#  $\Gamma$-convergence and stochastic homogenisation of phase-transition
functionals

Roberta Marziani  Angewandte Mathematik, WWU Münster, Germany
<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the asymptotics of singularly perturbed phase-
transition functionals of the form

$\mathscr{F}_{k}(u)=\frac{1}{\varepsilon_{k}}\int_{A}f_{k}(x,u,\varepsilon_{k}\nabla
u)\,{\rm d}x\,,$

where $u\in[0,1]$ is a phase-field variable, $\varepsilon_{k}>0$ a singular-
perturbation parameter; i.e., $\varepsilon_{k}\to 0$, as $k\to+\infty$, and
the integrands $f_{k}$ are such that, for every $x$ and every $k$,
$f_{k}(x,\cdot,0)$ is a double well potential with zeros at 0 and 1. We prove
that the functionals $\mathscr{F}_{k}$ $\Gamma$-converge (up to subsequences)
to a surface functional of the form

$\mathscr{F}_{\infty}(u)=\int_{S_{u}\cap A}f_{\infty}(x,\nu_{u})\,{\rm
d}\mathcal{H}^{n-1}\,,$

where $u\in BV(A;\\{0,1\\})$ and $f_{\infty}$ is characterised by the double
limit of suitably scaled minimisation problems. Afterwards we extend our
analysis to the setting of stochastic homogenisation and prove a
$\Gamma$-convergence result for _stationary random_ integrands.

###### Key words and phrases:

Keywords: Singular perturbation, phase-field approximation, free discontinuity
problems, $\Gamma$-convergence, deterministic and stochastic homogenisation.

###### 1991 Mathematics Subject Classification:

MSC 2010: 49J45, 49Q20, 74Q05.

##  1\. Introduction

The classical theory of phase transition is based on the assumption that at
the equilibrium two immiscible fluids in a container will separate to create a
sharp interface with minimal area. Alternatively Cahn and Hilliard in [19]
(see also [35]) proposed a model which describes the transition as a
continuous phenomenon concentrated on a thin layer, where a fine mixture of
the two fluids is allowed. The connection between the classical theory of
phase transition and the Cahn-Hilliard model was conjectured by Gurtin in [28]
and rigorously derived, via $\Gamma$-convergence, by Modica in [29], by
generalising a previous result together with Mortola [30] (see also [34], and
[2]).

The model considered in [29, 30] is formalised as follows. Representing by
$A\subset\mathbb{R}^{n}$, open bounded set with Lipschitz boundary, the
container of the two fluids, and by $u\colon A\to[0,1]$ the density of the
second fluid, the associated Modica-Mortola functionals are given by

$\mathcal{M}_{k}(u)=\int_{A}\left(\frac{W(u)}{\varepsilon_{k}}+\varepsilon_{k}^{p-1}|\nabla
u|^{p}\right)\,{\rm d}x\,.$ (1.1)

Here $\varepsilon_{k}\searrow 0$, as $k\to+\infty$, is an elliptic
regularisation parameter which corresponds to the characteristic length-scale
of a phase transition. Moreover, $p>1$ and $W\colon\mathbb{R}\to[0,+\infty)$
is a double well potential vanishing at 0 and 1. Clearly the functionals in
(2.8) are finite in $W^{1,p}(A)$. Note that if $(u_{k})\subset W^{1,p}(A)$,
$0\leq u_{k}\leq 1$, is a sequence satisfying
$\sup_{k}\mathcal{M}_{k}(u_{k})<+\infty$, then from the first term we may
deduce that $u_{k}\to u$ in measure with $u(x)\in\\{0,1\\}$ a.e. $x\in A$
(which corresponds to a separation of the two phases). Actually the condition
$0\leq u_{k}\leq 1$ implies that $u_{k}$ converges to $u$ strongly in
$L^{1}(A)$. On the other hand the second term penalizes spatial
inhomogeneities of $u_{k}$ or equivalently the occurrence of too many
transition regions. According to this, in [29] it was shown that the
functionals in (1.1) $\Gamma$-converge to the functional

$c_{p}\mathcal{H}^{n-1}(S_{u}),$ (1.2)

where now $u\in BV(A;\\{0,1\\})$ (the space of functions with bounded
variation taking values in $\\{0,1\\}$ a.e.). Here $S_{u}$ denotes the set of
discontinuity points of $u$, while the constant $c_{p}>0$ represents the
optimal cost to make a transition between 0 and 1. The quantity in (1.2) can
be interpreted as the surface tension between the two fluids. We stress also
that this result is stable under the volume constraint $\int_{A}u\,{\rm d}x=m$
with $0<m<\mathcal{L}^{n}(A)$ (that is if we fix the volume of the second
fluid).

Starting from the result in [30] phase-field models have been extended in
several directions including, among others, multi-phase models that describe
the behaviour of an arbitrary number of immiscible fluids [10] (see also [34,
27]), models for heterogeneous fluids that can also undergo a temperature
change [13], anisotropic models [11, 32], second order singular perturbations
[20, 26, 21], and models that allow to analyse the interaction between
singular perturbation and homogenisation [6, 12, 22, 23, 31]. These last
variants are those which are closest in the spirit to the present work.

Drawing some inspiration from [9], in the present paper, we aim first at
deriving $\Gamma$-convergence and integral representation results for a
general class of phase-transition functionals that in some sense unify several
of the variants mentioned above. Second, we will apply the abstract
$\Gamma$-convergence result to obtain a stochastic homogenisation result for a
sequence of phase-field functionals with random stationary integrands.
Precisely, in the first part we perform a $\Gamma$-convergence analysis, as
$k\to+\infty$, of a family of singularly perturbed phase-transition
functionals of the form

$\mathscr{F}_{k}(u)=\frac{1}{\varepsilon_{k}}\int_{A}f_{k}(x,u,\varepsilon_{k}\nabla
u)\,{\rm d}x\,,$ (1.3)

where $\varepsilon_{k}\searrow 0$ and $u\colon A\to[0,1]$. We assume that the
integrands
$f_{k}\colon\mathbb{R}^{n}\times[0,1]\times\mathbb{R}^{n}\to[0,+\infty)$
belong to a suitable class of functions denoted by $\mathcal{F}$ (see Section
2.2 for its definition). This in particular ensures that for every
$k\in\mathbb{N}$ and every $x\in\mathbb{R}^{n}$

$c_{1}\left(W(u)+|\xi|^{p}\right)\leq f_{k}(x,u,\xi)\leq
c_{2}\left(W(u)+|\xi|^{p}\right),$ (1.4)

for every $u\in[0,1]$ and $\xi\in\mathbb{R}^{n}$, and for some $0<c_{1}\leq
c_{2}<+\infty$. As a consequence $f_{k}(x,\cdot,0)$ is a double well potential
that vanishes at 0 and 1. In addition the functional $\mathscr{F}_{k}$ is
bounded from above and from below by the Modica-Mortola functional in (1.1).

Our first main result is contained in Theorem 3.1 which asserts the following:
if the integrands $f_{k}$ belong to $\mathcal{F}$, then (up to subsequences)
the functionals $\mathscr{F}_{k}$ $\Gamma$-converge to a surface-type
functional of the form

$\mathscr{F}_{\infty}(u)=\int_{S_{u}\cap A}f_{\infty}(x,\nu_{u})\,{\rm
d}\mathcal{H}^{n-1},$ (1.5)

where $u\in BV(A;\\{0,1\\})$ and $\nu_{u}$ denotes the external unit normal to
$S_{u}$. Moreover $f_{\infty}$ is characterised by a cell formula given by

$f_{\infty}(x,\nu)=\limsup_{\rho\to
0}\lim_{k\to+\infty}\inf\frac{1}{\varepsilon_{k}}\int_{Q_{\rho}^{\nu}(x)}f_{k}(y,u,\varepsilon_{k}\nabla
u)\,{\rm d}x,$ (1.6)

where $Q_{\rho}^{\nu}(x)$ is a cube centred at $x$ with side-length $\rho$
oriented in the direction $\nu$, and the infimum is taken over all $u\in
W^{1,p}(Q_{\rho}^{\nu}(x))$, $0\leq u\leq 1$, such that
$u=\bar{u}^{\nu}_{x,\varepsilon_{k}}$ near $\partial Q_{\rho}^{\nu}(x)$. Here
$\bar{u}^{\nu}_{x,\varepsilon_{k}}$ is a suitable regularisation (see Section
2.1 for the precise definition) of the jump function $u^{\nu}_{x}$ defined as

$u^{\nu}_{x}(y)=\begin{cases}1&\text{if }\;(y-x)\cdot\nu\geq 0\,,\cr
0&\text{if }\;(y-x)\cdot\nu<0\,.\end{cases}$

Notice that the presence of “regularised” boundary conditions is related to
the fact that $u$ belongs to $W^{1,p}(A)$ while $u^{\nu}_{x}$ jumps on the
hyperplane $\\{(x-y)\cdot\nu=0\\}$.

Due to the generality of our model the proof of Theorem 3.1 relies on an
abstract approach known as localisation method (cf. for instance [24]). This
argument consists of the following main steps: we first localise the
functionals $\mathscr{F}_{k}$ by introducing the dependence on the domain of
integration; then, by means of a fundamental estimate, we show that, up to
subsequence, the $\Gamma$-limit exists and admit a surface-integral
representation. Eventually we identify the surface energy density with the
cell formula in (1.6).

In the second part of the paper we are concerned with the case of stochastic
homogenisation. We namely consider functionals of type (1.3) which depend also
on a variable $\omega$ belonging to the set of events $\Omega$ of a
probability space $(\Omega,\mathcal{T},P)$. This dependence follows by
choosing integrands $f_{k}$ of type

$f_{k}(\omega,x,u,\xi)=f\left(\omega,\frac{x}{\varepsilon_{k}},u,\xi\right).$
(1.7)

Here $f$ is a stationary random variable such that
$f(\omega,\cdot,\cdot,\cdot)\in\mathcal{F}$ and is lower semicontinuous in the
variables $u$ and $\xi$ for every $\omega\in\Omega$ (see Section 7,
($F1$)-($F3$)). Our second main result is contained in Theorem 7.4 which
states that if $f_{k}$ are as in (1.7), then, almost surely, the functionals
$\mathscr{F}_{k}$ $\Gamma$-converge to a surface functional of type (1.5)
whose integrand is homogeneous in the space variable and is given by a cell
formula.
The proof of Theorem 7.4 is achieved by resorting on the abstract
$\Gamma$-convergence result Theorem 3.1 as follows. From Theorem 3.1 we first
deduce a homogenisation result (cf. Theorem 3.3) in the deterministic setting
without requiring any spatial periodicity of the integrands but instead
assuming the existence and spatial homogeneity of the limit of a certain cell
formula. Afterwards we show that in the above stochastic setting, almost
surely this limit indeed exists and is homogeneous almost surely. This second
step involves the combination of the subadditive ergodic Theorem (see e.g. [1,
25]) with a specific analysis for regularised surface functionals in the
spirit of [3, 18].
Eventually to complete our analysis we show that if we restrict to the
periodic setting, namely if $f_{k}$ are of the form
$f(\frac{x}{\varepsilon_{k}},u,\xi)$ with $f$ periodic in the space variable,
then the corresponding homogenisation result Theorem 8.1 can be deduced
without requiring any lower semicontinuity of $f$ in the variables $u$ and
$\xi$ (i.e., without assuming ($F3$)).

We notice that our stochastic/periodic homogenisation results (Theorem 7.4 and
Theorem 8.1) cover in particular the cases considered in [31, 23] and the
critical regime in [6]. As a final remark it would be interesting to fully
characterise the $\Gamma$-limit for integrands of type (1.7) when the
oscillation parameter $\varepsilon_{k}$ is replaced by a second scale
parameter $\delta_{k}\searrow 0$. We expect a different behaviour for the two
regimes $\varepsilon_{k}\ll\delta_{k}$ and $\varepsilon_{k}\gg\delta_{k}$.
However this type of analysis goes beyond the purpose of the present paper. A
similar analysis was performed in [6, 22, 23] in the deterministic setting
(see also [7, 8] for the corresponding analysis for Ambrosio-Tortorelli type
functionals).

### Outline of the paper

This paper is organised as follows. In Section 2 we collect the notation
adopted throughout the paper and introduce the functionals we will consider.
In Section 3 we state the first main result of the paper, that is, the
$\Gamma$-convergence and integral representation result (Theorem 3.1). We also
give a homogenisation result without any periodicity assumptions (Theorem
3.3). Sections 4-6 contain the proof of Theorem 3.1). In particular, in
Section 4 we implement the localisation method and prove a compactness and
integral representation result for the $\Gamma$-limit of $\mathscr{F}_{k}$
(Theorem 4.2); in Section 5 we characterise the surface integrand (Proposition
5.1); eventually in Section 6 we prove that the limit surface integrand
satisfies some properties (Proposition 6.2). In Section 7 we state and prove
our second main result, namely a stochastic homogenisation result for
stationary random integrands (Theorem 7.4). In Section 8 we prove a periodic
homogenisation result (Theorem 8.1). Eventually in the Appendix we prove some
technical lemmas which are used in Sections 7 8.

##  2\. Setting of the problem and preliminaries

In this section we collect some notation and we introduce the family of
functionals we will consider.

###  2.1. Notation

We start by listing the notation we will adopt throughout the paper.

  1. (a)

$n\geq 2$ is a fixed positive integer;

  2. (b)

$\mathbb{S}^{n-1}:=\\{\nu=(\nu_{1},\ldots,\nu_{n})\in\mathbb{R}^{n}\colon\nu_{1}^{2}+\cdots+\nu_{n}^{2}=1\\}$
and
$\widehat{\mathbb{S}}^{n-1}_{\pm}:=\\{\nu\in\mathbb{S}^{n-1}\colon\pm\nu_{i(\nu)}>0\\}$,
where $i(\nu):=\max\\{i\in\\{1,\ldots,n\\}\colon\nu_{i}\neq 0\\}$;

  3. (c)

$\mathcal{L}^{n}$ and and $\mathcal{H}^{n-1}$ denote the Lebesgue measure and
the $(n-1)$-dimensional Hausdorff measure on $\mathbb{R}^{n}$, respectively;

  4. (d)

for every $A\subset\mathbb{R}^{n}$ let $\chi_{A}$ denote the characteristic
function of the set $A$;

  5. (e)

$\mathcal{A}$ denotes the collection of all open and bounded subsets of
$\mathbb{R}^{n}$ with Lipschitz boundary. If $A,B\in\mathcal{A}$ by
$A\subset\subset B$ we mean that $A$ is relatively compact in $B$;

  6. (f)

$Q$ denotes the open unit cube in $\mathbb{R}^{n}$ with sides parallel to the
coordinate axis, centred at the origin; for $x\in\mathbb{R}^{n}$ and $r>0$ we
set $Q_{r}(x):=rQ+x$. Moreover, $Q^{\prime}$ denotes the open unit cube in
$\mathbb{R}^{n-1}$ with sides parallel to the coordinate axis, centred at the
origin, for every $r>0$ we set $Q_{r}^{\prime}:=rQ^{\prime}$;

  7. (g)

for every $\nu\in{\mathbb{S}}^{n-1}$ let $R_{\nu}$ denote an orthogonal
$(n\times n)$-matrix such that $R_{\nu}e_{n}=\nu$; we also assume that
$R_{-\nu}Q=R_{\nu}Q$ for every $\nu\in\mathbb{S}^{n-1}$,
$R_{\nu}\in\mathbb{Q}^{n\times n}$ if
$\nu\in\mathbb{S}^{n-1}\cap\mathbb{Q}^{n}$, and that the restrictions of the
map $\nu\mapsto R_{\nu}$ to $\widehat{{\mathbb{S}}}_{\pm}^{n-1}$ are
continuous. For an explicit example of a map $\nu\mapsto R_{\nu}$ satisfying
all these properties we refer the reader, e.g., to [17, Example A.1];

  8. (h)

for $x\in\mathbb{R}^{n}$, $r>0$, and $\nu\in\mathbb{S}^{n-1}$, we define
$Q^{\nu}_{r}(x):=R_{\nu}Q_{r}(x)$.

  9. (i)

for $x\in\mathbb{R}^{n}$ and $\nu\in{\mathbb{S}}^{n-1}$ we denote with
$u_{x}^{\nu}$ the piecewise constant function taking values $0,1$ and jumping
across the hyperplane
$\Pi^{\nu}(x):=\\{y\in\mathbb{R}^{n}\colon(y-x)\cdot\nu=0\\}$; i.e.,

$u^{\nu}_{x}(y):=\begin{cases}1&\text{if }\;(y-x)\cdot\nu\geq 0\,,\cr
0&\text{if }\;(y-x)\cdot\nu<0\,;\end{cases}$

  10. (j)

let ${\rm u}\in C^{1}(\mathbb{R})$ with $0\leq{\rm u}\leq 1$, be a one-
dimensional function such that

${\rm u}(t)=\chi_{(0,+\infty)}\quad\text{for}\quad|t|>1;$

  11. (k)

for $x\in\mathbb{R}^{n}$ and $\nu\in{\mathbb{S}}^{n-1}$ we set

$\bar{u}^{\nu}_{x}(y):={\rm u}((y-x)\cdot\nu)\,;$

  12. (l)

for $x\in\mathbb{R}^{n}$, $\nu\in\mathbb{S}^{n-1}$ and $\varepsilon>0$ we set

$\bar{u}^{\nu}_{x,\varepsilon}(y):={\rm
u}\big{(}\tfrac{1}{\varepsilon}(y-x)\cdot\nu)\,.$

We notice that in particular, $\bar{u}^{\nu}_{x,\varepsilon}(y)=u^{\nu}_{x}$
in $\\{y\colon|(y-x)\cdot\nu|>\varepsilon\\}$, and
$\bar{u}_{x,1}^{\nu}=\bar{u}_{x}^{\nu}$;

  13. (m)

for a given topological space $X$, $\mathcal{B}(X)$ denotes the Borel
$\sigma$\- algebra on $X$. If $X=\mathbb{R}^{d}$, with $d\in\mathbb{N}$,
$d\geq 1$ we simply write $\mathcal{B}^{d}$ in place of
$\mathcal{B}(\mathbb{R}^{d})$. For $d=1$ we write $\mathcal{B}$.

For $A\in\mathcal{A}$ we let $BV(A)$ be the space of functions with bounded
variation (see for instance [5] for a detailed exposition of the subject). We
recall that if $u\in BV(A)$, then its distributional derivative $Du$ is an
$\mathbb{R}^{n}$\- valued Radon measure on $A$. We denote by $|Du|(A)$ its
total variation.

Moreover a set $E\subset\mathbb{R}^{n}$ is a set of finite perimeter in $A$,
or a Caccioppoli set, if $\chi_{E}\in BV(A)$ and we let

$\mathcal{P}_{A}(E):=|D\chi_{E}|(A)\,,$

be the perimeter of $E$ in $A$. The family of sets with finite perimeter can
be identified with the space $BV(A;\\{0,1\\})$, namely, the space of functions
in $BV(A)$ taking values in $\\{0,1\\}$ almost everywhere. Indeed if $u\in
BV(A;\\{0,1\\})$ then $u=\chi_{E}$ where $E=\\{x\colon u(x)=1\\}$ and

$Du(B)=\int_{B\cap S_{u}}\nu_{u}\,\mathrm{d}\mathcal{H}^{n-1}\,,$

for every $B\in\mathcal{B}^{n}$, where $S_{u}$ is the set of approximate
discontinuity points of $u$ which coincides with the reduced boundary of $E$,
while $\nu_{u}$ is the external normal to $S_{u}$. Moreover

$\mathcal{P}_{A}(E)=|Du|(A)=\mathcal{H}^{n-1}(S_{u}\cap A)\,.$

Throughout the paper $C$ denotes a strictly positive constant which may vary
from line to line and within the same expression.

