# On minimizers of the 2D Ginzburg-Landau energy with tangential anchoring

Stan Alama, Lia Bronsard and Lee van Brussel  Department of Mathematics and
Statistics, McMaster University, Hamilton, ON, Canada<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We analyze Ginzburg–Landau minimization problems in two dimensions with either
a “strong or weak” tangential boundary condition. These problems are motivated
by experiments in liquid crystal with boundary defects. In the singular limit
when the correlation length tends to zero, we show that boundary defects will
be observed for weak anchoring, while both boundary and interior vortices are
possible for strong anchoring in the first order limit.

##  1\. Introduction

In this paper we study minimizers of two variational problems motivated by the
study of defects in a nematic liquid crystal. We consider a two-dimensional
setting, related to a thin-film reduction of the three dimensional Landau–de
Gennes model to two dimensions. The special feature we are interested in comes
from the work of Volovik and Lavrentovich [VL83] where nematic drops are
placed in an isotropic medium, allowing for the control of nematic boundary
behaviour. In this way, the liquid crystal and its associated defect dynamics
are studied as the nematic boundary molecules are transformed from having a
forced angle of $\alpha=\pi/2$ with respect to the unit normal $n$ to the
boundary of the droplet to $\alpha=0$. In this paper, we return to the well-
studied Ginzburg-Landau functional but with new tangential types of boundary
conditions, inspired by this physical phenomena.

We begin by describing the variational problem in mathematical terms, and
stating our main results in Theorems 1.1 and 1.2. We consider a two-
dimensional, bounded, simply connected domain
$\Omega\subset\mathbb{R}^{2}\cong\mathbb{C}$ representing the space occupied
by the liquid crystal with $C^{3,\alpha}$-smooth boundary
$\Gamma:=\partial\Omega$. Let $g:\Gamma\to\mathbb{S}^{1}$ be
$C^{3,\alpha}$-smooth boundary data with positive degree

$\mathcal{D}:=\deg(g;\Gamma)>0.$

A natural example is to choose $g$ to parametrize the (positively oriented)
unit tangent vector to $\partial\Omega$, but this need not be the case. In
order to force the order parameter $u$ to be parallel (or close to parallel)
with respect to $g$, we will be using two methods. The first method is to
impose that $u$ have zero projection along a vector orthogonal to $g$, that
is, impose the pointwise scalar product condition $\langle
u,g^{\perp}\rangle=0$ on $\Gamma$. With this, we consider the Ginzburg–Landau
energy defined for $H^{1}(\Omega;\mathbb{R}^{2})$ mappings

$E_{\varepsilon}(u):=\frac{1}{2}\int_{\Omega}\left(|\nabla
u|^{2}+\frac{1}{2\varepsilon^{2}}\left(1-|u|^{2}\right)^{2}\right)dx$

where $\varepsilon>0$ and observe the behaviour of solutions to the _strong
tangential minimization problem_

$\inf\left\\{E_{\varepsilon}(u):u\in\mathcal{H}_{g}(\Omega):=\\{u\in
H^{1}(\Omega;\mathbb{R}^{2}):\langle u,g^{\perp}\rangle=0\ \mbox{on}\
\Gamma\\}\right\\},$ (1.1)

in the limit as $\varepsilon\to 0$. In this way, we can observe the
topological defects associated to the limiting map by analyzing a sequence of
energy minimizing configurations $\\{u_{\varepsilon}\\}$ where the nematic
material is asked to be precisely (strongly) parallel to $g$ along $\Gamma$
for each $\varepsilon>0$. It turns out that asking such a condition to hold
does not quite translate to a standard Dirichlet or Neumann problem for the
associated Euler–Lagrange equations, but rather a mixture of the two within
appropriate coordinates. To see this, we make the additional assumption that
$g$ be defined on a tubular neighborhood

$\mathcal{N}_{\Gamma}:=\\{x\in\overline{\Omega}:\operatorname{dist}(x,\Gamma)<\delta\\}$

with $\delta>0$ is small. Using this assumption, there exists a natural
decomposition for functions $u$ in $\mathcal{N}_{\Gamma}$ using the
orthonormal frame $\\{g(x),g^{\perp}(x)\\}$ via

$u=u_{\parallel}g+u_{\perp}g^{\perp}$ (1.2)

where $u_{\parallel}:=\langle u,g\rangle$ and $u_{\perp}:=\langle
u,g^{\perp}\rangle$. In using this decomposition, we find that solutions to
the strong tangential problem (1.1) satisfy the Euler–Lagrange system

$\left\\{\begin{aligned} -\Delta
u&=\frac{1}{\varepsilon^{2}}(1-|u|^{2})u\quad&&\mbox{in}\ \Omega,\\\
u_{\perp}&=0\quad&&\mbox{on}\ \Gamma,\\\
\partial_{n}u_{\parallel}&=0\quad&&\mbox{on}\ \Gamma.\end{aligned}\right.$
(1.3)

The second method for enforcing parallelity is done through boundary energy
penalization (see e.g. Moser [Mos03]). Indeed, define

$E_{\varepsilon}^{g,s}(u):=E_{\varepsilon}(u)+\frac{1}{2\varepsilon^{s}}\int_{\Gamma}\langle
u,g^{\perp}\rangle^{2}\,ds$

where $s\in(0,1]$, so that solutions of the _weak tangential minimization
problem_

$\inf\left\\{E_{\varepsilon}^{g,s}(u):u\in
H^{1}(\Omega;\mathbb{R}^{2})\right\\}$ (1.4)

are energetically induced to decrease their projection along $g^{\perp}$. By
calculating the first variation for $E_{\varepsilon}^{g,s}$, it can be easily
shown that minimizers $u_{\varepsilon}$ satisfy the weak anchoring system

$\left\\{\begin{aligned} -\Delta
u&=\frac{1}{\varepsilon^{2}}(1-|u|^{2})u\quad&&\mbox{in}\ \Omega,\\\
\partial_{n}u&=-\frac{1}{\varepsilon^{s}}u_{\perp}g^{\perp}\quad&&\mbox{on}\
\Gamma.\end{aligned}\right.$ (1.5)

For either minimization problem, solutions to (1.1) and (1.4) are guaranteed
by the direct method from the calculus of variations. Moreover, it can also be
shown that strong tangential minimizers, in some sense, are weak limits of
solutions to a certain modified weakly tangential minimization problem.
Therefore, both problems are naturally connected and it is reasonable to
analyze the solutions of both.
It is well known from the literature (see Bethuel-Brezis-Hélein [BBH94], for
example) that the local winding behaviour of minimizers about vortices and the
global winding behaviour of the boundary data $g$ are directly linked to the
energy of a minimizing configuration. Thus, given that our interest is in the
observation of boundary defects, we must grasp, in some way, the winding
behaviour of minimizers near boundary vortices. Indeed, when a defect is
located in the interior, this winding is easily quantifiable by calculating
the standard topological degree of the minimizer’s normalization about a small
circle centered at the defect. However, since a closed curve cannot be made
about a boundary vortex, it is not immediately clear how one should proceed in
this case. To combat this, we develop a topological quantity called the
_boundary index_ , which essentially counts the net number of approximate
$\pi$-rotations that are made from one side of the vortex to the other on the
boundary. In this way, a boundary defect with an associated boundary index $d$
will resemble an interior vortex of degree $d$ cut in half, and thus carry a
“half-integer” degree (see Definition (4.3)). A rigorous construction of the
boundary index is given in Section 4. Using the notion of the boundary index,
the main results of this paper are summarized in Theorems 1.1 and 1.2.

######  Theorem 1.1.

Suppose $\\{u_{\varepsilon}\\}_{\varepsilon>0}$ is a sequence of solutions to
(1.1) with associated boundary function
$g:\mathcal{N}_{\Gamma}\to\mathbb{S}^{1}$ of degree
$\mathcal{D}=\deg(g;\Gamma)\geq 1$. Then there is a subsequence
$\varepsilon_{n}\to 0$, a finite number of point singularities
$\Sigma\subset\overline{\Omega}$ and a harmonic map $u_{0}\in
H^{1}(\overline{\Omega}\setminus\Sigma;\mathbb{S}^{1})$ such that

$u_{\varepsilon_{n}}\rightharpoonup u_{0}\quad\mbox{weakly in}\
H^{1}_{loc}(\overline{\Omega}\setminus\Sigma;\mathbb{R}^{2})$

with each defect contained in $\Sigma$ having either associated degree, or
boundary index, equal to one. In the particular case where $\mathcal{D}=1$,
then one and only one of the following scenarios hold:

  1. (1)

$\Sigma=\\{p\\}$ with $p\in\Omega$,

  2. (2)

$\Sigma=\\{q_{1},q_{2}\\}$ with $q_{1},q_{2}\in\Gamma$.

For the last part of Theorem 1.1, we remind the reader that our primary
motivation for studying this problem came from the topological observations
made on 3D samples, by Volovik and Lavrentovich in [VL83]. In particular, they
found experimentally single interior hedgehog defect when molecules are asked
to be normal to the boundary and a bipolar boojum pair when requiring
tangential conditions. In either case, the normal and tangential boundary data
are of degree one and thus our theoretical treatment of the problem weakly
recovers this observation. To get a complete picture, a renormalized energy
analysis would need to be conducted in order to show when one defect type is
preferred over another. To this end, in the last section of this work, we
provide a concrete example of strong tangential anchoring in the case
$\Omega=B_{1}(0)$, the unit disc, with $g=\tau$ the positively oriented unit
tangent vector to the boundary $\Gamma$, to highlight that the boundary vortex
pair may give the preferable energy minimizing configuration. Such a result in
2D would be a first step in obtaining theoretically results coinciding with
the found experimental data.

Next we state our result in case of weak tangential boundary conditions:

######  Theorem 1.2.

Suppose $\\{u_{\varepsilon}\\}_{\varepsilon>0}$ is a sequence of solutions to
(1.4) with associated boundary function
$g:\mathcal{N}_{\Gamma}\to\mathbb{S}^{1}$ of degree
$\mathcal{D}=\deg(g;\Gamma)\geq 1$. Then there is a subsequence
$\varepsilon_{n}\to 0$, a finite number of point singularities
$\Sigma\subset\overline{\Omega}$ and a harmonic map $u_{0}\in
H^{1}(\overline{\Omega}\setminus\Sigma;\mathbb{S}^{1})$ such that

$u_{\varepsilon_{n}}\rightharpoonup u_{0}\quad\mbox{weakly in}\
H^{1}_{loc}(\overline{\Omega}\setminus\Sigma;\mathbb{R}^{2})$

with each defect contained in $\Sigma$ having associated degree or boundary
index equal to one. If $s\in(0,1)$, it holds that $\Sigma\subset\Gamma$ with
$|\Sigma|=2\mathcal{D}$.

The primary takeaway of Theorem 1.2 comes from the observation that the
exponent $s\in(0,1]$ _almost_ completely dictates the allocation of defects in
$\overline{\Omega}$. In particular, the Theorem states that independent of the
winding behaviour of $g$ and the geometry of $\Omega$, giving vortices ‘more
room’ along the boundary (on the scale of $\varepsilon^{s}$ as opposed to
$\varepsilon$) is enough for boundary vortex pairs to _always_ be
energetically preferable when compared to interior vortices.

A related model is that of a thin ferromagnetic film as obtained in an
appropriate limiting regime by DeSimone, Kohn, Muller and Otto [DKMO02]. This
limiting ferromagnetic thin film was studied by Moser [Mos03], and by Kurzke
[Kur06] in certain settings. In those problems, they impose tangential weak
anchoring conditions with $g=\tau$ the unit tangent, and find critical
anchoring strength at which boundary vortices are favored over interior
vortices. In our case, we also consider a strong tangential anchoring, and
generalize their results to weak anchoring for any $g$. More recently, Ignat-
Kurzke [IK21] have obtained $\Gamma$-convergence results for the weak
tangential anchoring problem using a notion of global Jacobian in a different
limiting regime where interior vortices cost more energy than boundary
vortices. In the context of polarization-modulated orthogonal smectic liquid
crystal, Garcia-Cervera, Giorgi and Joo [GCGJ20] have studied boundary
vortices in a square domain with mixed weak and strong boundary conditions on
the edges.

In the work of Volovik and Lavrentovich [VL83] the topological dynamics of the
nematic material are observed as the boundary molecules are changed from being
parallel to the boundary to perpendicular, by varying the aperture $\alpha$ of
the cone formed by the molecule with the normal to the boundary. When the
angle $\alpha=\pi/2$, a bipolar structure is noticed with two point defects
occurring along the boundary called _boojums_. At the other extreme with
$\alpha=0$, a single interior hedgehog defect is realized. Using a vector-
valued order parameter $u$ for modelling the molecular alignment of the liquid
crystal, the authors note that a surface energy density proportional to

$[\langle u,n\rangle^{2}-\cos^{2}\alpha]^{2}$ (1.6)

can be used in comparison to an interior gradient energy for determining the
energy preference of boojum defects to hedgehog defects and vice versa. A
generalization of this setup was analyzed by Alama, Bronsard and Golovaty in
[ABG20] where they replaced the boundary’s normal vector $n$ in expression
(1.6) with general smooth $\mathbb{S}^{1}$-valued boundary data $g$,
possessing a positive associated winding number along the boundary, and
restricting $\alpha\in(0,\pi/2)$, the relative angle made between $g$ and the
order parameter $u$. In this present work, we aim to answer the question of
how this generalization operates for the specific case of $\alpha=\pi/2$ using
Ginzburg-Landau as a toy model for nematic material. In particular, we are
interested in obtaining conditions for which minimizing configurations prefer
boundary defects over interior defects in this setting.

The rest of the paper is organized as follows: in Section 2 we present upper
bounds for the energy of minimizers to each problem, as well as a priori
pointwise bounds for all solutions of the associated Euler-Lagrange equations,
adapted for our new settings. In Section 3, we present our $\eta$ compactness
results adapted from Struwe [Str94] to handle each type of boundary conditions
and use it to define the “bad balls” for each type, and show that they are
contained in a finite number of very small balls. Next in Section 4, we
analyze the winding behaviour of minimizers around boundary defects and
introduce our notion of boundary index and use it to obtain the important
“degree Proposition and Lemma” (Proposition 4.1 and Lemma 4.2) which will be
essential in proving the lower bound on the energy of boundary defects in
terms of the degree of the boundary data. In Section 5, we obtain an energy
lower bound for each type of tangential conditions over the appropriate ball
collections, and we put everything together and prove our two main theorems.
Finally in Section 6, we present an example with strong tangential anchoring
on the unit disc which suggest that two antipodal boundary defect would be
favored.

