$|\nabla\mathbf{M}_{\varepsilon}|^{2}\lesssim|\nabla\tilde{\mathbf{M}}_{\varepsilon}|^{2}$.
The latter can be estimated by differentiating both sides of (4.91), by the
BV-chain rule; this gives
$\|\nabla\tilde{\mathbf{M}}_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\sigma_{\varepsilon}})}\lesssim\|\nabla\tilde{\mathbf{Q}}_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\sigma_{\varepsilon}})}\lesssim\left|\log\varepsilon\right|$.
Let

$\eta_{\varepsilon}:=\frac{\sqrt{2}\beta+1}{u\left(\frac{\sigma_{\varepsilon}}{\varepsilon}\right)^{2}}$

We observe that $\eta_{\varepsilon}\to 1$ as $\varepsilon\to 0$, due to the
condition at infinity in (4.92). We have in $A_{\varepsilon}^{1}$:

$\frac{\varepsilon}{2}|\nabla\mathbf{M}_{\varepsilon}|^{2}=\frac{\eta_{\varepsilon}}{2}\left|u^{\prime}\left(\frac{|x_{2}|}{\varepsilon}\right)\right|^{2}+\mathrm{O}(\varepsilon|\nabla\tilde{\mathbf{M}}_{\varepsilon}|^{2})$

and therefore:

$\displaystyle\frac{\varepsilon}{2}\int_{A^{1}_{\varepsilon}}|\nabla\mathbf{M}_{\varepsilon}|^{2}\mathrm{d}x$
$\displaystyle\leq\mathrm{O}(\varepsilon|\log\varepsilon|)+\frac{\eta_{\varepsilon}}{2}\int_{A_{\varepsilon}^{1}}\left|u^{\prime}\left(\frac{|x_{2}|}{\varepsilon}\right)\right|^{2}\mathrm{d}x+\mathrm{O}(\varepsilon\|\nabla\tilde{\mathbf{M}}_{\varepsilon}\|^{2}_{L^{2}(\Omega_{\sigma_{\varepsilon}})})$
$\displaystyle=\mathrm{O}(\varepsilon|\log\varepsilon|)+\frac{\eta_{\varepsilon}}{2}\mathscr{H}^{1}(L_{1})\cdot\int_{-\sigma_{\varepsilon}}^{\sigma_{\varepsilon}}\left|u^{\prime}\left(\frac{|x_{2}|}{\varepsilon}\right)\right|^{2}\mathrm{d}x_{2}$
$\displaystyle=\mathrm{O}(\varepsilon|\log\varepsilon|)+\eta_{\varepsilon}\,\mathscr{H}^{1}(L_{1})\cdot\int_{0}^{\frac{\sigma_{\varepsilon}}{\varepsilon}}\left|u^{\prime}(t)\right|^{2}\,\mathrm{d}t.$

By repeating this argument on each $A^{j}_{\varepsilon}$, we deduce

(4.96)
$\frac{\varepsilon}{2}\int_{\Omega_{\sigma_{\varepsilon}}}|\nabla\mathbf{M}_{\varepsilon}|^{2}\mathrm{d}x\leq\mathrm{O}(\varepsilon|\log\varepsilon|)+\eta_{\varepsilon}\,\mathbb{L}(a_{1},\,\ldots,\,a_{2\left|d\right|})\cdot\int_{0}^{\frac{\sigma_{\varepsilon}}{\varepsilon}}\left|u^{\prime}(t)\right|^{2}\,\mathrm{d}t.$

Next, we estimate the potential term. On
$\Omega_{\sigma_{\varepsilon}}\setminus\bigcup_{j=1}^{\left|d\right|}A_{\varepsilon}^{j}$,
we have $|\mathbf{Q}_{\varepsilon}|=1+\kappa_{*}\varepsilon$ and
$|\mathbf{M}_{\varepsilon}|=(\sqrt{2}\beta+1)^{\frac{1}{2}}$. The identity
(4.91) can be written as

$\frac{\mathbf{Q}_{\varepsilon}}{1+\kappa_{*}\varepsilon}=\sqrt{2}\left(\frac{\mathbf{M}_{\varepsilon}\otimes\mathbf{M}_{\varepsilon}}{\sqrt{2}\beta+1}-\frac{\mathbf{I}}{2}\right)$

which implies

$\displaystyle\mathbf{Q}_{\varepsilon}\mathbf{M}_{\varepsilon}\cdot\mathbf{M}_{\varepsilon}$
$\displaystyle=\sqrt{2}(1+\kappa_{*}\varepsilon)\left(\frac{|\mathbf{M}_{\varepsilon}|^{4}}{\sqrt{2}\beta+1}-\frac{1}{2}|\mathbf{M}_{\varepsilon}|^{2}\right)=\frac{\sqrt{2}}{2}(1+\kappa_{*}\varepsilon)(\sqrt{2}\beta+1).$

In conclusion, at each point of
$\Omega_{\sigma_{\varepsilon}}\setminus\bigcup_{j=1}^{\left|d\right|}A_{\varepsilon}^{j}$
we have

(4.97)
$\begin{split}f_{\varepsilon}(\mathbf{Q}_{\varepsilon},\mathbf{M}_{\varepsilon})&=\frac{1}{4}(2\kappa_{*}\varepsilon+\kappa_{*}^{2}\varepsilon^{2})^{2}+\frac{\varepsilon\beta^{2}}{2}-\frac{\beta\varepsilon}{\sqrt{2}}(1+\kappa_{*}\varepsilon)(\sqrt{2}\beta+1)+\kappa_{\varepsilon}\\\
&=\mathrm{o}_{\varepsilon\to 0}(\varepsilon^{2})\end{split}$

by taking Lemma 3.1 into account. Therefore, the total contribution from the
potential on
$\Omega_{\sigma_{\varepsilon}}\setminus\bigcup_{j=1}^{\left|d\right|}A_{\varepsilon}^{j}$
is negligible. Let us compute the potential on $A_{\varepsilon}^{j}$.
Considering for simplicity the case $j=1$, again we have
$|\mathbf{Q}_{\varepsilon}|=1+\kappa_{*}\varepsilon$, but

$|\mathbf{M}_{\varepsilon}(x)|=\eta_{\varepsilon}^{1/2}\,u\left(\frac{|x_{2}|}{\varepsilon}\right)$

Then, (4.91) can be written as

$\mathbf{Q}_{\varepsilon}=\sqrt{2}(1+\kappa_{*}\varepsilon)\left(\frac{\mathbf{M}_{\varepsilon}\otimes\mathbf{M}_{\varepsilon}}{|\mathbf{M}_{\varepsilon}|^{2}}-\frac{\mathbf{I}}{2}\right)$

which implies

$\mathbf{Q}_{\varepsilon}\mathbf{M}_{\varepsilon}\cdot\mathbf{M}_{\varepsilon}=\frac{\sqrt{2}}{2}(1+\kappa_{*}\varepsilon)\,\eta_{\varepsilon}\,u^{2}\left(\frac{|x_{2}|}{\varepsilon}\right)$

At a generic point $x\in A_{\varepsilon}^{1}$, we have (writing
$v_{\varepsilon}:=\eta_{\varepsilon}^{1/2}u(|x_{2}|/\varepsilon)$ for
simplicity)

$\displaystyle
f_{\varepsilon}(\mathbf{Q}_{\varepsilon},\mathbf{M}_{\varepsilon})$
$\displaystyle=\frac{\kappa_{*}\varepsilon^{2}}{4}(2+\kappa_{*}\varepsilon)^{2}+\frac{\varepsilon}{4}(1-v_{\varepsilon}^{2})-\beta\varepsilon\frac{\sqrt{2}}{2}(1+\kappa_{*}\varepsilon)v_{\varepsilon}^{2}+\frac{1}{2}(\beta^{2}+\sqrt{2}\beta)\varepsilon+\kappa_{*}^{2}\varepsilon^{2}+\mathrm{o}(\varepsilon^{2})$
$\displaystyle=2\kappa_{*}^{2}\varepsilon^{2}+\frac{\varepsilon}{4}(1-v_{\varepsilon}^{2})^{2}-\frac{\beta\varepsilon}{\sqrt{2}}v_{\varepsilon}^{2}-\frac{\kappa_{*}\beta\varepsilon^{2}}{\sqrt{2}}v_{\varepsilon}^{2}+\frac{1}{2}(\beta^{2}+\sqrt{2}\beta)\varepsilon+\mathrm{o}(\varepsilon^{2})$
$\displaystyle=\mathrm{O}(\varepsilon^{2})+\varepsilon\left(h(v_{\varepsilon},0)-\frac{\beta^{2}+\sqrt{2}\beta}{2}\right)+\frac{1}{2}(\beta^{2}+\sqrt{2}\beta)\varepsilon$
$\displaystyle=\mathrm{O}(\varepsilon^{2})+\varepsilon h(v_{\varepsilon},0)$
$\displaystyle=\mathrm{O}(\varepsilon^{2})+\frac{\varepsilon}{2}H^{2}(v_{\varepsilon}).$

By repeating this argument on each $A^{j}_{\varepsilon}$, and taking the
integral over $A^{j}_{\varepsilon}$, we obtain

(4.98)
$\begin{split}\frac{1}{\varepsilon^{2}}\int_{\bigcup_{j=1}^{\left|d\right|}A_{\varepsilon}^{j}}f_{\varepsilon}(\mathbf{Q}_{\varepsilon},\mathbf{M}_{\varepsilon})\,\mathrm{d}x&=\int_{\bigcup_{j=1}^{\left|d\right|}A_{\varepsilon}^{j}}\mathrm{O}(1)+\frac{1}{2\varepsilon}H^{2}\left(\eta_{\varepsilon}^{1/2}\,u\left(\frac{|x_{2}|}{\varepsilon}\right)\right)\mathrm{d}x\\\
&=\mathrm{O}(\sigma_{\varepsilon})+\frac{1}{2\varepsilon}\sum_{j=1}^{\left|d\right|}\int_{A_{\varepsilon}^{j}}H^{2}\left(\eta_{\varepsilon}^{1/2}\,u\left(\frac{|x_{2}|}{\varepsilon}\right)\right)\mathrm{d}x\\\
&=\mathrm{o}_{\varepsilon\to
0}(1)+\mathbb{L}(a_{1},\cdots,a_{2{\left|d\right|}})\cdot\int_{0}^{\frac{\sigma_{\varepsilon}}{\varepsilon}}H^{2}(\eta_{\varepsilon}^{1/2}\,u(t))\,\mathrm{d}t.\end{split}$

By combining (4.90), (4.96), (4.97) and (4.98), keeping in mind that
$\eta_{\varepsilon}\to 1$, $\sigma_{\varepsilon}/\varepsilon\to+\infty$ as
$\varepsilon\to 0$, and applying Lebesgue’s dominated convergence theorem, we
obtain

(4.99)
$\begin{split}\mathscr{F}_{\varepsilon}(\mathbf{Q}_{\varepsilon},\mathbf{M}_{\varepsilon};\Omega\setminus\bigcup_{j=1}^{2\left|d\right|}B_{\sigma_{\varepsilon}}(a_{j}))&\leq\mathrm{o}_{\varepsilon\to
0}(1)+\mathbb{L}(a_{1},\cdots,a_{2d})\int_{0}^{+\infty}\left(u^{\prime
2}(t)+H^{2}(u(t))\right)\mathrm{d}t\\\
&\qquad+2\pi\left|d\right|\left|\log\varepsilon\right|+\mathbb{W}(a_{1},\cdots,a_{2\left|d\right|})+2\left|d\right|\gamma_{*}\\\
&\stackrel{{\scriptstyle\eqref{inte3}}}{{=}}2\pi\left|d\right|\left|\log\varepsilon\right|+\mathbb{W}_{\beta}(a_{1},\cdots,a_{2\left|d\right|})+2\left|d\right|\gamma_{*}+\mathrm{o}_{\varepsilon\to
0}(1).\end{split}$

It only remains to define $\mathbf{M}_{\varepsilon}$ in each ball
$B_{\sigma_{\varepsilon}}(a_{j})$. For each $j$, there exists
$\rho=\rho(j)\in(\sigma_{\varepsilon},2\sigma_{\varepsilon})$ such that

$\int_{\partial
B_{\rho}(a_{j})}|\nabla\mathbf{M}_{\varepsilon}|^{2}\,\mathrm{d}\mathscr{H}^{1}\leq\frac{1}{\sigma_{\varepsilon}}\int_{B_{2\sigma_{\varepsilon}}(a_{j})\setminus
B_{\sigma_{\varepsilon}}(a_{j})}|\nabla\mathbf{M}_{\varepsilon}|^{2}\,\mathrm{d}x=\mathrm{O}\left(\frac{\left|\log\varepsilon\right|}{\sigma_{\varepsilon}}\right)+\mathrm{O}(\frac{1}{\varepsilon})$