###  2.2. Setting of the problem

Let $p>1$, let $c_{1},c_{2}$ be given constants such that $0<c_{1}\leq
c_{2}<+\infty$. Let $W\colon\mathbb{R}\to[0,+\infty)$ be a double-well
potential, that is a continuous function vanishing only at 0 and 1. We denote
by $\mathcal{F}:=\mathcal{F}(W,p,c_{1},c_{2})$ the collection of all functions
$f\colon\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}\to[0,+\infty)$
satisfying the following conditions:

  1. ($f1$)

(measurability) $f$ is Borel measurable on
$\mathbb{R}^{n}\times\mathbb{R}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\times\mathbb{R}^{n}$;

  2. ($f2$)

(lower bound) for every $x\in\mathbb{R}^{n}$, every $u\in\mathbb{R}$, and
every $\xi\in\mathbb{R}^{n}$

$c_{1}\big{(}W(u)+|\xi|^{p}\big{)}\leq f(x,u,\xi)\,;$

  3. ($f3$)

(upper bound) for every $x\in\mathbb{R}^{n}$, every $u\in\mathbb{R}$, and
every $\xi\in\mathbb{R}^{n}$

$f(x,u,\xi)\leq c_{2}(W(u)+|\xi|^{p})\,.$

For $k\in\mathbb{N}$ let $(f_{k})\subset\mathcal{F}$ and let
$(\varepsilon_{k})\subset(0,1]$ be a decreasing sequence of real numbers
converging to zero, as $k\to+\infty$. We consider the sequence of singularly
perturbed phase-transitions functionals $\mathscr{F}_{k}\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\times\mathcal{A}\longrightarrow[0,+\infty]$
defined by

$\displaystyle\mathscr{F}_{k}(u,A):=\begin{cases}\displaystyle\frac{1}{\varepsilon_{k}}\int_{A}f_{k}(x,u,\varepsilon_{k}\nabla
u)\,{\rm d}x&\text{if}\ u\in W^{1,p}(A)\,,0\leq u\leq 1\,,\\\
+\infty&\text{otherwise}\,.\end{cases}$ (2.1)

It is convenient to introduce some further notation. For $A\in\mathcal{A}$,
$x\in\mathbb{R}^{n}$ and $\nu\in{\mathbb{S}}^{n-1}$ we consider the following
minimisation problem

$\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},A):=\inf\\{\mathscr{F}_{k}(u,A)\colon
u\in\mathscr{A}(\bar{u}_{x,\varepsilon_{k}}^{\nu},A)\\}\,,$ (2.2)

where

$\mathscr{A}(\bar{u}_{x,\varepsilon_{k}}^{\nu},A):=\big{\\{}u\in
W^{1,p}(A),\;0\leq u\leq 1\colon\,u=\bar{u}_{x,\varepsilon_{k}}^{\nu}\
\text{near}\ \partial A\ \big{\\}}\,.$ (2.3)

In (2.2) by “$u=\bar{u}^{\nu}_{x,\varepsilon_{k}}$ near $\partial A$” we mean
that the boundary datum is attained in a neighbourhood of $\partial A$.
Eventually for every $\rho>2\varepsilon_{k}$, every $x\in\mathbb{R}^{n}$, and
every $\nu\in\mathbb{S}^{n-1}$ we set

$f^{\prime}(x,\nu):=\limsup_{\rho\to
0}\frac{1}{\rho^{n-1}}\liminf_{k\to+\infty}\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q^{\nu}_{\rho}(x))\,,$
(2.4) $f^{\prime\prime}(x,\nu):=\limsup_{\rho\to
0}\frac{1}{\rho^{n-1}}\limsup_{k\to+\infty}\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q^{\nu}_{\rho}(x))\,.$
(2.5)

######  Remark 2.1.

  1. $(a)$

From ($f2$) and ($f3$) together with the assumptions on $W$ we have that

$f(x,u,\xi)=0\;\text{ if and only if }\;(u,\xi)\in\\{(0,0),(1,0)\\}\,,$ (2.6)

for every $x\in\mathbb{R}^{n}$. As a consequence $f(x,\cdot,0)$ is a double-
well potential vanishing at 0 and 1 for every $x\in\mathbb{R}^{n}$. Moreover
the functionals

$\frac{1}{\varepsilon_{k}}\int_{A}f(x,u,\varepsilon_{k}\nabla u)\,{\rm d}x\,,$

decrease under the transformation $u\to\min\\{\max\\{u,0\\},1\\}$. Thus it is
not restrictive to define $\mathscr{F}_{k}$ to be finite when $u\in
W^{1,p}(A)$ satisfies the bounds $0\leq u\leq 1$ and the $\Gamma$-convergence
for functionals $\mathscr{F}_{k}$ that are finite when $u\in W^{1,p}(A)$
readily follows from Theorem 3.1.

  2. $(b)$

Assumptions ($f2$)–($f3$) imply that for every $A\in\mathcal{A}$ and every
$u\in W^{1,p}(A)$ with $0\leq u\leq 1$ it holds

$\begin{split}c_{1}\mathcal{M}_{k}(u,A)\leq\mathscr{F}_{k}(u,A)\leq
c_{2}\mathcal{M}_{k}(u,A)\,;\end{split}$ (2.7)

where

$\mathcal{M}_{k}(u,A):=\int_{A}\bigg{(}\frac{W(u)}{\varepsilon_{k}}+\varepsilon_{k}^{p-1}|\nabla
u|^{p}\bigg{)}\,{\rm d}x\,,$ (2.8)

is the Modica-Mortola functional.

  3. $(c)$

From the assumption that $\mathscr{F}_{k}$ is finite when $u\in W^{1,p}(A)$,
$0\leq u\leq 1$, together with the bound in (2.7) and the result in [29] we
easily deduce the following compactness property: Let $(u_{k})\subset
W^{1,p}(A)$ be a sequence satisfying
$\sup_{k}\mathscr{F}_{k}(u_{k},A)<+\infty$. Then $u_{k}$ converges strongly in
$L^{1}(A)$ to some $u\in BV(A,\\{0,1\\})$.
If the function $W$ satisfies the following coercivity condition:

$W(u)\geq\varphi(|u|)\quad\forall u\in\mathbb{R}\,,$

with $\varphi\colon[0,+\infty)\to[0,+\infty)$ such that
$\lim_{t\to+\infty}\varphi(t)/t=+\infty$, then the same compactness property
holds if we define $\mathscr{F}_{k}$ to be finite when $u\in W^{1,p}(A)$ (cf.
[29]).

  4. $(d)$

Let $A\in\mathcal{A}$ be such that $A=A^{\prime}\times I$ with
$A^{\prime}\subset\mathbb{R}^{n-1}$ open and bounded and $I\subset\mathbb{R}$
open interval. Let $\nu\in\mathbb{S}^{n-1}$ and set $A_{\nu}:=R_{\nu}A$, with
$R_{\nu}$ as in (g). For every $k\in\mathbb{N}$ we have

$\begin{split}\mathcal{M}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},A_{\nu})\leq\int_{A^{\prime}}\int_{\mathbb{R}}\big{(}W({\rm
u}(t))+|{\rm u}^{\prime}(t)|^{p}\big{)}\,{\rm d}t\,{\rm d}y^{\prime}=C_{{\rm
u}}\mathcal{L}^{n-1}(A^{\prime})\,,\end{split}$ (2.9)

where

$C_{{\rm u}}:=\int_{\mathbb{R}}\big{(}W({\rm u}(t))+|{\rm
u}^{\prime}(t)|^{p}\big{)}\,{\rm d}t=\int_{0}^{1}\big{(}W({\rm u}(t))+|{\rm
u}^{\prime}(t)|^{p}\big{)}\,{\rm d}t<+\infty\,.$

In particular from ($f3$) and (2.9) we deduce

$\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},A_{\nu})\leq\mathscr{F}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},A_{\nu})\leq
c_{2}C_{{\rm u}}\mathcal{L}^{n-1}(A^{\prime})\,.$ (2.10)

##  3\. Statements of the main results

In this section we collect some of the results of this paper. We start by
stating our first main result (Theorem 3.1), that is a $\Gamma$-convergence
and an integral representation result. Next we provide a homogenisation result
without periodicity assumptions (Theorem 3.3). The latter, in particular, will
be crucial to obtain our second main result in the stochastic setting (see
Section 7).

###  3.1. $\Gamma$-convergence

The following result shows that, up to subsequences, the functionals
$\mathscr{F}_{k}$ $\Gamma$-converge to a surface integral functional.
Furthermore, the surface energy density can be characterised as a double limit
of a suitable minimisation problem.

######  Theorem 3.1 ($\Gamma$-convergence).

Let $(f_{k})\subset\mathcal{F}$ and let $\mathscr{F}_{k}$ be the functionals
as in (2.1). Then there exists a subsequence, not relabelled, such that for
every $A\in\mathcal{A}$ the functionals $\mathscr{F}_{k}(\cdot\,,A)$
$\Gamma$-converge in $L^{1}(A)$ to $\mathscr{F}_{\infty}(\cdot\,,A)$ with
$\mathscr{F}_{\infty}\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\times\mathcal{A}\longrightarrow[0,+\infty]$
given by

$\mathscr{F}_{\infty}(u,A):=\begin{cases}\displaystyle\int_{S_{u}\cap
A}f_{\infty}(x,\nu_{u})\,{\rm d}\mathcal{H}^{n-1}&\text{if}\ u\in
BV(A;\\{0,1\\})\,,\\\\[4.0pt] +\infty&\text{otherwise}\,,\end{cases}$

where $f_{\infty}\colon\mathbb{R}^{n}\times{\mathbb{S}}^{n-1}\to[0,+\infty)$
is a Borel function. Moreover for every $x\in\mathbb{R}^{n}$ and every
$\nu\in{\mathbb{S}}^{n-1}$ there hold

$f_{\infty}(x,\nu)=f^{\prime}(x,\nu)=f^{\prime\prime}(x,\nu)\,,$

and

$c_{1}c_{p}\leq f_{\infty}(x,\nu)\leq c_{2}c_{p}\,,$

with $f^{\prime}$, $f^{\prime\prime}$ as in (2.4) and (2.5), respectively and
$c_{p}:=p(p-1)^{\frac{1-p}{p}}\int_{0}^{1}W(t)^{\frac{p-1}{p}}\,{\rm d}t$.

For the reader’s convenience we divide the proof of Theorem 3.1 in three parts
which can be found in Sections 4, 5, and 6, respectively. Precisely, in
Section 4 we show that there is a sequence $(k_{j})$, with $k_{j}\to+\infty$
as $j\to+\infty$, such that for every $A\in\mathcal{A}$ the corresponding
functionals $\mathscr{F}_{k_{j}}(\,\cdot\,,A)$ $\Gamma$-converge to a
functional which is finite in $BV(A;\\{0,1\\})$ and is of the form

$\int_{S_{u}\cap A}\hat{f}(x,\nu_{u})\,{\rm d}\mathcal{H}^{n-1}\,,$

for some Borel function $\hat{f}$ (see Theorem 4.2). In Section 5 we identify
$\hat{f}$ by showing that it is equal both to $f^{\prime}$ and
$f^{\prime\prime}$ which in particular are shown to coincide along the
$\Gamma$-convergent subsequence (see Proposition 5.1). Eventually in Section 6
we prove that the function $f^{\prime}$ (respectively $f^{\prime\prime}$) is
Borel measurable and satisfies $c_{1}c_{p}\leq f^{\prime}(x,\nu)\leq
c_{2}c_{p}$ (respectively $c_{1}c_{p}\leq f^{\prime\prime}(x,\nu)\leq
c_{2}c_{p}$) for every $x\in\mathbb{R}^{n}$ and every $\nu\in\mathbb{S}^{n-1}$
(see Proposition 6.2).

By Theorem 3.1 and the Urysohn property of $\Gamma$-convergence [24,
Proposition 8.3] we can deduce the following useful property.

######  Corollary 3.2.

Let $(f_{k})\subset\mathcal{F}$ and let $\mathscr{F}_{k}$ be the functionals
as in (2.1). Let $f^{\prime}$, $f^{\prime\prime}$ be as in (2.4) and (2.5),
respectively. Assume that

$f^{\prime}(x,\nu)=f^{\prime\prime}(x,\nu)=:f_{\infty}(x,\nu),\;\text{ for
every $x\in\mathbb{R}^{n}$ and every $\nu\in{\mathbb{S}}^{n-1}$},$

for some Borel function
$f_{\infty}\colon\mathbb{R}^{n}\times{\mathbb{S}}^{n-1}\to[0,+\infty)$. Then,
for every $A\in\mathcal{A}$ the functionals $\mathscr{F}_{k}(\cdot\,,A)$
$\Gamma$-converge in $L^{1}(A)$ to $\mathscr{F}_{\infty}(\cdot\,,A)$ with
$\mathscr{F}_{\infty}\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\times\mathcal{A}\longrightarrow[0,+\infty]$
given by

$\mathscr{F}_{\infty}(u,A):=\begin{cases}\displaystyle\int_{S_{u}\cap
A}f_{\infty}(x,\nu_{u})\,{\rm d}\mathcal{H}^{n-1}&\text{if}\ u\in
BV(A;\\{0,1\\})\,,\\\\[4.0pt] +\infty&\text{otherwise}\,.\end{cases}$

###  3.2. Homogenisation

In this subsection we prove a general homogenisation theorem without assuming
any spatial periodicity of the integrands $f_{k}$. This theorem will be
employed to prove the stochastic homogenisation result Theorem 7.4.

We need to introduce some further notation. We fix $f\in\mathcal{F}$,
$A\in\mathcal{A}$, $u\in W^{1,p}(A)$, $z\in\mathbb{R}^{n}$,
$\nu\in{\mathbb{S}}^{n-1}$, and let $\bar{u}^{\nu}_{z}$ be as in (k) (with
$x=z$). Then we set

$\mathscr{F}(u,A):=\int_{A}f(x,u,\nabla u)\,{\rm d}x\,;$ (3.1)

and

$\mathbf{m}(\bar{u}^{\nu}_{z},A):=\inf\Bigl{\\{}\mathscr{F}(u,A)\colon
u\in\mathscr{A}(\bar{u}^{\nu}_{z},A)\Bigr{\\}}\,,$ (3.2)

where $\mathscr{A}(\bar{u}^{\nu}_{z},A)$ is as in (2.3), with
$\bar{u}^{\nu}_{x,\varepsilon_{k}}$ replaced by $\bar{u}^{\nu}_{z}$ (that is,
$x=z$ and $\varepsilon_{k}=1$).

When the integrands $f_{k}$ in the definition of $\mathscr{F}_{k}$ are of type

$f_{k}(x,u,\xi):=f\Big{(}\frac{x}{\varepsilon_{k}},u,\xi\Big{)}\,,$ (3.3)

we can deduce the following homogenisation result.

######  Theorem 3.3 (Deterministic homogenisation).

Let $f\in\mathcal{F}$ and let $\mathbf{m}(\bar{u}_{rx}^{\nu},Q^{\nu}_{r}(rx))$
be as in (3.2) with $z=rx$ and $A=Q_{r}^{\nu}(rx)$. Assume that for every
$x\in\mathbb{R}^{n}$, $\nu\in\mathbb{S}^{n-1}$ the following limit

$\lim_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{rx},Q^{\nu}_{r}(rx))}{r^{n-1}}=:f_{\rm
hom}(\nu)\,,$ (3.4)

exists and is independent of $x$. Then, for every $A\in\mathcal{A}$ the
functionals $\mathscr{F}_{k}(\cdot,A)$ defined in (2.1) with $f_{k}$ as in
(3.3) $\Gamma$-converge in $L^{1}(A)$ to the functional $\mathscr{F}_{\rm
hom}(\cdot,A)$, with $\mathscr{F}_{\rm hom}\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\times\mathcal{A}\longrightarrow[0,+\infty]$
given by

$\mathscr{F}_{\rm hom}(u,A):=\begin{cases}\displaystyle\int_{S_{u}\cap
A}f_{\rm hom}(\nu_{u})\,{\rm d}\mathcal{H}^{n-1}&\text{if}\ u\in
SBV^{p}(A)\,,u\in\\{0,1\\}\text{ a.e. }\\\\[4.0pt]
+\infty&\text{otherwise}\,.\end{cases}$

###### Proof.

Let $f^{\prime}$, $f^{\prime\prime}$ be as in (2.4), (2.5), respectively. By
Corollary 3.2 it suffices to show that

$f_{\mathrm{hom}}(\nu)=f^{\prime}(x,\nu)=f^{\prime\prime}(x,\nu)\,,$ (3.5)

for every $x\in\mathbb{R}^{n}$ and $\nu\in{\mathbb{S}}^{n-1}$.

Let $x\in\mathbb{R}^{n}$, $\nu\in\mathbb{S}^{n-1}$, $k\in\mathbb{N}$ and
$\rho>2\varepsilon_{k}$ be fixed and let
$u\in\mathscr{A}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))$. Define
$u_{k}\in
W^{1,p}(Q^{\nu}_{\frac{\rho}{\varepsilon_{k}}}(\frac{x}{\varepsilon_{k}}))$ as
$u_{k}(y):=u(\varepsilon_{k}y)$. Then clearly
$u_{k}\in\mathscr{A}(\bar{u}_{\frac{x}{\varepsilon_{k}}}^{\nu},Q^{\nu}_{\frac{\rho}{\varepsilon_{k}}}(\tfrac{x}{\varepsilon_{k}}))$
and a change of variables gives

$\mathscr{F}_{k}(u,Q^{\nu}_{\rho}(x))=\varepsilon_{k}^{n-1}\int_{Q^{\nu}_{\frac{\rho}{\varepsilon_{k}}}(\frac{x}{\varepsilon_{k}})}f(x,v_{k},\nabla
v_{k})\,{\rm d}x\,.$

Hence by setting $r_{k}:=\frac{\rho}{\varepsilon_{k}}$ we get

$\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))=\varepsilon_{k}^{n-1}\mathbf{m}\big{(}\bar{u}_{\frac{x}{\varepsilon_{k}}}^{\nu},Q_{\frac{\rho}{\varepsilon_{k}}}^{\nu}\big{(}\tfrac{x}{\varepsilon_{k}}\big{)}\big{)}=\frac{\rho^{n-1}}{r_{k}^{n-1}}\mathbf{m}\big{(}\bar{u}^{\nu}_{r_{k}\frac{x}{\rho}},Q_{r_{k}}\big{(}r_{k}\tfrac{x}{\rho}\big{)}\big{)}\,.$

Finally using (3.4) with $x/\rho$ in place of $x$ we obtain

$\lim_{k\to+\infty}\frac{\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))}{\rho^{n-1}}=\lim_{k\to+\infty}\frac{\mathbf{m}\big{(}\bar{u}^{\nu}_{r_{k}\frac{x}{\rho}},Q_{r_{k}}^{\nu}\big{(}r_{k}\tfrac{x}{\rho}\big{)}\big{)}}{r_{k}^{n-1}}=f_{\mathrm{hom}}(\nu)\,,$

and therefore
$f^{\prime}(x,\nu)=f^{\prime\prime}(x,\nu)=f_{\mathrm{hom}}(\nu)$, for every
$x\in\mathbb{R}^{n}$, $\nu\in{\mathbb{S}}^{n-1}$. ∎

##  4\. $\Gamma$-convergence and integral representation

In this section we prove that, up to subsequences, the functionals
$\mathscr{F}_{k}$ $\Gamma$-converge to a surface integral functional. In order
to do that we apply the so called localisation method of $\Gamma$-convergence.
In particular we closely follow the theory described in [24, Chapters 14-18]
and [16, Chapters 10, 11], for this reason we discuss here only the main
adaptations to our case.

We start by proving a preliminary result, namely we show that the functionals
$\mathscr{F}_{k}$ satisfy a fundamental estimate, uniformly in $k$ similarly
to [6, Lemma 3.2].

