Uniaxial versus biaxial character of nematic equilibria in three dimensions.

We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, t and some constraints on the material parameters, and we work in the t → ∞ limit that captures features of the widely used Lyuksyutov constraint (Kralj and Virga in J Phys A 34:829-838, 2001). We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map; (iii) estimates for the size of "strongly biaxial" regions in terms of the parameter t. We further show that global LdG minimizers in the restricted class of uniaxial Q -tensors cannot be stable critical points of the LdG energy in this limit.

Introduction

Nematic liquid crystals (LCs) are anisotropic liquids with long-range orientational order i.e. the constituent rod-like molecules have full translational freedom but align along certain locally preferred directions [11, 38]. The existence of distinguished directions renders nematics sensitive to light and external fields leading to unique electromagnetic, optical and rheological properties [11, 16, 30]. The analysis of nematic spatio–temporal patterns is a fascinating source of problems for mathematicians, physicists and engineers alike, especially in the context of defects or material singularities [13, 16].

In recent years, mathematicians have turned to the analysis of the celebrated Landau–de Gennes (LdG) theory for nematic liquid crystals, particularly in various asymptotic limits: see for example [8, 10, 15, 24] which is not an exhaustive list but are directly relevant to our paper. The LdG theory is a variational theory with an associated energy functional, defined in terms of a macroscopic order parameter, known as the -tensor order parameter [11, 21, 38]. Mathematically speaking, the LdG -tensor is a symmetric traceless matrix. The LdG energy typically comprises an elastic energy, convex in with several elastic constants, and a non-convex bulk potential, defined in terms of the temperature and the eigenvalues of -tensor [27]. With the one-constant approximation for the elastic energy density, the LdG energy functional has a similar structure to the Ginzburg–Landau (GL) functional for superconductivity [5, 9, 28] and we can borrow several ideas from GL theory to make qualitative predictions about global energy minimizers, at least away from singularities. However, there is an important distinction between the GL and LdG theories. In the GL-framework, researchers study maps, , (see e.g. [5, 25, 31]), whereas the LdG variable is a five-dimensional map, . A uniaxial -tensor has two degenerate non-zero eigenvalues and hence, three degrees of freedom and in this case, there is broader scope for transferable methodologies (see for example, [15]). A biaxial -tensor has three distinct eigenvalues with five degrees of freedom and maximal biaxiality corresponds to a vanishing eigenvalue. There are a plethora of open questions about how the two extra degrees of freedom manifest in the mathematics and physics of biaxial systems.

We re-visit questions related to the uniaxial versus biaxial structure of global LdG minimizers. We work with the simplest form of the LdG energy:with the one-constant Dirichlet elastic energy density and a quartic form of the bulk potentialwhere A is the re-scaled temperature, B and C are material-dependent constants [27]. The bulk potential, , is minimized by uniaxial tensors of a constant norm for low temperatures i.e. for , is minimized on the set of uniaxial tensors , with , and can be explicitly worked out in terms of A, B, C [4, 22, 27]. We work with large domains, , whose characteristic length D is much larger than the temperature-dependent biaxial correlation length, [17, 18] i.e. we assume that . We first present some heuristics (known in the literature) to motivate our rigorous results. Let denote the LdG energy of a bulk equilibrium i.e. a spatially homogeneous state that minimizes at a given . Scaling the -tensors according to , we obtainIn the language of length scales commonly adopted in the literature [17, 18], the uniaxial correlation length, , and if we work with , then the expression for above suggests that it is energetically preferable to have almost everywhere and to overcome topological frustration by allowing for biaxiality within small regions of size proportional to . In other words, we would expect LdG minimizers in this regime to have constant norm and defect cores to have biaxial structures with size dictated by .

We work with a dimensionless temperature and material-dependent parameterLet and a direct calculation shows that . Then the condition is equivalent to or , which in turn is equivalent to where is the isotropic-nematic transition temperature and is another critical temperature [17, 18]. In the literature, this is referred to as being “deep in the nematic phase” and this separation of scales can be achieved if B is an order of magnitude smaller than |A| and C (also see [14]). This assumption can be true for some commonly used liquid crystal materials [18]. Indeed, this assumption forms the basis of the popular Lyuksyutov constraint in the literature where authors work with -tensors of a fixed norm [17, 18] i.e.for sufficiently negative .