##  2\. Preliminary Facts for Minimizers

We begin by showing an important pointwise bound on solutions of the
Euler–Lagrange systems (1.3) and (1.5). The proof follows familiar lines, (see
[BBH94, ABGS15, ABG20]) and so we provide a sketch to highlight the
differences with previous papers.

######  Lemma 2.1.

Suppose $u$ is a solution of (1.3) or (1.5). Then $|u|\leq 1$ and there is a
constant $C_{0}>0$ independent of $\varepsilon$ for which $\varepsilon|\nabla
u|\leq C_{0}$ for all $x\in\Omega$.

###### Proof.

Define $V:=|u|^{2}-1$ and $V_{+}:=\max\\{V,0\\}$. Using the Euler–Lagrange
equations and integrating by parts over $\Omega$ we obtain

$0\leq\int_{\Omega}|u|^{2}V_{+}^{2}\,dx\leq\frac{1}{2}\int_{\Gamma}V_{+}\partial_{n}V\,ds-\frac{1}{2}\int_{\Omega}|\nabla
V_{+}|^{2}\,dx.$

If $u$ is a solution of (1.3), it follows that $V_{+}\equiv 0$. If $u$ is a
solution of (1.5), then

$\partial_{n}V=2\langle
u,\partial_{n}u\rangle=-\frac{2}{\varepsilon^{s}}(u_{\perp})^{2}\leq 0$

and we obtain $V_{+}\equiv 0$ again. Thus, $|u|\leq 1$ in $\Omega$.

The gradient bound can be obtained by contradiction: suppose that there exists
sequences $\varepsilon_{k}\to 0$ and $x_{k}\in\overline{\Omega}$ so that
$t_{k}:=|\nabla u_{k}(x_{k})|=\|\nabla u_{k}\|_{\infty}$ satisfies
$t_{k}\varepsilon_{k}\to\infty$ as $k\to\infty$. Let

$v_{k}(x):=u_{k}\left(x_{k}+\frac{x}{t_{k}}\right)$

which is defined whenever $y:=x_{k}+x/t_{k}\in\Omega$. Likewise, define
$h(x):=g(y)$ whenever $y\in\Gamma$. By the uniform bound on $u$ proven above
and the choice of scaling, we have $\|v_{k}\|_{\infty}\leq 1$ and

$|\nabla v_{k}(0)|=1\ \forall k.$ (2.1)

By the uniform bound $\|v_{k}\|_{\infty}\leq 1$ and using the fact that
$v_{k}$ solves

$-\Delta v_{k}=\frac{1}{(t_{k}\varepsilon_{k})^{2}}(1-|v_{k}|^{2})v_{k}\quad\
\mbox{for}\ x\in t_{k}[\Omega-x_{k}],$

we conclude $\Delta v_{k}\to 0$ uniformly on $\Omega$. There are two blow-up
cases to consider. If along some subsequence
$t_{k}\operatorname{dist}(x_{k},\Gamma)\to+\infty$, then $v_{k}\to v$ with $v$
bounded and harmonic in all of $\mathbb{R}^{2}$, and hence constant,
contradicting (2.1).

Suppose now $t_{k}\operatorname{dist}(x_{k},\Gamma)$ is bounded uniformly so
that the domains of $v_{k}$ converge to the half-space

$t_{k}[\Omega-x_{k}]\to\mathbb{R}^{2}_{+}\ \mbox{as}\ k\to\infty.$

For each $k$, the weak tangential problem becomes

$\left\\{\begin{aligned} -\Delta
v_{k}&=\frac{1}{(t_{k}\varepsilon_{k})^{2}}(1-|v_{k}|^{2})v_{k}\quad&&\mbox{in}\
t_{k}[\Omega-x_{k}],\\\\[5.0pt]
\partial_{n}v_{k}&=-\frac{1}{t_{k}\varepsilon_{k}^{s}}\langle
v_{k},h^{\perp}\rangle h^{\perp}\quad&&\mbox{on}\ t_{k}[\Gamma-
x_{k}],\end{aligned}\right.$ (2.2)

while the strong tangential problem is written

$\left\\{\begin{aligned} -\Delta
v_{k}&=\frac{1}{(t_{k}\varepsilon_{k})^{2}}(1-|v_{k}|^{2})v_{k}\quad&&\mbox{in}\
t_{k}[\Omega-x_{k}],\\\\[5.0pt] \langle
v_{k},h^{\perp}\rangle&=0\quad&&\mbox{on}\ t_{k}[\Gamma-x_{k}],\\\\[5.0pt]
\partial_{n}\langle v_{k},h\rangle&=0\quad&&\mbox{on}\ t_{k}[\Gamma-
x_{k}].\end{aligned}\right.$ (2.3)

Both problems yield a bounded harmonic limit $v$ defined on
$\mathbb{R}^{2}_{+}$ with $v_{k}\to v$ in $C^{k}_{loc}$. Since $v_{k}$ and $h$
are bounded uniformly, the normal derivative of system (2.2) has the limit
$\partial_{n}v_{k}\to 0$ as $k\to\infty$ and so the limiting harmonic map $v$
satisfies the Neumann condition $\partial_{n}v=0$ on
$\partial\mathbb{R}^{2}_{+}$. By the reflection principle, there exists a
bounded harmonic extension of $v$ to all of $\mathbb{R}^{2}$ which again
contradicts (2.1) by Liouville’s theorem.

For system (2.3), the boundary data $h$ converges to a constant vector field
on $\partial\mathbb{R}^{2}_{+}$ and the boundary conditions imply $\langle
v,h^{\perp}\rangle=\partial_{n}\langle v,h\rangle=0$ along
$\partial\mathbb{R}^{2}_{+}$. Let $\tilde{h}$ denote the extension of $h$ to
$\overline{\mathbb{R}^{2}_{+}}$ and note that $\langle v,\tilde{h}\rangle$ is
a harmonic scalar function defined on $\mathbb{R}^{2}_{+}$. By the reflection
principle and Liouville’s theorem, $\langle v,\tilde{h}\rangle$ extends to a
constant function on $\mathbb{R}^{2}$. Next, since $\langle
v,h^{\perp}\rangle=0$ on $\partial\mathbb{R}^{2}_{+}$ and $\langle
v,\tilde{h}\rangle$ is constant, it must be that $\langle
v,\tilde{h}\rangle=\pm|v|$ on all of $\mathbb{R}^{2}$ and thus $v$ is
constant. Condition (2.1) is contradicted once again giving
$\varepsilon|\nabla u|\leq C_{0}$ for all $x\in\Omega$ where $C_{0}$ is a
constant independent of $\varepsilon$. ∎

######  Proposition 2.2.

If $u_{\varepsilon}$ is a strongly tangential minimizer for $E_{\varepsilon}$,
then

$E_{\varepsilon}(u_{\varepsilon})\leq\pi\mathcal{D}|\ln\varepsilon|+C$ (2.4)

with $C>0$ a constant independent of $\varepsilon$. If $u_{\varepsilon}$ is a
weakly tangential minimizer for $E_{\varepsilon}^{g,s}$, then there is a
constant $C>0$ independent of $\varepsilon$ so that

$E_{\varepsilon}^{g,s}(u_{\varepsilon})\leq\pi
s\mathcal{D}|\ln\varepsilon|+C.$ (2.5)

The proof of this proposition utilizes a local polar coordinate system near
the boundary which is defined in [ABGS15, ABG20, Kur06] and which we will use
throughout the paper. For convenience, we provide a brief description here.
Let $x_{0}\in\overline{\Omega}$ and $R>0$. Set

$\omega_{R}(x_{0}):=B_{R}(x_{0})\cap\Omega$

and in the case where $x_{0}\in\Gamma$, define

$\Gamma_{R}(x_{0}):=\overline{\omega_{R}(x_{0})}\cap\Gamma.$

Whenever $x_{0}\in\Gamma$, $\tau(x_{0})$ will denote the positively oriented
unit tangent vector to $\Gamma$ at $x_{0}$. Using $\tau(x_{0})$ as a
reference, the polar coordinates $(r,\theta)$ centered at $x_{0}$ can be
defined so that $\theta$ is the angle measured from the ray defined by
$\tau(x_{0})$ and $r=|x-x_{0}|$. By the smoothness of $\Gamma$, note that $R$
can be chosen small enough so that

$\displaystyle\omega_{R}(x_{0})=\left\\{(r,\theta):\theta_{1}(r)<\theta<\theta_{2}(r),\
0<r<R\right\\}$

where $\theta_{1}(r)$ and $\theta_{2}(r)$ are smooth functions satisfying

$\displaystyle|\theta_{1}(r)|,\ |\pi-\theta_{2}(r)|\leq cr$ (2.6)

for some constant $c=c(\Gamma)\geq 0$. These coordinates allow us to
parametrize $\Gamma_{R}(x_{0})\setminus\\{x_{0}\\}$ in two pieces:

$\Gamma_{R}^{+}(x_{0}):=\\{(r,\theta_{1}(r)):0<r<R\\},\quad\Gamma_{R}^{-}(x_{0}):=\\{(r,\theta_{2}(r)):0<r<R\\}.$

Annular regions are defined similarly. For any $x_{0}\in\overline{\Omega}$ set

$A_{r_{1},r_{2}}(x_{0}):=\omega_{r_{2}}(x_{0})\setminus\overline{\omega_{r_{1}}(x_{0})},\quad
0<r_{1}<r_{2}.$

When $x_{0}\in\Gamma$ and $r_{2}>0$ is taken small enough, the intersection
$\overline{A_{r_{1},r_{2}}(x_{0})}\cap\Gamma$ consists of two disjoint smooth
arcs

$\begin{split}\Gamma_{r_{1},r_{2}}^{+}(x_{0})&=\\{(r,\theta_{1}(r)):r_{1}<r<r_{2}\\},\\\
\Gamma_{r_{1},r_{2}}^{-}(x_{0})&=\\{(r,\theta_{2}(r)):r_{1}<r<r_{2}\\},\end{split}$
(2.7)

where $\theta_{1}(r)$ and $\theta_{2}(r)$ are as in (2.6). For notational
convenience, we also set

$\Gamma_{r_{1},r_{2}}^{\pm}(x_{0}):=\overline{A_{r_{1},r_{2}}(x_{0})}\cap\Gamma=\Gamma_{r_{1},r_{2}}^{+}(x_{0})\cup\Gamma_{r_{1},r_{2}}^{-}(x_{0}).$

Lastly, we define a localized energy on subsets $\omega_{r}(x_{0})$ by

$\displaystyle E_{\varepsilon}(u;\omega_{r})$
$\displaystyle:=\frac{1}{2}\int_{\omega_{r}}\left(|\nabla
u|^{2}+\frac{1}{2\varepsilon^{2}}(1-|u|^{2})^{2}\right)dx,$ $\displaystyle
E_{\varepsilon}^{g,s}(u;\omega_{r})$
$\displaystyle:=E_{\varepsilon}(u;\omega_{r})+\frac{1}{2\varepsilon^{s}}\int_{\Gamma\cap\overline{\omega_{r}}}\langle
u,g^{\perp}\rangle^{2}\,ds.$

###### Proof of Proposition 2.2.

For inequality (2.4), the desired bound is a consequence of [Str94, Lemma
2.1]. Let $v_{\varepsilon}$ be a minimizer for $E_{\varepsilon}$ over
$H^{1}_{g}(\Omega)=\\{v\in H^{1}(\Omega;\mathbb{R}^{2}):v=g\ \mbox{on}\
\Gamma\\}$. The inclusion $H^{1}_{g}(\Omega)\subset\mathcal{H}_{g}(\Omega)$
implies $E_{\varepsilon}(u_{\varepsilon})\leq
E_{\varepsilon}(v_{\varepsilon})$ and applying [Str94, Lemma 2.1] to
$E_{\varepsilon}(v_{\varepsilon})$ yields

$E_{\varepsilon}(u_{\varepsilon})\leq
E_{\varepsilon}(v_{\varepsilon})\leq\pi\mathcal{D}|\ln\varepsilon|+C.$

For weakly tangential minimizers, a test function is constructed following
[ABGS15, Lemma 3.1] and [Kur06, Proposition 3.1]. Consider $2\mathcal{D}$ sets
of the form $\omega_{R}(q_{j})$ where $\\{q_{j}\\}_{j=1}^{2\mathcal{D}}$ are
well-separated points on $\Gamma$ and $R$ is chosen so that

$2\varepsilon^{s}<R<\frac{1}{2}|q_{i}-q_{i}|$

for all indices $i\neq j$. Assume that the points
$\\{q_{j}\\}_{j=1}^{2\mathcal{D}}$ are labeled such that $q_{j+1}$ is the
first point found by following the positively oriented tangent vector field
along $\Gamma$ starting from $q_{j}$. These points partition $\Gamma$ into
$2\mathcal{D}$ smooth segments $C_{j}$ in the sense that
$\Gamma=\cup_{j=1}^{2\mathcal{D}}C_{j}$ with $C_{j}$ being the curve
connecting $q_{j}$ and $q_{j+1}$. Let $\gamma$ be a lifting of $g$ on the
curve $\Gamma_{R}(q_{j})$, that is, $g=e^{i\gamma}$ on $\Gamma_{R}(q_{j})$,
and define

$\displaystyle h_{1}(r)$
$\displaystyle=\gamma\left(re^{i\theta_{1}(r)}\right)+(j-1)\pi,$
$\displaystyle h_{2}(r)$
$\displaystyle=\gamma\left(re^{i\theta_{2}(r)}\right)+j\pi,$
$\displaystyle\phi(r,\theta)$
$\displaystyle=\frac{h_{2}(r)-h_{1}(r)}{\theta_{2}(r)-\theta_{1}(r)}(\theta-\theta_{1}(r))+h_{1}(r),$

where $\theta_{1}(r)$ and $\theta_{2}(r)$ are as in (2.6). In this way we have
$e^{i\phi(r,\theta)}=g$ on $\Gamma_{R}^{+}(q_{j})$ for $j$ odd and on
$\Gamma_{R}^{-}(q_{j})$ for $j$ even. Similarly, we get
$e^{i\phi(r,\theta)}=-g$ for the opposite parities. Next, let
$\eta_{\varepsilon}(r)\in C^{\infty}$ be a cut-off function near $q_{j}$
satisfying $0\leq\eta_{\varepsilon}\leq 1$ for all $r$,
$\eta_{\varepsilon}(r)=1$ for $r\geq 2\varepsilon^{s}$ and
$\eta_{\varepsilon}(r)=0$ for $r<\varepsilon^{s}$. In setting