Define $\mathbf{M}_{\varepsilon}$ on $B_{\rho}(a_{j})$ as

(4.100)
$\mathbf{M}_{\varepsilon}(x):=\frac{|x-a_{j}|}{\rho}\mathbf{M}_{\varepsilon}\left(\frac{\rho(x-a_{j})}{|x-a_{j}|}\right)$

The vector field $\mathbf{M}_{\varepsilon}$ was already defined in
$B_{\rho}(a_{j})\setminus B_{\sigma_{\varepsilon}}(a_{j})$, but we disregard
its previous values and re-define it according to (4.100). We have

(4.101)
$\begin{split}\varepsilon\int_{B_{\rho}(a_{j})}|\nabla\mathbf{M}_{\varepsilon}|^{2}\,\mathrm{d}x&\leq\sigma_{\varepsilon}\int_{\partial
B_{\rho}(a_{j})}\varepsilon|\nabla\mathbf{M}_{\varepsilon}|^{2}\,\mathrm{d}\mathscr{H}^{1}+\varepsilon\int_{B_{\rho}(a_{j})}\mathrm{O}\left(\frac{1}{\rho^{2}}\right)\,\mathrm{d}x\\\
&\leq\mathrm{O}(\varepsilon|\log\varepsilon|)+\mathrm{O}(\sigma_{\varepsilon})+\mathrm{O}(\varepsilon)\to
0\end{split}$

and

(4.102)
$\frac{1}{\varepsilon^{2}}\int_{B_{\rho}(a_{j})}\left(f_{\varepsilon}(\mathbf{Q}_{\varepsilon},\mathbf{M}_{\varepsilon})-\frac{1}{4}(1-|\mathbf{Q}_{\varepsilon}|^{2})^{2}\right)\mathrm{d}x=\mathrm{O}(\frac{\sigma_{\varepsilon}^{2}}{\varepsilon}).$

If we choose $\varepsilon\ll\sigma_{\varepsilon}\ll\varepsilon^{\frac{1}{2}}$,
then the total contribution of $\mathbf{M}_{\varepsilon}$ to the energy on
each ball $B_{\rho}(a_{j})$ tends to zero as $\varepsilon\to 0$. ∎

######  Remark 4.2.

The proof of Proposition 4.19 carries over, with no essential modifications,
to the case we impose Dirichlet boundary conditions for the
$\mathbf{Q}$-component and Neumann boundary conditions for the
$\mathbf{M}$-component, as described in Remark 2.1. Indeed, while the
structure of the (orientable) boundary datum for $\mathbf{Q}$ is important to
the analysis, the boundary condition for $\mathbf{M}$ does not play a crucial
role; the coupling between $\mathbf{Q}$ and $\mathbf{M}$ is determined by the
potential $f_{\varepsilon}$ and not the boundary conditions.

We can now complete the proof of our main result, Theorem 2.1.

###### Conclusion of the proof of Theorem 2.1, proof of Proposition 4.13.

From Proposition 4.14 and Proposition 4.19, we deduce

(4.103)
$\begin{split}&\mathbb{W}(a^{*}_{1},\,\ldots,\,a^{*}_{2\left|d\right|})+c_{\beta}\,\mathscr{H}^{1}(\mathrm{S}_{\mathbf{M}^{*}})+\int_{\Omega}(\xi_{*}-\kappa_{*})^{2}\,\mathrm{d}x+2\left|d\right|\gamma_{*}\\\
&\hskip 56.9055pt\leq\liminf_{\varepsilon\to
0}\big{(}\mathscr{F}_{\varepsilon}(\mathbf{Q}^{*}_{\varepsilon},\,\mathbf{M}^{*}_{\varepsilon})-2\pi\left|d\right|\left|\log\varepsilon\right|\big{)}\\\
&\hskip 56.9055pt\leq\limsup_{\varepsilon\to
0}\big{(}\mathscr{F}_{\varepsilon}(\mathbf{Q}^{*}_{\varepsilon},\,\mathbf{M}^{*}_{\varepsilon})-2\pi\left|d\right|\left|\log\varepsilon\right|\big{)}\\\
&\hskip
56.9055pt\leq\mathbb{W}(a_{1},\,\ldots,\,a_{2\left|d\right|})+c_{\beta}\,\mathbb{L}(a_{1},\,\ldots,\,a_{2\left|d\right|})+2\left|d\right|\gamma_{*}\end{split}$

for any $(2\left|d\right|)$-uple of distinct points $a_{1}$, …,
$a_{2\left|d\right|}$ in $\Omega$. In particular, choosing $a_{j}=a_{j}^{*}$,
we obtain

(4.104)
$\mathscr{H}^{1}(\mathrm{S}_{\mathbf{M}^{*}})=\mathbb{L}(a^{*}_{1},\,\ldots,\,a^{*}_{2\left|d\right|}),\qquad\xi_{*}=\kappa_{*}$

and Proposition (4.13) follows. Moreover, Proposition 4.15 and (4.104) imply
that the jump set $\mathrm{S}_{\mathbf{M}^{*}}$ coincides (up to negligible
sets) with $\cup_{j=1}^{\left|d\right|}L_{j}$, where
$(L_{1},\,\ldots,\,L_{\left|d\right|})$ is a minimal connection for
$(a_{1},\,\ldots,\,a_{2\left|d\right|})$. Finally, from (4.103) and (4.104) we
deduce

(4.105)
$\mathbb{W}_{\beta}(a^{*}_{1},\,\ldots,\,a^{*}_{2\left|d\right|})\leq\mathbb{W}_{\beta}(a_{1},\,\ldots,\,a_{2\left|d\right|})$

for any $(2\left|d\right|)$-uple of distinct points $a_{1}$, …,
$a_{2\left|d\right|}$ in $\Omega$ — that is,
$(a^{*}_{1},\,\ldots,\,a^{*}_{2\left|d\right|})$ minimises
$\mathbb{W}_{\beta}$. ∎

##  5 Numerics

In this section, we numerically compute some stable critical points of the
ferronematic free energy, on square domains with topologically non-trivial
Dirichlet boundary conditions for $\mathbf{Q}$ and $\mathbf{M}$. These
numerical results do not directly support our main results on global energy
minimizers of (2.1) in the $\epsilon\to 0$ limit, since the numerically
computed critical points need not be global energy minimizers, and we expect
multiple local and global energy minimizers of (2.1) for $\epsilon>0$.

Instead of solving the Euler-Lagrange equations directly, we solve a
$L^{2}$-gradient flow associated with the effective re-scaled free energy for
ferronematics (2.1), given by

(5.1)
$\frac{\mathrm{d}}{\mathrm{d}t}\mathscr{F}_{\varepsilon}(\mathbf{Q},\mathbf{M})=-\int_{\Omega}(\eta_{1}|\partial_{t}\mathbf{Q}|^{2}+\eta_{2}|\partial_{t}\mathbf{M}|^{2})\mathrm{d}{\bf
x}.$

Here $\eta_{1}>0$ and $\eta_{2}>0$ are arbitrary friction coefficients. Due to
limited physical data, we do not comment on physically relevant values of
$\varepsilon$, $\beta$ and the friction coefficients. The system of
$L^{2}$-gradient flow equations for $Q_{11}$, $Q_{12}$ and the components,
$M_{1}$, $M_{2}$ of the magnetisation vector, can be written as

(5.2) $\begin{cases}&2\eta_{1}\,\partial_{t}Q_{11}=2\Delta
Q_{11}-\frac{1}{\varepsilon^{2}}(4Q_{11}(Q_{11}^{2}+Q_{12}^{2}-1/2)-\beta\varepsilon(M_{1}^{2}-M_{2}^{2}))\\\
&2\eta_{1}\,\partial_{t}Q_{12}=2\Delta
Q_{12}-\frac{1}{\varepsilon^{2}}(4Q_{12}(Q_{11}^{2}+Q_{12}^{2}-1/2)-2\beta\varepsilon
M_{1}M_{2})\\\ &\eta_{2}\,\partial_{t}M_{1}=\varepsilon\Delta
M_{1}-\frac{1}{\varepsilon^{2}}(\varepsilon(M_{1}^{2}+M_{2}^{2}-1)M_{1}-\beta\varepsilon(2Q_{11}M_{1}+2Q_{12}M_{2}))\\\
&\eta_{2}\,\partial_{t}M_{2}=-\varepsilon\Delta
M_{2}-\frac{1}{\varepsilon^{2}}(\varepsilon(M_{1}^{2}+M_{2}^{2}-1)M_{2}-\beta\varepsilon(-2Q_{11}M_{2}+2Q_{12}M_{1})),\\\
\end{cases}$

The stationary time-independent or equilibrium solutions of the
$L^{2}$-gradient flow satisfy the original Euler-Lagrange equations of (2.1).
For non-convex free energies as in (2.1), there are multiple critical points,
with many of them being unstable saddle points [47]. One can efficiently
compute stable critical points of such free energies by considering the
$L^{2}$-gradient flow associated with the non-convex free energies and these
gradient flows converge to a stable critical point, for a given initial
condition, thus avoiding the unstable saddle points. From a numerical
standpoint, the $L^{2}$-gradient flow can be more straightforward to solve
than the nonlinear coupled Euler-Lagrange equations, primarily due to the
inclusion of time relaxation in the $L^{2}$-gradient flow.

In the following simulations, we take $\eta_{1}=1$ and $\eta_{2}=\varepsilon$
and do not offer rigorous justifications for these choices, except as
numerical experiments to qualitatively support out theoretical results. We
impose the continuous degree $+k$ boundary condition

(5.3) $\mathbf{M}_{b}=(\sqrt{2}\beta+1)^{1/2}(\cos k\theta,\sin
k\theta),\quad{\bf Q}_{b}=\sqrt{2}\begin{pmatrix}\frac{1}{2}\cos
2k\theta&\frac{1}{2}\sin 2k\theta\\\\[5.0pt] \frac{1}{2}\sin
2k\theta&\frac{1}{2}\cos 2k\theta\\\ \end{pmatrix},$

where

(5.4) $\theta(x,y)={\rm
atan2}\left({y-0.5},\,{x-0.5}\right)-\pi/2,\quad(x,y)\in\partial\Omega.$

and ${\rm atan2}(y,x)$ is the 2-argument arctangent that computes the
principal value of the argument function applied to the complex number $x+iy$.
So $-\pi\leq{\rm atan2}(y,x)\leq\pi$. For example, if $x>0$, then ${\rm
atan2}(y,x)=\arctan\left(\frac{y}{x}\right)$. The initial condition is
prescribed to be

(5.5) $\mathbf{M}_{0}=(\sqrt{2}\beta+1)^{1/2}(\cos k\theta,\sin
k\theta),\quad{\bf Q}_{0}=\sqrt{2}\begin{pmatrix}\frac{1}{2}\cos
2k\theta&\frac{1}{2}\sin 2k\theta\\\\[5.0pt] \frac{1}{2}\sin
2k\theta&\frac{1}{2}\cos 2k\theta,\\\ \end{pmatrix}$

where

(5.6) $\theta(x,y)={\rm
atan2}\left({y-0.5},\,{x-0.5}\right)-\pi/2,\quad(x,y)\in(0,1)^{2}.$

We solve the $L^{2}$-gradient flow equation using standard central finite
difference methods [33]. For the temporal discretization, we employ a second-
order Crank-Nicolson method [33]. The grid size and temporal step size are
denoted by $h$ and $\tau$, respectively. In all our computations, we set
$h=1/50$ and $\tau=1/1000$.

Figure 1: Numerical results for the gradient flows (5.2) with (a)
$\beta=1,\varepsilon=0.05$ at $t=0.02$, $0.05$ and $1$ and (b)
$\beta=1,\varepsilon=0.02$ at $t=0.02$, $0.05$ and $1$ (Continuous degree +1
boundary condition, $h=1/50$, $\tau=1/1000$). In each sub-figure, the nematic
configuration is shown in the left panel, where the white bars represent
nematic field ${\bf n}$ (the eigenvector of $\mathbf{Q}$ associated with the
largest eigenvalue) and the color represents
$\operatorname{tr}\mathbf{Q}^{2}=2(Q_{11}^{2}+Q_{12}^{2})$; the
$\mathbf{M}$-profile is shown in the right panel, where the white bars
represent magnetic field ${\bf M}$ and the color bar represents $|{\bf
M}|^{2}=M_{1}^{2}+M_{2}^{2}$.  Figure 2: Numerical results for the gradient
flows (5.2) with (a) $\beta=1,\varepsilon=0.05$ at $t=0.02$, $0.05$ and $1$
and (b) $\beta=1,\varepsilon=0.02$ at $t=0.02$, $0.05$ and $1$ (Continuous
degree +2 boundary condition, $h=1/50$, $\tau=1/1000$). In each sub-figure,
the nematic configuration is shown in the left panel, where the white bars
represent nematic field ${\bf n}$ (the eigenvector of $\mathbf{Q}$ associated
with the largest eigenvalue) and the color represents
$\operatorname{tr}\mathbf{Q}^{2}=2(Q_{11}^{2}+Q_{12}^{2})$; the magnetic
configuration is shown in the right panel, where the white bars represent
magnetic field ${\bf M}$ and the color represents $|{\bf
M}|^{2}=M_{1}^{2}+M_{2}^{2}$.

