Symmetry and Multiplicity of Solutions in a Two-Dimensional Landau–de Gennes Model for Liquid Crystals

We consider a variational two-dimensional Landau–de Gennes model in the theory of nematic liquid crystals in a disk of radius R. We prove that under a symmetric boundary condition carrying a topological defect of degree k2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{k}{2}$$\end{document} for some given even non-zero integer k, there are exactly two minimizers for all large enough R. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points.


Introduction
The questions of symmetry and stability of critical points for the Landau-de Gennes energy functional on two dimensional domains have been recently raised in the mathematical liquid crystal community [4,15,20,23,24,27].The particular focus of these works was on analyzing special symmetric critical points and investigating their stability properties depending on multiple parameters of the problem.
In this paper we continue the study of the symmetry, stability and multiplicity of critical points of the Landau-de Gennes energy using the same mathematical setting.The main result we establish is the uniqueness (up to reflection) of the global minimizer in the most relevant physical regime of small elastic constant under the strong anchoring boundary condition which has a topological degree k 2 with even nonzero k (see (1.7)-(1.9)below).As a consequence of this uniqueness, the minimizers satisfy a k-fold O(2)-symmetry (see Definition 1.1) which has not been identified earlier.Additionally, we prove the existence of two other k-fold O(2)-symmetric critical points which are not minimizing.
We recall the (non-dimensional) Landau-de Gennes energy functional in the disk B R ⊂ R2 of radius R ∈ (0, ∞) centered at the origin: where S 0 is the set of Q-tensors: 2) The nonlinear bulk potential is given by where a 2 ≥ 0, b 2 , c 2 > 0 are appropriately scaled parameters and the normalizing constant f * is chosen such that the minimum value of f bulk over S 0 is zero.A direct computation gives The set of minimizers of f bulk , which we call the limit manifold, is given by the following set of uniaxial Q-tensors where I 3 is the 3 × 3 identity matrix.
The Euler-Lagrange equations satisfied by the critical points of F read The Landau-de Gennes energy describes the pattern formation in liquid crystal systems, in particular, the so-called defect patterns.A well-studied limit, relating the defects in the Landau-de Gennes framework with those in the Oseen-Frank framework, is that of small elastic constant (after a suitable non-dimensionalisation -see [16]), considered, for instance, in [1,4,10,18] in 2D and [11,29,30] in 3D.Qualitative properties of defects and their stability are studied, for example, in the case of one elastic constant in 2D domains in [15,23,24] and in 3D domains in [12,22,28].Numerical explorations of the defects in 2D domains and several elastic constants are available in [2,17,27].We couple the system (1.6) with the following strong anchoring boundary condition: where the map Q b : R 2 \ {0} → S 0 is defined, for some fixed k ∈ Z \ {0}, by n(r cos ϕ, r sin ϕ) := (cos kϕ 2 , sin kϕ 2 , 0), r > 0, 0 ≤ ϕ < 2π.
(1.9) 1 and, as a map from ∂B R ∼ = S 1 into RP 1 (see for instance formula (8.16) in [8]), has 1  2 Z-valued topological degree k 2 .It is worth pointing out the difference between the cases when k is even and odd.If k is even, the vector n defined in (1.9) is continuous at ϕ = 2π, however, if k is odd there is a jump discontinuity at ϕ = 2π.Nevertheless the boundary data Q b defined in terms of n in (1.8) is continuous for any k ∈ Z, but its topological features as a map from ∂B R ∼ = S 1 into RP 1 will depend on the parity of k, see Ball and Zarnescu [3], Bethuel and Chiron [7], Brezis, Coron and Lieb [8], Ignat and Lamy [21].In particular, this leads to major qualitative differences in the properties of the critical points; see [3,15,23,24] for analytical studies in 2D domains which involve only one elastic constant, and [17,27] for numerical studies for several elastic constants.Moreover, in the limit of small elastic constant the minimal Landau-de Gennes energy becomes infinite in the case of odd k (see [10,18]) and is finite in the case of even k (see [14]).This phenomenon leads to significant differences in the structure and distribution of defects depending on the parity of k (see Appendix A).
Two group actions on the space H 1 (B R , S 0 ).In the following we consider two types of symmetries induced by two group actions on the space H 1 (B R , S 0 ) which keep invariant both the energy functional F as well as the boundary condition (1.7)-(1.9).
Definition 1.1.Let k ∈ Z \ {0}.The subset of H 1 (B R , S 0 ) that is invariant under the group action (1.10) is called the set of k-fold O(2)-symmetric maps.Such a map Q ∈ H 1 (B R , S 0 ) is therefore characterized by Sometimes when k is clear (uniquely determined) from the context, we will omit "k-fold" and simply call the above property as O(2)-symmetry.The following proposition provides a characterization of k-fold O(2)-symmetric maps in the case of even k.Its proof is postponed until Section 2.
• Z 2 -symmetry.We introduce the group action of Z 2 on H 1 (B R , S 0 ): where J stands for the reflection with respect to the plane perpendicular to the (0, 0, 1)-direction. (1.18) Definition 1.3.The subset of H 1 (B R , S 0 ) that is invariant under the group action (1.17) is called the set of Z 2 -symmetric maps.Such a map Q ∈ H 1 (B R , S 0 ) is therefore characterized by We will see in Proposition 2.8 that a map Q ∈ H 1 (B R , S 0 ) is Z 2 -symmetric if and only if e 3 = (0, 0, 1) is an eigenvector of Q(x) for almost all x ∈ B R .We note an important difference between the definitions of k-fold O(2)-symmetry and Z 2 -symmetry: the O(2)-action on H 1 (B R , S 0 ) applies to both the domain and the target space while the Z 2 -action applies only to the target space.
It is clear that if Q is a minimizer (or a critical point) of F under the boundary condition (1.7) then the elements of its orbit under the k-fold O(2)-action as well as the Z 2 -action are also minimizers (or critical points, respectively).A natural question therefore arises: do minimizers/critical points of F (under (1.7)) have k-fold O(2)-symmetry, or Z 2 -symmetry, or both, or maybe none?Some partial answers are available in the literature.In a work of Bauman, Park and Phillips [4], which is not directly related to symmetry issues, it was shown that, for |k| = 0, 1 and as R → ∞, there exist none-O(2)-symmetric critical points.Their results might tempt one to extrapolate a lack of symmetry in general.This intuition would be also apparently supported by the numerical simulations in Hu, Qu and Zhang [20] which observed lack of symmetry for a certain radius.However, in [27] the k-fold O(2)-symmetry was numerically observed for a minimizer in the case of even k and large enough radius R (probably larger than in the examples explored numerically in [20]).
We will see later (in Section 2) that all Z 2 × O(2)-symmetric maps are of the form2 It is known from [23] that all Z 2 × O(2)-symmetric critical points of F coincide with the so-called k-radially symmetric critical points.See Section 2 for more details.Note that the boundary data -symmetric on ∂B R .However, we will prove that the minimizers of F [•; B R ] under the boundary condition (1.8) do not satisfy this symmetry (namely they are not Z 2 -symmetric).
The structure and stability properties of Z 2 × O(2)-symmetric critical points were investigated in [15,23,24].In particular, it was proved that • when b = 0 and R < ∞ they are minimizers of the Landau-de Gennes energy for all k ∈ Z\{0} (see [15]); • when b = 0, N ∋ |k| > 1 and R is large enough they are unstable (see [23]); • when b = 0 and k = ±1 they are locally stable for all R ≤ ∞ under suitable condition on w 0 and w 1 in (1.19) (see [24]). 3n this paper we focus on the case where the k-fold O(2)-symmetry does not imply in general the Z 2 -symmetry (some remarks on the case k odd are provided in Appendix A).Our main result states that for large enough radius R the Landau-de Gennes energy (1.1) under the boundary condition (1.7) has exactly two minimizers and these minimizers are k-fold O(2)-symmetric and Z 2 -conjugate to each other.