In what follows, we assume that and perform an asymptotic analysis of global LdG minimizers in the limit, for a fixed value of the ratio, , where is a characteristic material-dependent length scale often referred to as the bare biaxial correlation length [17]. We note that t cannot be unbounded but as with any asymptotic analysis, it is reasonable to expect that an analysis in the limit provides at least qualitative information for physically relevant large values of t, as have often been quoted in the literature. We can interpret this asymptotic limit in at least two different ways. One interpretation is to keep B, C, L fixed and vary the temperature, in terms of A. In the “deep nematic phase”, one could argue that the re-scaled temperature, |A|, is larger than B and C and therefore, t is large (say ; see [14]) and we might hope that our asymptotic analysis sheds qualitative insight into equilibrium properties for large but bounded values of t. For example, the celebrated biaxial torus solution on a three-dimensional droplet has been numerically reported in [14, 17, 18] for large values of t and our rigorous results reproduce some qualitative features of the biaxial torus in this asymptotic limit. In [14], the authors plot the eigenvalues of the biaxial torus solution away from the droplet centre and they find that the solution is uniaxial with positive order parameter at the droplet centre, followed by a torus of maximal biaxiality which contains an uniaxial ring of negative scalar order parameter; the biaxial torus then relaxes to match the imposed uniaxial Dirichlet radial boundary conditions. The uniaxial ring of negative order parameter contained within the biaxial torus is often referred to as a defect. Indeed private communication [16, 33] suggests that experimentalists have now observed the biaxial torus solution. We rigorously prove the co-existence of both maximal biaxiality and uniaxiality near each singular point (to be interpreted appropriately) of a global LdG minimizer on an arbitrary three-dimensional domain with uniaxial Dirichlet boundary conditions, and whilst this does not recover the structure of the biaxial torus solution, this is consistent with the picture of a torus of maximal biaxiality that contains a uniaxial ring of negative scalar order parameter.

We note that the eigenvalues of the minimizers of the bulk potential in (2) respect physical bounds if and only if [22, 38]If one were to work with i.e. with very low temperatures for fixed B and C, then the physicality constraints on the eigenvalues are violated. In this case, the bulk potential in (2) could be replaced by the Ball–Majumdar potential introduced in [2], the advantage being that the minimizers of the Ball–Majumdar potential respect the physical constraints on the eigenvalues for all temperatures. We expect the analysis in this paper to aid future work on these lines.

The other interpretation is to work with materials such that , where is some reference value for B, keeping |A| and C fixed. This condition is compatible with (4) and large values of t if one works with materials for which and , are fixed and consistent with (4). In particular, the physicality constraints on the eigenvalues can be respected with such an interpretation of the limit, for domains with .

Next, we briefly summarize the main results of this paper. We work with fixed topologically non-trivial Dirichlet boundary conditions, that are minimizers of the potential, , for . We first prove that global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting harmonic map, in this asymptotic limit. The singular set of a minimizing harmonic map is a finite set of isolated points [6, 34]. The uniform convergence follows from a Bochner-type inequality for the LdG energy density, first derived in [24] in the so-called vanishing elastic constant limit. We have two new cases to consider compared to [24], whilst deriving the Bochner inequality, see Cases II and III of Lemma 3.3, since we are studying a different asymptotic limit. The uniform convergence gives a fairly good picture away from the singular set of the limiting harmonic map. The next step is to adapt arguments from [10] to prove that the norm of a global LdG minimizer converges uniformly to a constant value, everywhere on (not merely away from singularities). In Theorem 1 (iii), we provide a rigorous proof of the norm convergence in the limit, yielding a rigorous justification of the common Lyuksyutov constraint in the literature. In this spirit, one may refer to our limit as the Lyuksyutov limit. In [10], the authors prove the global LdG minimizers have at least a point of maximal biaxiality, in the limit, but do not comment on the number or location of such points. We appeal to a topological result in [8] to deduce the existence of at least a point of maximal biaxiality and a point of uniaxiality near each singular point of the limiting map, in the limit; maximal biaxiality occurs when has a vanishing eigenvalue (see [26, 38]). We do not attempt to prove rigorous results about the Hausdorff measure or dimension of the regions of maximal biaxiality or uniaxiality i.e. do we have a biaxial torus or an uniaxial ring near each defect, since this is outside the scope of this paper. We make the notion of “strongly biaxial” regions in global minimizers more precise by computing estimates for the size of such regions in terms of t. The proof depends on scaling and blow-up arguments and well-established results in GL-theory and the theory of harmonic maps (e.g. [6, 34]). We consider all admissible scenarios and exclude all but one scenario, to find that the size of “strongly biaxial” regions scales as as . This implies that we only see strong biaxiality inside regions with radius proportional to the biaxial correlation length, , as would be expected on physical grounds [17, 18] (see Sect. 2 for precise statement).