In Figure 1, we plot the dynamical evolution of the solutions of the gradient
flow equations, for $k=1$ boundary conditions, with the initial condition
(5.5). The time-dependent solutions converge for $t\geq 1$, and we treat the
numerical solution at $t=1$ to the converged equilibrium state. We cannot
conclusively argue that the converged solution is an energy minimizer but it
is locally stable, the converged $\mathbf{Q}$-profile has two non-orientable
defects and the corresponding $\mathbf{M}$-profile has a jump set composed of
a straight line connecting the nematic defect pair, consistent with our
theoretical results on global energy minimizers. We consider two different
values of $\varepsilon$ and it is clear that the $\mathbf{Q}$-defects and the
jump set in $\mathbf{M}$ become more localised as $\varepsilon$ becomes
smaller, as expected from the theoretical- results. We have also investigated
the effects of $\beta$ on the converged solutions — the defects become closer
as $\beta$ increases. This is expected since the cost of the minimal
connection between the nematic defects increases as $\beta$ increases, and
hence the shorter connections require the defects to be closer to each other
(at least in a pairwise sense).

In Figure 2, we plot the dynamical evolution of the solutions of the gradient
flow equations, for $k=2$ boundary conditions, with the initial condition
(5.5), and we treat the numerical solution at $t=1$ to be the converged
equilibrium state. Again, the converged solution is locally stable, the
$\mathbf{Q}$-profile has four non-orientable defects,the $\mathbf{M}$-profile
has two distinct jump sets connecting two pairs of non-orientable nematic
defects, and the jump sets are indeed approximately straight lines. Smaller
values of $\epsilon$ correspond to the sharp interface limit which induces
more localised defects for $\mathbf{Q}$, straighter line defects for
$\mathbf{M}$ and larger values of $\beta$ push the defects closer together,
all in qualitative agreement with our theoretical results.

Theorem 2.1 is restricted to global minimizers of (2.1) in the $\epsilon\to 0$
limit, but the numerical illustrations in Figures 1 and 2 suggest that Theorem
2.1 may also partially apply to local energy minimizers of (2.1). In other
words, locally energy minimizing pairs,
$(\mathbf{Q}_{\varepsilon},\mathbf{M}_{\varepsilon})$, may also converge to a
pair $(\mathbf{Q}^{*},\mathbf{M}^{*})$, for which $\mathbf{Q}^{*}$ is a
canonical harmonic map with non-orientable point defects and $\mathbf{M}^{*}$
has a jump set connecting the non-orientable point defects of
$\mathbf{Q}^{*}$, with the location of the defects being prescribed by the
critical point(s) of the normalization energy in Theorem 2.1. The numerical
illustrations in Figures 1 and 2 cannot be directly related to Theorem 2.1,
since we have only considered two small and non-zero values of $\epsilon$ and
for a fixed $\beta>0$, there maybe multiple local and global energy minimizers
with different jump sets in $\mathbf{M}$ i.e. different choices of the minimal
connection of equal length, or different connections of different lengths
between the nematic defect pairs. For example, it is conceivable that a
locally stable $\mathbf{M}$-profile also connects the nematic defects by means
of straight lines, but this connection is not minimal. There may also be non
energy-minimising critical points with orientable point defects in
$\mathbf{M}$ tailored by the non-orientable nematic defects. Similarly, there
may be non energy-minimising critical points with non-orientable and
orientable nematic defects, whose locations are not minimisers but critical
points of the modified renormalised energy in Theorem 2.1. We defer these
interesting questions to future work.

##  6 Conclusions

We study a simplified model for ferronematics in two-dimensional domains, with
Dirichlet boundary conditions, building on previous work in [14]. The model is
only valid for dilute ferronematic suspensions and we do not expect
quantitative agreement with experiments. Further, the experimentally relevant
choices for the boundary conditions for $\mathbf{M}$ are not well established
and our methods can be adapted to other choices of boundary conditions e.g.
Neumann conditions for the magnetisation vector. Similarly, it is not clear if
topologically non-trivial Dirichlet conditions can be imposed on the nematic
directors, for physically relevant experimental scenarios. Having said that,
our model problem is a fascinating mathematical problem because of the
tremendous complexity of ferronematic solution landscapes, the multiplicity of
the energy minimizers and non energy-minimizing critical points, and the
multitude of admissible coupled defect profiles for the nematic and magnetic
profiles. There are several forward research directions, some of which could
facilitate experimental observations of the theoretically predicted
morphologies in this manuscript. For example, one could study the
experimentally relevant generalisation of our model problem with Dirichlet
conditions for $\mathbf{Q}$ and Neumann conditions for $\mathbf{M}$, or study
different asymptotic limits of the ferronematic free energy in (1.1), a prime
candidate being the $\varepsilon\to 0$ limit for fixed $\xi$ and $c_{0}$
(independent of $\varepsilon$). This limit, although relevant for dilute
suspensions, would significantly change the vacuum manifold $\mathscr{N}$ in
the $\varepsilon\to 0$ limit. In fact, we expect to observe stable point
defects in the energy-minimizing $\mathbf{M}$-profiles for this limit, where
$\xi$ and $c_{0}$ are independent of $\varepsilon$, as $\varepsilon\to 0$.
Further, there is the interesting question of how this ferronematic model can
be generalised to non-dilute suspensions or to propose a catalogue of magneto-
nematic coupling energies for different kinds of MNP-MNP interactions and MNP-
NLC interactions. The physics of ferronematics is complex, and it is
challenging to translate the physics to tractable mathematical problems with
multiple order parameters, and we hope that our work is solid progress in this
direction with bright interdisciplinary prospects.

Taxonomy: GC, BS and AM conceived the project based on a model developed by AM
and her ex-collaborators. GC and BS led the analysis, followed by AM. YW
performed the numerical simulations, as advised by AM and GC. All authors
contributed to the scientific writing.

Acknowledgements: GC, BS and AM gratefully acknowledge support from the CIRM-
FBK (Trento) Research in Pairs grant awarded in 2019, when this collaboration
was initiated. GC, BS and AM gratefully acknowledge support from an ICMS
Research in Groups grant awarded in 2020, which supported the completion of
this project and submission of this manuscript. AM gratefully acknowledges the
hospitality provided by the University of Verona in December 2019, GC
gratefully acknowledges the hospitality provided by the University Federico II
(Naples) under the PRIN project 2017TEXA3H, and AM, BS and GC gratefully
acknowledge support from the Erwin Schrodinger Institute in Vienna in December
2019, all of which facilitated this collaboration. AM acknowledges support
from the Leverhulme Trust and the University of Strathclyde New Professor’s
Fund. We thank the referee for their careful reading of the manuscript and
comments.

Data Availability Statement: Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

Conflict of interest statement. The authors have no competing interests to
declare that are relevant to the content of this article.

##  Appendix A Lifting of a map with non-orientable singularities

The aim of this section is to prove Proposition 4.15. We reformulate the
problem in a slightly more general setting.

Let $a\in\mathbb{R}^{2}$, and let $\mathbf{Q}\in
W^{1,2}_{\operatorname{loc}}(\mathbb{R}^{2}\setminus\\{a\\},\,\mathscr{N})$.
By Fubini theorem and Sobolev embedding, the restriction of $\mathbf{Q}$ on
the circle $\partial B_{\rho}(a)$ is well-defined and continuous for a.e.
$\rho>0$. Therefore, it makes sense to define the topological degree of
$\mathbf{Q}$ on $\partial B_{\rho}(a)$ as an half-integer,
$\deg(\mathbf{Q},\,a)\in\frac{1}{2}\mathbb{Z}$. As the notation suggests, the
degree is independent of the choice of $\rho$: for a.e. $0<\rho_{1}<\rho_{2}$,
the degrees of $\mathbf{Q}$ on $\partial B_{\rho_{1}}(a)$ and $\partial
B_{\rho_{2}}(a)$ are the same. If $\mathbf{Q}$ is smooth, this is a
consequence of the homotopy lifting property; for more general $\mathbf{Q}\in
W^{1,2}_{\operatorname{loc}}(\mathbb{R}^{2}\setminus\\{a\\},\,\mathscr{N})$,
this follows from an approximation argument (based on [44, Proposition p.
267]). We will say that $a$ is a non-orientable singularity of $\mathbf{Q}$ if
$\deg(\mathbf{Q},\,a)\in\frac{1}{2}\mathbb{Z}\setminus\mathbb{Z}$.

Given an open set $\Omega\subseteq\mathbb{R}^{2}$, a map
$\mathbf{Q}\colon\Omega\to\mathscr{N}$ and a unit vector field
$\mathbf{M}\colon\Omega\to\SS^{1}$, we say that $\mathbf{M}$ is a lifting for
$\mathbf{Q}$ if

(A.1)
$\mathbf{Q}(x)=\sqrt{2}\left(\mathbf{M}(x)\otimes\mathbf{M}(x)-\frac{\mathbf{I}}{2}\right)\qquad\textrm{for
a.e. }x\in\Omega.$

Any map $\mathbf{Q}\in\operatorname{BV}(\Omega,\,\mathscr{N})$ admits a
lifting $\mathbf{M}\in\operatorname{BV}(\Omega,\,\SS^{1})$ (see e.g. [32]).
The vector field $\mathbf{M}^{*}$ given by Theorem 2.1 is not a lifting of
$\mathbf{Q}^{*}$, according to the definition above, because
$\left|\mathbf{M}^{*}\right|\neq 1$. However, $\left|\mathbf{M}^{*}\right|$ is
still a positive constant (see Proposition 4.11), so we can construct a
lifting of unit-norm simply by rescaling.

We focus on properties of the lifting for $\mathbf{Q}$-tensors of a particular
form, namely, we assume that $\mathbf{Q}$ has an even number of non orientable
singularities at distinct points $a_{1}$, …, $a_{2d}$. We recall that a
_connection_ for $\\{a_{1},\,\ldots,\,a_{2d}\\}$ as a finite collection of
straight line segments $\\{L_{1},\,\ldots,\,L_{d}\\}$, with endpoints in
$\\{a_{1},\,\ldots,\,a_{2d}\\}$, such that each $a_{i}$ is an endpoint of one
of the segments $L_{j}$. We recall that

(A.2)
$\mathbb{L}(a_{1},\,\ldots,\,a_{2d}):=\min\left\\{\sum_{i=1}^{d}\mathscr{H}^{1}(L_{i})\colon\\{L_{1},\,\ldots,\,L_{d}\\}\textrm{
is a connection for }\\{a_{1},\,\ldots,\,a_{2d}\\}\right\\}\\!.$

A minimal connection for $\\{a_{1},\,\ldots,\,a_{2d}\\}$ is a connection that
attains the minimum in the right-hand side of (A.2). Given two sets $A$, $B$,
we denote their symmetric difference as $A\Delta B:=(A\setminus
B)\cup(B\setminus A)$.

######  Proposition A.1.

Let $\Omega\subseteq\mathbb{R}^{2}$ be a bounded, convex domain, let $d\geq 1$
be an integer, and let $a_{1}$, …, $a_{2d}$ be distinct points in $\Omega$.
Let $\mathbf{Q}\in W^{1,1}(\Omega,\,\mathscr{N})\cap
W^{1,2}_{\operatorname{loc}}(\Omega\setminus\\{a_{1},\,\ldots,a_{2d}\\},\,\mathscr{N})$
be a map with a non-orientable singularity at each $a_{j}$. If
$\mathbf{M}\in\operatorname{SBV}(\Omega,\,\SS^{1})$ is a lifting for
$\mathbf{Q}$ such that $\mathrm{S}_{\mathbf{M}}\subset\\!\subset\Omega$, then

$\mathscr{H}^{1}(\mathrm{S}_{\mathbf{M}})\geq\mathbb{L}(a_{1},\,\ldots,\,a_{2d})$

The equality holds if and only if there exists a minimal connection
$\\{L_{1},\,\ldots,\,L_{d}\\}$ for $\\{a_{1},\,\ldots,a_{d}\\}$ such that
$\mathscr{H}^{1}(\mathrm{S}_{\mathbf{M}}\Delta\cup_{j=1}^{d}L_{j})=0$.

Proposition 4.15 is an immediate consequence of Proposition A.1. The proof of
Proposition A.1 is based on classical results in Geometric Measure Theory, but
we provide it in full detail for the reader’s convenience. Before we prove
Proposition A.1, we state a few preliminary results.

