Energies of <span class="mathjax-formula">${S}^{2}$</span>-valued harmonic maps on polyhedra with tangent boundary conditions

Energies of S ² -valued harmonic maps on polyhedra with tangent boundary conditions

A. Majumdar

, J.M. Robbins

, M. Zyskin

aSchool of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK bHewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK

Received 17 May 2006; accepted 13 November 2006 Available online 28 December 2006

Abstract

A unit-vector fieldn:P →S²on a convex polyhedronP ⊂R³satisfies tangent boundary conditions if, on each face ofP, ntakes values tangent to that face. Tangent unit-vector fields are necessarily discontinuous at the vertices ofP. We consider fields which are continuous elsewhere. We derive a lower boundE⁻_P(h)for the infimum Dirichlet energyE^inf_P (h)for such tangent unit- vector fields of arbitrary homotopy typeh.E_P⁻(h)is expressed as a weighted sum of minimal connections, one for each sector of a natural partition ofS²induced byP. ForP a rectangular prism, we derive an upper bound forE^inf_P (h)whose ratio to the lower bound may be bounded independently ofh. The problem is motivated by models of nematic liquid crystals in polyhedral geometries. Our results improve and extend several previous results.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:35A15; 35A20; 35J50; 35J65; 55P15; 58E20; 82D30

Keywords:Harmonic maps with defects; Tangent boundary conditions; Minimal connection; Liquid crystals; Bistability

1. Introduction

S²-valued harmonic maps on three-dimensional domains with holes were studied in a well known paper by Brezis, Coron and Lieb [4]. As a simple representative example, consider the domainΩ=R³− {r¹, . . . ,rⁿ}(for which the holes are points), and letn:Ω→S²denote a unit-vector field onΩ. For∇nsquare-integrable, we define the Dirichlet energy ofnto be

E(n)=

Ω

(∇n)²dV . (1.1)

* Corresponding author.

E-mail addresses:a.majumdar@bristol.ac.uk (A. Majumdar), j.robbins@bristol.ac.uk (J.M. Robbins), m.zyskin@bristol.ac.uk (M. Zyskin).

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.11.003

Continuous unit-vector fields onΩ may be classified up to homotopy by their degrees,d =(d¹, . . . , dⁿ)∈Zⁿ, on spheres about each of the excluded points (the restriction ofnto such a sphere may be regarded as a map fromS²into itself). In order thatnhave finite energy, we must have that

j

d^j=0. (1.2)

LetCΩ(d)denote the homotopy class of continuous unit-vector fields with degreesdsatisfying (1.2), and letHC_Ω(d) denote the elements ofCΩ(d)with finite Dirichlet energy. LetE^inf_Ω(d)denote the infimum of the energy overHC_Ω(d),

E_Ω^inf(d)= inf

n∈HCΩ(d)E(n). (1.3)

It turns out thatE_Ω^inf(d)is just 8π times the length of aminimal connectiononΩ. We recall the definition of a minimal connection. Given twom-tuples of points inR³,P=(a¹, . . . ,a^m)andN=(b¹, . . . ,b^m)(whose points need not be distinct), a connection is a pairing(a^j,b^{π(j )})of points inP andN, specified here in terms of a permutation π∈S_m(Smdenotes the symmetric group). The length of a connection is the sum of the distances between the paired points, and a minimal connection is a connection with minimum length. Let

L(P,N)= min

π∈Sm

m

j=1

a^j−b^{π(j )} (1.4)

denote the length of a minimal connection betweenPandN. Let|d| =¹₂

j|d^j|. Then

Theorem 1.1.[4]The infimumE_Ω^inf(d)of the Dirichlet energy of continuous unit-vector fields on the domainΩ= R³− {r¹, . . . ,rⁿ}of given degreesd^j about the excluded pointsr^j is given by

E_Ω^inf(d)=8π L

P(d),N(d)

, (1.5)

whereP(d)is the|d|-tuple of excluded points of positive degree, withr^j includedd^jtimes, andN (d)is the|d|-tuple of excluded points of negative degree, withr^k included|d^k|times.

In this paper we consider a natural variant of this problem which emerges from a boundary-value problem of some physical and technological interest; the domain is taken to be a polyhedron on whichnis required to satisfytangent boundary conditions. LetP denote a convex bounded polyhedron inR³, including the interior of the polyhedron but excluding its vertices. Letn:P →S² denote a unit-vector field onP. We say thatnsatisfies tangent boundary conditions, or is tangent, if, on each face ofP,ntakes values tangent to that face. (It is clear that this condition could not be satisfied at the vertices of the polyhedron, which belong to three or more faces.)

