Analysis of a Ginzburg–Landau type energy model for smectic C* liquid crystals with defects

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Ann. I. H. Poincaré – AN 30 (2013) 1009–1026

www.elsevier.com/locate/anihpc

Analysis of a Ginzburg–Landau type energy model for smectic C*

liquid crystals with defects

Sean Colbert-Kelly, Daniel Phillips

^∗

Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, IN, 47907, United States Received 2 July 2012; accepted 10 December 2012

Available online 14 January 2013

Abstract

This work investigates properties of a smectic C* liquid crystal film containing defects that cause distinctive spiral patterns in the film’s texture. The phenomena are described by a Ginzburg–Landau type model and the investigation provides a detailed analysis of minimal energy configurations for the film’s director field. The study demonstrates the existence of a limiting location for the defects (vortices) so as to minimize a renormalized energy. It is shown that if the degree of the boundary data is positive then the vortices each have degree+1 and that they are located away from the boundary. It is proved that the limit of the energies for a sequence of minimizers minus the sum of the energies around their vortices, as the G–L parameterεtends to zero, is equal to the renormalized energy for the limiting state.

Résumé

Dans ce travail on étudie les propriétés d’un film de cristaux liquides smectiques C* contenant des défauts qui produisent des motifs en spirale dans la texture du film. Les phénomènes sont décrits par un modèle de type Ginzburg–Landau dans un domaine borné du plan, et cet article fournit une analyse détaillée des configurations d’énergie minimale du champ de directions du film. On montre l’existence d’une configuration limite pour les défauts (tourbillons) qui minimise une énergie renormalisée. On démontre que si le degré du champ sur le bord du domaine est positif, alors les tourbillons sont dans l’intérieur du domaine et sont chacun de degré+1. On prouve que quand le paramètreεde Ginzburg–Landau tend vers zéro, pour une suite de minimiseurs, la limite de l’énergie moins la somme des énergies autour des tourbillons est égale à l’énergie renormalisée de l’état limite.

1. Introduction

We study the occurrence of point defects in a thin ferroelectric smectic C* (Sm C^∗) liquid crystal by using a director field description based on the Ginzburg–Landau theory. The unknown functionuis a vector field inR². When convenient, for ease of notation, we view it as aC-valued function such thatu=u¹(x₁, x₂)+iu²(x₁, x₂)for(x₁, x₂) in a bounded domainΩ inR². We assumeΩ has a smooth(C³)boundary in the plane and thatΩ represents the reference configuration of a very thin liquid crystal material.

* Corresponding author.

E-mail addresses:scolbert@math.purdue.edu(S. Colbert-Kelly),phillips@math.purdue.edu(D. Phillips).

http://dx.doi.org/10.1016/j.anihpc.2012.12.010

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We analyze minimizers for a Ginzburg–Landau type energy, Jε(u)=1

2

Ω

ks(divu)²+kb(curlu)²+ 1 2ε²

1− |u|²2 dx=

Ω

jε(u,∇u) dx, (1.1)

where k_s and k_b represent the two-dimensional splay and bend moduli for the film respectively, with k_s, k_b>0.

Here J_ε(·)is defined foru∈H_g¹(Ω;R²), consisting of fieldsu∈H¹(Ω;R²)with Dirichlet boundary conditions, u|∂Ω=g∈C³(∂Ω;S¹)such that deg(g, ∂Ω)=d >0. The variableε >0 represents the radius of the defect cores.

Previous work has considered the case where k_s=k_b, reducing to the classical Ginzburg–Landau functional[1–5].

Our work focuses on the cases wherek_s=k_b. The elastic energy term of (1.1) is used to model thin film liquid crystals with chirality, such as an Sm C^∗material. The resulting pattern consists of a family of point defects in the film forming vortices in the molecular texture that spiral in a fashion determined by the relative values fork_s andk_b; see[6].