$\psi_{j}(r,\theta):=\eta_{\varepsilon}(r)\phi(r,\theta)+(1-\eta_{\varepsilon}(r))(\gamma(q_{j})+(j-1)\pi)$

we may define the $\mathbb{S}^{1}$-valued test function
$v_{\varepsilon}^{(j)}=e^{i\psi_{j}(r,\theta)}$ on $\omega_{R}(q_{j})$, which
by construction, simulates a half-vortex in the annular region
$A_{2\varepsilon^{s},R}(q_{j})$. Using the properties of the cut-off function
along with Cauchy-Schwarz and the fact that $v_{\varepsilon}^{(j)}$ is
$\mathbb{S}^{1}$-valued,

$\displaystyle\frac{1}{2\varepsilon^{2}}\int_{\omega_{R}(q_{j})}(1-|v_{\varepsilon}|^{2})^{2}\,dx=0\quad\mbox{and}\quad\frac{1}{2\varepsilon^{s}}\int_{\Gamma_{R}(q_{j})}\langle
v_{\varepsilon},g^{\perp}\rangle^{2}\,ds\leq C$

where $C>0$ is a constant independent of $\varepsilon$. The Dirichlet energy
of $v_{\varepsilon}^{(j)}$ on $\omega_{R}(q_{j})$, can be conveniently
estimated using polar coordinates: it is a straightforward calculation to
confirm that the radial component
$\int_{\omega_{R}(q_{j})}|\partial_{r}v_{\varepsilon}|^{2}\ dx$ is uniformly
bounded in $\varepsilon$. To bound the angular energy over
$\omega_{2\varepsilon^{s}}(q_{j})$, we note that by (2.6) and the smoothness
of $\gamma$, there is a constant $c>0$ so that

$\displaystyle|h_{2}(r)-h_{1}(r)|\leq\pi+cr,\quad\mbox{and}\quad|\theta_{2}(r)-\theta_{1}(r)|\geq\pi-
cr.$

Thus, we have:

$\int_{\omega_{2\varepsilon^{s}}(q_{j})}\frac{1}{r^{2}}|\partial_{\theta}v|^{2}\,dx\leq\frac{(\pi+cR)^{2}}{(\pi-
cR)}\ln 2<\infty.$

Therefore it must be that the primary energy contribution comes from the
subset $A_{2\varepsilon^{s},R}(q_{j})$. Using the same estimates as above,

$\int_{\omega_{R}(q_{j})}\frac{1}{r^{2}}|\partial_{\theta}v_{\varepsilon}|^{2}\,dx\leq\pi
s|\ln\varepsilon|+C$

for $C>0$ independent of $\varepsilon$ and so

$E_{\varepsilon}^{g,s}(v_{\varepsilon};\omega_{R}(q_{j}))\leq\frac{\pi}{2}s|\ln\varepsilon|+C$

on each $\omega_{R}(q_{j})$. Finally, we must connect these local test
functions in a way that is independent of $\varepsilon$. Consider the
punctured domain

$\tilde{\Omega}:=\Omega\setminus\bigcup_{j=1}^{2\mathcal{D}}\omega_{R}(q_{j})$

with boundary given by

$\tilde{\Gamma}:=\partial\tilde{\Omega}=\left(\Gamma\setminus\cup_{j=1}^{2\mathcal{D}}\Gamma_{R}(q_{j})\right)\bigcup\left(\cup_{j=1}^{2\mathcal{D}}\partial
B_{R}(q_{j})\cap\Omega\right).$

We set the orientation of $\tilde{\Gamma}$ to match that of $\Gamma$ where
they coincide. With this orientation, the function
$\tilde{g}:\tilde{\Gamma}\to\mathbb{S}^{1}$ defined by

$\displaystyle\tilde{g}:=\left\\{\begin{array}[]{cl}g&\mbox{on}\
\tilde{\Gamma}\cap C_{j}\ \mbox{for}\ j\ \mbox{odd}\\\\[5.0pt] -g&\mbox{on}\
\tilde{\Gamma}\cap C_{j}\ \mbox{for}\ j\ \mbox{even}\\\\[5.0pt]
v_{\varepsilon}^{(j)}&\mbox{on}\ \partial B_{R}(q_{j})\cap\Omega\ \mbox{for
each}\ j=1,\ldots,2\mathcal{D},\end{array}\right.$

satisfies $\deg(\tilde{g};\tilde{\Gamma})=0$ by construction and therefore we
may let $V$ be the $\mathbb{S}^{1}$-valued harmonic extension of $\tilde{g}$
to $\tilde{\Omega}$ which has uniformly bounded Dirichlet energy. Setting

$\displaystyle H_{\varepsilon}=\left\\{\begin{array}[]{ll}V&\mbox{in}\
\tilde{\Omega},\\\\[5.0pt] v_{\varepsilon}^{(j)}&\mbox{in}\ \omega_{R}(q_{j})\
\mbox{for each}\ j=1,\ldots,2\mathcal{D},\end{array}\right.$

we obtain a bound on $E_{\varepsilon}^{g,s}(u_{\varepsilon})$ via

$E_{\varepsilon}^{g,s}(u_{\varepsilon})\leq
E_{\varepsilon}^{g,s}(H_{\varepsilon})=\sum_{j=1}^{2\mathcal{D}}E_{\varepsilon}^{g,s}(v_{\varepsilon}^{(j)};\omega_{R}(q_{j}))+E_{\varepsilon}^{g,s}(V;\tilde{\Omega})\leq\pi
s\mathcal{D}|\ln\varepsilon|+C$

as desired. ∎

##  3\. $\eta$-Compactness and Related Consequences

In this section, we prove an $\eta$-compactness result which allows one to
relate an energy bound to the non-existence of vortices. The idea here is that
for two concentric balls, if the energy on the larger ball is small enough,
then it is impossible for a vortex to exist in the smaller ball. This fact is
pivotal in proving that the set of points $x\in\overline{\Omega}$ for which
$|u_{\varepsilon}|$ is small can be covered by a finite set of
$\varepsilon$-balls whose number is bounded independent of $\varepsilon$. We
begin by stating a Pohosaev-type identity for solutions of (1.3) or (1.5)
which is obtained via integrating by parts against a smooth function. This
identity will be needed to obtain an $\eta$-compactness result which will be
developed in the next section.

Define

$e_{\varepsilon}(u):=\frac{1}{2}|\nabla
u|^{2}+\frac{1}{4\varepsilon^{2}}\left(1-|u|^{2}\right)^{2}.$

######  Proposition 3.1.

Let $\psi\in C^{2}(\Omega;\mathbb{R}^{2})$. If $u$ is a solution of (1.5) or
(1.3), then

$\int_{\partial\omega_{r}}\left\\{e_{\varepsilon}(u)\langle\psi,n\rangle-\langle\partial_{n}u,\psi\cdot\nabla
u\rangle\right\\}ds\\\
=\int_{\omega_{r}}\left\\{e_{\varepsilon}(u)\operatorname{div}\psi-\sum_{j,l}\psi_{x_{j}}^{l}\langle\partial_{x_{j}}u,\partial_{x_{l}}u\rangle\right\\}dx.$
(3.1)

######  Theorem 3.2 ($\eta$-Compactness).

[Strong Tangental Case] Let $\frac{3}{4}\leq\beta<\gamma<1$. There exists
constants $\eta$, $\tilde{C}$, $\varepsilon_{0}>0$ such that for any solution
$u_{\varepsilon}$ of (1.3) with $\varepsilon\in(0,\varepsilon_{0})$, if
$x_{0}\in\overline{\Omega}$ and
$E_{\varepsilon}(u_{\varepsilon};\omega_{2\varepsilon^{\beta}}(x_{0}))\leq\eta|\ln\varepsilon|$,
then

$\displaystyle|u_{\varepsilon}|\geq\frac{1}{2}\quad\mbox{in}\
\omega_{\varepsilon^{\gamma}}(x_{0}),$ (3.2)
$\displaystyle\frac{1}{4\varepsilon^{2}}\int_{\omega_{\varepsilon^{\gamma}}(x_{0})}(1-|u_{\varepsilon}|^{2})^{2}\,dx\leq\tilde{C}\eta.$
(3.3)

[Weak Tangential Case] Let $\frac{3}{4}s\leq\beta<\gamma<s\leq 1$. There
exists constants $\eta$, $\tilde{C}$, $\varepsilon_{0}>0$ such that for any
solution $u_{\varepsilon}$ of (1.5) with $\varepsilon\in(0,\varepsilon_{0})$,
if $x_{0}\in\overline{\Omega}$ and
$E_{\varepsilon}^{g,s}(u_{\varepsilon};\omega_{2\varepsilon^{\beta}}(x_{0}))\leq\eta|\ln\varepsilon|$,
then

$\displaystyle|u_{\varepsilon}|\geq\frac{1}{2}\quad\mbox{in}\
\omega_{\varepsilon^{\gamma}}(x_{0}),$ (3.4) $\displaystyle|\langle
u_{\varepsilon},g^{\perp}\rangle|\leq\frac{1}{4}\quad\mbox{on}\
\Gamma\cap\overline{\omega_{\varepsilon^{\gamma}}(x_{0})},$ (3.5)
$\displaystyle\frac{1}{4\varepsilon^{2}}\int_{\omega_{\varepsilon^{\gamma}}(x_{0})}(1-|u_{\varepsilon}|^{2})^{2}\,dx+\frac{1}{2\varepsilon^{s}}\int_{\Gamma\cap\overline{\omega_{\varepsilon^{\gamma}}(x_{0})}}\langle
u_{\varepsilon},g^{\perp}\rangle^{2}\,ds\leq\tilde{C}\eta.$ (3.6)

The proof for Theorem 3.2 is heavily dependent on a crucial estimate. For
$x_{0}\in\overline{\Omega}$, define as in [Str94, Mos03] the functions

$\displaystyle F(r)=F(r;x_{0},u,\varepsilon):=r\int_{\partial
B_{r}(x_{0})\cap\Omega}e_{\varepsilon}(u)\,ds,\quad
F_{\Gamma}(r):=F(r)+\frac{r}{2\varepsilon^{s}}\sum_{x\in\partial\Gamma_{r}(x_{0})}\langle
u,g^{\perp}\rangle^{2}$

where the second function above is defined when $x_{0}\in\Gamma$.

######  Lemma 3.3.

Let $x_{0}\in\overline{\Omega}$. There exists constants $C>0$ and $r_{0}>0$
such that for $\varepsilon\in(0,1)$ and $r\in(0,r_{0})$ we have:

  1. (1)

If $x_{0}\in\Omega$, $\overline{\omega_{r}(x_{0})}\cap\Gamma=\varnothing$ and
$u$ is a solution of either (1.5) or (1.3), then

$\frac{1}{4\varepsilon^{2}}\int_{\omega_{r}}(1-|u|^{2})^{2}\,dx\leq
r\int_{\omega_{r}}\frac{1}{2}|\nabla u|^{2}dx+F(r).$

  2. (2)

If $x_{0}\in\Gamma$ and $u$ is a solution of (1.3), then

$\frac{1}{4\varepsilon^{2}}\int_{\omega_{r}}(1-|u|^{2})^{2}\,dx\leq
C\left[r\int_{\omega_{r}}\frac{1}{2}|\nabla
u|^{2}\,dx+F(r)+\frac{r^{2}}{\varepsilon}\right].$ (3.7)

  3. (3)

If $x_{0}\in\Gamma$ and $u$ is a solution of (1.5), then

$\frac{1}{4\varepsilon^{2}}\int_{\omega_{r}}(1-|u|^{2})^{2}\,dx+\frac{1}{2\varepsilon^{s}}\int_{\Gamma_{r}}\langle
u,g^{\perp}\rangle^{2}\,ds\leq C\left[r\int_{\omega_{r}}\frac{1}{2}|\nabla
u|^{2}\,dx+F_{\Gamma}(r)+\frac{r^{2}}{\varepsilon^{s}}\right].$ (3.8)

###### Proof.

The proof for case (1) is shown in [Str94, Lemma 2]. Inequality (3.8) follows
from [Mos03, Lemma 5.4] or by changing every instance of $|u-g|^{2}$ with
$\langle u,g^{\perp}\rangle^{2}$ throughout [ABGS15, Lemma 4.2]. Thus, it only
remains to prove (3.7). Let $r_{0}>0$ be chosen small enough so that
$\Gamma\cap B_{r}(x_{0})$ consists of a single smooth arc satisfying
$|\Gamma_{r}|\leq Cr$ for all $0<r\leq r_{0}$. As in [ABGS15] we let
$\mathcal{N}$ be a $2r_{0}$-neighbourhood of $\Gamma$, and by taking $r_{0}$
smaller if necessary, there exists a vector field $X\in
C^{2}(\mathcal{N};\mathbb{R}^{2})$ satisfying

$\displaystyle\langle X,n\rangle=X_{n}=0\quad\mbox{for all}\ x\in\Gamma_{r},$
(3.9) $\displaystyle|X-(x-x_{0})|\leq C|x-x_{0}|^{2}\quad\mbox{for all}\
x\in\omega_{r},$ (3.10) $\displaystyle|\partial_{x_{i}}X^{j}-\delta_{ij}|\leq
C|x-x_{0}|\quad\mbox{for all}\ x\in\omega_{r},$ (3.11)

for a constant $C>0$ and for any $x_{0}\in\Gamma$. To obtain inequality (3.7)
we consider the Pohosaev-type identity (3.1) with $\psi=X$ and estimate. Using
the boundary decomposition $\partial\omega_{r}=\Gamma_{r}\cup(\partial
B_{r}(x_{0})\cap\Omega)$, it will be convenient to perform these estimates on
$\Gamma_{r}$ and $\partial B_{r}(x_{0})\cap\Omega$ separately.

