ξ ∈ C ∞ 0 (B 1 \ {0}) satisfying E ′′ 0 (ξv;H) < 0 for ε > 0 small enough, which proves the first claim of the theorem. In order to discuss the case λ > 0 we rescale the radial function ξ above into ξ δ (x) = ξ(x/δ), for 0 < δ < 1 to be chosen later. Computing the second variation of E λ along the vector field AsH is degree-zero homogeneous, a simple rescaling gives Since by construction E ′′ 0 (Φ;H) < 0, the conclusion follows for δ > 0 small enough. Finally we consider the radial hedgehog H µ λ as the uniaxial critical point of the functional F λ,µ of the form (1.20) discussed in the introduction. Recall that such critical point is the unique minimizer of F λ,µ in the class of O(3)-equivariant maps in W 1,2 (B 1 ; S 0 ) which agree withH on the boundary (see [29,Theorem 1.4]). Moreover, arguing as in the proof of Theorem 1.3 above, it is not difficult to show that H µ λ →H strongly in W 1,2 as µ → ∞ (convergence of minimizers in the class of O(3)-equivariant maps), and moreover the convergence is locally uniform away from the origin because |H µ λ | = 2/3 s µ λ → 1 locally uniformly away from the origin as µ → ∞. Exploiting the aforementioned convergence of H µ λ to its constant norm counterpart we are going to infer instability property of H µ λ from the corresponding one forH passing to the limit in the second variations of the energies F λ,µ and using Proposition 4.7. In order to do this we first set for any Ψ ∈ C ∞ 0 (B 1 ; S 0 ) Simple calculations based on (1.10) now yield and We have the following instability result for the radial hedgehog in the Lyuksyutov regime.
Since by construction H µ λ : Φ T ≡ 0, we obtain Recall that H µ λ →H strongly in W 1,2 (B 1 ; S 0 ) and locally uniformly away from the origin as µ → ∞. As a consequence dominated convergence theorem yields lim µ→∞ B1 On the other hand, since H µ λ is a critical point of F λ,µ , using (4.17) with the choice of vector field Since ∇H :H ≡ 0 and ∇W (H) :H = (1 −β(H))/ √ 6 ≡ 0, letting µ → ∞ in the previous formula, as H µ λ →H strongly in W 1,2 (B 1 ; S 0 ) and uniformly on the support of ξ we conclude that  . Hence, if we choosev such that R tv R =v for any R ∈ S 1 , then each map H µ λ +tξv is S 1 -equivariant for any t ∈ R. As a consequence, according to Theorem 4.8 the radial hedgehog is an unstable critical point of F λ,µ also in the restricted class of S 1 -equivariant maps (a similar conclusion is valid forH as critical point of E λ in view of Proposition 4.7).
In the final remark we discuss the role of the biaxial phase in the instability results.
Expanding around the value t = 0 and using stationarity ofH and H µ λ both forβ and for the energy functionals, as t → 0 we infer together with As a consequence of (4.22) and (4.23) we see that for t sufficiently small biaxial escape occurs for the perturbed maps in the set where Φ T = 0; moreover, if Φ = ξv and µ is large enough then Proposition 4.7 and Theorem 4.8 show that this escape is energetically more favourable because the second variations of the energy functionals in (4.22) and (4.23) are negative.

Topology of minimizers
In this section, we discuss topological properties of field configurations Q satisfying assumptions (HP 0 ) − (HP 3 ), and restricting to energy minimizing configurations we will obtain in particular the proof of Theorem 1.6.
In connection with assumption (HP 2 ), we start recalling the following auxiliary result which characterizes simple connectivity of any smooth bounded domain Ω ⊆ R 3 . Then Ω is simply connected if and only if its boundary can be written as ∂Ω = ∪ N i=1 S i and each surface S i is diffeomorphic to the standard sphere S 2 ⊆ R 3 .
As already recalled in the Introduction, by assumption (HP 1 ) the maximal eigenvalue λ max (x) of Q(x) is simple on ∂Ω and there is a well defined smooth eigenspace map V max : ∂Ω → RP 2 . In addition, as Ω is simply connected, in view of Lemma 5.1 there exists a smooth lifting v ∈ C 1 (∂Ω; S 2 ) such that, under the inclusion Notice that as in (1.21), the caseβ = 1 corresponds to Q : ∂Ω → RP 2 ⊆ S 4 , up to normalization; in this case λ max (x) ≡ 2 3 on ∂Ω and still in view of (HP 2 ) there exists a map v ′ ∈ C 1 (∂Ω; S 2 ) such that under the inclusion RP 2 ⊆ S 4 we have Q(x) = |Q(x)| 3/2(v ′ (x) ⊗ v ′ (x) − 1 3 I) for all x ∈ ∂Ω. Hence, under this further assumption onβ one has Q ≡ |Q|V max on ∂Ω.
Recall also that assumption (HP 3 ) on the lifting v of the map V max : ∂Ω → RP 2 , namely that the total degree deg(v, ∂Ω) = N i=1 deg(v, S i ) is odd, does not depend on the chosen lifting since on each spherical component S i from Lemma 5.1 the lifting exists by simple connectivity of S i and is unique up a sign, so that each deg(v, S i ) may only change by a sign passing to a different lifting. Now we discuss properties of the biaxiality regions defined in (1.23). The first result we have shows that the biaxial escape observed in the introduction is indeed topological in nature and every possible value of the biaxiality is attained.