1.1. Main results

By denotingk=min{k_s, k_b}, we can express (1.1) as Jε(u)=Jε(u)+k

Ω

det∇u dx=Jε(u)+kπ d (1.2)

foru∈H_g¹(Ω;R²)where J_ε(u)=

Ω

j¯_ε(u,∇u) dx=

⎧⎨

⎩

1 2

Ω(k_s|∇u|²+(k_b−k_s)(curlu)²+_2ε¹2(1− |u|²)²) dx ifk=k_s,

1 2

Ω(k_b|∇u|²+(k_s−k_b)(divu)²+_2ε¹2(1− |u|²)²) dx ifk=k_b. (1.3) Thenu∈H_g¹(Ω;R²)is a minimizer ofJ_ε(u)if and only ifuis a minimizer forJ_ε(u). Hence, it suffices to consider the minimizers of Eq. (1.3) and analyze this functional. In this way, by the strict convexity of the integral in (1.3), we have the existence of a minimizeru_εfor eachε; see[7].

We need a detailed description ofΩ. LetD⊂R²be a bounded, simply connected domain with aC³boundaryΓ0. For=1, . . . , kletΛ⊂Dbe pair-wise disjoint, simply connected sets withC³boundariesΓ. Consider the domain Ω =D\k

=1Λ where we take the natural orientation for ∂Ω =k

=0∂Γ, such that Γ0 is oriented counter- clockwise andΓ are oriented clockwise for 1k. For eachg∈C³(∂Ω;S¹)setd:=winding number ofg|Γ

with respect to the curve’s orientation, and denote the degreed(g, ∂Ω):=d=_k

=0d. We fixk points,y∈Λ, and setw(x)=k

=1(_|^x_x⁻₋^y_y

|)⁻^d=e^{iζ (x)}forx∈Ω. Thusζ is a multi-valued, harmonic expression such that∇ζ (x) is point-wise well defined. We use w(x) to fix specific representations of functions having boundary values with winding numbers d with respect to Γ for 1k. The minimizers of the energy functional over H_g¹ have a number of structural properties that lead to the first main result of the paper.

Theorem A. Let {u_ε} be a sequence of minimizers for J_ε(u) over H_g¹ such that ε→0. Then there is a subse- quence{u_ε}, a functionh∈H¹(Ω)andd points{a₁, . . . , a_d} ∈Ω such that

|uε| →1 uniformly on compact subsets ofΩ\ {a1, . . . , ad},

and more generallyu_ε(x)→u_∗(x)=e^i(h(x)⁺^{ζ (x)}⁺^dⁿ⁼¹^θ^an^(x)) (1.4) inC_loc^α (Ω\ {a1, . . . , ad})and inC_loc^m(Ω\ {a1, . . . , ad})for every0< α <1and integerm0, in whichθa_n=θa_n(x) denotes the polar angle ofx with respect to the centera_n.

The d-tuplea=(a₁, . . . , a_d)∈Ω^d represents the point defects within the domainΩ. The energy functional, just as in[1], has a renormalized form,

kW (b)+H (b, ks, k_b) forb∈Ω^d (1.5)

here

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W (b)=1 2

∂Ω

2Gb(g×∂_τg)−(∂_νG_b)G_b

dσ+π d

−

m=n

πln

|b_n−b_m| +

d n=1

k =1

π dln

|b_n−y|

(1.6) and

H (b, ks, k_b)=inf

φ H(b, φ, ks, k_b)= infφ1

2

Ω(ks|∇φ|²+(kb−ks)(curlv)²) dx ifk=ks, infφ1

2

Ω(kb|∇φ|²+(ks−kb)(divv)²) dx ifk=kb

(1.7) whereGb(x)=d

n=1ln(|x−bn|)−k

=1dln(|x−y|),v(x)=d n=1

x−b_n

|x−b_n|e^{i(φ (x)}⁺^{ζ (x))}, andH(b, φ, ks, kb)is minimized over the class of functionsφ∈H¹(Ω)such thatv=gon∂Ω. The expression (1.5) is a variant of the renormalized energy from[1]. Moreover the two agree for the case whereks=kbandΩis simply connected. By the definition of (1.5), we have that in order for the renormalized energy to be finite,bn=bmforn=mandbn∈/∂Ω for everyn. For each such set of configurationsb, there exists a functionh_b, in a particular class of functions, such that H (b, ks, k_b)=H(b, hb, k_s, k_b), which leads to the second main theorem of this work.