It is clear that the Euler-Lagrange equation (1.6) for Q ± R then reduces to a system of ODEs for (w 0 , w 1 , 0, ±w 3 , 0) with the boundary condition w 0 The idea of the proof of Theorem 1.5 is presented in Section 3.1.An assumption in the above theorem concerns the radius of the domain which is taken to be large enough.This is a physically relevant assumption, capturing the most interesting physical regime of small elastic constant (as explained in [16] and studied, for instance, in [1,4,10,14,18] in 2D and [11,29,30] in 3D).
We would like to draw the attention to our related uniqueness results in a Ginzburg-Landau settings [25,26] where the bulk potential satisfies a suitable global convexity assumption.In these articles, we established a link between the so-called non-escaping phenomenon and uniqueness of minimizers.In the context of Q-tensors, a non-escaping phenomenon would mean the existence of O(2)-symmetric critical point Q such that Q •E 3 does not change sign.While it is not hard to prove the existence of such critical points for large R (see the last paragraph in the proof of Theorem 3.1), the method in [25,26] does not apply to the present setting as our bulk potential f bulk does not satisfy the relevant global convexity.In a sequel to the present article, we will apply the method developed here to prove a similar uniqueness result for minimizers of a Ginzburg-Landau type energy functional where the bulk potential satisfies only a local convexity property near the limit manifold.
Our second result concerns the multiplicity of k-fold O(2)-symmetric critical points of F .This is coherent with the numerical simulations in [27,Section 3.2] for k = 2 and [20, Section 2.2], which observed, for large enough R, that there can be several distinct solutions, corresponding to boundary conditions (1.7).
Theorem 1.6.Let a 2 ≥ 0, b 2 , c 2 > 0 be any fixed constants and k ∈ 2Z \ {0}.There exists some The rough idea of proving Theorem 1.6 is the following: Theorem 1.5 gives us two global minimizers Q ± R .By the mountain pass theorem, there is a mountain pass critical point, denoted The main point in the proof of Theorem 1.6 is to show that the mountain pass solution Q mp R does not coincide with the k-radially symmetric critical point Q str R constructed in [23].This is done by an energy estimate showing in particular the existence of paths between Q ± R for which the energy is uniformly bounded with respect to R, see (4.6).As the maps Q ± R , after suitably rescaled, converge to two S * -valued minimizing harmonic maps of different topological nature, this highlights the difficulty of constructing that path; see Section 4 for a more detailed discussion.Moreover, we show that the mountain pass critical point is not Z 2 -symmetric, thus its Z 2 -conjugate Qmp R is also a critical point, thus yielding five different critical points.
In Table 1, we summarize the properties of the critical points from Theorem 1.6. 4he two global minimizers Q ± R as well as the two mountain pass critical points 5 The subscript k in the little o-terms indicates that the rate of convergence may depend on k.The subscript k in the big O-terms indicates that the implicit constant may depend on k.The energy of Q ± R is bounded from above by and converges as R → ∞ to the Dirichlet energy of the S * -valued minimal harmonic map(s) on B 1 , which is 4πs The estimate for the energy of the mountain pass solutions is given in (4.5).In addition to these solutions, we also have the non-O(2)-symmetric solutions constructed in [4], which have energy O(|k| ln R) for large R, see [4,Theorem B].
To dispel confusion, we note that the Ginzburg-Landau counterpart for our model is the 2D − 3D Ginzburg-Landau model (see [26,Theorem 1.1]).In particular, the minimal energy remains bounded as R → ∞, which is contrary to the 2D − 2D Ginzburg-Landau case where the minimal energy grows like ln R as R → ∞ (see e.g. the seminal book of Béthuel, Brezis and Hélein [6] or [31]).In the 2D − 3D case, it was shown in [26] that, for every k ∈ 2Z \{0} and under the boundary condition (1.9), there exists R * > 0 such that the Ginzburg-Landau energy functional has a unique critical point for R ≤ R * and has exactly two minimizers which 'escape in the third dimension' for R > R * .
The paper is organized as follows: In Section 2 we present some basic facts about the two types of symmetry induced by the O(2)-and Z 2 -group actions, and, in particular, about k-fold SO(2)-symmetric minimizers of the Landau-de Gennes energy.Section 3 contains the main part of the paper, namely, the proof of Theorem 1.5.The overall idea and main mathematical set-up of the proof are described in the Sections 3.1 -3.3.Sections 3.4 -3.7 contain formulations and proofs of the auxiliary results used in Sections 3.8 -3.9 to prove Theorem 1.5.In Section 4 we prove the existence of multiple critical points for large enough domains, namely Theorem 1.6.In Appendix A we provide a couple of remarks on the minimal energy and the symmetry properties of minimizers of F [•; B R ] for odd k.Finally, in the Appendices B, C, D we put some technical details required to prove our results.
2 Structure of symmetric maps.Proof of Proposition 1.2 We work with a moving (i.e., x-dependent) orthonormal basis of the space S 0 (defined in (1.2)), which is compatible with the boundary condition (1.7).We use polar coordinates in R 2 , i.e., x = (r cos ϕ, r sin ϕ) with r > 0 and ϕ ∈ [0, 2π).Let {e i } 3 i=1 be the standard basis of R 3 , and let We endow S 0 with the Frobenius scalar product of symmetric matrices Q • P = tr(Q P ) and the induced norm |Q| = (Q • Q) 1/2 .We define, for x ∈ R 2 , the following orthonormal basis of S 0 : Recall that I 3 is the 3 × 3 identity matrix and I 2 = I 3 − e 3 ⊗ e 3 .It should be noted that this choice of basis elements for S 0 differs slightly from [23,24] where both even and odd values of k are considered.This is due to the fact that E 3 and E 4 are continuous when we identify ϕ = 0 with ϕ = 2π if and only if k is even.We identify a map Q : B R → S 0 with a map w = (w 0 , . . ., w 4 ) : where we have used the following identities for even k The Landau-de Gennes energy (1.1) becomes The boundary condition (1.7) becomes In the introduction, we defined a group action of O(2) on H 1 (B R , S 0 ).There we viewed O(2) as a direct product of {0, 1} and SO(2).This naturally induces two group actions of {0, 1} ∼ = Z 2 and of SO(2), as subgroups of O(2), on H 1 (B R , S 0 ). (2.6) Here Q α,ψ is defined by (1.11).
Note that the groups Z 2 and {0, 1} are isomorphic, but we have deliberately distinguished the notations to avoid confusion with the Z 2 -action defined in the introduction.Moreover, the nature of the two group actions are somewhat different.The Z 2 -action is related to the reflection along e 3 direction of the target, while the {0, 1}-action is related to the reflection along the e 2 direction in both the domain and the target.
Remark 2.7.Let R ∈ (0, ∞], k be an odd integer and where w 0 ∈ H 1 ((0, R); r dr) and We also have the following characterization of Z 2 -symmetry: Proof.We know that any Therefore we obtain w 3 = w 4 = 0, Q(x) = 2 i=0 w i (x)E i and hence e 3 is an eigenvector of Q(x) for a.e.x ∈ B R .
Step 2. We prove (2 =⇒ 3).Assume now that e 3 is an eigenvector of Q(x) for a.e.x ∈ B R .Therefore there exists λ(x) such that Since e 3 , n and m form an orthonormal basis of R 3 , it is clear that w 3 (x) = w 4 (x) = 0 a.e.x ∈ B R .
For k ∈ 2Z\{0}, we next note a connection of SO(2)-symmetric and O(2)-symmetric minimizers: Under an O(2)-symmetric boundary condition, in particular (1.7), SO(2)-symmetric minimizers of F are in fact O(2)-symmetric.We do not know however if this remains true for all SO(2)-symmetric critical points.See [23, Proposition 1.3 and Remark 1.4] for a related statement that i=0 w i (r) E i as in Proposition 2.3 and let w = (w 0 , . . ., w 4 ).We will only consider the case where w 1 (R) ≥ 0 and w 3 (R) ≥ 0. (Note that the O(2)-symmetry of the boundary data implies that w 2 (R) = w 4 (R) = 0.) The other cases are treated similarly.
We recall that boundary conditions are k-fold O(2)-symmetric and therefore we have w 2 (R) = w 4 (R) = 0.
In Table 2, we summarize the characterization of various symmetries that we introduced for maps in H 1 (B R , S 0 ) (in particular, critical points or minimizers of F ) in terms of components w 0 , . . ., w 4 and k = 0 even.Table 2: Characterization of symmetries in the components w 0 , . . ., w 4 and k

