# Hölder regularity and convergence for a non-local model
of nematic liquid crystals in the large-domain limit

Giacomo Canevari  and Jamie M. Taylor  Dipartimento di Informatica —
Università di Verona, Strada le Grazie 15, 37134 Verona, Italy.
_E-mail address_<EMAIL_ADDRESS>— Basque Center for Applied
Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain.
_E-mail address_<EMAIL_ADDRESS>
###### Abstract

We consider a non-local free energy functional, modelling a competition
between entropy and pairwise interactions reminiscent of the second order
virial expansion, with applications to nematic liquid crystals as a particular
case. We build on previous work on understanding the behaviour of such models
within the large-domain limit, where minimisers converge to minimisers of a
quadratic elastic energy with manifold-valued constraint, analogous to
harmonic maps. We extend this work to establish Hölder bounds for
(almost-)minimisers on bounded domains, and demonstrate stronger convergence
of (almost)-minimisers away from the singular set of the limit solution. The
proof techniques bear analogy with recent work of singularly perturbed energy
functionals, in particular in the context of the Ginzburg-Landau and Landau-de
Gennes models.

##  1 Introduction

###  1.1 Variational models of liquid crystals

Liquid crystalline systems are those which sit outside of the classical solid-
liquid-gas trichotomy. While there are a plethora of different systems
classified as liquid crystals, they can be broadly described as fluid systems
where molecules admit a long range order of certain degrees of freedom. This
is in contrast to classical fluids, which lack long range correlations between
molecules. The fluidity of the systems makes them “soft”, that is, easily
susceptible to influence by external influences such as fields or stresses,
whilst the long range ordering permits anisotropic electrostatic and optical
behaviour. These two properties combined make them ideal for a variety of
technological applications, as their anisotropy is exploitable whilst their
softness makes them easy to manipulate.

The simplest liquid crystalline system is that of a nematic liquid crystal.
These are systems of elongated molecules, often idealised as having axial
symmetry, which form phases with no long range positional order, but where the
long axes of molecules is generally well aligned over larger length scales.
Even in the well studied case of nematics there are a variety of models that
one may use to study their theoretical behaviour, where the choice of model is
usually dependent on the length scales considered and the type of defects one
wishes to observe.

One of the earliest and most studied free energy functionals one may consider
in continuum modelling is the Oseen-Frank model [16]. In the simplest
formulation, we consider a prescribed domain $\Omega\subseteq\mathbb{R}^{3}$
and a unit vector field as our continuum variable
$n\colon\Omega\to\mathbb{S}^{2}$, interpreted as the local alignment axis of
molecules. As molecules are assumed to be (statistically) head-to-tail
symmetric, we interpret the configurations $n$, $-n$ as equivalent. In the
simplified one-constant approximation, we look for minimisers of the free
energy

(1.1) $\int_{\Omega}\frac{K}{2}\left|\nabla n(x)\right|^{2}\,\mathrm{d}x,$

subject to certain boundary conditions, although more general formulations are
possible. The problem has attracted interest not only from the liquid crystal
community, but also from the mathematical community as the prototypical
harmonic map problem. If a prescribed Dirichlet boundary condition admits non-
zero degree, then by necessity any $n$ satisfying it must admit
discontinuities, meaning that defects/singularities are an unavoidable part of
the model’s study.

More generally, one may consider an Oseen-Frank energy where different modes
of deformation are penalised to different extents. Neglecting the saddle-splay
null-Lagrangian term, this gives a free energy of the form

(1.2) $\int_{\Omega}\frac{K_{1}}{2}(\nabla\cdot
n(x))^{2}+\frac{K_{2}}{2}(n(x)\cdot\nabla\times
n(x))^{2}+\frac{K_{3}}{2}|n(x)\times\nabla\times n(x)|^{2}\,\mathrm{d}x.$

The constants $K_{1}$, $K_{2}$, $K_{3}$ are known as the Frank constants, and
represent the penalisations of splay, twist, and bend deformations
respectively. In the case where $K_{1}=K_{2}=K_{3}=K$, we reclaim the one-
constant approximation (1.1).

It is natural to ask if such a free energy can be justified. While the
original formulation was more phenomenological in nature and based solely on
symmetry arguments and a small-deformation assumption, attempts have been made
to identify the Oseen-Frank model as a large-domain limit of a more
fundamental model, the Landau-de Gennes model [12, 36]. In the Landau-de
Gennes model, the continuum variable is the Q-tensor, corresponding to the
normalised second moment of a one-particle distribution function. Explicitly,
if the distribution of the long axes of molecules in a small neighbourhood of
a point $x\in\Omega$ are described by a probability distribution
$f(x,\,\cdot)\colon\mathbb{S}^{2}\to[0,\,+\infty)$, we define the Q-tensor at
the point $x$ as

(1.3) $Q(x)=\int_{\mathbb{S}^{2}}f(x,\,p)\left(p\otimes
p-\frac{1}{3}I\right)\,\mathrm{d}p.$

As molecules are assumed to be head-to-tail symmetric, a molecule is as likely
to have orientation $p\in\mathbb{S}^{2}$ as $-p$, so that
$f(x,\,p)=f(x,\,-p)$. For this reason the first moment of $f(x,\,\cdot)$ will
always vanish, making the Q-tensor the first non-trivial moment, containing
information on molecular alignment. Q-tensors are, following their definition,
traceless, symmetric, $3\times 3$ matrices. We denote this set as

(1.4) $\text{Sym}_{0}(3)=\left\\{Q\in\mathbb{R}^{3}\colon
Q=Q^{T},\,\text{Trace}(Q)=0\right\\}.$

The Q-tensor contains more information than the director field, namely that it
does not force the interpretation of axially symmetric ordering about an axis
(less symmetric configurations are permitted), and the degree of orientational
ordering is permitted to vary.

Depending on their eigenvalues, they come in one of three varieties.

  * •

If all eigenvalues are equal, $Q=0$, and we say that $Q$ is isotropic, and
representative of a disordered system. In particular, if $f$ is a uniform
distribution on $\mathbb{S}^{2}$, $Q=0$.

  * •

If two eigenvalues are equal and the third is distinct, we say $Q$ is
uniaxial. A uniaxial Q-tensor can be written as $Q=s\left(n\otimes
n-\frac{1}{3}I\right)$, for a scalar $s$ and unit vector $n$. We interpret $n$
as the favoured direction of alignment, and $s$ as a measure of the degree of
ordering molecules about $n$.

  * •

If all three eigenvalues are distinct, we say that $Q$ is biaxial.

The corresponding free energy to be minimised is

(1.5) $\int_{\Omega}\psi_{b}(Q(x))+W(Q(x),\nabla Q(x))\,\mathrm{d}x.$

The function $\psi_{b}\colon\text{Sym}_{0}(3)\to\mathbb{R}\cup\\{+\infty\\}$
is a frame indifferent bulk potential, which may be taken as a polynomial or
the Ball-Majumdar singular potential. Its main characteristic is that, in the
cases considered, it is minimised on the set

(1.6) $\mathscr{N}=\left\\{Q\in\text{Sym}_{0}(3)\colon\textrm{there exists
}n\in\mathbb{S}^{2}\textrm{ such that }Q=s_{0}\left(n\otimes
n-\frac{1}{3}I\right)\right\\},$

with $s_{0}$ a temperature, concentration and material dependent constant. The
elastic energy $W$ is minimised when $\nabla Q=0$. While many forms are
possible, by symmetry the only frame-indifferent, quadratic energy that only
depends on the gradient of $Q$ is of the form

(1.7) $W(\nabla
Q)=\frac{L_{1}}{2}Q_{ij,k}Q_{ij,k}+\frac{L_{2}}{2}Q_{ij,k}Q_{ik,j}+\frac{L_{3}}{2}Q_{ij,j}Q_{ik,k},$

where Einstein summation notation is used. While Oseen-Frank represents
nematic defects as discontinuities in the continuum variables, the Landau de-
Gennes approach admits a different description, where nematic defects point
defects are typically described as a melting of nematic order, that is $Q=0$.
This permits smooth configurations to describe defects.

In an appropriate large-domain limit of a rescaled problem, the contributions
of the bulk energy become overwhelming, and we expect the minimisers to
converge to minimisers of a constrained problem, where we minimise the elastic
energy

(1.8) $\int_{\Omega}W(\nabla Q)\,\mathrm{d}x,$

subject to the constraint that $Q(x)\in\mathscr{N}$ almost everywhere. In the
case where $Q=s_{0}\left(n\otimes n-\frac{1}{3}I\right)$ almost everywhere for
some $n\in W^{1,2}(\Omega,\mathbb{S}^{2})$, we say that $Q$ is orientable, and
the problem in the presence of Dirichlet boundary conditions that are
$\mathscr{N}$-valued almost everywhere becomes equivalent to that of the
minimising the energy (1.2) for $n$. The constants $L_{i}$ and $K_{i}$ are
related in the case of Dirichlet boundary conditions where null-Lagrangian
terms may be neglected as

(1.9) $\begin{split}\frac{1}{s_{0}^{2}}K_{1}=&2L_{1}++L_{2}+L_{3},\\\
\frac{1}{s_{0}^{2}}K_{2}=&2L_{1},\\\
\frac{1}{s_{0}^{2}}K_{3}=&2L_{1}.\end{split}$

An energy purely quadratic in $\nabla Q$ cannot give rise to three independent
elastic constants in the Oseen-Frank model, with the so-called “cubic term”
$Q_{ij}Q_{kl,i}Q_{kl,j}$ often being used to fill the degeneracy. Such a term
does not arise from the model we will consider, although a more complex
variant taking into account molecular length scales has been proposed to avoid
this issue [11].

Studying the convergence of minimisers of Landau-de Gennes towards the Oseen-
Frank limit has attracted interest, with Majumdar and Zarnescu showing global
$W^{1,2}$ convergence and uniform convergence away from singular sets in the
one-constant case [35], Nguyen and Zarnescu proving convergence results in
stronger topologies [37], Contreras, Lamy and Rodiac generalising the approach
to other harmonic-map problems [10], and further extensions by Contreras and
Lamy [9] and Canevari, Majumdar and Stroffolini [7] to more general elastic
energies. In other settings, the $W^{1,2}$-convergence does not hold globally
but only locally, away from the singular sets, due to topological obstructions
carried by the boundary data and/or the domain (see e.g. [2, 19, 6, 23]).
Recently, Di Fratta, Robbins, Slastikov and Zarnescu found higher-order
Landau-de Gennes corrections to the Oseen-Frank functional, in two dimensional
domains, by studying the $\Gamma$-expansion of the Landau-de Gennes functional
in the large-domain limit [14]. The problem holds many parallels to the now-
classical Ginzburg-Landau problem [4]. Other singular limits and qualitative
features of Landau-de Gennes solutions have been studied too; see, for
instance, [8, 13, 22, 25, 29, 24, 26, 27] and the references therein.

While Landau-de Gennes has proven an effective model in many situations, there
are still open questions as to how one may justify the model in a rigorous
way. While one may use Landau-de Gennes, in appropriate situations, to justify
Oseen-Frank, a rigorous justification of Landau-de Gennes itself is lacking.
Historically it was justified on a phenomenological basis, but other work has
been able to provide Landau-de Gennes as a gradient expansion of a non-local
mean field model [17, 20]. Justification by formal gradient expansions leaves
open the question as to the consistency minimisers of the original free energy
with minimisers of its approximation. To this end, recent work has been
focused on rigorous asymptotic analysis of non-local free energies, which
similarly produce the Oseen-Frank model in a large-domain limit [31, 32, 42,
43]. These approaches “bypass” the intermediate and non-rigorous derivation of
Landau-de Gennes. This is analogous to recent investigations into
peridynamics, a formulation of elasticity based on non-local interactions.
These formulations of elasticity bear mathematical similarity with the mean-
field theory approach, where stress-strain relations are described in terms of
non-local operators on the deformation map, rather than derivatives as in the
more classical formulations of elasticity [3, 40]. The classical density
functional theory we will consider in this work is based on a simplified
competition between an entropic contribution to the energy, favouring
disorder, and an interaction energy, favouring order. The models themselves
are justified as a second order truncation of the virial expansion in the
dilute regime based on long-range attractive interactions in the style of
Maier and Saupe [34] and with mathematical similarity to the model of Onsager
[38]. Explicitly, given the one-particle distribution function $f(x,\cdot)$ in
a neighbourhood of $x$, we define a free energy functional

(1.10)
$\begin{split}&k_{B}T\rho_{0}\int_{\Omega\times\mathbb{S}^{2}}f(x,\,p)\ln
f(x,\,p)\,\mathrm{d}x\,\mathrm{d}p\\\
&\qquad\qquad\qquad-\frac{\rho_{0}^{2}}{2}\int_{\Omega\times\mathbb{S}^{2}}\int_{\Omega\times\mathbb{S}^{2}}f(x,\,p)f(y,\,p)\mathcal{K}(x-y,\,p,\,q)\,\mathrm{d}x\,\mathrm{d}p\,\mathrm{d}y\,\mathrm{d}q.\end{split}$

$\rho_{0}>0$ is the number density of particles in space, $k_{B}$ the
Boltzmann constant, $T>0$ temperature and $\mathcal{K}(z,\,p,\,q)$ denotes the
interaction energy of particles with orientations $p$, $q$ and with centres of
mass separated by a vector $z$. The entropic term on the left is convex and
readily shown to be minimised at a uniform distribution, that is, an isotropic
disordered system. The nature of the pairwise interaction energy on the right
is that nearby particles will prefer to be aligned with each other. We see
that temperature and concentration mediate the competition between these
opposing mechanisms. Recent work has established the Oseen-Frank energy (1.2)
in terms of a large-domain limit of the energy (1.10) under certain
assumptions, in which the elastic constants $K_{i}$ can be related to second
moments of the interaction kernel. Previous work has established weaker modes
of convergence, while in this work we will establish stronger convergence of
minimisers away from defect sets, analogous to the approach taken by Majumdar
and Zarnescu for the Landau-de Gennes model [35].

###  1.2 Simplification of the model and non-dimensionalisation

Here and throughout the sequel, we consider the more general case where
molecules admit an internal degree of freedom $p$ in a manifold $\mathcal{M}$.
We will employ a macroscopic order parameter $u\in\mathbb{R}^{m}$ to emphasise
the analysis is not limited to the concrete case of nematic liquid crystals.

Through most of the paper, we consider the case where $f$ is prescribed on
$\left(\mathbb{R}^{3}\setminus\frac{1}{{\varepsilon}}\Omega\right)\times\mathcal{M}$,
where $\Omega$ is a non-dimensional reference domain and ${\varepsilon}>0$ is
a small parameter, representative of the inverse of a large length scale of
the domain. In Section 5, we relax this assumption and study a minimisation
problem where $f$ is prescribed only in a neighbourhood of the domain, of
suitable thickness. We consider the free energy

(1.11)
$\begin{split}&\tilde{\mathcal{G}}_{\varepsilon}(f)=k_{B}T\rho_{0}\int_{\frac{1}{{\varepsilon}}\Omega\times\mathcal{M}}f(x,\,p)\ln
f(x,\,p)\,\mathrm{d}x\,\mathrm{d}p\\\
&\qquad\qquad\qquad-\frac{\rho_{0}^{2}}{2}\int_{\mathbb{R}^{3}\times\mathcal{M}}\int_{\mathbb{R}^{3}\times\mathcal{M}}f(x,\,p)f(y,\,q)\mathcal{K}(x-y,\,p,\,q)\,\mathrm{d}x\,\mathrm{d}p\,\mathrm{d}y\,\mathrm{d}q.\end{split}$

For simplification of the problem, we take the interaction energy to be of the
form

(1.12) $\mathcal{K}(z,\,p,\,q)=K(z)\sigma(p)\cdot\sigma(q),$

where $\sigma\in L^{\infty}(\mathcal{M},\mathbb{R}^{m})$ is some “microscopic
order parameter”, and $K\colon\mathbb{R}^{3}\to\mathbb{R}^{m\times m}$ is a
symmetric tensor field, which will satisfy certain technical conditions (see
(K1)–(K6) in Section 2). By applying Fubini we may then introduce a
“macroscopic order parameter”, $u\in
L^{\infty}(\mathbb{R}^{3},\mathbb{R}^{m})$ by

(1.13) $u(x)=\int_{\mathcal{M}}f(x,\,p)\sigma(p)\,\mathrm{d}p,$

and re-write the interaction energy as

(1.14)
$\begin{split}&-\frac{\rho_{0}^{2}}{2}\int_{\mathbb{R}^{3}\times\mathcal{M}}\int_{\mathbb{R}^{3}\times\mathcal{M}}f(x,\,p)f(y,\,p)\mathcal{K}(x-y,\,p,\,q)\,\mathrm{d}x\,\mathrm{d}p\,\mathrm{d}y\,\mathrm{d}q\\\
&\qquad\qquad\qquad=-\frac{\rho_{0}^{2}}{2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}K(x-y)u(x)\cdot
u(y)\,\mathrm{d}x\,\mathrm{d}y.\end{split}$

While it is not possible to write the entropic term explicitly in terms of
$u$, we may provide a lower bound by means of the singular Ball-
Majumdar/Katriel potential and its extensions [1, 28, 41] by

(1.15) $\int_{\frac{1}{{\varepsilon}}\Omega\times\mathcal{M}}f(x,\,p)\ln
f(x,\,p)\,\mathrm{d}x\,\mathrm{d}p\geq\int_{\frac{1}{{\varepsilon}}\Omega}\psi_{s}(u(x))\,\mathrm{d}x,$

where the function
$\psi_{s}\colon\mathbb{R}^{m}\to\mathbb{R}\cup\\{+\infty\\}$ is defined by

(1.16) $\psi_{s}(u)=\min\left\\{\int_{\mathcal{M}}f(p)\ln
f(p)\,\mathrm{d}p\colon f\geq 0\textrm{
a.e.,}\,\int_{\mathcal{M}}f(p)\,\mathrm{d}p=1,\,\int_{\mathcal{M}}f(p)\sigma(p)\,\mathrm{d}p=u\right\\}\\!,$

where by convention $\psi_{s}(u)=+\infty$ when the constraint set is empty.
Note that the minimisation problem (1.16) is strictly convex, thus solutions
are necessarily unique, and we may define $f_{u}$ to be the corresponding
minimiser for $u\in\mathcal{Q}=\left\\{u:\psi_{s}(u)<+\infty\right\\}$. That
is,

(1.17) $f_{u}=\text{arg min}\left\\{\int_{\mathcal{M}}f(p)\ln
f(p)\,\mathrm{d}p\colon f\geq 0\textrm{
a.e.,}\,\int_{\mathcal{M}}f(p)\,\mathrm{d}p=1,\,\int_{\mathcal{M}}f(p)\sigma(p)\,\mathrm{d}p=u\right\\}.$

The precise definition of $\psi_{s}$ will be unimportant in this work, and we
employ any function $\psi_{s}$ satisfying certain technical assumptions in the
sequel (see (H1)–(H6) in Section 2)

We in fact have the result that $f^{*}$ is a minimiser of
$\tilde{\mathcal{G}}_{\varepsilon}$ if and only if, for
$u^{*}(x)=\int_{\mathbb{S}^{2}}f^{*}(x,\,p)\sigma(p)\,\mathrm{d}p$, $u^{*}$ is
a minimiser of