######  Proposition 4.1 (Fundamental estimate).

Let $\mathscr{F}_{k}$ be as in (2.1). Then there exist $K>0$ depending only on
$c_{1},c_{2},p$ and $W$ and functions $\omega_{k}\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})^{2}\times\mathcal{A}^{3}\to[0,+\infty)$
$($depending only on $k)$ such that for every
$A,\,A^{\prime},\,B\in\mathcal{A}$ with $A\subset\subset A^{\prime}$ the
following hold:

  1. $(i)$

For every $u\in W^{1,p}(A^{\prime})$, $v\in W^{1,p}(B)$, $0\leq u,v\leq 1$,
there exists $w\in W^{1,p}(A\cup B)$ with $0\leq w\leq 1$ such that

$w=u\quad\text{ a.e.\ in }\quad A\quad\text{ and }\quad w=v\quad\text{ a.e.\
in }\quad B\setminus\overline{A}^{\prime}\,,$

and

$\displaystyle\mathscr{F}_{k}(w,A\cup
B)\leq(1+K\varepsilon_{k})(\mathscr{F}_{k}(u,A^{\prime})+\mathscr{F}_{k}(v,B))+\omega_{k}(u,v,A,A^{\prime},B)\,;$

  2. $(ii)$

For every $u\in W^{1,p}(A^{\prime})$, $v\in W^{1,p}(B)$ $0\leq u,v\leq 1$ and
every $\widetilde{S}\subset(A^{\prime}\setminus\overline{A})\cap B$ with

$u(x)=v(x)\in\\{0,1\\}\quad\text{ a.e. in }\quad\widetilde{S}\,;$

we have

$\begin{split}\omega_{k}(u,v,A,A^{\prime},B)\leq
K\mathcal{L}^{n}((A^{\prime}\setminus\overline{A})\cap
B\setminus\widetilde{S})\,;\end{split}$

  3. $(iii)$

For every $(u_{k}),(v_{k})\subset L^{1}_{\rm loc}(\mathbb{R}^{n})$ having the
same limit as $k\to\infty$ in $L^{1}((A^{\prime}\setminus\overline{A})\cap B)$
and satisfying

$\sup_{k\in\mathbb{N}}\left(\mathscr{F}_{k}(u_{k},A^{\prime})+\mathscr{F}_{k}(v_{k},B)\right)<+\infty\,,$

it holds

$\lim_{k\to\infty}\omega_{k}(u_{k},v_{k},A,A^{\prime},B)=0\,.$ (4.1)

###### Proof.

Let $A,\,A^{\prime},\,B\in\mathcal{A}$ with $A\subset\subset A^{\prime}$ be
fixed.

Step 1: we show $(i)$. Let $d:=\operatorname{dist}(A;\mathbb{R}^{n}\setminus
A^{\prime})$, $N_{k}:=\lfloor 2\varepsilon_{k}^{-1}\rfloor$ (here $\lfloor
a\rfloor$ denotes the integer part of $a$), and let
$A_{1},\ldots,A_{N_{k}+1}\in\mathcal{A}$ with

$A\subset\subset A_{1}\subset\subset\dots\subset\subset
A_{N_{k}+1}\subset\subset A^{\prime}\,,$

and

$\operatorname{dist}(A_{i};\mathbb{R}^{n}\setminus
A_{i+1})\geq\frac{d}{N_{k}+2}\quad i=1,\dots,N_{k}\,.$

For each $i=1,\ldots,N_{k}$ let $\varphi_{i}$ be a smooth cut-off function
between $A_{i}$ and $A_{i+1}$ such that

$M_{k}:=\max_{1\leq i\leq
N_{k}}\|\nabla\varphi_{i}\|_{\infty}\leq\frac{2(N_{k}+2)}{d}\,.$

Let $u$ and $v$ be as in the statement. For $i=1,\ldots,N_{k}$ we define
$w^{i}\in W^{1,p}(A\cup B)$ as follows

$w^{i}:=\varphi_{i}u+(1-\varphi_{i})v\,.$ (4.2)

By setting $S_{i}:=(A_{i+1}\setminus\overline{A}_{i})\cap B$ we obtain

$\begin{split}\mathscr{F}_{k}(w^{i},A\cup
B)&\leq\mathscr{F}_{k}(u,A_{i})+\mathscr{F}_{k}(v,B\setminus\overline{A}_{i+1})+\mathscr{F}_{k}(w^{i},S_{i})\\\
&\leq\mathscr{F}_{k}(u,A^{\prime})+\mathscr{F}_{k}(v,B)+\mathscr{F}_{k}(w^{i},S_{i})\,.\end{split}$
(4.3)

Now from ($f3$) we deduce

$\begin{split}\mathscr{F}_{k}(w^{i},S_{i})\leq c_{2}\int_{S_{i}\cap
B}\bigg{(}\frac{W(w^{i})}{\varepsilon_{k}}+\varepsilon_{k}^{p-1}|\nabla
w^{i}|^{p}\bigg{)}\,{\rm d}x\,.\end{split}$ (4.4)

Moreover a convexity argument gives

$\begin{split}|\nabla w_{i}|^{p}&\leq 2^{p-1}\left(|\nabla u|^{p}+|\nabla
v|^{p}+|\nabla\varphi_{i}|^{p}|u-v|^{p}\right)\\\ &\leq 2^{p-1}\left(|\nabla
u|^{p}+|\nabla v|^{p}+M_{k}^{p}|u-v|^{p}\right)\,.\end{split}$ (4.5)

From (4.4), (4.5) and ($f2$) we obtain

$\begin{split}\mathscr{F}_{k}(w^{i},S_{i})&\leq\frac{2^{p-1}c_{2}}{c_{1}}\big{(}\mathscr{F}_{k}(u,S_{i})+\mathscr{F}_{k}(v,S_{i})\big{)}\\\
&+2^{p-1}c_{2}M_{k}^{p}\varepsilon_{k}^{p-1}\int_{S_{i}}|u-v|^{p}\,{\rm
d}x+c_{2}\int_{S_{i}}\frac{W(w^{i})}{\varepsilon_{k}}\,{\rm d}x\,.\end{split}$
(4.6)

Now summing up in (4.3) over all $i$, by averaging we find an index
$i^{*}\in\\{1,\ldots,N_{k}\\}$ such that

$\displaystyle\mathscr{F}_{k}(w^{i^{*}},A\cup
B)\leq\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\mathscr{F}_{k}(w^{i},A\cup
B)\leq\mathscr{F}_{k}(u,A^{\prime})+\mathscr{F}_{k}(v,B)+\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\mathscr{F}_{k}(w^{i},S_{i})\,.$
(4.7)

By (4.6) we have

$\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\mathscr{F}_{k}(w^{i},S_{i})\leq\frac{\widehat{M}}{N_{k}}\left(\mathscr{F}_{k}(u,A^{\prime})+\mathscr{F}_{k}(v,B)+M_{k}^{p}\varepsilon_{k}^{p-1}\int_{S}|u-v|^{p}\,{\rm
d}x+\sum_{i=1}^{N_{k}}\int_{S_{i}}\frac{W(w^{i})}{\varepsilon_{k}}\,{\rm
d}x\right)\,,$ (4.8)

with $\widehat{M}:=\max\\{\frac{2^{p-1}c_{2}}{c_{1}},2^{p-1}c_{2},c_{2}\\}$
and $S:=(A^{\prime}\setminus\overline{A})\cap B$. Using that
$\frac{2-\varepsilon_{k}}{\varepsilon_{k}}\leq
N_{k}\leq\frac{2}{\varepsilon_{k}}$ we have

$\frac{\widehat{M}}{N_{k}}\leq\frac{\widehat{M}\varepsilon_{k}}{2-\varepsilon_{k}}\leq\widehat{M}\varepsilon_{k}\,,\quad\text{
and }\quad\frac{\widehat{M}}{N_{k}}M_{k}^{p}\varepsilon_{k}^{p-1}\leq
C\varepsilon_{k}^{p}\left(\frac{1}{\varepsilon_{k}}+1\right)^{p}\leq C\,.$

This together with (4.7) and (4.8) give

$\mathscr{F}_{k}(w^{i},A\cup
B)\leq(1+\widehat{M}\varepsilon_{k})\left(\mathscr{F}_{k}(u,A^{\prime})+\mathscr{F}_{k}(v,B)\right)+\omega_{k}(u,v,A,A^{\prime},B)$

where

$\omega_{k}(u,v,A,A^{\prime},B):=C\left(\int_{S}|u-v|^{p}\,{\rm
d}x+\sum_{i=1}^{N_{k}}\int_{S_{i}}W(w^{i})\,{\rm d}x\right)\,,$ (4.9)

for some positive constant $C>0$ independent of $k$.

Step 2: we show $(ii)$. Assume that $0\leq u,v\leq 1$ are such that

$u(x)=v(x)\in\\{0,1\\}\quad\text{ a.e. in }\quad\widetilde{S}\subset S\,.$

Then clearly $\int_{\widetilde{S}}|u-v|^{p}\,{\rm d}x=0$ and

$W(w^{i})=0\quad\text{ a.e.\ in }\quad S_{i}\cap\widetilde{S}\,.$

Moreover, since $0\leq w^{i}\leq 1$ for all $i=1,...,N_{k}$, by the continuity
of $W$

$W(w_{i})\leq M\quad\text{ a.e.\ in }\quad S_{i}\setminus\widetilde{S}\,,\
\forall i=1,...,N_{k}\,.$

Hence (4.9) readily implies

$\omega_{k}(u,v,A,A^{\prime},B)\leq
C(2^{p}+M)\mathcal{L}^{n}(S\setminus\widetilde{S})\,,$

and by setting $K:=\max\\{\widehat{M},C(2^{p}+M)\\}$ we deduce ($(ii)$).

Step 3: we show $(iii)$. Let $(u_{k}),(v_{k})\subset L^{1}_{\rm
loc}(\mathbb{R}^{n})$, $0\leq u_{k},v_{k}\leq 1$, be two sequences converging
to $w$ in $L^{1}(S)$ and such that

$\sup_{k\in\mathbb{N}}\left(\mathscr{F}_{k}(u_{k},A^{\prime})+\mathscr{F}_{k}(v_{k},B)\right)<C\,.$

Since $0\leq u_{k},v_{k}\leq 1$ we immediately deduce that $u_{k}$ and $v_{k}$
converge to $w$ also in $L^{p}(S)$.

Thus (4.1) follows if we show that

$\lim_{k\to\infty}\sum_{i=1}^{N_{k}}\int_{S_{i}}W(w_{k}^{i})\,{\rm d}x=0\,,$

with $w^{i}_{k}$ defined as in (4.2) with $u_{k}$ and $v_{k}$ in place of $u$
and $v$ respectively. Let

$w_{k}:=\begin{cases}w^{i}_{k}&\text{ if }x\in S_{i}\,,i=1,\dots N_{k}\,,\\\
w&\text{ otherwise}\,;\end{cases}$

then clearly $w_{k}$ converges to $w$ in $L^{p}(S)$, moreover since $W$ is
continuous

$0\leq\lim_{k\to\infty}\sum_{i=1}^{N_{k}}\int_{S_{i}}W(w_{k}^{i})\,{\rm
d}x\leq\lim_{k\to\infty}\int_{S}W(w_{k})\,{\rm
d}x=\lim_{k\to\infty}\int_{S}W(u_{k})\,{\rm
d}x\leq\lim_{k\to\infty}\varepsilon_{k}C=0\,.$

∎

With the help of Proposition 4.1, we can now prove the following
$\Gamma$-convergence result.

######  Theorem 4.2.

Let $\mathscr{F}_{k}$ be as in (2.1). Then there exist a subsequence
$(\mathscr{F}_{k_{j}})$ of $(\mathscr{F}_{k})$ and a functional
$\mathscr{F}_{\infty}\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\times\mathcal{A}\longrightarrow[0,+\infty]$
such that for every $A\in\mathcal{A}$ the functionals
$\mathscr{F}_{k_{j}}(\cdot\,,A)$ $\Gamma$-converge in $L^{1}(A)$ to
$\mathscr{F}_{\infty}(\cdot\,,A)$. Moreover, $\mathscr{F}_{\infty}$ is given
by

$\mathscr{F}_{\infty}(u,A):=\begin{cases}\displaystyle\int_{S_{u}\cap
A}\hat{f}(x,\nu_{u})\,{\rm d}\mathcal{H}^{n-1}\,&\text{if}\ u\in BV(A)\,,\
v\in\\{0,1\\}\,\text{a.e.\ in }A\,,\\\ +\infty&\text{otherwise}\,,\end{cases}$

with ${\hat{f}}\colon\mathbb{R}^{n}\times\mathbb{S}^{n-1}\to[0,+\infty)$
defined as

$\hat{f}(x,\nu):=\limsup_{\rho\to
0}\frac{1}{\rho^{n-1}}\mathbf{m}(u_{x}^{\nu},Q_{\rho}^{\nu}(x))\,,$ (4.10)

for every $x\in\mathbb{R}^{n}$ and $\nu\in{\mathbb{S}}^{n-1}$, where for
$A\in\mathcal{A}$ and $\bar{u}\in BV(A,\\{0,1\\})$

$\displaystyle\mathbf{m}(\bar{u},A):=\inf\\{\mathscr{F}_{\infty}(u,A)\colon
u\in BV(A;\\{0,1\\})\,,\ u=\bar{u}\ \text{near}\ \partial A\\}\,.$

###### Proof.

The proof is rather standard, thus here we only sketch it and refer to [24]
for more details. By [24, Theorem 16.9] there exists a subsequence $(k_{j})$,
with $k_{j}\to+\infty$ as $j\to+\infty$, such that the inner regular envelopes
of the $\Gamma$-lower limit and of the $\Gamma$-upper limit coincide. More
precisely for every $u\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ and every
$A\in\mathcal{A}$

$\sup_{A^{\prime}\subset\subset
A,\,A^{\prime}\in\mathcal{A}}\mathscr{F}^{\prime}(u,A^{\prime})\quad=\sup_{A^{\prime}\subset\subset
A,\,A^{\prime}\in\mathcal{A}}\mathscr{F}^{\prime\prime}(u,A^{\prime})=:\mathscr{F}_{\infty}(u,A)\,,$
(4.11)

with $\mathscr{F}^{\prime},\mathscr{F}^{\prime\prime}\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\times\mathcal{A}\longrightarrow[0,+\infty]$
defined as

$\mathscr{F}^{\prime}(\,\cdot\,,A):=\Gamma\hbox{-}\liminf_{k_{j}\to+\infty}\mathscr{F}_{k_{j}}(\,\cdot\,,A)\quad\text{and}\quad\mathscr{F}^{\prime\prime}(\,\cdot\,,A):=\Gamma\hbox{-}\limsup_{k_{j}\to+\infty}\mathscr{F}_{k_{j}}(\,\cdot\,,A)\,.$

Now by recalling Remark $(b)$ and [30, Theorem 1] we can find a constant $C>0$
such that

$\begin{split}\frac{1}{C}\mathcal{H}^{n-1}(S_{u}\cap
A)\leq\mathscr{F}^{\prime}(u,A)\leq\mathscr{F}^{\prime\prime}(u,A)\leq
C\mathcal{H}^{n-1}(S_{u}\cap A)\,,\end{split}$ (4.12)

for every $A\in\mathcal{A}$ and every $u\in BV(A;\\{0,1\\})$ and

$\mathscr{F}^{\prime}(u,A)=\mathscr{F}^{\prime\prime}(u,A)=+\infty\quad\text{if
}\;u\notin BV(A;\\{0,1\\})\,.$ (4.13)

In particular, (4.12) together with [24, Proposition 18.6] and Proposition 4.1
imply that

$\mathscr{F}_{\infty}(u,A)=\mathscr{F}^{\prime}(u,A)=\mathscr{F}^{\prime\prime}(u,A)\quad\text{if}\quad
u\in BV(A;\\{0,1\\})\,,$

while (4.11) and (4.13) yield

$\mathscr{F}_{\infty}(u,A)=\mathscr{F}^{\prime}(u,A)=\mathscr{F}^{\prime\prime}(u,A)=+\infty\quad\text{if
}\;u\notin BV(A;\\{0,1\\})\,.$

Hence, $\mathscr{F}_{\infty}(\cdot,A)$ coincides with the $\Gamma$-limit of
$\mathscr{F}_{k_{j}}(\cdot,A)$ on $L^{1}(\mathbb{R}^{n})$, for every
$A\in\mathcal{A}$.

We now observe that the functional $\mathscr{F}_{\infty}$ defined in (4.11)
satisfies the following properties: the functional
$\mathscr{F}_{\infty}(\cdot,A)$ is $L^{1}(\mathbb{R}^{n})$ lower
semicontinuous [24, Propositions 6.7–6.8 and Remark 15.10] and local [24,
Remark 15.25 and Proposition 16.15], while the set function
$\mathscr{F}_{\infty}(u,\cdot)$ is inner regular, increasing and superadditive
[24, Propositions 6.7, 16.12 and Remark 15.10]. Finally by means of the
fundamental estimate Proposition 4.1 we can appeal to [24, Proposition 18.4]
to deduce that $\mathscr{F}_{\infty}(u,\cdot)$ is also a subadditive set
function. This together with the measure-property criterion of De Giorgi and
Letta (see e.g., [24, Theorem 14.23]) imply that for every $u\in
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ the set function
$\mathscr{F}_{\infty}(u,\cdot)$ is the restriction to $\mathcal{A}$ of a Borel
measure.
Thus we can invoke [4, Theorem 3.1] (see also [14, Theorem 3]) and conclude
that that for every $A\in\mathcal{A}$ and $u\in BV(A;\\{0,1\\})$ we can
represent the $\Gamma$-limit $\mathscr{F}_{\infty}$ can be represented in an
integral form as

$\mathscr{F}_{\infty}(u,A)=\int_{S_{u}\cap A}\hat{f}(x,\nu_{u})\,{\rm
d}\mathcal{H}^{n-1}\,,$

with $\hat{f}$ given by (4.10). ∎

##  5\. Identification of the surface integrand

In this section we identify the surface integrand $\hat{f}$. Namely, we show
that $\hat{f}$ coincides with both $f^{\prime}$ and $f^{\prime\prime}$,
defined as in (2.4) and (2.5) with respect to the $\Gamma$-converging
subsequence.

######  Proposition 5.1.

Let $(f_{k})\subset\mathcal{F}$ ; let $(k_{j})$ and $\hat{f}$ be as in Theorem
4.2. Then, it holds

$\displaystyle\hat{f}(x,\nu)=f^{\prime}(x,\nu)=f^{\prime\prime}(x,\nu)\,,$

for every $x\in\mathbb{R}^{n}$ and $\nu\in{\mathbb{S}}^{n-1}$, where
$f^{\prime}$ and $f^{\prime\prime}$ are, respectively, as in (2.4) and (2.5),
with $k$ replaced by $k_{j}$.

###### Proof.

For notational simplicity, in what follows we still denote with $k$ the index
of the subsequence provided by Theorem 4.2.

By definition we have $f^{\prime}\leq f^{\prime\prime}$, hence to conclude we
need to show that $\hat{f}\leq f^{\prime}$ and $\hat{f}\geq f^{\prime\prime}$.
We divide the proof into two steps.

_Step 1:_ In this step we show that $\hat{f}(x,\nu)\leq f^{\prime}(x,\nu)$ for
every $x\in\mathbb{R}^{n}$ and $\nu\in{\mathbb{S}}^{n-1}$.