Our second theorem generalizes the results in [15] to arbitrary 3D geometries with topologically non-trivial Dirichlet conditions. There exists a global LdG energy minimizer in the restricted class of uniaxial -tensors, for all material constants and temperatures. We show that these restricted minimizers cannot be stable critical points of the LdG energy for sufficiently large t in (3), for fixed L in (1). Appealing to topological arguments, we prove that any uniaxial critical point of the LdG energy, subject to the hypotheses of Proposition 2.1, has an isotropic point (with ) near each singular point of the limiting harmonic map, as . We then proceed with a local version of the global analysis in [15], equipped with certain energy quantization results for harmonic maps [6] and blow-up techniques, to deduce the local radial-hedgehog (RH) profile near each isotropic point and the instability of the RH profile for large t suffices to conclude the argument. In [20], the author rules out uniaxial critical points in one and two dimensional domains but the RH solution is an example of a semi-explicit uniaxial critical point (excluding an isolated isotropic point) on a 3D droplet and hence, uniaxial critical points in 3D remain of interest.

The paper is organized as follows. In Sect. 2, we review the theoretical background and state our main results. In Sect. 3, we give the proofs and in Sect. 4, we conclude with future perspectives.

Statement of results

Let be an arbitrary simply-connected 3D domain with smooth boundary. Let be the set of unit vectors in and let denote the set of symmetric, traceless matrices i.e.where is the set of matrices. The corresponding matrix norm is defined to be [24]and we use the Einstein summation convention throughout the paper.

We work with the Landau–de Gennes (LdG) theory for nematic liquid crystals [11] whereby a LC state is described by a macroscopic order parameter: the -tensor order parameter. The -tensor is a macroscopic measure of the LC anisotropy. Mathematically, the LdG -tensor order parameter is a symmetric, traceless matrix in the space in (5). A LC state is said to be (i) isotropic (disordered with no orientational ordering) when , (ii) uniaxial when has two degenerate non-zero eigenvalues and (iii) biaxial when has three distinct eigenvalues. A uniaxial -tensor can be written asfor some and some unit-vector , for a.e. . We include in our definition although corresponds to the isotropic phase. The unit-vector, , is the director or equivalently, the single distinguished direction of molecular alignment in the sense that all directions orthogonal to are physically equivalent for an uniaxial nematic. We recall the definition of the biaxiality parameter [26, 38],The definition (8) is commonly accepted in the literature and works well to differentiate biaxial phases from uniaxial phases since if and only if is uniaxial i.e. if and only if [4] and we have maximal biaxiality if and only if which necessarily requires a vanishing eigenvalue.

The LdG theory is a variational theory and has an associated LdG free energy. The LdG energy density is a nonlinear function of and its spatial derivatives [11, 27]. We work with the simplest form of the LdG energy functional that allows for a first-order nematic-isotropic phase transition and spatial inhomogeneities as shown below [24, 27]:Here, is a small material-dependent elastic constant, (note that ) with is an elastic energy density and is the bulk energy density that dictates the preferred nematic phase: isotropic/uniaxial/biaxial. For our purposes, we take to be a quartic polynomial in the -tensor invariants:where with ; ; are material-dependent constants, T is the absolute temperature and is a characteristic temperature below which the isotropic phase, , loses its stability [22, 27]. One can readily verify that is bounded from below and attains its minimum on the set of -tensors given by Majumdar [22, 23] is the identity matrix andThe set, (11), is the set of uniaxial -tensors with constant order parameter, . The physically relevant range is often defined to be [2, 22] and the physicality constraint is clearly violated for and , for fixed B and C.

In what follows, we take the Dirichlet boundary condition to befor some arbitrary smooth unit-vector field, , with topological degree (see, e.g., [7] and [6] for the definition and the main properties of the topological degree). In particular, this means that has no continuous -valued extension inside . The corresponding admissible space iswhere is the Soboblev space of square-integrable -tensors with square-integrable first derivatives [12], with normIn what follows, we identify the degree of a uniaxial -tensor in on the boundary, with the degree of the director field, on the boundary, , which is well defined because is smooth. The existence of a global minimizer of in the admissible space, , is immediate from the direct method in the calculus of variations [12]; the details are omitted for brevity. It follows from standard arguments in elliptic regularity that all global minimizers are smooth and real analytic solutions of the Euler–Lagrange equations associated with on ,where is a Lagrange multiplier accounting for the tracelessness constraint [24].