Theorem B.Let {u}be a sequence of minimizers for J_ε, for which a=(a₁, . . . , a_d)is a limiting configuration of distinct defects asε→0 as described in TheoremAand h∈H¹(Ω)is as in TheoremA. Then it holds that H (a, ks, k_b)=H(a, h, ks, k_b)and

lim→∞

J_ε(u)−kπ dln 1

ε

=kW (a)+H (a, ks, k_b)+dγ

whereγis a fixed constant associated with each of the defect core’s energy. Moreover, the renormalized energy attains its minimum amongb∈Ω^dwith distinct components atb=a.

The termkπ dln(¹_ε)represents the energy around the vortices to leading order. The limit asε tends to zero of the difference between this term and (1.1) gives the remaining energy over the domainΩ minus the vortices, with their locationa, minimizing (1.5). For the casek_s =k_b the proofs for Theorems A and B follow from the results in[1]and[3]ifΩ is simply connected. Moreover their proofs can be extended if the domain is multiply connected.

Theorem B allows us to characterize the limiting pattern,u_∗(x), near eacha_m. This follows from the fact thath minimizes (1.7). For the casek_s=k_bthis implies thathis a harmonic function such thatv=gon∂Ω. Thus

u_∗(ρy+am)→βmy

asρ→0 for eachy∈∂B₁(0)whereβ_m=e^i(h(a^m⁾⁺^{ζ (a}^m⁾⁺ⁿ⁼^m^θ^an^(a^m⁾⁾. Fork_s=k_bwe find a much different structure.

In this case the integral in (1.7) involving either the term curlvor divvmust be finite, and as a result pins the values ofhnear eacha_mso that

u_∗(ρy+a_m)→

±y ifk_s< k_b,

±iy ifk_b< k_s

inL²(∂B1(0);C)asρ→0. Thus ifks < kbthe limiting textureu_∗has a pure splay pattern near each defect and if kb< ks thenu_∗asymptotically has a pure bend pattern near eacham.

1.2. Applications

Smectic C materials are made of layers of liquid crystal molecules that pack so that their long axes form a fixed angle 0< θ₀< π/2 with the layer normal. The pattern is described using the layer structure and a director fieldn(x) for the liquid crystal. The director is a unit vector field that lies parallel to the local average of the molecular long axes atx. One can then expressn(x)=cos(θ0)ν(x)+sin(θ0)c(x)whereν(x)andc(x)each are unit vector fields that are respectively parallel and perpendicular to the layer normal atx. These two fields are the fundamental unknowns that are used to characterize the material’s configuration[8]. For the case of a thin film the layers are planer, given by the

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domainΩ⊂R²such thatν(x)= 0,0,1andc(x)= c¹(x), c²(x),0withx=(x₁, x₂). The film can be just several layers thick and the elastic energy for the liquid crystal pattern is given by the Oseen–Frank energy. Each layer in a smectic C (Sm C) liquid crystal can be represented as a two-dimensional liquid[9]and the integral is taken over the film,

1 2

Ω

k_s(divc)²+k_b(curlc)²

dx. (1.8)

An Sm C^∗ liquid crystal additionally forms a spontaneous polarization field that produces elastic and electro-static contributions to the energy. The polar field generates an elastic stress on the film whose effect is modeled by intro- ducing boundary values forc(x)=g(x)on∂Ω; see[6]. The field induces an electro-static contribution that appears in our energy by increasing the splay constantk_babove its bare elastic value[10]and this is a motivation for studying the casek_s=k_b.

If a smoke or dust particle lands on a free-standing film a defect forms in the film’s texture. The particle induces a singularity in the spontaneous polarization field that in turn causes an island, several layers thicker than the film, to form around the defect. The island’s shape eventually stabilizes and the island migrates within the film so as to reduce the total energy. Various experiments have been conducted and models put forward to investigate this phenomenon.