Let $\rho>0$ and $\eta>0$ be fixed, and for every $k$ choose
$u_{k}\in\mathscr{A}(\bar{u}_{x,\varepsilon_{k}}^{\nu},A)$ satisfying

$\displaystyle\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))$
$\displaystyle\leq\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))+\eta\rho^{n-1}$
(5.1) $\displaystyle\leq(c_{2}C_{{\rm u}}+\eta)\rho^{n-1}\,.$ (5.2)

Here the last inequality follows from (2.10). Up to estracting a subsequence
we may assume that

$\lim_{k\to+\infty}\mathscr{F}_{{k}}(u_{k},Q_{\rho}^{\nu}(x))=\liminf_{k\to+\infty}\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))\,.$

Then extend $u_{k}$ to a $W^{1,p}_{\mathrm{loc}}(\mathbb{R}^{n})$-function by
setting

$\displaystyle w_{k}:=\begin{cases}u_{k}&\text{in }Q_{\rho}^{\nu}(x)\,,\\\
\bar{u}_{x,\varepsilon_{k}}^{\nu}&\text{in }\mathbb{R}^{n}\setminus
Q_{\rho}^{\nu}(x)\,.\end{cases}$

From (5.2), (2.7) and [30, Lemma 4.1] there exists a subsequence (not
relabelled) such that

$w_{k}\to u\quad\text{in}\quad L^{p}_{\mathrm{loc}}(\mathbb{R}^{n})\,,$

for some $u\in L^{p}_{\mathrm{loc}}(\mathbb{R}^{n};\mathbb{R}^{m})\cap
BV(Q^{\nu}_{(1+\eta)\rho}(x);\\{0,1\\})$ such that $u=u_{x}^{\nu}$ in a
neighbourhood of $\partial Q^{\nu}_{(1+\eta)\rho}(x)$. Hence by
$\Gamma$-convergence, (2.10) and (5.1) we obtain

$\displaystyle\mathscr{F}_{\infty}(u,Q^{\nu}_{(1+\eta)\rho}(x))$
$\displaystyle\leq\liminf_{k\to+\infty}\mathscr{F}_{{k_{j}}}(w_{k},Q_{(1+\eta)\rho}^{\nu}(x))$
$\displaystyle\leq\lim_{k\to+\infty}\mathscr{F}_{k_{j}}(u_{k},Q_{\rho}^{\nu}(x))+c_{2}C_{\rm
u}\big{(}(1+\eta)^{n-1}-1)\rho^{n-1}$
$\displaystyle\leq\liminf_{k\to+\infty}\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))+\eta\rho^{n-1}+c_{2}C_{\rm
u}\big{(}(1+\eta)^{n-1}-1\big{)}\rho^{n-1}\,.$

Combining the above inequality with

$\mathbf{m}(u^{\nu}_{x},Q^{\nu}_{(1+\eta)\rho}(x))\leq\mathscr{F}_{\infty}(u,Q^{\nu}_{(1+\eta)\rho}(x))\,,$

dividing by $\rho^{n-1}$ and passing to the limsup as $\rho\to 0$ we infer

$(1+\eta)^{n-1}{\hat{f}}(x,\nu)\leq f^{\prime}(x,\nu)+\eta+c_{2}C_{\rm
u}\big{(}(1+\eta)^{n-1}-1\big{)}\,.$

Eventually we conclude by sending $\eta\to 0$.

_Step 2:_ In this step we show that $\hat{f}(x,\nu)\geq
f^{\prime\prime}(x,\nu)$ for every $x\in\mathbb{R}^{n}$,
$\nu\in{\mathbb{S}}^{n-1}$, and $\rho>0$.
We fix $\eta>0$ and choose $u\in BV(Q_{\rho}^{\nu}(x);\\{0,1\\})$ with
$u=u_{x}^{\nu}$ near $\partial Q_{\rho}^{\nu}(x)$ and

$\mathscr{F}_{\infty}(u,Q_{\rho}^{\nu}(x))\leq\mathbf{m}(u_{x}^{\nu},Q_{\rho}^{\nu}(x))+\eta\,.$
(5.3)

We extend $u$ to the whole $\mathbb{R}^{n}$ to the jump function $u_{x}^{\nu}$
in $\mathbb{R}^{n}\setminus Q_{\rho}^{\nu}(x)$. By $\Gamma$-convergence there
exists a sequence $u_{k}$ converging to $u$ in
$L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ such that

$\lim_{k\to+\infty}\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))=\mathscr{F}_{\infty}(u,Q_{\rho}^{\nu}(x))\,.$
(5.4)

We next properly modify the sequence $u_{k}$ in order to obtain a new sequence
$\hat{u}_{k}\in\mathscr{A}(\bar{u}_{x,\varepsilon_{k}}^{\nu},A)$. To this end
let $0<\rho^{\prime\prime}<\rho^{\prime}<\rho$ be such that $u=u_{x}^{\nu}$ on
$Q_{\rho}^{\nu}(x)\setminus\overline{Q}_{\rho^{\prime\prime}}^{\nu}(x)$. Then
we apply Proposition 4.1 with $A=Q_{\rho^{\prime\prime}}^{\nu}(x)$,
$A^{\prime}=Q_{\rho^{\prime}}^{\nu}(x)$,
$B=Q_{\rho}^{\nu}(x)\setminus\overline{Q}_{\rho^{\prime\prime}}^{\nu}(x)$ and
$u=u_{k}$, $v=\bar{u}^{\nu}_{x,\varepsilon_{k}}$. In this way to get a new
sequence $\hat{u}_{k}\subset W^{1,p}_{\mathrm{loc}}(\mathbb{R}^{n})$
converging to $u$ in $L^{p}(Q_{\rho}^{\nu}(x))$ such that

$\hat{u}_{k}=u_{k}\quad\text{in}\quad
Q_{\rho^{\prime\prime}}^{\nu}(x),\quad\hat{u}_{k}=\bar{u}_{x,\varepsilon_{k}}^{\nu}\quad\text{in}\quad
Q_{\rho}^{\nu}(x)\setminus\overline{Q}_{\rho^{\prime}}^{\nu}(x),$

and

$\begin{split}\limsup_{k\to+\infty}\mathscr{F}_{k}(\hat{u}_{k},Q_{\rho}^{\nu}(x))&\leq\limsup_{k\to+\infty}\Big{(}\mathscr{F}_{k}(u_{k},Q_{\rho^{\prime}}^{\nu}(x))+\mathscr{F}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x)\setminus\overline{Q}_{\rho^{\prime\prime}}^{\nu}(x))\Big{)}\\\
&\leq\mathscr{F}_{\infty}(u,Q_{\rho}^{\nu}(x))+c_{2}C_{\rm
u}\mathcal{L}^{n-1}\big{(}Q_{\rho}^{\prime}\setminus\overline{Q^{\prime}}_{\\!\\!\rho^{\prime\prime}}\big{)}\,.\end{split}$
(5.5)

Here the second inequality follows from (5.4) and (2.10) together with (2.6).

Recalling that $\hat{u}_{k}$ is a test function for the definition of
$\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))$,
combining (5.3) and (5.5) we get

$\displaystyle\limsup_{k\to+\infty}\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$
$\displaystyle\leq\mathbf{m}(u_{x}^{\nu},Q_{\rho}^{\nu}(x))+\eta+C(\rho^{n-1}-(\rho^{\prime\prime})^{n-1})\,.$

We divide the above inequality by $\rho^{n-1}$ and conclude by passing t to
the limit first as $\rho^{\prime\prime}\to\rho$, second as $\eta\to 0$ and
eventually to the limsup as $\rho\to 0$. ∎

##  6\. Properties of $f^{\prime},f^{\prime\prime}$

This section is devoted to prove some properties satisfied by the functions
$f^{\prime}$, and $f^{\prime\prime}$ defined in (2.4) and (2.5) respectively.
To this purpose is convenient to characterise them in an alternative way. Let
us fix first some notation. For $\rho,\delta,\varepsilon_{k}>0$ with
$\rho>\delta>2\varepsilon_{k}$ we set

$\mathbf{m}_{k}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x)):=\inf\\{\mathscr{F}_{k}(u,Q_{\rho}^{\nu}(x))\colon
u\in\mathscr{A}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))\\},$

where

$\mathscr{A}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x)):=\\{u\in
W^{1,p}(Q_{\rho}^{\nu}(x)),\;0\leq u\leq 1\colon
u=\bar{u}_{x,\varepsilon_{k}}^{\nu}\ \text{in}\
Q_{\rho}^{\nu}(x)\setminus\overline{Q}_{\rho-\delta}^{\nu}(x)\\}\,.$

Moreover, let
$f^{\prime}_{\rho},f^{\prime\prime}_{\rho}:\mathbb{R}^{n}\times\mathbb{S}^{n-1}\to[0,+\infty]$
be the functions defined as

$\begin{split}f^{\prime}_{\rho}(x,\nu)&:=\inf_{\delta>0}\liminf_{k\to+\infty}\mathbf{m}_{k}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))=\lim_{\delta\to
0}\liminf_{k\to+\infty}\mathbf{m}_{k}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))\,,\\\\[4.0pt]
f^{\prime\prime}_{\rho}(x,\nu)&:=\inf_{\delta>0}\limsup_{k\to+\infty}\mathbf{m}_{k}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))=\lim_{\delta\to
0}\limsup_{k\to+\infty}\mathbf{m}_{k}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))\,.\end{split}$
(6.1)

Next we prove the following technical Lemma.

######  Lemma 6.1.

Let $f^{\prime}_{\rho},f^{\prime\prime}_{\rho}$ be as in (6.1). Then the
following hold:

  1. $(i)$

The restrictions of $f^{\prime}_{\rho}$, $f^{\prime\prime}_{\rho}$ to the sets
$\mathbb{R}^{n}\times\widehat{\mathbb{S}}^{n-1}_{+}$ and
$\mathbb{R}^{n}\times\widehat{\mathbb{S}}^{n-1}_{-}$ are upper semicontinuous;

  2. $(ii)$

The functions $\rho\to(f^{\prime}_{\rho}(x,\nu)-c_{2}C_{\rm u}\rho^{n-1})$ and
$\rho\to(f^{\prime\prime}_{\rho}(x,\nu)-c_{2}C_{\rm u}\rho^{n-1})$ are
nonincreasing on $(0,+\infty)$;

  3. $(iii)$

For every $x\in\mathbb{R}^{n}$ and every $\nu\in\mathbb{S}^{n-1}$ there hold

$f^{\prime}(x,\nu)=\limsup_{\rho\to
0}\frac{1}{\rho^{n-1}}f_{\rho}^{\prime}(x,\nu)\;\text{ and
}\;f^{\prime\prime}(x,\nu)=\limsup_{\rho\to
0}\frac{1}{\rho^{n-1}}f_{\rho}^{\prime\prime}(x,\nu)\,,$

with $f^{\prime}$ and $f^{\prime\prime}$ as in (2.4) and (2.5) respectively.

###### Proof.

We prove the statement only for $f^{\prime}_{\rho}$ and we show $(i)$ only for
its restriction to the set
$\mathbb{R}^{n}\times\widehat{\mathbb{S}}^{n-1}_{+}$, the other cases can be
treated arguing similarly.

Step 1: we show the validity of $(i)$. Let $\rho>0$, $x\in\mathbb{R}^{n}$, and
$\nu\in\widehat{\mathbb{S}}^{n-1}_{+}$ be fixed. Let
$(x_{j},\nu_{j})\subset\mathbb{R}^{n}\times\widehat{\mathbb{S}}^{n-1}_{+}$ be
a sequence converging to $(x,\nu)$. We want to show that

$\displaystyle\limsup_{j\to+\infty}f_{\rho}^{\prime}(x_{j},\nu_{j})\leq
f_{\rho}^{\prime}(x,\nu)\,.$ (6.2)

To this aim we notice that (6.1) implies the following: for fixed $\eta>0$
there exists $\delta_{\eta}>0$ such that

$\liminf_{k\to+\infty}\mathbf{m}_{k}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))\leq
f_{\rho}^{\prime}(x,\nu)+\eta,$ (6.3)

for every $\delta\in(0,\delta_{\eta})$. Hence we can choose $\delta_{0}$ such
that $3\delta_{0}\in(0,{\delta_{\eta}})$ and (6.3) holds true with
$\delta=3\delta_{0}$. Let also
$u_{k}\in\mathscr{A}^{3\delta_{0}}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))$
that satisfies

$\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))\leq\mathbf{m}_{k}^{3\delta_{0}}(u_{x}^{\nu},Q_{\rho}^{\nu}(x))+\eta\,.$
(6.4)

In what follows we modify $u_{k}$ in order to obtain a new sequence
$\widetilde{u}_{k}\in\mathscr{A}^{\delta}(\bar{u}_{x_{j},\varepsilon_{k}}^{\nu_{j}},Q_{\rho}^{\nu_{j}}(x_{j}))$
for $j$ large enough and any $\delta\in(0,\delta_{0})$ without essentially
increasing the energy. We define

$h_{k,j}:=\varepsilon_{k}+|x-x_{j}|+\frac{\sqrt{n}}{2}\rho|\nu-\nu_{j}|$

and

$R_{k,j}:=R_{\nu}\Big{(}\big{(}Q^{\prime}_{\rho-2\delta_{0}}\setminus\overline{Q}^{\prime}_{\rho-3\delta_{0}}\big{)}\times\big{(}-h_{k,j},h_{k,j}\big{)}\Big{)}+x\,.$

Since the map $\nu\mapsto R_{\nu}$ is continuous, there exists
$\hat{\jmath}=\hat{\jmath}(\delta_{0})\in\mathbb{N}$ such that

$Q_{\rho-5\delta_{0}}^{\nu_{j}}(x_{j})\subset
Q_{\rho-4\delta_{0}}^{\nu}(x)\subset Q_{\rho-2\delta_{0}}^{\nu}(x)\subset
Q_{\rho-\delta_{0}}^{\nu_{j}}(x_{j})\,,$ (6.5)

and

$R_{k,j}\subset
Q_{\rho-2\delta_{0}}^{\nu}(x)\setminus\overline{Q}^{\nu}_{\rho-3\delta_{0}}(x),$

for every $j\geq\hat{\jmath}$. For later convenience we observe that

$u_{k}(y)=\bar{u}^{\nu}_{x,\varepsilon_{k}}(y)=\bar{u}^{\nu_{j}}_{x_{j},\varepsilon_{k}}(y)\in\\{0,1\\}\;\text{
for
a.e.}\;y\in\Big{(}Q_{\rho-2\delta_{0}}^{\nu}(x)\setminus\overline{Q}_{\rho-3\delta_{0}}^{\nu}(x)\Big{)}\setminus\overline{R}_{k,j}\,.$
(6.6)

Indeed we have

$\bar{u}^{\nu}_{x,\varepsilon_{k}}=u^{\nu}_{x}=u^{\nu_{j}}_{x_{j}}=\bar{u}^{\nu_{j}}_{x_{j},\varepsilon_{k}}\;\text{
in
}\;\\{(y-x)\cdot\nu\geq\varepsilon_{k}\\}\cap\\{(y-x_{j})\cdot\nu_{j}\geq\varepsilon_{k}\\}\,.$
(6.7)

On the other hand for
$y\in\Big{(}Q_{\rho-3\delta_{0}+2\varepsilon_{k}}^{\nu}(x)\setminus\overline{Q}_{\rho-3\delta_{0}}^{\nu}(x)\Big{)}\setminus\overline{R}_{k,j}$
it holds

$|(y-x)\cdot\nu|=|R_{\nu}^{T}(y-x)\cdot e_{n}|>h_{k,j}>\varepsilon_{k}\,,$

and by applying the triangular inequality twice we have

$\begin{split}|(y-x_{j})\cdot\nu_{j}|&>h_{k,j}-|x-x_{j}|-|y-x||\nu-\nu_{j}|\\\
&\geq\varepsilon_{k}+(\frac{\sqrt{n}}{2}\rho-|y-x|)|\nu-\nu_{j}|\geq\varepsilon_{k}\,,\end{split}$

where the last inequality is a consequence of $Q_{\rho}^{\nu}(x)\subset
B_{\frac{\sqrt{n}}{2}\rho}(x)$.
Next we apply Proposition 4.1 with

$A:=Q^{\nu}_{\rho-3\delta_{0}}(x)\,,\quad
A^{\prime}:=Q^{\nu}_{\rho-2\delta_{0}}(x)\,,\quad
B:=Q_{\rho}^{\nu_{j}}(x_{j})\setminus\overline{Q}^{\nu_{j}}_{\rho-5\delta_{0}}(x_{j})\,,$

and $u:=u_{k}$, $v:=\bar{u}^{\nu_{j}}_{x_{j},\varepsilon_{k}}$. Note that
thanks to (6.5) the following hold

$A\cup B=Q_{\rho}^{\nu_{j}}(x_{j})\,,\quad
B\setminus\overline{A}^{\prime}=Q_{\rho}^{\nu_{j}}(x_{j})\setminus\overline{Q}^{\nu}_{\rho-2\delta_{0}}(x)\,,$

and

$(A^{\prime}\setminus\overline{A})\cap
B=A^{\prime}\setminus\overline{A}=Q^{\nu}_{\rho-2\delta_{0}}(x)\setminus\overline{Q}^{\nu}_{\rho-3\delta_{0}}(x)\,.$
(6.8)

In particular Proposition 4.1 $(i)$ provides us with $\widetilde{u}_{k}\in
W^{1,p}(Q_{\rho}^{\nu_{j}}(x_{j}))$, $0\leq\widetilde{u}_{k}\leq 1$ with

$\widetilde{u}_{k}=u_{k}\quad\text{ a.e. in
}Q^{\nu}_{\rho-3\delta_{0}}(x)\,,\quad\widetilde{u}_{k}=\bar{u}^{\nu_{j}}_{x_{j},\varepsilon_{k}}\quad\text{
a.e. in
}Q_{\rho}^{\nu_{j}}(x_{j})\setminus\overline{Q}^{\nu}_{\rho-2\delta_{0}}(x)\,,$
(6.9)

and such that

$\begin{split}\mathscr{F}_{k}(\widetilde{u}_{k},Q_{\rho}^{\nu_{j}}(x_{j}))\leq(1+K\varepsilon_{k})\Big{(}\mathscr{F}_{k}(u_{k},Q^{\nu}_{\rho}(x))+\mathscr{F}_{k}(\bar{u}^{\nu_{j}}_{x_{j},\varepsilon_{k}}&,Q_{\rho}^{\nu_{j}}(x_{j})\setminus\overline{Q}^{\nu_{j}}_{\rho-5\delta_{0}}(x_{j}))\Big{)}\\\
&+\omega_{k}(u_{k},u^{\nu_{j}}_{x_{j},\varepsilon_{k}},A,A^{\prime},B)\,.\end{split}$
(6.10)

Moreover from (6.8), property (6.6) becomes

$u_{k}(y)=u^{\nu_{j}}_{x_{j},\varepsilon_{k}}(y)\in\\{0,1\\}\quad\text{ for
a.e. $y$ in }((A^{\prime}\setminus\overline{A})\cap B)\setminus R_{k,j}\,.$

Thus, applying Proposition 4.1 $(ii)$ with
$\widetilde{S}=((A^{\prime}\setminus\overline{A})\cap B)\setminus R_{k,j}$
yields

$\omega_{k}(u_{k},u^{\nu_{j}}_{x_{j},\varepsilon_{k}},A,A^{\prime},B)\leq
K\mathcal{L}^{n}(R_{k,j})\,.$ (6.11)

Now (2.10) together with (2.6) give

$\mathscr{F}_{k}(\bar{u}^{\nu_{j}}_{x_{j},\varepsilon_{k}},Q_{\rho}^{\nu_{j}}(x_{j})\setminus\overline{Q}^{\nu_{j}}_{\rho-5\delta_{0}}(x_{j}))\leq
c_{2}C_{{\rm
u}}\mathcal{L}^{n-1}\Big{(}Q^{\prime}_{\rho}\setminus\overline{Q^{\prime}}_{\\!\\!\rho-5\delta_{0}}\big{)}\Big{)}\leq
C\delta_{0}\rho^{n-2}\,,$ (6.12)

while

$\mathcal{L}^{n}(R_{k,j})\leq C\delta_{0}\rho^{n-2}h_{k,j}\,.$ (6.13)

By (6.9) we have
$\widetilde{u}_{k}\in\mathscr{A}^{\delta}(\bar{u}_{x_{j},\varepsilon_{k}}^{\nu_{j}},Q_{\rho}^{\nu_{j}}(x_{j}))$
for any $j\geq\hat{\jmath}$ and any $\delta\in(0,\delta_{0})$. Hence gathering
(6.4), (6.10), (6.11),(6.12), and (6.13) we get

$\begin{split}\mathbf{m}_{k}^{\delta}(\bar{u}_{x_{j},\varepsilon_{k}}^{\nu_{j}},Q_{\rho}^{\nu_{j}}(x_{j}))\leq(1+C\varepsilon_{k})(\mathbf{m}_{k}^{3\delta_{0}}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))+\eta)+C\rho^{n-2}\delta_{0}(1+h_{k,j})\,.\end{split}$
(6.14)

Using (6.3) with $\delta=3\delta_{0}$ and passing to the limit in (6.14) first
as $k\to+\infty$, then as $\delta\to 0$, and finally as $j\to+\infty$ gives

$\displaystyle\limsup_{j\to+\infty}f_{\rho}^{\prime}(x_{j},\nu_{j})\leq
f_{\rho}^{\prime}(x,\nu)+2\eta+C\delta_{0}\rho^{n-2}\,,$

since $\lim_{j}\lim_{k}h_{k,j}=0$. By letting $\delta_{0}\to 0$ (6.2) and then
$\eta\to 0$ we finally conclude.