######  Lemma A.2.

If $\\{L_{1},\,\ldots,\,L_{d}\\}$ is a minimal connection for
$\\{a_{1},\,\ldots,\,a_{2d}\\}$, then the $L_{j}$’s are pairwise disjoint.

###### Proof.

Suppose, towards a contradiction, that $\\{L_{1},\,\ldots,\,L_{d}\\}$ is a
minimal connection with $L_{1}\cap L_{2}\neq\emptyset$. The intersection
$L_{1}\cap L_{2}$ must be either a non-degenerate sub-segment of both $L_{1}$
and $L_{2}$ or a point. If $L_{1}\cap L_{2}$ is non-degenerate, then
$(L_{1}\cup L_{2})\setminus(L_{1}\cap L_{2})$ can be written as the disjoint
union of two straight line segments, $K_{1}$ and $K_{2}$, and

$\mathscr{H}^{1}(K_{1})+\mathscr{H}^{1}(K_{2})=\mathscr{H}^{1}((L_{1}\cup
L_{2})\setminus(L_{1}\cap
L_{2}))<\mathscr{H}^{1}(L_{1})+\mathscr{H}^{1}(L_{2})$

This contradicts the minimality of $\\{L_{1},\,\ldots,\,L_{d}\\}$. Now,
suppose that $L_{1}\cap L_{2}$ is a point. By the pigeon-hole principle,
$L_{1}\cap L_{2}$ cannot be an endpoint for either $L_{1}$ or $L_{2}$. Say,
for instance, that $L_{1}$ is the segment of endpoints $a_{1}$, $a_{2}$, while
$L_{2}$ is the segment of endpoints $a_{3}$, $a_{4}$. Let $H_{1}$, $H_{2}$ be
the segments of endpoints $(a_{1},\,a_{3})$, $(a_{2},\,a_{4})$ respectively.
Then, by the triangular inequality,

$\mathscr{H}^{1}(H_{1})+\mathscr{H}^{1}(H_{2})<\mathscr{H}^{1}(L_{1})+\mathscr{H}^{1}(L_{2}),$

which contradicts again the minimality of $\\{L_{1},\,\ldots,L_{d}\\}$. ∎

######  Lemma A.3.

Let $\Omega\subseteq\mathbb{R}^{2}$ be a bounded, convex domain and let
$a_{1}$, …, $a_{2d}$ be distinct points in $\Omega$. Let $\mathbf{Q}\in
W^{1,1}(\Omega,\,\mathscr{N})\cap
W^{1,2}_{\operatorname{loc}}(\Omega\setminus\\{a_{1},\,\ldots,a_{2d}\\},\,\mathscr{N})$
be a map with a non-orientable singularity at each $a_{j}$. If
$\\{L_{1},\,\ldots,\,L_{d}\\}$ is a minimal connection for
$\\{a_{1},\,\ldots,\,a_{2d}\\}$, then there exists a lifting
$\mathbf{M}^{*}\in\operatorname{SBV}(\Omega,\,\SS^{1})$ such that
$\mathscr{H}^{1}(\mathrm{S}_{\mathbf{M}^{*}}\Delta\cup_{j=1}^{d}L_{j})=0$.

###### Proof.

For any $\rho>0$ and $j\in\\{1,\,\ldots,\,d\\}$, we define

$U_{j,\rho}:=\left\\{x\in\mathbb{R}^{2}\colon\operatorname{dist}(x,\,L_{j})<\rho\right\\}\\!.$

and

$\Omega_{\rho}:=\Omega\setminus\bigcup_{j=1}^{d}U_{j,\rho}.$

Since $\Omega$ is convex, $L_{j}\subseteq\Omega$ for any $j$ and hence,
$U_{j,\rho}\subseteq\Omega$ for any $j$ and $\rho$ small enough. Each
$U_{j\,\rho}$ is a simply connected domain with piecewise smooth boundary.
Moreover, for $\rho$ fixed and small, the sets $U_{j,\rho}$ are pairwise
disjoint, because the $L_{j}$’s are pairwise disjoint (Lemma A.2). The trace
of $\mathbf{Q}$ on $\partial U_{j,\rho}$ is orientable, because $\partial
U_{j,\rho}$ contains exactly two non-orientable singularities of $\mathbf{Q}$.
Then, for any $\rho>0$ small enough, $\mathbf{Q}_{|\Omega_{\rho}}$ has a
lifting $\mathbf{M}^{*}_{\rho}\in W^{1,2}(\Omega_{\rho},\,\SS^{1})$ [8,
Proposition 7]. In fact, the lifting is unique up to the choice of the sign
[8, Proposition 2]; in particular, if $0<\rho_{1}<\rho_{2}$ then we have
either $\mathbf{M}^{*}_{\rho_{2}}=\mathbf{M}^{*}_{\rho_{1}}$ a.e. in
$\Omega_{\rho_{2}}$ or $\mathbf{M}^{*}_{\rho_{2}}=-\mathbf{M}^{*}_{\rho_{1}}$
a.e. in $\Omega_{\rho_{2}}$. As a consequence, for any sequence
$\rho_{k}\searrow 0$, we can choose liftings $\mathbf{M}^{*}_{\rho_{k}}\in
W^{1,2}(\Omega_{\rho_{k}},\,\SS^{1})$ of $\mathbf{Q}^{*}_{|\Omega_{\rho_{k}}}$
in such a way that $\mathbf{M}^{*}_{\rho_{k+1}}=\mathbf{M}^{*}_{\rho_{k}}$
a.e. in $\Omega_{\rho_{k}}$. By glueing the $\mathbf{M}^{*}_{\rho_{k}}$’s, we
obtain a lifting

$\mathbf{M}^{*}\in
W^{1,2}_{\operatorname{loc}}(\Omega\setminus\cup_{j}L_{j},\,\SS^{1})$

of $\mathbf{Q}$. By differentiating the identity (A.1), we obtain
$\sqrt{2}\left|\nabla\mathbf{M}^{*}\right|=\left|\nabla\mathbf{Q}\right|$ a.e.
and, since $\nabla\mathbf{Q}\in
L^{1}(\Omega,\,\mathbb{R}^{2}\otimes\mathbb{R}^{2\times 2})$ by assumption, we
deduce that so $\mathbf{M}^{*}\in
W^{1,1}(\Omega\setminus\cup_{j}L_{j},\,\SS^{1})$. The set $\cup_{j}L_{j}$ has
finite length and $\mathbf{M}^{*}$ is bounded, so we also have
$\mathbf{M}^{*}\in\operatorname{SBV}(\Omega,\,\SS^{1})$ (see [3, Proposition
4.4]).

By construction, we have $\mathrm{S}_{\mathbf{M}^{*}}\subseteq\cup_{j}L_{j}$.
Therefore, it only remains to prove that $\mathrm{S}_{\mathbf{M}^{*}}$
contains $\mathscr{H}^{1}$-almost all of $\cup_{j}L_{j}$. Consider, for
instance, the segment $L_{1}$; up to a rotation and traslation, we can assume
that $L_{1}=[0,\,b]\times\\{0\\}$ for some $b>0$. Given a small parameter
$\rho>0$ and $t\in(0,\,b)$, we define
$K_{\rho,t}:=(-\rho,\,t)\times(-\rho,\,\rho)$. Fubini theorem implies that,
for a.e. $\rho$ and $t$, $\mathbf{Q}$ restricted to $\partial K_{\rho,t}$
belongs to $W^{1,2}(\partial K_{\rho,t},\,\mathscr{N})$ and hence, by Sobolev
embedding, is continuous. Since the segments $L_{j}$ are pairwise disjoint by
Lemma A.2, for $\rho$ small enough there is exactly one non-orientable
singularity of $\mathbf{Q}$ inside $K_{\rho,t}$. Therefore, $\mathbf{Q}$ is
non-orientable on $\partial K_{\rho,t}$ for a.e. $t\in(0,\,b)$ and a.e.
$\rho>0$ small enough; in particular, there is no continuous lifting of
$\mathbf{Q}$ on $\partial K_{\rho,t}$. Since $\mathbf{M}^{*}$ is continuous on
$\partial K_{\rho,t}\setminus L_{1}$ for a.e. $\rho$ and $t$, we conclude that
$\mathrm{S}_{\mathbf{M}^{*}}$ contains $\mathscr{H}^{1}$-almost all of
$L_{1}$. ∎

Given a countably $1$-rectifiable set $\Sigma\subseteq\mathbb{R}^{2}$ and a
$\mathscr{H}^{1}$-measurable unit vector field
${\boldsymbol{\tau}}\colon\Sigma\to\SS^{1}$, we say that ${\boldsymbol{\tau}}$
is an orientation for $\Sigma$ if ${\boldsymbol{\tau}}(x)$ spans the
(approximate) tangent line of $\Sigma$ at $x$, for $\mathscr{H}^{1}$-a.e.
$x\in\Sigma$. In case $\Sigma$ is the jump set of an $\operatorname{SBV}$-map
$\mathbf{M}$, ${\boldsymbol{\tau}}\colon\mathrm{S}_{\mathbf{M}}\to\SS^{1}$ is
an orientation for $\mathrm{S}_{\mathbf{M}}$ if and only if
${\boldsymbol{\tau}}(x)\cdot{\boldsymbol{\nu}}_{\mathbf{M}}(x)=0$ for
$\mathscr{H}^{1}$-a.e. $x\in\mathrm{S}_{\mathbf{M}}$.

######  Lemma A.4.

Let $\Omega\subseteq\mathbb{R}^{2}$ be a bounded, convex domain and let
$a_{1}$, …, $a_{2d}$ be distinct points in $\Omega$. Let $\mathbf{Q}\in
W^{1,1}(\Omega,\,\mathscr{N})\cap
W^{1,2}_{\operatorname{loc}}(\Omega\setminus\\{a_{1},\,\ldots,a_{2d}\\},\,\mathscr{N})$
be a map with a non-orientable singularity at each $a_{j}$. Let
$\\{L_{1},\,\ldots,\,L_{d}\\}$ be a minimal connection for
$\\{a_{1},\,\ldots,\,a_{2d}\\}$. Up to relabelling, we assume that $L_{j}$ is
the segment of endpoints $a_{2j-1}$, $a_{2j}$, for any
$j\in\\{1,\,\ldots,\,d\\}$. Let
$\mathbf{M}\in\operatorname{SBV}(\Omega,\,\SS^{1})$ be a lifting for
$\mathbf{Q}$ such that $\mathrm{S}_{\mathbf{M}}\subset\\!\subset\Omega$. Then,
there exist $\mathscr{H}^{1}$-measurable sets $T_{j}\subseteq L_{j}$ and an
orientation ${\boldsymbol{\tau}}_{\mathbf{M}}$ for $\mathrm{S}_{\mathbf{M}}$
such that, for any $\varphi\in C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$, there
holds

$\int_{\mathrm{S}_{\mathbf{M}}}\nabla\varphi\cdot{\boldsymbol{\tau}}_{\mathbf{M}}\,\mathrm{d}\mathscr{H}^{1}=\sum_{j=1}^{d}\left(\varphi(a_{2j-1})-\varphi(a_{2j})\right)-2\sum_{j=1}^{d}\int_{T_{j}}\nabla\varphi\cdot\frac{a_{2j-1}-a_{2j}}{\left|a_{2j-1}-a_{2j}\right|}\,\mathrm{d}\mathscr{H}^{1}$

###### Proof.

Let $\mathbf{M}^{*}\in\operatorname{SBV}(\Omega,\,\SS^{1})$ be the lifting of
$\mathbf{Q}$ given by Lemma A.3. By construction,
$\mathrm{S}_{\mathbf{M}^{*}}$ coincides with
$\cup_{j}L_{j}\subset\\!\subset\Omega$ up to $\mathscr{H}^{1}$-negligible
sets. Since we have assumed that
$\mathrm{S}_{\mathbf{M}}\subset\\!\subset\Omega$, there exists a neighbourhood
$U\subseteq\overline{\Omega}$ of $\partial\Omega$ in $\overline{\Omega}$ such
that $\mathbf{M}\in W^{1,1}(U,\,\SS^{1})$, $\mathbf{M}^{*}\in
W^{1,1}(U,\,\SS^{1})$. A map that belongs to $W^{1,1}(U,\,\mathscr{N})$ has at
most two different liftings in $W^{1,1}(U,\,\SS^{1})$, which differ only for
the sign [8, Proposition 2]. Therefore, since both $\mathbf{M}$ and
$\mathbf{M}^{*}$ are liftings of $\mathbf{Q}$ in $U$, we have that either
$\mathbf{M}=\mathbf{M}^{*}$ a.e. in $U$ or $\mathbf{M}=-\mathbf{M}^{*}$ a.e.
in $U$. Changing the sign of $\mathbf{M}^{*}$ if necessary, we can assume that
$\mathbf{M}=-\mathbf{M}^{*}$ a.e. in $U$. Then, the set

$A:=\\{x\in\Omega\colon\mathbf{M}(x)\cdot\mathbf{M}^{*}(x)=1\\}$

is compactly contained in $\Omega$.