One motivation for the problem comes from liquid crystals applications, in which n describes the mean local orientation of a nematic liquid crystal, and the Dirichlet energy (1.6) coincides with the elastic or Frank–Oseen en- ergy in the so-called one-constant approximation (see, e.g., [6,26,14,25]). Polyhedral cells have been proposed as a mechanism for engendering bistability – they may support two nematic configurations with distinct optical properties, both of which are local minima of the elastic energy [11,23,12,7,5,21]. In many cases of interest the orientation at interfaces is well described by tangent boundary conditions, and low-energy local minimisers appear to have different topologies. We also remark that harmonic maps between Riemannian polyhedra have been studied by Gromov and Shoen [9] and Eells and Flugede [8], in particular in cases where the target manifold has nonpositive curvature.

Here we will restrict our attention to continuous tangent unit-vector fields onP. (Let us note that, while nematic orientation is, in general, described by a director, orRP²-valued field, a continuous director field on a simply con- nected domain such asP can be lifted to a continuous unit-vector field.) Continuous tangent unit-vector fields onP can be partitioned into homotopy classesCP(h)labelled by a complete set of homotopy invariants denoted collectively byh. A full account of this classification is given in [24] (see also [28]). Below we reprise the results we need here.

General discussions of topological defects in liquid crystals are given in [22,13,14].

For∇nsquare integrable onP, let E(n)=

P

(∇n)²dV (1.6)

denote its Dirichlet energy. LetHCP(h)denote the elements ofCP(h)with finite Dirichlet energy, and let E_P^inf(h)= inf

n∈HCP(h)E(n) (1.7)

denote the infimum of the energy overHC_P(h). Our first result (Theorem 1.2 below) is a lower bound forE_P^inf(h).

This is expressed in terms of certain homotopy invariants called wrapping numbers, which we now define. Let f denote the number of faces ofP,F^c thecth face ofP, andF^c the outward normal onF^c, where 1cf. For each face we consider the great circle onS²containing the unit-vectorsstangent to it, i.e.F^c·s=0. Thesef (not necessarily distinct) great circles partitionS² into open spherical polygons, which we callsectors. The sectors are characterised by sgn(F^c·s). We write

S^σ =

s∈S²|sgn s·F^c

=σ^c , (1.8)

whereσ=(σ¹, . . . , σ^f)is anf-tuple of signs. It should be noted that most of theS^σ’s are empty; indeed, reckoning based on Euler’s formula for polygonal partitions of the sphere (|vertices| − |edges| + |faces| =2) shows that there are at most (and generically)f²−f+2 nonempty sectors.

Next, letC^adenote a smooth surface inP which separates the vertexv^afrom the others. For definiteness, takeC^a to be oriented so thatv^alies on the positive side ofC^a. We callC^aacleaved surface. Givenn∈CP(h), letn^adenote its restriction toC^a. The wrapping numberw^aσ is the number of timesn^acoversS^σ, counted with orientation. Forn differentiable, this is given by

w^aσ= 1 A^σ

C^a

n^∗ χ^σω

, (1.9)

whereωis the area two-form onS², normalised to have integral 4π,χ^σ is the characteristic function ofS^σ ⊂S², and A^σ = |

S²χ^σω|is the area ofS^σ. Alternatively,w^aσ can be expressed as the index of a regular values∈S^σ ofn^a, i.e.

w^aσ=

r|n^a(r)=s

sgn det dn^a(r). (1.10)

One can show that w^aσ does not depend on the choice of s∈S^σ nor on the choice of cleaving surface C^a, that the definition (1.9) can be extended to continuous n, and that its value depends only on the homotopy class ofn [24]. In fact, the wrapping numbers constitute a complete set of invariants, as is shown in Appendix A. They are not independent, however. For example, continuity in the interior ofP (absence of singularities) implies that, for allσ,

a

w^aσ=0, (1.11)

where the sum is taken over verticesv^a. Continuity on the faces and edges ofP implies additional constraints. We will say thath= {w^aσ}is anadmissible topologyif it can be realised by some continuous configurationn:P →S². Theorem 1.2.Leth= {w^aσ}be an admissible topology for continuous tangent unit-vector fields on a polyhedronP. Then

E_P^inf(h)E_P⁻(h):=

σ

2A^σL

P^σ(h),N^σ(h)

, (1.12)

whereP^σ (resp.N^σ)contains the vertices ofP for whichw^aσ is positive(resp. negative), each such vertex included with multiplicity|w^aσ|.

Thus, to each sectorσ may be associated a constellation of point defects at the verticesv^a of degreesw^aσ. The lower boundE_P⁻(h)is a sum of the lengths of minimal connections for these constellations weighted by the areas of the sectors.

Theorem 1.2 is proved using arguments similar to those used to show thatE_Ω^inf(d)8π L(P(d),N(d))in the proof of Theorem 1.1. In Theorem 1.1, one obtains an equality forE_Ω^inf(d), rather than just a lower bound, by constructing a sequencen^{(j )}whose energies approach 8π L(P(d),N(d)). It can be shown that a subsequencen^(k)approaches a

constant away from a minimal connection while|∇n^(k)|²approaches a singular measure supported on the minimal connection [4]. In the present case, tangent boundary conditions preclude such a construction;nis required to vary across the faces and, therefore, throughout the interior ofP. However, forP a rectangular prism, we can show that E⁻_P(h)correctly describes the dependence of the infimum energy on homotopy type.