(1.18)
$\tilde{\mathcal{F}}_{\varepsilon}(u)=k_{B}T\rho_{0}\int_{\frac{1}{{\varepsilon}}\Omega}\psi_{s}(u(x))\,\mathrm{d}x-\frac{\rho_{0}^{2}}{2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}K(x-y)u(x)\cdot
u(y)\,\mathrm{d}x\,\mathrm{d}y,$

with $f^{*}=f_{u^{*}}$. This is readily seen by writing the minimisation as a
two-step process, first minimising over all $f$ such that $f_{u}=f$, and later
minimising over $u$ and noting that the first minimisation may be performed
pointwise almost-everywhere in $\mathbb{R}^{3}$, as in [42]. That is to say,
we have a simpler, macroscopic energy with equivalent minimisers. By
introducing a change of variables,

$x=\frac{x^{\prime}}{{\varepsilon}},\quad
y=\frac{y^{\prime}}{{\varepsilon}},\quad
u^{\prime}(x^{\prime})=u(x),\quad{\varepsilon}^{\prime}:=\frac{{\varepsilon}}{\rho_{0}^{1/3}},\quad
K^{\prime}(x^{\prime})=\frac{1}{k_{B}T}K({\varepsilon}^{\prime}x),$

and a (non-dimensional) constant $C_{{\varepsilon}^{\prime}}$ to be specified
later, we rescale the domain and obtain the free energy we will consider for
the remainder of this work, so that

(1.19)
$\begin{split}E_{{\varepsilon}^{\prime}}(u^{\prime})&:=\frac{{\varepsilon}}{k_{B}T\rho_{0}^{1/3}}\tilde{\mathcal{F}}_{\varepsilon}(u)+C_{{\varepsilon}^{\prime}}\\\
&=\frac{1}{{{\varepsilon}^{\prime}}^{2}}\int_{\Omega}\psi_{s}(u^{\prime}(x^{\prime}))\,\mathrm{d}x^{\prime}-\frac{1}{2{{\varepsilon}^{\prime}}^{5}}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}K^{\prime}\left(\frac{x^{\prime}-y^{\prime}}{{\varepsilon}^{\prime}}\right)u^{\prime}(x^{\prime})\cdot
u^{\prime}(y^{\prime})\,\mathrm{d}x^{\prime}\,\mathrm{d}y^{\prime}+C_{{\varepsilon}^{\prime}}.\end{split}$

The additive constant $C_{{\varepsilon}^{\prime}}$ is irrelevant for the
purpose of minimisation; however, we will make a specific choice of
$C_{{\varepsilon}^{\prime}}$ (see Equation (2.6) below) for analytical
convenience. We will consider the regime as ${\varepsilon}^{\prime}\to 0$ in
this work. From the definition of ${\varepsilon}^{\prime}$, this may be
interpreted in two forms, one in which the characteristic length scale of the
domain, $\frac{1}{{\varepsilon}}$, becomes large, and one in which the density
$\rho_{0}$ becomes large. However, as the energy we consider is based on the
second order virial expansion which is explicitly a model for dilute regimes,
we interpret the limit ${\varepsilon}^{\prime}\to 0$ as the former, that is, a
large-domain limit. In the sequel we omit the primes and consider (1.19) as
our free energy functional to be minimised at scale ${\varepsilon}>0$.

##  2 Technical assumptions and main results

Let $\text{Sym}_{0}(m)$ be the space of $(m\times m)$-symmetric matrices, with
real coefficients. Given an interaction kernel
$K\colon\mathbb{R}^{3}\to\text{Sym}_{0}(m)$ and ${\varepsilon}>0$, we define
$K_{\varepsilon}(z):={\varepsilon}^{-3}K({\varepsilon}^{-1}z)$ for any
$z\in\mathbb{R}^{3}$. Then, we may rewrite the functional (1.19) as

(2.1)
$\begin{split}E_{\varepsilon}(u):=-\frac{1}{2{\varepsilon}^{2}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}K_{\varepsilon}(x-y)u(x)\cdot
u(y)\,\mathrm{d}x\,\mathrm{d}y+\frac{1}{{\varepsilon}^{2}}\int_{\Omega}\psi_{s}(u(x))\,\mathrm{d}x+C_{\varepsilon},\end{split}$

where $u\colon\mathbb{R}^{3}\to\mathbb{R}^{m}$ is the macroscopic order
parameter, $\Omega\subseteq\mathbb{R}^{3}$ is a bounded, smooth domain, and
$\psi_{s}\colon\mathbb{R}^{m}\to[0,\,+\infty]$ is any convex potential that
satisfies the assumptions (H1)–(H6) below (for instance, the Ball-
Majumdar/Katriel potential defined by (1.16)).

#### Assumptions on the kernel $K$.

Our assumptions on the kernel $K$ are reminiscent of [42]. We define
$g(z):=\lambda_{\min}(K(z))$ for any $z\in\mathbb{R}^{3}$, where
$\lambda_{\min}(K)$ denotes the minimum eigenvalue of $K$.

  1. (K1)

$K\in W^{1,1}(\mathbb{R}^{3},\,\text{Sym}_{0}(m))$.

  2. (K2)

$K$ is even, that is $K(z)=K(-z)$ for a.e. $z\in\mathbb{R}^{m}$.

  3. (K3)

$g\geq 0$ a.e. on $\mathbb{R}^{3}$, and there exist positive numbers
$\rho_{1}<\rho_{2}$, $k$ such that $g\geq k$ a.e. on $B_{\rho_{2}}\setminus
B_{\rho_{1}}$.

  4. (K4)

$g\in L^{1}(\mathbb{R}^{3})$ and has finite second moment, that is
$\int_{\mathbb{R}^{3}}g(z)\left|z\right|^{2}\mathrm{d}z<+\infty$.

  5. (K5)

There exists a positive constant $C$ such that $\lambda_{\max}(K(z))\leq
Cg(z)$ for a.e. $z\in\mathbb{R}^{3}$ (where $\lambda_{\max}(K)$ denotes the
maximum eigenvalue of $K$).

  6. (K6)

There holds

$\int_{\mathbb{R}^{3}}\left\|\nabla
K(z)\right\|\left|z\right|^{3}\mathrm{d}z<+\infty,$

where $\left\|\nabla
K(z)\right\|^{2}:=\partial_{\alpha}K_{ij}(z)\,\partial_{\alpha}K_{ij}(z)$.

In the case of physically meaningful systems the tensor $K$ will have to
respect frame invariance. In the case of nematic liquid crystals, where the
order parameter is a traceless symmetric matrix $Q$, frame indifference
implies that the bilinear form must necessarily be of the form

(2.2) $K(z)Q_{1}\cdot Q_{2}=f_{1}(|z|)Q_{1}\cdot Q_{2}+f_{2}(|z|)Q_{1}z\cdot
Q_{2}z+f_{3}(|z|)(Q_{1}z\cdot z)(Q_{2}z\cdot z),$

for all $Q_{1},Q_{2}\in\text{Sym}_{0}(3)$, where $f_{1}$, $f_{2}$, $f_{3}$ are
real-valued functions defined on $[0,\,+\infty)$ [42]. It is clear that by
appropriate choices of $f_{1}$, $f_{2}$, $f_{3}$ which are $C^{1}$ and with
sufficient decay at infinity the previous assumptions can be satisfied. This
family of bilinear forms includes the simplified interaction energy

(2.3) $K(z)Q_{1}\cdot Q_{2}=C\frac{\varphi(|z|)}{|z|^{6}}Q_{1}\cdot Q_{2}$

for a suitable cutoff function $\varphi$, zero near the origin, as found in
[5, Equation (3.43)]. The cutoff of the energy in a vicinity of $|z|=0$ is
reflective of the energy being derived as an approximation of a long-ranged,
attractive interaction.

#### Assumptions on the singular potential $\psi_{s}$.

  1. (H1)

$\psi_{s}\colon\mathbb{R}^{m}\to[0,\,+\infty]$ is a convex function.

  2. (H2)

The domain of $\psi_{s}$,
$\mathcal{Q}:=\psi_{s}^{-1}[0,\,+\infty)\subseteq\mathbb{R}^{m}$, is a non-
empty, bounded open set and $\psi_{s}\in C^{2}(\mathcal{Q})$.

  3. (H3)

There exists a constant $c>0$ such that
$\nabla^{2}\psi_{s}(y)\chi\cdot\chi\geq c\left|\chi\right|^{2}$ for any
$y\in\mathcal{Q}$ and any $\chi\in\mathbb{R}^{m}$.

  4. (H4)

There holds $\psi_{s}(y)\to+\infty$ as
$\operatorname{dist}(y,\,\partial\mathcal{Q})\to 0$.

We define the “bulk potential” $\psi_{b}\colon\mathcal{Q}\to\mathbb{R}$ in
terms of $K$ and $\psi_{s}$, as

(2.4)
$\psi_{b}(y):=\psi_{s}(y)-{\frac{1}{2}}\left(\int_{\mathbb{R}^{3}}K(z)\,\mathrm{d}z\right)y\cdot
y+c_{0}\quad\textrm{for any }y\in\mathcal{Q},$

where $c_{0}\in\mathbb{R}$ is a constant, uniquely determined by imposing that
$\inf\psi_{b}=0$. We make the following assumptions on $\psi_{b}$:

  1. (H5)

The set $\mathscr{N}:=\psi_{b}^{-1}(0)\subseteq\mathcal{Q}$ is a compact,
smooth, connected manifold without boundary.

  2. (H6)

For any $y\in\mathscr{N}$ and any unit vector $\xi\in\mathbb{S}^{m-1}$ that is
orthogonal to $\mathscr{N}$ at $y$, we have
$\nabla^{2}\psi_{b}(y)\xi\cdot\xi>0$.

######  Remark 2.1.

If the norm of $\int_{\mathbb{R}^{3}}K(z)\,\mathrm{d}z$ is smaller than the
constant $c$ given by (H3), then the function $\psi_{b}$ is convex and hence,
its zero-set $\mathscr{N}$ reduces to a point. This happens, for example, in
the sufficiently high temperature regime, independently of the precise form of
$K$. Nevertheless, our arguments remain valid in this case, too.

######  Remark 2.2.

The Ball-Majumdar/Katriel potential, defined by (1.16), satisfies the
conditions (H1)–(H6). (H1), (H2), (H4), and (H5) follow from [1], apart from
the $C^{2}$ smoothness of $\psi_{s}$ which is implicitly proven in [28] via an
inverse function theorem argument, although not stated. (H3) is proven in
[41]. With this choice of the potential, the set
$\mathscr{N}:=\psi_{b}^{-1}(0)$ is either a point or the manifold given by
(1.6) (see [1, Section 4]). In both cases, (H6) is satisfied (see [30,
Proposition 4.2]).

#### The admissible class and an equivalent expression for the free energy.

We complement the minimisation of the functional (2.1) by prescribing $u$ on
$\mathbb{R}^{3}\setminus\Omega$. We take a map $u_{\mathrm{bd}}\in
H^{1}(\mathbb{R}^{3},\,\mathbb{R}^{m})$ such that

(BD) $u_{\mathrm{bd}}(x)\in\mathcal{Q}\quad\textrm{for a.e.
}x\in\mathbb{R}^{3}\setminus\Omega,\qquad
u_{\mathrm{bd}}(x)\in\mathscr{N}\quad\textrm{for a.e. }x\in\Omega,$

and we define the admissible class

(2.5) $\mathscr{A}:=\left\\{u\in
L^{2}(\mathbb{R}^{3},\,\mathcal{Q})\colon\psi_{s}(u)\in L^{1}(\Omega),\
u=u_{\mathrm{bd}}\textrm{ a.e. on }\mathbb{R}^{3}\setminus\Omega\right\\}\\!.$

In the class $\mathscr{A}$, the functional $E_{\varepsilon}$ has an alterative
expression. For any $y\in\mathbb{R}^{m}$, we use the abbreviated notation
$y^{\otimes 2}:=y\otimes y$. We choose

(2.6)
$C_{\varepsilon}:=\frac{c_{0}}{{2}{\varepsilon}^{2}}\left|\Omega\right|+\frac{1}{{2}{\varepsilon}^{2}}\int_{\mathbb{R}^{3}\setminus\Omega}\left(\int_{\mathbb{R}^{3}}K(z)\,\mathrm{d}z\right)\cdot
u_{\mathrm{bd}}(x)^{\otimes 2}\,\mathrm{d}x,$

where $\left|\Omega\right|$ denotes the volume of $\Omega$ and
$c_{0}\in\mathbb{R}$ is the same number as in (2.4). The constant
$C_{\varepsilon}$ only depends on ${\varepsilon}$, $\Omega$, $K$ and
$u_{\mathrm{bd}}$, so it is does not affect minimisers of the functional. By
applying the algebraic identity

$-2K(x-y)u(x)\cdot u(y)=K(x-y)\cdot(u(x)-u(y))^{\otimes 2}-K(x-y)\cdot
u(x)^{\otimes 2}-K(x-y)\cdot u(y)^{\otimes 2}$

and using (2.4), (2.6), we re-write (2.1) as

(2.7)
$\begin{split}E_{\varepsilon}(u)=\frac{1}{4{\varepsilon}^{2}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}K_{\varepsilon}(x-y)\cdot\left(u(x)-u(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y+\frac{1}{{\varepsilon}^{2}}\int_{\Omega}\psi_{b}(u(x))\,\mathrm{d}x\end{split}$

for any $u\in\mathscr{A}$. We note that the free energy admits parallels to
the Landau-de Gennes energy, with the right-hand term being a corresponding
bulk energy and the left-hand term acting as a non-local analogue of the
elastic energy, which we shall see is reclaimed in a precise way in the
asymptotic limit as ${\varepsilon}\to 0$. Let $L$ be the unique symmetric
fourth-order tensor that satisfies

(2.8) $L\xi\cdot\xi:=\frac{1}{4}\int_{\mathbb{R}^{3}}K(z)\cdot(\xi z)^{\otimes
2}\,\mathrm{d}z\qquad\textrm{for any }\xi\in\mathbb{R}^{m\times 3}.$

Coordinate-wise, $L$ is defined by

$L_{ij\alpha\beta}=\frac{1}{4}\int_{\mathbb{R}^{3}}K_{\alpha\beta}(z)\,z_{i}\,z_{j}\,\mathrm{d}z$

for any $i$, $j\in\\{1,\,2,\,3\\}$ and $\alpha$,
$\beta\in\\{1,2,\,\ldots,\,m\\}$. Let $E_{0}\colon\mathscr{A}\to[0,\,+\infty]$
be given as

(2.9) $E_{0}(u):=\begin{cases}\displaystyle\int_{\Omega}L\nabla u\cdot\nabla
u&\textrm{if }u\in H^{1}(\Omega,\,\mathscr{N})\cap\mathscr{A}\\\
+\infty&\textrm{otherwise.}\end{cases}$

By assumption (BD), the set $H^{1}(\Omega,\,\mathscr{N})\cap\mathscr{A}$ is
non-empty and hence, the functional $E_{0}$ is not identically equal to
$+\infty$. Taylor [42] proved that, as ${\varepsilon}\to 0$, the functional
$E_{\varepsilon}$ $\Gamma$-converges to $E_{0}$ with respect to the
$L^{2}$-topology. In particular, up to subsequences, minimisers
$u_{\varepsilon}$ of $E_{\varepsilon}$ in the class $\mathscr{A}$ converge
$L^{2}$-strongly to a minimiser $u_{0}$ of $E_{0}$ in $\mathscr{A}$. Our aim
is to prove a convergence result for minimisers, in a stronger topology.

#### Main results.

Given a Borel set $G\subseteq\mathbb{R}^{3}$ and $u\in
L^{\infty}(G,\,\mathcal{Q})$, we define

(2.10) $F_{\varepsilon}(u,\,G):=\frac{1}{4{\varepsilon}^{2}}\int_{G\times
G}K_{\varepsilon}(x-y)\cdot\left(u(x)-u(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y+\frac{1}{{\varepsilon}^{2}}\int_{G}\psi_{b}(u(x))\,\mathrm{d}x.$

For any $\mu\in(0,\,1)$, we denote the $\mu$-Hölder semi-norm of $u$ on $G$ as

$[u]_{C^{\mu}(G)}:=\sup_{x,\,y\in G,\ x\neq
y}\frac{\left|u(x)-u(y)\right|}{\left|x-y\right|^{\mu}}.$

######  Theorem A (Uniform $\eta$-regularity).

Assume that the conditions (K1)–(K6), (H1)–(H6) and (BD) are satisfied. Then,
there exist positive numbers $\eta$, ${\varepsilon}_{*}$, $M$ and
$\mu\in(0,\,1)$ such that for any ball $B_{r_{0}}(x_{0})\subseteq\Omega$, any
${\varepsilon}\in(0,\,{\varepsilon}_{*}r_{0})$, and any minimiser
$u_{\varepsilon}$ of $E_{\varepsilon}$ in $\mathscr{A}$, there holds

$r_{0}^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{r_{0}}(x_{0}))\leq\eta\qquad\Longrightarrow\qquad
r_{0}^{\mu}\,[u_{\varepsilon}]_{C^{\mu}(B_{r_{0}/2}(x_{0}))}\leq M.$

As a corollary, we deduce a convergence result for minimisers of
$E_{\varepsilon}$, in the locally uniform topology. We recall that any
minimiser $u_{0}$ for the limit functional (2.9) in $\mathscr{A}$ is smooth in
$\Omega\setminus S[u_{0}]$, where

(2.11) $S[u_{0}]:=\left\\{x\in\Omega\colon\liminf_{\rho\to
0}\rho^{-1}\int_{B_{\rho}(x)}\left|\nabla u_{0}\right|^{2}>0\right\\}\\!.$

Moreover, $S[u_{0}]$ is a closed set of zero total length (see e.g. [21, 33]).

######  Theorem B.

Assume that the conditions (K1)–(K6), (H1)–(H6) and (BD) are satisfied. Let
$u_{\varepsilon}$ be a minimiser of $E_{\varepsilon}$ in $\mathscr{A}$. Then,
up to extraction of a (non-relabelled) subsequence, we have

$u_{\varepsilon}\to u_{0}\qquad\textrm{locally uniformly in }\Omega\setminus
S[u_{0}],$

where $u_{0}$ is a minimiser of the functional (2.9) in $\mathscr{A}$.

The strategy of the proof for Theorem A is inspired by [9]. Under the
assumption $F_{\varepsilon}(u_{\varepsilon},\,B_{1})\leq\eta$, we obtain an
algebraic decay for the mean oscillation of $u_{\varepsilon}$, that is

(2.12)
$\fint_{B_{\rho}}\left|u_{\varepsilon}-\fint_{B_{\rho}}u_{\varepsilon}\right|^{2}\leq
C\rho^{2\mu}$

for any $\rho\in(0,\,1)$ and some positive constants $C$, $\mu$ that do not
depend on $\rho$, ${\varepsilon}$. If the radius $\rho$ is large enough, i.e.
$\rho\geq\lambda_{1}{\varepsilon}$ for some ${\varepsilon}$-independent
constant $\lambda_{1}$, we exploit the decay properties for the limit
functional $E_{0}$ (see e.g. [21, 33]) to obtain an algebraic decay for
$F_{\varepsilon}(u_{\varepsilon},\,B_{\rho})$ as a function of $\rho$; then,
we deduce (2.12) via a suitable Poincaré inequality (Proposition 3.4). On the
other hand, if $\rho\leq\lambda_{1}{\varepsilon}$ we obtain (2.12) from the
Euler-Lagrange equations for $E_{\varepsilon}$ (Proposition 3.1). The
inequality (2.12) immediately implies the desired bound on the Hölder norm of
$u_{\varepsilon}$, by Campanato embedding. Once Theorem A is proven, Theorem B
follows, via the Ascoli-Arzelà theorem.