Estimates Along $\Gamma_{r}$:
By (3.9) we may write $X=\langle X,\tau\rangle\tau=X_{\tau}\tau$ where $\tau$
is the unit tangent vector to $\Gamma_{r}$ and so $X\cdot\nabla
u=X_{\tau}\partial_{\tau}u$ on $\Gamma_{r}$. With this, the lefthand side of
(3.1) becomes

$\int_{\Gamma_{r}}\left\\{e_{\varepsilon}(u)X_{n}-\langle\partial_{n}u,X\cdot\nabla
u\rangle\right\\}ds=-\int_{\Gamma_{r}}\langle\partial_{n}u,X_{\tau}\partial_{\tau}u\rangle\,ds.$

Using the derivative representations

$\displaystyle\partial_{n}u$
$\displaystyle=\partial_{n}(u_{\parallel}g+u_{\perp}g^{\perp})=u_{\parallel}\partial_{n}g+\partial_{n}u_{\parallel}g+u_{\perp}\partial_{n}g^{\perp}+\partial_{n}u_{\perp}g^{\perp},$
$\displaystyle\partial_{\tau}u$
$\displaystyle=\partial_{\tau}(u_{\parallel}g+u_{\perp}g^{\perp})=u_{\parallel}\partial_{\tau}g+\partial_{\tau}u_{\parallel}g+u_{\perp}\partial_{\tau}g^{\perp}+\partial_{\tau}u_{\perp}g^{\perp},$

with the known conditions
$u_{\perp}=\partial_{n}u_{\parallel}=\partial_{\tau}u_{\perp}=0$, we obtain

$\displaystyle\langle\partial_{n}u,X_{\tau}\partial_{\tau}u\rangle$
$\displaystyle=X_{\tau}\langle
u_{\parallel}\partial_{n}g+\partial_{n}u_{\perp}g^{\perp},u_{\parallel}\partial_{\tau}g+\partial_{\tau}u_{\parallel}g\rangle$
$\displaystyle=X_{\tau}((u_{\parallel})^{2}\langle\partial_{n}g,\partial_{\tau}g\rangle+u_{\parallel}\partial_{n}u_{\perp}\langle
g^{\perp},\partial_{\tau}g\rangle).$

Applying Lemma 2.1 and Cauchy-Schwarz,

$\displaystyle|(u_{\parallel})^{2}\langle\partial_{n}g,\partial_{\tau}g\rangle|$
$\displaystyle\leq|\partial_{n}g||\partial_{\tau}g|\leq|\nabla g|^{2}\leq
C=C(g),$ $\displaystyle|u_{\parallel}\partial_{n}u_{\perp}\langle
g^{\perp},\partial_{\tau}g\rangle|$
$\displaystyle\leq|\partial_{n}u_{\perp}||g^{\perp}||\partial_{\tau}g|\leq|\nabla
u||\nabla g|\leq CC_{0}\varepsilon^{-1}.$

Therefore, there is a constant $c$ for which

$|\langle\partial_{n}u,X_{\tau}\partial_{\tau}u\rangle|\leq|X_{\tau}|\frac{c}{\varepsilon}.$

Moreover since $|X_{\tau}|\leq Cr$ and $|\Gamma_{r}|\leq Cr$ we have another
constant $C$ (independent of $\varepsilon$) so that

$\left|\int_{\Gamma_{r}}\langle\partial_{n}u,X\cdot\nabla
u\rangle\,ds\right|\leq\int_{\Gamma_{r}}|X_{\tau}|\frac{c}{\varepsilon}\,ds\leq\frac{Cr^{2}}{\varepsilon}.$

Estimates Along $\partial B_{r}(x_{0})\cap\Omega$:
The lefthand side of (3.1) along $\partial B_{r}(x_{0})\cap\Omega$ can be
written as the sum of integrals $I_{1}+I_{2}$ which we estimate separately.
First, by (3.10), $|X_{n}|$, $|X_{\tau}|\leq Cr$ and by applying Cauchy-
Schwarz we get

$\displaystyle I_{1}$ $\displaystyle=\int_{\partial
B_{r}(x_{0})\cap\Omega}\left\\{\frac{1}{2}|\nabla
u|^{2}X_{n}-X_{n}|\partial_{n}u|^{2}-X_{\tau}\langle\partial_{n}u,\partial_{\tau}u\rangle\right\\}ds$
$\displaystyle\leq Cr\int_{\partial
B_{r}(x_{0})\cap\Omega}\left\\{\frac{1}{2}|\partial_{\tau}u|^{2}+\frac{1}{2}|\partial_{n}u|^{2}+\frac{1}{2}|\partial_{n}u|^{2}+\frac{1}{2}|\partial_{\tau}u|^{2}\right\\}ds$
$\displaystyle=Cr\int_{\partial B_{r}(x_{0})\cap\Omega}|\nabla u|^{2}\,ds.$

An easy estimate for $I_{2}$ is given by

$I_{2}=\frac{1}{4\varepsilon^{2}}\int_{\partial
B_{r}(x_{0})\cap\Omega}(1-|u|^{2})^{2}X_{n}\,ds\leq\frac{Cr}{4\varepsilon^{2}}\int_{\partial
B_{r}(x_{0})\cap\Omega}(1-|u|^{2})^{2}\,ds.$

Thus, for $C>0$ large enough

$\displaystyle I_{1}+I_{2}\leq Cr\int_{\partial
B_{r}(x_{0})\cap\Omega}\frac{1}{2}\left\\{|\nabla
u|^{2}+\frac{1}{2\varepsilon^{2}}(1-|u|^{2})^{2}\right\\}ds=CF(r)$

and so the lefthand side of (3.1) has the estimate

$\displaystyle\int_{\partial\omega_{r}}\left\\{e_{\varepsilon}(u)X_{n}-\langle\partial_{n}u,X\cdot\nabla
u\rangle\right\\}ds$
$\displaystyle=I_{1}+I_{2}-\int_{\Gamma_{r}}\langle\partial_{n}u,X\cdot\nabla
u\rangle\,ds$ $\displaystyle\leq
C\left[F(r)+\frac{r^{2}}{\varepsilon}\right].$

Estimates on $\omega_{r}$:
The righthand side of (3.1) can be written as the sum of integrals
$J_{1}+J_{2}$ which again we estimate separately. By (3.11) and Cauchy-
Schwarz,

$\sum_{j,l}X_{x_{j}}^{l}\langle\partial_{x_{j}}u,\partial_{x_{l}}u\rangle\leq|\nabla
u|^{2}+2Cr|\nabla u|^{2}$

and since $\operatorname{div}X\geq 2-2Cr$ we have

$\displaystyle J_{1}$
$\displaystyle=\int_{\omega_{r}}\left\\{\frac{1}{2}|\nabla
u|^{2}\operatorname{div}X-\sum_{j,l}X_{x_{j}}^{l}\langle\partial_{x_{j}}u,\partial_{x_{l}}u\rangle\right\\}dx$
$\displaystyle\geq\int_{\omega_{r}}\left\\{\frac{1}{2}|\nabla
u|^{2}\operatorname{div}X-|\nabla u|^{2}-2Cr|\nabla u|^{2}\right\\}dx$
$\displaystyle\geq-Cr\int_{\omega_{r}}|\nabla u|^{2}\,dx.$

For the integral $J_{2},$ one can choose $r_{0}$ smaller if necessary so that
$\operatorname{div}X\geq 2-2Cr\geq 1$ and

$\displaystyle
J_{2}=\frac{1}{4\varepsilon^{2}}\int_{\omega_{r}}(1-|u|^{2})^{2}\operatorname{div}X\,dx\geq\frac{1}{4\varepsilon^{2}}\int_{\omega_{r}}(1-|u|^{2})^{2}\,dx.$

Putting these estimates together,

$\frac{1}{4\varepsilon^{2}}\int_{\omega_{r}}(1-|u|^{2})^{2}\,dx-
Cr\int_{\omega_{r}}\frac{1}{2}|\nabla u|^{2}\,dx\leq
J_{1}+J_{2}=I_{1}+I_{2}\leq C\left[F(r)+\frac{r^{2}}{\varepsilon}\right]$

which completes the proof for inequality (3.7). ∎

We now prove Theorem 3.2.

###### Proof of Theorem 3.2..

The case where $x_{0}\in\Omega$ and
$\omega_{2\varepsilon^{\beta}}(x_{0})\cap\Gamma=\varnothing$ is shown in
[Str94, Lemma 2.3]. Therefore, it is sufficient to prove the result for
$x_{0}\in\Gamma$. We begin by proving (3.3) for strong tangential solutions.
Inequality (3.6) for weak tangential solutions is done similarly. Using the
mean value theorem for integrals, there exists
$r_{\varepsilon}\in(2\varepsilon^{\gamma},2\varepsilon^{\beta})$ such that
$F(r_{\varepsilon})\leq\eta(\gamma-\beta)^{-1}$. Using the radius
$r=r_{\varepsilon}$ in (3.7) gives

$\displaystyle\frac{1}{4\varepsilon^{2}}\int_{\omega_{r_{\varepsilon}}(x_{0})}(1-|u_{\varepsilon}|^{2})^{2}\,dx$
$\displaystyle\leq
C\left[r_{\varepsilon}\int_{\omega_{r_{\varepsilon}}(x_{0})}\frac{1}{2}|\nabla
u|^{2}\,dx+F(r_{\varepsilon})+\frac{r_{\varepsilon}^{2}}{\varepsilon}\right]$
$\displaystyle\leq
C\left[2\varepsilon^{3/4}\eta|\ln\varepsilon|+\frac{\eta}{\gamma-\beta}+4\sqrt{\varepsilon}\right].$

Then for $\varepsilon<\varepsilon_{0}$ with appropriately chosen
$\varepsilon_{0}>0$, we get (3.3). Inequalities (3.2) and (3.4) can be
obtained using a contradiction argument. By assuming there is some
$x_{1}\in\omega_{\varepsilon^{\gamma}}(x_{0})$ such that $|u(x_{1})|<1/2$,
standard methods involving the mean value theorem and smoothness properties of
$\Gamma$ [BBH94] allow us to conclude that there is a radius $r=c\varepsilon$
and a constant $c^{\prime}>0$ independent of $\eta$ and $\varepsilon$ such
that

$\tilde{C}\eta\geq\frac{1}{4\varepsilon^{2}}\int_{\omega_{2\varepsilon^{\gamma}}(x_{0})}(1-|u|^{2})^{2}dx\geq\frac{1}{4\varepsilon^{2}}\int_{\omega_{c\varepsilon}(x_{1})}(1-|u|^{2})^{2}dx\geq
c^{\prime}.$

Taking $\eta$ smaller if necessary gives the contradiction. For inequality
(3.5), suppose $r_{0}>0$ is taken small enough so that $\omega_{r}(x_{0})$ is
strictly starshaped with respect to some point $x_{2}\in\omega_{r}(x_{0})$. By
setting $\psi=x-x_{2}$ in (3.1) and following [ABGS15, Proposition 4.1] or
[ABG20, Proposition 4.1], one can find the estimate

$|u_{\varepsilon}(x)-u_{\varepsilon}(y)|\leq C\sqrt{|x-y|}\varepsilon^{-s/2}$
(3.12)

holding for all $x,y\in\Gamma_{r_{\varepsilon}}$ with $C$ a constant
independent of $\varepsilon$ and $x_{0}$ . As in the interior case, suppose
there is some point $x_{3}\in\Gamma_{r_{\varepsilon}}$ such that $|\langle
u_{\varepsilon}(x_{3}),g^{\perp}(x_{3})\rangle|>1/4$. Then by the triangle
inequality, Cauchy-Schwarz and the uniform bound of Lemma 2.1:

$|\langle
u_{\varepsilon}(x),g^{\perp}(x)\rangle|>1/4-|u_{\varepsilon}(x)-u_{\varepsilon}(x_{3})|-|g^{\perp}(x)-g^{\perp}(x_{3})|.$

By (3.12) and the smoothness of $g$, a radius $\rho$ proportional to
$\varepsilon^{s}$ can be chosen so that

$|u_{\varepsilon}(x)-u_{\varepsilon}(x_{3})|,\
|g^{\perp}(x)-g^{\perp}(x_{3})|<\frac{1}{16}$

for all $x\in\Gamma_{r_{\varepsilon}}\cap B_{\rho}(x_{3})$ and so $|\langle
u_{\varepsilon}(x),g^{\perp}(x)\rangle|>1/8$ on this set. Applying inequality
(3.6) along with the estimate $|\Gamma_{r_{\varepsilon}}\cap
B_{\rho}(x_{3})|\geq c^{\prime\prime}\varepsilon^{s}$ with $c^{\prime\prime}$
independent of $\varepsilon$ and $\eta$, we have

$\tilde{C}\eta\geq\frac{1}{2\varepsilon^{s}}\int_{\Gamma_{r_{\varepsilon}}\cap
B_{\rho}(x_{3})}\langle
u_{\varepsilon},g^{\perp}\rangle^{2}\,ds>\frac{c^{\prime\prime}}{128}.$

As before, we choose $\eta$ small enough to obtain a contradiction. ∎

Define the family of bad sets

$\displaystyle S_{\varepsilon}$
$\displaystyle:=\left\\{x\in\overline{\Omega}:|u_{\varepsilon}(x)|<\frac{1}{2}\right\\},$
$\displaystyle S_{\varepsilon}^{g,s}$
$\displaystyle:=S_{\varepsilon}\cup\left\\{x\in\Gamma:|\langle
u_{\varepsilon}(x),g^{\perp}(x)\rangle|>\frac{1}{4}\right\\}.$

######  Proposition 3.4.

[Strong Tangential Case] There exists $\tilde{N}\in\mathbb{N}$ depending only
on $\Omega$, a constant $\lambda>1$ independent of $\varepsilon$ and points
$p_{\varepsilon,1},\ldots,p_{\varepsilon,I_{\varepsilon}}\in
S_{\varepsilon}\cap\Omega$,
$q_{\varepsilon,1},\ldots,q_{\varepsilon,J_{\varepsilon}}\in
S_{\varepsilon}\cap\Gamma$ such that

  1. (i)

$I_{\varepsilon}+J_{\varepsilon}\leq\tilde{N}$,

  2. (ii)

$S_{\varepsilon}\subset\bigcup_{i=1}^{I_{\varepsilon}}B_{\lambda\varepsilon}(p_{\varepsilon,i})\cup\bigcup_{j=1}^{J_{\varepsilon}}B_{\lambda\varepsilon}(q_{\varepsilon,j})$,

  3. (iii)

$\\{B_{\lambda\varepsilon}(p_{\varepsilon,i}),B_{\lambda\varepsilon}(q_{\varepsilon,j})\\}_{1\leq
i\leq I_{\varepsilon},1\leq j\leq J_{\varepsilon}}$ are mutually disjoint with
centers satisfying

$|p_{\varepsilon,i}-p_{\varepsilon,j}|,\
|q_{\varepsilon,i}-q_{\varepsilon,j}|,\
|p_{\varepsilon,i}-q_{\varepsilon,j}|>8\lambda\varepsilon,$

  4. (iv)

$\overline{B_{\lambda\varepsilon}(p_{\varepsilon,i})}\cap\Gamma=\varnothing$
for all $i=1,\ldots,I_{\varepsilon}$.