Theorem 1.3.Let P denote a rectangular prism with sides of lengthL_xL_yL_z and largest aspect ratioκ= L_x/L_z. Then

E_P^inf(h)Cκ³E_P⁻(h) (1.13)

for some constantCindependent ofhandL_x,L_y,L_z.

The upper bound of Theorem 1.3 is obtained by estimating the energy of explicitly constructed tangent unit-vector fields which satisfy the Euler–Lagrange equations near each vertex.

The general form of the Frank–Oseen energy is given by [6,26,14,25]

E_FO(n)=

P

K₁(divn)²+K₂(n·curl n)²+K₃(n×curl n)²+K₄div

(n·∇)n−(divn)n

dV , (1.14) where the elastic constantsK_j are material-dependent. It is easily shown that tangent boundary conditions imply that the contribution from theK₄-term in (1.14) vanishes. The elastic constantsK₁,K₂ andK₃ are constrained to be nonnegative, and the one-constant approximation (1.6) follows from takingK₁=K₂=K₃=1. We remark that Theorems 1.2 and 1.3 imply the following bounds for the Frank–Oseen energy:

K₋E_P⁻(h) inf

n∈HC_P(h)EFO(n)CK₊κ³E_P⁻(h), (1.15)

whereK₋(resp.K₊) is the smallest (resp. largest) of the elastic constantsK1,K2andK3.

Theorems 1.2 and 1.3 improve and extend several earlier results. In [19] we obtained a lower bound forE_P^inf(h)of the form

2 max

ξ^a

a

ξ^a

σ

A^σw^aσ

, (1.16)

where the maximum is taken overξ^a’s such that |ξ^a−ξ^b||v^a−v^b|. The quantity (1.16) is generally less than the lower bound given by Theorem 1.2, in particular because it allows for cancellations between wrapping numbers of opposite sign. For example, for a regular tetrahedron with sides of unit length, (1.16) gives a lower bound of

aσA^σw^aσ, whereas Theorem 1.2 gives the lower bound

aσA^σ|w^aσ|. For a rectangular prism, Theorem 1.3 does not hold if (1.16) is substituted forE_P⁻(h). ((1.16) can be directly compared to (2.15) below, which gives an equivalent (dual) expression forE_P⁻(h).) A restricted example of Theorem 1.2 was given in [20] for the caseh is areflection- symmetric topology. These are the topologies of configurations which are invariant under reflections through the midplanes of the prism.

Results related to Theorem 1.3 were obtained for the special case of reflection-symmetric topologies in [18,20].

The constructions and estimates are simpler in this case, and one can show that the ratio of the upper and lower bounds scales linearly with the aspect ratioκ, rather than asκ³. Indeed, for conformal and anticonformal reflection-symmetric topologies (for which the wrapping numbersw^aσ about a given vertex have the same sign), one can show that the ratio is bounded by(L²_x+L²_y+L²_z)^1/2/L_z. It is not clear that for general prism topologies theκ³ dependence in Theorem 1.3 is optimal.

An important question is whether within a given homotopy class the infimum Dirichlet energy is achieved. The homotopy classesHCP(h)are not weakly closed with respect to the Sobolev norm, so it is not automatically the case that the infimum is achieved. However, while a givenHC_P(h)may not be weakly closed, it may still contain a local (smooth) minimiser of the Dirichlet energy. Indeed, there is some numerical evidence and heuristics to suggest that, in the case of a rectangular prism, for the simplest topologies a smooth minimiser always exists, while for others a smooth local minimiser may or may not exist depending on the aspect ratios [18,17]. It would be interesting to establish for which topologies there exist smooth local minimisers, also from the point of view of device applications.

Such configurations would of course satisfy the bounds established here.

There is an extensive literature on S²-valued harmonic maps with fixed (Dirichlet) boundary data; reviews are given in [10,3]. Problems related to the one considered here concern liquid crystal droplets [15,26], in which one seeks configurationsnon a three-dimensional regionΩ which minimise the elastic energy [16]. In case of tangent boundary conditions, there are necessarily singularities on the surface of Ω, e.g. ‘boojums’ [27]; in a polyhedral domain, these singularities are pinned at the vertices.

The remainder of the paper is organised as follows. Theorem 1.2 is proved in Section 2, and Theorem 1.3 in Section 3 modulo two lemmas concerning the explicit construction of and estimates for the representative prism configurations (Sections 4 and 5). In Appendix A it is shown that homotopy classes of continuous tangent unit-vector fields onP are classified by wrapping numbers.

2. Lower bound for general polyhedra

Proof of Theorem 1.2. In [19] we show that smooth n’s are dense in HC_P(h) with respect to the Sobolev W^(1,2)-norm. Therefore, it suffices to establish the lower bound (1.12) fornsmooth.