##  3 Preliminary results

###  3.1 The Euler-Lagrange equations

Throughout the paper, we denote by $C$ several constants that depend only on
$\Omega$, $K$, $m$, $\psi_{s}$ and $u_{\mathrm{bd}}$. We write $A\lesssim B$
as a short-hand for $A\leq CB$. We also define
$g_{\varepsilon}(z):={\varepsilon}^{-3}g({\varepsilon}^{-1}z)$ for
$z\in\mathbb{R}^{3}$ (where, we recall, $g(z)$ is the minimum eigenvalue of
$K(z)$) and

(3.1) $\Lambda:=\nabla\psi_{s}\colon\mathcal{Q}\to\mathbb{R}^{m}.$

######  Proposition 3.1.

Consider the free energy $E_{\varepsilon}$, given by (2.1), with
$u=u_{\mathrm{bd}}$ on $\mathbb{R}^{3}\setminus\Omega$. Then there exists a
minimiser $u_{\varepsilon}\in L^{\infty}(\Omega,\,\mathcal{Q})$ (identified
with its extension by $u_{\mathrm{bd}}$ to $\mathbb{R}^{3}$), and it satisfies
the Euler-Lagrange equation,

(3.2)
$\Lambda(u_{\varepsilon}(x))=\int_{\mathbb{R}^{3}}K_{\varepsilon}(x-y)u_{\varepsilon}(y)\,\mathrm{d}y$

for a.e. $x\in\Omega$.

###### Proof.

By neglecting the additive constant in (2.1), and multiplying by
${\varepsilon}^{2}$, without loss of generality we may consider the functional

$\mathcal{F}(u):=\int_{\Omega}\psi_{s}(u(x))\,\mathrm{d}x-\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}K_{\varepsilon}(x-y)u(x)\cdot
u(y)\,\mathrm{d}x\,\mathrm{d}y$

instead of $E_{\varepsilon}$. To show existence, we use a direct method
argument. First we show that the bilinear form admits a global lower bound. As
$u_{\mathrm{bd}}\in L^{2}(\mathbb{R}^{3},\,\mathcal{Q})$ and $u$ admits
uniform $L^{\infty}$-bounds on $\Omega$, we have that $u\in
L^{2}(\mathbb{R}^{3},\,\overline{\mathcal{Q}})$,
$\left\|u\right\|_{L^{2}(\mathbb{R}^{3})}$ is bounded uniformly. We thus have
the estimate that

(3.3)
$\begin{split}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}&|K_{\varepsilon}(x-y)u(x)\cdot
u(y)|\,\mathrm{d}x\,\mathrm{d}y\\\
&\lesssim\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}g_{\varepsilon}(x-y)|u(x)||u(y)|\,\mathrm{d}x\,\mathrm{d}y\\\
&=\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\left(g_{\varepsilon}(x-y)^{\frac{1}{2}}|u(x)|\right)\left(g_{\varepsilon}(x-y)^{\frac{1}{2}}|u(y)|\right)\,\mathrm{d}x\,\mathrm{d}y\\\
&\lesssim\left(\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}g_{\varepsilon}(x-y)|u(x)|^{2}\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}g_{\varepsilon}(x-y)|u(y)|^{2}\,\mathrm{d}x\,\mathrm{d}y\right)^{\frac{1}{2}}\\\
&=\left\|g_{\varepsilon}\right\|_{L^{1}(\mathbb{R}^{3})}\left\|u\right\|_{L^{2}(\mathbb{R}^{3})}^{2}=\left\|g\right\|_{L^{1}(\mathbb{R}^{3})}\left\|u\right\|_{L^{2}(\mathbb{R}^{3})}^{2}\end{split}$

The singular function $\psi_{s}$ admits a lower bound pointwise, hence the
functional $\mathcal{F}$ admits a global lower bound. To show the admissible
set is non empty, simply take $u(x)=u_{0}\in\mathcal{Q}$ for all $x\in\Omega$,
so that $\psi_{s}(u(x))$ is a non-infinite constant.

The uniform $L^{\infty}$ bounds on $u$ imply that we have $L^{\infty}$ weak-*
compactness of a minimising sequence. As $\psi_{s}$ is strictly convex, we
have weak-* lower semicontinuity of the entropic term. It suffices to show
weak-* lower semicontinuity of the bilinear term. First we split the bilinear
term into the “boundary” and “bulk” contributions. That is, we write
$u=u_{\mathrm{bd}}\,\chi_{\mathbb{R}^{3}\setminus\Omega}+u\,\chi_{\Omega}$,
where $\chi_{\mathbb{R}^{3}\setminus\Omega}$ and $\chi_{\Omega}$ are the
characteristic functions of $\mathbb{R}^{3}\setminus\Omega$ and $\Omega$
respectively. As
$K_{\varepsilon}*(u_{\mathrm{bd}}\,\chi_{\mathbb{R}^{3}\setminus\Omega})\in
L^{1}(\Omega)$, if $u_{j}\overset{*}{\rightharpoonup}u$,

(3.4)
$\int_{\Omega}u_{j}(x)\,K_{\varepsilon}*(u_{\mathrm{bd}}\,\chi_{\mathbb{R}^{3}\setminus\Omega})(x)\,\mathrm{d}x\to\int_{\Omega}u(x)\,K_{\varepsilon}*(u_{\mathrm{bd}}\,\chi_{\mathbb{R}^{3}\setminus\Omega})(x)\,\mathrm{d}x.$

The second term requires a little more care. Following [15, Corollary 4.1],
the map $L^{\infty}(\Omega)\ni u\mapsto K_{\varepsilon}*(u\,\chi_{\Omega})$ is
$L^{\infty}$-to-$L^{1}$ compact if and only if the set
$\left\\{K_{\varepsilon}(x-\cdot)\,\chi_{\Omega}\colon x\in\Omega\right\\}$ is
relatively $L^{1}$-compact. This is immediate however as $\Omega$ is a bounded
set and $K_{\varepsilon}$ is integrable. Therefore the map

(3.5) $u\mapsto\int_{\Omega}\int_{\Omega}K_{\varepsilon}(x-y)u(x)\cdot
u(y)\,\mathrm{d}x\,\mathrm{d}y$

is in fact continuous with the weak-* $L^{\infty}$ topology, and therefore the
entire bilinear term is continuous also. Therefore the energy functional is
lower semicontinuous and minimisers exist by the direct method.

To show that minimisers satisfy the Euler-Lagrange equation, we note that if
$u$ has finite energy, then the measure of the set
$\\{x\in\Omega:u(x)\in\partial\mathcal{Q}\\}$ is zero. In particular, we may
define $U_{\delta}=\left\\{x\in\Omega:\psi_{s}(u(x))<1/\delta\right\\}$, and
we have that

(3.6) $\Omega=\Gamma\cup\bigcup\limits_{\delta>0}U_{\delta},$

where $\Gamma$ is a null set. By Assumption (H4), for every $\delta>0$, there
exists some $\gamma>0$ so that if $\psi_{s}(\tilde{u})<1/\delta$, then
$\operatorname{dist}(\tilde{u},\,\partial\mathcal{Q})>\gamma$. In particular,
for $\phi\in L^{\infty}(\mathbb{R}^{3},\,\mathbb{R}^{m})$ supported on
$U_{\delta}$ and $\eta$ sufficiently small, $u+\eta\phi$ is bounded away from
$\partial\mathcal{Q}$ on $U_{\delta}$. Therefore we may take variations
without issue, as

$\begin{split}\frac{1}{\eta}\left(\mathcal{F}(u+\eta\phi)-\mathcal{F}(u)\right)&=\int_{U_{\delta}}\frac{1}{\eta}\left(\psi_{s}(u(x)+\eta\phi(x))-\psi_{s}(u(x))\right)\,\mathrm{d}x\\\
&-\frac{1}{2}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}K_{\varepsilon}(x-y)\cdot\left(2\phi(x)\otimes
u(y)+\eta\phi(x)\otimes\phi(y)\right)\,\mathrm{d}x\,\mathrm{d}y.\end{split}$

Now we have no issue taking the limit as $\eta\to 0$, as $\psi_{s}$ is $C^{2}$
away from $\partial\mathcal{Q}$, to give

$\begin{split}\lim\limits_{\eta\to
0}\frac{1}{\eta}\left(\mathcal{F}(u+\eta\phi)-\mathcal{F}(u)\right)=&\int_{U_{\delta}}\Lambda(u(x))\cdot\phi(x)\,\mathrm{d}x-\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}K_{\varepsilon}(x-y)u(y)\,\mathrm{d}y\cdot\phi(x)\,\mathrm{d}x\\\
=&\int_{U_{\delta}}\left(\Lambda(u(x))-\int_{\mathbb{R}^{3}}K_{\varepsilon}(x-y)u(y)\,\mathrm{d}y\right)\cdot\phi(x)\,\mathrm{d}x,\end{split}$

recalling that $\phi(x)=0$ outside of $U_{\delta}$. As $\phi$ was arbitrary,
this implies that $u$ satisfies

(3.7)
$\Lambda(u(x))=\int_{\mathbb{R}^{3}}K_{\varepsilon}(x-y)u(y)\,\mathrm{d}y$

on $U_{\delta}$, and since $\delta$ was arbitrary, this implies that $u$
satisfies the Euler-Lagrange equation outside of $\Gamma$, which is of measure
zero. ∎

The Euler-Lagrange equations are particularly useful when used in combination
with the following property.

######  Lemma 3.2.

The map $\Lambda\colon\mathcal{Q}\to\mathbb{R}^{m}$ is invertible and its
inverse is of class $C^{1}$. Moreover,

(3.8) $\sup_{z\in\mathbb{R}^{m}}\left\|\nabla(\Lambda^{-1})(z)\right\|\leq
c^{-1},$

where $c$ is the constant given by (H3), and

(3.9) $\left|\Lambda(y)\right|\to+\infty\qquad\textrm{as
}\operatorname{dist}(y,\,\partial\mathcal{Q})\to 0.$

###### Proof.

To prove (3.9), it suffices to note that as $\psi_{s}$ is a closed proper
convex function which is $C^{1}$ on an open domain, so by applying classical
results from convex analysis [39, Theorem 25.1, Theorem 26.1], we see that
$\psi_{s}$ satisfies the property of essential smoothness, which implies
(3.9). More so, as $\psi_{s}$ is also strictly convex on a bounded domain,
this implies $\psi_{s}$ is a Legendre-type function which provides the results
that $\Lambda(\mathcal{Q})=\mathbb{R}^{m}$ [39, Corollary 13.3.1], and that
$\Lambda$ is a $C^{0}$ bijection from $\mathcal{Q}\to\Lambda(Q)$ [39, Theorem
26.5]. The $C^{1}$ regularity of $\Lambda^{-1}$ follows immediately from the
inverse function theorem, as $\psi_{s}$ is strongly convex. ∎

The Euler-Lagrange equation (3.2) and Lemma 3.2 have important consequences in
terms of regularity and “strict physicality” of minimisers — that is, the
image of $u_{\varepsilon}$ does not touch the boundary of the physically
admissible set $\mathcal{Q}$.

######  Proposition 3.3.

Minimisers $u_{\varepsilon}$ of the functional $E_{\varepsilon}$ in the class
$\mathscr{A}$ are Lipschitz-continuous on $\Omega$, with $\left\|\nabla
u_{\varepsilon}\right\|_{L^{\infty}(\Omega)}\lesssim{\varepsilon}^{-1}$.
Moreover, there exists a number $\delta>0$ such that for any ${\varepsilon}>0$
and any $x\in\Omega$,

(3.10)
$\operatorname{dist}(u_{\varepsilon}(x),\,\partial\mathcal{Q})\geq\delta.$

###### Proof.

The minimiser $u_{\varepsilon}$ takes values in the bounded set $\mathcal{Q}$
and hence, $\|u_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{3})}\leq C$, where the
constant $C$ does not depend on ${\varepsilon}$. Moreover,
$\|K_{\varepsilon}\|_{L^{1}(\mathbb{R}^{3})}=\|K\|_{L^{1}(\mathbb{R}^{3})}<+\infty$.
Then, by applying Young’s inequality to (3.2), we obtain

$\left\|\Lambda(u_{\varepsilon})\right\|_{L^{\infty}(\Omega)}\leq\|K_{\varepsilon}\|_{L^{1}(\mathbb{R}^{3})}\|u_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{3})}\leq
C.$

On the other hand, we have $\left|\Lambda(z)\right|\to+\infty$ as
$z\to\partial\mathcal{Q}$ by (3.9) and hence, (3.10) follows. Since we have
assumed that $K\in W^{1,1}(\mathbb{R}^{3},\,\text{Sym}_{0}(m))$, from the
Euler-Lagrange equation (3.2) we deduce

$\left\|\nabla(\Lambda\circ
u_{\varepsilon})\right\|_{L^{\infty}(\Omega)}=\left\|\nabla
K_{\varepsilon}*u_{\varepsilon}\right\|_{L^{\infty}(\Omega)}\leq{\varepsilon}^{-1}\left\|\nabla
K\right\|_{L^{1}(\Omega)}\left\|u_{\varepsilon}\right\|_{L^{\infty}(\Omega)}<+\infty.$

By Lemma 3.2, we conclude that $\left\|\nabla
u_{\varepsilon}\right\|_{L^{\infty}(\Omega)}\lesssim{\varepsilon}^{-1}$. ∎

###  3.2 A Poincaré-type inequality for $F_{\varepsilon}$

The goal of this section is to prove the following inequality on
$F_{\varepsilon}$. We recall that the functional $F_{\varepsilon}$ is defined
in (2.10).

######  Proposition 3.4.

There exists ${\varepsilon}_{1}>0$ such that, for any $u\in
L^{\infty}(\mathbb{R}^{3},\,\mathbb{R}^{m})$, any $\rho>0$, any
$x_{0}\in\mathbb{R}^{3}$ and any
${\varepsilon}\in(0,\,{\varepsilon}_{1}\rho]$, there holds

$\fint_{B_{\rho/2}(x_{0})}\left|u-\fint_{B_{\rho/2}(x_{0})}u\right|^{2}\lesssim\rho^{-1}F_{\varepsilon}(u,\,B_{\rho}(x_{0})).$

To simplify the proof of Proposition 3.4, we will take advantage of the
scaling properties of $F_{\varepsilon}$: if $u_{\rho}\colon
B_{1}\to\mathbb{R}^{m}$ is defined by $u_{\rho}(x):=u(\rho x+x_{0})$ for $x\in
B_{1}$, then a change of variables gives

(3.11)
$\rho^{-1}F_{\varepsilon}(u,\,B_{\rho}(x_{0}))=F_{{\varepsilon}/\rho}(u_{\rho},\,B_{1}).$

In the proof of Proposition 3.4, we will adapt arguments from [42]. By
assumption (K3), there exist positive numbers $\rho_{1}<\rho_{2}$, $k$ such
that $g\geq k$ a.e. on $B_{\rho_{2}}\setminus B_{\rho_{1}}$. Let $\varphi\in
C^{\infty}_{\mathrm{c}}(B_{\rho_{2}}\setminus B_{\rho_{1}})$ be a non-
negative, radial function (i.e. $\varphi(z)=\tilde{\varphi}(\left|z\right|)$
for $z\in\mathbb{R}^{3}$) such that
$\int_{\mathbb{R}^{3}}\varphi(z)\,\mathrm{d}z=1$. Since $g$ is bounded away
from zero on the support of $\varphi$, there holds

$\varphi+\left|\nabla\varphi\right|\leq Cg\qquad\textrm{pointwise
a.e.\leavevmode\nobreak\ on }\mathbb{R}^{3},$

for some constant $C$ that depends on $g$ and $\varphi$; however, $\varphi$ is
fixed once and for all, and so is $C$. We define
$\varphi_{\varepsilon}(z):={\varepsilon}^{-3}\varphi({\varepsilon}^{-1}z)$ for
any $z\in\mathbb{R}^{3}$ and ${\varepsilon}>0$. Then,
$\varphi_{\varepsilon}\in C^{\infty}_{\mathrm{c}}(\mathbb{R}^{3})$ is non-
negative, even, satisfies
$\int_{\mathbb{R}^{3}}\varphi_{\varepsilon}(z)\,\mathrm{d}z=1$ and

(3.12)
$\varphi_{\varepsilon}+{\varepsilon}\left|\nabla\varphi_{\varepsilon}\right|\leq
Cg_{\varepsilon}\qquad\textrm{pointwise a.e.\leavevmode\nobreak\ on
}\mathbb{R}^{3}.$

######  Lemma 3.5.

There exists ${\varepsilon}_{2}>0$ such that, for any $u\in
L^{\infty}(B_{1},\,\mathbb{R}^{m})$ and any
${\varepsilon}\in(0,\,{\varepsilon}_{2}]$, there holds

$\int_{B_{1/2}}\left|\nabla(\varphi_{\varepsilon}*u)\right|^{2}\lesssim{\varepsilon}^{-2}\int_{B_{1}\times
B_{1}}K_{\varepsilon}(x-y)\cdot\left(u(x)-u(y)\right)^{\otimes
2}\mathrm{d}x\,\mathrm{d}y.$

###### Proof.

We adapt the arguments from [42, Lemma 2.1 and Proposition 2.1]. We define

$I(y,\,z):=\int_{B_{1/2}}\nabla\varphi_{\varepsilon}(x-y)\cdot\nabla\varphi_{\varepsilon}(x-z)\,\mathrm{d}x\qquad\textrm{for
}y,\,z\in\mathbb{R}^{3}.$

We express the gradient of $\varphi_{\varepsilon}*u$ as
$\nabla(\varphi_{\varepsilon}*u)=(\nabla\varphi_{\varepsilon})*u$. By applying
the identity $2a\cdot
b=-\left|a-b\right|^{2}+\left|a\right|^{2}+\left|b\right|^{2}$, we obtain

$\begin{split}\int_{B_{1/2}}\left|\nabla(\varphi_{\varepsilon}*u)(x)\right|^{2}\mathrm{d}x&=\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}u(y)\cdot
u(z)\,I(y,\,z)\,\mathrm{d}y\,\mathrm{d}z\\\
&=\underbrace{-\frac{1}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\left|u(y)-u(z)\right|^{2}I(y,\,z)\,\mathrm{d}y\,\mathrm{d}z}_{=:I_{1}}\\\
&+\underbrace{\frac{1}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\left|u(y)\right|^{2}I(y,\,z)\,\mathrm{d}y\,\mathrm{d}z}_{=:I_{2}}+\underbrace{\frac{1}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\left|u(z)\right|^{2}I(y,\,z)\,\mathrm{d}y\,\mathrm{d}z}_{=:I_{3}}\end{split}$