-symmetry on large disks
In this section, we provide the proof of Theorem 1.5.Instead of working directly with the functional We work with a new parameter ε = 1 R and the following rescaled Landau-de Gennes energy functional defined on the set with Q b given by (1.8).Throughout the section k is an even non-zero integer.The Euler-Lagrange equation for F ε reads The statement on the uniqueness up to Z 2 -conjugation for minimizers of F [•; B R ] in Theorem 1.5 is equivalent to the following.Theorem 3.1.Let a 2 ≥ 0, b 2 , c 2 > 0 be any fixed constants and k ∈ 2Z \ {0}.There exists some

Towards the proof of Theorem 3.1
Using standard arguments it is straightforward to show that as ε → 0 the minimizers of F ε converge, along subsequences, in H 1 Q b (D, S 0 ) to the minimizers of the harmonic map problem where and S * defined in (1.5) is the set of global minimizers of f bulk (Q), see e.g.[5,29].Due to the explicit form of Q b and the fact that k is even, the minimizers of [3]), where n * minimizes the problem It is well known, see e.g.[9, Lemma A.2], that minimizers of (3.4) are conformal and have images in either the upper or lower hemisphere.The compositions of these minimizers with the stereographic projections of the upper and lower hemispheres of S 2 onto the unit disk are the complex maps z → z k/2 or z → zk/2 , respectively.Therefore F OF has exactly two minimizers in H 1 n (D, S 2 ) which are given by The corresponding minimizers of F * are We note that Q ± * are smooth and O(2)-symmetric but not Z 2 -symmetric and we can explicitly write Q ± * in terms of the basis tensors {E i } (see (2.2)) as where It is possible to show that from any sequence of minimizers can appear as limits of the sequences of minimizers of F ε .Now, restrict F ε to the set of k-fold O(2)-symmetric tensors Arguing as before but restricting F ε to A rs it is straightforward to show that any sequence of minimizers Q rs ε k of F ε k in A rs has a subsequence which converges in C 1,α ( D) and C j loc (D) for any j ≥ 2 to a minimizer of F * in the set of O(2)-symmetric tensors in H 1 Q b (D, S * ), which clearly must be either Q + * or Q − * as these are O(2)-symmetric.Based on the above we can construct two sequences of critical points of F ε converging to , not as yet guaranteeing k-fold O(2)-symmetry, and another consisting of minimizers of F ε A rs , which are k-fold O(2)-symmetric.Therefore, to prove Theorem 3.1 we need to show that if ε is small enough these sequences coincide, in particular, the minimizers of F ε are O(2)-symmetric.One possible approach is to employ the contraction mapping theorem or the implicit function theorem to show that there are "neighborhoods" N ± of Q ± * such that when ε is small enough F ε admits at most one critical point in each of N ± .
( †) In this approach typically the neighborhoods N ± are set up in relatively stronger norms than the energy-associated norm.In addition, the norm and thus the neighborhood are dependent on ε.A delicate point is the competition between the size of the neighborhood where one can prove uniqueness and the rate of convergence of minimizing sequences to the limit Q ± * (so that one can squeeze all minimizers into the designed neighborhood).
Below we present a roadmap to the proof of Theorem 3.1.Since Q ± * are equivalent up to a Z 2 -conjugation, it suffices to construct one such neighborhood, say N + of Q + * .For simplicity, we will in the sequel drop the superindex +, so that Q * = Q + * , n * = n + * , etc.In Subsection 3.2, we will provide a parameterization of suitable neighbourhoods N ± where we have a decomposition with ε 2 P being a "transversal component" of Q and Q ♯ being a (non-orthogonal) "projection" onto the limit manifold S * .In Subsection 3.3 we employ the above parameterization to obtain a new representation of the Euler-Lagrange equations (3.2) in terms of the variables ψ and P .In particular, we will derive a coupled system of equations for ψ and P with the following properties: 1.One equation is of the form where the operator L = −∆ − |∇n * | 2 is the linearized harmonic map operator at the minimizer n * of the problem (3.4).See (3.16) for the exact equation.