LetB (v^a)denote the -ball aboutv^a, and let P =P−

a

B v^a

∩P

(2.1) denote the domain obtained by excising these balls fromP. Clearly

E(n)=

P

(∇n)²dV

P

(∇n)²dV . (2.2)

Letχ^σ denote the characteristic function of the sectorS^σ⊂S². It will be useful to introduce smooth approximations

˜

χ^σ toχ^σ, such thatχ˜^σ has support inS^σ and satisfies 0χ˜^σ χ^σ. Then

E(n)

σ

P

χ˜^σ◦n

(∇n)²dV . (2.3)

Using the inequality [4]

(∇n)² 2 n^∗ω , (2.4)

wheren^∗ωdenotes the pullback ofωbynand · denotes the norm on forms induced by the standard metrics onR³ andS², we get that

E(n)2

σ

P

n^∗

˜

χ^σωdV . (2.5)

For eachσ, letξ^σ denote a continuous piecewise-differentiable function onP with

dξ^σ1. (2.6)

Then for arbitrarya,b,c, we have that n^∗ω dV (a,b,c)

dξ^σ∧n^∗ω

(a,b,c), (2.7)

wheredV is here regarded as the Euclidean volume form onR³. But dξ^σ∧n^∗

˜ χ^σω

=d ξ^σn^∗

˜ χ^σω

, (2.8)

since d(n^∗(χ˜^σω))=n^∗d(χ˜^σω)=0 (χ˜^σωis a two-form onS²). Therefore, E(n)2

σ

P

d ξ^σn^∗

˜ χ^σω

. (2.9)

From Stokes’ theorem, (2.9) implies that E(n)2

σ

∂P

ξ^σn^∗

˜ χ^σω

. (2.10)

The boundary ofP consists of (i) the faces ofP with points inB (v^a)removed, and (ii) the intersections of the two-spheres∂B (v^a)withP. The latter, denoted byC^a=∂B (v^a)∩P, we callcleaved surfaces. Tangent boundary conditions imply thatn^∗ωvanishes on the faces ofP (since the values ofnon a face are restricted to a great circle inS²). Therefore, only the cleaved surfaces contribute to the integral in (2.10). We obtain

E(n)2

σ

a

C^a

ξ^σn^∗

˜ χ^σω

. (2.11)

We can replaceχ˜^σ byχ^σ in (2.11) (by the bounded convergence theorem). Taking the limit →0, we obtain E(n)2

σ

a

ξ^σ v^a

lim→0

C^a

n^∗ χ^σω

. (2.12)

From (1.9), the integral overC^ayieldsA^σ times the wrapping numberw^aσ, which depends only on the homotopy type ofn. Thus,

E_P^inf(h)2

σ

A^σ

a

ξ^σ v^a

w^aσ. (2.13)

The remainder of the argument proceeds as in [4]. We note that (2.13) holds for any choice ofξ^σ’s consistent with the constraints (2.6). These constraints imply that

ξ^σ v^a

−ξ^σ

v^bv^a−v^b. (2.14)

Conversely, given any set of valuesξ^aσ for which|ξ^aσ−ξ^bσ||v^a−v^b|, we can find functionsξ^σ which satisfy the constraints (2.6) and assume these values at the vertices (for example, takeξ^σ(r)=maxa(ξ^aσ − |r−v^a|)). Thus, we obtain a lower bound forE_P(h)in terms of the solutions of a finite number of linear optimisation problems, one for each sector,

E_P^inf(h)2

σ

A^σ

|ξ^aσ−ξmax^bσ||v^a−v^b|

a

ξ^aσw^aσ

. (2.15)

A simpler characterisation is provided by the dual formulation, E_P^inf(h)2

σ

A^σ

min

Ω^ab,σ

a,b

v^a−v^bΩ^ab,σ

, (2.16)

where theΩ^ab,σ’s are constrained by

a

Ω^ab,σ = −w^bσ,

b

Ω^ab,σ =w^aσ. (2.17)

Let us fixσ. Without loss of generality, we can restrictΩ^ab,σ to be nonnegative and equal to zero unlessw^aσ >0 andw^bσ <0. Suppose first that the nonzero wrapping numbers are either+1 or−1; by (1.11) there are an equal number,msay, of each. Therefore, the nonvanishing elements ofΩ^ab,σ may be identified with an m×mmatrix, which we denote byM. (2.17) implies thatMis doubly stochastic. By a theorem of Birkhoff [2],Mcan be expressed as a convex linear combination of permutation matrices. Then the minimum in (2.16) is necessarily achieved at an extremal point, i.e. forMa permutation matrix corresponding to a minimal connection. In this case,

min

Ω^ab,σ

a,b

v^a−v^bΩ^ab,σ=L

P^σ(h),N^σ(h)

, (2.18)

whereP^σ(h)(resp.N^σ(h)) contains verticesv^afor whichw^aσ equals+1 (resp.−1). The case of general nonzero wrapping numbers values is treated by includingv^awith multiplicity|w^aσ|in eitherP^σ(h)(forw^aσ >0) orN^σ(h) (forw^aσ <0). The same argument applies in every sector (there is a separate minimal connection for eachσ), and (1.12) follows. 2