We first consider the term $I_{2}$. Since $\varphi_{\varepsilon}$ is compactly
supported, we have
$\int_{\mathbb{R}^{3}}\nabla\varphi_{\varepsilon}(z)\,\mathrm{d}z=0$.
Therefore,

$I_{2}=\frac{1}{2}\int_{B_{1/2}\times\mathbb{R}^{3}}\left|u(y)\right|^{2}\nabla\varphi_{\varepsilon}(x-y)\cdot\left(\int_{\mathbb{R}^{3}}\nabla\varphi_{\varepsilon}(x-z)\,\mathrm{d}z\right)\mathrm{d}x\,\mathrm{d}y=0,$

and likewise $I_{3}=0$. Now, we consider $I_{1}$. The gradient
$\nabla\varphi_{\varepsilon}$ is supported in a ball of radius
$C{\varepsilon}$, where $C$ is an ${\varepsilon}$-independent constant. This
implies

$\begin{split}I_{1}&=\frac{1}{2}\int_{B_{1/2+C{\varepsilon}}\times
B_{1/2+C{\varepsilon}}}\left|u(y)-u(z)\right|^{2}\left(\int_{B_{1/2}}\nabla\varphi_{\varepsilon}(x-y)\cdot\nabla\varphi_{\varepsilon}(x-z)\,\mathrm{d}x\right)\,\mathrm{d}y\,\mathrm{d}z\\\
&\leq\int_{B_{1/2}\times B_{1/2+C{\varepsilon}}\times
B_{1/2+C{\varepsilon}}}\left|u(y)-u(x)\right|^{2}\left|\nabla\varphi_{\varepsilon}(x-y)\right|\left|\nabla\varphi_{\varepsilon}(x-z)\right|\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\\\
&\qquad+\int_{B_{1/2}\times B_{1/2+C{\varepsilon}}\times
B_{1/2+C{\varepsilon}}}\left|u(x)-u(z)\right|^{2}\left|\nabla\varphi_{\varepsilon}(x-y)\right|\left|\nabla\varphi_{\varepsilon}(x-z)\right|\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\\\
&\leq
2\left\|\nabla\varphi_{\varepsilon}\right\|_{L^{1}(\mathbb{R}^{3})}\int_{B_{1/2}\times
B_{1/2+C{\varepsilon}}}\left|u(y)-u(x)\right|^{2}\left|\nabla\varphi_{\varepsilon}(y-x)\right|\mathrm{d}x\,\mathrm{d}y\end{split}$

Thanks to (3.12), we obtain

$\begin{split}I_{1}\lesssim{\varepsilon}^{-2}\left\|g\right\|_{L^{1}(\mathbb{R}^{3})}\int_{B_{1/2}\times
B_{1/2+C{\varepsilon}}}\left|u(y)-u(x)\right|^{2}g_{\varepsilon}(y-x)\,\mathrm{d}x\,\mathrm{d}y.\end{split}$

For ${\varepsilon}$ sufficiently small we have $1/2+C{\varepsilon}<1$, and the
lemma follows. ∎

Given two sets $A\subseteq\mathbb{R}^{3}$,
$A^{\prime}\subseteq\mathbb{R}^{3}$, we write $A\subset\\!\subset A^{\prime}$
when the _closure_ of $A$ is contained in $A^{\prime}$.

######  Lemma 3.6.

Let $A$, $A^{\prime}$ be open sets such that $A\subset\\!\subset
A^{\prime}\subseteq\mathbb{R}^{3}$. Then, there exists
${\varepsilon}_{3}={\varepsilon}_{3}(A,\,A^{\prime})$ such that, for any $u\in
L^{\infty}(A^{\prime},\,\mathbb{R}^{m})$ and any
${\varepsilon}\in(0,\,{\varepsilon}_{3}]$, there holds

$\int_{A}\left|u-\varphi_{\varepsilon}*u\right|^{2}\lesssim\int_{A^{\prime}\times
A^{\prime}}K_{\varepsilon}(x-y)\cdot\left(u(x)-u(y)\right)^{\otimes
2}\mathrm{d}x\,\mathrm{d}y.$

###### Proof.

Since $\int_{\mathbb{R}^{3}}\varphi_{\varepsilon}(z)\,\mathrm{d}z=1$, we have

$I:=\int_{A}\left|u(x)-(\varphi_{\varepsilon}*u)(x)\right|^{2}\mathrm{d}x=\int_{A}\left|\int_{\mathbb{R}^{3}}\varphi_{\varepsilon}(x-y)\left(u(x)-u(y)\right)\mathrm{d}y\right|^{2}\mathrm{d}x.$

We apply Jensen inequality with respect to the probability measure
$\varphi_{\varepsilon}(x-y)\,\mathrm{d}y$:

$I\leq\int_{A}\left(\int_{\mathbb{R}^{3}}\varphi_{\varepsilon}(x-y)\left|u(x)-u(y)\right|^{2}\mathrm{d}y\right)\mathrm{d}x.$

Because the support of $\varphi_{\varepsilon}$ is contained in a ball of
radius $C{\varepsilon}$, where $C$ is an ${\varepsilon}$-independent constant,
the integrand is equal to zero if $x\in A$,
$\operatorname{dist}(y,\,A)>C{\varepsilon}$. By applying (3.12), we obtain

$I\leq\int_{A\times\\{y\in\mathbb{R}^{3}\colon\operatorname{dist}(y,\,A)\leq
C{\varepsilon}\\}}g_{\varepsilon}(x-y)\left|u(x)-u(y)\right|^{2}\mathrm{d}x\,\mathrm{d}y$

and, if ${\varepsilon}\leq C^{-1}\operatorname{dist}(A,\,\partial
A^{\prime})$, the lemma follows. ∎

###### Proof of Proposition 3.4.

Due to the scaling property (3.11), it suffices to prove that

(3.13) $\fint_{B_{1/2}}\left|u-\fint_{B_{1/2}}u\right|^{2}\lesssim
F_{{\varepsilon}/\rho}(u,\,B_{1})$

for any $u\in L^{\infty}(\mathbb{R}^{3},\,\mathbb{R}^{m})$ and any
${\varepsilon}$, $\rho$ with ${\varepsilon}/\rho$ sufficiently small. The
triangle inequality and the elementary inequality $(a+b+c)^{2}\leq
3(a^{2}+b^{2}+c^{2})$ imply

$\fint_{B_{1/2}}\left|u-\fint_{B_{1/2}}u\right|^{2}\leq
6\fint_{B_{1/2}}\left|u-\varphi_{{\varepsilon}/\rho}*u\right|^{2}+3\fint_{B_{1/2}}\left|\varphi_{{\varepsilon}/\rho}*u-\fint_{B_{1/2}}\varphi_{{\varepsilon}/\rho}*u\right|^{2}$

Thanks to the Poincaré inequality, we obtain

$\fint_{B_{1/2}}\left|u-\fint_{B_{1/2}}u\right|^{2}\lesssim\int_{B_{1/2}}\left|u-\varphi_{{\varepsilon}/\rho}*u\right|^{2}+\int_{B_{1/2}}\left|\nabla(\varphi_{{\varepsilon}/\rho}*u)\right|^{2}.$

If ${\varepsilon}/\rho$ is sufficiently small, Lemma 3.5 and Lemma 3.6 give

$\fint_{B_{1/2}}\left|u-\fint_{B_{1/2}}u\right|^{2}\lesssim\left(({\varepsilon}/\rho)^{2}+1\right)F_{{\varepsilon}/\rho}(u,\,B_{1}),$

so (3.13) follows. ∎

###  3.3 Localised $\Gamma$-convergence for the non-local term

The $\Gamma$-convergence of the functional $E_{\varepsilon}$, as
${\varepsilon}\to 0$, was studied in [42]. In this section, we adapt the
arguments of [42] to prove a localised $\Gamma$-convergence result. We focus
on the interaction part of the free energy only, since this is all we need in
the proof of Theorem A.

######  Proposition 3.7.

Let $\rho>0$, and let $v_{\varepsilon}\in
L^{\infty}(B_{\rho}(x_{0}),\,\mathbb{R}^{m})$ be a sequence of maps such that

$\sup_{{\varepsilon}>0}\left(\frac{1}{{\varepsilon}^{2}}\int_{B_{\rho}(x_{0})\times
B_{\rho}(x_{0})}K_{\varepsilon}(x-y)\cdot\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y+\left\|v_{\varepsilon}\right\|_{L^{\infty}(B_{\rho}(x_{0}))}\right)<+\infty.$

Then, there exist a map $v_{0}\in(L^{\infty}\cap
H^{1})(B_{\rho/2}(x_{0}),\,\mathbb{R}^{m})$ and a (non-relabelled) subsequence
such that $v_{\varepsilon}\to v_{0}$ strongly in $L^{2}(B_{\rho/2}(x_{0}))$ as
${\varepsilon}\to 0$. Moreover, for any open set $G\subseteq
B_{\rho/2}(x_{0})$ we have

(3.14) $\int_{G}L\nabla v_{0}(x)\cdot\nabla
v_{0}(x)\,\mathrm{d}x\leq\liminf_{{\varepsilon}\to
0}\frac{1}{4{\varepsilon}^{2}}\int_{G\times
G}K_{\varepsilon}(x-y)\cdot\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y.$

######  Proposition 3.8.

Let $\rho>0$. Let $v_{\varepsilon}\in(L^{\infty}\cap
H^{1})(B_{\rho}(x_{0}),\,\mathbb{R}^{m})$ be a sequence such that
$v_{\varepsilon}\to v_{0}$ strongly in $H^{1}(B_{\rho}(x_{0}))$ and
$\sup_{\varepsilon}\left\|v_{\varepsilon}\right\|_{L^{\infty}(B_{\rho}(x_{0}))}<+\infty$.
Then

$\limsup\limits_{{\varepsilon}\to
0}\frac{1}{4{\varepsilon}^{2}}\int_{B_{\rho}(x_{0})\times
B_{\rho}(x_{0})}K_{\varepsilon}(x-y)\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y\leq\int_{B_{\rho}(x_{0})}L\nabla
v_{0}(x)\cdot\nabla v_{0}(x)\,\mathrm{d}x.$

In the proofs of Proposition 3.7 and 3.8, we will use the following notation.
Given a vector $w\in\mathbb{R}^{3}\setminus\\{0\\}$ and a function $u$ defined
on a subset of $\mathbb{R}^{3}$, we define the difference quotient

$D_{w}u(x):=\frac{u(x+w)-u(x)}{\left|w\right|}$

for any $x$ in the domain of $u$ such that $x+w$ belongs to the domain of $u$.
If $\left|w\right|\leq h$, $u\in H^{1}(B_{\rho+h})$, and $|\cdot|_{*}$ is any
seminorm on $\mathbb{R}^{m}$, then

(3.15)
$\int_{B_{\rho}}|D_{{\varepsilon}w}u(x)|_{*}^{2}\,\mathrm{d}x\leq\int_{B_{\rho+{\varepsilon}h}}|(\hat{w}\cdot\nabla)u(x)|_{*}^{2}\,\mathrm{d}x$

where $\hat{w}:=w/\left|w\right|$. This follows from the same technique as,
e.g., [18, Lemma 7.23], for the case in which we have the standard Euclidean
norm. However, we realise the proof only relies on the convexity of the
seminorm, and no further structure. For convenience, we give the proof of
Proposition 3.8 first.

###### Proof of Proposition 3.8.

We assume that $x_{0}=0$. Using a reflection across the boundary of $B_{\rho}$
and a cut-off function, we define $v_{\varepsilon}$ and $v_{0}$ on
$\mathbb{R}^{3}\setminus B_{\rho}$, in such a way that
$v_{\varepsilon}\in(L^{\infty}\cap H^{1})(\mathbb{R}^{3},\,\mathbb{R}^{m})$,
$v_{\varepsilon}\to v_{0}$ strongly in $H^{1}(\mathbb{R}^{3})$ and
$\sup_{\varepsilon}\left\|v_{\varepsilon}\right\|_{L^{\infty}(\mathbb{R}^{3})}<+\infty$.
Let $t>0$ be a parameter. We have

$\begin{split}&\frac{1}{4{\varepsilon}^{2}}\int_{B_{\rho}}\int_{B_{\rho}}K_{\varepsilon}(x-y)\cdot\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\frac{1}{4}\int_{B_{\rho}}\int_{\mathbb{R}^{3}}|z|^{2}K(z)\cdot\left(D_{{\varepsilon}z}v_{\varepsilon}(x)\right)^{\otimes
2}\,\mathrm{d}z\,\mathrm{d}x\\\
&=\frac{1}{4}\int_{B_{\rho}}\int_{B_{\frac{t}{{\varepsilon}}}}|z|^{2}K(z)\cdot\left(D_{{\varepsilon}z}v_{\varepsilon}(x)\right)^{\otimes
2}\,\mathrm{d}z\,\mathrm{d}x+\frac{1}{4}\int_{B_{\rho}}\int_{\mathbb{R}^{3}\setminus
B_{\frac{t}{{\varepsilon}}}}|z|^{2}K(z)\cdot\left(D_{{\varepsilon}z}v_{\varepsilon}(x)\right)^{\otimes
2}\,\mathrm{d}z\,\mathrm{d}x.\end{split}$

To estimate the first integral at the right-hand side, we exchange the order
of integration and, for any $z$, we apply (3.15) to the seminorm
$\left|\xi\right|_{*}^{2}:=\left|z\right|^{2}K(z)\cdot\xi^{\otimes 2}$; for
the second integral, we apply (K5):

(3.16)
$\begin{split}&\frac{1}{4{\varepsilon}^{2}}\int_{B_{\rho}}\int_{B_{\rho}}K_{\varepsilon}(x-y)\cdot\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\frac{1}{4}\int_{\mathbb{R}^{3}}\int_{B_{\rho+t}}K(z)\cdot\left((z\cdot\nabla)v_{\varepsilon}(x)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}z+C\int_{B_{\rho}}\int_{\mathbb{R}^{3}\setminus
B_{\frac{t}{{\varepsilon}}}}g(z)|z|^{2}|D_{{\varepsilon}z}v_{\varepsilon}(x)|^{2}\,\mathrm{d}z\,\mathrm{d}x\\\
&\stackrel{{\scriptstyle\eqref{L}}}{{=}}\int_{B_{\rho+t}}L\nabla
v_{\varepsilon}(x)\cdot\nabla
v_{\varepsilon}(x)\,\mathrm{d}x+C\int_{B_{\rho}}\int_{\mathbb{R}^{3}\setminus
B_{\frac{t}{{\varepsilon}}}}g(z)|z|^{2}|D_{{\varepsilon}z}v_{\varepsilon}(x)|^{2}\,\mathrm{d}z\,\mathrm{d}x.\end{split}$

We now estimate the latter summand independently. For $z\in
B_{\frac{t}{{\varepsilon}}}$, $|{\varepsilon}z|^{2}>t^{2}$, so

$|D_{{\varepsilon}z}v_{\varepsilon}(x)|^{2}\leq\frac{4}{t^{2}}\left\|v_{\varepsilon}\right\|_{L^{\infty}(\mathbb{R}^{3})}^{2}\\!.$

Therefore we may estimate

(3.17) $\begin{split}\int_{B_{\rho}}\int_{\mathbb{R}^{3}\setminus
B_{\frac{t}{{\varepsilon}}}}g(z)|z|^{2}|D_{{\varepsilon}z}v_{\varepsilon}(x)|^{2}\,\mathrm{d}z\,\mathrm{d}x&\leq\int_{B_{\rho}}\int_{\mathbb{R}^{3}\setminus
B_{\frac{t}{{\varepsilon}}}}\frac{4}{t^{2}}g(z)|z|^{2}\left\|v_{\varepsilon}\right\|_{L^{\infty}(\mathbb{R}^{3})}^{2}\,\mathrm{d}z\,\mathrm{d}x\\\
&=\frac{4\left|B_{\rho}\right|\left\|v_{\varepsilon}\right\|_{L^{\infty}(\mathbb{R}^{3})}^{2}}{t^{2}}\int_{\mathbb{R}^{3}\setminus
B_{\frac{t}{{\varepsilon}}}}g(z)|z|^{2}\,\mathrm{d}z\end{split}$

As $g$ has finite second moment and we have assumed that
$\left\|v_{\varepsilon}\right\|_{L^{\infty}(\Omega)}\leq C$, for fixed $t$ we
must have that

(3.18) $\lim\limits_{{\varepsilon}\to
0}\frac{4\left|B_{\rho}\right|\left\|v_{\varepsilon}\right\|_{L^{\infty}(\mathbb{R}^{3})}^{2}}{t^{2}}\int_{\mathbb{R}^{3}\setminus
B_{\frac{t}{{\varepsilon}}}}g(z)|z|^{2}\,\mathrm{d}z=0.$

Combining (3.16), (3.17) and (3.18) gives

$\limsup\limits_{{\varepsilon}\to
0}\frac{1}{4{\varepsilon}^{2}}\int_{B_{\rho}}\int_{B_{\rho}}K_{\varepsilon}(x-y)\cdot\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}y\,\mathrm{d}x\leq\limsup\limits_{{\varepsilon}\to
0}\int_{B_{\rho+t}}L\nabla v_{\varepsilon}(x)\cdot\nabla
v_{\varepsilon}(x)\,\mathrm{d}x.$

As $v_{\varepsilon}\to v_{0}$ in $H^{1}(\mathbb{R}^{3})$, this implies

$\lim\limits_{{\varepsilon}\to 0}\int_{B_{\rho+t}}L\nabla
v_{\varepsilon}(x)\cdot\nabla
v_{\varepsilon}(x)\,\mathrm{d}x=\int_{B_{\rho+t}}L\nabla v_{0}(x)\cdot\nabla
v_{0}(x)\,\mathrm{d}x.$

Therefore we have

$\limsup\limits_{{\varepsilon}\to
0}\frac{1}{4{\varepsilon}^{2}}\int_{B_{\rho}}\int_{B_{\rho}}K_{\varepsilon}(x-y)\cdot\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}y\,\mathrm{d}x\leq\int_{B_{\rho+t}}L\nabla v_{0}(x)\cdot\nabla
v_{0}(x)\,\mathrm{d}x,$

and passing to the limit as $t\to 0$ in the right-hand side gives the desired
result. ∎

###### Proof of Proposition 3.7.

Again, we assume that $x_{0}=0$. Let $(\varphi_{\varepsilon})$ be the sequence
of mollifiers given by (3.12). By Lemma 3.5, the sequence
$\varphi_{\varepsilon}*v_{\varepsilon}$ is bounded in $H^{1}(B_{\rho/2})$ and
hence, up to extraction of a (non-relabelled) subsequence,
$\varphi_{\varepsilon}*v_{\varepsilon}\rightharpoonup v_{0}$ weakly in
$H^{1}(B_{\rho/2})$ and strongly in $L^{2}(B_{\rho/2})$. By Lemma 3.6,
$\varphi_{\varepsilon}*v_{\varepsilon}-v_{\varepsilon}\to 0$ strongly in
$L^{2}(B_{\rho/2})$. Therefore, $v_{\varepsilon}\to v_{0}$ strongly in
$L^{2}(B_{\rho/2})$. Let $G\subseteq B_{\rho/2}$ be open and
$G^{\prime}\subset\\!\subset G$. Then we may write that