The other equation is of the form
with the linear operator for the exact form.
We will see that, although the nonlinear operators F and G are second ordered in the fields ψ and P , they are 'super-linear' and are 'small' when ψ and P are suitably 'small'.Subsection 3.4 is devoted to study the operator L and in Subsection 3.5 we concentrate on the operator L ε,⊥ .In Subsection 3.6, using previously derived properties of L , we revisit (3.8) and study the dependence of its solution ψ ε (with zero Dirichlet boundary condition) as a map of P .In order to balance the rate of convergence of the sequences of minimizers and the size of the neighborhoods N ± it will be convenient to measure the size of P with respect to an ε-dependent H 2 norm, specifically defined as: (3.10) We first show that, for ).Furthermore, when P is measured with respect to the norm • ε above, ψ ε is Lipschitz with respect to P with Lipschitz constant O(ε) (see Proposition 3.12).Using the Lipschitz estimate above, we show that the map This proves the uniqueness statement formulated informally above in relation ( †).See Proposition 3.14 in Subsection 3.7.In Subsection 3.8, using convergence results from [30] and the results presented above we prove Theorem 3.1.Finally, the proof of Theorem 1.5 is done in Subsection 3.9.

A parametrization in small H
) sufficiently close to a minimizer Q * of F * can be decomposed in a special way (see (3.11)) that takes into account the geometry of the limit manifold S * and the way it embeds into the space S 0 of Q-tensors.
Since S * is a smooth compact submanifold of S 0 , we can find a neighborhood N (S * ) of S * in S 0 , such that for every B ∈ N (S * ), there exists a unique B ort ∈ S * such that |B − B ort | = dist(B, S * ) where | • | stands for the norm associated to the Frobenius scalar product.Furthermore the projection B → B ort is a smooth map from N (S * ) onto S * .See Figure 1.
Although the above orthogonal projection suffices for many purposes, it is somewhat more convenient in our current setting to work with a different projection which is more adapted to Q * .Let n * = n + * be as in (3.5) and S * the tangent space to the limit manifold S * at Q * (x) and by (T Q * S * ) ⊥ its orthogonal complement in S 0 ≈ R 5 , which is normal to S * at Q * (x).It is known that (T Q * S * ) ⊥ consists of all matrices in S 0 commuting with Q * ; see [30,Eq. (3.2)].In particular, all matrices in (T Q * S * ) ⊥ admits n * as an eigenvector. 9e want to show that every Q in a "sufficiently small neighborhood" of Q * decomposes as ⊥ , which will be useful later.See Figure 1.We specify the result in the following lemma.).There exist γ > 0 and some large C 0 > 0 such that for every ε > 0 and every where Q ♯ and P satisfy • and Furthermore, there exists a unique ψ ∈ H Remark 3.3.In the above lemma, we have deliberately written the "transversal" component of Q as ε 2 P even though ε plays no role at the moment.In [30], it is shown that, in a similar setting, if Q ε is a minimizer for F ε , then its "transversal" contribution is of size ε 2 in some appropriate topology.In the setting of the present paper, we will show this holds in the L 2 (D, S 0 )-topology; see (3.44) below.This rate of convergence however does not hold in the H 2 (D, S 0 )-topology. 10This is related to the comment we made earlier on the fact that the sets N ± in ( †) are ε-dependent.
Remark 3.4.It should be noted that the map ψ appearing in the representation of Q ♯ belongs to a linear space (as ψ is orthogonal to n * ) as opposed to Q ♯ that belongs to a nonlinear set (as its values being constrained in S * ).
Proof.Since S * is a smooth submanifold of S 0 , there exists for every point B * ∈ S * a neighborhood U B * of B * in S 0 such that S * ∩ U B * is a graph over the tangent plane T B * S * .We then select local Cartesian-type coordinates {x 1 , . . ., x 5 } of S 0 ≈ R 5 such that B * corresponds to the origin, T B * S * coincides with {(x 1 , x 2 , 0, 0, 0) : for some open set U ⊂ R 2 and some smooth function u = (u 1 , u 2 , u 3 ) : U → R 3 with u(0) = 0 and ∇u(0) = 0. Define a projection P B * from U B * to S * by One can check that P B * (B) is well-defined (i.e.independent of local charts) and smooth as a function of two variables B ∈ S 0 and B * ∈ S * .Furthermore, P B * is the unique projection with the property As D is two dimensional, maps in H 2 (D, S 0 ) are continuous.Thus, there exists some large constant C 0 > 0 such that whenever there holds Q(x) ∈ U Q * (x) for all x ∈ D. The decomposition (3.12) is achieved by We now proceed to check the desired properties of Q ♯ and P .First, we have Using the smoothness of P in both variables, we obtain the claimed control of We turn to the second part of the lemma.Note that Q ♯ ∈ H 2 (D, S * ) is continuous.As D is simply connected and S * can be topologically identified with a projective plane, a standard result in topology about covering spaces implies that there is a unique continuous function v ∈ C 0 (D, S 2 ) such that Furthermore, by [3, Theorem 2], we have v ∈ W 1,p (D, S 2 ) for any p ≥ 2. Note that , and so, as |v| = 1, from which one can easily deduce that v ∈ H 2 (D, S 2 ).Next note that, as Observe that the above is equivalent to ψ • n * = 0 and v = n * +ψ |n * +ψ| , which gives the uniqueness of ψ.On the other hand, one has Recalling relations (3.15) we can represent . This concludes the proof.