[Weak Tangential Case] There exists $\tilde{N}\in\mathbb{N}$ depending only on
$\Omega$, a constant $\lambda>1$ independent of $\varepsilon$ and points
$p_{\varepsilon,1},\ldots,p_{\varepsilon,I_{\varepsilon}}\in
S_{\varepsilon}^{g,s}\cap\Omega$,
$q_{\varepsilon,1},\ldots,q_{\varepsilon,J_{\varepsilon}}\in
S_{\varepsilon}^{g,s}\cap\Gamma$ such that

  1. (i)

$I_{\varepsilon}+J_{\varepsilon}\leq\tilde{N}$,

  2. (ii)

$S_{\varepsilon}^{g,s}\subset\bigcup_{i=1}^{I_{\varepsilon}}B_{\lambda\varepsilon}(p_{\varepsilon,i})\cup\bigcup_{j=1}^{J_{\varepsilon}}B_{\lambda\varepsilon^{s}}(q_{\varepsilon,j})$,

  3. (iii)

$\\{B_{\lambda\varepsilon}(p_{\varepsilon,i}),B_{\lambda\varepsilon^{s}}(q_{\varepsilon,j})\\}_{1\leq
i\leq I_{\varepsilon},1\leq j\leq J_{\varepsilon}}$ are mutually disjoint with
centers satisfying

$|p_{\varepsilon,i}-p_{\varepsilon,j}|>8\lambda\varepsilon,\quad|q_{\varepsilon,i}-q_{\varepsilon,j}|>8\lambda\varepsilon^{s},\quad
and\quad|p_{\varepsilon,i}-q_{\varepsilon,j}|>8\lambda\varepsilon^{s},$

  4. (iv)

$\overline{B_{\lambda\varepsilon}(p_{\varepsilon,i})}\cap\Gamma=\varnothing$
for all $i=1,\ldots,I_{\varepsilon}$.

The proof is exactly as in [ABG20, Lemma 4.4] which is based on the method of
[Str94, Section 3] and a ball merging method which is presented in [BBH94,
Theorem IV.1]. As a consequence of Balzano-Weierstrass, we also have a result
which states that the bad sets $S_{\varepsilon}$, $S_{\varepsilon}^{g,s}$ can
eventually be covered by a static ball covering (along a subsequence
$\varepsilon_{n}\to 0$).

######  Proposition 3.5.

[Strong Tangential Case] For any sequence of $\varepsilon\to 0$ there is a
subsequence $\varepsilon_{n}\to 0$, a constant $\sigma_{0}>0$ and a finite
collection of points $\\{p_{1},\ldots,p_{I}\\}\subset\Omega$,
$\\{q_{1},\ldots,q_{J}\\}\subset\Gamma$ such that for any
$0<\sigma<\sigma_{0}$ and for all $n\in\mathbb{N}$, the collection of sets

$\mathcal{S}_{\sigma}:=\\{B_{\sigma}(p_{i})\\}_{i=1}^{I}\cup\\{B_{\sigma}(q_{j})\\}_{j=1}^{J}$
(3.13)

are mutually disjoint and cover $S_{\varepsilon_{n}}$.
[Weak Tangential Case] The same result holds for the bad set
$S_{\varepsilon_{n}}^{g,s}$ but with $\mathcal{S}_{\sigma}$ replaced by

$\mathcal{S}_{\sigma}^{g,s}:=\\{B_{\sigma}(p_{i})\\}_{i=1}^{I}\cup\\{B_{\sigma^{s}}(q_{j})\\}_{j=1}^{J}.$
(3.14)

##  4\. Local Orientation and Defect Windings

Now that the bad sets $S_{\varepsilon}$ and $S_{\varepsilon}^{g,s}$ have been
shown to have finite bad ball coverings, we are in a position to analyze the
winding behaviour of minimizers around defects. When dealing with interior bad
balls, we may quantify the winding of $u_{\varepsilon}$ on $\partial
B_{\lambda\varepsilon}(p_{i,\varepsilon})$ in the usual way since
$|u_{\varepsilon}|\geq 1/2$ on this curve. In particular, we define the degree
of $u_{\varepsilon}$ around $\partial
B_{\lambda\varepsilon}(p_{i,\varepsilon})$ to be the degree of the
normalization of $u_{\varepsilon}$ about $\partial
B_{\lambda\varepsilon}(p_{i,\varepsilon})$:

$d_{i}=d_{i,\varepsilon}=\deg(u_{\varepsilon};\partial
B_{\lambda\varepsilon}(p_{i,\varepsilon})):=\deg(u_{\varepsilon}/|u_{\varepsilon}|;\partial
B_{\lambda\varepsilon}(p_{i,\varepsilon})).$

Analyzing the winding of $u_{\varepsilon}$ about boundary bad balls is
slightly more subtle, however. As mentioned in the introduction, we define the
notion of a _boundary index_ , whose function is analogous to the degree of
interior defects. Specifically, the boundary index aims to quantify the
turning behaviour of $u_{\varepsilon}$ along circular arcs lying in the
interior $\Omega$ that connect two nearby points on $\Gamma$. To begin
constructing this quantity, we place focus on strong tangential solutions and
then show the necessary modifications for weak tangential solutions.

Consider again the local polar coordinate system found in Section 2 as defined
by the angular bounds (2.6) with center point $q_{\varepsilon,j}\in\Gamma$.
Let $B_{\lambda\varepsilon}(q_{\varepsilon,j})$ be some fixed boundary bad
ball for a strong tangential solution $u_{\varepsilon}$ and fix
$R>\lambda\varepsilon$ so that for all $\rho\in[\lambda\varepsilon,R]$ the
closure of $\omega_{\rho}(q_{\varepsilon,j})$ does not intersect the closure
of any other bad ball. Since $|u_{\varepsilon}|\geq 1/2$ outside
$B_{\lambda\varepsilon}(q_{\varepsilon,j})$, there is a single-valued function
$\psi$ with

$\frac{u_{\varepsilon}}{|u_{\varepsilon}|}=e^{i\psi}\quad\mbox{on}\
\overline{A_{\lambda\varepsilon,R}(q_{\varepsilon,j})}.$

Likewise, there is a lifting $\gamma$ of $g$ for which $g=e^{i\gamma}$ on
$\overline{A_{\lambda\varepsilon,R}(q_{\varepsilon,j})}$ since
$g:\mathcal{N}_{\Gamma}\to\mathbb{S}^{1}$. Looking along the curve
$\Gamma^{+}_{\lambda\varepsilon,R}(q_{\varepsilon,j})$ and using the
definition $\mathcal{H}_{g}(\Omega)$, it holds that either

$|\psi(\rho,\theta_{1}(\rho))-\gamma(\rho,\theta_{1}(\rho))|=0\quad\mbox{or}\quad|\psi(\rho,\theta_{1}(\rho))-\gamma(\rho,\theta_{1}(\rho))\pm\pi|=0.$
(4.1)

These two possible conditions comes down the observation that
$u_{\varepsilon}$ has either a phase equal to that of $g$ or to $-g$. This
fact induces a sense of orientation for $u_{\varepsilon}$ with respect to $g$
along boundary components outside of bad balls. In particular, we say that
$u_{\varepsilon}$ is positively oriented with respect to $g$ at $x\in\Gamma$
(p.o.) provided $\langle u(x),g(x)\rangle>0$ (which corresponds to the first
condition of (4.1)) and that $u_{\varepsilon}$ is negatively oriented (n.o.)
in the opposite case $\langle u(x),g(x)\rangle<0$ (the second condition of
(4.1)).

Figure 1. Depiction of negative orientation with respect to $g$ for a strong
tangential solution $u$ at a point $x_{0}\in\Gamma$ away from bad balls.

Upon starting at the point

$q_{\varepsilon,j}^{+}=q_{\varepsilon,j}^{+}(\rho):=\partial
B_{\rho}(q_{\varepsilon,j})\cap\Gamma_{\lambda\varepsilon,R}^{+}(q_{\varepsilon,j})$

and traveling along the arc $\partial B_{\rho}(q_{\varepsilon,j})\cap\Omega$,
one arrives at the point

$q_{\varepsilon,j}^{-}=q_{\varepsilon,j}^{-}(\rho):=\partial
B_{\rho}(q_{\varepsilon,j})\cap\Gamma_{\lambda\varepsilon,R}^{-}(q_{\varepsilon,j})$

for which $u_{\varepsilon}$ has accumulated an approximate net number of
$\pi$-rotations. More precisely, there exists a unique integer
$k\in\mathbb{Z}$ such that

$|\psi(\rho,\theta_{2}(\rho))-\gamma(\rho,\theta_{2}(\rho))-k\pi|=0\quad\mbox{on}\
\Gamma_{\lambda\varepsilon,R}^{-}(q_{\varepsilon,j}).$ (4.2)

With this information, we may now define the boundary index. Let
$w:A_{\lambda\varepsilon,R}(q_{\varepsilon,j})\to\mathbb{S}^{1}$ be defined by

$w:=\left(\frac{u_{\varepsilon}}{|u_{\varepsilon}|}\right)^{2}=e^{2i\psi}.$

Using (4.1) and (4.2),

$\displaystyle\left\\{\begin{array}[]{ll}|\arg(w)-2\gamma|=2|\psi-\gamma|=0&\mbox{on
$\Gamma_{\lambda\varepsilon,R}^{+}(q_{\varepsilon,j})$ if $u_{\varepsilon}$ is
p.o.,}\\\\[5.0pt] |\arg(w)-2\gamma\pm 2\pi|=2|\psi-\gamma\pm\pi|=0&\mbox{on
$\Gamma_{\lambda\varepsilon,R}^{+}(q_{\varepsilon,j})$ if $u_{\varepsilon}$ is
n.o.,}\\\\[5.0pt] |\arg(w)-2\gamma-2\pi k|=2|\psi-\gamma-k\pi|=0&\mbox{on
$\Gamma_{\lambda\varepsilon,R}^{-}(q_{\varepsilon,j})$},\end{array}\right.$

and so $w$ has preserved orientation on
$\Gamma^{\pm}_{\lambda\varepsilon,R}(q_{\varepsilon,j})$ with respect to
$g^{2}$. Therefore we may extend $w$ to
$x\in\Gamma_{\lambda\varepsilon}(q_{\varepsilon,j})$ as an
$\mathbb{S}^{1}$-valued, piecewise $C^{2}$ map by simply setting $w=g^{2}$
along $\Gamma_{\lambda\varepsilon}(q_{\varepsilon,j})$. The reader can refer
to Figure 2 for an illustration.

(a) Possible profile of a strong tangential
solution $u$ outside a boundary bad ball

(b) Corresponding profile of $w$ outside a
boundary bad ball with interior extension

Figure 2. Relationship between $u$, $w$, $g$ and $g^{2}$.

Next, for any $\rho\in[\lambda\varepsilon,R]$, form the closed curve

$C_{\rho}:=(\partial
B_{\rho}(q_{\varepsilon,j})\cap\Omega)\cup\Gamma_{\rho}(q_{\varepsilon,j})$

with positive orientation. By construction of $w$ we may define

$D_{j}=D_{j}(q_{\varepsilon,j}):=\deg(w;C_{\rho})$ (4.3)

whose value is independent of $\rho\in[\lambda,R]$ for any particular
extension of $w$ by properties of the degree. Returning to the local polar
coordinate system $(\rho,\theta)$ centered at $q_{\varepsilon,j}$, we note
that

$\arg(w)=2\psi=2D_{j}\theta+\tilde{\phi}(\rho,\theta)$

where $\tilde{\phi}$ is a single-valued function in
$A_{\lambda\varepsilon,R}(q_{\varepsilon,j})$. Therefore
$u_{\varepsilon}=f(\rho,\theta)e^{i\psi}$ with

$\psi(\rho,\theta)=D_{j}\theta+\phi(\rho,\theta)$ (4.4)

and $\phi$ a single-valued function in
$\overline{A_{\lambda\varepsilon,R}(q_{\varepsilon,j})}$. The integer
$D_{j}\in\mathbb{Z}$ is what we define as the boundary index. In some cases,
we will use the notation

$D_{j}:=\operatorname{ind}(u_{\varepsilon};\partial
B_{\rho}(q_{\varepsilon,j})\cap\Omega).$

The boundary index for weakly tangential solutions can be constructed in the
same way, but now with boundary bad balls having radii
$\rho=\lambda\varepsilon^{s}$. However, the phase of $u_{\varepsilon}$ along
$\Gamma_{\lambda\varepsilon^{s},R}^{\pm}(q_{\varepsilon,j})$ no longer
satisfies the strict conditions of (4.1) and (4.2). In this case, the
definition of $S_{\varepsilon}^{g,s}$ can be used to show that

$|\psi(\rho,\theta_{1}(\rho))-\gamma(\rho,\theta_{1}(\rho))|<\frac{\pi}{6}\quad\mbox{or}\quad|\psi(\rho,\theta_{1}(\rho))-\gamma(\rho,\theta_{1}(\rho))\pm\pi|<\frac{\pi}{6}$
(4.5)

on $\Gamma_{\lambda\varepsilon^{s},R}^{+}(q_{\varepsilon,j})$ (depending on
the orientation of $u$ with respect to $g$), and that there is a unique
integer $k\in\mathbb{Z}$ such that

$|\psi(\rho,\theta_{2}(\rho))-\gamma(\rho,\theta_{2}(\rho))-k\pi|<\frac{\pi}{6}\quad\mbox{on}\
\Gamma_{\tilde{r},R}^{-}(q_{\varepsilon,j}).$ (4.6)

Figure 3. Illustration of positive orientation with respect to $g$ for a
weakly tangential solution $u$ at a point $x_{0}\in\Gamma$ away from bad
balls. By definition of $S_{\varepsilon}^{g,s}$, the vector $u(x_{0})$ may
only reside within the shaded double cone with axis defined by $g$.