(3.19)
$\begin{split}\frac{1}{4{\varepsilon}^{2}}\int_{G}\int_{G}&K_{\varepsilon}(x-y)\cdot\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}y\,\mathrm{d}x\\\
&\geq\frac{1}{4{\varepsilon}^{2}}\int_{G^{\prime}}\int_{G}K_{\varepsilon}(x-y)\cdot\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}y\,\mathrm{d}x\\\
&=\frac{1}{4}\int_{G^{\prime}}\int_{\frac{G-x}{{\varepsilon}}}|z|^{2}K(z)\cdot\left(D_{{\varepsilon}z}v_{\varepsilon}(x)\right)^{\otimes
2}\,\mathrm{d}z\,\mathrm{d}x.\end{split}$

Let $G^{c}:=\mathbb{R}^{3}\setminus G$ and
$\delta:=\operatorname{dist}(G^{\prime},\,G^{c})>0$. We note that

$\begin{split}\left|\int_{G^{\prime}}\int_{\left(\frac{G-x}{{\varepsilon}}\right)^{c}}|z|^{2}K(z)\cdot\left(D_{{\varepsilon}z}v_{\varepsilon}(x)\right)^{\otimes
2}\,\mathrm{d}y\,\mathrm{d}x\right|\lesssim\int_{G^{\prime}}\int_{B_{\frac{\delta}{{\varepsilon}}}^{c}}g(z)|z|^{2}|D_{{\varepsilon}z}v_{\varepsilon}(x)|^{2}\,\mathrm{d}z\,\mathrm{d}x,\end{split}$

which by previous estimates (see (3.17), (3.18)) we have seen converges to
zero as ${\varepsilon}\to 0$. This means

(3.20) $\begin{split}\liminf\limits_{{\varepsilon}\to
0}&\int_{G^{\prime}}\int_{\frac{G-x}{{\varepsilon}}}|z|^{2}K(z)\cdot\left(D_{{\varepsilon}z}v_{\varepsilon}(x)\right)^{\otimes
2}\,\mathrm{d}z\,\mathrm{d}x\\\ &=\liminf\limits_{{\varepsilon}\to
0}\int_{G^{\prime}}\int_{\mathbb{R}^{3}}|z|^{2}K(z)\cdot\left(D_{{\varepsilon}z}v_{\varepsilon}(x)\right)^{\otimes
2}\,\mathrm{d}z\,\mathrm{d}x\\\ \end{split}$

Furthermore, we note this can be written as an $L^{2}$ norm, by defining
$w_{\varepsilon}:G^{\prime}\times\mathbb{R}^{3}\to\mathbb{R}^{m}$ by
$w_{\varepsilon}(z,x):=|z|K^{\frac{1}{2}}(z)D_{{\varepsilon}z}v_{\varepsilon}(x)$.
We immediately see that $w_{\varepsilon}$ is $L^{2}$-bounded, so must admit an
$L^{2}$-weakly converging subsequence $w_{j}:=w_{{\varepsilon}_{j}}$ with
${\varepsilon}_{j}\to 0$ and $w_{j}$ has weak-$L^{2}$ limit $w_{0}$.
Furthermore, we take

(3.21) $\liminf\limits_{{\varepsilon}\to
0}\int_{G^{\prime}}\int_{\mathbb{R}^{3}}|z|^{2}K(z)\cdot\left(D_{{\varepsilon}z}v_{\varepsilon}(x)\right)^{\otimes
2}\,\mathrm{d}z\,\mathrm{d}x=\liminf\limits_{j\to\infty}\left\|w_{j}\right\|^{2}_{L^{2}(G^{\prime}\times\mathbb{R}^{3})}\geq\left\|w_{0}\right\|^{2}_{L^{2}(G^{\prime}\times\mathbb{R}^{3})}\\!.$

It remains to identify the limit $w_{0}$. We may do this by integrating
against test functions. Let $\phi\in
C^{\infty}_{\mathrm{c}}(G^{\prime}\times\mathbb{R}^{3})$. There exists some
$R_{0}>0$ such that, for any $(y,\,z)\in\mathbb{R}^{3}\times\mathbb{R}^{3}$
with $|z|>R_{0}$, $\phi(y,\,z)=0$. Furthermore, there exists some $\delta>0$
so that if $\operatorname{dist}(y,\,(G^{\prime})^{c})<\delta$, then
$\phi(y,\,z)=0$. In particular, if ${{\varepsilon}_{j}}<\frac{\delta}{R_{0}}$
and $(x-{\varepsilon}_{j}z,\,z)\in\mathrm{supp}(\phi)$, then $x\in
G^{\prime}$. Therefore

$\begin{split}\langle
w_{j},\,\phi\rangle&=\int_{G^{\prime}}\int_{\mathbb{R}^{3}}\phi(x,\,z)|z|K^{\frac{1}{2}}(z)D_{{{\varepsilon}_{j}}z}v_{{\varepsilon}_{j}}(x)\,\mathrm{d}z\,\mathrm{d}x\\\
&=\frac{1}{{{\varepsilon}_{j}}}\int_{G^{\prime}}\int_{\mathbb{R}^{3}}\Big{(}\phi(x-{{\varepsilon}_{j}}z,\,z)-\phi(x,\,z)\Big{)}K^{\frac{1}{2}}(z)v_{{\varepsilon}_{j}}(x)\,\mathrm{d}z\,\mathrm{d}x,\end{split}$

and we may exploit the fact that

$\frac{1}{{{\varepsilon}_{j}}}\Big{(}\phi(x-{{\varepsilon}_{j}}z,\,z)-\phi(x,\,z)\Big{)}\to(-z\cdot\nabla_{x})\phi(x,\,z)\qquad\textrm{uniformly
on }G^{\prime}\textrm{ as }j\to+\infty,$

with the assumed $L^{2}$ convergence of $v_{{\varepsilon}_{j}}\to v_{0}$, to
give that

$\begin{split}\lim\limits_{j\to\infty}\langle
w_{j},\,\phi\rangle&=\lim\limits_{j\to\infty}\frac{1}{{\varepsilon}_{j}}\int_{G^{\prime}}\int_{\mathbb{R}^{3}}\Big{(}\phi(x-{{\varepsilon}_{j}}z,\,z)-\phi(x,\,z)\Big{)}K^{\frac{1}{2}}(z)v_{{\varepsilon}_{j}}(x)\,\mathrm{d}z\,\mathrm{d}x,\\\
&=\int_{G^{\prime}}\int_{\mathbb{R}^{3}}(-z\cdot\nabla_{x})\phi(x,\,z)K^{\frac{1}{2}}(z)v_{0}(x)\,\mathrm{d}z\,\mathrm{d}x\\\
&=\int_{G^{\prime}}\int_{\mathbb{R}^{3}}\phi(x,\,z)K^{\frac{1}{2}}(z)(z\cdot\nabla)v_{0}(x)\,\mathrm{d}z\,\mathrm{d}x=\langle
w_{0},\,\phi\rangle.\end{split}$

Therefore $w_{0}(x,\,z)=K^{\frac{1}{2}}(z)(z\cdot\nabla)v_{0}(x)$, and by
(3.19), (3.20), (3.21) we have

$\begin{split}\liminf\limits_{{\varepsilon}\to
0}&\frac{1}{4{\varepsilon}^{2}}\int_{G}\int_{G}K_{\varepsilon}(x-y)\cdot\left(v_{\varepsilon}(x)-v_{\varepsilon}(y)\right)^{\otimes
2}\,\mathrm{d}y\,\mathrm{d}x\\\
&\geq\frac{1}{4}\liminf\limits_{j\to\infty}\left\|w_{j}\right\|^{2}_{L^{2}(G^{\prime}\times\mathbb{R}^{3})}\\\
&\geq\frac{1}{4}\left\|w_{0}\right\|^{2}_{L^{2}(G^{\prime}\times\mathbb{R}^{3})}\\\
&=\frac{1}{4}\int_{G^{\prime}}\int_{\mathbb{R}^{3}}K(z)\cdot\Big{(}(z\cdot\nabla)v_{0}(x)\Big{)}^{\otimes
2}\,\mathrm{d}z\,\mathrm{d}x\\\
&\stackrel{{\scriptstyle\eqref{L}}}{{=}}\int_{G^{\prime}}L\nabla
v_{0}(x)\cdot\nabla v_{0}(x)\,\mathrm{d}x.\end{split}$

As the set $G^{\prime}\subset\\!\subset G$ was arbitrary, by monotonicity the
lower bound (3.14) holds. ∎

##  4 Proof of the main results

###  4.1 A compactness result for $\omega$-minimisers

The goal of this section is to prove a compactness result for minimisers of
$E_{\varepsilon}$, subject to variable “boundary conditions”, as
${\varepsilon}\to 0$. For later convenience, we state our result in terms of
“almost minimisers” — or, more precisely, $\omega$-minimisers, as defined
below. This will be useful to study variants of our original minimisation
problem, as we will do in Section 5.

######  Definition 4.1.

Let $\Omega\subseteq\mathbb{R}^{3}$ be a bounded domain. Let
$\omega\colon[0,\,+\infty)\to[0,\,+\infty)$ be an increasing function such
that $\omega(s)\to 0$ as $s\to 0$. We say that a function $u\in
L^{\infty}(\mathbb{R}^{3},\,\mathcal{Q})$ is an $\omega$-minimiser of
$E_{\varepsilon}$ in $\Omega$ if, for any ball
$B_{\rho}(x_{0})\subseteq\Omega$ and any $v\in
L^{\infty}(\mathbb{R}^{3},\,\mathcal{Q})$ such that $v=u$ a.e. on
$\mathbb{R}^{3}\setminus B_{\rho}(x_{0})$, there holds

$E_{\varepsilon}(u)\leq E_{\varepsilon}(v)+\omega({\varepsilon})\,\rho.$

By definition, a minimiser for $E_{\varepsilon}$ in the class $\mathscr{A}$
defined by (2.5) is also a $\omega$-minimiser in $\Omega$, for any $\omega\geq
0$. $\omega$-minimisers behave nicely with respect to scaling. Given $u\in
L^{\infty}(\mathbb{R}^{3},\,\mathcal{Q})$, an increasing function
$\omega\colon[0,\,+\infty)\to[0,\,+\infty)$, $x_{0}\in\mathbb{R}^{3}$ and
$\rho>0$, we define $u_{\rho}\colon\mathbb{R}^{3}\to\mathcal{Q}$ and
$\omega_{\rho}\colon[0,\,+\infty)\to[0,\,+\infty)$ as
$u_{\rho}(y):=u(x_{0}+\rho y)$ for $y\in\mathbb{R}^{3}$ and
$\omega_{\rho}(s):=\omega(\rho s)$ for $s\geq 0$, respectively. A scaling
argument implies

######  Lemma 4.1.

If $u$ is an $\omega$-minimiser for $E_{\varepsilon}$ in a bounded domain
$\Omega\subseteq\mathbb{R}^{3}$, then $u_{\rho}$ is an
$\omega_{\rho}$-minimiser for $E_{{\varepsilon}/\rho}$ in $(\Omega-
x_{0})/\rho$.

The goal of this section is to prove the following

######  Proposition 4.2.

Let $\Omega\subseteq\mathbb{R}^{3}$ be a bounded domain. Let
$\omega\colon[0,\,+\infty)\to[0,\,+\infty)$ be an increasing function such
that $\omega(s)\to 0$ as $s\to 0$. Let $u_{\varepsilon}$ be a sequence of
$\omega$-minimisers of $E_{\varepsilon}$ in $\Omega$, and let
$B_{\rho}(x_{0})\subseteq\Omega$ be a ball such that
$\sup_{{\varepsilon}>0}F_{\varepsilon}(u_{\varepsilon},\,B_{\rho}(x_{0}))<+\infty$.
Then, up to extraction of a non-relabelled subsequence, $u_{\varepsilon}$
converge $L^{2}(B_{\rho/2}(x_{0}))$-strongly to a map $u_{0}\in
H^{1}(B_{\rho/2}(x_{0}),\,\mathscr{N})$, which minimises the functional

$w\in
H^{1}(B_{\rho/2}(x_{0}),\,\mathscr{N})\mapsto\int_{B_{\rho/2}(x_{0})}L\nabla
w\cdot\nabla w$

subject to its own boundary conditions. Moreover, for any $s\in(0,\,\rho/2)$
there holds

(4.1) $\lim_{{\varepsilon}\to
0}F_{\varepsilon}(u_{\varepsilon},\,B_{s}(x_{0}))=\int_{B_{s}(x_{0})}L\nabla
u_{0}\cdot\nabla u_{0}.$

Proposition 4.2 differs from the results in [42] in that no “boundary
condition” is prescribed: each $u_{\varepsilon}$ minimises the functional
$E_{\varepsilon}$ (possibily up to a small error, which is quantified by the
function $\omega$) subject to its own “boundary condition”. The main
ingredient in the proof of Proposition 4.2 is the following extension lemma.

######  Lemma 4.3 ([33, 10]).

For any $M>0$, there exists $\eta=\eta(M)>0$ such that the following statement
holds. Let $\mathcal{Q}_{0}\subset\\!\subset\mathcal{Q}$ be an open set that
contains $\mathscr{N}$. Let $\rho$, $\lambda$ be positive numbers with
$\lambda<\rho$, and let $u\in H^{1}(\partial B_{\rho},\,\mathcal{Q}_{0})$,
$v\in H^{1}(\partial B_{\rho},\,\mathscr{N})$ be such that

$\int_{\partial B_{\rho}}\left(\left|\nabla u\right|^{2}+\left|\nabla
v\right|^{2}\right)\mathrm{d}\mathscr{H}^{2}\leq M,\qquad{\int_{\partial
B_{\rho}}\left|u-v\right|^{2}\,\mathrm{d}\mathscr{H}^{2}\leq\eta\lambda^{2}.}$

Then, there exists a map $w\in H^{1}(B_{\rho}\setminus
B_{\rho-\lambda},\,\mathcal{Q}_{0})$ such that $w(x)=u(x)$ for
$\mathscr{H}^{2}$-a.e. $x\in\partial B_{\rho}$, $w(x)=v(\rho
x/(\rho-\lambda))$ for $\mathscr{H}^{2}$-a.e. $x\in\partial B_{\rho-\lambda}$,
and

$\displaystyle\int_{B_{\rho}\setminus B_{\rho-\lambda}}\left|\nabla
w\right|^{2}\lesssim\lambda\int_{\partial B_{\rho}}\left(\left|\nabla
u\right|^{2}+\left|\nabla
v\right|^{2}+\frac{\left|u-v\right|^{2}}{\lambda^{2}}\right)\mathrm{d}\mathscr{H}^{2}$
$\displaystyle\int_{B_{\rho}\setminus
B_{\rho-\lambda}}\psi_{b}(w)\lesssim\lambda\int_{\partial
B_{\rho}}\psi_{b}(u)\,\mathrm{d}\mathscr{H}^{2}$

######  Remark 4.1.

Lemma 4.3, in case $\psi_{b}=0$, was first proven by Luckhaus [33, Lemma 1].
Up to a scaling, the statement given here is essentially the same as [10,
Lemma B.2]. However, in [10] the potential is assumed to be finite and smooth
on the whole of $\mathbb{R}^{m}$, while our potential $\psi_{b}$ is singular
out of $\mathcal{Q}$. Nevertheless, the proof carries over to our setting.
Indeed, the map $w$ constructed in [10] takes values in a neighbourhood of
$\mathscr{N}$, whose thickness can be made arbitrarily small by choosing
$\eta$ small (see also [33, Lemma 1]). In particular, we can make sure that
the image of $w$ is contained in the set $\mathcal{Q}_{0}$, where the function
$\psi_{b}$ is finite and smooth, and the arguments of [10] carry over.
Incidentally, Lemma 4.3 crucially depends on the non-degeneracy assumption
(H6) for the bulk potential $\psi_{b}$.

We state a few other technical results, which will be useful in the proof of
Proposition 4.2.

######  Lemma 4.4.

Let $G$, $G^{\prime}$ be open sets in $\mathbb{R}^{3}$, with
$G\subset\\!\subset G^{\prime}$. Given
$0<\theta<\operatorname{dist}(G,\,\partial G^{\prime})$, define
$\partial_{\theta}G:=\\{x\in\mathbb{R}^{3}\colon\operatorname{dist}(x,\,\partial
G)<\theta\\}$. Then, for any ${\varepsilon}>0$ and $u\in
L^{\infty}(\mathbb{R}^{3},\,\mathbb{R}^{m})$, there holds

$\begin{split}F_{\varepsilon}(u,\,G^{\prime})\leq
F_{\varepsilon}(u,\,G)&+F_{\varepsilon}(u,\,G^{\prime}\setminus
G)+F_{\varepsilon}(u,\,\partial_{\theta}G)\\\
&+\frac{C}{\theta^{2}}\left\|u\right\|^{2}_{L^{\infty}(\mathbb{R}^{3})}\left|G\right|\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z.\end{split}$

###### Proof.

Since $K_{\varepsilon}$ is even (by (K2)), we have

$\begin{split}F_{\varepsilon}(u,\,G^{\prime})\leq
F_{\varepsilon}(u,\,G)&+F_{\varepsilon}(u,\,G^{\prime}\setminus G)\\\
&+\frac{1}{4{\varepsilon}^{2}}\int_{\partial_{\theta}G\times\partial_{\theta}G}K_{\varepsilon}(x-y)\cdot\left(u(x)-u(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y\\\
&+\frac{1}{2{\varepsilon}^{2}}\int_{\\{(x,\,y)\in
G\times\mathbb{R}^{3}\colon\left|x-y\right|\geq\theta\\}}K_{\varepsilon}(x-y)\cdot\left(u(x)-u(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y\end{split}$

Keeping in mind that $u$ is bounded, and using (K5), we obtain

$\begin{split}F_{\varepsilon}(u,\,G^{\prime})\leq
F_{\varepsilon}(u,\,G)&+F_{\varepsilon}(u,\,G^{\prime}\setminus
G)+F_{\varepsilon}(u,\,\partial_{\theta}G)\\\
&+\frac{C}{{\varepsilon}^{2}}\left\|u\right\|_{L^{\infty}(\mathbb{R}^{3})}^{2}\underbrace{\int_{\\{(x,y)\in
G\times\mathbb{R}^{3}\colon\left|x-y\right|\geq\theta\\}}g_{\varepsilon}(x-y)\,\mathrm{d}x\,\mathrm{d}y}_{=:I}\end{split}$

We bound the term $I$ by making the change of variable $y=x+{\varepsilon}z$:

$\begin{split}I=\int_{G}\left(\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\,\mathrm{d}z\right)\mathrm{d}x\leq\frac{{\varepsilon}^{2}}{\theta^{2}}\left|G\right|\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z.\end{split}$

The right-hand side is finite, because of (K4). The lemma follows. ∎

Another variant of Lemma 4.4 is the following “gluing lemma” for the non-local
functional $E_{\varepsilon}$.