The Euler-Lagrange equations
In this subsection we rewrite the Euler-Lagrange equations (3.2) for F ε in terms of the variables ψ and P introduced in Lemma 3.2.This new form of the Euler-Lagrange equations will be used in the subsequent analysis.
) be a critical point of F ε for some ε > 0, and n * and Q * be given by (3.5) and (3.6).Suppose that Q − Q * H 2 (D) is sufficiently small and let Q ♯ , P and ψ be as in Lemma 3.2.Then ψ and P satisfy the following equations where λ ε (x) is a Lagrange multiplier accounting for the constraint ψ In Lemma 3.5 above, we do not provide exact form of A, B ε , C ε nor indicate their explicit dependence on x as we show later (see the proof of Lemma 3.5) that these are lower order terms that do not play a role in our analysis.We will only use their properties summarized in the following proposition.
Proposition 3.6.Let ε ∈ (0, 1), n * and Q * be given by (3.5) and (3.6), and let A, B ε and C ε be the operators appearing in Lemma 3.5, defined in (B.9),(B.10),(B.11) in Appendix B. Then, for we have the following: with C denoting various constants independent of ε and the functions appearing in the inequalities.
The proofs of Lemma 3.5 and Proposition 3.6 are lengthy though elementary.We postpone them to Appendix B.

The linearized harmonic map problem
In this subsection we briefly study the properties of the operator L = −∆ − |∇n * | 2 appearing on the left hand side of (3.16), i.e. the linearized harmonic map operator at n * given by (3.5), as well as its inverse L −1 .Proposition 3.7.For every f ∈ L 2 (D, R 3 ), the minimization problem admits a minimizer which is the unique solution to the problem where λ is a Lagrange multiplier.11 Using Proposition 3.7 we can define the inverse operator L −1 .
) to be the unique solution to (3.25).
The proof of Proposition 3.7 is a standard argument using Lax-Milgram's theorem and the strict stability of n * .For completeness, we give the proof in Appendix C.
In the following lemma we prove some useful properties of L −1 required for our analysis.
Lemma 3.9.The range of the operator L −1 over L 2 -data is Furthermore, there exists some positive constant C such that, for f ∈ L 2 (D, R 3 ),

The transversal linearized problem
In this section we study the linear operator appearing on the left hand side of (3.17) (3.28) As in the previous subsection we would like to define the inverse operator L −1 ε,⊥ and prove some properties required for our analysis.
We claim that P → b 2 s Indeed, recall that n * is an eigenvector of P ∈ (T Q * S * ) ⊥ .Thus, in some orthonormal basis of R 3 , P takes the form diag(λ 1 , λ 2 , −λ 1 − λ 2 ) with P n * = λ 1 n * .It is not hard to see that this implies one can easily show that, for every q ∈ L 2 (D, S 0 ), there exists a unique solution where F (x) ∈ T Q * S * is a Lagrange multiplier accounting for the constraint P ∈ (T Q * S * ) ⊥ a.e. in D. Therefore we have the following definition.
We summarise properties of the operator L −1 ε,⊥ in the following lemma.
Proof.In the proof C will denote some generic constant which varies from line to line but is always independent of ε.
Let us fix some q ∈ L 2 (D, S 0 ) and let P be the solution of (3.30).Testing (3.30) against P and using (3.29), we obtain Next, we would like to show that P ∈ H 2 (D, S 0 ).Let Π = Π x be the orthogonal projection of S 0 onto T Q * (x) S * .Then, the first equation of (3.30) is equivalent to Here, we naturally extended Π to distributions, in particular to ∆P ∈ H To this end we use the following formula for Π which was computed in [30, Eq. (3.4)]:13 for all A ∈ S 0 .
Since P (x) ∈ (T Q * S * ) ⊥ in D, Π(P ) = 0 in D and so ∆Π(P ) = 0 in D. On the other hand, as Q * is smooth, it follows from the above formula for Π, applied to P and ∆P , that Combining the above two facts, we obtain (3.34) and hence (3.33).It follows that F ∈ L 2 (D, S 0 ) and P ∈ H 2 (D, S 0 ).Moreover, for 0 < ε < 1, Returning to the first equation in (3.30), elliptic estimates yield 3.6 Solution of (3.16) for given P In this section we solve equation (3.16) for given P .The properties of the map P → ψ ε (P ) obtained in this section will be used later in proving uniqueness of the critical point of F ε in a small neighbourhood of Q * using fixed point arguments.
Proof.Let us fix some C 1 > 0. In this proof C will denote some generic constant which may depend on C 1 , a 2 , b 2 , c 2 but is independent of ε (and C 2 and P which will appear below).For P ∈ Y , define an operator K ε,P : X → X by where L −1 is given in Definition 3.8, and A and B ε are the operators appearing on the right hand side of (3.16).
Proof of (i): It suffices to show that, for sufficiently large C 2 and all Observe that, in view of (3.37) and Poincaré's inequality in H 1 0 (D, S 0 ), one has for all sufficiently large C 2 that < 1 for all P ∈ U .
Estimates (3.20) and (3.22) imply, for ψ, ψ ∈ O and P ∈ U , Therefore, by Lemma 3.9, we have for ψ, ψ ∈ O and P ∈ U that Also, by (3.19), (3.21) and Lemma 3.9, for all P ∈ U . (3.38) From the above two estimates, we deduce that there exist a large constant C 2 > 1 and a small constant ε 0 > 0 such that, for every ε ∈ (0, ε 0 ) and for every P ∈ U , K ε,P is a contraction from O into O and so has a unique fixed point ψ ε (P ) ∈ O.
Proof of (ii) and (iii): We now fix C 2 so that ψ ε is defined on U as above and K ε,P is a contraction from O into itself.It follows from (3.38) and Lemma 3.13 (see below) that the unique fixed point which proves (ii).Next, Lemma 3.9 and estimate (3.23) imply that Taking ψ = ψ ε (P ) and using (ii), we find that ≤ Cε P − P ε for all P, P ∈ U .
This proves (iii) and completes the proof.
We used the following simple lemma whose proof is omitted.