Again, by defining the $\mathbb{S}^{1}$-valued function $w=(u/|u|)^{2}$ on
$A_{\lambda\varepsilon^{s},R}(q_{\varepsilon,j})$, from (4.5) and (4.6) we
have

$\displaystyle\left\\{\begin{array}[]{ll}|\arg(w)-2\gamma|=2|\psi-\gamma|<\frac{\pi}{3}&\mbox{on
$\Gamma_{\lambda\varepsilon^{s},R}^{+}(q_{\varepsilon,j})$ if
$u_{\varepsilon}$ is p.o.,}\\\\[5.0pt] |\arg(w)-2\gamma\pm
2\pi|=2|\psi-\gamma\pm\pi|<\frac{\pi}{3}&\mbox{on
$\Gamma_{\lambda\varepsilon^{s},R}^{+}(q_{\varepsilon,j})$ if
$u_{\varepsilon}$ is n.o.,}\\\\[5.0pt] |\arg(w)-2\gamma-2\pi
k|=2|\psi-\gamma-k\pi|<\frac{\pi}{3}&\mbox{on
$\Gamma_{\lambda\varepsilon^{s},R}^{-}(q_{\varepsilon,j})$},\end{array}\right.$

which as before, shows that the orientation of $w$ with respect to $g^{2}$ is
preserved on $\Gamma_{\lambda\varepsilon^{s},R}^{\pm}(q_{\varepsilon,j})$. We
can now extend $w$ to $x\in\Gamma_{\lambda\varepsilon^{s}}(q_{\varepsilon,j})$
as an $\mathbb{S}^{1}$-valued, piecewise $C^{2}$ map satisfying
$|\arg(w)-2\gamma|<\pi/3$ which can be done via interpolating the phase
linearly across $\Gamma_{\lambda\varepsilon^{s}}(q_{\varepsilon,j})$, for
example. The boundary index can now be defined in the same way as the strong
tangential case.

The first main identity we obtain from this definition connects the sum of all
associated bad ball boundary indices with $\mathcal{D}=\deg(g;\Gamma)$ and the
sum of degrees for the interior bad balls.

######  Proposition 4.1.

[Strong Tangential Case] Suppose $u_{\varepsilon}$ is a solution of (1.3) with
associated bad ball covering
$\\{B_{\lambda\varepsilon}(p_{\varepsilon,i}),B_{\lambda\varepsilon}(q_{\varepsilon,j})\\}_{1\leq
i\leq I_{\varepsilon},1\leq j\leq J_{\varepsilon}}$. Let

$\displaystyle d_{i}=\deg(u_{\varepsilon};\partial
B_{\lambda\varepsilon}(p_{\varepsilon,i}))\quad\mbox{and}\quad
D_{j}=\operatorname{ind}(u_{\varepsilon};\partial
B_{\lambda\varepsilon}(q_{\varepsilon,j})\cap\Omega)$

be the degrees and boundary indices for $u_{\varepsilon}$ about its interior
and boundary bad balls respectively. Then

$\mathcal{D}=\sum_{i=1}^{I_{\varepsilon}}d_{i}+\frac{1}{2}\sum_{j=1}^{J_{\varepsilon}}D_{j}.$
(4.7)

[Weak Tangential Case] Identity (4.7) holds for solutions of (1.5) and its
associated bad ball covering
$\\{B_{\lambda\varepsilon}(p_{\varepsilon,i}),B_{\lambda\varepsilon^{s}}(q_{\varepsilon,j})\\}_{1\leq
i\leq I_{\varepsilon},1\leq j\leq J_{\varepsilon}}$.

The proof for the weak tangential case is done identically to that of the
strong tangential case simply by replacing the radii $\lambda\varepsilon$ of
the boundary bad balls with $\lambda\varepsilon^{s}$. Thus, we provide a proof
only for strong tangential solutions.

###### Proof.

Define the domain

$\tilde{\Omega}:=\Omega\setminus\left\\{\bigcup_{j=1}^{J_{\varepsilon}}\overline{\omega_{\lambda\varepsilon}(q_{\varepsilon,j})}\right\\}$

and let $\tilde{\Gamma}=\partial\tilde{\Omega}$,
$C(q_{\varepsilon,j})=\partial\omega_{\lambda\varepsilon}(q_{\varepsilon,j})\cap\Omega$.
As in the definition of boundary index (4.3), the function
$w=(u_{\varepsilon}/|u_{\varepsilon}|)^{2}$ is defined on $\tilde{\Omega}$ and
$\tilde{\Gamma}$ and can be extended across each segment
$\Gamma_{\lambda\varepsilon}(q_{\varepsilon,j})$. By the construction of the
extension $w$, we have $\deg(w;\Gamma)=\deg(g^{2};\Gamma)=2\mathcal{D}$ and so
by the definition of boundary index,

$\displaystyle\deg(w;\tilde{\Gamma})$
$\displaystyle=\frac{1}{2\pi}\int_{\Gamma\setminus\cup_{j}\Gamma_{\lambda\varepsilon}(q_{j})}(iw,\partial_{\tau}w)\,ds+\sum_{j=1}^{J_{\varepsilon}}\frac{1}{2\pi}\int_{C(q_{j})}(iw,\partial_{\tau}w)\,ds$
$\displaystyle=\frac{1}{2\pi}\int_{\Gamma}(iw,\partial_{\tau}w)\,ds-\sum_{j=1}^{J_{\varepsilon}}\frac{1}{2\pi}\int_{\partial\omega_{\lambda\varepsilon}(q_{j})}(iw,\partial_{\tau}w)\,ds$
$\displaystyle=\deg(w;\Gamma)-\sum_{j=1}^{J_{\varepsilon}}D_{j}=2\mathcal{D}-\sum_{j=1}^{J_{\varepsilon}}D_{j}.$

Lastly, the vortices $p_{\varepsilon,i}$ are contained inside $\tilde{\Gamma}$
and so $\deg(w;\tilde{\Gamma})=\sum_{i=1}^{I_{\varepsilon}}2d_{i}$ where we
note that the degree along each interior bad ball is doubled by the definition
of $w$. Thus, we obtain (4.7) by equating the two quantities for
$\deg(w;\tilde{\Gamma})$ and dividing by $2$. ∎

We also have a local summation property holding between degrees and boundary
indices:

######  Lemma 4.2.

[Strong Tangential Case] Let $\mathcal{I}$ and $\mathcal{J}$ be sets of
indices for a collection of bad balls
$\\{B_{\lambda\varepsilon}(p_{\varepsilon,i})\\}_{i\in\mathcal{I}}\cup\\{B_{\lambda\varepsilon}(q_{\varepsilon,j})\\}_{j\in\mathcal{J}}$
for a strongly tangential solution $u$ and suppose there is a point
$y_{0}\in\Gamma$ and radius $R>0$ such that the ball $\mathscr{B}_{R}(y_{0})$
satisfies

$\left(\bigcup_{i\in\mathcal{I}}B_{\lambda\varepsilon}(p_{\varepsilon,i})\cup\bigcup_{j\in\mathcal{J}}B_{\lambda\varepsilon}(q_{\varepsilon,j})\right)\subset\mathscr{B}_{R}(y_{0})$

where $\overline{\mathscr{B}_{R}(y_{0})}$ does not intersect the closure of
any other bad ball. Then if
$\mathscr{D}=\operatorname{ind}(u_{\varepsilon};\partial\mathscr{B}_{R}(y_{0})\cap\Omega)$,
$d_{i}=\deg(u_{\varepsilon};\partial
B_{\lambda\varepsilon}(p_{\varepsilon,i}))$ and
$D_{j}=\operatorname{ind}(u_{\varepsilon};\partial
B_{\lambda\varepsilon}(q_{\varepsilon,j})\cap\Omega)$, we have

$\mathscr{D}=\sum_{j\in\mathcal{J}}D_{j}+2\sum_{i\in\mathcal{I}}d_{i}.$

[Weak Tangential Case] Under the same hypotheses but with boundary bad ball
radii replaced by $\lambda\varepsilon^{s}$, the above identity also holds for
a collection of bad balls
$\\{B_{\lambda\varepsilon}(p_{\varepsilon,i})\\}_{i\in\mathcal{I}}\cup\\{B_{\lambda\varepsilon^{s}}(q_{\varepsilon,j})\\}_{j\in\mathcal{J}}$
for weakly tangential solutions.

The proof of this Lemma follows the same lines as Proposition 4.1, but can
also be shown using longer methods found in [vB22, Lemma 4.7]. As above, the
proof for the weak tangential case is done identically to that of the strong
tangential case by replacing the radii scaling of the boundary bad balls
accordingly. To this end, we proceed with the strong tangential case only.

###### Proof.

Let
$\tilde{\Omega}=\mathscr{B}_{R}(y_{0})\setminus\cup_{j\in\mathcal{J}}\overline{\omega_{\lambda\varepsilon}(q_{j})}$,
$\tilde{\Gamma}=\partial\tilde{\Omega}$ and
$C(q_{j})=\partial\omega_{\lambda\varepsilon}(q_{j})\cap\Omega$. Since
$|u|\geq 1/2$ in $\tilde{\Omega}$, the function $w=(u/|u|)^{2}$ is defined on
$\overline{\tilde{\Omega}}$. As in the construction of the boundary index, $w$
can be appropriately extended across each segment
$\Gamma_{\lambda\varepsilon}(q_{j})$, $j\in\mathcal{J}$, and so we take $w$ to
be defined on
$\tilde{\Gamma}\cup_{j\in\mathcal{J}}\Gamma_{\lambda\varepsilon}(q_{j})$. In
particular, this extension of $w$ is defined on
$\partial\mathscr{B}_{R}(y_{0})$ and so by definition of the boundary index,

$\mathscr{D}=\operatorname{ind}(u;\partial\mathscr{B}_{R}(y_{0})\cap\Omega)=\deg(w;\partial\mathscr{B}_{R}(y_{0})).$
(4.8)

Calculating the degree of $w$ along $\tilde{\Gamma}$, we have by (4.8)

$\displaystyle\deg(w;\tilde{\Gamma})$
$\displaystyle=\frac{1}{2\pi}\int_{\partial\mathscr{B}_{R}(y_{0})\setminus\cup_{j}\Gamma_{\lambda\varepsilon}(q_{j})}(iw,\partial_{\tau}w)\,ds+\sum_{j\in\mathcal{J}}\frac{1}{2\pi}\int_{C(q_{j})}(iw,\partial_{\tau}w)\,ds$
$\displaystyle=\frac{1}{2\pi}\int_{\partial\mathscr{B}_{R}(y_{0})}(iw,\partial_{\tau}w)\,ds-\sum_{j\in\mathcal{J}}\frac{1}{2\pi}\int_{\partial\omega_{\lambda\varepsilon}(q_{j})}(iw,\partial_{\tau}w)\,ds$
$\displaystyle=\mathscr{D}-\sum_{j\in\mathcal{J}}D_{j}.$

On the other hand, each circle $\partial\omega_{\lambda\varepsilon}(p_{i})$,
$i\in\mathcal{I}$, is contained inside $\tilde{\Gamma}$ and so

$\deg(w;\tilde{\Gamma})=\sum_{i\in\mathcal{I}}\deg(w;\partial\omega_{\lambda\varepsilon}(p_{i}))=2\sum_{i\in\mathcal{I}}d_{i}$

proving the desired identity. ∎

##  5\. Lower Bounds for the Energy and Convergence

In this section, we display the details needed to complete the proof of
Theorems 1.1 and 1.2, which mainly comes down to providing a lower bound for
the energies $E_{\varepsilon}$ and $E_{\varepsilon}^{g,s}$ on the ball
collections $\mathcal{S}_{\sigma}$ and $\mathcal{S}_{\sigma}^{g,s}$
respectively. This result is given in Lemma 5.3 at the end of this section.
The first step in our analysis is to calculate the cost of a vortex locally on
annular regions $A_{r,R}$ with $r<R$. To do this, it is useful to begin by
characterizing solutions of (1.3) and (1.5) in terms of a local polar
representation with central point $x_{0}\in\overline{\Omega}$. Depending on
whether $x_{0}\in\Omega$ or $x_{0}\in\Gamma$, the representation for the phase
of $u_{\varepsilon}$ will look slightly different. In any case, the general
form for $u_{\varepsilon}$ on $A_{r,R}(x_{0})$, $x_{0}\in\overline{\Omega}$
can be given by

$u_{\varepsilon}(\rho,\theta)=f(\rho,\theta)e^{i\psi(\rho,\theta)}\ \mbox{on}\
A_{r,R}(x_{0}),\ \rho\in[r,R]$

where $\rho=|x-x_{0}|$, $\theta$ is an appropriately chosen polar angle and
$f(\rho,\theta)=|u_{\varepsilon}|$. In the specific case when $x_{0}\in\Gamma$
there are four general scenarios which can occur for solutions on
$\Gamma^{\pm}_{r,R}(x_{0})$ when $\langle u,g\rangle\neq 0$:

  1. (a)

$u$ is p.o. on $\Gamma_{r,R}^{+}$ and n.o. on $\Gamma_{r,R}^{-}$,

  2. (b)

$u$ is p.o. on $\Gamma_{r,R}^{+}$ and on $\Gamma_{r,R}^{-}$,

  3. (c)

$u$ is n.o. on $\Gamma_{r,R}^{+}$ and p.o. on $\Gamma_{r,R}^{-}$,

  4. (d)

$u$ is n.o. on $\Gamma_{r,R}^{+}$ and on $\Gamma_{r,R}^{-}$.