To see the first statement, we argue by contradiction and suppose that min Ω β • Q > −1. Then the maximal eigenvalue λ max (x) of Q(x) is always simple in Ω, hence C 1 , and there is a well defined eigenspace mapV : Ω → RP 2 which extends V max from the boundary of Ω to its interior andV ∈ C 1 (Ω; RP 2 ). Since Ω is simply connected this map can be lifted toṽ ∈ C 1 (Ω; S 2 ) which has to satisfy deg(ṽ, This result and the corollary below are the key points where analyticity is used.
Proof. Sinceβ • Q ∈ C ω (Ω), by Sard's theorem for analytic functions (see [58]) the set of singular value is finite on each compact set K ⊆ Ω, hence all but countably many t ∈ (−1,β) are regular for β•Q in Ω and for such t the level set {x ∈ Ω s.t.β•Q(x) = t} is contained in Ω by definition ofβ and it is a finite union of analytic, connected, orientable and boundaryless surfaces. However, since the singular values are finite on compact sets and in view of the definition ofβ the only accumulation point for the singular values can beβ. Otherwise there would be a countably many distinct singular value β n → β * ∈ [−1,β) and corresponding distinct critical points x n ∈ {β = β n } ⊆ Ω such that up to subsequences x n → x * ∈ {β = β * }. Notice that x * ∈ ∂Ω, otherwise x * would be a critical point as well and β * would be a singular value, with coutably many singular values attained in a neighborhood of x * , which contradicts Sard's Theorem. Thus x * ∈ {β = β * } ∩ ∂Ω, which however is impossible by definition ofβ. To conclude the proof, notice that the set of regular value is open, so for a regular value t choosing t ′ sufficiently close the conclusion 1) (resp 2)) follows easily by standard retraction following the gradient (resp. negative gradient) flow associated toβ • Q in Ω in a neighboorhood of {β = t} ⊆ Ω. Actually the same argument works even for any singular value t; such value being isolated by the previous part, the conclusion follows from real analyticity and the retraction theorem of Lojasiewicz (see [38,Theorem 5]).
Proof. The proof is similar to the one of Lemma 5.3 so it will be just sketched. In view of the analytic regularity up to the boundary the tensor Q has an analytic extensionQ (just by power series) to a larger open set Ω ⊆Ω, so that the functionβ =β •Q is analytic inΩ with finitely many critical values in Ω again by Sard's theorem. Clearly β = 1 is a critical value (maximum) hence choosing a slightly smaller regular value t ′ the conclusion still follows from [38] retracting the set {β t ′ } ⊆ Ω onto {β = 1} by the gradient flow of β • Q in Ω.
The first information on the topology of the biaxiality regions is contained in the following result. Proof. In view of Lemma 5.3 it is enough to prove claim 1) and 2) when t ∈ (−1,t) is a regular value to conclude the general case because (non)simple connectivity passes to deformation retracts. A similar argument applies to claim 3); since t = −1 is a singular vale (minimum) and by Lemma 5.3 it is isolated, combining claim 2) for regular values t ′ close to −1 the set {β t ′ } is not simply connected, hence its deformation retract {β = −1} is also nonsimply connected. Proof of claim 1) and 4). We assume that t ∈ (−1,β) is a fixed regular value ofβ • Q ∈ C ω (Ω) therefore the set {β t} is the closure of the open set Ω ∩ {β > t} which is open, bounded and with smooth boundary; in addition, {β t} and Ω ∩ {β > t} are homotopically equivalent (by inward-retracting both sets along the normal direction in a small neighborhood of the boundary). So it is enough to show thatΩ := Ω ∩ {β > t} is not simply connected. Observe that in view of the regularity of t and smoothness of the boundaries we can write ∂ Ω as a disjoint union where each S i is diffeomorphic to S 2 and eachS j is compact, analytic, connected, orientable and boundaryless surface because {β = t} ⊆ Ω. Now we claim that there exists j such thatS j has positive genus, whence claim 4) holds and the open setΩ is not simply connected in view of Lemma 5.1, i.e., claim 1) is completely proved.
To prove claim 2) we fix t ∈ (−1,β) a regular value and recall that ∂{β t} = {β = t} ⊆ Ω is a finite union of surfaces which are C 1 (indeed analytic), disjoint, embedded, connected and boundaryless. Notice that ∂{β t} = ∂Ω ∪ {β = t} is also a finite union of surfaces which are C 1 , disjoint, embedded, connected and boundaryless. Moreover, since Ω is simply connected by assumption and {β t} is not because of claim 1), by Lemma 5.1 one of the component of {β = t} has positive genus. Applying again Lemma 5.1 to {β < t} ⊆ Ω we see that {β < t} is not simply connected because the total genus of its boundary is positive, hence {β t} is also not simply connected because the two sets are homotopically equivalent.
Finally, the proof of claim 5) follows from claim 1) for regular values t ∈ (−1, 1) combined with the homotopic equivalence property stated in Corollary 5.4.
As a direct consequence of the previous proposition we have the linking property for biaxiality sets. In the final result of this section, which contains Theorem 1.6 as a particular case, we summarize the topological information we have as a straightforward combination of Lemma 5.2, Lemma 5.3, Corollary 5.4, Proposition 5.5 and Proposition 5.6.