Define the re-scaled maps, . Letwhere D is a characteristic geometrical length scale related to the size of . Then and we work with the re-scaled LdG energy given byIn (17), we have added a re-scaled constant () to the energy density to ensure that the energy density is non-negative.

We point out that can be interpreted as the ratio of the bare biaxial correlation length, , (see [17, 18]) and D i.e. . The regime studied in this paper corresponds to and simultaneously. For clarity and simplicity, we work with fixed but small values of ; there is no loss of generality since all the arguments are valid when varies with t but satisfies . Further, the bulk potential has two contributions in (17), both of which are nonnegative in view of (8). The first term vanishes for and the second term vanishes if and only if is uniaxial with unit norm [from (8) again]. The two contributions scale differently as i.e. the uniaxiality preferred by the second term is a lower order effect compared to the preference for unit norm, as enforced by the first term in the bulk potential. The re-scaled boundary condition is . In what follows, all statements are to be understood in terms of the re-scaled variables and we drop the bars from the variables for brevity. We recall the definition of a minimizing limiting harmonic map.

Definition 1

A (minimizing) limiting harmonic map with respect to the re-scaled Dirichlet condition in (13), is a uniaxial map of the formwhere is a minimizer of the Dirichlet energyin the admissible space [34].

In particular, is a harmonic map into , i.e., a solution of the harmonic map equations . The singular set of , denoted by , is a finite set of points [34, 35].

We have two main theorems.

Theorem 1

Let be as above. Let with and let be a corresponding global minimizer of the LdG energy in (17), in the admissible space Then (up to a subsequence), we have the following results.

converges to a limiting harmonic map, defined in (18), strongly in and uniformly away from , as .

Let where denotes a ball of radius centered at , and , for a fixed and . Then for j large enough.

uniformly on as .

For each , we have for j large enough, and .

For each and , we have for j sufficiently large.

An immediate consequence is that global energy minimizers cannot be purely uniaxial, as also stated in [10] where the authors prove that global LdG minimizers must have at least a point of maximal biaxiality.

Theorem 2

Let be as above. Let be such that as . For each , let be a global minimizer of the LdG energy (17) in the restricted class of uniaxial -tensors of the form (7). Then has non-negative scalar order parameter and satisfies the following energy boundwith and as in (19), for each . For j sufficiently large, cannot be a stable critical point of the LdG energy in (17) in the admissible space,

Theorem 2 is a consequence of Proposition 2.1 below and Propositions 3 and 8 of [15].

Proposition 2.1

Let be as above. Let be such that as . Let be a sequence of uniaxial critical points of the re-scaled LdG energy in (17) with non-negative scalar order parameter and satisfying the energy bound (22) for all . Then, passing to a subsequence (still indexed by j), the sequence converges uniformly to a (minimizing) limiting harmonic map, as , everywhere away from the singular set of . We have that

for each , there exists such that for all and and

given any sequence, , such that , there exists a subsequence and an orthogonal transformation (which may depend on the subsequence) such that the shifted maps converge to in for all , where is the unique, monotonically increasing solution, with , of the boundary-value problem

Proposition 2.1 provides a local description of the structural profile near a set of isotropic points in the uniaxial critical points, , in terms of the well-known RH solution. The RH solution is a rare example of a semi-explicit critical point of the LdG energy simply given bywhere h is defined as in (24). The boundary-value problem (24) has been studied in detail, see for example in [19, 23]. The RH solution is locally unstable with respect to biaxial perturbations, as has been demonstrated in [22, 26].

Proof of the theorems

Recall that the re-scaled LdG energy is given bywithand that for all , the potential is minimized on the setDenote the LdG energy density byIn Theorem 1 we consider global minimizers of (26) in the admissible space, , for each , the existence of which is guaranteed by the direct method of the calculus of variations. Standard elliptic regularity arguments (presented in [24, Prop. 13]) show that each minimizer is a real analytic solution of the Euler-Lagrange equationswhereTheorem 2 is proved by assuming that a sequence of minimizers in the restricted class of uniaxial -tensors is composed of stable critical points of the LdG energy and then reaching a contradiction. In both cases, we consider classical solutions of (30) that satisfy the energy bound (22) (this follows from the fact that any minimizing limiting harmonic map belongs to , so it can be used as a trial function). As done in [10, 15], the arguments in [24, Lemmas 2 and 3; Props. 3, 4, and 6] can be adapted to prove the following preliminary results.

Proposition 3.1

Let and, for each , let be a classical solution of the corresponding equations (30), satisfying the energy bound (22). Then, passing to a subsequence,

converges strongly to a (minimizing) limiting harmonic map in ,

and for some C independent of j,

for all and so that ,

for any compact , where denotes the singular set of , uniformly in K.