Uniqueness of critical points in a neighborhood of Q *
In this subsection we show the uniqueness of critical points of F ε in a small neighbourhood of Q * ∈ {Q ± * } given in (3.6).In particular, we prove the following version of the informal statement ( †) formulated in Subsection 3.1: Proposition 3.14.For every C 1 > 0, there exist large C 2 > 1 and small ε 0 > 0 such that, for all 0 < ε ≤ ε 0 , F ε has at most one critical point Q ε , represented by (ψ ε , P ε ) as in Lemma 3.5, with In the proof of Proposition 3.14, the exact form of n * is used only to have the tubular neighborhood representation (Lemma 3.2) and the stability inequality (Lemma C.2). Therefore, provided these are true, the statement of Proposition 3.14 will hold for more general domains and boundary conditions.
Proof.Let X and Y be defined by (3.26) and (3.31).Let L −1 ε,⊥ be as in Definition 3.10 and By Lemma 3.11, as n * is smooth, we have for every ε ∈ (0, 1): for some constant C 0 independent of ε, and where • ε is as defined in (3.10).Fix some C 1 > 0 and let ε 0 ∈ (0, 1) and C 2 be as in Proposition 3.12.By shrinking ε 0 if necessary, we have for 0 < ε ≤ ε 0 that the solution ψ ε (P ) to (3.16) is defined for all given P ∈ U := U ε,C 1 ,C 2 (see (3.36)) and where we have used (3.37).Here and below, C denotes some constant which may depend on C 1 , C 2 , a 2 , b 2 , c 2 but is always independent of ε.For P ∈ U we define where Cε [ψ, P ] = C ε [ψ, P ]− 1 3 tr(C ε [ψ, P ])I 3 and C ε is the operator appearing on the right hand side of (3.17).It should be clear that if P is a fixed point of K ε,⊥ , then (ψ ε (P ), P ) solves (3.16)- (3.18), and so the map Q ε corresponding to (ψ ε (P ), P ) in the representation Lemma 3.2 is a critical point of F ε .Therefore, to reach the conclusion, it suffices to show that for all sufficiently small ε, the map K ε,⊥ has at most one fixed point in U .In fact, we show that, for all small ε, K ε,⊥ is contractive on U with respect to the norm • ε .
In the following, we will use Ladyzhenskaya's inequality in two dimensions: In particular, it holds that Using the estimate (3.24) and inequality (3.40), we have + Cε for all P, P ∈ U and ψ, ψ ∈ X with ψ H 2 (D) , ψ H 2 (D) ≤ 1.Thus, by Proposition 3.12(ii) and (iii), we get In view of Lemma 3.11, it follows that This implies that, for all sufficiently small ε, K ε,⊥ has at most one fixed point in U , which concludes the proof.
3.8 Proof of Theorem 3.1 Let n ± * be given by (3.5) and It is well known that (see e.g.[5,13]), if ε m → 0 and Q εm ∈ C εm , then Q εm converges along a subsequence in H 1 (D, S 0 ) to either , it holds for all small ε > 0 that It should be clear that C ± ε = JC ∓ ε J.To conclude, it is enough to show that, for all sufficiently small ε, C + ε consists of a single map which is O(2)-symmetric.
Step 1.We prove that sup In fact, it suffices to show that sup D) for any σ ∈ (0, 1) and in C 2 loc (D) (note that in the cited paper the results are in 3D domains but one can easily check that those convergences also hold in 2D domains).Furthermore, by [30,Corollary 2], ∆Q εm is bounded in L ∞ (D).By Lebesgue's dominated convergence theorem and so ∆Q εm converges to ∆Q * in L 2 (D, S 0 ).By elliptic estimates, we conclude that Q εm converges to Q * in H 2 (D, S 0 ), which gives a contradiction.We have thus established (3.42).
In view of (3.42) and Lemma 3.2, for all sufficiently small ε and Q ε ∈ C + ε , we can represent where ψ ε • n * = 0 and P ε ∈ (T Q * S * ) ⊥ .We let C + ε denote the set of (ψ, P ) representing elements of C + ε as above: By (3.42) and Lemma 3.2, sup Step 2. In view of (3.43) and Proposition 3.14, in order to prove that C + ε consists of a single point for all sufficiently small ε, it suffices to show that there exist ε 1 > 0 and C 1 > 0 such that, for all ε ∈ (0, ε 1 ), sup We recall some results from [30].Let Q ε ∈ C + ε and (ψ ε , P ε ) ∈ C + ε be its corresponding representation as above.We consider the tensor: (The polynomial on the right hand side is a multiple of the minimal polynomial of matrices belonging to the limit manifold S * .)By [30, Proposition 4], Let End(S 0 ) be the set of linear endomorphisms of S 0 and define µ ε : D → End(S 0 ) by for all x ∈ D and M ∈ S 0 .Then (3.45) is equivalent to where µ ε P ε stands for the map x → µ ε (x)(P ε (x)).By definition, P ε ∈ (T Q * S * ) ⊥ and so by [30, Lemma 2], P ε commutes with Q * .It follows that Note that (n * ⊗ n * − 1 2 I 3 ) is invertible when considered as an endomorphism of S 0 and that lim ε→0 µ ε C 0 ( D) = 0 (in view of (3.43)).Thus, as X ε is bounded in C 0 ( D), we have that P ε is also bounded in C 0 ( D), and in particular in L 2 (D).The assertion in (3.44) is established.By Proposition 3.14, we hence have for all sufficiently small ε that C + ε consists of a single element and C ε consists of exactly two distinct Z 2 -conjugate elements.
Step 3: Recall that we denoted in (3.7) by A rs the set of O(2)-symmetric maps in H 1 (D, S 0 ).Let C rs ε denote the set of minimizers of F ε | A rs .The same argument as above shows that, for all small ε, This completes the proof.

Proof of Theorem 1.5
Proof.Using Theorem 3.1 and a simple scaling argument, we find R 0 = R 0 (a 2 , b 2 , c 2 , k) > 0 such that for all R > R 0 , there exist exactly two global minimizers Q ± of F [•; B R ] subjected to the boundary condition (1.7) and these minimizers are k-fold O(2)-symmetric and are Z 2 -conjugate to each other.By Proposition 1.2, we can express Q ± in the form It is clear that (w 0 , w 1 , 0, ±w 3 , 0) satisfies (2.8)-(2.12).Now, note that in view of formula (2.4), (1.7).By the above uniqueness up to Z 2 -conjugation, we may assume that w 3 ≥ 0 in B R .Also, as Q + = Q − , w 3 ≡ 0. Recalling equation (2.11) and noting that w 2 = w 4 = 0, we can apply the strong maximum principle to conclude that w 3 > 0. The proof is complete.