The Leibnitz rule for BV-functions (see e.g. [3, Example 3.97]) implies that
$\mathbf{M}\cdot\mathbf{M}^{*}\in\operatorname{SBV}(\Omega;\,\\{-1,\,1\\})$ As
a consequence, $A$ has finite perimeter in $\Omega$ (see e.g. [3, Theorem
3.40]); since $A\subset\\!\subset\Omega$, $A$ has also finite perimeter in
$\mathbb{R}^{2}$. By the Gauss-Green formula (see e.g. [3, Theorem 3.36, Eq.
(3.47)]), for any $\varphi\in C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$ we have

(A.3)
$0=\int_{A}\operatorname{curl}\nabla\varphi=\int_{\partial^{*}A}\nabla\varphi\cdot{\boldsymbol{\tau}}_{A}\,\mathrm{d}\mathscr{H}^{1},$

where $\partial^{*}A$ is the reduced boundary of $A$ and
${\boldsymbol{\tau}}_{A}$ is an orientation for $\partial^{*}A$. Up to
$\mathscr{H}^{1}$-negligible sets, $\partial^{*}A$ coincides with
$\mathrm{S}_{\mathbf{M}\cdot\mathbf{M}^{*}}$ (see e.g. [3, Example 3.68 and
Theorem 3.61]). By the Leibnitz rule for BV-functions,
$\mathrm{S}_{\mathbf{M}\cdot\mathbf{M}^{*}}$ coincides with
$\mathrm{S}_{\mathbf{M}}\Delta\mathrm{S}_{\mathbf{M}^{*}}$ up to
$\mathscr{H}^{1}$-negligible sets, so

(A.4)
$\mathscr{H}^{1}\left(\partial^{*}A\,\Delta\,(\mathrm{S}_{\mathbf{M}}\Delta\cup_{j=1}^{d}L_{j})\right)=0$

For any $j\in\\{1,\,\ldots,\,d\\}$, let
${\boldsymbol{\tau}}_{j}:=(a_{2j-1}-a_{2j})/|a_{2j-1}-a_{2j}|$. We define an
orientation ${\boldsymbol{\tau}}_{\mathbf{M}}$ for $\mathrm{S}_{\mathbf{M}}$
as ${\boldsymbol{\tau}}_{\mathbf{M}}:={\boldsymbol{\tau}}_{A}$ on
$\mathrm{S}_{\mathbf{M}}\setminus(\cup_{j}L_{j})$ (observing that, by (A.4),
$\mathscr{H}^{1}$-almost all of
$\mathrm{S}_{\mathbf{M}}\setminus(\cup_{j}L_{j})$ is contained in
$\partial^{*}A$) and
${\boldsymbol{\tau}}_{\mathbf{M}}:={\boldsymbol{\tau}}_{j}$ on
$\mathrm{S}_{\mathbf{M}}\cap L_{j}$, for any $j$. Then, (A.3) and (A.4) imply

(A.5)
$\begin{split}\int_{\mathrm{S}_{\mathbf{M}}}\nabla\varphi\cdot{\boldsymbol{\tau}}_{\mathbf{M}}\,\mathrm{d}\mathscr{H}^{1}-\sum_{j=1}^{d}\int_{L_{j}}\nabla\varphi\cdot{\boldsymbol{\tau}}_{j}\,\mathrm{d}\mathscr{H}^{1}+\sum_{j=1}^{d}\int_{L_{j}\setminus\mathrm{S}_{\mathbf{M}}}(1+{\boldsymbol{\tau}}_{A}\cdot{\boldsymbol{\tau}}_{j})\nabla\varphi\cdot{\boldsymbol{\tau}}_{j}\,\mathrm{d}\mathscr{H}^{1}=0,\end{split}$

On $\mathscr{H}^{1}$-almost all of $L_{j}\setminus\mathrm{S}_{\mathbf{M}}$,
both ${\boldsymbol{\tau}}_{j}$ and ${\boldsymbol{\tau}}_{A}$ are tangent to
$L_{j}$. Therefore, for $\mathscr{H}^{1}$-a.e. $x\in
L_{j}\setminus\mathrm{S}_{\mathbf{M}}$ we have
${\boldsymbol{\tau}}_{A}(x)\cdot{\boldsymbol{\tau}}_{j}(x)\in\\{-1,\,1\\}$. If
we define $T_{j}:=\\{x\in
L_{j}\setminus\mathrm{S}_{\mathbf{M}}\colon{\boldsymbol{\tau}}_{A}(x)\cdot{\boldsymbol{\tau}}_{j}(x)=1\\}$,
then the lemma follows from (A.5). ∎

Lemma A.4 can be reformulated in terms of currents. We recall a few basic
definitions in the theory of currents, because they will be useful to complete
the proof of Proposition A.1. Actually, we will only work with currents of
dimension $0$ or $1$. We refer to, e.g., [27, 45] for more details.

A $0$-dimensional current, or $0$-current, in $\mathbb{R}^{2}$ is just a
distribution on $\mathbb{R}^{2}$, i.e. an element of the topological dual of
$C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$ (where
$C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$ is given a suitable topology). A
$1$-dimensional current, or $1$-current, in $\mathbb{R}^{2}$ is an element of
the topological dual of
$C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2};\,(\mathbb{R}^{2})^{\prime})$, where
$(\mathbb{R}^{2})^{\prime}$ denotes the dual of $\mathbb{R}^{2}$ and
$C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2};\,(\mathbb{R}^{2})^{\prime})$ is given
a suitable topology, in much the same way as
$C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$. In other words, a $1$-dimensional
current is an $\mathbb{R}^{2}$-valued distribution. The boundary of a
$1$-current $T$ is the $0$-current $\partial T$ defined by

$\langle\partial T,\,\varphi\rangle:=\langle
T,\,\mathrm{d}\varphi\rangle\qquad\textrm{for any }\varphi\in
C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2}).$

The mass of a $1$-current $T$ is defined as

$\mathbb{M}(T):=\sup\left\\{\langle T,\,\omega\rangle\colon\omega\in
C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2};\,(\mathbb{R}^{2})^{\prime}),\
\left|\omega(x)\right|\leq 1\quad\textrm{for any
}x\in\mathbb{R}^{2}\right\\}\\!;$

the mass of a $0$-current is defined analogously.

We single out a particular subset of currents, called integer-multiplicity
rectifiable currents or rectifiable currents for short. A rectifiable
$0$-current is a current of the form

(A.6) $T=\sum_{k=1}^{p}n_{k}\,\delta_{b_{k}},$

where $k\in\mathbb{N}$, $n_{k}\in\mathbb{Z}$ and $b_{k}\in\mathbb{R}^{2}$. A
rectifibiable $0$-current has finite mass: for the current $T$ given by (A.6),
we have $\mathbb{M}(T)=\sum_{k=1}^{p}\left|n_{k}\right|$. A $1$-current is
called rectifiable if there exist a countably $1$-rectifiable set
$\Sigma\subseteq\mathbb{R}^{2}$ with $\mathscr{H}^{1}(\Sigma)<+\infty$, an
orientation ${\boldsymbol{\tau}}\colon\Sigma\to\SS^{1}$ for $\Sigma$ and an
integer-valued, $\mathscr{H}^{1}$-integrable function
$\theta\colon\Sigma\to\mathbb{Z}$ such that

(A.7) $\langle
T,\,\omega\rangle=\int_{\Sigma}\theta(x)\langle{\boldsymbol{\tau}}(x),\,\omega(x)\rangle\,\mathrm{d}\mathscr{H}^{1}(x)\qquad\textrm{for
any }\omega\in
C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2};\,(\mathbb{R}^{2})^{\prime}).$

The current $T$ defined by (A.7) is called the rectifiable $1$-current carried
by $\Sigma$, with multiplicity $\theta$ and orientation ${\boldsymbol{\tau}}$;
it satisfies

$\mathbb{M}(T)=\int_{\Sigma}\left|\theta(x)\right|\,\mathrm{d}\mathscr{H}^{1}(x)<+\infty.$

The set of rectifiable $0$-currents, respectively rectifiable $1$-currents, is
denoted by $\mathscr{R}_{0}(\mathbb{R}^{2})$, respectively
$\mathscr{R}_{1}(\mathbb{R}^{2})$.

Given a Lipschitz, injective map $\mathbf{f}\colon[0,\,1]\to\mathbb{R}^{2}$,
we denote by $\mathbf{f}_{\\#}I$ the rectifiable $1$-current carried by
$\mathbf{f}([0,\,1])$, with unit multiplicity and orientation given by
$\mathbf{f}^{\prime}$. The mass of $\mathbf{f}_{\\#}I$ is the length of the
curve parametrised by $\mathbf{f}$ and
$\partial(\mathbf{f}_{\\#}I)=\delta_{\mathbf{f}(1)}-\delta_{\mathbf{f}(0)}$;
in particular, $\partial(\mathbf{f}_{\\#}I)=0$ if
$\mathbf{f}(1)=\mathbf{f}(0)$. The assumption that $\mathbf{f}$ is injective
can be relaxed; for instance, if the curve parametrised by $\mathbf{f}$ has
only a finite number of self-intersections, then $\mathbf{f}_{\\#}I$ is still
well-defined and the properties above remain valid.

We take a bounded, convex domain $\Omega\subseteq\mathbb{R}^{2}$, distinct
points $a_{1}$, …$a_{2d}$ and a map $\mathbf{Q}\in
W^{1,1}(\Omega,\,\mathscr{N})\cap
W^{1,2}_{\operatorname{loc}}(\Omega\setminus\\{a_{1},\,\ldots,\,a_{2d}\\},\,\mathscr{N})$
with a non-orientable singularity at each $a_{i}$. Let
$\mathbf{M}\in\operatorname{SBV}(\Omega,\,\SS^{1})$ be a lifting of
$\mathbf{Q}$ such that $\mathrm{S}_{\mathbf{M}}\subset\\!\subset\Omega$. By
Federer-Vol’pert theorem (see e.g. [3, Theorem 3.78]), the set
$\mathrm{S}_{\mathbf{M}}$ is countably $1$-rectifiable. We claim that
$\mathscr{H}^{1}(\mathrm{S}_{\mathbf{M}})<+\infty$. Indeed, since $\mathbf{Q}$
has no jump set, by the BV-chain rule (see e.g. [3, Theorem 3.96]) we deduce
that $\mathbf{M}^{+}(x)=-\mathbf{M}^{-}(x)$ at $\mathscr{H}^{1}$-a.e. point
$x\in\mathrm{S}_{\mathbf{M}}$. This implies

$\mathscr{H}^{1}(\mathrm{S}_{\mathbf{M}})\leq\frac{1}{2}\int_{\mathrm{S}_{\mathbf{M}}}|\mathbf{M}^{+}-\mathbf{M}^{-}|\,\mathrm{d}\mathscr{H}^{1}\lesssim\left|\mathrm{D}\mathbf{M}\right|(\Omega)<+\infty,$

as claimed. In particular, there is a well-defined, rectifiable $1$-current
carried by $\mathrm{S}_{\mathbf{M}}$, with unit multiplicity and orientation
${\boldsymbol{\tau}}_{\mathbf{M}}$ given by Lemma A.4; we denote it by
$\llbracket\mathrm{S}_{\mathbf{M}}\rrbracket$. Lemma A.4 provides information
on the boundary of $\llbracket\mathrm{S}_{\mathbf{M}}\rrbracket$. More
precisely, Lemma A.4 implies

(A.8)
$\partial\llbracket\mathrm{S}_{\mathbf{M}}\rrbracket=\sum_{i=1}^{2d}\delta_{a_{i}}+2\,\partial
Q$

where $Q$ is a rectifiable $1$-chain, defined as

(A.9) $\langle
Q,\,\psi\rangle:=\sum_{j=1}^{d}\int_{T_{j}}\left\langle\psi(x),\frac{a_{2j-1}-a_{2j}}{\left|a_{2j-1}-a_{2j}\right|}\right\rangle\,\mathrm{d}\mathscr{H}^{1}(x)$

for any $\psi\in
C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2},\,(\mathbb{R}^{2})^{\prime})$. The
$T_{j}$’s are $1$-rectifiable sets that depend only on $\mathbf{M}$, not on
$\psi$, as given by Lemma A.4.