However, this only ensures that uniformly as , away from . By contrast, in [24], the authors could prove that for a fixed , the global minimizers uniformly approach , everywhere away from , as in (1). The analysis in [24] would carry over to the limit i.e. when the domain is large compared to a characteristic material-dependent correlation length. We want to prove the following result.

Proposition 3.2

Under the hypotheses of Proposition 3.1, converges uniformly to away from , as .

The key step is to prove a Bochner inequality of the form

Lemma 3.3

There exist and a constant independent of t such that if is a solution of (30), thenfor all satisfying

This inequality is proven in [24, Lemmas 5–7] in the case when is close to the manifold , defined in (28), which does not necessarily hold in our case as explained below.

Proof of Lemma 3.3

The quantity plays an important role and we note the elementary inequalityThis quantity controls the biaxiality parameter as shown below: since from (33) by assumption.

As in [24], we use the Euler–Lagrange equations (30) to derive the following inequality:Moreover, we haveandWe consider three separate cases according to the sign of and the magnitude of .

Case I . This, when combined with (33), implies that and that the biaxiality parameter, , so that is approximately uniaxial with unit norm for sufficiently small. Equivalently, for some (where depends on ). In this case, we can repeat all the arguments in [24, Lemmas 5–7]; we state the key steps for completeness.

We denote the eigenvectors of by respectively and let and denote the corresponding eigenvalues. DefineFrom the inequality with , we necessarily have thatOne can repeat the arguments of [24, Lemma 5] verbatim to show that there exists a positive constant such that:for all with where , the positive constant C being independent of t.

The proof of the Bochner inequality now follows from the chain of inequalities: In the above, we have followed Equations of [24] to deduce thatfor some positive constant independent of t, for sufficiently close to , as is the case here. In Eq. (41) above, we have carried out a Taylor series expansion of the right-hand side of (37) about , is the remainder term in the Taylor series expansion which is well-controlled and the constants and are independent of t but dependent on (which does not matter since is fixed). For sufficiently small, we can absorb the -term on the right by the -contribution on the left so thatfor sufficiently small [from (40)], yielding the Bochner inequalityfor independent of t, as required.

Case II and . The key difference between Case I and Cases II and III is that need not be small for Cases II and III i.e. need not be close to the manifold, , rendering these cases outside the scope of the results in [24].

We refer to the relations (37)–(39) and use the Cauchy–Schwarz inequality in (38) to see thatFor and chosen as above, we havefor positive constants independent of t and is fixed for our purposes. Finally, we appeal to (39) to obtainFor sufficiently small, we can absorb the term in (44) by the term in (43). Choosing small enough (and independent of t), recalling (37), the lower bound (43) and the upper bound (44), we havewhich is precisely the Bochner inequality.

Case III so that .

A large part of the computations for Case II carry over to Case III. In particular, (44) is unchanged and it remains to note that for , the bulk potential . In particular,Define and to bewhere by assumption. The second, third and fifth terms in (38) are positive, henceSince , we get and therefore, the Bochner inequality (32) then follows from (37), (44), (48), (46).

The uniform convergence result in Proposition 3.1 (also see the maximum principle in [22]) ensures that away from the singularities of and hence, (33) is satisfied for all sufficiently large. Thus, Bochner’s inequality holds away from for large t, allowing us to deduce the following -regularity property, exactly as in [24, Lemma 7]:

Lemma 3.4

Let be a compact subset that does not contain any singularity of . Then there exist and constants (independent of j) so that if for and , we havethenfor all .

Proof of Proposition 3.2

The normalized energy, , can be controlled away from , by simply (i) using the strong convergence of the sequence, to as in and (ii) the fact that is bounded away from , independently of . Thus, the uniform convergence, , away from as , follows immediately from Lemma 3.4, combining (49) and (50) and Ascoli–Arzelá Theorem.

We are almost ready to prove the main theorems. It only remains to state a standard result from homotopy theory and to recall that the Landau–de Gennes energy functional has a Ginzburg–Landau like structure by blowing-up at scale and working in the limit.

Lemma 3.5

Let for some , where B is a ball . Suppose that is homotopic in [see (28)] to for some . Then .

Proof

Since has a continuous -valued extension inside , it is homotopic in to the constant tensor . Hence, combining the two homotopies, we deduce that is homotopic to a constant in .

Since is simply-connected and is a universal cover of , the latter homotopy lifts to , implying that is homotopic to a constant in and hence, , as needed.