See[6,10–13]. In these papers the notion of thec-director is generalized to allow for defects. The experiments reported in[6,10–12]indicate that a stable island is disk-like and that thec-director is tangential at the island’s edge, so that the winding number ofcon the edge of the disk is+1. In[6], Lee et al. model the island-defect ensemble by setting Ω =B_R(0)\B_ε(0)with the defect represented by theεvoid at the origin and investigate numerically the stability of rotationally invariant equilibria for (1.8). Their simulations for the casek_s< k_b andεsufficiently small, indicate that minimizers for (1.8) overH¹(Ω;S¹)subject to tangential boundary conditions on∂B_R(0)form a simple spiral, turning from the tangential pattern at the edge of the disk to radial near the defect at the center.

In this paper we follow an order parameter approach as in [13]for the energy (1.1). The unknown fieldu(x)is taken to be a generalization of thecdirector. In this caseuneed not have unit length and vanishes at a defect where the smectic order is allowed to melt. This description, in contrast to the one above, does not presuppose the nature or location of individual defects. We can apply Theorem A to obtain information on minimizers for the problem of an island,Ω=B_R(0)with the tangential boundary valuesg(x)= ±^x_R^⊥ (wherex^⊥=(−x₂, x₁)≡ix). Ifk_s< k_bandεis taken sufficiently small it follows that a minimizer has one defect with degree+1 inB_R(0), moreover the minimizer’s pattern is near radial in a neighborhood of the defect. This is consistent with what was observed in the experiments and simulations from[6].

In [13], Silvestre et al. investigate a different aspect of the problem. They consider a free-standing Sm C^∗ film occupying a simply connected regionDcontainingd disjoint circular islands{B_R_j(x_j); 1j d}and they investigate the texture in the background film Ω≡D\_d

j=1B_R_j(x_j). In this case∂Ω hasd+1 components,∂Dand

∂B_R_j(x_j)for 1jd. On∂Dthey takeg=const. andg= ±^(x⁻_R^x_j^j⁾^⊥ on∂B_R_j(x_j)for 1jd. It follows that deg(g, ∂Ω)= −d. The simulations and experiments in[13]exhibitdtopological defects inΩ, each with degree−1, as companions to the d chiral islands. Our results do not directly apply to this setting. For the case of the classic Ginzburg–Landau energy (2.7), structure proved for the case deg(g, ∂Ω) >0 directly translates to the same information for the case deg(g, ∂Ω) <0. This is not obvious for the energy (1.1). Nevertheless the present analysis should be useful for investigating the case of boundary data with negative degree.

Our paper is organized as follows. In Section 2, we prove Theorem A, developing a number of qualitative features for the minimizers ofJε. We express the integral in the form of (1.2)–(1.3). Written in this way one sees that minimizers form a family of low energy states for the Ginzburg–Landau functional (2.7), that is a family{uε} ⊂H_g¹(Ω) satisfying (2.9) for a fixed constantK. It is proved in[4,5,14]that such a family has a number of structure and compactness properties, in particular it is shown that for a sequenceεk→0, there exists a subsequence {uε_k()}and a functionu_∗ such thatuε_k()→u_∗. The analysis in[14]due to Fanghua Lin, contains the detailed description ofu_∗ that is needed here and we expand on this work in Proposition 2.3. We then use the fact that theuεare minimizers to refine the notion of convergence away from the defects ofu_∗. Our work here builds on the investigation of minimizers for the Ginzburg–Landau energy (2.7) carried out by Brezis, Bethuel and Hélein in[1]. Their work however relies on a priori estimates for sup|u_ε|, which they obtain by applying a maximum principle that is not available in the case at

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hand. Here we proceed as in[15], obtaining bounds on the bulk term of the energy, giving us a prioriL⁴estimates for the sequence of minimizers. A uniform bound on sup|u_ε|follows from this and elliptic estimates. In Section 3 we analyze a class of polar functions for a particular set of pointsb=(b₁, . . . , b_d)and show that the spherical average of these functions aroundb_ntend to either 0 or^π₂, modπ, depending on the relative values ofk_s andk_b. In Section 4, we construct the renormalized energy in (1.5). We show that for eacha, there exists a polar functionh_athat minimizes the renormalized energy fora. In Section 5, we prove Theorem B.