To accommodate for these four cases, we define a polar representation for
$u_{\varepsilon}$ on $A_{r,R}(x_{0})$ whose phase depends on the orientation
with respect to $g$ when near the boundary. Let $\gamma(x)$ be such that
$g(x)=e^{i\gamma(x)}$ along $\Gamma_{R}(x_{0})$ with
$\gamma_{0}=\gamma(x_{0})$ provided $x_{0}\in\Gamma$. By modifying the single-
valued function $\phi$ from (4.4) if necessary, the phase $\psi$ for
$u_{\varepsilon}$ can be given as

$\displaystyle\psi(\rho,\theta)=\left\\{\begin{array}[]{ll}d\theta+\phi(\rho,\theta)&\mbox{if
$B_{R}(x_{0})\subset\Omega$}\\\\[5.0pt]
D\theta+\gamma_{0}+\phi(\rho,\theta)&\mbox{if $x_{0}\in\Gamma$ and $u$ is
p.o.\@ on $\Gamma_{r,R}^{+}$},\\\\[5.0pt]
D\theta+\gamma_{0}+\phi(\rho,\theta)\pm\pi&\mbox{if $x_{0}\in\Gamma$ and $u$
is n.o.\@ on $\Gamma_{r,R}^{+}$}.\end{array}\right.$ (5.4)

In this form, $\phi$ is a smooth, single-valued function defined on
$A_{r,R}(x_{0})$ and can be thought of strictly as a function of $\rho>0$ on
$\Gamma_{r,R}^{\pm}$ by the choice of coordinates given in (2.7). That is,
$\phi=\phi(\rho,\theta(\rho))$ on $\Gamma_{r,R}^{\pm}$. The integers
$d=\deg(u;\partial B_{\rho}(x_{0}))$, $D=\operatorname{ind}(u;\partial
B_{\rho}(x_{0})\cap\Omega)\in\mathbb{Z}$ are the associated degree and
boundary index for $u_{\varepsilon}$ respectively. Through representation
(5.4), the boundary index $D$ determines the orientation of $u$ along
$\Gamma_{r,R}^{-}$. Indeed, when $R$ is taken to be small and $D$ is even, the
phase difference across $\Gamma_{r,R}^{\pm}$ will be approximately an even
multiple of $\pi$. In this case, the orientation of $u_{\varepsilon}$ with
respect to $g$ will be maintained along $\Gamma_{r,R}^{\pm}$ (cases (b) and
(d)). When $D$ is odd, the orientation of $u_{\varepsilon}$ with respect to
$g$ changes sign, giving cases (a) and (c).

The function $\phi$ plays an important role in estimating the energy
contribution of a defect and it is critical to show that it is appropriately
bounded. The following proposition is needed for this estimation process.

######  Proposition 5.1.

Let $\phi$ be as defined in (5.4) and suppose $|u|\geq 1/2$, $|\langle
u,g^{\perp}\rangle|\leq 1/4$ on $\Gamma_{r,R}^{\pm}$. Then there exists a
constant $C>0$ for which $|\phi|\leq C(|\langle u,g^{\perp}\rangle|+\rho)$. In
the special case that $\langle u,g^{\perp}\rangle=0$ on $\Gamma_{r,R}^{\pm}$,
we have the simplified bound $|\phi|\leq C\rho$.

The result of Proposition 5.1 is claimed in [Mos03] for $g^{\perp}=n$ in the
weak tangential case, but is not explicitly shown. We prove it here for
completeness.

###### Proof.

Observe the inner product

$|\langle
u,g^{\perp}\rangle|=|u||\cos(\psi-(\gamma-\pi/2))|=|u||\sin(\psi-\gamma)|=|u||\sin(\psi-\gamma\pm\pi)|.$

When $|\langle u,g^{\perp}\rangle|\leq 1/4$ and using the bound $|u|\geq 1/2$,
we obtain $|\psi-\gamma|\leq\pi/6$ or $|\psi-\gamma\pm\pi|\leq\pi/6$ for all
$x\in\Gamma^{\pm}_{r,R}$ depending on orientation. If $\langle
u,g^{\perp}\rangle=0$, then we precisely get $|\psi-\gamma|=0$ or
$|\psi-\gamma\pm\pi|=0$ which again depends on the orientation of $u$ with
respect to $g$. By considering the four possible orientations for
$u_{\varepsilon}$ (cases (a)–(d)) separately, it can be shown that there is a
constant $c>0$ such that $|\phi|\leq\pi/6+c\rho$ on $\Gamma^{\pm}_{r,R}$ in
the weak tangential case and $|\phi|\leq c\rho$ in the strong tangential case.
To see this, we analyze case (a) from above and claim the other cases follow
similarly. When $u_{\varepsilon}$ is positively oriented on $\Gamma_{r,R}^{+}$
and negatively oriented on $\Gamma_{r,R}^{-}$ there is $\xi\in[-\pi/6,\pi/6]$
such that,

$\psi-\gamma=D\theta+\gamma_{0}-\gamma+\phi=\xi\ \quad\mbox{on}\
\Gamma_{r,R}^{+}$

with $D\in 2\mathbb{Z}+1$. The triangle inequality gives

$\displaystyle|\phi|\leq|\xi|+|D\theta|+|\gamma_{0}-\gamma|\leq\frac{\pi}{6}+c\rho.$

On $\Gamma_{r,R}^{-}$, we have $D\theta+\gamma_{0}-\gamma+\phi=D\pi+\xi$ and a
similar estimate yields

$|\phi|\leq|\xi|+|D||\pi-\theta|+|\gamma_{0}-\gamma|\leq\frac{\pi}{6}+c\rho.$

The strong tangential condition corresponds to the scenario where $\xi=0$ for
each of the four cases (a)–(d), and so the estimate above can be reduced to
$|\phi|\leq C\rho$ in this case, which finishes the proof for solutions
satisfying $\langle u,g^{\perp}\rangle=0$ on $\Gamma_{r,R}^{\pm}$.

Assume $R$ is chosen small enough such that $c\rho\leq\pi/12$, for example, so
that $|\phi|\leq\pi/4$ on $\Gamma^{\pm}_{r,R}$. Returning to the inner product
and omitting the cases where we consider $\pm\pi$ in the argument, the reverse
triangle inequality gives

$|\langle
u,g^{\perp}\rangle|=|u||\sin(\psi-\gamma)|\geq\frac{1}{2}|\sin(\phi)||\cos(D\theta+\gamma_{0}-\gamma)|-\frac{1}{2}|\cos(\phi)||\sin(D\theta+\gamma_{0}-\gamma)|$

which holds on $\Gamma^{\pm}_{r,R}$ for any of the four orientation scenarios.
Since $\Gamma$ and $\gamma$ are smooth, for $R$ taken small enough it holds
that

$|\sin(D\theta+\gamma_{0}-\gamma)|,\ |1-|\cos(D\theta+\gamma_{0}-\gamma)||\leq
C\rho,$

and so we may assume $|\cos(D\theta+\gamma_{0}-\gamma)|\geq 1/2$ on
$\Gamma^{\pm}_{r,R}$. Thus,

$|\langle u,g^{\perp}\rangle|\geq\frac{1}{4}|\sin(\phi)|-\frac{1}{2}C\rho.$

Finally, since $|\phi|\leq\pi/4$ we have

$|\langle u,g^{\perp}\rangle|\geq\frac{1}{8}|\phi|-\frac{1}{2}C\rho$

which finishes the proof. ∎

As described in much of the surrounding literature, the energy contribution of
a non-trivial interior defect for solutions of the Ginzburg–Landau equations
on an annulus $A_{r,R}$ is known to be logarithmic in the ratio $R/r$ and
depends on the _square_ of the degree $d$ of $u$ around the vortex. A similar
result holds for boundary defects with associated boundary index $D$. This
result is given in Theorem 5.2 below.

######  Theorem 5.2.

[Strong Tangential Case] Suppose $x_{0}\in\overline{\Omega}$ and assume that
$1/2\leq|u|\leq 1$ in $A_{r,R}(x_{0})$. Additionally, suppose $\langle
u,g^{\perp}\rangle=0$ on $\Gamma_{r,R}^{\pm}(x_{0})$ and that there is some
number $K$ such that

$\displaystyle E_{\varepsilon}(u)\leq K|\ln\varepsilon|+K,$
$\displaystyle\frac{1}{\varepsilon^{2}}\int_{\omega_{\varepsilon^{\gamma}}(x_{0})}(1-|u|^{2})^{2}\,dx\leq
K,$

where $\varepsilon^{\gamma}$ is as in Theorem 3.2. Then there exists a
constant $C$ depending only on $\Omega$, $\gamma$ and $K$ such that:

  1. (i)

If $B_{R}(x_{0})\subset\Omega$, $\varepsilon\leq r<R\leq r_{0}$ and
$d=\deg(u;\partial B_{r}(x_{0}))\neq 0$,

$\int_{A_{r,R}(x_{0})}|\nabla u|^{2}\,dx\geq
2d^{2}\pi\ln\left(\frac{R}{r}\right)-C.$ (5.5)

  2. (ii)

If $x_{0}\in\Gamma$, $\varepsilon\leq r<R\leq r_{0}$ and
$D=\operatorname{ind}(u;\partial B_{r}(x_{0})\cap\Omega)\neq 0$,

$\int_{A_{r,R}(x_{0})}|\nabla u|^{2}\,dx\geq
D^{2}\pi\ln\left(\frac{R}{r}\right)-C.$ (5.6)

[Weak Tangential Case] Suppose $x_{0}\in\overline{\Omega}$ and assume that
$1/2\leq|u|\leq 1$ in $A_{r,R}(x_{0})$. Additionally, suppose $|\langle
u,g^{\perp}\rangle|\leq 1/4$ on $\Gamma_{r,R}^{\pm}$ and that there is some
number $K$ such that

$\displaystyle E_{\varepsilon}^{g,s}(u)\leq K|\ln\varepsilon|+K,$
$\displaystyle\frac{1}{\varepsilon^{2}}\int_{\omega_{\varepsilon^{\gamma}}(x_{0})}(1-|u|^{2})^{2}\,dx+\frac{1}{\varepsilon^{s}}\int_{\Gamma_{\varepsilon^{\gamma}}}\langle
u,g^{\perp}\rangle^{2}\,ds\leq K,$

where $\varepsilon^{\gamma}$ is as in Theorem 3.2. Then there exists a
constant $C$ depending only on $\Omega$, $\gamma$ and $K$ such that:

  1. (i)

If $B_{R}(x_{0})\subset\Omega$, $\varepsilon\leq r<R\leq r_{0}$ and
$d=\deg(u;\partial B_{r}(x_{0}))\neq 0$, then (5.5) holds.

  2. (ii)

If $x_{0}\in\Gamma$, $\varepsilon^{s}\leq r<R\leq r_{0}$ and
$D=\operatorname{ind}(u;\partial B_{r}(x_{0})\cap\Omega)\neq 0$, then (5.6)
holds.

###### Proof.

The proof of inequality (5.5) is omitted since it follows identically to that
of [Str94, Proposition 3.4] and [Str95, Proposition 3.4’]. For (5.6), we
provide a brief sketch to show how the boundary index appears and how the
boundary conditions are handled. With this, we assume $x_{0}\in\Gamma$. Using
the polar representation for $u$ on $A_{r,R}(x_{0})$,

$\displaystyle\int_{A_{r,R}}|\nabla u|^{2}\,dx$
$\displaystyle=\int_{A_{r,R}}\left(f^{2}|\nabla\psi|^{2}+|\nabla
f|^{2}\right)dx$ $\displaystyle\geq\int_{A_{r,R}}f^{2}|\nabla
D\theta+\nabla\phi|^{2}\,dx$
$\displaystyle=\int_{A_{r,R}}\frac{D^{2}f^{2}}{\rho^{2}}\,dx+\int_{A_{r,R}}\frac{2Df^{2}}{\rho^{2}}\partial_{\theta}\phi\,dx+\int_{A_{r,R}}f^{2}|\nabla\phi|^{2}\,dx$
$\displaystyle=I_{1}+I_{2}+I_{3}.$

The lower estimates for $I_{1}$ and $I_{3}$ primarily follow [Str94,
Proposition 3.4] and [Str95, Proposition 3.4’], with details regarding the
boundary given in [Mos03, Proposition 5.6] and [ABGS15, Proposition 4.3].
Specifically,

$I_{1}\geq D^{2}\pi^{2}\ln\left(\frac{R}{r}\right)-C,\quad
I_{3}\geq\frac{1}{4}\int_{A_{r,R}(x_{0})}|\nabla\phi|^{2}\,dx$

where $C$ is a constant independent of $\varepsilon$. For the integral
$I_{2}$, Proposition 5.1 implies

$|\phi(\rho,\theta_{2})-\phi(\rho,\theta_{1})|\leq
2C\left(\sum_{x\in\partial\Gamma_{\rho}^{\pm}}|u_{\perp}(\rho,\theta_{i}(\rho))|+\rho\right)$

with $u_{\perp}=0$ for strong tangential solutions. Applying bounding methods
found in [Mos03, Proposition 5.6] and [Str94, Proposition 3.4],

$\displaystyle|I_{2}|$
$\displaystyle\leq\left|\int_{A_{r,R}}\frac{2D}{\rho^{2}}\partial_{\theta}\phi\,dx\right|+2\left|\int_{A_{r,R}}\frac{D(1-f^{2})}{\rho^{2}}\partial_{\theta}\phi\,dx\right|$
$\displaystyle\leq\int_{r}^{R}\frac{2|D||\phi(\rho,\theta_{2})-\phi(\rho,\theta_{1})|}{\rho}\,d\rho+\frac{1}{4}\int_{A_{r,R}}|\nabla\phi|^{2}\,dx+C$
$\displaystyle\leq 4|D|C\int_{\Gamma_{r,R}^{\pm}}\frac{|\langle
u,g^{\perp}\rangle|}{\rho}\,d\rho+\frac{1}{4}\int_{A_{r,R}}|\nabla\phi|^{2}\,dx+C^{\prime}.$

If $u$ is a strong tangential solution, the first integral in the last line
above does not appear and so the estimate ends there. If $u$ is a weak
tangential solution, the proof of [Mos03, Proposition 5.6] can be followed
with $|f\cdot\nu|$ replaced by $|\langle u,g^{\perp}\rangle|$ throughout,
giving

$|I_{2}|\leq C+\frac{1}{4}\int_{A_{r,R}(x_{0})}|\nabla\phi|^{2}\,dx.$

The desired lower bound is then estimated by

$\int_{A_{r,R}}|\nabla u|^{2}\,dx\geq I_{1}-|I_{2}|+I_{3}\geq
D^{2}\pi\ln\left(\frac{R}{r}\right)-C.$

∎

At this point, we are ready to describe a lower bound for the energy on the
sets comprising $\mathcal{S}_{\sigma}$ and $\mathcal{S}_{\sigma}^{g,s}$ as
defined in (3.13) and (3.14) respectively.