3. Upper bound for rectangular prisms

LetP denote a rectangular prism centred at the origin of three-dimensional Euclidean space. We take the edges of P to be parallel to the coordinate axes and of lengthsL_x,L_y,L_z, oriented so thatL_xL_yL_z. It will be convenient to introduce the half-lengths

lj=L_j

2 (3.1)

(here and in what follows, the indexj takes valuesx,yorz). Then the vertices ofP are of the form

v^a=(±l_x,±l_y,±l_z), (3.2)

where the vertex labeladesignates the signs in (3.2).

Let O^a ⊂S² denote the spherical octant of directions about v^a which are contained in P. E.g., for v^a = (−l_x,−l_y,−l_z),O^ais the positive octant{s∈S²|s_j0}. The boundary ofO^a,∂O^a, contains the directions which lie in the faces atv^a, and is composed of quarter-segments of the great circles aboutx,ˆ yˆandz. Letˆ ∂O_j^adenote the segment aboutˆj.

Choose l so that 0< llz. Then v^a+lO^a is contained in P, so that v^a+lO^a is a cleaved surface. Given n∈CP(h), we can define a unit-vector fieldν^aonO^aby

ν^a(s)=n v^a+ls

. (3.3)

Tangent boundary conditions imply that, fors∈∂O_j^a,ν^a(s)is orthogonal toˆj. Denote the set ofν^a’s collectively byν. The wrapping numbers ofn, and hence its homotopy type, are determined byν.νis an example of anoctant configuration, which we define generally as follows:

Definition 3.1.An octant configurationνwith admissible topologyh= {w^aσ}is a set of continuous piecewise-smooth mapsν^a:O^a→S²satisfying tangent boundary conditions,

s∈∂O_j^a⇒ν(s)· ˆj=0, (3.4)

such that

O^a

ν^∗ χ^σω

=w^aσA^σ. (3.5)

The Dirichlet energy of an octant configurationνon the octantO^ais defined by E₍₂₎^a (ν)=

O^a

∇ν^a2

(s)dΩ^a. (3.6)

Here and in what follows, it will be convenient to regard∇ν_j^a(s)(the gradient of thejth component ofν^a) as a vector inR³which is tangent toO^aats. dΩ^ain (3.6) denotes the area element onO^a(normalised so thatO^ahas areaπ/2).

The Dirichlet energy on the octant edge∂O_j^ais given by

E_(1)j^a (ν)= π/2

0

d dαν^a

s^a_j(α)²

dα. (3.7)

Here,s^a_j(α)denotes the parameterisation of∂O_j^aby arclength (i.e., angle)α. For example, ifv^a=(−l_x,−l_y,−l_z), s^a_j(α)=cosαkˆ+sinαˆl, 0απ

2, (3.8)

where(j, k, l)denote a triple of distinct indices.

By anextensionof an octant configurationν, we mean a continuous, piecewise-smooth unit-vector fieldnonP such thatn(v^a+ls)=ν^a(s)for alls∈O^a. Obviously, ifνhas topologyh, so has its extensionn. We introduce the following notation: Given functionsf andgon a domainW, we writef gto mean there exists a constantCsuch that|f|C|g|onW. In this case, we say thatf is dominated byg.

Lemma 3.1.Letν be an octant configuration with admissible topologyh. Thenν can be extended to a continuous piecewise-differentiable configurationn∈HCP(h)such that

E(n)κ³Lz a

E₍₂₎^a (ν)+

aj

E_(1)j^a (ν)1/2

. (3.9)

Theorem 1.3 is proved by constructing octant configurations whose Dirichlet energies on the octants and their edges scale appropriately with the wrapping numbers. These configurations are provided by the following:

Lemma 3.2.Given an admissible topologyh= {w^aσ}, there exists an octant configurationνwith topologyhsuch

that

a

E^a₍₂₎(ν)

aσ

w^aσ, (3.10)

aj

E^a_(1)j(ν)

aσ

w^aσ2. (3.11)

The proofs of Lemmas 3.1 and 3.2, which involve explicit constructions and estimates, are given in Sections 4 and 5 respectively.