Step 2: we show $(ii)$. Let $0<\rho\leq\rho^{\prime}$, $x\in\mathbb{R}^{n}$,
$\nu\in S^{n-1}$ be fixed. We show that

$f^{\prime}_{\rho^{\prime}}(x,\nu)-c_{2}C_{\rm u}(\rho^{\prime})^{n-1}\leq
f^{\prime}_{\rho}(x,\nu)-c_{2}C_{\rm u}(\rho)^{n-1}\,.$ (6.15)

Let $\eta>0$ be fixed and let $\delta_{\eta}>0$ be such that (6.3) holds true
for every $\delta\in(0,\delta_{\eta})$. For fixed $\delta\in(0,\delta_{\eta})$
let also
$u_{k}\in\mathscr{A}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho^{\prime}}^{\nu}(x))$
that satisfies

$\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))\leq\mathbf{m}_{k}^{\delta}(u_{x}^{\nu},Q_{\rho}^{\nu}(x))+\eta\,.$
(6.16)

we now extend $u_{k}$, without relabelling it, to
$\bar{u}^{\nu}_{x,\varepsilon_{k}}$ in
$Q^{\nu}_{\rho^{\prime}}(x)\setminus\overline{Q}^{\nu}_{\rho}(x)$ so that it
belongs to the class
$\mathscr{A}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho^{\prime}}^{\nu}(x))$.
Then (2.6) and (2.10) yield

$\begin{split}\mathscr{F}_{k}(u_{k},Q_{\rho^{\prime}}^{\nu}(x))&\leq\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))+\mathscr{F}_{k}\big{(}\bar{u}_{x,\varepsilon_{k}}^{\nu},(Q_{\rho^{\prime}}^{\nu}(x)\setminus\overline{Q}_{\rho}^{\nu}(x)\big{)}\\\
&\leq\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))+c_{2}C_{\rm
u}((\rho^{\prime})^{n-1}-(\rho)^{n-1})\,.\end{split}$ (6.17)

Finally letting first $k\to\infty$ and then $\delta\to 0$ from (6.3) and
(6.16) we get

$f^{\prime}_{\rho^{\prime}}(x,\nu)\leq f^{\prime}_{\rho}(x,\nu)+c_{2}C_{\rm
u}((\rho^{\prime})^{n-1}-(\rho)^{n-1})+2\eta\,.$

Eventually by sending $\eta\to 0$ we deduce (6.15).

Step 3: we show $(iii)$. One inequality follows immediately. Indeed as
$\mathscr{A}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))\subset\mathscr{A}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))$
for every $\delta\in(0,\rho)$, we have

$f^{\prime}(x,\nu)\leq\limsup_{\rho\to
0}\frac{1}{\rho^{n-1}}f_{\rho}^{\prime}(x,\nu)\,.$

It remains to prove the opposite inequality. For fixed $\rho>0$,
$x\in\mathbb{R}^{n}$, and $\nu\in\mathbb{S}^{n-1}$ and for every
$k\in\mathbb{N}$ such that $\varepsilon_{k}\in(0,\frac{\rho}{2})$ let
$u_{k}\in\mathscr{A}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))$
satisfy

$\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))\leq\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))+\rho^{n}\,,$
(6.18)

Fix $\alpha>0$, let $\rho_{\alpha}:=(1+\alpha)\rho$, and extend $u_{k}$,
without relabelling it, to $\bar{u}_{x,\varepsilon_{k}}^{\nu}$ in
$Q_{\rho_{\alpha}}^{\nu}(x)\setminus\overline{Q}_{\rho}^{\nu}(x)$. In this way
$u_{k}\in\mathscr{A}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho_{\alpha}}^{\nu}(x))$
for any $\delta\in(0,\alpha\rho)$ and similarly to (6.17) it holds

$\begin{split}\mathscr{F}_{k}(u_{k},Q_{\rho_{\alpha}}^{\nu}(x))&\leq\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))+\mathscr{F}_{k}\big{(}\bar{u}_{x,\varepsilon_{k}}^{\nu},(Q_{\rho_{\alpha}}^{\nu}(x)\setminus\overline{Q}_{\rho}^{\nu}(x)\big{)}\\\
&\leq\mathscr{F}_{k}(u_{k},Q_{\rho}^{\nu}(x))+c_{2}C_{\rm
u}((1+\alpha)^{n-1}-1))\rho^{n-1}\,.\end{split}$ (6.19)

Now (6.18) and (6.19) give

$\inf_{\delta>0}\liminf_{k\to+\infty}\mathbf{m}_{k}^{\delta}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{(1+\alpha)\rho}^{\nu}(x))\leq\liminf_{k\to+\infty}\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q_{\rho}^{\nu}(x))+\rho^{n}+c_{2}C_{\rm
u}((1+\alpha)^{n-1}-1))\rho^{n-1}\,.$

Rescaling the above inequality by $((1+\alpha)\rho)^{n-1}$ and passing to the
limsup as $\rho\to 0$ we infer

$(1+\alpha)^{n-1}\limsup_{\rho\to
0}\frac{1}{\rho^{n-1}}f_{\rho}^{\prime}(x,\nu)\leq
f^{\prime}(x,\nu)+c_{2}C_{\rm u}((1+\alpha)^{n-1}-1))\,.$

We finally conclude by the arbitrariness of $\alpha>0$. ∎

We now state the main result of this section:

######  Proposition 6.2.

Let $(f_{k})\subset\mathcal{F}$; then the functions $f^{\prime}$ and
$f^{\prime\prime}$ defined, respectively, as in (2.4) and (2.5) are Borel
measurable and satisfy the following: for every $x\in\mathbb{R}^{n}$ and every
$\nu\in{\mathbb{S}}^{n-1}$ it holds

$c_{1}c_{p}\leq f^{\prime}(x,\nu)\leq c_{2}c_{p},\quad c_{1}c_{p}\leq
f^{\prime\prime}(x,\nu)\leq c_{2}c_{p},$ (6.20)

with

$c_{p}:=p(p-1)^{\frac{1-p}{p}}\int_{0}^{1}W(t)^{\frac{p-1}{p}}\,{\rm d}t\,.$

###### Proof.

We prove the statement only for $f^{\prime}$, as the proof for
$f^{\prime\prime}$ can be achieved in a similar way.

Step 1: we show that $f^{\prime}$ is Borel measurable. Let $\rho>0$ and let
$f^{\prime}_{\rho}$ be the function defined in (6.1). By Lemma 6.1 $(ii)$ we
have that the function $\rho\to f^{\prime}_{\rho}(x,\nu)-c_{2}C_{\rm
u}\rho^{n-1}$ is nonincreasing on $(0,+\infty)$, hence in particular

$\lim_{\rho^{\prime}\to\rho^{-}}f^{\prime}_{\rho^{\prime}}(x,\nu)\geq
f^{\prime}_{\rho}(x,\nu)\geq\lim_{\rho^{\prime}\to\rho^{+}}f^{\prime}_{\rho^{\prime}}(x,\nu),$

for every $x\in\mathbb{R}^{n}$, $\nu\in{\mathbb{S}}^{n-1}$, and every
$\rho>0$. This together with Lemma 6.1 $(iii)$ imply that

$f^{\prime}(x,\nu)=\limsup_{\rho\to
0}\frac{1}{\rho^{n-1}}f^{\prime}_{\rho}(x,\nu)=\limsup_{\begin{subarray}{c}\rho\to
0\\\ \rho\in D\end{subarray}}\frac{1}{\rho^{n-1}}f^{\prime}_{\rho}(x,\nu)$

where $D$ is any countable dense subset of $(0,+\infty)$. Since by Lemma 6.1
$(i)$ the function $(x,\nu)\mapsto f^{\prime}_{\rho}(x,\nu)$ is Borel
measurable for every $\rho>0$, and the limit on a countable set of Borel
functions is Borel we conclude.

Step 2: we show that $f^{\prime}$ is bounded. For every $x\in\mathbb{R}^{n}$,
$\nu\in\mathbb{S}^{n-1}$, and $\rho>0$ we define the minimisation problem

$\mathbf{m}_{k,\rho}(x,\nu):=\min\bigg{\\{}\mathcal{M}_{k}(u,Q^{\nu}_{\rho}(x))\colon
u\in\mathscr{A}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q^{\nu}_{\rho}(x))\bigg{\\}},$

where $\mathcal{M}_{k}$ is defined as in (2.8). From Remark 2.1 $(b)$ we have

$c_{1}\mathbf{m}_{k,\rho}(x,\nu)\leq\mathbf{m}_{k}(\bar{u}_{x,\varepsilon_{k}}^{\nu},Q^{\nu}_{\rho}(x))\leq
c_{2}\mathbf{m}_{k,\rho}(x,\nu).$

Therefore to prove that $f^{\prime}$ satisfies the bounds in (6.20) it is
enough to show that

$\lim_{k\to+\infty}\mathbf{m}_{k,\rho}(x,\nu)=c_{p}\rho^{n-1}.$ (6.21)

By the homogeneity and rotation invariance of the Modica-Mortola functional
there holds

$\mathbf{m}_{k,\rho}(x,\nu)=\mathbf{m}_{k,\rho}(0,e_{n}).$

Let $\eta>0$ be arbitrary fixed. By the $\Gamma$-convergence result in [30] we
can find a sequence $(u_{k})\subset W^{1,p}(Q_{\rho}(0))$, $0\leq u_{k}\leq 1$
such that $u_{k}$ converges to $u_{0}^{e_{n}}$ in $L^{p}(Q_{\rho}(0))$ and

$\limsup_{k\to+\infty}\mathcal{M}_{k}(u_{k},Q_{\rho}(0))=(c_{p}+\eta)\rho^{n-1}\,.$

Up to suitably modifying $u_{k}$ (for example by using Proposition 4.1), we
can assume that $u_{k}=u_{0}^{e_{n}}$ in $\\{|y_{n}|>\varepsilon_{k}\\}$ so
that in particular
$(u_{k})\subset\mathscr{A}(\bar{u}_{0,\varepsilon_{k}}^{e_{n}},Q_{\rho}(0))$.
In particular, by the arbitrariness of $\eta>0$ we deduce that

$\limsup_{k\to+\infty}\mathbf{m}_{k,\rho}(x,\nu)=\limsup_{k\to+\infty}\mathbf{m}_{k,\rho}(0,e_{n})\leq
c_{p}\rho^{n-1}\,.$ (6.22)

On the other hand by Fubini’s Theorem we have

$\begin{split}\mathcal{M}_{k}(u,Q_{\rho}(0))&=\int_{Q^{\prime}_{\rho}}\int_{-\frac{\rho}{2}}^{\frac{\rho}{2}}\bigg{(}\frac{W(u(x^{\prime},x_{n}))}{\varepsilon_{k}}+\varepsilon_{k}^{p-1}|\nabla
u(x^{\prime},x_{n})|^{p}\bigg{)}\,{\rm d}x_{n}\,{\rm d}x^{\prime}\\\
&\geq\int_{Q^{\prime}_{\rho}}\int_{-\frac{\rho}{2}}^{\frac{\rho}{2}}\bigg{(}\frac{W(u(x^{\prime},x_{n}))}{\varepsilon_{k}}+\varepsilon_{k}^{p-1}\Big{|}\frac{\partial
u(x^{\prime},x_{n})}{\partial x_{n}}\Big{|}^{p}\bigg{)}\,{\rm d}x_{n}\,{\rm
d}x^{\prime}\,,\end{split}$ (6.23)

and by Young’s inequality

$\begin{split}\int_{-\frac{\rho}{2}}^{\frac{\rho}{2}}\Bigg{(}\frac{W(u(x^{\prime},x_{n}))}{\varepsilon_{k}}&+\varepsilon_{k}^{p-1}\bigg{|}\frac{\partial
u(x^{\prime},x_{n})}{\partial x_{n}}\bigg{|}^{p}\Bigg{)}\,{\rm d}x_{n}\\\
\geq&\Big{(}\frac{p}{p-1}\Big{)}^{\frac{p-1}{p}}p^{\frac{1}{p}}\int_{-\frac{\rho}{2}}^{\frac{\rho}{2}}\Bigg{(}(W(u(x^{\prime},x_{n})))^{\frac{p-1}{p}}\bigg{|}\frac{\partial
u(x^{\prime},x_{n})}{\partial x_{n}}\bigg{|}\Bigg{)}\,{\rm
d}x_{n}\,,\end{split}$ (6.24)

for $\mathcal{L}^{n-1}$-a.e. $x^{\prime}\in Q^{\prime}_{\rho}$. Now if
$u\in\mathscr{A}((\bar{u}_{0,\varepsilon_{k}}^{e_{n}},Q_{\rho}(0))$, then it
coincides with $u_{0}^{e_{n}}$ in a neighbourhood of

$\partial^{\pm}Q_{\rho}(0):=\Big{\\{}(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}\colon
x^{\prime}\in\overline{Q^{\prime}_{\rho}},\;x_{n}=\pm\frac{\rho}{2}\Big{\\}}\,.$

Therefore, for $\mathcal{L}^{n-1}$-a.e. $x^{\prime}\in Q^{\prime}_{\rho}$, the
function $u_{x^{\prime}}(t):=u(x^{\prime},t)$ belongs to
$W^{1,p}(-\frac{\rho}{2},\frac{\rho}{2})$ and satisfies
$u_{x^{\prime}}(-\frac{\rho}{2})=0$, and $u_{x^{\prime}}(\frac{\rho}{2})=1$,
and a change of variable in (6.24) yields

$\displaystyle\int_{-\frac{\rho}{2}}^{\frac{\rho}{2}}\bigg{(}\frac{W(u(x^{\prime},x_{n}))}{\varepsilon_{k}}+\varepsilon_{k}^{p-1}\bigg{|}\frac{\partial
u(x^{\prime},x_{n})}{\partial x_{n}}\bigg{|}^{p}\bigg{)}\,{\rm
d}x_{n}\geq\Big{(}\frac{p}{p-1}\Big{)}^{\frac{p-1}{p}}p^{\frac{1}{p}}\int_{0}^{1}W(t)^{\frac{p-1}{p}}\,{\rm
d}t=c_{p}\,.$ (6.25)

Thus, combining (6.23) with (6.25) we obtain

$\mathcal{M}_{k}(u,Q_{\rho}(0))\geq c_{p}\rho^{n-1},$

for every $u\in\mathscr{A}((\bar{u}_{0,\varepsilon_{k}}^{e_{n}},Q_{\rho}(0))$.
Passing to the infimum on $u$ and to the liminf as $k\to+\infty$ we get

$\liminf_{k\to+\infty}\mathbf{m}_{k,\rho}(x,\nu)=\liminf_{k\to+\infty}\mathbf{m}_{k,\rho}(0,e_{n})\geq
c_{p}\rho^{n-1},$ (6.26)

for every $\rho>0$. Eventually, gathering (6.22) and (6.26) we get (6.21), and
hence the thesis. ∎

We now have all the ingredients to prove the main result of this paper,
namely, Theorem 3.1.

###### Proof of Theorem 3.1.

The proof follows by combining Theorem 4.2, Proposition 5.1 and Proposition
6.2. ∎

##  7\. Stochastic homogenisation

In this section we derive a $\Gamma$-convergence result for functionals of
type $\mathscr{F}_{k}$ when $f_{k}$ are random integrands of the form

$f_{k}(\omega,x,u,\xi)=f\Big{(}\omega,\frac{x}{\varepsilon_{k}},u,\xi\Big{)},$

where $\omega$ belongs to the sample space $\Omega$ of a complete probability
space $(\Omega,\mathcal{T},P)$.

In order to do that we need to recall some useful definitions.

######  Definition 7.1 (Group of $P$-preserving transformations).

Let $d\in\mathbb{N}$, $d\geq 1$. A group of $P$-preserving transformations on
$(\Omega,\mathcal{T},P)$ is a family $(\tau_{z})_{z\in\mathbb{Z}^{d}}$ of
mappings $\tau_{z}\colon\Omega\to\Omega$ satisfying the following properties:

  1. (1)

(measurability) $\tau_{z}$ is $\mathcal{T}$-measurable for every
$z\in\mathbb{Z}^{d}$;

  2. (2)

(invariance) $P(\tau_{z}(E))=P(E)$, for every $E\in\mathcal{T}$ and every
$z\in\mathbb{Z}^{d}$;

  3. (3)

(group property) $\tau_{0}={\rm id}_{\Omega}$ and
$\tau_{z+z^{\prime}}=\tau_{z}\circ\tau_{z^{\prime}}$ for every
$z,z^{\prime}\in\mathbb{Z}^{d}$.

If, in addition, every $(\tau_{z})_{z\in\mathbb{Z}^{d}}$-invariant set (i.e.,
every $E\in\mathcal{T}$ with $\tau_{z}(E)=E$ for every $z\in\mathbb{Z}^{d}$)
has probability 0 or 1, then $(\tau_{z})_{z\in\mathbb{Z}^{d}}$ is called
ergodic.

Let $a:=(a_{1},\dots,a_{d}),\,b:=(b_{1},\dots,b_{d})\in\mathbb{Z}^{d}$ with
$a_{i}<b_{i}$ for all $i\in\\{1,\ldots,d\\}$; we define the $d$-dimensional
interval

$[a,b):=\\{x\in\mathbb{Z}^{d}\colon a_{i}\leq
x_{i}<b_{i}\,\,\text{for}\,\,i=1,\ldots,d\\}$

and we set

$\mathcal{I}_{d}:=\\{[a,b)\colon
a,b\in\mathbb{Z}^{d}\,,\,a_{i}<b_{i}\,\,\text{for}\,\,i=1,\ldots,d\\}\,.$

######  Definition 7.2 (Subadditive process).

A discrete subadditive process with respect to a group
$(\tau_{z})_{z\in\mathbb{Z}^{d}}$ of $P$-preserving transformations on
$(\Omega,\mathcal{T},P)$ is a function
$\mu\colon\Omega\times\mathcal{I}_{d}\to\mathbb{R}$ satisfying the following
properties:

  1. (1)

(measurability) for every $A\in\mathcal{I}_{d}$ the function
$\omega\mapsto\mu(\omega,A)$ is $\mathcal{T}$-measurable;

  2. (2)

(covariance) for every $\omega\in\Omega$, $A\in\mathcal{I}_{d}$, and
$z\in\mathbb{Z}^{d}$ we have $\mu(\omega,A+z)=\mu(\tau_{z}(\omega),A)$;

  3. (3)

(subadditivity) for every $A\in\mathcal{I}_{d}$ and for every finite family
$(A_{i})_{i\in I}\subset\mathcal{I}_{d}$ of pairwise disjoint sets such that
$A=\cup_{i\in I}A_{i}$, we have

$\mu(\omega,A)\leq\sum_{i\in I}\mu(\omega,A_{i})\quad\text{for
every}\,\,\omega\in\Omega\,;$

  4. (4)

(boundedness) there exists $c>0$ such that $0\leq\mu(\omega,A)\leq
c\mathcal{L}^{d}(A)$ for every $\omega\in\Omega$ and $A\in\mathcal{I}_{d}$.

######  Definition 7.3 (Stationarity).

Let $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ be a group of $P$-preserving
transformations on $(\Omega,\mathcal{T},P)$. We say that
$f\colon\Omega\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}\to[0,+\infty)$
is stationary with respect to $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ if

$f(\omega,x+z,u,\xi)=f(\tau_{z}(\omega),x,u,\xi)$

for every $\omega\in\Omega$, $x\in\mathbb{R}^{n}$, $z\in\mathbb{Z}^{n}$,
$u\in[0,1]$ and $\xi\in\mathbb{R}^{n}$.