######  Lemma 4.5.

Given a number $\theta>0$, a Borel set $G\subseteq\mathbb{R}^{3}$, and maps
$u_{1}$, $u_{2}\in L^{\infty}(\mathbb{R}^{3},\,\mathbb{R}^{m})$, define the
map

$u:=\begin{cases}u_{1}&\textrm{on }G\\\ u_{2}&\textrm{on
}\mathbb{R}^{3}\setminus G\end{cases}$

and
$\partial_{\theta}G:=\\{x\in\mathbb{R}^{3}\colon\operatorname{dist}(x,\,\partial
G)<\theta\\}$, as above. Then, for any ${\varepsilon}>0$, there holds

$\begin{split}E_{\varepsilon}(u)\leq
F_{\varepsilon}(u_{1},\,G)&+F_{\varepsilon}(u_{2},\,\mathbb{R}^{3}\setminus
G)+2F_{\varepsilon}(u_{2},\,\partial_{\theta}G)+\frac{C}{{\varepsilon}^{2}}\int_{\partial_{\theta}G}\left|u_{1}-u_{2}\right|^{2}\\\
&+\frac{C}{\theta^{2}}\left\|u\right\|^{2}_{L^{\infty}(\mathbb{R}^{3})}\left|G\right|\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z.\end{split}$

###### Proof.

We repeat the arguments of Lemma 4.4, with $G^{\prime}=\mathbb{R}^{3}$:

$\begin{split}E_{\varepsilon}(u)\leq
F_{\varepsilon}(u,\,G)&+F_{\varepsilon}(u,\,G^{\prime}\setminus G)\\\
&+\underbrace{\frac{1}{4{\varepsilon}^{2}}\int_{\partial_{\theta}G\times\partial_{\theta}G}K_{\varepsilon}(x-y)\cdot\left(u(x)-u(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y}_{=:J}\\\
&+\frac{1}{2{\varepsilon}^{2}}\int_{\\{(x,\,y)\in
G\times\mathbb{R}^{3}\colon\left|x-y\right|\geq\theta\\}}K_{\varepsilon}(x-y)\cdot\left(u(x)-u(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y\end{split}$

The last term at the right-hand side can be bounded exactly as in Lemma 4.4.
The triangle inequality and the elementary inequality $(a+b)^{2}\leq
2(a^{2}+b^{2})$ imply

$\begin{split}J&\leq\frac{1}{2{\varepsilon}^{2}}\int_{\partial_{\theta}G\times\partial_{\theta}G}K_{\varepsilon}(x-y)\cdot\left(u_{1}(x)-u_{2}(x)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y\\\
&\qquad\qquad\qquad+\frac{1}{2{\varepsilon}^{2}}\int_{\partial_{\theta}G\times\partial_{\theta}G}K_{\varepsilon}(x-y)\cdot\left(u_{2}(x)-u_{2}(y)\right)^{\otimes
2}\,\mathrm{d}x\,\mathrm{d}y\\\
&\leq\frac{1}{2{\varepsilon}^{2}}\left\|K\right\|_{L^{1}(\mathbb{R}^{3})}\int_{\partial_{\theta}G}\left|u_{1}(x)-u_{2}(x)\right|^{2}\mathrm{d}x+2F_{\varepsilon}(u_{2},\,\partial_{\theta}G)\end{split}$

The lemma follows. ∎

Finally, we will need an inequality on the bulk potential $\psi_{b}$.

######  Lemma 4.6.

For any $\delta>0$, there exists a constant $C_{\delta}>0$ such that, for any
$y_{1}\in\mathcal{Q}$, $y_{2}\in\mathcal{Q}$ with
$\operatorname{dist}(y_{2},\,\partial\mathcal{Q})\geq\delta$, we have

(4.2) $\psi_{b}(y_{2})\leq
C_{\delta}\left(\psi_{b}(y_{1})+\left|y_{1}-y_{2}\right|^{2}\right).$

###### Proof.

The assumption (H6) implies, via a Taylor expansion and a compactness
argument, that there exist $\gamma>0$, $\kappa_{1}>0$, $\kappa_{2}>0$ so that
if $\operatorname{dist}(y,\,\mathscr{N})<\gamma$, then

(4.3)
$\kappa_{1}\operatorname{dist}^{2}(y,\,\mathscr{N})\leq\psi_{b}(y)\leq\kappa_{2}\operatorname{dist}^{2}(y,\,\mathscr{N}).$

To prove the result we exhaust three cases,

  1. 1.

$\operatorname{dist}(y_{1},\,\mathscr{N})\geq\frac{1}{2}\gamma$.

  2. 2.

$\operatorname{dist}(y_{1},\,\mathscr{N})<\frac{1}{2}\gamma$,
$\operatorname{dist}(y_{2},\,\mathscr{N})\geq\gamma$.

  3. 3.

$\operatorname{dist}(y_{1},\,\mathscr{N})<\frac{1}{2}\gamma$,
$\operatorname{dist}(y_{2},\,\mathscr{N})<\gamma$.

In the case of (1), we have that such $y_{1}$ satisfy $\psi_{b}(y_{1})>c_{1}$
for a constant $c_{1}>0$ (that depends on $\gamma$), as $y_{1}$ is bounded
away from the minimising manifold. We furthermore have that
$\psi_{b}(y_{2})\leq c_{2}$ because
$\operatorname{dist}(y_{2},\,\partial\mathcal{Q})>\delta$ (and the constant
$c_{2}$ will depend on $\delta$). Therefore the inequality (4.2) holds
trivially with $C_{\delta}=\frac{c_{1}}{c_{2}}$.

In the case of (2), since
$\operatorname{dist}(y_{1},\,\mathscr{N})<\frac{1}{2}\gamma$,
$\operatorname{dist}(y_{2},\,\mathscr{N})\geq\gamma$, we must have
$|y_{1}-y_{2}|^{2}\geq\frac{1}{4}\gamma^{2}$, then we use the upper bound on
$\psi_{b}(y_{2})$ as before.

In the case of (3), we note that since $y_{1},y_{2}$ are both sufficiently
close to $\mathscr{N}$,

$\begin{split}\psi_{b}(y_{2})&\stackrel{{\scriptstyle\eqref{nondeg}}}{{\lesssim}}\operatorname{dist}^{2}(y_{2},\,\mathscr{N})\lesssim\operatorname{dist}^{2}(y_{1},\,\mathscr{N})+|y_{1}-y_{2}|^{2}\stackrel{{\scriptstyle\eqref{nondeg}}}{{\lesssim}}\psi_{b}(y_{1})+|y_{1}-y_{2}|^{2}.\qed\end{split}$

###### Proof of Proposition 4.2.

By a scaling argument (see Equation (3.11) and Lemma 4.1), we can assume
without loss of generality that $\rho=1$ and $x_{0}=0$.

######  Step 1 (Compactness).

Let $\varphi_{\varepsilon}\in C^{\infty}_{\mathrm{c}}(\mathbb{R}^{3})$ be
defined as in Section 3.2. Lemma 3.5 and Lemma 3.6 imply that

(4.4)
$\int_{B_{1/2}}\left|\nabla(\varphi_{\varepsilon}*u_{\varepsilon})\right|^{2}\lesssim
F(u_{\varepsilon},\,B_{1}),\qquad\int_{B_{1/2}}\left|\varphi_{\varepsilon}*u_{\varepsilon}-u_{\varepsilon}\right|^{2}\lesssim{\varepsilon}^{2}F(u_{\varepsilon},\,B_{1})$

for ${\varepsilon}$ small enough. Since $F(u_{\varepsilon},\,B_{1})$ is
bounded, we can extract a (non-relabelled) subsequence so that
$\varphi_{\varepsilon}*u_{\varepsilon}\rightharpoonup u_{0}$ weakly in
$H^{1}(B_{1/2})$, $u_{\varepsilon}\to u_{0}$ strongly in $L^{2}(B_{1/2})$. We
must show that

(4.5) $\int_{B_{1/2}}L\nabla u_{0}\cdot\nabla u_{0}\leq\int_{B_{1/2}}L\nabla
v\cdot\nabla v$

for any $v\in H^{1}(B_{1/2},\,\mathscr{N})$ such that $v=u_{0}$ on $\partial
B_{1/2}$. By an approximation argument, it suffices to prove (4.5) in case
$v=u_{0}$ in a neighbourhood of $\partial B_{1/2}$. Therefore, we fix
$s\in(0,\,1/2)$ and we take a map $v\in H^{1}(B_{1/2},\,\mathscr{N})$ such
that $v=u_{0}$ on $B_{1/2}\setminus\bar{B}_{s}$. The map $v$ is not an
admissible competitor for $u_{\varepsilon}$, because in general $u_{0}\neq
u_{\varepsilon}$ on $\mathbb{R}^{3}\setminus B_{1}$. In the next step, we will
modify $v$ near the boundary of $B_{s}$, so to obtain an admissible
competitor.

######  Step 2 (Construction of a competitor for $u_{\varepsilon}$).

Let $N\geq 1$ be an integer number, and let $\bar{s}:=\max(s,\,1/4)$. We
consider the annulus $B_{1/2}\setminus\bar{B}_{\bar{s}}$ and divide it into
$N$ concentric sub-annuli:

$A_{i}:=B_{\bar{s}+i\frac{1/2-\bar{s}}{N}}\setminus\bar{B}_{\bar{s}+(i-1)\frac{1/2-\bar{s}}{N}}\qquad\textrm{for
}i=1,\,2,\,\ldots,\,N.$

We have $\sum_{i=1}^{N}F_{\varepsilon}(u_{\varepsilon},\,A_{i})\leq
F_{\varepsilon}(u_{\varepsilon},\,B_{1})$ and hence, for any ${\varepsilon}$
we can choose an index $i({\varepsilon})$ such that

(4.6)
$F_{\varepsilon}(u_{\varepsilon},\,A_{i({\varepsilon})})\leq\frac{F_{\varepsilon}(u_{\varepsilon},\,B_{1})}{N}.$

Passing to a subsequence, we may also assume that all the indices
$i({\varepsilon})$ are the same, so from now on, we write $i$ instead of
$i({\varepsilon})$. We take positive numbers $a<b$ such that
$A^{\prime}:=B_{b}\setminus\bar{B}_{a}\subset\\!\subset A_{i}$. Then, Lemma
3.6 gives

(4.7)
$\frac{1}{{\varepsilon}^{2}}\int_{A^{\prime}}\left|\varphi_{\varepsilon}*u_{\varepsilon}-u_{\varepsilon}\right|^{2}\lesssim
F_{\varepsilon}(u_{\varepsilon},\,A_{i})\lesssim\frac{F_{\varepsilon}(u_{\varepsilon},\,B_{1})}{N}$

for ${\varepsilon}$ small enough. From Proposition 3.3 and Lemma 4.6, we
deduce

(4.8)
$\begin{split}\frac{1}{{\varepsilon}^{2}}\int_{A^{\prime}}\psi_{b}(\varphi_{\varepsilon}*u_{\varepsilon})&\lesssim\frac{1}{{\varepsilon}^{2}}\int_{A^{\prime}}\left(\psi_{b}(u_{\varepsilon})+\left|\varphi_{\varepsilon}*u_{\varepsilon}-u_{\varepsilon}\right|^{2}\right)\stackrel{{\scriptstyle\eqref{comp2}}}{{\lesssim}}\frac{F_{\varepsilon}(u_{\varepsilon},\,B_{1})}{N}\end{split}$

Using Fatou’s lemma, we see that

$\begin{split}&\int_{a}^{b}\left(\liminf_{{\varepsilon}\to 0}\int_{\partial
B_{r}}\left|\nabla(\varphi_{\varepsilon}*u_{\varepsilon})\right|^{2}+\frac{1}{{\varepsilon}^{2}}\psi_{b}(\varphi_{\varepsilon}*u_{\varepsilon})\,\mathrm{d}\mathscr{H}^{2}\right)\mathrm{d}r\\\
&\qquad\qquad\qquad\leq\liminf_{{\varepsilon}\to
0}\int_{A^{\prime}}\left(\left|\nabla(\varphi_{\varepsilon}*u_{\varepsilon})\right|^{2}+\frac{1}{{\varepsilon}^{2}}\psi_{b}(\varphi_{\varepsilon}*u_{\varepsilon})\right)\stackrel{{\scriptstyle\eqref{energybd},\eqref{comp3}}}{{\leq}}C.\end{split}$

Since $A^{\prime}\subseteq B_{1/2}\setminus\bar{B}_{s}$, we have $v=u_{0}$ on
$A^{\prime}$ and hence, $\varphi_{\varepsilon}*u_{\varepsilon}\to v$ strongly
in $L^{2}(A^{\prime})$. Therefore, by Fubini theorem, there exists a radius
$r\in(a,\,b)\subseteq(1/4,\,1/2)$ and a (non-relabelled) subsequence
${\varepsilon}\to 0$ such that

(4.9) $\displaystyle\int_{\partial B_{r}}\left(\left|\nabla
v\right|^{2}+\left|\nabla(\varphi_{\varepsilon}*u_{\varepsilon})\right|^{2}+\frac{1}{{\varepsilon}^{2}}\psi_{b}(\varphi_{\varepsilon}*u_{\varepsilon})\right)\,\mathrm{d}\mathscr{H}^{2}\leq\frac{C}{b-a}$
(4.10) $\displaystyle\int_{\partial
B_{r}}\left|\varphi_{\varepsilon}*u_{\varepsilon}-v\right|^{2}\mathrm{d}\mathscr{H}^{2}\to
0\qquad\textrm{as }{\varepsilon}\to 0.$

Let

$\lambda_{\varepsilon}:={\left({\varepsilon}+\int_{\partial
B_{r}}\left|\varphi_{\varepsilon}*u_{\varepsilon}-v\right|^{2}\mathrm{d}\mathscr{H}^{2}\right)^{1/4}}>0$

Thanks to Proposition 3.3, (4.9) and (4.10), we can apply Lemma 4.3 to
construct a map $w_{\varepsilon}\in H^{1}(B_{r}\setminus
B_{r-\lambda_{\varepsilon}},\,\mathcal{Q})$ such that
$w_{\varepsilon}(x)=(\varphi*u_{\varepsilon})(x)$ for $x\in\partial B_{r}$,
$w_{\varepsilon}(x)=v(rx/(r-\lambda_{\varepsilon}))$ for $x\in\partial
B_{r-\lambda_{\varepsilon}}$, and

(4.11) $\int_{B_{r}\setminus B_{r-\lambda_{\varepsilon}}}\left(\left|\nabla
w_{\varepsilon}\right|^{2}+\frac{1}{{\varepsilon}^{2}}\psi_{b}(w_{\varepsilon})\right)\lesssim\frac{\lambda_{\varepsilon}}{b-a}+\frac{1}{\lambda_{\varepsilon}}\int_{\partial
B_{r}}\left|\varphi_{\varepsilon}*u_{\varepsilon}-v\right|^{2}\,\mathrm{d}\mathscr{H}^{2}$

The right-hand side of (4.11) converges to zero as ${\varepsilon}\to 0$, due
to (4.10). Finally, we take $c\in(r,\,b)$ and we define

$v_{\varepsilon}(x):=\begin{cases}u_{\varepsilon}(x)&\textrm{if
}x\in\mathbb{R}^{3}\setminus B_{c}\\\
(\varphi_{\varepsilon}*u_{\varepsilon})(x)&\textrm{if }B_{c}\setminus B_{r}\\\
w_{\varepsilon}(x)&\textrm{if }x\in B_{r}\setminus
B_{r-\lambda_{\varepsilon}}\\\
v\left(\dfrac{rx}{r-\lambda_{\varepsilon}}\right)&\textrm{if }x\in
B_{r-\lambda_{\varepsilon}}\end{cases}$

The map $v_{\varepsilon}$ is an admissible competitor for $u_{\varepsilon}$,
because $v_{\varepsilon}\in L^{\infty}(\mathbb{R}^{3},\,\mathcal{Q})$ and
$v_{\varepsilon}=u_{\varepsilon}$ a.e. on $\mathbb{R}^{3}\setminus B_{1}$.

######  Step 3 (Bounds on $E_{\varepsilon}(v_{\varepsilon})$).

We observe that $\partial B_{c}\subset\\!\subset
A^{\prime}\setminus\bar{B}_{r}$ and hence, there exists $\theta>0$ such that
the $B_{c+\theta}\setminus B_{c-\theta}\subseteq
A^{\prime}\setminus\bar{B}_{r}$. Then, Lemma 4.5 gives

$\begin{split}E_{\varepsilon}(v_{\varepsilon})\leq
F_{\varepsilon}(v_{\varepsilon},\,B_{c})&+F_{\varepsilon}(u_{\varepsilon},\,\mathbb{R}^{3}\setminus
B_{c})+CF_{\varepsilon}(u_{\varepsilon},\,A^{\prime})+\frac{C}{{\varepsilon}^{2}}\int_{A^{\prime}\setminus\bar{B}_{r}}\left|u_{\varepsilon}-\varphi_{\varepsilon}*u_{\varepsilon}\right|^{2}\\\
&+\frac{C}{\theta^{2}}\left\|v_{\varepsilon}\right\|^{2}_{L^{\infty}(\mathbb{R}^{3})}\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z\end{split}$

The map $v_{\varepsilon}$ takes values in the bounded set $\mathcal{Q}$. Due
to (4.6) and (4.7), we deduce

(4.12) $\begin{split}E_{\varepsilon}(v_{\varepsilon})\leq
F_{\varepsilon}(v_{\varepsilon},\,B_{c})&+F_{\varepsilon}(u_{\varepsilon},\,\mathbb{R}^{3}\setminus
B_{c})+\frac{C}{N}+\frac{C}{\theta^{2}}\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z.\end{split}$

We bound $F_{\varepsilon}(v_{\varepsilon},\,B_{c})$ in a similar way, with the
help of Lemma 4.4. Reducing if necessary the value of $\theta$, so to have
$B_{r+\theta}\setminus B_{r-\theta}\subset\\!\subset A^{\prime}$, and
observing that $B_{c}\setminus\bar{B}_{r}\subseteq A^{\prime}$, we obtain

$\begin{split}F_{\varepsilon}(v_{\varepsilon},\,B_{c})&\leq
F_{\varepsilon}(v_{\varepsilon},\,B_{r})+CF_{\varepsilon}(\varphi_{\varepsilon}*u_{\varepsilon},\,A^{\prime})+\frac{C}{\theta^{2}}\left\|v_{\varepsilon}\right\|^{2}_{L^{\infty}(\mathbb{R}^{3})}\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z\\\
&\stackrel{{\scriptstyle\eqref{comp1},\,\eqref{comp3}}}{{\leq}}F_{\varepsilon}(v_{\varepsilon},\,B_{r})+\frac{C}{N}+\frac{C}{\theta^{2}}\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z.\end{split}$

Together with (4.12), this implies

(4.13) $\begin{split}E_{\varepsilon}(v_{\varepsilon})\leq
F_{\varepsilon}(v_{\varepsilon},\,B_{r})+F_{\varepsilon}(u_{\varepsilon},\,\mathbb{R}^{3}\setminus
B_{c})+\frac{C}{N}+\frac{C}{\theta^{2}}\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z.\end{split}$

######  Step 4 ($u_{0}$ is a minimiser: proof of (4.5)).

Since $u_{\varepsilon}$ is an $\omega$-minimiser for $E_{\varepsilon}$, from
(4.13) we deduce

$\begin{split}E_{\varepsilon}(u_{\varepsilon})\leq
F_{\varepsilon}(v_{\varepsilon},\,B_{r})+F_{\varepsilon}(u_{\varepsilon},\,\mathbb{R}^{3}\setminus
B_{c})+\frac{C}{N}+\frac{C}{\theta^{2}}\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z+\omega({\varepsilon}).\end{split}$

On the other hand,
$F_{\varepsilon}(u_{\varepsilon},\,B_{r})+F_{\varepsilon}(u_{\varepsilon},\,\mathbb{R}^{3}\setminus
B_{c})\leq E_{\varepsilon}(u_{\varepsilon})$ and hence,

(4.14) $\begin{split}F_{\varepsilon}(u_{\varepsilon},\,B_{r})\leq
F_{\varepsilon}(v_{\varepsilon},\,B_{r})+\frac{C}{N}+\frac{C}{\theta^{2}}\int_{\mathbb{R}^{3}\setminus
B_{\theta/{\varepsilon}}}g(z)\left|z\right|^{2}\,\mathrm{d}z+\omega({\varepsilon}).\end{split}$

Using (4.11), a routine computation shows that $v_{\varepsilon}\to v$ strongly
in $H^{1}(B_{r})$ as ${\varepsilon}\to 0$. Then, we can pass to the limit as
${\varepsilon}\to 0$ in (4.14), using Proposition 3.7, Proposition 3.8, (4.8)
and (K4). We obtain

(4.15) $\int_{B_{r}}L\nabla u_{0}\cdot\nabla u_{0}\leq\int_{B_{r}}L\nabla
v\cdot\nabla v+\frac{C}{N}$

We have chosen $v$ in such a way that $v=u_{0}$ on
$B_{1/2}\setminus\bar{B}_{s}$; moreover, by construction,
$r>\bar{s}:=\max(s,\,1/4)\geq s$. Therefore, (4.15) implies

$\int_{B_{s}}L\nabla u_{0}\cdot\nabla u_{0}\leq\int_{B_{s}}L\nabla
v\cdot\nabla v+\frac{C}{N}$

and, letting $N\to+\infty$, (4.5) follows.