Proof of Theorem 1.3. Given an admissible topologyh= {w^aσ}, we choose an octant configurationνwith topology has in Lemma 3.2, and extend it to a unit-vector fieldnonP as in Lemma 3.1. From the Cauchy–Schwartz inequality, (3.11) implies that

aj

E^a_(1)j(ν)1/2

aσ

w^aσ. (3.12)

Together, (3.9), (3.10) and (3.12) provide an estimate forE(n), and therefore an upper bound forE^inf_P (h), E_P^inf(h)E(n)κ³L_z

aσ

w^aσ. (3.13)

From Theorem 1.2, a lower bound forE^inf_P (h)is given by E_P⁻(h)=

σ

2π 2L

P^σ(h), N^σ(h)

(3.14) (A^σ=π/2 for a rectangular prism). The minimum distance between vertices ofP isL_z. As the number of elements ofP^σ(h)(and ofN^σ(h)) is¹₂

a|w^aσ|, it follows that L

P^σ(h), N^σ(h)

1

2Lz

a

w^aσ. (3.15)

From (3.13) and (3.15), we conclude that

E_P^inf(h)κ³E⁻_P(h). 2 (3.16)

We remark that the octant configurations of Lemma 3.2 must be chosen with some care, as the following example illustrates (details may be found in [17]). For simplicity, takeP to be the unit cube. In the (positive) octant about v^a=(−¹₂,−¹₂,−¹₂), let

ν^a(θ, φ)=(sinαcosβ,sinαsinβ,cosα), (3.17)

where 0θ, φπ/2 andα=(4M+1)θ,β=(4N+1)φfor integersMandN. Given(x, y, z)∈P withx, y, z0, let(θ, φ)denote the polar angles of(x, y, z)with respect tov^a, and letn(x, y, z)=ν^a(θ, φ). We definenelsewhere vian(±x, y, z)=n(x,±y, z)=n(x, y,±z)=n(x, y, z)(so thatnis a reflection-symmetric configuration [18,20]).

Thennis continuous and satisfies tangent boundary conditions. Denote its homotopy class byhMN. It is straightfor- ward to compute the wrapping numbers (it turns out that they scale linearly withMandN), and, from Theorem 1.2, to obtain the following lower bound:

E_P⁻(h_MN)=

2 max(M+2N,2M+N )+1π

4. (3.18)

It turns out thatE(n)can be evaluated exactly as a finite sum of Appell hypergeometric functions. For largeM andN, the energy is given asymptotically by

E(n)∼4√ 3

(4M+1)²+1

2lnM(4N+1)² π

2. (3.19)

ClearlyE(n)does not scale linearly withMandN, so is not dominated by the lower boundE_P⁻(hMN).

4. Extending octant configurations

Let us specify the geometry of the prismP in greater detail. Let

m^a_{(j )}=v^a−v_j^aˆj (4.1)

denote the midpoint of the edge throughv^aalongˆj(here,v_j^ais thejth component ofv^a). LetC^adenote the triangle whose vertices are the midpoints of the edges coincident atv^a,

C^a=

r=τxm^a_(x)+τym^a_(y)+τzm^a_(z)τj0,

j

τj=1

. (4.2)

We callC^aacleaved face(see Fig. 1).c∈C^asatisfies

C^a·c=2, (4.3)

where C^a=

v_x^a₋1

, v_y^a₋1

, v_z^a₋1

(4.4) is an (unnormalised) outward normal onC^a. Lethdenote the distance fromC^ato the origin. Then

2

3lzh= 2

|C^a|<2lz. (4.5)

Let F^{j τ}, whereτ = ±1, denote face of the prism which lies in the plane {r_j =τ l_j}. Let F^{j τ} ⊂F^{j τ} denote the rhombus whose vertices lie at the midpoints of the edges ofF^{j τ}. We callF^{j τ} atruncated face(see Fig. 1).

We partitionP into three sets of pyramids, denotedX^a,Y^aandZ^{j τ}.X^aandY^ahave the cleaved faceC^aas their (shared) base.X^a has its apex atv^a, whileY^ahas its apex at the origin.Z^{j τ} has the truncated faceF^{j τ} as its base and its apex at the origin. Every point ofP belongs either to the interior of just one of these pyramids or else to the boundary between two or more of them (see Fig. 1).

In the proof of Lemma 3.1, we definen, the extension of the octant configurationν, to be constant along rays from C^ato the origin (apart from a small neighbourhood thereof) and along rays tov^a. We then show that the energy ofn inX^aandY^ais proportional toE^a₍₂₎(ν). The construction ofninZ^{j τ} is more complicated. On∂F^{j τ} (the boundary of the base),nis determined byν, and in the interior ofF^{j τ}, we definenby a simple interpolation which respects tangent boundary conditions. In the interior ofZ^{j τ}, we donottakento be constant along rays from the apex to the base, as this would give rise to an energy proportional to

v^a∈F^{j τ}E_(1)j^a (ν), which, for the octant configurations of Lemma 3.2, would scale as the square of the wrapping numbers. Instead, along such rays, and over a distance

σ=

π

v^a∈F^{j τ}

E_(1)j^a (ν)^−1/2

, (4.6)

nis rotated toward the normalˆj. This leads to an energy inZ^{j τ} proportional to 1/σ.

(a) (b) (c)

(d) (e)

Fig. 1. (a) The cleaved planeC^a, (b) the pyramidX^a, (c) the pyramidY^a, (d) the truncated faceF^{j τ}, (e) the pyramidZ^{j τ}.