2. Qualitative properties of minimizers

We begin by developing a number of structural properties for minimizers of (1.1) for eachε. As stated before, the minimizers inH_g¹(Ω;R²)of (1.1) are also minimizers inH_g¹(Ω;R²)of (1.3). In translating the variational problem into minimizing the energy (1.3) inH_g¹(Ω;R²), we have a strictly convex integrand (in the gradient), giving us the existence of a minimizer to our problem[7]. Then, taking the first variation of (1.3), we get that the minimizeruε

satisfies

Ω

k_su¹_x

1

v¹_x

1+

k_su¹_x

2+(k_b−k_s) u¹_x

2−u²_x

1

v¹_x

2+

k_su²_x

1+(k_b−k_s) u²_x

1−u¹_x

2

v²_x

1

+ k_su²_x

2

v_x²

2+ 1

ε² u

1− |u|²

·v

dx=0 ifk=k_s,

Ω

k_bu¹_x

1+(k_s−k_b) u¹_x

1+u²_x

2

v_x¹

1+

k_bu¹_x

2

v_x¹

2+

k_bu²_x

1

v_x²

1

+ kbu²_x

2+(ks−kb) u¹_x

1+u²_x

2

v_x²

2+ 1

ε² u

1− |u|²

·v

dx=0 ifk=kb (2.1)

forv∈H₀¹(Ω;R²). From regularity theory[16,17], we haveu_ε∈C^∞(Ω)∩C^2,α(Ω)for each 0< α <1 andε >0, since∂Ω isC³,g∈C³, and the coefficients of the elliptic elastic energy term are constant. Thus we conclude that

|u_ε|_2,α_;_Ω C(ε). Using integration by parts, we have that the Euler–Lagrange equation the minimizer satisfies is

−k_su¹−(k_b−k_s) u¹_x

2x2−u²_x

1x2

= 1 ε²u¹

1− |u|² ,

−k_su²−(k_b−k_s) u²_x

1x1−u¹_x

2x1

= 1 ε²u²

1− |u|²

(2.2) ifk=k_s, and

−k_bu¹−(k_s−k_b) u¹_x

1x1+u²_x

2x1

= 1 ε²u¹

1− |u|² ,

−k_bu²−(k_s−k_b) u²_x

2x2+u¹_x

1x2

= 1 ε²u²

1− |u|²

(2.3) ifk=k_b inΩ, withu=gon∂Ω. For eachε >0, there is an upper and a lower bound that can be obtained for the integral (1.3).

Proposition 2.1.

J_ε(u_ε)kπ dln 1

ε

+C₂(Ω, g, d, k_s, k_b) (2.4)

for a minimizeru_εand

Jε(u)kπ dln 1

ε

−C1(Ω, g, d, ks, kb) (2.5)

for any functionu∈H_g¹whereC₁andC₂are positive constants.

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Proof. For the upper estimate, this proof is similar to the proof of Lemma 2.1 in[3]. Consider the case whenΩ= B_R(0)=B_Randg(x)=β_|^x_x_|,β= ±1 ifk=k_s orβ= ±iifk=k_b. We drop theεfor notational purposes. Denote the values

Iβ(ε, R)= inf

u∈H_g¹ BR(0)

j¯ε(u,∇u) dx

. (2.6)

LetI_β(t)=I_β(t,1). By using a change of variables,I_β(ε, R)=I_β(1,^R_ε)=I_β(_R^ε). Then, using the same method as in the proof of Theorem 3.1 in[1], noting that div(_|^ix_x_|)=curl(_|^x_x_|)=0, we haveI_β(t₁)kπln(^t_t²

1)+I_β(t₂)for all t₁t₂.

Fixd points,a₁, a₂, . . . , a_d, a_n=a_mforn=minΩandR >0 such that B_R(a_n)⊂Ω for eachn and B_R(a_n)∩B_R(a_m)= ∅ for everyn=m.