Mountain pass critical points
In this section, we give the proof of Theorem 1.6, which asserts the existence of at least five O(2)symmetric critical points satisfying the boundary condition (1.7) for F R := F [•; B R ] for all large enough R.
We denote by A rs R and A str R the sets of k-fold O(2)-symmetric and Z 2 × O(2)-symmetric maps, respectively, satisfying the boundary conditions (1.7): By the characterization of symmetric maps (see Propositions 1.2 and 2.9), we can express the sets A rs R and A str R in terms of the basis components defined in Section 2 as follows: ).To prove Theorem 1.6, we use the fact that F R has two global minimizers in A rs R (due to Theorem 1.5) and the mountain pass theorem.An energetic consideration is needed to show that the obtained mountain pass critical point does not coincide with critical points of F R in A str R .We start with an estimate for the minimal energy of F R in A str R .Lemma 4.1.There exists some C > 0 depending only on a 2 , b 2 and c 2 such that, for all δ ∈ (0, 1), k ∈ Z \ {0} and R > max(1, Cek 2 δ 2 ), there holds As a consequence, Remark 4.2.In [4], it was shown that F R has critical points whose energies are of order k ln R; and these are not SO(2)-symmetric for k = ±1.
Proof.By (2.4), we have and f * is given by (1.3).We note (see e.g.[24,Lemma 5.1]) that h(x, y) ≥ 0 and equality holds if and only if (x, y) belongs to the set {(± s + √ 2 , − s + √ 6 ), (0, 2s + √ 6 )}.Furthermore, the Hessian of h is positive definite at these critical points.In particular, one has ) 2 for all (x, y) satisfying where, here and below, C denotes some positive constant (that may change from line to line) which depends only on a 2 , b 2 and c 2 , and in particular is always independent of R, k and δ.
Step 2: Proof of the lower bound for α R in (4.1).
Let (w 0 , w 1 ) be a minimizer of E R the existence of which is guaranteed by the direct method of the calculus of variations.We fix some δ ∈ (0, 1).Due to the fact that w 1 is continuous, w 1 (0) = 0 and w 1 (R) = s + √ 2 , there exists the largest number R 1 ∈ (0, R) such that w 1 (R 1 ) = s + (1+2δ) √ 2 .By the same arguments, there exists the smallest number On the other hand, by the definition of R 1 and R 2 , we have . Also, by [23,Eq. (3.12)], w 2 0 + w 2 1 ≤ 2 3 s 2 + .Thus, using (4.3), we have Therefore, it follows that and hence, in view of (4.4), This leads to, by Cauchy-Schwarz' inequality, .