######  Lemma A.5.

Let $\Omega$, $\mathbf{Q}$ be as above. Let
$\mathbf{M}\in\operatorname{SBV}(\Omega,\,\SS^{1})$ be a lifting of
$\mathbf{Q}$ with $\mathrm{S}_{\mathbf{M}}\subset\\!\subset\Omega$. Then,
there exist countably may Lipschitz functions
$\mathbf{f}_{j}\colon[0,\,1]\to\mathbb{R}^{2}$, with finitely many self-
intersections, a rectifiable $1$-current $R\in\mathscr{R}_{1}(\mathbb{R}^{2})$
and a permutation $\sigma$ of the indices $\\{1,\,\ldots,\,2d\\}$ such that
the following properties hold:

(A.10) $\displaystyle\llbracket\mathrm{S}_{\mathbf{M}}\rrbracket=\sum_{j\geq
1}\mathbf{f}_{j,\\#}I+2R$ (A.11)
$\displaystyle\mathbb{M}(\llbracket\mathrm{S}_{\mathbf{M}}\rrbracket)=\sum_{j\geq
1}\mathbb{M}\left(\mathbf{f}_{j,\\#}I\right)$ (A.12)
$\displaystyle\partial(\mathbf{f}_{j,\\#}I)=\delta_{\sigma(2j)}-\delta_{\sigma(2j-1)}\quad\textrm{if
}j\in\\{1,\,\ldots,\,d\\},\qquad\partial(\mathbf{f}_{j,\\#}I)=0\quad\textrm{otherwise.}$

Figure 3: A decomposition of the graph $\mathscr{G}$, as defined in the proof
of Lemma A.5, into edge-disjoint trails $\mathscr{E}_{1}$ (in red) and
$\mathscr{E}_{2}$ (in blue). In addition to the edges of $\mathscr{G}$, there
may be other cycles, carried by the curves $\mathbf{g}_{j}([0,\,1])$ with
$j\geq q+1$; they are shown in black.

###### Proof.

By applying, e.g., [48, Theorem 6.3] or [4, Corollary 4.2], we find
rectifiable $1$-currents $T$, $R\in\mathscr{R}_{1}(\mathbb{R}^{2})$ such that
$\mathbb{M}(T)=\mathbb{M}(\llbracket\mathrm{S}_{\mathbf{M}}\rrbracket)=\mathscr{H}^{1}(\mathrm{S}_{\mathbf{M}})$,
$\partial T\in\mathscr{R}_{0}(\mathbb{R}^{2})$ and

(A.13) $T=\llbracket\mathrm{S}_{\mathbf{M}}\rrbracket+2R.$

By taking the boundary of both sides of (A.13), and applying (A.8), we obtain

(A.14) $\partial T=\sum_{i=1}^{2d}\delta_{a_{i}}+2P$

with $P:=\partial(R+Q)$ (and $Q$ as in (A.9)). The current $2P=\partial
T-\sum_{i=1}^{2d}\delta_{a_{i}}$ is rectifiable, so $\mathbb{M}(P)<+\infty$.
Moreover, $P$ isthe boundary of a rectifiable $1$-current. Then, Federer’s
closure theorem [27, 4.2.16] implies that $P$ itself is rectifiable. As a
consequence, we can re-write (A.14) as

(A.15) $\partial
T=\sum_{i=1}^{2d}\delta_{a_{i}}+2\sum_{k=1}^{p}n_{k}\,\delta_{b_{k}},$

for some integers $n_{k}$ and some distinct points $b_{k}\in\mathbb{R}^{2}$.
By applying [27, 4.2.25], we find countably many Lipschitz, injective maps
$\mathbf{g}_{j}\colon[0,\,1]\to\mathbb{R}$ such that

(A.16) $T=\sum_{j\geq 1}\mathbf{g}_{j,\\#}I,\qquad\sum_{j\geq
1}\left(\mathbb{M}(\mathbf{g}_{j,\\#}I)+\mathbb{M}(\partial(\mathbf{g}_{j,\\#}I))\right)=\mathbb{M}(T)+\mathbb{M}(\partial
T)<+\infty.$

For any $j$, we have either $\partial(\mathbf{g}_{j,\\#}I)=0$ (if
$\mathbf{g}_{j,\\#}(1)=\mathbf{g}_{j,\\#}(0)$) or
$\mathbb{M}(\partial(\mathbf{g}_{j,\\#}I))=2$ (otherwise). Therefore, by
(A.16), there are only finitely many indices $j$ such that
$\mathbf{g}_{j,\\#}(1)\neq\mathbf{g}_{j,\\#}(0)$. Up to a relabelling of the
$\mathbf{g}_{j}$’s, we assume that there is an integer $q$ such that
$\mathbf{g}_{j,\\#}(1)\neq\mathbf{g}_{j,\\#}(0)$ if and only if $j\leq q$.

Now, the problem reduces to a combinatorial, or graph-theoretical, one. We
consider the finite (multi-)graph $\mathscr{G}$ whose edges are the curves
parametrised by $\mathbf{g}_{1}$, …, $\mathbf{g}_{q}$, and whose vertices are
the endpoints of such curves. There can be two or more edges that join the
same pair of vertices. However, we can disregard the orientation of the edges:
changing the orientation of the curve parametrised by $\mathbf{g}_{j}$
corresponds to passing from the current $\mathbf{g}_{j,\\#}I$ to the current
$-\mathbf{g}_{j,\\#}I$; the difference
$\mathbf{g}_{j,\\#}I-(-\mathbf{g}_{j,\\#}I)=2\mathbf{g}_{j,\\#}I$ can be
absorbed into the term $2R$ that appears in (A.10).

We would like to partition the set of edges of $\mathscr{G}$ into $d$ disjoint
subsets $\mathscr{E}_{1}$, …$\mathscr{E}_{d}$, where each $\mathscr{E}_{j}$ is
a trail (i.e., a sequence of distinct edges such that each edge is adjacent to
the next one) and, for a suitable permutation $\sigma$ of
$\\{1,\,\ldots,\,2d\\}$, the trail $\mathscr{E}_{j}$ connects
$a_{\sigma(2j-1)}$ with $a_{\sigma(2j)}$. If we do so, then we can define
$\mathbf{f}_{j}\colon[0,\,1]\to\mathbb{R}^{2}$ for $j\in\\{1,\,\ldots,d\\}$ as
a Lipschitz map that parameterises the trail $\mathscr{E}_{j}$, with suitable
orientations of each edge; for $j\geq d+1$, we define
$\mathbf{f}_{j}:=\mathbf{g}_{q+j-d}$. With this choice of $\mathbf{f}_{j}$,
the lemma follows. It is possible to find $\mathscr{E}_{1}$,
…$\mathscr{E}_{d}$ as required because the graph $\mathscr{G}$ has the
following property: any $a_{i}$ is an endpoint of an _odd_ number of edges of
$\mathscr{G}$; conversely, any vertex of $\mathscr{G}$ other than the
$a_{i}$’s is an endpoint of an _even_ number of edges of $\mathscr{G}$. This
property follows from (A.15). Then, we can construct $\mathscr{E}_{1}$,
…$\mathscr{E}_{d}$ by reasoning along the lines of, e.g., [16, Theorem 12]. ∎

We can now conclude the proof of Proposition A.1.

###### Proof of Proposition A.1.

We consider the decomposition of $\llbracket\mathrm{S}_{\mathbf{M}}\rrbracket$
given by Lemma A.5. Thanks to (A.12), for any $j\in\\{1,\,\ldots,\,d\\}$ the
curve parametrised by $\mathbf{f}_{j}$ joins $a_{\sigma(2j-1)}$ with
$a_{\sigma(2j)}$. Then,

$\mathscr{H}^{1}(\mathrm{S}_{\mathbf{M}})=\mathbb{M}(\llbracket\mathrm{S}_{\mathbf{M}}\rrbracket)\geq\sum_{j=1}^{d}\mathbb{M}(\mathbf{f}_{j,\\#}I)\geq\sum_{j=1}^{d}\left|a_{\sigma(2j)}-a_{\sigma(2j-1)}\right|\geq\mathbb{L}(a_{1},\,\ldots,\,a_{2d}).$

The equality can only be attained if there are exactly $d$ maps
$\mathbf{f}_{j}$ and each of them parametrises a straight line segment. ∎

##  Appendix B Properties of $f_{\varepsilon}$

The aim of this section is to prove Lemma 3.1. We first of all, we
characterise the zero-set of the potential $f_{\varepsilon}$, in terms of the
(unique) solution to an algebraic system depending on $\varepsilon$ and
$\beta$.

######  Lemma B.1.

For any $\varepsilon>0$, the algebraic system

(B.1)
$\begin{cases}X(X-1-\beta^{2}\varepsilon)^{2}=\dfrac{\beta^{2}\varepsilon^{2}}{2}\\\
X>1+\beta^{2}\varepsilon\end{cases}$

admits a unique solution $X_{\varepsilon}$, which satisfies

$X_{\varepsilon}=1+\frac{1}{\sqrt{2}}\left(\sqrt{2}\beta+1\right)\beta\varepsilon-\frac{1}{4}\left(\sqrt{2}\beta+1\right)\beta^{2}\varepsilon^{2}+\mathrm{o}(\varepsilon^{2})\qquad\textrm{as
}\varepsilon\to 0.$

###### Proof.

The function $P(X):=X(X-1-\beta^{2}\varepsilon)^{2}$ is continuous and
strictly increasing in the interval $[1+\beta^{2}\varepsilon,\,+\infty)$,
because
$P^{\prime}(X)=(X-1-\beta^{2}\varepsilon)(3X-1-\beta^{2}\varepsilon)>0$ for
$X>1+\beta^{2}\varepsilon$. Moreover, $P(1+\beta^{2}\varepsilon)=0$ and
$P(X)\to+\infty$ as $X\to+\infty$. Therefore, the system (B.1) admits a unique
solution. Let $Y_{\varepsilon}>0$ be such that

$X_{\varepsilon}=1+\beta^{2}\varepsilon+\beta\varepsilon\,Y_{\varepsilon}.$

Then, (B.1) can be rewritten as

(B.2)
$Y_{\varepsilon}^{2}=\frac{1}{2+2\beta^{2}\varepsilon+2\beta\varepsilon\,Y_{\varepsilon}},$

which implies $Y_{\varepsilon}\to 1/\sqrt{2}$ as $\varepsilon\to 0$. Using
(B.2) again, we obtain

$Y_{\varepsilon}=\frac{1}{\left(2+2\beta^{2}\varepsilon+\sqrt{2}\beta\varepsilon+\mathrm{o}(\varepsilon)\right)^{1/2}}=\frac{1}{\sqrt{2}}-\frac{1}{4}\left(\sqrt{2}\beta+1\right)\beta\varepsilon+\mathrm{o}(\varepsilon)$

as $\varepsilon\to 0$, and the lemma follows. ∎

For any $\varepsilon>0$, we define

(B.3)
$s_{\varepsilon}:=X_{\varepsilon}^{1/2},\qquad\lambda_{\varepsilon}:=\left(\frac{X_{\varepsilon}-1}{X_{\varepsilon}-1-\beta^{2}\varepsilon}\right)^{1/2}.$

Lemma B.1 implies, via routine algebraic manipulations, that

(B.4) $\displaystyle
s_{\varepsilon}=1+\frac{1}{2\sqrt{2}}\left(\sqrt{2}\beta+1\right)\beta\varepsilon+\mathrm{o}(\varepsilon),\qquad\lambda_{\varepsilon}^{2}=\sqrt{2}\beta+1+\frac{1}{2}\left(\sqrt{2}\beta+1\right)\beta^{2}\varepsilon+\mathrm{o}(\varepsilon)$

as $\varepsilon\to 0$.