LetΩ_R=Ω\d

n=1B_R(a_n)and considerg(x)¯ :∂Ω_R→S¹such that

¯ g(x)=

⎧⎨

⎩

g(x), ifx∈∂Ω,

eîθ, ifx=a_j+Reîθ∈∂B_R(a_j)for somej andk=k_s, ieîθ, ifx=a_j+Reîθ∈∂B_R(a_j)for somej andk=k_b.

Note that deg(g, ∂Ω¯ R)=0, hence there exists a smooth function v:ΩR→S¹ such thatv|∂Ω_R = ¯g. Then by the above claim for 0< ε < R,

J_ε(u)

Ω_R

j¯_ε(v,∇v) dx+ d i=1

I_β(ε, R)

kπ dln 1

ε

+C₂. Define

Fε(u)=Fε(u;Ω)=1 2

Ω

k|∇u|²+ 1 2ε²

1− |u|²2

dx. (2.7)

Now, letu∈H_g¹be any function. Note thatJ_ε(u)F_ε(u). The lower bound (2.5) holds for a minimizerv_ε∈H_g¹ forFε(·)and is proved in[1]for the case thatΩ is star shaped. The general case follows from[3]and[1]. (See[4]

and[5]for alternative proofs.) It follows that the lower bound holds for anyu∈H_g¹. 2

The following corollary is a direct result of the previous proposition, utilizing a method described in[5].

Corollary 2.1.Ifuis a minimizer forJ_ε(u), then there existsC₄=C₄(g, Ω, k_s, k_b, d)such that

Ω

(kb−ks)(curlu)²+ 1 2ε²

1− |u|²2

dxC4 ifk=ks,

Ω

(k_s−k_b)(divu)²+ 1 2ε²

1− |u|²2

dxC₄ ifk=k_b. (2.8)

Proof. For clarity, we provide a short proof. From Proposition 2.1, we have that for any minimizeruε∈H_g¹,Jε(uε) kπ dln(ε⁻¹)+C2. LetF2ε(u)be as defined in the proof of Proposition 2.1. ThenF2ε(uε)kπ dln(ε⁻¹)−C3. Thus (2.8) follows from considering the expressionJ_ε(u_ε)−F_2ε(u_ε). 2

Using the estimate from the corollary and the strong ellipticity of the system in (2.2) and (2.3), we obtain several bounds for minimizers over the entire domain for smallε. Letx∈Ωand set

Ω˜ = {y: εy+x∈Ω} and u(y)˜ =u_ε(εy+x) fory∈ ˜Ω.

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Then from (2.8) we have that ˜u₄_{; ˜}_Ω_∩_B₁₍₀₎CwhereCis independent ofx∈Ω and 0< ε1. If we express (2.2) and (2.3) asLku=ε⁻²f(u), wheref(u)=u(1− |u|²), then Lk is a second order strongly elliptic operator with constant coefficients. Moreover we have that Lku˜ =f(u)˜ on Ω˜ ∩B₁(0). Based on the L⁴ a priori estimate, the ellipticity of the operatorLk, and the smoothness of both the boundary data and∂Ω, the proof of the next proposition follows just as the proof for Lemma 3.1 in[15].

Proposition 2.2.There exists a constantC5=C5(g, Ω, ks, kb, d)so that ifuεis a minimizer toJε(u), then

|u_ε|, ε|∇u_ε|C₅ inΩ,for0< ε <1.

From Proposition 2.1 and the definition ofj¯_ε(u,∇u), we get that F_ε(u_ε)kπ dln

1 ε

+K (2.9)

for every 0< ε <1 and minimizeru_εtoJ_ε(·). With this estimate we are able to apply a structure and compactness theorem to the sequence of minimizers forJ_ε(u).

Proposition 2.3.There exist constantsδ >0andCdepending onK,k,g, andΩso that for any sequence of functions u_ε∈H_g¹(Ω;R²)withε↓0satisfying(2.9)there exists a subsequence{u_ε}, points{a₁, . . . , a_d} ⊂Ω, and a function h(x)∈H¹(Ω)so that

min

dist(am, ∂Ω), |a_m−a_n|, m=n, 1m, nd

δ, h_H¹C,

and

u_ε→ d m=1

x−a_m

|x−a_m|e^i(h(x)⁺^{ζ (x))}

where the convergence of{u_ε}is weak inH_loc¹ (Ω\ {a₁, . . . , a_d};C)and strong inL²(Ω;C).