Lemma 3.6

Let and, for each , let be a classical solution of (30). Suppose that converges strongly in to a minimizing limiting harmonic map . Let be a sequence of points converging to some in the singular set of . Then (up to a subsequence) the rescaled mapsconverge in for all to a smooth solution of the Ginzburg–Landau equations , in , which satisfies the energy bound

The proof follows from the celebrated energy quantization result for minimizing harmonic maps at singular points, established in [6]:We begin by noting that , thereforefor every small .

By the monotonicity formula, Proposition 3.1 (iii), for every fixed , every small , and every j sufficiently large, we have that (we have used the inequality above). This combined with (54) and (53) yields the following inequalityfor every .

Using the energy bound above, we can extract a diagonal subsequence, converging weakly in , to a limit map satisfying the energy bound (52). One can check that solves the weak form of the Ginzburg–Landau equations in (write the weak form of the partial differential equations for and pass to the limit when ). Standard arguments in elliptic regularity then show that is a classical solution of the Ginzburg–Landau equations and that the diagonal subsequence converges in to .

Proof of Theorem 1

(i) It follows from Propositions 3.1 and 3.2.

(ii) This is an immediate consequence of the uniform convergence, as , away from the singular set, of . is purely uniaxial by definition i.e. [see (8) for the definition of the biaxiality parameter, ]. The map is continuous for and the conclusion, , follows for any fixed , provided j is large enough.

(iii) This can be proven as in [10], where the authors prove that on , for j large enough. We argue by contradiction and we assume that there exist points such that (for some independent of j), for all j in the sequence.

In view of part (i), we may assume for some and repeat the arguments in Lemmas 3.1 and 4.1 of [10] i.e. perform a blow-up analysis of the re-scaled maps, . By Lemma 3.6 and [10, Lemma 3.1], the rescaled minimizers converge locally smoothly to a minimizer, , of the Ginzburg–Landau energy,(on open sets with compact closure with respect to its own boundary conditions) with the energy growth as . In addition we have because of the normalization. We can then use the same blow-down analysis as in [10] to show that converges strongly in to a -valued minimizing harmonic map, labelled by , as . Indeed, one can use the well-known Luckhaus interpolation Lemma as in [29], Proposition 4.4, still for a sequence of functionals converging to the Dirichlet integral for maps into a manifold, showing that minimality persist in the limit and the convergence is actually strong in .

From the monotonicity formula for the Ginzburg–Landau energy, is a degree-zero homogeneous harmonic map, hence it is smooth away from the origin by partial regularity theory [34]. Since the latter is constant by Schoen [36], the GL minimizer is also a constant matrix of norm one from the monotonicity formula for the GL energy. Thus, which yields the desired contradiction.

(iv) Let be fixed. From part (i), (ii) and since is fixed and arbitrary, we necessarily havefor j sufficiently large, (depending only on ). From (iii) above, we have that uniformly on as .

Thus, if we define the setfor each and let , we have the following:Suppose, for a contradiction, that and let . Then the composition of the aforementioned retraction with yields a map whose trace is homotopic in to . By Lemma 3.5 we would conclude that , a contradiction with the fact that near each singular point, for small enough fixed at the beginning.

The restriction of to the boundary, for j large enough (depending only on [by (61)] and (in view of the inclusion ).

For , the maps and are homotopic in (thanks to the uniform convergence; composing pointwise with the affine homotopy in keeps the images inside for j large enough).

retracts homotopically onto for every , see [8, Lemma 3.10]; see also [10], Corollary 1.2 and Section 5 therein.

Similarly, assume that for infinitely many j in the sequence. Then is purely biaxial for all [recall that there are no isotropic points from part (iii)]. Let be the eigenvector corresponding to the maximum eigenvalue (which is uniquely defined up to a sign). Recall that is continuous in , hence is continuous (if and then clearly maximizes in and the maximal eigenvalue is simple). As a consequence, we choose to be a continuous lifting so that .

Now, converges uniformly to on . Therefore, and uniformly on . This implies that uniformly on sinceWe conclude that is homotopic to , first in for some small (composing pointwise with the affine homotopy in ) and then in (composing with the retraction from to ). Since has a continuous extension inside , we recall Lemma 3.5 and obtain a contradiction with the fact that for every small enough.

The uniaxial set has zero Lebesgue-measure, as has already been established in [24, Prop. 14].

(v) For each and fixed, consider the biaxiality set, , around and its diameter, . We have as from (i) and (ii) above.

We claim that as , which follows by blowing up , at scale , and excluding remaining decay rates. Firstly, let such that and let . Clearly and as . Then by defining and by (ii), we immediately have on , at two antipodal points on , and , for j large enough.