######  Step 5 (Proof of (4.1)).

We choose $v=u_{0}$. Passing to the limit as ${\varepsilon}\to 0$ in both
sides of (4.14), with the help of Proposition 3.8 and (K4), we obtain

(4.16) $\begin{split}\limsup_{{\varepsilon}\to
0}F_{\varepsilon}(u_{\varepsilon},\,B_{r})\leq\int_{B_{r}}L\nabla
u_{0}\cdot\nabla u_{0}+\frac{C}{N}\end{split}$

On the other hand, Proposition 3.7 implies

(4.17) $\begin{split}\int_{B_{r}\setminus\bar{B}_{s}}L\nabla u_{0}\cdot\nabla
u_{0}\leq\liminf_{{\varepsilon}\to
0}F_{\varepsilon}(u_{\varepsilon},\,B_{r}\setminus\bar{B}_{s})\end{split}$

Combining (4.16) with (4.17) and Proposition 3.7, we deduce

$\begin{split}\int_{B_{s}}L\nabla u_{0}\cdot\nabla
u_{0}\leq\liminf_{{\varepsilon}\to
0}F_{\varepsilon}(u_{\varepsilon},\,B_{s})\leq\limsup_{{\varepsilon}\to
0}F_{\varepsilon}(u_{\varepsilon},\,B_{s})\leq\int_{B_{s}}L\nabla
u_{0}\cdot\nabla u_{0}+\frac{C}{N}\end{split}$

Letting $N\to+\infty$, (4.1) follows.∎

###  4.2 A decay lemma for $F_{\varepsilon}$

The aim of this section is to prove a decay property for $F_{\varepsilon}$:

######  Lemma 4.7.

Let $\Omega\subseteq\mathbb{R}^{3}$ be a bounded domain. Let
$\omega\colon[0,\,+\infty)\to[0,\,+\infty)$ be an increasing function such
that $\lim_{s\to 0}\omega(s)=0$. Then, there exist numbers $\eta>0$,
$\theta\in(0,\,1)$ and ${\varepsilon}_{*}>0$ such that, for any ball
$B_{\rho}(x_{0})\subseteq\Omega$, any
${\varepsilon}\in(0,\,{\varepsilon}_{*}\rho)$ and any $\omega$-minimiser
$u_{\varepsilon}$ of $E_{\varepsilon}$ in $\Omega$, there holds

$\rho^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{\rho}(x_{0}))\leq\eta\quad\Longrightarrow\quad(\theta\rho)^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{\theta\rho}(x_{0}))\leq\frac{1}{2}\,\rho^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{\rho}(x_{0})).$

We will deduce Lemma 4.7 from the analogous property satisfied by the limit
functional, (2.9). To this end, we will first need to check that the limit
functional is elliptic. Recall that the tensor $L$ is defined by (2.8).

######  Proposition 4.8.

There exists a constant $\lambda>1$ so that

$\lambda^{-1}|\xi|^{2}\leq L\xi\cdot\xi\leq\lambda|\xi|^{2}$

for all $\xi\in\mathbb{R}^{m\times 3}$.

###### Proof.

The upper bound comes trivially, as

$\begin{split}4L\xi\cdot\xi=\int_{\mathbb{R}^{3}}K(z)(\xi z)\cdot(\xi
z)\,\mathrm{d}z\lesssim\int_{\mathbb{R}^{3}}g(z)|\xi
z|^{2}\,\mathrm{d}z\lesssim\left(\int_{\mathbb{R}^{3}}g(z)|z|^{2}\,\mathrm{d}z\right)|\xi|^{2}\end{split}$

and the constant at the right-hand side is finite, due to (K4). For the lower
bound, recall that $g$ is non-negative and satisfies $g(z)\geq k$ for
$\rho_{1}<|z|<\rho_{2}$. Then we have that

$\begin{split}4L\xi\cdot\xi=\int_{\mathbb{R}^{3}}K_{ij}(z)z_{\alpha}z_{\beta}\,\xi_{i,\alpha}\,\xi_{j,\beta}\,\mathrm{d}z&\geq\int_{\mathbb{R}^{3}}g(z)z_{\alpha}z_{\beta}\,\xi_{i,\alpha}\,\xi_{i,\beta}\,\mathrm{d}z\\\
&\geq k\int_{B_{\rho_{2}}\setminus
B_{\rho_{1}}}z_{\alpha}z_{\beta}\,\mathrm{d}z\,\xi_{i,\alpha}\,\xi_{i,\beta}\end{split}$

We may evaluate the inner integral as

$\begin{split}\int_{B_{\rho_{2}}\setminus
B_{\rho_{1}}}z_{\alpha}z_{\beta}\,\mathrm{d}z=&\int_{\rho_{1}}^{\rho_{2}}\int_{\mathbb{S}^{2}}r^{2}p_{\alpha}p_{\beta}\,\mathrm{d}p\,\mathrm{d}r\\\
=&\int_{\rho_{1}}^{\rho_{2}}r^{2}\,\mathrm{d}r\int_{\mathbb{S}^{2}}p_{\alpha}p_{\beta}\,\mathrm{d}p\\\
=&\left(\frac{\rho_{2}^{3}-\rho_{1}^{3}}{3}\right)\frac{4\pi}{3}\delta_{\alpha\beta}\end{split}$

This gives a lower bound on the bilinear form as

$L\xi\cdot\xi\geq\frac{k\pi(\rho_{2}^{3}-\rho_{1}^{3})}{9}\delta_{\alpha\beta}\xi_{i\alpha}\xi_{i\beta}=\frac{k\pi(\rho_{2}^{3}-\rho_{1}^{3})}{9}|\xi|^{2}\qed$

###### Proof of Lemma 4.7.

Since $L$ is elliptic (Proposition 4.8), the limit functional (2.9) satisfies
a decay property: there exist numbers $\eta\in(0,\,+\infty)$,
$\theta\in(0,\,1/4)$ such that, for any minimiser $u_{0}\in
W^{1,2}(B_{1/4},\,\mathscr{N})$ of $E_{0}$ subject to its own boundary
conditions, there holds

(4.18) $\int_{B_{1/4}}L\nabla u_{0}\cdot\nabla
u_{0}\leq\eta\quad\Longrightarrow\quad\theta^{-1}\int_{B_{\theta}}L\nabla
u_{0}\cdot\nabla u_{0}\leq\frac{1}{4}\int_{B_{1/4}}L\nabla u_{0}\cdot\nabla
u_{0}$

(see e.g. [33, 21]). We claim that there exists ${\varepsilon}_{*}>0$ such
that, for any ball $B_{\rho}(x_{0})\subseteq\Omega$, any
${\varepsilon}\in(0,\,{\varepsilon}_{*}\rho)$ and any $\omega$-minimiser
$u_{\varepsilon}$ of $E_{\varepsilon}$,

(4.19)
$\rho^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{\rho}(x_{0}))\leq\eta\quad\Longrightarrow\quad(\theta\rho)^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{\theta\rho}(x_{0}))\leq\frac{1}{2}\,\rho^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{\rho/4}(x_{0})).$

Once (4.19) is proven, the lemma follows. Suppose, towards a contradiction,
that (4.19) does not hold. Then, we find a sequence
$(\bar{{\varepsilon}}_{j},\,\bar{\rho}_{j},\,\bar{x}_{j},\,\bar{u}_{j})_{j\in\mathbb{N}}$,
where $\bar{{\varepsilon}}_{j}/\bar{\rho}_{j}\to 0$ and $\bar{u}_{j}$ is a
$\omega$-minimiser of $E_{\bar{{\varepsilon}}_{j}}$, such that
$B_{\bar{\rho}_{j}}(\bar{x}_{j})\subseteq\Omega$ and

(4.20)
$\bar{\rho}_{j}^{-1}F_{\bar{{\varepsilon}}_{j}}(\bar{u}_{j},\,B_{\bar{\rho}_{j}}(\bar{x}_{j}))\leq\eta,\quad(\theta\bar{\rho}_{j})^{-1}F_{\bar{{\varepsilon}}_{j}}(\bar{u}_{j},\,B_{\theta\bar{\rho}_{j}}(\bar{x}_{j}))>\frac{1}{2}\,\bar{\rho}_{j}^{-1}F_{\bar{{\varepsilon}}_{j}}(\bar{u}_{j},\,B_{\bar{\rho}_{j}/4}(\bar{x}_{j})).$

We scale the space variables, defining
$u_{j}(y):=\bar{u}_{j}(\bar{x}_{j}+\bar{\rho}_{j}y)$ for $y\in\mathbb{R}^{3}$
and ${\varepsilon}_{j}:=\bar{{\varepsilon}}_{j}/\bar{\rho}_{j}\to 0$. By Lemma
4.1, $u_{j}$ is an $\omega_{j}$-minimser for $E_{{\varepsilon}_{j}}$, where
$\omega_{j}(s):=\omega(\bar{\rho}_{j}s)$ for $s\geq 0$. However, the radii
$\bar{\rho}_{j}$ are bounded by a constant that depends on $\Omega$ only, say
$\bar{\rho}_{j}\leq R_{0}$ for any $j$. Let us define
$\omega_{0}(s):=\omega(R_{0}s)$ for any $s\geq 0$. Since $\omega$ is
increasing, we deduce that $u_{j}$ is an $\omega_{0}$-minimiser of
$E_{{\varepsilon}_{j}}$, for any $j$. Moreover, (3.11) and (4.20) imply

(4.21)
$F_{{\varepsilon}_{j}}(u_{j},\,B_{1})\leq\eta,\qquad\theta^{-1}F_{{\varepsilon}_{j}}(u_{j},\,B_{\theta})>\frac{1}{2}\,F_{{\varepsilon}_{j}}(u_{j},\,B_{1/4}).$

As a consequence, we can apply Proposition 4.2 to the sequence $u_{j}$. Up to
extraction of a (non-relabelled) subsequence, we obtain that
$u_{\varepsilon}\to u_{0}$ in $L^{2}(B_{1/2})$, where $u_{0}\in
H^{1}(B_{1/2},\,\mathscr{N})$ minimises the limit functional (2.9) subject to
its own boundary condition; moreover,

(4.22)
$\lim_{j\to+\infty}F_{{\varepsilon}_{j}}(u_{j},\,B_{s})=\int_{B_{s}}L\nabla
u_{0}\cdot\nabla u_{0}\qquad\textrm{for any }s\in(0,\,1/2).$

Due to (4.21) and (4.22), we have $\int_{B_{1/4}}L\nabla u_{0}\cdot\nabla
u_{0}\leq\eta$ and hence, by (4.18),

(4.23) $\theta^{-1}\int_{B_{\theta}}L\nabla u_{0}\cdot\nabla
u_{0}\leq\frac{1}{4}\int_{B_{1/4}}L\nabla u_{0}\cdot\nabla u_{0}.$

On the other hand, from (4.21) and (4.22) we obtain

$\theta^{-1}\int_{B_{\theta}}L\nabla u_{0}\cdot\nabla
u_{0}\geq\frac{1}{2}\int_{B_{1/4}}L\nabla u_{0}\cdot\nabla u_{0},$

which contradicts (4.23). Therefore, (4.19) follows. ∎

###  4.3 Proof of Theorem A and Theorem B

###### Proof of Theorem A.

Let $\eta$, $\theta$ and ${\varepsilon}_{*}$ be given by Lemma 4.7. Let
$B_{r_{0}}(x_{0})\subset\\!\subset\Omega$ be a ball,
${\varepsilon}\in(0,\,{\varepsilon}_{*}r_{0})$, and let $u_{\varepsilon}$ be a
minimiser of $E_{\varepsilon}$ in $\mathscr{A}$ such that

(4.24)
$r_{0}^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{r_{0}}(x_{0}))\leq\eta.$

By a scaling argument, using (3.11), we can assume without loss of generality
that $x_{0}=0$ and $r_{0}=1$.

######  Step 1 (Campanato estimate for radii $\rho\gtrsim{\varepsilon}$).

Thanks to (4.24), we can apply Lemma 4.7 iteratively, and we deduce that

$\theta^{-n}F_{\varepsilon}(u_{\varepsilon},\,B_{\theta^{n}})\leq
2^{-n}F_{\varepsilon}(u_{\varepsilon},\,B_{1})\stackrel{{\scriptstyle\eqref{Holder0}}}{{\leq}}2^{-n}\eta$

for any integer $n\geq 1$ such that
$\theta^{n}{\varepsilon}_{*}\geq{\varepsilon}$. As a consequence, there exist
positive numbers $\alpha$ (depending on $\theta$ only) and $C_{1}$ (depending
on $\eta$, $\theta$, ${\varepsilon}_{*}$ only) such that

$\rho^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{\rho})\leq
C_{1}\rho^{\alpha}\qquad\textrm{for any }\rho\in({\varepsilon},\,1).$

By applying Proposition 3.4, and possibly modifying the value of $C_{1}$, we
obtain

(4.25)
$\fint_{B_{\rho}}\left|u_{\varepsilon}-\fint_{B_{\rho}}u_{\varepsilon}\right|^{2}\leq
C_{1}\rho^{\alpha}\qquad\textrm{for any
}\rho\in(\lambda_{1}{\varepsilon},\,1)$

where $\lambda_{1}:=\max\\{1/2,\,1/(2{\varepsilon}_{1})\\}$ and
${\varepsilon}_{1}$ is given by Proposition 3.4.

######  Step 2 (Campanato estimate for radii $\rho\lesssim{\varepsilon}$).

We need to show that an estimate similar to (4.25) holds for
$\rho<\lambda_{1}{\varepsilon}$ as well. To this end, we define

(4.26) $p:=\frac{2\alpha}{3}+3,\qquad\beta:=\frac{2\alpha+9}{3\alpha+9}.$

We have $p>3$, $0<\beta<1$. Let

$m_{\varepsilon}:=\fint_{B_{2{\varepsilon}^{\beta}}}u_{\varepsilon}$

and let $\chi_{\varepsilon}$ be the characteristic function of the ball
$B_{{\varepsilon}^{\beta}}$. Since $\nabla K_{\varepsilon}$ has zero average,
from the Euler-Lagrange equation (Proposition 3.1) we obtain

$\begin{split}\nabla(\Lambda\circ u_{\varepsilon})&=(\nabla
K_{\varepsilon})*(u_{\varepsilon}-m_{\varepsilon})\\\
&=(\chi_{\varepsilon}\nabla
K_{\varepsilon})*(u_{\varepsilon}-m_{\varepsilon})+\left((1-\chi_{\varepsilon})\nabla
K_{\varepsilon}\right)*(u_{\varepsilon}-m_{\varepsilon}).\end{split}$

Let $\tilde{\chi}_{\varepsilon}$ be the characteristic function of the ball
$B_{2{\varepsilon}^{\beta}}$. Since $\chi_{\varepsilon}\nabla K_{\varepsilon}$
is supported on $B_{{\varepsilon}^{\beta}}$, we deduce

$\begin{split}\nabla(\Lambda\circ u_{\varepsilon})&=(\chi_{\varepsilon}\nabla
K_{\varepsilon})*(\tilde{\chi}_{\varepsilon}(u_{\varepsilon}-m_{\varepsilon}))+\left((1-\chi_{\varepsilon})\nabla
K_{\varepsilon}\right)*(u_{\varepsilon}-m_{\varepsilon})\quad\textrm{in
}B_{{\varepsilon}^{\beta}}.\end{split}$

We apply Hölder’s inequality, and then Young’s inequality for the convolution:

(4.27) $\begin{split}\left\|\nabla(\Lambda\circ
u_{\varepsilon})\right\|_{L^{p}(B_{{\varepsilon}^{\beta}})}&\lesssim\|(\chi_{\varepsilon}\nabla
K_{\varepsilon})*(\tilde{\chi}_{\varepsilon}(u_{\varepsilon}-m_{\varepsilon}))\|_{L^{p}(\mathbb{R}^{3})}\\\
&\qquad\qquad+{\varepsilon}^{3\beta/p}\left\|((1-\chi_{\varepsilon})\nabla
K_{\varepsilon})*(u_{\varepsilon}-m_{\varepsilon})\right\|_{L^{\infty}(\mathbb{R}^{3})}\\\
&\lesssim\left\|\nabla
K_{\varepsilon}\right\|_{L^{1}(B_{{\varepsilon}^{\beta}})}\left\|u_{\varepsilon}-m_{\varepsilon}\right\|_{L^{p}(B_{2{\varepsilon}^{\beta}})}\\\
&\qquad\qquad+{\varepsilon}^{3\beta/p}\left\|\nabla
K_{\varepsilon}\right\|_{L^{1}(\mathbb{R}^{3}\setminus
B_{{\varepsilon}^{\beta}})}\left\|u_{\varepsilon}-m_{\varepsilon}\right\|_{L^{\infty}(\mathbb{R}^{3})}\end{split}$

We bound the terms at the right-hand side. For ${\varepsilon}$ small enough
(so that $2{\varepsilon}^{\beta}\geq\lambda_{1}{\varepsilon}$), the inequality
(4.25) implies

$\left\|u_{\varepsilon}-m_{\varepsilon}\right\|^{2}_{L^{2}(B_{2{\varepsilon}^{\beta}})}\lesssim{\varepsilon}^{(3+\alpha)\beta}.$

Since $\|u_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{3})}\leq C$, by
interpolation we obtain