Rearranging, we obtain Λe
The conclusion of the result is immediate.
In order to use the mountain pass theorem to show that F R has more than two critical points, we need to exhibit a path γ connecting the two minimizers where α R is the minimal energy of F R | A str R .The existence of such a path is a priori not clear.Indeed, note that, as R → ∞ and after a suitable rescaling, Q ± R tend to Q ± * (see equation (3.6) in the previous section).As maps from D into S * , Q + * and Q − * belong to different homotopy classes and so cannot be connected by a continuous path in H 1 (D, S * ).The desired path γ must therefore necessarily leave the limit manifold S * .In particular, the contribution of the bulk energy potential f bulk to F R [γ(t)] cannot be neglected.
Our construction of the path γ is of a completely different flavor.We exploit the conformal invariance of the Dirichlet energy in 2D to connect Q ± R to Q ± R 0 for some fixed R 0 by using Q ± r (with variable r) and their inverted copies, and then finally connect As a result we obtain a mountain path with energy O k (1) (see (4.5)), which is clearly less than α R for large R.
Proof of Theorem 1.6.In the proof, C denotes some positive constant which is always independent of R. As denoted earlier, critical points of By Theorem 1.5, there exists R 0 > 0 such that, for R ≥ R 0 , F R has two distinct minimizers in A rs R which are O(2)-symmetric but not Z 2 × O(2)-symmetric (in fact, they are Z 2 -conjugate).We label these minimizers as Q ± R and claim that, for any 0 Assume by contradiction that there exists a sequence Without loss of generality, we can also assume that Q m is weakly convergent in H 1 (B R , S 0 ) and strongly convergent in L p (B R , S 0 ) for any p ∈ [1, ∞).The limit of Q m is then a minimizer of F R , and thus, by Theorem 1.5 and our assumption on d, must coincide with The claim is proved.It is standard to check that F R satisfies the Palais-Smale condition.Indeed, if Q m is a Palais-Smale sequence for F R , then as which, by the compact embedding theorem, maps bounded sets of H 1 (B R , S 0 ) into relatively compact sets of H −1 (B R , S 0 ).Thus, up to extracting a subsequence, we may assume that Applying the mountain pass theorem (see e.g.[32, Theorem 6.1]), we conclude for R ≥ R 0 that F R has a mountain pass critical point in A rs R connecting Q ± R , which will be denoted by Q mp R .Claim: There exists some C > 0 independent of k such that To this end, it suffices to construct a continuous path γ : where C is independent of R, k and t.
Proof of Claim.Let n ± * be defined by (3.5).Its rescaled version to B R is given by We define Step 1.We first construct γ [−2,−1]∪ [1,2] .For that, let r 1 , r 2 : [−2, 2] → [R 0 , R] be given by To dispel confusion, we note that on the lower two cases (i.e., r 1 (t) < |x| < R), we are using the "plus" minimizing branch Q + R .Since for any r > 0 we have , the inner and outer traces of γ(t) at ∂B r 1 (t) coincide and so γ(t) belongs to A rs R .See Figure 2. The continuity of γ with respect to t is a consequence of the uniqueness part of Theorem 1.5.
Let us check that (4.6) holds for 1 ≤ |t| ≤ 2. In view of (4.7) and the fact that the integrand of F is non-negative, we have Therefore, we only need to show that Indeed, by a change of variable y = r 2 (t) 2 |x| 2 x , we have in view of (4.7): Step 2. We continue the argument by letting γ (−1,1) be the linear interpolation between γ(±1), A standard argument using the maximum principle (see the proof of [29, Proposition 3]) shows that This together with the convexity of the Dirichlet energy, the non-negativity of the integrand of F , and (4.7) gives Recalling (4.6) for t = 1, we conclude the proof of the claim.
Let us prove now that Q mp R / ∈ A rs R for sufficiently large R. Indeed, we take R 1 > max(R 0 , 4C 1 ek 2 ) such that for all R > R 1 and k ∈ 2Z \ {0}, we have where C 1 is the constant from (4.1) corresponding to δ = 1/2, and the constants C 2 and R 0 are the ones from (4.5).Then the mountain pass critical point Q mp R (which belongs to A rs R ) has the energy F R (Q mp R ) bounded from above by C 2 |k| and thus does not belong to A str R thanks to Lemma 4.1.In other words, We have thus shown that F R has at least five k-fold O(2)-symmetric critical points in A R , at least four of which are not k-fold Z 2 × O(2)-symmetric.
Remark A.2.For the unit disk Ω = D, the following can be stated in the case the boundary data Q b is given by (1.8)-(1.9).We have seen that when k is even the minimizers of When k is odd the symmetry of minimizers is more delicate and so far is unknown.For k = ±1, we conjecture that there exists a unique minimizer and this minimizer is Z 2 × O(2)symmetric -see [24,25] for some supporting evidence.In the case of odd |k| > 1 the above energy bounds forbid minimizers to have Z 2 × SO(2)-symmetry (in view of Lemma 4.1) as well as the configuration of k-vortices of degree ± 1 2 constructed in [4].We would like now to briefly outline how one can obtain energy bounds (A.2) and (A.3).When D is a disk and k is odd, these bounds were established by Canevari [10,11].We will see below that (A.2) and the upper bound in (A.3) can be established by elementary arguments using some extension property for S planar * -valued maps of degree zero.The proof of the lower bound in (A.3) is more substantial and draws on the corresponding aforementioned estimate for disks [11].
Let us start with (A.2) when k is even.The lower bound C 1 > 0 can be taken to be the minimal Dirichlet energy under the given (non-constant) boundary data Q b (since f bulk ≥ 0).For the upper bound, we construct a test function by splitting the domain Ω into a disk B R ⊂ Ω, which without loss of generality is assumed to be centered at the origin, of some small radius R, and the complement Ω \ B R .On the boundary of the disk, we impose the test function to have boundary data of the form (1.8)-(1.9).We define Q test by joining together minimizers of F ε [•; B R ] and F ε [•; Ω \ B R ] with respect to the indicated boundary data on each subdomain.It is clear that the minimal energy satisfies the following bound min where the two minimizations on the right hand side are under the constraint that Q = Q test on the respected boundary.We know (see Table 1) that the first term on the right hand side of (A.4) is bounded uniformly in ε.Since the degree of the map is zero we can use an H 1 extension -see Lemma A.3 below -to show that the second term on the right hand side of (A.4) is bounded uniformly in ε.Estimate (A.2) follows.
Proof.As S planar * is diffeomorphic to a circle, this result is well known (see [6,Section I.2]).See also [19, p.126] for results on continuous extensions in any dimension.
We now consider the upper bound in (A.3) when k is odd.We select two disjoint disks B R (x 1 ) and B R (x 2 ) inside Ω.On the boundary of these disks, we consider the test function Q test : where we set for Re iϕ = (R cos ϕ, R sin ϕ) ∈ ∂B R (0), 0 ≤ ϕ < 2π: In Ω\(B R (x 1 )∪B R (x 2 )) and B R (x 2 ), the test function Q test is constructed by minimizing F ε under the indicated boundary conditions.In Using Lemma 4.1 (namely the upper bound in (4.1)) and arguing as in the previous case, we arrive at the upper bound in (A.3).We turn to the lower bound in (A.3) when k is odd.Take some large disk B R ′ (0) ⊃ Ω.We impose on ∂B R ′ (0) a boundary condition of the form (A.5) where By [11,Proposition 15], we have within the above boundary condition (B.C.) on ∂B R ′ (0): min and by (A.2), within the above boundary condition The lower bound in (A.3) follows from the above two estimates and the inequality min B The Euler-Lagrange equations near Q * In this appendix, we give the proof of Lemma 3. The expression ∇(n * ⊗ ψ + ψ ⊗ n * ) : ∇P on the right hand side of (B.3) contains some terms which are quadratic in the derivatives of P and ψ.However, we can eliminate this quadratic character by using some specific geometric information as follows: we note that P ∈ (T Q * S * ) ⊥ and n * ⊗ ψ + ψ ⊗ n * ∈ T Q * S * .Indeed, by [ It is readily seen that the integral on the right hand side is linear in the derivatives of P and ψ.This cancellation will play a role on our later analysis: its contribution to the Euler-Lagrange equations of F ε is of first order rather than second order.
We now put together all the previous expressions, to get a new form of the full energy.Using the expression of |∇v| is the gradient of h with respect to P ∈ S 0 ,14 λ ε is a Lagrange multiplier accounting for the constraint ψ • n * = 0, F ε (x) ∈ T Q * S * is a Lagrange multiplier accounting for the constraint P (x) ∈ (T Q * S * ) ⊥ , and This finishes the proof of Lemma 3.5.
We have |C (2) [ψ] − C (2)  As for C In particular, n * is a stable harmonic map.Equality holds if and only if ζ = t(1−r k ) 1+r k for some t ∈ R.
is a Lagrange multiplier accounting for the constraint P ∈ (T Q * S * ) ⊥ , and maps A, B ε , C ε are defined in equations (B.9), (B.10) and (B.11) in Appendix B.

1 2 P
− P ε for all P, P ∈ U .
0 ).Therefore it suffices to work with F R A rs R .To simplify the notation in what follows we still use F R instead of F R A rs R .

Table 1 :
Properties of critical points for even k = 0 and large radius R .15) Taking C 0 large enough in (3.14) and using equality (3.15), we obtain (v • n * ) 2 ≥ 1 4 .Since both v and n * are continuous and coincide at the boundary we deduce v • n * ≥ 1 2 .Therefore we can define The inequality(3.29)follows in view of the identity −a 2 − b 2 3 s + + 2c 2 0 ) : P ∈ (T Q * S * ) ⊥ a.e. in D}, is necessarily of mountain pass type).