######  Lemma 5.3.

[Strong Tangential Case] Suppose $\varepsilon_{n}$ is the subsequence taken in
Proposition 3.5 and let $d_{i}=\deg(u_{\varepsilon_{n}};\partial
B_{\sigma}(p_{i}))$ and $D_{j}=\operatorname{ind}(u_{\varepsilon_{n}};\partial
B_{\sigma}(q_{j})\cap\Omega)$. There exists a constant $C$, independent of
$\varepsilon_{n}$ and $\sigma$ such that:

$\displaystyle
E_{\varepsilon_{n}}(u_{\varepsilon_{n}};B_{\sigma}(p_{i}))\geq\pi|d_{i}|\ln\left(\frac{\sigma}{\varepsilon_{n}}\right)-C,\quad
i=1,\ldots,I,$ $\displaystyle
E_{\varepsilon_{n}}(u_{\varepsilon_{n}};B_{\sigma}(q_{j}))\geq\frac{\pi}{2}|D_{j}|\ln\left(\frac{\sigma}{\varepsilon_{n}}\right)-C,\quad
j=1,\ldots,J.$

[Weak Tangential Case] Suppose $\varepsilon_{n}$ is the subsequence taken in
Proposition 3.5 and let $d_{i}=\deg(u_{\varepsilon_{n}};\partial
B_{\sigma}(p_{i}))$ and $D_{j}=\operatorname{ind}(u_{\varepsilon_{n}};\partial
B_{\sigma^{s}}(q_{j})\cap\Omega)$. There exists a constant $C$, independent of
$\varepsilon_{n}$ and $\sigma$ such that:

$\displaystyle
E_{\varepsilon_{n}}^{g,s}(u_{\varepsilon_{n}};B_{\sigma}(p_{i}))\geq\pi|d_{i}|\ln\left(\frac{\sigma}{\varepsilon_{n}}\right)-C,\quad
i=1,\ldots,I,$ $\displaystyle
E_{\varepsilon_{n}}^{g,s}(u_{\varepsilon_{n}};B_{\sigma^{s}}(q_{j}))\geq\frac{\pi
s}{2}|D_{j}|\ln\left(\frac{\sigma}{\varepsilon_{n}}\right)-C,\quad
j=1,\ldots,J.$

The proof for this lemma comes from a result developed by Sandier [San98]
(Jerrard [Jer99] gives a similar result) which uses techniques involving the
logarithmic lower bound as found in Theorem 5.2. The method involves a two-
step approach where balls containing subsets of $S_{\varepsilon}$ (or
$S_{\varepsilon}^{g,s}$) are expanded and fused such that the energy on these
balls can be estimated from below while preserving the natural scaling by
$\varepsilon$. A fundamental difference between our work and that of Sandier’s
are details regarding boundary data. Indeed, Sandier’s work assumes Dirichlet
boundary conditions and thus one does not obtain boundary vortices in this
case. For our problem, boundary vortices are expected and thus some extra care
needs to be taken when one performs the ball expansion and fusion argument. We
refer the reader to [ABM20] and [ABG20, Lemma 7.1] for a proof on how to
modify Sandier’s result to accommodate for boundary vortices. In particular,
the proof not only removes the assumption of Dirichlet boundary data, but also
explains how one can deal with the different radial scalings $\varepsilon$ and
$\varepsilon^{s}$ of the bad balls. However, it is worth noting that the proof
of [ABG20, Lemma 7.1] is done in a global sense due to the way boojums must be
dealt with. For our case, thanks to Lemma 4.2, the arguments of [ABG20, Lemma
7.1] can be applied to each $\sigma$-ball separately which results in Lemma
5.3.
As a consequence of Lemma 5.3 and Proposition 2.2, we have

$\pi\left(\sum_{i=1}^{I}|d_{i}|+\frac{1}{2}\sum_{j=1}^{J}|D_{j}|\right)|\ln\varepsilon_{n}|-C\leq
E_{\varepsilon}(u_{\varepsilon_{n}};\mathcal{S}_{\sigma})\leq\pi
s\mathcal{D}|\ln\varepsilon|+C$ (5.7)

for strong tangential solutions and

$\pi\left(\sum_{i=1}^{I}|d_{i}|+\frac{s}{2}\sum_{j=1}^{J}|D_{j}|\right)|\ln\varepsilon_{n}|-C\leq
E_{\varepsilon}^{g,s}(u_{\varepsilon_{n}};\mathcal{S}_{\sigma}^{g,s})\leq\pi
s\mathcal{D}|\ln\varepsilon|+C$ (5.8)

for weak tangential solutions. Using these estimates, we find that each degree
$d_{i}$ and boundary index $D_{j}$ are uniformly bounded in $\varepsilon$ and
therefore can be taken to be constant along a subsequence $\varepsilon_{n}\to
0$. It is also clear from (5.7) and (5.8) that all $\sigma$-balls constituting
$\mathcal{S}_{\sigma}$ and $\mathcal{S}_{\sigma}^{g,s}$ respectively, which
satisfy $d_{i}=D_{j}=0$ do not contribute substantial energy. Therefore, the
associated balls can be seen to belong to the set where $u_{\varepsilon_{n}}$
converges. By relabeling the approximate vortices if necessary, we define

$\Sigma:=\\{p_{1},\ldots,p_{I}\\}\cup\\{q_{1},\ldots,q_{J}\\}$

to be the collection of all $\sigma$-ball centers with non-trivial degree or
boundary index. Upon dividing by $\pi|\ln\varepsilon|$ and taking
$\varepsilon\to 0$ in (5.7) and (5.8), it holds that

$\sum_{i=1}^{I}|d_{i}|+\frac{1}{2}\sum_{j=1}^{J}|D_{j}|\leq\mathcal{D}$

for strong tangential solutions and

$\sum_{i=1}^{I}|d_{i}|+\frac{s}{2}\sum_{j=1}^{J}|D_{j}|\leq s\mathcal{D}$

for weak tangential solutions. Using identity (4.7) in combination with the
above inequalities shows all integers $d_{i}$ and $D_{j}$ must be positive
(since we’ve assumed $\mathcal{D}>0$). In fact, we have the equalities

$\mathcal{D}=\sum_{i=1}^{I}d_{i}+\frac{1}{2}\sum_{j=1}^{J}D_{j}$ (5.9)

and

$s\mathcal{D}=\sum_{i=1}^{I}d_{i}+\frac{s}{2}\sum_{j=1}^{J}D_{j}$ (5.10)

for the strong and weak cases respectively. This allows us to conclude the
following: Let

$\Omega_{\sigma}:=\Omega\setminus\overline{\mathcal{S}_{\sigma}},\quad\Omega_{\sigma}^{g,s}:=\Omega\setminus\overline{\mathcal{S}_{\sigma}^{g,s}}.$

######  Corollary 5.4.

[Strong Tangential Case] For any $\sigma\in(0,\sigma_{0})$, there exists a
constant $C$ independent of $\varepsilon$ and $\sigma$ such that

$E_{\varepsilon_{n}}(u_{\varepsilon_{n}};\Omega_{\sigma})\leq\pi\mathcal{D}|\ln\sigma|+C.$

Moreover, there is a constant $C^{\prime}$ independent of $\varepsilon$ such
that

$\frac{1}{4\varepsilon_{n}^{2}}\int_{\Omega}\left(1-|u_{\varepsilon_{n}}|^{2}\right)^{2}dx\leq
C^{\prime}.$

[Weak Tangential Case] For any $\sigma\in(0,\sigma_{0})$, there exists a
constant $C$ independent of $\varepsilon$ and $\sigma$ such that

$E_{\varepsilon_{n}}^{g,s}(u_{\varepsilon_{n}};\Omega_{\sigma}^{g,s})\leq\pi
s\mathcal{D}|\ln\sigma|+C.$

There is also a constant $C^{\prime}$ independent of $\varepsilon$ such that

$\frac{1}{4\varepsilon_{n}^{2}}\int_{\Omega}\left(1-|u_{\varepsilon_{n}}|^{2}\right)^{2}dx+\frac{1}{2\varepsilon_{n}^{s}}\int_{\Gamma}\langle
u_{\varepsilon_{n}},g^{\perp}\rangle^{2}\,ds\leq C^{\prime}.$

Upon taking an appropriate subsequence $\sigma_{n}\to 0$, Corollary 5.4 and
following the methods of [BBH94] and [Str94] allows us to conclude that
$u_{\varepsilon_{n}}\rightharpoonup u_{0}$ weakly in
$H^{1}_{loc}(\overline{\Omega}\setminus\Sigma;\mathbb{R}^{2})$ as
$\varepsilon_{n}\to 0$ where $u_{0}\in
H^{1}(\overline{\Omega}\setminus\Sigma;\mathbb{S}^{1})$ is harmonic. By
observing annular regions in $\Omega_{\sigma}$ (and $\Omega_{\sigma}^{g,s}$)
and applying Corollary 5.4 and Theorem 5.2, it is easy to show that vortices
of degree or boundary index larger than $1$ require too much energy, and
therefore we conclude $d_{i}=D_{j}=1$ for all $1\leq i\leq I$, $1\leq j\leq
J$. In light of equation (5.9), making the further assumption that
$\mathcal{D}=1$ forces that either $\Sigma=\\{p_{1}\\}\subset\Omega$ or
$\Sigma=\\{q_{1},q_{2}\\}\subset\Gamma$ which finishes the proof of Theorem
1.1. Moreover, equation $\eqref{sD}$ can be rewritten

$s\mathcal{D}=\sum_{i=1}^{I}d_{i}+\frac{s}{2}\sum_{j=1}^{J}D_{j}=(1-s)\sum_{i=1}^{I}d_{i}+s\mathcal{D}$

which immediately implies $\Sigma\subset\Gamma$ whenever $0<s<1$. The fact
that each $D_{j}=1$ also implies $|\Sigma|=J=2\mathcal{D}$ which then
completes the proof of Theorem 1.2.

##  6\. Defect Locations on a Disc when $g=\tau$

This final section is dedicated to analyzing a strong tangential anchoring
example. The primary point of studying this case is to shed light on the fact
that certain domain geometries may exist for which boundary vortices could
still be energetically preferable to those in the interior, even when all
vortices are given equal scaling. We consider the special case where $g=\tau$,
the positively oriented unit tangent vector to $\Gamma$, and take
$\Omega=B_{1}(0)$ to be the unit disc for simplicity. In this scenario,
$\mathcal{D}=\deg(\tau,\Gamma)=1$ and so in light of equation (4.7) there are
only two possibilities for defect locations, exactly one in the interior or
exactly two along the boundary. To investigate this further, we observe a
renormalized energy.
One Defect in $\Omega$
Let $p\in\Omega$ denote the interior singularity and assume the limiting
harmonic map $u_{0}=\tau$ on $\Gamma$. Following [ABGS15, Section 6] and
[Riv99], consider the solution $\Phi_{p}$ to

$\displaystyle\left\\{\begin{array}[]{ll}\Delta\Phi_{p}=2\pi\delta_{p}(x)&\mbox{in}\
\Omega,\\\\[5.0pt] \dfrac{\partial\Phi_{p}}{\partial n}=g\times
g_{\tau}&\mbox{on}\ \Gamma,\end{array}\right.$

with associated asymptotic energy expansion

$E_{\varepsilon}^{g,1}(u_{\varepsilon})=\pi|\ln\varepsilon|+W(p)+c_{\Omega}+o(1)$

where

$W(p)=\lim_{\rho\to
0}\left(\frac{1}{2}\int_{\Omega_{\rho}}|\nabla\Phi_{p}|^{2}\,dx-\pi\ln\left(\frac{1}{\rho}\right)\right)$

is the renormalized energy and $c_{\Omega}$ is the vortex core energy
associated to $p$. It can be shown (see [BBH94] for example) that the
renormalized energy $W$ has a minimum value of zero at the origin $p=0$.
Two Defects on $\Gamma$
Let $q_{1},q_{2}\in\Gamma$ be the boundary singularities and consider the PDE

$\displaystyle\left\\{\begin{array}[]{ll}\Delta\Phi_{q}=0&\mbox{in}\
\Omega,\\\\[5.0pt] \dfrac{\partial\Phi_{q}}{\partial
n}=g\times\partial_{\tau}g-\pi(\delta_{q_{1}}(x)+\delta_{q_{2}}(x))&\mbox{on}\
\Gamma\end{array}\right.$

which has solution $\Phi_{q}(x)=\ln|x-q_{1}|+\ln|x-q_{2}|$. The energy
expansion and renormalized energy are

$\displaystyle E_{\varepsilon}^{g,1}(u_{\varepsilon})$
$\displaystyle=\pi|\ln\varepsilon|+W(q_{1},q_{2})+2c_{\Gamma}+o(1),$
$\displaystyle W(q_{1},q_{2})$ $\displaystyle=\lim_{\rho\to
0}\left(\frac{1}{2}\int_{\Omega_{\rho}}|\nabla\Phi_{q}|^{2}\,dx-\pi\ln\left(\frac{1}{\rho}\right)\right),$

and $c_{\Gamma}$ represents the vortex core energy associated to each $q_{j}$.
By the identity

$\int_{\Omega_{\rho}}|\nabla\Phi_{q}|^{2}\,dx=\sum_{j=1}^{2}\int_{\partial
B_{\rho}(q_{j})\cap\Omega}\Phi_{q}\frac{\partial\Phi_{q}}{\partial
n_{q_{j}}}ds+\int_{\Gamma\setminus(\Gamma_{\rho}(q_{1})\cup\Gamma_{\rho}(q_{2}))}\Phi_{q}\frac{\partial\Phi_{q}}{\partial
n}ds$

it can be shown via direct calculation that

$W(q_{1},q_{2})=-\pi\ln|q_{1}-q_{2}|$

which is minimized whenever $q_{1}$ and $q_{2}$ are antipodal. In particular,

$\min_{q_{1},q_{2}\in\Gamma}W(q_{1},q_{2})=-\pi\ln|2q_{1}|<0=\min_{p\in\Omega}W(p).$

This calculation suggests that the case where $\Sigma=\\{q_{1},q_{2}\\}$ gives
the energetically preferable singularity allocation. To conclude that this is
indeed the case, it must be shown that the core energy associated to a
boundary vortex is not too large compared to $c_{\Omega}/2$. Although we do
not have a rigorous proof for this, we believe it is possible to show using
estimates such as those found in [ABM20].

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