(4.28)
$\begin{split}\left\|u_{\varepsilon}-m_{\varepsilon}\right\|_{L^{p}(B_{2{\varepsilon}^{\beta}})}&\lesssim\left\|u_{\varepsilon}-m_{\varepsilon}\right\|_{L^{2}(B_{2{\varepsilon}^{\beta}})}^{2/p}\lesssim{\varepsilon}^{(3+\alpha)\beta/p}.\end{split}$

By a change of variable, we have

(4.29) $\left\|\nabla
K_{\varepsilon}\right\|_{L^{1}(B_{{\varepsilon}^{\beta}})}\leq{\varepsilon}^{-1}\left\|\nabla
K\right\|_{L^{1}(\mathbb{R}^{3})}$

and

(4.30) $\begin{split}\left\|\nabla
K_{\varepsilon}\right\|_{L^{1}(\mathbb{R}^{3}\setminus
B_{{\varepsilon}^{\beta}})}&={\varepsilon}^{-1}\int_{\mathbb{R}^{3}\setminus
B_{{\varepsilon}^{\beta-1}}}\left\|\nabla K(z)\right\|\mathrm{d}z\\\
&\leq{\varepsilon}^{-1}\int_{\mathbb{R}^{3}\setminus
B_{{\varepsilon}^{\beta-1}}}\left\|\nabla
K(z)\right\|\frac{\left|z\right|^{3}}{{\varepsilon}^{3\beta-3}}\,\mathrm{d}z\\\
&\leq{\varepsilon}^{2-3\beta}\int_{\mathbb{R}^{3}}\left\|\nabla
K(z)\right\|\left|z\right|^{3}\mathrm{d}z,\end{split}$

where the integral at the right-hand side is finite by Assumption (K6).
Combining (4.27), (4.28), (4.29) and (4.30), and using that $u_{\varepsilon}$
is bounded in $L^{\infty}(\mathbb{R}^{3})$, we obtain

$\left\|\nabla(\Lambda\circ
u_{\varepsilon})\right\|_{L^{p}(B_{{\varepsilon}^{\beta}})}\lesssim{\varepsilon}^{(3+\alpha)\beta/p-1}+{\varepsilon}^{2-3\beta+3\beta/p}$

By simple algebra, from (4.26) we obtain

$\frac{(3+\alpha)\beta}{p}-1=2-3\beta+\frac{3\beta}{p}=0$

so $\|\nabla(\Lambda\circ
u_{\varepsilon})\|_{L^{p}(B_{{\varepsilon}^{\beta}})}$ is bounded. Thanks to
(3.8), we deduce that $\|\nabla
u_{\varepsilon}\|_{L^{p}(B_{{\varepsilon}^{\beta}})}$ is bounded too. Since
$p>3$, by Sobolev embedding we conclude that

(4.31) $[u_{\varepsilon}]_{C^{\mu}(B_{{\varepsilon}^{\beta}})}\leq C,$

where

(4.32) $\mu:=1-\frac{3}{p}=\frac{2\alpha}{2\alpha+9}.$

Now, let us fix $\rho\leq\lambda_{1}{\varepsilon}$. We have $\beta<1$ and
hence, $\rho\leq\lambda_{1}{\varepsilon}\leq{\varepsilon}^{\beta}$ for
${\varepsilon}$ small enough. Then, (4.31) implies

(4.33)
$\begin{split}\fint_{B_{\rho}}\left|u_{\varepsilon}-\fint_{B_{\rho}}u_{\varepsilon}\right|^{2}&\lesssim\rho^{2\mu}\qquad\textrm{for
any }\rho\in(0,\,\lambda_{1}{\varepsilon}].\end{split}$

######  Step 3 (Conclusion).

By combining (4.25), (4.32) and (4.33), we deduce that

$\fint_{B_{\rho}}\left|u_{\varepsilon}-\fint_{B_{\rho}}u_{\varepsilon}\right|^{2}\leq
C_{2}\,\rho^{\min(\alpha,\,2\mu)}=C_{2}\,\rho^{2\mu}$

for any radius $\rho\in(0,\,1)$ and for some constant $C_{2}>0$ that does not
depend on ${\varepsilon}$, $\rho$. Then, Campanato embedding gives an
${\varepsilon}$-independent bound on the $\mu$-Hölder semi-norm of
$u_{\varepsilon}$ on $B_{1/2}$. This completes the proof. ∎

###### Proof of Theorem B.

Let $u_{\varepsilon}$ be a minimiser of $E_{\varepsilon}$ in $\mathscr{A}$. By
the results of [42], there exists a (non-relabelled) subsequence such that
$u_{\varepsilon}\to u_{0}$ strongly in $L^{2}(\Omega)$, where $u_{0}$ is a
minimiser of the limit functional (2.9). Take a point $x_{0}\in\Omega\setminus
S[u_{0}]$, where $S[u_{0}]$ is defined by (2.11). By definition of $S[u_{0}]$,
there exists a number $r_{0}>0$ such that

$r_{0}^{-1}\int_{B_{r_{0}}(x_{0})}L\nabla u_{0}\cdot\nabla
u_{0}\leq\frac{\eta}{2},$

where $\eta$ is given by Theorem A. Proposition 4.2 implies

$r_{0}^{-1}F_{\varepsilon}(u_{\varepsilon},\,B_{r_{0}}(x_{0}))\leq\eta$

for any ${\varepsilon}$ small enough and hence, by Theorem A,
$[u_{\varepsilon}]_{C^{\mu}(B_{r_{0}}(x_{0}))}$ is uniformly bounded. Then,
Ascoli-Arzelà’s theorem implies that $u_{\varepsilon}\to u_{0}$ uniformly in
$B_{r_{0}}(x_{0})$. ∎

##  5 Generalisation to finite-thickness boundary conditions

In this section, we discuss a variant of the minimisation problem, where we
prescribe $u$ in a neighbourhood of $\partial\Omega$ only. Let
$\Omega_{\varepsilon}\supset\\!\supset\Omega$ be a larger domain, possibly
depending on ${\varepsilon}$. We consider the functional

(5.1)
$\begin{split}\tilde{E}_{\varepsilon}(u):=-\frac{1}{2{\varepsilon}^{2}}\int_{\Omega_{\varepsilon}\times\Omega_{\varepsilon}}K_{\varepsilon}(x-y)u(x)\cdot
u(y)\,\mathrm{d}x\,\mathrm{d}y+\frac{1}{{\varepsilon}^{2}}\int_{\Omega}\psi_{s}(u(x))\,\mathrm{d}x+C_{\varepsilon},\end{split}$

where $C_{\varepsilon}$ is given by (2.6). As before, we take a map
$u_{\mathrm{bd}}\in H^{1}(\mathbb{R}^{3},\,\mathcal{Q})$ that satisfies (BD)
and define the admissible class

(5.2) $\tilde{\mathscr{A}}_{\varepsilon}:=\left\\{u\in
L^{2}(\Omega_{\varepsilon},\,\mathbb{R}^{m})\colon\psi_{s}(u)\in
L^{1}(\Omega),\ u=u_{\mathrm{bd}}\textrm{ a.e. on
}\Omega_{\varepsilon}\setminus\Omega\right\\}\\!.$

The thickness of the boundary layer $\Omega_{\varepsilon}\setminus\Omega$ must
be related to the decay properties of the kernel $K$. More precisely, in
addition to (K1)–(K6), (H1)–(H6), we assume that

  1. (K′)

There exist $q\geq 2$, $\tau>0$ such that

$\displaystyle\int_{\mathbb{R}^{3}}g(z)\left|z\right|^{q}\mathrm{d}z<+\infty$

and
$\operatorname{dist}(\Omega,\,\partial\Omega_{\varepsilon})\geq\tau{\varepsilon}^{1-2/q}$
for any ${\varepsilon}>0$.

######  Remark 5.1.

In case the kernel $K$ is compactly supported, we can allow for a boundary
layer of thickness proportional to ${\varepsilon}$. More precisely, we can
replace the assumption (K′) with the following: there exists $R_{0}>0$ such
that $\mathrm{supp}(K)\subseteq B_{R_{0}}$ and
$\operatorname{dist}(\Omega,\,\partial\Omega_{\varepsilon})\geq
R_{0}{\varepsilon}$ for any ${\varepsilon}>0$. The proofs in this case remain
essentially unchanged.

Under these assumptions, we can prove the analogues of Theorems A and B.

######  Theorem 5.1.

Assume that the conditions (K1)–(K6), (H1)–(H6), (BD) and (K′) are satisfied.
Then, there exist positive numbers $\eta$, ${\varepsilon}_{*}$, $M$ and
$\mu\in(0,\,1)$ such that, for any ball
$B_{r_{0}}(x_{0})\subset\\!\subset\Omega$, any
${\varepsilon}\in(0,\,{\varepsilon}_{*}r_{0})$, and any minimiser
$\tilde{u}_{\varepsilon}$ of $\tilde{E}_{\varepsilon}$ in
$\tilde{\mathscr{A}}_{\varepsilon}$, there holds

$r_{0}^{-1}F_{\varepsilon}(\tilde{u}_{\varepsilon},\,B_{r_{0}}(x_{0}))\leq\eta\qquad\Longrightarrow\qquad
r_{0}^{\mu}\,[\tilde{u}_{\varepsilon}]_{C^{\mu}(B_{r_{0}/2}(x_{0}))}\leq M.$

######  Theorem 5.2.

Assume that the conditions (K1)–(K6), (H1)–(H6), (BD) and (K′) are satisfied.
Let $\tilde{u}_{\varepsilon}$ be a minimiser of $\tilde{E}_{\varepsilon}$ in
$\tilde{\mathscr{A}}_{\varepsilon}$. Then, up to extraction of a (non-
relabelled) subsequence, we have

$\tilde{u}_{\varepsilon}\to u_{0}\qquad\textrm{locally uniformly in
}\Omega\setminus S[u_{0}],$

where $u_{0}$ is a minimiser of the functional (2.9) in $\mathscr{A}$ and
$S[u_{0}]$ is defined by (2.11).

The proofs of Theorem 5.1 and 5.2 are largely similar to those of Theorem A
and B. They rely on the following results:

######  Lemma 5.3.

For any ${\varepsilon}$, there exists a minimiser $\tilde{u}_{\varepsilon}$
for $\tilde{E}_{\varepsilon}$ in $\tilde{\mathscr{A}}_{\varepsilon}$ and it
satisfies the Euler-Lagrange equation,

(5.3)
$\Lambda(\tilde{u}_{\varepsilon}(x))=\int_{\Omega_{\varepsilon}}K_{\varepsilon}(x-y)\tilde{u}_{\varepsilon}(y)\,\mathrm{d}y$

for a.e. $x\in\Omega$.

The proof of Lemma 5.3 is identical to that of Proposition 3.1, so we skip it
for brevity. We remark that the equation (5.3) can be written as

$\Lambda(\tilde{u}_{\varepsilon})=K_{\varepsilon}*(\tilde{u}_{\varepsilon}\chi_{\varepsilon})\qquad\textrm{a.e.
on }\Omega,$

where $\chi_{\varepsilon}$ is the characteristic function of
$\Omega_{\varepsilon}$. In particular, the uniform strict physicality of
$\tilde{u}_{\varepsilon}$ follows from (5.3), exactly as in Proposition 3.3.

######  Lemma 5.4.

Let $\tilde{u}_{\varepsilon}$ be a minimiser for $\tilde{E}_{\varepsilon}$ in
$\tilde{\mathscr{A}}_{\varepsilon}$, identified with its extension by
$u_{\mathrm{bd}}$ to $\mathbb{R}^{3}$. Then, $\tilde{u}_{\varepsilon}$ is an
$\omega$-minimiser for $E_{\varepsilon}$ in $\Omega$, where
$\omega\colon[0,\,+\infty)\to[0,\,+\infty)$ is an increasing function that
depends only on $\Omega$, $K$, $\mathcal{Q}$ and satisfies $\lim_{s\to
0}\omega(s)=0$.

Once Lemma 5.4 is proven, Theorem 5.1 and Theorem 5.2 follow by the same
arguments as before. Before we give the proof of Lemma 5.4, we introduce the
auxiliary function

(5.4)
$H_{\varepsilon}(x):=\frac{1}{{\varepsilon}^{2}}\int_{\mathbb{R}^{3}\setminus(\Omega_{\varepsilon}-x)/{\varepsilon}}K(z)\,\mathrm{d}z\qquad\textrm{for
any }x\in\mathbb{R}^{3}.$

######  Lemma 5.5.

Under the assumption (K′), $H_{\varepsilon}\to 0$ uniformly in $\Omega$, as
${\varepsilon}\to 0$.

###### Proof.

For any $x\in\Omega$, we have
$B_{\tau{\varepsilon}^{1-2/q}}(x)\subseteq\Omega_{\varepsilon}$ by (K′), and
hence
$B_{\tau{\varepsilon}^{-2/q}}\subseteq(\Omega_{\varepsilon}-x)/{\varepsilon}$,
$\mathbb{R}^{3}\setminus(\Omega_{\varepsilon}-x)/{\varepsilon}\subseteq\mathbb{R}^{3}\setminus
B_{\tau{\varepsilon}^{-2/q}}$. Then, the definition (5.4) of $H_{\varepsilon}$
gives

$\begin{split}\left\|H_{\varepsilon}(x)\right\|&\leq\frac{1}{{\varepsilon}^{2}}\int_{\mathbb{R}^{3}\setminus(\Omega_{\varepsilon}-x)/{\varepsilon}}\left\|K(z)\right\|\mathrm{d}z\\\
&\leq\frac{1}{{\varepsilon}^{2}}\int_{\mathbb{R}^{3}\setminus
B_{\tau{\varepsilon}^{-2/q}}}\left\|K(z)\right\|\mathrm{d}z\\\
&\leq\int_{\mathbb{R}^{3}\setminus
B_{\tau{\varepsilon}^{-2/q}}}\left\|K(z)\right\|\frac{\left|z\right|^{q}}{\tau^{q}}\,\mathrm{d}z\end{split}$

and the right-hand side tends to zero as ${\varepsilon}\to 0$, due to (K′). ∎

###### Proof of Lemma 5.4.

We write
$\Omega_{\varepsilon}^{c}:=\mathbb{R}^{3}\setminus\Omega_{\varepsilon}$. Let
$\tilde{u}_{\varepsilon}$ be a minimiser for $\tilde{E}_{\varepsilon}$ in
$\tilde{\mathscr{A}}_{\varepsilon}$. Let $B:=B_{\rho}(x_{0})\subseteq\Omega$
be a ball, and let $v\in L^{\infty}(\mathbb{R}^{3},\,\mathcal{Q})$ be such
that $v=\tilde{u}_{\varepsilon}$ a.e. on $\mathbb{R}^{3}\setminus B$. By
comparing (2.1) with (5.1), and using (K2), we obtain

(5.5) $\begin{split}&E_{\varepsilon}(v)-\tilde{E}_{\varepsilon}(v)\\\
&=-\frac{1}{{\varepsilon}^{2}}\int_{\Omega_{\varepsilon}\times\Omega_{\varepsilon}^{c}}K_{\varepsilon}(x-y)v(x)\cdot
v(y)\,\mathrm{d}x\,\mathrm{d}y-\frac{1}{2{\varepsilon}^{2}}\int_{\Omega_{\varepsilon}^{c}\times\Omega_{\varepsilon}^{c}}K_{\varepsilon}(x-y)v(x)\cdot
v(y)\,\mathrm{d}x\,\mathrm{d}y\\\
&=-\frac{1}{{\varepsilon}^{2}}\int_{B\times\Omega_{\varepsilon}^{c}}K_{\varepsilon}(x-y)v(x)\cdot
u_{\mathrm{bd}}(y)\,\mathrm{d}x\,\mathrm{d}y-\frac{1}{{\varepsilon}^{2}}\int_{(\Omega\setminus
B)\times\Omega_{\varepsilon}^{c}}K_{\varepsilon}(x-y)\tilde{u}_{\varepsilon}(x)\cdot
u_{\mathrm{bd}}(y)\,\mathrm{d}x\,\mathrm{d}y\\\
&\qquad\qquad\qquad-\frac{1}{2{\varepsilon}^{2}}\int_{\Omega_{\varepsilon}^{c}\times\Omega_{\varepsilon}^{c}}K_{\varepsilon}(x-y)u_{\mathrm{bd}}(x)\cdot
u_{\mathrm{bd}}(y)\,\mathrm{d}x\,\mathrm{d}y\end{split}$

The second and third integral at the right-hand side are independent of $v$.
We bound the first integral by making the change of variable
$y=x+{\varepsilon}z$ and applying Fubini theorem:

$\begin{split}\frac{1}{{\varepsilon}^{2}}\left|\int_{B\times\Omega_{\varepsilon}^{c}}K_{\varepsilon}(x-y)v(x)\cdot
u_{\mathrm{bd}}(y)\,\mathrm{d}x\,\mathrm{d}y\right|\leq\left\|u_{\mathrm{bd}}\right\|_{L^{\infty}(\mathbb{R}^{3})}\int_{B}\left\|H_{\varepsilon}(x)\right\|\left|v(x)\right|\,\mathrm{d}x\end{split}$

where $H_{\varepsilon}$ is defined by (5.4). Since $u_{\mathrm{bd}}$ and $v$
both take values in the bounded set $\mathcal{Q}$, and since
$\left|B\right|\lesssim\rho^{3}\lesssim\rho$, by Lemma 5.5 there exists an
increasing function $\omega\colon[0,\,+\infty)\to[0,\,+\infty)$, depending
only on $H_{\varepsilon}$ and $\mathcal{Q}$, such that $\lim_{s\to
0}\omega(s)=0$ and

(5.6)
$\begin{split}\frac{1}{{\varepsilon}^{2}}\left|\int_{B\times\Omega_{\varepsilon}^{c}}K_{\varepsilon}(x-y)v(x)\cdot
u_{\mathrm{bd}}(y)\,\mathrm{d}x\,\mathrm{d}y\right|\leq\frac{\omega({\varepsilon})\,\rho}{2}.\end{split}$

From (5.5) and (5.6), we deduce

(5.7)
$\begin{split}E_{\varepsilon}(\tilde{u}_{\varepsilon})-\tilde{E}_{\varepsilon}(\tilde{u}_{\varepsilon})\leq
E_{\varepsilon}(v)-\tilde{E}_{\varepsilon}(v)+\omega({\varepsilon})\,\rho.\end{split}$

On the other hand,

(5.8)
$\tilde{E}_{\varepsilon}(\tilde{u}_{\varepsilon})\leq\tilde{E}_{\varepsilon}(v),$

because $\tilde{u}_{\varepsilon}$ is a minimiser for $\tilde{E}_{\varepsilon}$
and $v\in\tilde{\mathscr{A}}_{\varepsilon}$. Combining (5.7) and (5.8), the
lemma follows. ∎

#### Acknowledgements.

The authors are grateful to Arghir D. Zarnescu, who brought the problem to
their attention. Part of this work was carried out when the authors were
visiting the _Centro Internazionale per la Ricerca Matematica_ (CIRM) in
Trento (Italy), supported by the Research-in-Pairs program. The authors would
like to thank the CIRM for its hospitality. This research has been partially
supported by the Basque Government through the BERC 2018-2021 program; and by
Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo
Ochoa excellence accreditation SEV-2017-0718 and through project
MTM2017-82184-R funded by (AEI/FEDER, UE) and acronym “DESFLU”. G. C. was
partially supported by GNAMPA-INdAM.

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