We consider random integrands
$f\colon\Omega\times\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}\to[0,+\infty)$
satisfying the following assumptions:

  1. ($F1$)

$f$ is
$(\mathcal{T}\otimes\mathcal{B}^{n}\otimes\mathcal{B}\otimes\mathcal{B}^{n})$-measurable;

  2. ($F2$)

$f(\omega,\,\cdot\,,\,\cdot\,,\,\cdot\,)\in\mathcal{F}$ for every
$\omega\in\Omega$;

  3. ($F3$)

For every $\omega\in\Omega$, and every $x\in\mathbb{R}^{n}$ the map
$(u,\xi)\mapsto f(\omega,x,u,\xi)$ is lower semicontinuous.

For any $f$ as above we define the sequence of random phase-field functionals
$\mathscr{F}_{k}(\omega)\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\times\mathcal{A}\longrightarrow[0,+\infty]$
given by

$\mathscr{F}_{k}(\omega)(u,A):=\frac{1}{\varepsilon_{k}}\int_{A}f\left(\omega,\frac{x}{\varepsilon_{k}},u,\varepsilon_{k}\nabla
u\right)\,{\rm d}x\,,$ (7.1)

if $u\in W^{1,p}(A)$, $0\leq u\leq 1$ and extended to $+\infty$ otherwise. For
$\omega\in\Omega$, $A\in\mathcal{A}$ we define

$\mathscr{F}(\omega)(u,A):=\int_{A}f(\omega,x,u,\nabla u)\,{\rm d}x\,,$

and

$\mathbf{m}_{\omega}(\bar{u}^{\nu}_{z},A):=\inf\Bigl{\\{}\mathscr{F}(\omega)(u,A)\colon
u\in\mathscr{A}(\bar{u}^{\nu}_{z},A)\Bigr{\\}}\,,$ (7.2)

where $\mathscr{A}(\bar{u}^{\nu}_{z},A)$ is as in (2.3), with
$\bar{u}^{\nu}_{x,\varepsilon_{k}}$ replaced by $\bar{u}^{\nu}_{z}$; i.e.,
with $\varepsilon_{k}=1$. Moreover for every $A\subset\mathbb{R}^{n}$ with
$\rm int\,A\in\mathcal{A}$ we set

$\mathbf{m}_{\omega}(\bar{u}_{x}^{\nu},A):=\mathbf{m}_{\omega}(\bar{u}_{x}^{\nu},\rm
int\,A),\quad\text{and}\quad\mathscr{A}(\bar{u}_{0}^{\nu},A):=\mathscr{A}(\bar{u}_{0}^{\nu},\rm
int\,A).$

Finally we are able to state the main result of this section.

######  Theorem 7.4 (Stochastic homogenisation).

Let $f$ be a random integrand satisfying ($F1$)-($F3$). Assume also that $f$
is stationary with respect to a group $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ of
$P$-preserving transformations on $(\Omega,\mathcal{T},P)$. For every
$\omega\in\Omega$ let $\mathscr{F}_{k}(\omega)$ be as in (7.1) and
$\mathbf{m}_{\omega}$ be as in (7.2). Then there exists
$\Omega^{\prime}\in\mathcal{T}$, with $P(\Omega^{\prime})=1$ such that for
every $\omega\in\Omega^{\prime}$, $x\in\mathbb{R}^{n}$,
$\nu\in{\mathbb{S}}^{n-1}$ the limit

$\lim_{r\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}^{\nu}_{rx},Q^{\nu}_{r}(rx))}{r^{n-1}}=\lim_{r\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))}{r^{n-1}}=:f_{\mathrm{hom}}(\omega,\nu)$
(7.3)

exists and is independent of $x$. The function
$f_{\mathrm{hom}}\colon\Omega\times{\mathbb{S}}^{n-1}\to[0,+\infty)$ is
$(\mathcal{T}\otimes\mathcal{B}({\mathbb{S}}^{n-1}))$-measurable.

Moreover, for every $\omega\in\Omega^{\prime}$ and for every $A\in\mathcal{A}$
the functionals $\mathscr{F}_{k}(\omega)(\cdot,A)$ $\Gamma$-converge in
$L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ to the functional
$\mathscr{F}_{\mathrm{hom}}(\omega)(\cdot,A)$ with
$\mathscr{F}_{\mathrm{hom}}(\omega)\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\times\mathcal{A}\longrightarrow[0,+\infty]$
given by

$\mathscr{F}_{\rm hom}(\omega)(u,A):=\begin{cases}\displaystyle\int_{S_{u}\cap
A}f_{\rm hom}(\omega,\nu_{u})\,{\rm d}\mathcal{H}^{n-1}&\text{if}\ u\in
BV(A;\\{0,1\\})\\\\[4.0pt] +\infty&\text{otherwise}\,.\end{cases}$

If, in addition, $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ is ergodic, then
$f_{\mathrm{hom}}$ is independent of $\omega$ and

$\displaystyle f_{\rm hom}(\nu)$
$\displaystyle=\lim_{r\to+\infty}\frac{1}{r^{n-1}}\int_{\Omega}\mathbf{m}_{\omega}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))\,{\rm
d}P(\omega)\,,$

thus, $\mathscr{F}_{\mathrm{hom}}$ is deterministic.

######  Remark 7.5.

The notion of stationarity given in Definition 7.3 generalises the notion of
spatial periodicity. Therefore Theorem 7.4 applies also to the case of
deterministic periodic homogenisation. However we will show later (see Section
8) that condition ($F3$) is not needed if we restrict to the periodic setting.

The rest of the section is dedicated to prove Theorem 7.4.

###  7.1. Existence of the limit

The almost sure $\Gamma$-convergence result in Theorem 7.4 readily follows by
Theorem 3.3 if there exists a $\mathcal{T}$-measurable set
$\Omega^{\prime}\subset\Omega$, with $P(\Omega^{\prime})=1$, such that for
every $\omega\in\Omega^{\prime}$ the limit in (7.3) exists and is independent
of $x$, and if $f_{\mathrm{hom}}$ is
$(\mathcal{T}\otimes\mathcal{B}({\mathbb{S}}^{n-1}))$-measurable. We will
prove that this is actually the case when the integrand $f$ satisfies
($F1$)-($F3$) and is stationary in the sense of Definition 7.3.

Since the boundary conditions given in the definition of
$\mathbf{m}_{\omega}(\bar{u}_{rx}^{\nu},Q_{r}^{\nu}(rx))$ depend on $x$, the
proof of the existence and $x$-homogeneity of the limit in (7.3) does not
follow by a direct application of the Subadditive Ergodic Theorem [1, Theorem
2.4]. For this reason we need to follow a more technical argument, in the same
way as in [3, 18, 15], which can be divided into three main steps. In the
first step we prove that when $x=0$ the minimisation problem (7.2) defines a
subadditive process on $\Omega\times\mathcal{I}_{n-1}$ (see Proposition 7.6).
In the second step we prove the almost sure existence of the limit in (7.3)
when $x=0$ (see Proposition 7.7). Eventually in the third one we show that the
same holds for an arbitrary $x\in\mathbb{R}^{n}$ (see Proposition 7.8).

For $\nu\in{\mathbb{S}}^{n-1}$ we let $R_{\nu}$ be an orthogonal matrix as in
(g). Note that $\\{R_{\nu}e_{i}\colon i=1\,,\ldots\,,n-1\\}$ is an orthonormal
basis for $\Pi^{\nu}$, and $R_{\nu}\in\mathbb{Q}^{n\times n}$, if
$\nu\in{\mathbb{S}}^{n-1}\cap\mathbb{Q}^{n}$. Let $M_{\nu}>2$ be an integer
such that $M_{\nu}R_{\nu}\in\mathbb{Z}^{n\times n}$; therefore
$M_{\nu}R_{\nu}(z^{\prime},0)\in\Pi^{\nu}\cap\mathbb{Z}^{n}$ for every
$z^{\prime}\in\mathbb{Z}^{n-1}$.

Let $I\in\mathcal{I}_{n-1}$; i.e., $I=[a,b)$ with $a,b\in\mathbb{Z}^{n-1}$.
For every $\nu\in\mathbb{S}^{n-1}$ and every $I\in\mathcal{I}_{n-1}$ we define
the $n$-dimensional interval $I_{\nu}$ as

$I_{\nu}:=M_{\nu}R_{\nu}\big{(}I\times[-c,c)\big{)}\quad\text{where}\quad
c:=\frac{1}{2}\max_{i=1,\ldots,n-1}(b_{i}-a_{i})\,.$ (7.4)

Eventually, for fixed $\nu\in{\mathbb{S}}^{n-1}\cap\mathbb{Q}^{n}$ we consider
the function $\mu_{\nu}\colon\Omega\times\mathcal{I}_{n-1}\to\mathbb{R}$ given
by

$\mu_{\nu}(\omega,I):=\frac{1}{M_{\nu}^{n-1}}\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},I_{\nu})\,,$
(7.5)

where $\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},I_{\nu})$ is as in (7.2) with
$x=0$ and $A=I_{\nu}$. Then the following result holds true.

######  Proposition 7.6.

Let $f$ satisfy ($F1$)-($F3$) and assume that it is stationary with respect to
a group $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ of $P$-preserving transformations on
$(\Omega,\mathcal{T},P)$. Let $\nu\in{\mathbb{S}}^{n-1}\cap\mathbb{Q}^{n}$ and
let $\mu_{\nu}\colon\Omega\times\mathcal{I}_{n-1}\mapsto\mathbb{R}$ be as in
(7.5). Then there exists a group of $P$-preserving transformations
$(\tau_{z^{\prime}}^{\nu})_{z^{\prime}\in\mathbb{Z}^{n-1}}$ on
$(\Omega,\mathcal{T},P)$ such that $\mu_{\nu}$ is a subadditive process on
$(\Omega,\mathcal{T},P)$ with respect to
$(\tau_{z^{\prime}}^{\nu})_{z^{\prime}\in\mathbb{Z}^{n-1}}$. Moreover, it
holds

$0\leq\mu_{\nu}(\omega,I)\leq c_{2}C_{\rm u}\mathcal{L}^{n-1}(I),$ (7.6)

for $P$-a.e. $\omega\in\Omega$ and for every $I\in\mathcal{I}_{n-1}$.

###### Proof.

Let $\nu\in\mathbb{S}^{n-1}\cap\mathbb{Q}^{n}$ be fixed; below we show that
there exists a group of $P$-preserving transformations
$(\tau_{z^{\prime}}^{\nu})_{z^{\prime}\in\mathbb{Z}^{n-1}}$ for which
$\mu_{\nu}$ satisfies conditions (1)–(4) of Definition 7.2.

Step 1: measurability. The measurabilty can be achieved by suitably adapting
the proof of [33, Lemma C.1.], for this reason here we only sketch it. Let
$I\in\mathcal{I}_{n-1}$ and let $I_{\nu}\subset\mathbb{R}^{n}$ be as in (7.4).
For every $n\in\mathbb{N}$ we let $f^{n}$ be the Moreau-Yosida-regularisation
of $f$ with respect to the last two variables, that is

$f^{n}(\omega,x,u,\xi):=\inf_{(v,\zeta)\in\mathbb{R}\times\mathbb{R}^{n}}\left\\{f(\omega,x,v,\zeta)+n|(v,\zeta)-(u,\xi)|\right\\}\,.$

It is known that $f^{n}$ is $n$-Lipschitz in $(u,\xi)$. Then arguing as in the
proof of [33, Lemma C.1.] it can be shown that the map
$\mathscr{F}^{n}(\omega)\colon W^{1,p}(I_{\nu})\to[0,+\infty)$ given by

$\mathscr{F}^{n}(\omega)(u):=\int_{I_{\nu}}f^{n}(\omega,x,u,\nabla u)\,{\rm
d}x\,,$

is well defined for every $\omega\in\Omega$ and that
$(\omega,u)\mapsto\mathscr{F}^{n}(\omega)(u)$ is
$\mathcal{T}\otimes\mathcal{B}(W^{1,p}(I_{\nu}))$-measurable. By ($F3$)
$f^{n}\nearrow f$ pointwise so that in particular $\mathscr{F}^{n}(\omega)(u)$
converges to $\mathscr{F}(\omega)(u,I_{\nu})$ pointwise. This in turn implies
that $(\omega,u)\mapsto\mathscr{F}(\omega)(u,I_{\nu})$ is
$\mathcal{T}\otimes\mathcal{B}(W^{1,p}(I_{\nu}))$-measurable as well. Now
recall that $(\Omega,\mathcal{T},P)$ is a complete probability space, while
the set

$\mathscr{A}(\bar{u}_{0}^{\nu},I_{\nu})=\\{u\in W^{1,p}(I_{\nu})\,,\;0\leq
u\leq 1\,,\ u=\bar{u}^{\nu}_{0}\ \text{ near }\ \partial I_{\nu}\\}\subset
W^{1,p}(I_{\nu})$

defines a separable metric space, when endowed with the distance induced by
the $W^{1,p}(I_{\nu})$-norm. Moreover from ($F3$) the map
$u\mapsto\mathscr{F}(\omega)(u,I_{\nu})$ is lower semicontinuous and not
constantly $+\infty$ since from (2.10) we have
$\mathscr{F}(\omega)(\bar{u}_{0}^{\nu},I_{\nu})<+\infty$. Hence we can apply
[33, Lemma C.2.] to deduce the $\mathcal{T}$-measurability of
$\omega\mapsto\mathbf{m}_{\omega}(\bar{u}^{\nu}_{0},I_{\nu})$ and in
particular of $\omega\mapsto\mu_{\nu}(\omega,I)$.

Step 2: covariance. Let $z^{\prime}\in\mathbb{Z}^{n-1}$ be fixed and let
$I\in\mathcal{I}_{n-1}$. Note that (7.4) implies the following equality

$(I+z^{\prime})_{\nu}=I_{\nu}+M_{\nu}R_{\nu}(z^{\prime},0)=I_{\nu}+z^{\prime}_{\nu}\,,$

where
$z^{\prime}_{\nu}:=M_{\nu}R_{\nu}(z^{\prime},0)\in\mathbb{Z}^{n}\cap\Pi^{\nu}$.
Therefore we have

$\mu_{\nu}(\omega,I+z^{\prime})=\frac{1}{M_{\nu}^{n-1}}\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},I_{\nu}+z^{\prime}_{\nu})\,.$
(7.7)

Let $u\in\mathscr{A}(\bar{u}_{0}^{\nu},I_{\nu}+z_{\nu}^{\prime})$ and define
$\widetilde{u}(x):=u(x+z^{\prime}_{\nu})$. Clearly
$\widetilde{u}\in\mathscr{A}(\bar{u}_{0}^{\nu},I_{\nu})$. Indeed since
$z_{\nu}^{\prime}\in\Pi^{\nu}$, we have
$\widetilde{u}=\bar{u}_{0}^{\nu}(\cdot+z^{\prime}_{\nu})=\bar{u}_{0}^{\nu}$
near $\partial I_{\nu}$. Further by a change of variables and using the
stationarity of $f$ we obtain the following

$\begin{split}\mathscr{F}(\omega)(u,{\rm
int}\,(I_{\nu}+z^{\prime}_{\nu}))=&\int_{I_{\nu}+z^{\prime}_{\nu}}f(\omega,x,u,\nabla
u)\,{\rm
d}x=\int_{I_{\nu}}f(\omega,x+z^{\prime}_{\nu},\widetilde{u},\nabla\widetilde{u})\,{\rm
d}x\\\
=&\int_{I_{\nu}}f(\tau_{z^{\prime}_{\nu}}(\omega),x,\widetilde{u},\nabla\widetilde{u})\,{\rm
d}x=\mathscr{F}(\tau_{z^{\prime}_{\nu}}(\omega))(\widetilde{u},{\rm
int}\,I_{\nu})\,.\end{split}$

Let
$(\tau^{\nu}_{z^{\prime}})_{z^{\prime}\in\mathbb{Z}^{n-1}}:=(\tau_{z^{\prime}_{\nu}})_{z^{\prime}\in\mathbb{Z}^{n-1}}$;
then $(\tau^{\nu}_{z^{\prime}})_{z^{\prime}\in\mathbb{Z}^{n-1}}$ is well
defined since $z^{\prime}_{\nu}\in\mathbb{Z}^{n}$ and it defines a group of
$P$-preserving transformations on $(\Omega,\mathcal{T},P)$. In particular the
equality above becomes

$\mathscr{F}(\omega)(u,{\rm
int}\,(I_{\nu}+z^{\prime}_{\nu}))=\mathscr{F}(\tau_{z^{\prime}}^{\nu}(\omega))(\widetilde{u},{\rm
int}\,I_{\nu})\,.$ (7.8)

Eventually, gathering (7.7) and (7.8), by the arbitrariness of $u$ we infer

$\mu_{\nu}(\omega,I+z^{\prime})=\mu_{\nu}(\tau^{\nu}_{z^{\prime}}(\omega),I)\,,$

and the covariance of $\mu_{\nu}$ with respect to
$(\tau^{\nu}_{z^{\prime}})_{z^{\prime}\in\mathbb{Z}^{n-1}}$ is shown.

Step 3: subadditivity. Let $\omega\in\Omega$, $I\in\mathcal{I}_{n-1}$, and let
$\\{I_{1},\dots,I_{N}\\}\subset\mathcal{I}_{n-1}$ be a finite family of
pairwise disjoint sets such that $I=\bigcup_{i}I_{i}$. For $\eta>0$ fixed and
for every $i=1,\dots,N$ we let $u_{i}\in W^{1,p}((I_{i})_{\nu})$ be admissible
for $\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},(I_{i})_{\nu})$ such that

$\mathscr{F}(\omega)(u_{i},{\rm
int}\,(I_{i})_{\nu})\leq\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},(I_{i})_{\nu})+\eta\,.$
(7.9)

We let

$u:=\begin{cases}u_{i}&\text{in }\;(I_{i})_{\nu}\,,\ i=1\,,\ldots,N\,,\\\
\bar{u}_{0}^{\nu}&\text{in
}\;I_{\nu}\setminus\bigcup_{i}(I_{i})_{\nu}\,,\end{cases}$

in this way $u\in W^{1,p}(I_{\nu};\mathbb{R}^{m})$ and $u=\bar{u}_{0}^{\nu}$
near $\partial I_{\nu}$. Hence in particular
$u\in\mathscr{A}(\bar{u}_{0}^{\nu},I_{\nu})$ and

$\mathscr{F}(\omega)(u,{\rm
int}\,I_{\nu})=\sum_{i=1}^{N}\mathscr{F}(\omega)(u_{i},{\rm
int}\,(I_{i})_{\nu})+\mathscr{F}(\omega)\big{(}\bar{u}^{\nu}_{0},{\rm
int}\,(I_{\nu}\setminus\bigcup_{i=1}^{N}(I_{i})_{\nu})\big{)}\,.$ (7.10)

We observe that

$\mathscr{F}(\omega)\big{(}\bar{u}^{\nu}_{0},{\rm
int}\,(I_{\nu}\setminus\bigcup_{i=1}^{N}(I_{i})_{\nu})\big{)}=0\,.$ (7.11)

Indeed, being $M_{\nu}>2$ and $c\geq\frac{1}{2}$ in (7.4) we have that
$\\{y\in I_{\nu}\colon|y\cdot\nu|\leq 1\\}\subset\bigcup_{i}(I_{i})_{\nu}$ and
thus $\bar{u}_{0}^{\nu}\equiv u_{0}^{\nu}$ in
$I_{\nu}\setminus\bigcup_{i}(I_{i})_{\nu}$. Now recalling (2.6) we derive
(7.11). Putting together (7.9)–(7.11) we obtain

$\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},I_{\nu})\leq\mathscr{F}(\omega)(u,{\rm
int}\,I_{\nu})=\sum_{i=1}^{N}\mathscr{F}(\omega)(u_{i},{\rm
int}\,(I_{i})_{\nu})\leq\sum_{i=1}^{N}\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},(I_{i})_{\nu})+N\eta\,,$

and the subadditivity of $\mu_{\nu}$ follows by letting $\eta\to 0$-

Step 4: boundedness. Let $\omega\in\Omega$ and $I\in\mathcal{I}_{n-1}$. From
(7.4) and (2.10) we easily deduce

$0\leq\mu_{\nu}(\omega,I)=\frac{1}{M_{\nu}^{n-1}}\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},I_{\nu})\leq
c_{2}C_{\rm u}\mathcal{L}^{n-1}(I).$

∎

Proposition 7.6 is the key ingredient to prove that almost surely the limit
defining $f_{\mathrm{hom}}$ exists when $x=0$.