Define and we get, up to a sequence of rotations which we do not specify explicitly,on the unit ball . The the rescaled maps are defined on the family of expanding domains, and are local minimizers on compact subdomains of the functionalswith as . Taking into account the Euler–Lagrange equations [corresponding to (64)], we can exclude the following regimes:

(a) since we easily deduce that (up to subsequences) in for by the uniform -bound and elliptic regularity. Indeed, for , the nonlinear terms in the Euler–Lagrange equations vanish as . Thus is bounded and harmonic, hence constant (of norm one from (iii) above) by Liouville’s Theorem and this fact contradicts (63) which holds for the limiting map by uniform convergence.

(b) ; this regime has already been discussed in item (iii) above (when proving norm convergence generalizing results in [10]) and hence, up to a subsequence, in for . Here is a bounded Ginzburg–Landau local minimizer on the whole of such that as . Arguing as in Lemma 3.6 and item (iii) above, we infer that is a constant matrix of norm one, contradicting (63) which still passes to the limit under smooth convergence and clearly cannot hold for constant maps.

(c) . Here (up to a subsequence), the limiting map is a local minimizer of among -valued maps. Indeed the sequence is locally bounded in by the monotonicity formula and hence converges weakly in (up to a subsequence). The limiting map is clearly -valued, as can be seen by applying Fatou’s Lemma to (64). Additionally, we can prove strong convergence to the limiting map and the minimality of the limiting map, arguing as in item (iii) above, exactly as in Lemma 4.1 of [10]. We omit the details for brevity.

Therefore, in and is a locally minimizing harmonic map. At this point, we appeal to two powerful and celebrated results in [36] (also see [34]) (i) If , then every minimizing map from a manifold M of dimension n into is smooth in the interior of M and in our case and and (ii) for and , there is no non-constant minimizing map u from into . This shows that by the constancy of stable tangent maps into spheres proven in [36]. We use the constancy of stable tangent maps into spheres from [36] and the monotonicity formula, arguing by analogy with case (b), to infer that is a constant matrix of norm one. In view of this constancy property, we can improve the convergence in to a smooth convergence [we just need to use the argument based on the Bochner inequality from (i) above]. Since biaxiality is constant for constant maps, we contradict (63).

(d) Finally, we consider the regime . Here, the limiting energy is again the Dirichlet energy, , for -valued maps in , as can be seen by applying Fatou’s Lemma to (64). We again have in , arguing similarly to part (c) above. However, from the uniaxiality of the limiting tensor and the lifting results in [3], we lift to an -valued minimizing harmonic map . From the classification result for harmonic unit-vector fields, such as , in [1, Thm. 2.2], we either have or . This is in contrast to case (c) above where all minimizing maps from to are constant. Again as in step (c), we can improve the convergence in to locally smooth convergence except at most at one point (combining the smoothness of the limiting map with small energy regularity to infer smooth convergence). This contradicts (63) since everywhere except possibly for the origin, since is uniaxial for .

Proof of Theorem 2

We can prove the existence of a global LdG minimizer , of the re-scaled energy (17), in the restricted class of uniaxial -tensors, for each , from the direct methods in the calculus of variations. It suffices to note that the uniaxiality constraint, is weakly closed and the existence result follows immediately.

The limiting harmonic map is uniaxial and hence, the energy bound (22) follows immediately since the upper bound is simply the re-scaled LdG energy of . The uniaxial map, , necessarily has non-negative scalar order parameter, . Indeed, note that by uniaxiality, (resp. ) iff has positive (resp. negative) scalar order parameter and also that iff at any . We set , which is an open subset (possibly empty), since is globally Lipschitz in . If , then we define the uniaxial admissible perturbationand one can easily check that is globally Lipschitz in and , contradicting the assumed global minimality of in the restricted class of uniaxial -tensors. We can then appeal to Proposition 2.1 and proceed by contradiction. We assume that the global LdG-minimizers, , in the restricted class of uniaxial -tensors, are stable critical points of the LdG energy, for j large enough. The sequence, , then satisfies the hypothesis of Proposition 2.1, for large j. We thus, conclude that each , has a set of isotropic points (at least one near each singular point of ) and is asymptotically described by the RH-profile near each isotropic point as in the sense of Proposition 2.1. Recall that the RH-solution, (23) is known to be unstable with respect to biaxial perturbations localized around the origin [22, 26], for large t. This suffices to prove that global minimizers in the restricted class of uniaxial -tensors cannot be stable critical points of the LdG energy in this limit, since stability of would pass to the limit under smooth convergence.