Proof of Lemma 3.1. Given an octant configurationν, we define an extensionnon the pyramidsX^a,Y^a andZ^{j τ} (Steps 1–3) with Dirichlet energiesEX^a(n),EY^a(n)andE_Zj τ(n)bounded as follows:

E_X^a(n)1

2κL_zE^a₍₂₎(ν), (4.7a)

EY^a(n)κ³LzE₍₂₎^a (ν), (4.7b)

E_Zj τ(n)κ²L_z

v^a∈F^{j τ}

E_(1)j^a (ν)1/2

. (4.7c)

Then

E(n)=

a

E_X^a(n)+E_Y^a(n)

+

j τ

E_Zj τ(n)

κ³L_z

a

E₍₂₎^a (ν)+

aj

E_(1)j^a (ν)1/2

. (4.8)

To ensure continuity (Step 4) we modify the construction ofnnear the origin while preserving the bounds (4.7).

Step 1. Construction inX^a. From (4.2) and (4.3), points inX^aare of the formv^a+rs, wheres∈O^aand 0< r r^a(s)with

r^a(s)= 1

|C^a·s|; (4.9)

r^a(s)is the distance fromv^atoC^aalongs. The maximal distance is half the length of the longest edge, so that

r^a(s)r^a(x)ˆ =l_x. (4.10)

We defineninX^aby n

v^a+rs

=ν^a(s), s∈O^a,0< rr^a(s). (4.11)

Then, from (3.6) and (4.10), E_X^a(n):=

X^a

(∇n)²dV =

O^a

r^a(s)

∇ν^a(s)2

dΩ^al_xE^a₍₂₎(ν)1

2κL_zE₍₂₎^a (ν), (4.12)

as in (4.7a).

Step 2. Construction inY^a. Points inY^a(excluding the origin) are of the formλc, wherec∈C^aand 0< λ1.

We defineninY^aby

n(λc)=n(c). (4.13)

Note thatn(c)is fixed in Step 1, sinceC^abelongs toX^aas well asY^a.

To estimateE_Y^a(n), we resolve∇ninto components tangent and normal to the cleaved faceC^a,

∇n=∇tn+∇nn, (4.14)

so that(C^a·∇t)n=0 and(t·∇n)n=0 forttangent toC^a. From (4.13),∇tn(λc)=λ⁻¹∇tn(c). Therefore, ∇tn(λc)2

= 1 λ²

∇tn(c)2

. (4.15)

To estimate(∇nn)², we note that(c·∇)n(λc)=0 (nis constant along rays inY^a through the origin) and resolvec into componentsct andcntangent and normal toC^ato obtain

∇nn(λc)2

|ct|²

|cn|²

∇tn(λc)2

|ct|²

|cn|² 1 λ²

∇tn(c)2

. (4.16)

Since(∇tn(c))²(∇n(c))², (4.15) and (4.16) together give ∇n(λc)2

= |c|²

|cn|² 1 λ²

∇n(c)2

. (4.17)

Clearly|c|l_xwhile|cn|is justh, the distance fromC^ato the origin, andh >2lz/3 (cf. (4.5)), so that ∇n(λc)2

9

4κ² 1 λ²

∇n(c)2

. (4.18)

The volume element onY^ais given by

dV =hλ²d²cdλ, (4.19)

where d²cis the Euclidean area element onC^a. Sinceh <2lz, E_Y^a(n)=

Y^a

(∇n)²dV <9 2κ²l_z

C^a

(∇n)²(c)d²c. (4.20)

Lettings=(c−v)/|c−v|, we can write the preceding as an integral overO^a. We have that d²c= |c|²

|s·C^a|/|C^a|dΩ^a,

∇n(c)2

= 1

|c|²

∇ν^a(s)2

, (4.21)

and

|C^a|

|s·C^a|<3/ lz

1/ lx

=3κ, (4.22)

so that (4.20) becomes E_Y^a(n)27

2 κ³l_z

O^a

∇ν^a(s)2

dΩ^aκ³L_zE₂^a(ν), (4.23)

as in (4.7b).

Step 3. Construction inZ^{j τ}. To simplify the discussion and the notation, let us fix our attention on the top face of the prism, withj =zandτ =1, and henceforth drop the designationj τ, writingZ forZ^{j τ},FforF^{j τ}, etc, in what follows (the other faces are handled similarly).∂Fmay be parameterised asR(φ)=(R(φ)cosφ, R(φ)sinφ, l_z), where

R(φ)= l_xl_y

ly|cosφ| +lx|sinφ| (4.24)

andφis the polar angle about thez-axis. On∂F(which is also contained in the cleaved faces),nis defined in Step 1.

It follows thatnis continuous on∂F(including the midpoints of the edges ofF, which belong to two cleaved faces, asνhas an admissible topology) and satisfies tangent boundary conditions there. Tangent boundary conditions imply that

n R(φ)

= cos Θ(φ)

ˆ y+sin

Θ(φ) ˆ

x, (4.25)

whereΘ(φ)may be taken to be continuous and piecewise smooth, withΘ(2π )−Θ(0)equal to a multiple of 2π.