######  Lemma B.2.

A pair $(\mathbf{Q},\,\mathbf{M})\in\mathscr{S}_{0}^{2\times
2}\times\mathbb{R}^{2}$ satisfies $f_{\varepsilon}(\mathbf{Q},\,\mathbf{M})=0$
if and only if

$\begin{split}\left|\mathbf{M}\right|=\lambda_{\varepsilon},\qquad\mathbf{Q}=\sqrt{2}\,s_{\varepsilon}\left(\frac{\mathbf{M}\otimes\mathbf{M}}{\lambda_{\varepsilon}^{2}}-\frac{\mathbf{I}}{2}\right)\\!.\end{split}$

###### Proof.

By imposing that the gradient of $f_{\varepsilon}$ is equal to zero, we obtain
the system

(B.5)
$\displaystyle(\left|\mathbf{Q}\right|^{2}-1)\mathbf{Q}=\beta\varepsilon\left(\mathbf{M}\otimes\mathbf{M}-\dfrac{\left|\mathbf{M}\right|^{2}}{2}\mathbf{I}\right)$
(B.6)
$\displaystyle(\left|\mathbf{M}\right|^{2}-1)\mathbf{M}=2\beta\,\mathbf{Q}\mathbf{M}.$

Suppose first that $\mathbf{M}=0$. Then, Equation (B.5) implies that either
$\mathbf{Q}=0$ or $\left|\mathbf{Q}\right|=1$. The pair $\mathbf{Q}=0$,
$\mathbf{M}=0$ is not a minimiser for $f_{\varepsilon}$, because
$\nabla_{\mathbf{Q}}^{2}f_{\varepsilon}(0,\,0)=-\mathbf{I}<0$. If
$\left|\mathbf{Q}\right|=1$, $\mathbf{M}=0$, then
$\nabla_{\mathbf{M}}^{2}f_{\varepsilon}(\mathbf{Q},\,0)=-\varepsilon(\mathbf{I}+2\beta\mathbf{Q})$.
Since $\mathbf{Q}$ is non-zero, symmetric and trace-free, there exists
$\mathbf{n}\in\SS^{1}$ such that $\mathbf{Q}\mathbf{n}\cdot\mathbf{n}>0$.
Then,
$\nabla_{\mathbf{M}}^{2}f_{\varepsilon}(\mathbf{Q},\,0)\mathbf{n}\cdot\mathbf{n}<0$,
so the pair $(\mathbf{Q},\,\mathbf{M}=0)$ is not a minimiser of
$f_{\varepsilon}$. It remains to consider the case $\mathbf{M}\neq 0$. In this
case, we have $\mathbf{Q}\neq 0$ and $\left|\mathbf{Q}\right|\neq 1$, due to
(B.5). Solving (B.5) for $\mathbf{Q}$, and then substituting in (B.6), we
obtain

$\displaystyle\left|\mathbf{M}\right|^{2}-1=\frac{\beta^{2}\varepsilon\left|\mathbf{M}\right|^{2}}{\left|\mathbf{Q}\right|^{2}-1}$

and hence, solving for $\left|\mathbf{M}\right|^{2}$,

(B.7)
$\displaystyle\left|\mathbf{M}\right|^{2}=\frac{\left|\mathbf{Q}\right|^{2}-1}{\left|\mathbf{Q}\right|^{2}-1-\beta^{2}\varepsilon}.$

By taking the squared norm of both sides of (B.5), we obtain

$(\left|\mathbf{Q}\right|^{2}-1)^{2}\left|\mathbf{Q}\right|^{2}=\frac{\beta^{2}\varepsilon^{2}}{2}\left|\mathbf{M}\right|^{4}$

and hence, using (B.7),

(B.8)
$\left|\mathbf{Q}\right|^{2}=\frac{\beta^{2}\varepsilon^{2}}{2(\left|\mathbf{Q}\right|^{2}-1-\beta^{2}\varepsilon)^{2}}$

We either have $\left|\mathbf{Q}\right|^{2}<1$ or
$\left|\mathbf{Q}\right|^{2}>1+\beta^{2}\varepsilon$, because of (B.7). On the
other hand, by imposing that the second derivative of $f_{\varepsilon}$ with
respect to $\mathbf{Q}$ is non-negative, we obtain
$\left|\mathbf{Q}\right|^{2}\geq 1$. Therefore, we conclude that
$\left|\mathbf{Q}\right|^{2}=X_{\varepsilon}$ is the unique solution to the
system (B.1) and, taking (B.7) into account, the proposition follows. ∎

We can now prove Lemma 3.1. For convenience, we recall the statement here.

######  Lemma B.3.

The potential $f_{\varepsilon}$ satisfies the following properties.

  1. (i)

The constant $\kappa_{\varepsilon}$ in (2.2), uniquely defined by imposing the
condition $\inf f_{\varepsilon}=0$, satisfies

(B.9)
$\kappa_{\varepsilon}=\frac{1}{2}\left(\beta^{2}+\sqrt{2}\beta\right)\varepsilon+\kappa_{*}^{2}\,\varepsilon^{2}+\mathrm{o}(\varepsilon^{2})$

In particular, $\kappa_{\varepsilon}\geq 0$ for $\varepsilon$ small enough.

  2. (ii)

If $(\mathbf{Q},\,\mathbf{M})\in\mathscr{S}_{0}^{2\times
2}\times\mathbb{R}^{2}$ is such that

(B.10)
$\left|\mathbf{M}\right|=(\sqrt{2}\beta+1)^{1/2},\qquad\mathbf{Q}=\sqrt{2}\left(\frac{\mathbf{M}\otimes\mathbf{M}}{\sqrt{2}\beta+1}-\frac{\mathbf{I}}{2}\right)$

then
$f_{\varepsilon}(\mathbf{Q},\,\mathbf{M})=\kappa_{*}\,\varepsilon^{2}+\mathrm{o}(\varepsilon^{2})$.

  3. (iii)

If $\varepsilon$ is sufficiently small, then

(B.11)
$\displaystyle\frac{1}{\varepsilon^{2}}f_{\varepsilon}(\mathbf{Q},\,\mathbf{M})$
$\displaystyle\geq\frac{1}{4\varepsilon^{2}}(\left|\mathbf{Q}\right|^{2}-1)^{2}-\frac{\beta}{\sqrt{2}\varepsilon}\left|\mathbf{M}\right|^{2}\,\left|\left|\mathbf{Q}\right|-1\right|$
(B.12)
$\displaystyle\frac{1}{\varepsilon^{2}}f_{\varepsilon}(\mathbf{Q},\,\mathbf{M})$
$\displaystyle\geq\frac{1}{8\varepsilon^{2}}(\left|\mathbf{Q}\right|^{2}-1)^{2}-\beta^{2}\left|\mathbf{M}\right|^{4}$

for any $(\mathbf{Q},\,\mathbf{M})\in\mathscr{S}_{0}^{2\times
2}\times\mathbb{R}^{2}$.

_Proof of Statement (i)_. Let
$(\mathbf{Q}^{*}_{*},\,\mathbf{M}^{*})\in\mathscr{S}_{0}^{2\times
2}\times\mathbb{R}^{2}$ be a minimiser for $f_{\varepsilon}$, i.e.
$f_{\varepsilon}(\mathbf{Q}^{*}_{*},\,\mathbf{M}^{*})=0$. By Lemma B.2, we
have

$\begin{split}\kappa_{\varepsilon}&=-\frac{1}{4}(|\mathbf{Q}^{*}_{*}|^{2}-1)^{2}-\frac{\varepsilon}{4}(|\mathbf{M}^{*}|^{2}-1)^{2}+\beta\varepsilon\,\mathbf{Q}^{*}_{*}\mathbf{M}^{*}\cdot\mathbf{M}^{*}\\\
&=-\frac{1}{4}(s_{\varepsilon}^{2}-1)^{2}-\frac{\varepsilon}{4}(\lambda_{\varepsilon}^{2}-1)^{2}+\frac{\beta\varepsilon}{\sqrt{2}}\,s_{\varepsilon}\lambda_{\varepsilon}^{2}\end{split}$

We expand $s_{\varepsilon}$, $\lambda_{\varepsilon}$ in terms of
$\varepsilon$, as given by (B.4). Equation (B.9) then follows by standard
algebraic manipulations.

_Proof of Statement (ii)_. The assumption (B.10) implies

$\left|\mathbf{Q}\right|=1,\qquad\mathbf{Q}\mathbf{M}\cdot\mathbf{M}=\sqrt{2}\left(\frac{\left|\mathbf{M}\right|^{4}}{\sqrt{2}\beta+1}-\frac{1}{2}\left|\mathbf{M}\right|^{2}\right)=\frac{\sqrt{2}}{2}\left(\sqrt{2}\beta+1\right)=\beta+\frac{\sqrt{2}}{2}$

Therefore,

$\begin{split}f_{\varepsilon}(\mathbf{Q},\,\mathbf{M})=\frac{\varepsilon\,\beta^{2}}{2}-\beta\varepsilon\left(\beta+\frac{\sqrt{2}}{2}\right)+\kappa_{\varepsilon}=-\frac{\varepsilon}{2}\left(\beta^{2}+\sqrt{2}\beta\right)+\kappa_{\varepsilon}\stackrel{{\scriptstyle\eqref{kepsapp}}}{{=}}\kappa_{*}\,\varepsilon^{2}+\mathrm{o}(\varepsilon^{2})\end{split}$

_Proof of Statement (iii)_. When $\mathbf{Q}=0$, we have
$f_{\varepsilon}(0,\,\mathbf{M})\geq 1/4+\kappa_{\varepsilon}$ and
$\kappa_{\varepsilon}>0$ is positive for $\varepsilon$ small enough, due to
(B.9). Then, (B.11) follows. When $\mathbf{Q}\neq 0$, it is convenient to make
the change of variables we have introduced in Section 3. We write

$\mathbf{Q}=\frac{\left|\mathbf{Q}\right|}{\sqrt{2}}\left(\mathbf{n}\otimes\mathbf{n}-\mathbf{m}\otimes\mathbf{m}\right)$

where $(\mathbf{n},\,\mathbf{m})$ is an orthonormal basis of eigenvalues for
$\mathbf{Q}$. We define $\mathbf{u}=(u_{1},\,u_{2})\in\mathbb{R}^{2}$ as
$u_{1}:=\mathbf{M}\cdot\mathbf{n}$, $u_{2}:=\mathbf{M}\cdot\mathbf{m}$. The
potential $f_{\varepsilon}$ can be expressed in terms of $\mathbf{Q}$,
$\mathbf{u}$ as (see Equation (3.14)),

$\begin{split}\frac{1}{\varepsilon^{2}}f_{\varepsilon}(\mathbf{Q},\,\mathbf{M})&=\frac{1}{4\varepsilon^{2}}(\left|\mathbf{Q}\right|^{2}-1)^{2}+\frac{1}{\varepsilon}h(\mathbf{u})+\frac{\beta}{\sqrt{2}\,\varepsilon}(1-\left|\mathbf{Q}\right|)\,(u_{1}^{2}-u_{2}^{2})\\\
&\qquad\qquad+\frac{\kappa_{\varepsilon}}{\varepsilon^{2}}-\frac{1}{2\varepsilon}(\beta^{2}+\sqrt{2}\beta)\end{split}$

where $h$ is defined in (3.8). By Lemma (3.4), we know that $h\geq 0$.
Moreover, Equation (B.9) implies

$\frac{\kappa_{\varepsilon}}{\varepsilon^{2}}-\frac{1}{2\varepsilon}(\beta^{2}+\sqrt{2}\beta)=\kappa_{*}^{2}+\mathrm{o}(1)\geq
0$

for $\varepsilon$ small enough. Then, (B.11) follows. Equation (B.12) follows
from (B.11), as

$\begin{split}\frac{\beta}{\sqrt{2}\varepsilon}\left|\mathbf{M}\right|^{2}\left|\left|\mathbf{Q}\right|-1\right|\leq\beta^{2}\left|\mathbf{M}\right|^{4}+\frac{1}{8\varepsilon^{2}}(\left|\mathbf{Q}\right|-1)^{2}\leq\beta^{2}\left|\mathbf{M}\right|^{4}+\frac{1}{8\varepsilon^{2}}(\left|\mathbf{Q}\right|^{2}-1)^{2}\end{split}$

∎

##  Appendix C Proof of Lemma 4.5

The aim of this section is to prove Lemma C.1, which we recall here for the
convenience of the reader. We recall that
$g_{\varepsilon}\colon\mathscr{S}_{0}^{2\times 2}\to\mathbb{R}$ is the
function defined in (3.7).

######  Lemma C.1.

Let $B=B_{r}(x_{0})\subseteq\Omega$ be an open ball. Suppose that
$\mathbf{Q}^{*}_{\varepsilon}\rightharpoonup\mathbf{Q}^{*}$ weakly in
$W^{1,2}(\partial B)$ and that

(C.1) $\begin{split}\int_{\partial
B}\left(\frac{1}{2}\left|\nabla\mathbf{Q}^{*}_{\varepsilon}\right|^{2}+g_{\varepsilon}(\mathbf{Q}^{*}_{\varepsilon})\right)\mathrm{d}\mathscr{H}^{1}\leq
C\end{split}$

for some constant $C$ that may depend on the radius $r$, but not on
$\varepsilon$. Then, there exists a map $\mathbf{Q}_{\varepsilon}\in
W^{1,2}(B,\,\mathscr{S}_{0}^{2\times 2})$ such that

(C.2)
$\displaystyle\mathbf{Q}_{\varepsilon}=\mathbf{Q}_{\varepsilon}^{*}\quad\textrm{on
}\partial
B,\qquad\left|\mathbf{Q}_{\varepsilon}\right|\geq\frac{1}{2}\quad\textrm{in
}B$ (C.3)
$\displaystyle\int_{B}\left(\frac{1}{2}\left|\nabla\mathbf{Q}_{\varepsilon}\right|^{2}+g_{\varepsilon}(\mathbf{Q}_{\varepsilon})\right)\mathrm{d}x\to\frac{1}{2}\int_{B}\left|\nabla\mathbf{Q}^{*}\right|^{2}\,\mathrm{d}x$

Lemma C.1 is inspired by interpolation results in the literature on harmonic
maps (see e.g. [39, Lemma 1]). As we work in a two-dimensional domain, we can
simplify some points of the proof in [39]. On the other hand, we need to
estimate the contributions from the term
$g_{\varepsilon}(\mathbf{Q}_{\varepsilon})$, which is not present in [39].