######  Proposition 7.7 (Homogenised surface integrand for $x=0$).

Let $f$ satisfy ($F1$)-($F3$) and assume that it is stationary with respect to
a group $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ of $P$-preserving transformations on
$(\Omega,\mathcal{T},P)$. For $\omega\in\Omega$ let $\mathbf{m}_{\omega}$ be
as in (7.2). Then there exist $\widetilde{\Omega}\in\mathcal{T}$ with
$P(\widetilde{\Omega})=1$ and a
$(\mathcal{T}\otimes\mathcal{B}({\mathbb{S}}^{n-1}))$-measurable function
$f_{\mathrm{hom}}:\Omega\times{\mathbb{S}}^{n-1}\to[0,+\infty)$ such that

$\lim_{r\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r}^{\nu}(0))}{r^{n-1}}=f_{\mathrm{hom}}(\omega,\nu)$
(7.12)

for every $\omega\in\widetilde{\Omega}$ and every $\nu\in{\mathbb{S}}^{n-1}$.
Moreover, $\widetilde{\Omega}$ and $f_{\mathrm{hom}}$ are
$(\tau_{z})_{z\in\mathbb{Z}^{n}}$-translation invariant; i.e.,
$\tau_{z}(\widetilde{\Omega})=\widetilde{\Omega}$ for every
$z\in\mathbb{Z}^{n}$ and

$f_{\mathrm{hom}}(\tau_{z}(\omega),\nu)=f_{\mathrm{hom}}(\omega,\nu),$ (7.13)

for every $z\in\mathbb{Z}^{n}$, for every $\omega\in\widetilde{\Omega}$, and
every $\nu\in{\mathbb{S}}^{n-1}$. Eventually, if
$(\tau_{z})_{z\in\mathbb{Z}^{n}}$ is ergodic then $f_{\mathrm{hom}}$ is
independent of $\omega$ and given by

$f_{\rm
hom}(\nu)=\lim_{r\to+\infty}\frac{1}{r^{n-1}}\int_{\Omega}\mathbf{m}_{\omega}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))\,{\rm
d}P(\omega).$ (7.14)

###### Proof.

We divide the proof into three steps.

Step 1: existence of the limit for
$\nu\in{\mathbb{S}}^{n-1}\cap\mathbb{Q}^{n}$.  In this step we show that there
exists $\widetilde{\Omega}\in\mathcal{T}$ with $P(\widetilde{\Omega})=1$ such
that the following holds: for every
$\nu\in{\mathbb{S}}^{n-1}\cap\mathbb{Q}^{n}$ there is a
$\mathcal{T}$-measurable function $f_{\nu}\colon\Omega\to[0,+\infty)$ such
that

$f_{\nu}(\omega)=\lim_{r\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r}^{\nu}(0))}{r^{n-1}}\,,\quad\forall\omega\in\widetilde{\Omega}\,.$

We fix $\nu\in{\mathbb{S}}^{n-1}\cap\mathbb{Q}^{n}$. By Proposition 7.6 we
know that the map $\mu_{\nu}$ given in (7.5) is a subadditive process on
$(\Omega,\mathcal{T},P)$ with respect to some group of $P$-preserving
transformations. This allows us to apply the Subadditive Ergodic Theorem [1,
Theorem 2.4] and deduce the following: let $I=[-1,1)^{n-1}$, so that
$I_{\nu}=2M_{\nu}Q^{\nu}(0)$; then there exists a set
$\Omega_{\nu}\in\mathcal{T}$, with $P(\Omega_{\nu})=1$, and a
$\mathcal{T}$-measurable function $f_{\nu}\colon\Omega\to[0,+\infty)$ such
that

$f_{\nu}(\omega)=\lim_{j\to+\infty}\frac{{\mu_{\nu}}(\omega,jI)}{(2j)^{n-1}}=\lim_{j\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},j2M_{\nu}Q^{\nu}(0))}{(j2M_{\nu})^{n-1}}\,.$
(7.15)

for every $\omega\in\Omega_{\nu}$.

Let now $(r_{j})$ be a sequence of strictly positive real numbers with
$r_{j}\to+\infty$, as $j\to+\infty$ and define

$r_{j}^{-}:=2M_{\nu}\Big{(}\Big{\lfloor}\frac{r_{j}}{2M_{\nu}}\Big{\rfloor}-1\Big{)}\quad\text{and}\quad
r_{j}^{+}:=2M_{\nu}\Big{(}\Big{\lfloor}\frac{r_{j}}{2M_{\nu}}\Big{\rfloor}+2\Big{)}\,.$

For $j$ sufficiently large $r_{j}>4(1+M_{\nu})$, and thus $r_{j}^{-}>4$,
moreover

$Q^{\nu}_{r_{j}^{-}+2}(0)\subset\subset Q_{r_{j}}^{\nu}(0)\subset\subset
Q_{r_{j}+2}^{\nu}(0)\subset\subset Q_{r_{j}^{+}}^{\nu}(0)\,.$

We then can apply Lemma A.1 twice: the first time with $x=\widetilde{x}=0$,
$r=r_{j}^{-}$, and $\widetilde{r}=r_{j}$ and the second one with
$x=\widetilde{x}=0$, $r=r_{j}^{+}$, and $\widetilde{r}=r_{j}$ and get the two
following estimates

$\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r_{j}}^{\nu}(0))}{r_{j}^{n-1}}\leq\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r_{j}^{-}}^{\nu}(0))}{(r_{j}^{-})^{n-1}}+\frac{L(r_{j}-r_{j}^{-}+1)}{r_{j}}\,,$
(7.16)
$\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r_{j}}^{\nu}(0))}{r_{j}^{n-1}}\geq\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r_{j}^{+}}^{\nu}(0))}{(r_{j}^{+})^{n-1}}-\frac{L(r_{j}^{+}-r_{j}+1)}{r_{j}}\,.$
(7.17)

Using that $r_{j}^{+}-r_{j}\leq 4M_{\nu}$ and $r_{j}-r_{j}^{-}\leq 4M_{\nu}$,
thus thanks to (7.15), passing to the limsup in (7.16) and to the liminf in
(7.17) as $j\to+\infty$ yield

$\limsup_{j\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r_{j}}^{\nu}(0))}{r_{j}^{n-1}}\leq
f_{\nu}(\omega)\quad\forall\omega\in\Omega_{\nu},$ (7.18)

and

$\liminf_{j\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r_{j}}^{\nu}(0))}{r_{j}^{n-1}}\geq
f_{\nu}(\omega)\quad\forall\omega\in\Omega_{\nu}.$ (7.19)

From (7.18) and (7.19) we have that for every $\omega\in\Omega_{\nu}$ the
limit along $(r_{j})$ exists and satisfies

$\lim_{j\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r_{j}}^{\nu}(0))}{r_{j}^{n-1}}=f_{\nu}(\omega).$

Eventually to conclude we define

$\widetilde{\Omega}:=\bigcap_{\nu\in\mathbb{S}^{n-1}\cap\mathbb{Q}^{n}}\Omega_{\nu},$

which satisfies the desired properties.

Step 2: existence of the limit for
$\nu\in{\mathbb{S}}^{n-1}\setminus\mathbb{Q}^{n}$. In this step we prove that
there is a $(\mathcal{T}\otimes\mathcal{B}({\mathbb{S}}^{n-1}))$-measurable
function $f_{\mathrm{hom}}:\Omega\times{\mathbb{S}}^{n-1}\to[0,+\infty)$ such
that (7.12) holds for every $\omega\in\widetilde{\Omega}$ and every
$\nu\in{\mathbb{S}}^{n-1}$.

Let
$\underline{f},\overline{f}\colon\widetilde{\Omega}\times{\mathbb{S}}^{n-1}\to[0,+\infty]$
be given by

$\underline{f}(\omega,\nu):=\liminf_{r\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r}^{\nu}(0))}{r^{n-1}}\,,\quad\overline{f}(\omega,\nu):=\limsup_{r\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{r}^{\nu}(0))}{r^{n-1}}.$

We first observe that $\widehat{{\mathbb{S}}}_{\pm}^{n-1}\cap\mathbb{Q}^{n}$
is dense in $\widehat{{\mathbb{S}}}_{\pm}^{n-1}$. Moreover in Step 1 we showed
that $\underline{f}(\omega,\nu)=\overline{f}(\omega,\nu)=f_{\nu}(\omega)$, for
every $\omega\in\widetilde{\Omega}$ and for every
$\nu\in{\mathbb{S}}^{n-1}\cap\mathbb{Q}^{n}$. Therefore, in order to obtain
the same equality for every for every $\omega\in\widetilde{\Omega}$ and every
$\nu\in{\mathbb{S}}^{n-1}$, it is enough to show that the restrictions of the
functions $\nu\mapsto\underline{f}(\omega,\nu)$ and
$\nu\mapsto\overline{f}(\omega,\nu)$ to the sets
$\widehat{\mathbb{S}}_{\pm}^{n-1}$ are continuous. Further once we have that
we can also deduce the following:

$\omega\mapsto\overline{f}(\omega,\nu)\;\text{ is $\mathcal{T}$-measurable in
$\widetilde{\Omega}$, for every $\nu\in{\mathbb{S}}^{n-1}$}$

together with

$\nu\mapsto\overline{f}(\omega,\nu)\;\text{ is continuous in
$\widehat{\mathbb{S}}^{n-1}_{\pm}$, for every
$\omega\in\widetilde{\Omega}$}\,,$

readily imply that the restriction of $\overline{f}$ to
$\widetilde{\Omega}\times\widehat{\mathbb{S}}^{n-1}_{\pm}$ is measurable with
respect to the $\sigma$-algebra induced in
$\widetilde{\Omega}\times\widehat{\mathbb{S}}^{n-1}_{\pm}$ by
$\mathcal{T}\otimes\mathcal{B}({\mathbb{S}}^{n-1})$. Hence the claim follows
by setting

$f_{\mathrm{hom}}(\omega,\nu):=\begin{cases}\overline{f}(\omega,\nu)&\text{if
}\;\omega\in\widetilde{\Omega},\cr c_{2}c_{p}&\text{if
}\;\omega\in\Omega\setminus\widetilde{\Omega}.\end{cases}$ (7.20)

We now prove that $\overline{f}(\omega,\cdot)$ is continuous in
$\widehat{\mathbb{S}}^{n-1}_{+}$. The other proofs are analogous and therefore
are left to the readers. Let $\nu\in\widehat{\mathbb{S}}_{+}^{n-1}$,
$(\nu_{j})\subset\widehat{\mathbb{S}}_{+}^{n-1}$ be such that $\nu_{j}\to\nu$,
as $j\to+\infty$. Then, for every $\alpha\in(0,\frac{1}{2})$ we find
$j_{\alpha}\in\mathbb{N}$ such that (A.7) holds true for every $j\geq
j_{\alpha}$. Hence we can apply Lemma A.2 with $x=0$ and
$\widetilde{\nu}=\nu_{j}$ and get

$\mathbf{m}_{\omega}(\bar{u}^{\nu_{j}}_{0},Q^{\nu_{j}}_{(1+\alpha)r}(0))-c_{\alpha}r^{n-1}\leq\mathbf{m}_{\omega}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))\leq\mathbf{m}_{\omega}(\bar{u}^{\nu_{j}}_{0},Q^{\nu_{j}}_{(1-\alpha)r}(0))+c_{\alpha}r^{n-1},$

where $c_{\alpha}\to 0$, as $\alpha\to 0$. Now dividing the above inequality
by $r^{n-1}$ and passing to the limsup as $r\to+\infty$ we get

$\displaystyle(1+\alpha)^{n-1}\,\overline{f}(\omega,\nu_{j})$
$\displaystyle\leq\overline{f}(\omega,\nu)+c_{\alpha}\,,$ (7.21)
$\displaystyle(1-\alpha)^{n-1}\,\overline{f}(\omega,\nu_{j})$
$\displaystyle\geq\overline{f}(\omega,\nu)-c_{\alpha}\,.$ (7.22)

Eventually passing to the limsup as $j\to+\infty$ in (7.21) and to the liminf
as $j\to+\infty$ in (7.22), and letting $\alpha\to 0$ we have

$\limsup_{j\to+\infty}\overline{f}(\omega,\nu_{j})\leq\overline{f}(\omega,\nu)\leq\liminf_{j\to+\infty}\overline{f}(\omega,\nu_{j})\,,$

and the proof of step 2 is achieved.

Step 3: $(\tau_{z})_{z\in\mathbb{Z}^{n}}$-translation invariance. In this step
we show that $\widetilde{\Omega}$ and $f_{\mathrm{hom}}$ are
$(\tau_{z})_{z\in\mathbb{Z}^{n}}$-translation invariant.

Let $z\in\mathbb{Z}^{n}$, $\omega\in\widetilde{\Omega}$, and
$\nu\in\mathbb{S}^{n-1}$ be fixed. Let $r>4$ and
$u\in\mathscr{A}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))$ satisfying

$\mathscr{F}(\omega)(u,Q_{r}^{\nu}(0))\leq\mathbf{m}_{\omega}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))+1\,.$
(7.23)

Setting $\widetilde{u}(y):=u(y+z)$, then since $f$ is stationary there holds

$\mathscr{F}(\omega)(u,Q_{r}^{\nu}(0))=\mathscr{F}(\tau_{z}(\omega))(\widetilde{u},Q_{r}^{\nu}(-z)).$

This together with (7.23) and the fact that
$\widetilde{u}\in\mathscr{A}(\bar{u}^{\nu}_{-z},Q^{\nu}_{r}(-z))$ yield

$\mathbf{m}_{\tau_{z}(\omega)}(\bar{u}^{\nu}_{-z},Q^{\nu}_{r}(-z))\leq\mathbf{m}_{\omega}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))+1\,.$
(7.24)

We choose $r,\widetilde{r}$ such that $\widetilde{r}>r$ and

$Q^{\nu}_{r+2}(-z)\subset\subset Q^{\nu}_{\widetilde{r}}(0)\;\text{ and
}\;\operatorname{dist}(0,\Pi^{\nu}(-z))\leq\frac{r}{4}.$

We next apply Lemma A.1 twice: once with $x=-z$ and $\widetilde{x}=0$ to the
minimisation problem $\mathbf{m}_{\tau_{z}(\omega)}$ and once with $x=z$ and
$\widetilde{x}=0$ to the minimisation problem $\mathbf{m}_{\omega}$ and get

$\mathbf{m}_{\tau_{z}(\omega)}(\bar{u}_{0}^{\nu},Q_{\widetilde{r}}^{\nu}(0))\leq\mathbf{m}_{\tau_{z}(\omega)}(\bar{u}_{-z}^{\nu},Q_{r}^{\nu}(-z))+L\big{(}|z|+|r-\widetilde{r}|+1\big{)}(\widetilde{r})^{n-2}\,,$
(7.25)

and

$\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{\widetilde{r}}^{\nu}(0))\leq\mathbf{m}_{\omega}(\bar{u}_{z}^{\nu},Q_{r}^{\nu}(z))+L\big{(}|z|+|r-\widetilde{r}|+1\big{)}(\widetilde{r})^{n-2}\,.$
(7.26)

Hence, combining (7.24) with (7.25) and (7.26) we have

$\frac{\mathbf{m}_{\tau_{z}(\omega)}(\bar{u}_{0}^{\nu},Q_{\widetilde{r}}^{\nu}(0))}{\widetilde{r}^{n-1}}\leq\frac{\mathbf{m}_{\omega}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))+1}{r^{n-1}}+\frac{L\big{(}|z|+|r-\widetilde{r}|+1\big{)}}{\widetilde{r}}\,,$
(7.27)

and

$\frac{\mathbf{m}_{\omega}(\bar{u}_{0}^{\nu},Q_{\widetilde{r}}^{\nu}(0))}{\widetilde{r}^{n-1}}\leq\frac{\mathbf{m}_{\tau_{z}(\omega)}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))+1}{r^{n-1}}+\frac{L\big{(}|z|+|r-\widetilde{r}|+1\big{)}}{\widetilde{r}}\,.$
(7.28)

We now take in (7.27) the limsup as $\widetilde{r}\to+\infty$ and the limit as
$r\to+\infty$ and find

$\limsup_{\widetilde{r}\to+\infty}\frac{\mathbf{m}_{\tau_{z}(\omega)}(\bar{u}_{0}^{\nu},Q_{\widetilde{r}}^{\nu}(0))}{\widetilde{r}^{n-1}}\leq
f_{\mathrm{hom}}(\omega,\nu).$ (7.29)

Similarly we take in (7.28) the limit as $\widetilde{r}\to+\infty$ and the
liminf as $r\to+\infty$ and obtain

$f_{\mathrm{hom}}(\omega,\nu)\leq\liminf_{r\to+\infty}\frac{\mathbf{m}_{\tau_{z}(\omega)}(\bar{u}_{0}^{\nu},Q_{r}^{\nu}(0))}{r^{n-1}}.$
(7.30)

Gathering (7.29) and (7.30) we deduce that
$\tau_{z}(\omega)\in\widetilde{\Omega}$ and that

$f_{\mathrm{hom}}(\tau_{z}(\omega),\nu)=f_{\mathrm{hom}}(\omega,\nu)\,,$

for every $z\in\mathbb{Z}$, $\omega\in\widetilde{\Omega}$, and
$\nu\in{\mathbb{S}}^{n-1}$. Notice also that thanks to the group properties of
$(\tau_{z})_{z\in\mathbb{Z}^{n}}$ we also have that
$\omega\in\tau_{z}(\widetilde{\Omega})$, for every $z\in\mathbb{Z}^{n}$.

We conclude by observing that if $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ is ergodic,
then the independence of $\omega$ of the function of $f_{\mathrm{hom}}$
follows by (7.13) (cf. [18, Corollary 6.3]) and (7.14) follows by integrating
(7.12) on $\Omega$ and by using the Dominated Convergence Theorem, thanks to
(7.6) (see also (7.5)). ∎

We conclude this section by establishing the last crucial result which extends
Proposition 7.7 to the case of an arbitrary $x\in\mathbb{R}^{n}$. More
precisely, Proposition 7.8 below states that the limit in (7.12) exists when
$x=0$ is replaced by any $x\in\mathbb{R}^{n}$ and that it is $x$-independent,
and hence it coincides with (7.12).

The proof of the following proposition can be obtained arguing exactly as in
[18, Theorem 6.1] (see also [3, Theorem 5.5]), now appealing to Proposition
7.7, Lemma A.1, and Lemma A.2. For this reason we skip its proof here.

######  Proposition 7.8 (Homogenised surface integrand).

Let $f$ satisfy ($F1$)-($F3$) and assume that it is stationary with respect to
a group $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ of $P$-preserving transformations on
$(\Omega,\mathcal{T},P)$. For $\omega\in\Omega$ let $\mathbf{m}_{\omega}$ be
as in (7.2). Then there exists $\Omega^{\prime}\in\mathcal{T}$ with
$P(\Omega^{\prime})=1$ such that

$\lim_{r\to+\infty}\frac{\mathbf{m}_{\omega}(\bar{u}_{rx}^{\nu},Q_{r}^{\nu}(rx))}{r^{n-1}}=f_{\mathrm{hom}}(\omega,\nu)$
(7.31)

for every $\omega\in\Omega^{\prime}$, every $x\in\mathbb{R}^{n}$, and every
$\nu\in{\mathbb{S}}^{n-1}$, where $f_{\mathrm{hom}}$ is given by (7.12). In
particular, the limit in (7.31) is independent of $x$. Moreover, if
$(\tau_{z})_{z\in\mathbb{Z}^{n}}$ is ergodic, then $f_{\mathrm{hom}}$ is
independent of $\omega$ and given by (7.14).