Sinceνhas an admissible topology, we must haveΘ(2π )=Θ(0). = ±1 can be chosen so that

Θ(0)=Θ(2π )=0. (4.26)

We introduce polygonal cylindrical coordinates(h, ξ, φ)onZ, defined by

x=hξ R(φ)cosφ, y=hξ R(φ)sinφ, z=l_zh, 0< h1, 0ξ1,0φ <2π. (4.27) ξ, the radial coordinate, is scaled to equal 1 on the sides ofZand 0 along thez-axis. Then let

N(2)(ξ, φ)= cos(ξ Θ)yˆ+sin(ξ Θ)x.ˆ (4.28)

N(2)gives a continuous extension ofnto the interior ofFwhich satisfies tangent boundary conditions. A continuous extension to the interior ofZis given by

N(h, ξ, φ)=cosγN(2)+sinγz,ˆ (4.29)

whereγ=γ (h, ξ )is given by γ (h, ξ )=Φ

1−h σ

Φ

1−ξ σ

π

2, Φ(s)=

s, s <1,

1, s1, (4.30)

and 0< σ <1. Thus,γ vanishes on the boundary ofZ and has a constant value,π/2, at interior points sufficiently far from the boundary.σ, which determines how far, will be specified below. We definenonZas

n

x(h, ξ, φ), y(h, ξ, φ), z(h, ξ, φ)

=N(h, ξ, φ). (4.31)

It is readily checked that (4.31) agrees with (4.13) at points on the boundary ofZexcept at the origin.

The energy ofninZis given byE_Z(n)as follows. In terms of the coordinates(h, ξ, φ), we have that E_Z(n)=

Z

(∇n)²dV

= 1 0

dh 1 0

dξ 2π 0

dφ(∇ξ×∇φ)·∇h⁻¹(Nh∇h+Nξ∇ξ+Nφ∇φ)², (4.32) whereNh=∂N/∂h,Nξ=∂N/∂ξ, etc. From (4.28) and (4.29), we get that

Nh=γh(cosγzˆ−sinγN(2)),

Nξ=γ_ξ(cosγzˆ−sinγN(2))+ ΘcosγN(2)× ˆz,

Nφ= ξ ΘcosγN(2)× ˆz. (4.33)

From (4.27), we get that

∇h=(0,0,1/ lz),

∇ξ=ξ

(Rcosφ+Rsinφ)/(ρR), (Rsinφ−Rcosφ)/(ρR),−1/(lzh) ,

∇φ=(−sinφ,cosφ,0)/ρ, (4.34)

whereρ=(x²+y²)^1/2=hξ R. Straightforward calculation then gives an expression forEZ(n)of the form E_Z(n)=

5 i=1

Ei= 5 i=1

1 0

dh 1 0

dξ 2π 0

dφ Ii, (4.35)

where the integrands for the separate contributionsEiare given by I1=l_zcos²γ ξ Θ², I2=h²

l_zγ_h²ξ R², I3= −2h

l_zγ_hγ_ξξ²R², I₄= −2 lzcos²γ ξR

RΘΘ, I₅=

l_zξ

1+R² R²

+ξ³ lz

R²

cos²γ Θ²+γ_ξ²

. (4.36)

We consider these contributions in turn.

ConcerningE1, since cos²γ vanishes for 0< ξ, h <1−σ (cf. (4.30)), it follows that E12lzσ

2π 0

Θ²(φ)dφ. (4.37)

The integral ofΘ²(φ)can be related to the Dirichlet edge energiesE^a_(1)z(ν)for the verticesv^awhich lie onF. For convenience, label these anticlockwise bya=0,1,2,3 so that

n R(φ)

=ν^a s^a_z

α(φ)

, (a−1)π/2< φ < aπ/2, (4.38)

wheres^a_z(α)parameterises the octant edge∂O_z^aas in (3.8), andα(φ)is given by tanα=

(l_x/ l_y)²tanφ, 0φ < π/2 orπφ <3π/2,

(l_y/ l_x)²tanφ, π/2φ < πor 3π/2φ <2π. (4.39) (φis the angle with respect to the centre ofF andα, with 0< α < π/2, the angle with respect consecutive vertices.

(4.39) gives the elementary relation between them. See also Fig. 2.) Recalling (4.25), we get that Θ²(φ)=

d dφn

R(φ)²

= d

dαν^a

s^a_z(α)² α=α(φ)

dα dφ

2

. (4.40)

It follows that 2π

0

Θ²(φ)dφ=

v^a∈F

π/2

0

d dαν^a

s^a_z(α)² dφ dα

₋1

dα. (4.41)

From (4.39) one has that

dφ dα

₋1 l²_x

l²_y κ². (4.42)

Therefore, 2π 0

Θ²(φ)dφκ²

v^a∈F

E_(1)z^a (ν). (4.43)

Fig. 2. The anglesαandφ.