###### Proof of Lemma C.1.

Without loss of generality, we can assume that $x_{0}=0$. By assumption, we
have $\mathbf{Q}_{\varepsilon}^{*}\rightharpoonup\mathbf{Q}^{*}$ weakly in
$W^{1,2}(\partial B)$ and hence, by Sobolev embedding, uniformly on $\partial
B$. In particular, $\left|\mathbf{Q}^{*}_{\varepsilon}\right|\to 1$ uniformly
on $\partial B$. Let $\lambda_{\varepsilon}>0$ be a small number, to be chosen
later on. We consider the decomposition $B=A_{\varepsilon}^{1}\cup
A^{2}_{\varepsilon}\cup A^{3}_{\varepsilon}$, where

$A^{1}_{\varepsilon}:=B_{r}\setminus\bar{B}_{r-\lambda_{\varepsilon}r},\qquad
A^{2}_{\varepsilon}:=\bar{B}_{r-\lambda_{\varepsilon}r}\setminus\bar{B}_{r-2\lambda_{\varepsilon}r},\qquad
A^{3}_{\varepsilon}:=\bar{B}_{r-2\lambda_{\varepsilon}r}$

We define the map $\mathbf{Q}_{\varepsilon}$ using polar coordinates
$(\rho,\,\theta)$, as follows. If $x=\rho e^{i\theta}\in A^{1}_{\varepsilon}$,
we define

$\mathbf{Q}_{\varepsilon}(x):=t_{\varepsilon}(\rho)\,\mathbf{Q}_{\varepsilon}^{*}(re^{i\theta})+(1+\kappa_{*}\varepsilon)(1-t_{\varepsilon}(\rho))\,\dfrac{\mathbf{Q}_{\varepsilon}^{*}(re^{i\theta})}{\left|\mathbf{Q}^{*}_{\varepsilon}(re^{i\theta})\right|}$

where $t_{\varepsilon}\colon\mathbb{R}\to\mathbb{R}$ is an affine function
such that $t_{\varepsilon}(r)=1$,
$t_{\varepsilon}(r-\lambda_{\varepsilon}r)=0$. If $x=\rho e^{i\theta}\in
A^{2}_{\varepsilon}$, we define

$\mathbf{Q}_{\varepsilon}(x):=(1+\kappa_{*}\varepsilon)\,\frac{s_{\varepsilon}(\rho)\,\mathbf{Q}^{*}_{\varepsilon}(re^{i\theta})+(1-s_{\varepsilon}(\rho))\,\mathbf{Q}^{*}(re^{i\theta})}{\left|s_{\varepsilon}(\rho)\,\mathbf{Q}^{*}_{\varepsilon}(re^{i\theta})+(1-s_{\varepsilon}(\rho))\,\mathbf{Q}^{*}(re^{i\theta})\right|}$

where $s_{\varepsilon}\colon\mathbb{R}\to\mathbb{R}$ is an affine function
such that $s_{\varepsilon}(r-\lambda_{\varepsilon}r)=1$,
$s_{\varepsilon}(r-2\lambda_{\varepsilon}r)=0$. Finally, if $x\in
A_{\varepsilon}^{3}$, we define

$\mathbf{Q}_{\varepsilon}(x):=(1+\kappa_{*}\varepsilon)\,\mathbf{Q}^{*}\left(\frac{x}{1-2\lambda_{\varepsilon}}\right)$

The map $\mathbf{Q}_{\varepsilon}$ is well-defined in $B$, beacuse
$\left|\mathbf{Q}_{\varepsilon}\right|\to 1$ uniformly on $\partial B$.
Moreover, we have $\left|\mathbf{Q}_{\varepsilon}\right|\geq 1/2$ for
$\varepsilon$ small enough, $\mathbf{Q}_{\varepsilon}\in
W^{1,2}(B,\,\mathscr{S}_{0}^{2\times 2})$ (at the interfaces between
$A_{\varepsilon}^{1}$, $A_{\varepsilon}^{2}$, $A^{3}_{\varepsilon}$, the
traces of $\mathbf{Q}_{\varepsilon}$ on either side of the interface match),
and $\mathbf{Q}_{\varepsilon}=\mathbf{Q}^{*}_{\varepsilon}$ on $\partial B$.

It only remains to prove (C.3). First, we estimate the integral of
$g_{\varepsilon}(\mathbf{Q}_{\varepsilon})$. On $A^{2}_{\varepsilon}\cup
A^{3}_{\varepsilon}$, we have
$|\mathbf{Q}_{\varepsilon}|=1+\kappa_{*}\varepsilon$ and hence, substituting
in (3.7),

(C.4)
$g_{\varepsilon}(\mathbf{Q}_{\varepsilon})=\kappa_{*}^{2}\left(\frac{1}{4}(2+\kappa_{*}\varepsilon)^{2}-1\right)=\kappa_{*}^{2}\left(\kappa_{*}\varepsilon+\kappa_{*}^{2}\varepsilon^{2}\right)=\mathrm{O}(\varepsilon)$

We consider the annulus $A_{\varepsilon}^{1}$. By Lemma 3.3, we have

$\begin{split}g_{\varepsilon}(\mathbf{Q}_{\varepsilon})\leq\left(\frac{1}{\varepsilon}(|\mathbf{Q}_{\varepsilon}|-1)-\kappa_{*}\right)^{2}+\frac{C}{\varepsilon^{2}}(|\mathbf{Q}_{\varepsilon}|-1)^{2}\end{split}$

For $x\in A^{1}_{\varepsilon}$, we have
$|\mathbf{Q}_{\varepsilon}(x)|=t_{\varepsilon}\left|\mathbf{Q}^{*}_{\varepsilon}(rx/\left|x\right|)\right|+(1-t_{\varepsilon})(1+\kappa_{*}\varepsilon)$,
with $t_{\varepsilon}=t_{\varepsilon}(\rho)\in[0,\,1]$. As a consequence,

(C.5)
$\begin{split}\int_{A^{1}_{\varepsilon}}g_{\varepsilon}(\mathbf{Q}_{\varepsilon})\,\mathrm{d}x&\lesssim\lambda_{\varepsilon}\int_{\partial
B}\left(\frac{1}{\varepsilon}(\left|\mathbf{Q}^{*}_{\varepsilon}\right|-1)-\kappa_{*}\right)^{2}\mathrm{d}\mathscr{H}^{1}+\frac{\lambda_{\varepsilon}}{\varepsilon^{2}}\int_{\partial
B}(\left|\mathbf{Q}^{*}_{\varepsilon}\right|-1)^{2}\,\mathrm{d}\mathscr{H}^{1}+\lambda_{\varepsilon}\kappa_{*}^{2}\end{split}$

On the other hand, as $\left|\mathbf{Q}^{*}_{\varepsilon}\right|\to 1$
uniformly on $\partial B$, from Lemma 3.3 we deduce that

(C.6)
$\begin{split}g_{\varepsilon}(\mathbf{Q}^{*}_{\varepsilon})\geq\left(\frac{1}{\varepsilon}(|\mathbf{Q}^{*}_{\varepsilon}|-1)-\kappa_{*}\right)^{2}-\frac{3}{4\varepsilon^{2}}(|\mathbf{Q}^{*}_{\varepsilon}|-1)^{2}\geq\frac{1}{8\varepsilon^{2}}(|\mathbf{Q}^{*}_{\varepsilon}|-1)^{2}-7\kappa_{*}^{2}\end{split}$

at any point of $\partial B$, for $\varepsilon$ small enough. Combining (C.5)
and (C.6), we obtain

(C.7)
$\int_{A^{1}_{\varepsilon}}g_{\varepsilon}(\mathbf{Q}_{\varepsilon})\,\mathrm{d}x\lesssim\lambda_{\varepsilon}\int_{\partial
B}g_{\varepsilon}(\mathbf{Q}^{*}_{\varepsilon})\,\mathrm{d}\mathscr{H}^{1}+\lambda_{\varepsilon}\kappa_{*}^{2}\stackrel{{\scriptstyle\eqref{hp:interp-
app}}}{{\lesssim}}\lambda_{\varepsilon}$

If we choose $\lambda_{\varepsilon}$ in such a way that
$\lambda_{\varepsilon}\to 0$ as $\varepsilon\to 0$, then (C.4) and (C.7) imply

(C.8) $\int_{B}g_{\varepsilon}(\mathbf{Q}_{\varepsilon})\,\mathrm{d}x\to
0\qquad\textrm{as }\varepsilon\to 0.$

Finally, we estimate the gradient term. An explicit computation shows that

$\begin{split}\int_{A^{1}_{\varepsilon}\cup
A^{2}_{\varepsilon}}\left|\nabla\mathbf{Q}_{\varepsilon}\right|^{2}\,\mathrm{d}x&\lesssim\lambda_{\varepsilon}\int_{\partial
B}\left(\left|\nabla\mathbf{Q}^{*}_{\varepsilon}\right|^{2}+\left|\nabla\mathbf{Q}^{*}\right|^{2}+\frac{1}{\lambda_{\varepsilon}^{2}}\left|\mathbf{Q}^{*}_{\varepsilon}-\mathbf{Q}^{*}\right|^{2}+\frac{1}{\lambda_{\varepsilon}^{2}}\left(\left|\mathbf{Q}^{*}_{\varepsilon}\right|-1-\kappa_{*}\varepsilon\right)^{2}\right)\mathrm{d}\mathscr{H}^{1}\\\
&\stackrel{{\scriptstyle\eqref{interp6}}}{{\lesssim}}\lambda_{\varepsilon}\int_{\partial
B}\left(\left|\nabla\mathbf{Q}^{*}_{\varepsilon}\right|^{2}+\left|\nabla\mathbf{Q}^{*}\right|^{2}+\frac{1}{\lambda_{\varepsilon}^{2}}\left|\mathbf{Q}^{*}_{\varepsilon}-\mathbf{Q}^{*}\right|^{2}+\frac{\varepsilon^{2}}{\lambda_{\varepsilon}^{2}}g_{\varepsilon}(\mathbf{Q}^{*}_{\varepsilon})+\frac{\varepsilon^{2}\kappa_{*}}{\lambda_{\varepsilon}^{2}}\right)\mathrm{d}\mathscr{H}^{1}\\\
\end{split}$

By the assumption (C.1), we deduce

(C.9) $\begin{split}\int_{A^{1}_{\varepsilon}\cup
A^{2}_{\varepsilon}}\left|\nabla\mathbf{Q}_{\varepsilon}\right|^{2}\,\mathrm{d}x&\stackrel{{\scriptstyle\eqref{interp6}}}{{\lesssim}}\lambda_{\varepsilon}+\frac{\varepsilon^{2}}{\lambda_{\varepsilon}}+\frac{1}{\lambda_{\varepsilon}}\int_{\partial
B}\left|\mathbf{Q}^{*}_{\varepsilon}-\mathbf{Q}^{*}\right|^{2}\,\mathrm{d}\mathscr{H}^{1}\\\
\end{split}$

We take

(C.10) $\lambda_{\varepsilon}:=\varepsilon+\left(\int_{\partial
B}\left|\mathbf{Q}^{*}_{\varepsilon}-\mathbf{Q}^{*}\right|^{2}\,\mathrm{d}\mathscr{H}^{1}\right)^{1/2}$

By assumption, we have
$\mathbf{Q}^{*}_{\varepsilon}\rightharpoonup\mathbf{Q}^{*}$ weakly in
$W^{1,2}(\partial B)$, hence strongly in $L^{2}(\partial B)$. Therefore,
$\lambda_{\varepsilon}\to 0$ as $\varepsilon\to 0$. Moreover, (C.9) and (C.10)
imply

(C.11) $\begin{split}\int_{A^{1}_{\varepsilon}\cup
A^{2}_{\varepsilon}}\left|\nabla\mathbf{Q}_{\varepsilon}\right|^{2}\,\mathrm{d}x\to
0\qquad\textrm{as }\varepsilon\to 0,\end{split}$

On the other hand, we have

(C.12)
$\begin{split}\int_{A^{3}_{\varepsilon}}\left|\nabla\mathbf{Q}_{\varepsilon}\right|^{2}\,\mathrm{d}x=\int_{B}\left|\nabla\mathbf{Q}^{*}\right|^{2}\,\mathrm{d}x\end{split}$

for any $\varepsilon$. Therefore, (C.3) follows from (C.8), (C.11) and (C.12).
∎

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