Proposition 2.3 is due to Fanghua Lin[18,14,19,20]ifΩis simply connected. In this case the limit takes the form d

m=1 x−am

|x−am|e^ih(x). We modify his arguments below to prove it for generalΩ. We first need a lemma.

Lemma 2.1.There is a constantK, depending only onK,k,g, andΩso that ifusatisfies(2.9)thenw^∗(x)u(x)= e⁻^{iζ (x)}u(x)satisfies

F_ε w^∗u

kπ dln 1

ε

+K.

Proof. Fora∈Cwe denotea^∗as the complex conjugate ofa. Writingwu=e^iζu. We have F_ε(wu)=F_ε(u)+k

Ω

Im

u^∗∇ζ · ∇u dx+k

2

Ω

|∇ζ|²|u|²dx.

Since deg(wu;∂Ω)=d we have the lower bound F_ε(wu)kπ dln

1 ε

−C

whereC depends onwu|∂Ω=wg. Second, sinceζ is smooth and fixed inΩwe have that

Ω

|∇ζ|²|u|²dxM

Ω

|u|²dxM₁

ε²ln 1

ε

+1

.

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With these two estimates, together with our hypothesis onF_ε(u)we get

−

Ω

Im

u^∗∇ζ· ∇u dxC

whereCdepends only onK,k,g, andΩ. Finally we can expand and estimate F_ε

w^∗u

=F_ε(u)−k

Ω

Im

u^∗∇ζ · ∇u dx+k

2

Ω

|∇ζ|²|u|²dx kπ dln

1 ε

+K. 2

Proof of Proposition 2.3. Letu_ε∈H_g¹(Ω;R²)withε↓0 satisfying (2.9). Setz_ε=w^∗u_ε. Then the winding number for z_ε|Γ_j is 0 for 1j k and we can extend z_ε ontoΛ_j as a function in H¹(Λ_j;S¹)that is independent of ε for eachj. We also have that the winding number forz_ε|Γ₀ isd. Thus settingz_ε|Γ₀ = ˜g, we have that the sequence {z_ε} ⊂H_g¹_˜(D;C)and that

F_ε(z_ε;D)kπ dln 1

ε

+K

whereKis independent ofε. We can apply the proposition for the simply connected domainD; see[14]. We find a subsequence{z_ε}, points{a₁, . . . , a_d} ⊂D, andh∈H¹(D)so that

zε→z_∗(x)= d m=1

x−a_m

|x−a_m|e^ih(x) inD.

Thus sinceu_ε=wz_εwe have that u_ε→

d m=1

x−a_m

|x−a_m|e^i(h(x)⁺^{ζ (x))} inΩ.

We know that the{a1, . . . , ad}are uniformly bounded away from each other andΓ0. It remains to show that they are bounded away from_k

j=1Λ_j. If this is not so, then we can find a case witha∈Λ_j for someandj. We choose r >0 sufficiently small so thatBr(a)∩ {an: n=} = ∅. By constructionzε(x)forx∈Λj are independent ofε

and inH¹(Λj). Thusz_∗∈H¹(Λj). On the other hand we have z_∗(x)= x−a

|x−a|z(x)˜ such that z(x)˜ =1 forx∈Br(a). Moreoverz˜∈H¹ Br(a)

. Thusz_∗∈/H¹(Λ_j)and this is a contradiction. 2

Forρ >0 setΩ_ρ=Ω\_d

m=1B_ρ(a_m).

Proposition 2.4.Let{u_ε}be a sequence of minimizers converging tou_∗(x)=_d

m=1 x−am

|x−am|e^i(h(x)⁺^{ζ (x))} inL²(Ω).

Then for eachρ >0the convergence is inH¹(Ω_ρ)andε⁻²

Ωρ(1− |u_ε|²)²dx→0as→ ∞. Moreover,u_∗is a local minimizer for the limiting energy inH¹(Ω_ρ,S¹).