Proof of Proposition 2.1

Proof of (i): By Propositions 3.1 and 3.2, after extracting a subsequence, we have that converges strongly in and uniformly away from the singular set , to a (minimizing) limiting harmonic map, . We prove that for each and every fixed sufficiently small, there exists such that for every , the map has an isotropic point, , in . The stated conclusion then follows by a diagonal argument on . Suppose, for a contradiction, that we can find a subsequence, , such that for all . Since is purely uniaxial for all j by assumption, we have that is continuous on and the uniform convergence to implies that converges uniformly to on . Arguing as in the proof of Theorem 1 (iv) we obtain a contradiction.

Proof of (ii): The aim is to prove that has a RH type of profile, (23), near each singular point in , for j sufficiently large. The proof follows from Lemma 3.6 and Propositions 4 and 8 in [15]. We begin by noting that for each in , we can extract a sequence, , such that and as . By Lemma 3.6, the rescaled maps (51) converges in to a classical solution of the Ginzburg–Landau equations satisfying the energy growth (52). Moreover, it can be seen that is uniaxial and has a non-negative scalar order parameter. Finally because for each k, by assumption. We conclude that the hypotheses of [15, Prop. 8] are satisfied. We reproduce the statement of [15, Prop. 8] below, for completeness.

Proposition 3.7

(Proposition 8, [15]) Let be a uniaxial solution of with and non-negative scalar order parameter, satisfying the energy bound (52). Let h denote the unique solution for the boundary-value problem (24). Then there exists an orthogonal matrix such that

This yields the conclusion of Proposition 2.1.

Conclusions

Theorem 1 focuses on global LdG minimizers on arbitrary 3D domains, with arbitrary topologically non-trivial Dirichlet conditions, subject to fixed in (17), in the limit. We prove that global minimizers are well described by a (minimizing) limiting harmonic map, , away from the singular set of . Further, we prove that the norm of a global minimizer converges uniformly to a constant everywhere and there is at least a point of maximal biaxiality (with ) and a point of uniaxiality (with ) near each singular point of , in this asymptotic limit. As explained in the introduction, these results provide a rigorous justification for the widely used Lyuksyutov constraint for low temperatures [17, 18]. Further, they give us quantitative information about the expected number and location of points (or regions) of maximal biaxiality, which was previously not proven in the literature. In other words, a precise knowledge of the limiting map yields accurate information about both the far-field behaviour, the location of the defect cores and some partial information about the structure of the defect cores, which appears to be independent of the geometry. Numerical results in [17, 18] support our analysis that defect structures are independent of the geometry for “large” domains.

We briefly explain how our results are related to the celebrated biaxial torus structure of nematic defect cores reported in [14, 17, 32, 37]. The biaxial torus has been largely reported in 3D droplets although one could conjecture that it is a generic defect structure independent of geometry. The biaxial torus usually describes a uniaxial ring of negative scalar order parameter enclosed by a torus of maximal biaxiality, as has been suggested by the exhaustive work in [14, 17, 32, 37]. There is no rigorous proof of the existence of such a biaxial torus and our results are only a first step in that direction. In Theorem 1, we prove the co-existence of maximal biaxiality and uniaxiality near each singular point of , which is at least consistent with the biaxial torus picture. Indeed, rigorous analytical results such as ours could motivate experimentalists to probe into defect cores, in quest of the biaxial torus or indeed other defect structures, which may have the qualitative properties proven in Theorem 1.

Theorem 2 is a statement on global LdG minimizers of (17) within the restricted class of uniaxial -tensors in the limit. These constrained uniaxial minimizers exist although they need not be critical points of the LdG energy. Indeed, uniaxial solutions of (15) are, in general, difficult to find but the RH solution is a 3D uniaxial critical point of the LdG energy i.e. is a solution of the system (15) of the form (7) with for on a 3D droplet [14, 15, 20]. Indeed, one could imagine a continuous uniaxial perturbation of the RH solution that remains a solution of the system (15), at least in an approximate sense which needs to be carefully defined. In the absence of a rigorous exclusion of generic uniaxial critical points in 3D, Theorem 2 and Proposition 2.1 have a twofold purpose: (i) firstly, they rule out the stability of such uniaxial critical points, if they can be constructed and (ii) secondly, they establish the universal RH-type defect profiles for such uniaxial critical points with specific properties as in Theorem 2 (if they exist). We hope to make rigorous studies of uniaxial and biaxial defect profiles, in differents asymptotic limits, in future work.