Proof. The proof is similar to the proof of Lemma 3.9 in[15]. For notational purposes we write{u} = {u_ε}, where ε is a subsequence of ε andε→0. By Proposition 2.3,u u_∗ inH¹(Ω_ρ) for everyρ >0 andu is a local minimizer for

Ωρj¯_ε(u,∇u) dx. As in the proof of Lemma 3.9 of[15], choosex¯∈Ω\ {a₁, . . . , a_d}, assuming first thatx /¯∈∂Ωand letd¯= ¯d(x)¯ be such thatB₂_d_¯=B₂_d_¯(x)¯ ⊂Ω\ {a1, . . . , ad}. Setω(x)=h(x)+ζ (x)+d

n=1θa_n(x) where _|^x_x⁻₋^a_a^m

m| =e^iθ^am^(x). Then with out loss of generalityωis single valued inB₂_d_¯. Furthermore ω∈H¹(B₂_d_¯)by

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Proposition 2.3 andu_∗(x)=e^iω(x) onB₂_d_¯. From Corollary 2.1, Proposition 2.3, and as in the proof of Lemma 3.9 of[15], for a subsequence{u}that we do not relabel, there exists a radiusdsuch thatd¯d2d¯and for which

1 2

∂Bd

k|∂_τu|²+ 1 2ε²

1− |u|²2

dσC₁. (2.10)

It follows that{u}converges tou_∗uniformly on∂B_d and weakly inH¹(∂B_d). Note that degu_∗|∂Bd =0. Therefore degu|∂B_d =0 for sufficiently large. Thusu(x)can be expressed asu(x)= |u(x)|e^iω^(x)such that|u(x)| =0 for everyx∈∂B_dwhereωconverges toωuniformly on∂B_d and weakly inH¹(∂B_d)as well.

Define, for each, the functionΦ(r, θ ):=φ(r)(ω(θ )−ω(d, θ )), wheredr= |x− ¯x|, φ(r)=

1 ifr=d, 0 ifr < r< d,

φis a smooth cut-off function and|∇φ|_d₋¹_r, whereris a sequence of radii such thatr→d so that

∂B_d

|ω−ω|²

(r−d) dσ→0.

By construction and the fact thatω ωinH¹(∂B_d), we get thatΦ→0 inH¹(B_d)andΦ→0 uniformly inB_d. Letu¯ minimize

B_dj (u,¯ ∇u) dxin the set{u∈H¹(B_d;S¹): u=u_∗on∂B_d}. Consider the functione^iΦu. Then we¯ havee^iΦu¯→ ¯uinH¹(B_d), and uniformly inB_d. Now, we construct comparison functions

ˆ

u= | ˆu|e^iΦu¯ onBd

where

| ˆu| =1 onB_d₋_ε,

| ˆu| = |u| on∂Bd,

and for eachθ, define | ˆu|(|x − ¯x|, θ )to be linear ford −ε|x− ¯x|d. This gives thatuˆ=u on∂B_d. By construction, we get thatuˆ→ ¯uuniformly inB_d,uˆ→ ¯uinH¹(B_d), andε⁻²

Bd(1− | ˆu|²)²dx→0 as→ ∞.

These limits imply that

lim→0

B_d

j¯ε(uˆ,∇ ˆu) dx=

B_d

j (¯u,¯ ∇ ¯u) dx:=J (u)¯

where

j (u,¯ ∇u)=

k_s|∇u|²+(k_b−k_s)(curlu)² ifk=k_s, k_b|∇u|²+(k_s−k_b)(divu)² ifk=k_b.

This notion forj (u,¯ ∇u)will be used throughout the rest of the work. Then by the lower semicontinuity of the integral

Ωj (u,¯ ∇u) dx,

Bd

j (u¯ _∗,∇u_∗) dxlim inf

→∞

Bd

j¯_ε(u,∇u) dxlim sup

→∞

Bd

j¯_ε(u,∇u) dx

lim

→∞

Bd

j¯_ε(uˆ,∇ ˆu) dx=

Bd

j (¯u,¯ ∇ ¯u) dx

Bd

j (u¯ _∗,∇u_∗) dx.