We conclude this section with the proof of Theorem 7.4.

###### Proof of Theorem 7.4.

The proof follows by Theorem 3.3 now invoking Proposition 7.8. ∎

##  8\. Periodic Homogenisation

In this last section we prove a periodic homogenisation result without
requiring any lower semicontinuity on the integrand $f$ (i.e., without
assuming ($F3$)).

A careful inspection of the proof of Theorem 7.4 shows that condition ($F3$)
is used only once in the proof of Proposition 7.6 to establish the
$\mathcal{T}$-measurability of the map $\mu_{\nu}$ defined in (7.5). This
suggests that in the periodic setting Theorem 7.4 should still be true if we
drop condition ($F3$).

Let $f\in\mathcal{F}$ be such that for every $u\in\mathbb{R}$ and every
$\xi\in\mathbb{R}^{n}$ $f(\cdot,u,\xi)$ is $Q$-periodic. Let $f_{k}$ and
$\mathscr{F}_{k}$ be defined as in (3.3) and (2.1) accordingly. We now state
and prove the following result.

######  Theorem 8.1 (Periodic homogenisation).

Let $f\in\mathcal{F}$ be such that for every $u\in\mathbb{R}$ and every
$\xi\in\mathbb{R}^{n}$ $f(\cdot,u,\xi)$ is $Q$-periodic. Let $\mathbf{m}$ be
as in (3.2). Then the limit

$\lim_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{rx},Q^{\nu}_{r}(rx))}{r^{n-1}}=\lim_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))}{r^{n-1}}=:f_{\mathrm{hom}}(\nu)$
(8.1)

exists and is independent of $x$. The function
$f_{\mathrm{hom}}\colon{\mathbb{S}}^{n-1}\to[0,+\infty)$ is
$\mathcal{B}({\mathbb{S}}^{n-1})$-measurable.

Moreover, for every $A\in\mathcal{A}$, the functionals
$\mathscr{F}_{k}(\cdot,A)$, defined in (2.1) with $f_{k}$ as in (3.3),
$\Gamma$-converge in $L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ to the functional
$\mathscr{F}_{\mathrm{hom}}(\cdot,A)$ with $\mathscr{F}_{\mathrm{hom}}\colon
L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\times\mathcal{A}\longrightarrow[0,+\infty]$
given by

$\mathscr{F}_{\rm hom}(u,A):=\begin{cases}\displaystyle\int_{S_{u}\cap
A}f_{\rm hom}(\nu_{u})\,{\rm d}\mathcal{H}^{n-1}&\text{if}\ u\in
BV(A;\\{0,1\\})\\\\[4.0pt] +\infty&\text{otherwise}\,.\end{cases}$

###### Proof.

From Theorem 3.3 it is sufficient to show that the limit in (8.1) exists and
is independent of $x$ and that $f_{\mathrm{hom}}$ is
$\mathcal{B}({\mathbb{S}}^{n-1})$-measurable. We divide the proof into a nuber
of steps.

Step 1: existence of the limit for $x=0$.  In this step we show that for $x=0$
and for every $\nu\in{\mathbb{S}}^{n-1}$ the following limit in (8.1) exists.
Let $\nu\in{\mathbb{S}}^{n-1}$ and let $s>r>0$ be fixed. By reasoning as in
step 2 of the proof of Proposition 7.7 we can assume without loss of
generality that $\nu\in{\mathbb{S}}^{n-1}\cap\mathbb{Q}^{n}$. Next we choose
$u_{r}\in\mathscr{A}(\bar{u}^{\nu}_{0},Q_{r}^{\nu}(0))$ such that

$\mathscr{F}(u_{r},Q_{r}^{\nu}(0))\leq\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))+1\,.$
(8.2)

In what follows we extend $u_{r}$ to $Q^{\nu}_{s}(0)$ to get a new function
$u_{s}\in\mathscr{A}(\bar{u}^{\nu}_{0},Q_{s}^{\nu}(0))$ without essentially
increasing the energy. For every $z\in\mathbb{Z}^{n-1}\times\\{0\\}$ we set

$z_{r}^{\nu}:=\left(\left\lfloor\frac{r}{M_{\nu}}\right\rfloor+1\right)M_{\nu}R_{\nu}z\,,\quad\widetilde{Q}_{z}:=Q_{r}^{\nu}(0)+z_{r}^{\nu}\,,$

with $R_{\nu}$ as in (g) and $M_{\nu}$ defined as in section 7.1, and let

$I_{s}:=\left\\{z\in\mathbb{Z}^{n-1}\times\\{0\\}\colon\widetilde{Q}_{z}\subset
Q_{s}^{\nu}(0)\right\\}\,.$

Note that by definition of $M_{\nu}$ and $R_{\nu}$ we have
$z_{r}^{\nu}\in\mathbb{Z}^{n}\cap\Pi_{\nu}$ for every
$z\in\mathbb{Z}^{n-1}\times\\{0\\}$. Moreover a direct computation shows that

$s^{n-1}\left(\frac{1}{r+1}-\frac{1}{s}\right)^{n-1}\leq\\#(I_{s})\leq\frac{s^{n-1}}{r^{n-1}}\,.$
(8.3)

Let us define

$u_{s}(y):=\begin{cases}u_{r}(y-z_{r}^{\nu})&\text{ if
}y\in\widetilde{Q}_{z},\,z\in I_{s}\,,\\\ \bar{u}^{\nu}_{0}(y)&\text{
otherwise in }Q^{\nu}_{s}(0)\,.\end{cases}$

Clearly $u_{s}\in\mathscr{A}(\bar{u}^{\nu}_{0},Q_{s}^{\nu}(0))$. Furthermore a
change of variable together with the the $Q$-periodicity of $f$ yield

$\begin{split}\mathscr{F}(u_{s},Q_{s}^{\nu}(0))\leq\\#(I_{s})\mathscr{F}(u_{r},Q_{r}^{\nu}(0))+\mathscr{F}(\bar{u}^{\nu}_{0},Q_{s}^{\nu}(0)\setminus\cup_{z\in
I_{s}}\widetilde{Q}_{z})\,.\end{split}$ (8.4)

Moreover by (2.10) we may deduce that

$\mathscr{F}(\bar{u}^{\nu}_{0},Q_{s}^{\nu}(0)\setminus\cup_{z\in
I_{s}}\widetilde{Q}_{z})\leq c_{2}C_{\rm u}(s^{n-1}-\\#(I_{s})r^{n-1})\,.$
(8.5)

Gathering together (8.3)-(8.5) we obtain

$\begin{split}\frac{\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{s}(0))}{s^{n-1}}\leq\frac{\mathscr{F}(u_{s},Q_{s}^{\nu}(0))}{s^{n-1}}\leq\frac{\mathscr{F}(u_{r},Q_{r}^{\nu}(0))}{r^{n-1}}+c_{2}C_{\rm
u}\left(1-r^{n-1}\Big{(}\frac{1}{r+1}-\frac{1}{s}\Big{)}^{n-1}\right)\,.\end{split}$
(8.6)

Finally combining (8.6) with (8.2) and passing first to the limsup as
$s\to+\infty$ and then to the liminf as $r\to+\infty$ we get

$\limsup_{s\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{s}(0))}{s^{n-1}}\leq\liminf_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))}{r^{n-1}}\,,$

and, being the converse inequality trivial, the proof of Step 1 is achieved.

Step 2: existence of the limit for every $x$.  In this step we show that for
every $x\in\mathbb{R}^{n}$ the limit in (8.1) exists and is $x$-independent.
Let $x\neq 0$, and let $r\geq 4\sqrt{n}$ be fixed. Let
$u_{r}\in\mathscr{A}(\bar{u}^{\nu}_{0},Q_{r}^{\nu}(0))$ be such that (8.2)
holds. Define $x_{r}:=\lfloor(r+3)x\rfloor$ (i.e., the vector whose componets
are the integer parts of the components of $(r+3)x$) so that
$|(r+3)x-x_{r}|\leq\sqrt{n}$ and moreover

$Q_{r+2}^{\nu}(x_{r})\subset\subset Q_{r+3}^{\nu}((r+3)x)\,,\quad\text{ and
}\quad\operatorname{dist}((r+3)x,\Pi^{\nu}(x_{r}))\leq\frac{r}{4}\,.$

The function $\hat{u}_{r}(y):=u_{r}(y-x_{r})$ belongs to
$\mathscr{A}(\bar{u}^{\nu}_{x_{r}},Q_{r}^{\nu}(x_{r}))$. Moreover by
$Q$-periodicity of $f$ and (8.2) we have

$\mathbf{m}(\bar{u}^{\nu}_{x_{r}},Q^{\nu}_{r}(x_{r}))\leq\mathscr{F}(\hat{u}_{r},Q_{r}^{\nu}(x_{r}))=\mathscr{F}(u_{r},Q_{r}^{\nu}(0))\leq\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))+1\,.$
(8.7)

Next we invoke Lemma A.1 and get

$\mathbf{m}(\bar{u}^{\nu}_{(r+3)x},Q^{\nu}_{r+3}((r+3)x))\leq\mathbf{m}(\bar{u}^{\nu}_{x_{r}},Q^{\nu}_{r}(x_{r}))+L\big{(}\sqrt{n}+3+1\big{)}(r+3)^{n-2}\,.$
(8.8)

Combining (8.7) with (8.8) and rescaling by $(r+3)^{n-1}$ we have

$\frac{\mathbf{m}(\bar{u}^{\nu}_{(r+3)x},Q^{\nu}_{r+3}((r+3)x))}{(r+3)^{n-1}}\leq\frac{r^{n-1}}{(r+3)^{n-1}}\frac{\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))}{r^{n-1}}+\frac{C}{r+3}\,,$

hence, up to replacing $r+3$ with $r$, passing to the limsup as $r\to+\infty$
we find

$\limsup_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{rx},Q^{\nu}_{r}(rx))}{r^{n-1}}\leq\limsup_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))}{r^{n-1}}=\lim_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))}{r^{n-1}}\,.$

In a similar way one can show that also

$\lim_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))}{r^{n-1}}=\liminf_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{0},Q^{\nu}_{r}(0))}{r^{n-1}}\leq\liminf_{r\to+\infty}\frac{\mathbf{m}(\bar{u}^{\nu}_{rx},Q^{\nu}_{r}(rx))}{r^{n-1}}\,.$

The above two inequalities conclude step 2.

Conclusions.  From step 2 we deduce (8.1). Moreover arguing as in the step 2
of the proof of Proposition 7.7 we may also deduce that $f_{\mathrm{hom}}$ is
$\mathcal{B}({\mathbb{S}}^{n-1})$-measurable.

∎

######  Remark 8.2.

We conclude this section by observing that our analysis in the
stochastic/periodic setting (and the corresponding results Theorem 7.4 and
Theorem 8.1) covers, in particular, the case of functionals considered in [31,
23] and the critical regime of [6].

## Acknowledgments

The author wishes to thank Caterina Ida Zeppieri for having drawn her
attention to this problem and for many helpful discussions and suggestions,
and Annika Bach for useful comments on a preliminary version. The present work
was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) project 3160408400 and under the Germany Excellence Strategy EXC
2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure.

## Appendix

In this last section we state and prove two technical lemmas which are used in
Subsection 7.1.

For $A\in\mathcal{A}$, $x\in\mathbb{R}^{n}$, and $\nu\in{\mathbb{S}}^{n-1}$,
in what follows $\mathbf{m}(\bar{u}^{\nu}_{x},A)$ denotes the infimum value
given by (3.2).

######  Lemma A.1.

Let $f\in\mathcal{F}$; let $\nu\in{\mathbb{S}}^{n-1}$,
$x,\widetilde{x}\in\mathbb{R}^{n}$, and $\widetilde{r}>r>4$ be such that

${\rm(i)}\ Q_{r+2}^{\nu}(x)\subset\subset
Q_{\widetilde{r}}^{\nu}(\widetilde{x})\,,\qquad{\rm(ii)}\
\operatorname{dist}(\widetilde{x},\Pi^{\nu}(x))\leq\frac{r}{4}\,.$

Then there exists a constant $L>0$ (independent of
$\nu,x,\widetilde{x},r,\widetilde{r}$) such that

$\mathbf{m}(\bar{u}_{\widetilde{x}}^{\nu},Q_{\widetilde{r}}^{\nu}(\widetilde{x}))\leq\mathbf{m}(\bar{u}_{x}^{\nu},Q_{r}^{\nu}(x))+L\big{(}|x-\widetilde{x}|+|r-\widetilde{r}|+1\big{)}\widetilde{r}^{n-2}\,.$
(A.1)

###### Proof.

Let $\nu\in S^{n-1}$, $\eta>0$ be fixed and let
$u\in\mathscr{A}(\bar{u}^{\nu}_{x},Q^{\nu}_{r}(x))$ with

$\mathscr{F}(u,Q^{\nu}_{r}(x))\leq\mathbf{m}(\bar{u}^{\nu}_{x},Q^{\nu}_{r}(x))+\eta\,.$
(A.2)

Then $u=\bar{u}^{\nu}_{x}$ a.e. in $U$, where $U$ is a neighbourhood of
$\partial Q^{\nu}_{r}(x)$. Let moreover $\beta\in(0,1)$ be such that
$Q^{\nu}_{r}(x)\setminus\overline{Q}^{\nu}_{r-\beta}(x)\subset U$. Let
$\widetilde{x}\in\mathbb{R}^{n}$ and $\widetilde{r}>r>4$ satisfy (i)-(ii). We
set

$R:=R_{\nu}\biggl{(}\big{(}Q^{\prime}_{r}\setminus\overline{Q^{\prime}}_{r-\beta}\big{)}\times\Big{(}-1-\frac{|(x-\widetilde{x})\cdot\nu|}{2}\,,1+\frac{|(x-\widetilde{x})\cdot\nu|}{2}\Big{)}\biggr{)}+x+\frac{(\widetilde{x}-x)\cdot\nu}{2}\nu\,,$

where $R_{\nu}$ be as in (g). Let $\varphi$ be a smooth cutoff between
$Q^{\nu}_{r-\beta}(x)$ and $Q^{\nu}_{\widetilde{r}}(\widetilde{x})$ and define

$\widetilde{u}:=\varphi u+(1-\varphi)\bar{u}^{\nu}_{\widetilde{x}}\,,$

which clearly belongs to
$\mathscr{A}(\bar{u}^{\nu}_{\widetilde{x}},Q^{\nu}_{\widetilde{r}}(\widetilde{x}))$
and moreover it satisfies

$\mathscr{F}(\widetilde{u},Q^{\nu}_{\widetilde{r}}(\widetilde{x}))\leq\mathscr{F}(u,Q_{r}^{\nu}(x))+\mathscr{F}(\widetilde{u},Q^{\nu}_{\widetilde{r}}(\widetilde{x})\setminus\overline{Q}^{\nu}_{r-\beta}(\widetilde{x}))\,.$
(A.3)

Notice that

$\widetilde{u}=\varphi\bar{u}^{\nu}_{x}+(1-\varphi)\bar{u}^{\nu}_{\widetilde{x}}\quad\text{
in }\quad
Q^{\nu}_{\widetilde{r}}(\widetilde{x})\setminus\overline{Q}^{\nu}_{r-\beta}(x)\,,$

and in particular

$\widetilde{u}={u}^{\nu}_{x}=u^{\nu}_{\widetilde{x}}\in\\{0,1\\}\quad\text{
a.e. in
}\quad(Q^{\nu}_{\widetilde{r}}(\widetilde{x})\setminus\overline{Q}^{\nu}_{r-\beta}(x))\setminus\overline{R}\,.$

This together with (2.6) and ($f3$) imply

$\begin{split}\mathscr{F}(\widetilde{u},Q^{\nu}_{\widetilde{r}}(\widetilde{x})\setminus\overline{Q}^{\nu}_{r-\beta}(\widetilde{x}))&\leq
c_{2}\int_{R}\left(W(\widetilde{u})+|\nabla\widetilde{u}|^{p}\right)\,{\rm
d}y\\\ &\leq
C\mathcal{L}^{n}(R)+C\int_{R}\left(|\nabla\bar{u}^{\nu}_{x}|^{p}+|\nabla\bar{u}^{\nu}_{\widetilde{x}}|^{p}+|\nabla\varphi|^{p}|\bar{u}^{\nu}_{\widetilde{x}}-\bar{u}^{\nu}_{x}|^{p}\right)\,{\rm
d}y\\\ &\leq
C\mathcal{L}^{n}(R)+C\mathcal{L}^{n-1}(Q^{\prime}_{r}\setminus\overline{Q}^{\prime}_{r-\beta})\,,\end{split}$
(A.4)

where the last two inequality follow by recalling that $W$ is bounded on
compact sets and by (2.10). Now we observe that

$\mathcal{L}^{n-1}\big{(}Q^{\prime}_{\widetilde{r}}\setminus\overline{Q}^{\prime}_{r-\beta}\big{)}\leq
C|r-\widetilde{r}|\widetilde{r}^{n-2}\,,$ (A.5)

while

$\mathcal{L}^{n}(R)\leq C\beta r^{n-2}(|x-\widetilde{x}|+1)\,.$ (A.6)

Finally gathering (A.2), (A.3), (A.4), (A.5) and (A.6) we obtain

$\mathbf{m}(\bar{u}^{\nu}_{\widetilde{x}},Q^{\nu}_{\widetilde{r}}(\widetilde{x}))\leq\mathscr{F}\big{(}\widetilde{u},Q^{\nu}_{\widetilde{r}}(\widetilde{x})\big{)}\leq\mathbf{m}(\bar{u}^{\nu}_{x},Q^{\nu}_{r}(x))+L\big{(}|x-\widetilde{x}|+|r-\widetilde{r}|+1\big{)}\widetilde{r}^{n-2}+\eta\,,$

for some $L>0$ independent of $x,\widetilde{x},r,\widetilde{r},\nu$, thus
(A.1) follows by the arbitrariness of $\eta>0$. ∎

######  Lemma A.2.

Let $f\in\mathcal{F}$; let $\alpha\in(0,\frac{1}{2})$ and
$\nu,\widetilde{\nu}\in{\mathbb{S}}^{n-1}$ be such that

$\max_{1\leq i\leq
n-1}|R_{\nu}e_{i}-R_{\widetilde{\nu}}e_{i}|+|\nu-\widetilde{\nu}|<\frac{\alpha}{\sqrt{n}},$
(A.7)

where $R_{\nu}$ and $R_{\widetilde{\nu}}$ are orthogonal $(n\times
n)$-matrices as in (g). Then there exists a constant $c_{\alpha}>0$
(independent of $\nu,\widetilde{\nu}$), with $c_{\alpha}\to 0$ as $\alpha\to
0$, such that for every $x\in\mathbb{R}^{n}$ and every $r>2$ we have

$\begin{split}\mathbf{m}\big{(}\bar{u}_{rx}^{\widetilde{\nu}},Q_{(1+\alpha)r}^{\widetilde{\nu}}(rx)\big{)}-c_{\alpha}r^{n-1}&\leq\mathbf{m}\big{(}\bar{u}_{rx}^{\nu},Q_{r}^{\nu}(rx)\big{)}\\\
&\leq\mathbf{m}\big{(}\bar{u}_{rx}^{\widetilde{\nu}},Q_{(1-\alpha)r}^{\widetilde{\nu}}(rx)\big{)}+c_{\alpha}r^{n-1}\,.\end{split}$
(A.8)

###### Proof.

We show only that

$\mathbf{m}\big{(}\bar{u}_{rx}^{\widetilde{\nu}},Q_{(1+\alpha)r}^{\widetilde{\nu}}(rx)\big{)}-c_{\alpha}r^{n-1}\leq\mathbf{m}\big{(}\bar{u}_{rx}^{\nu},Q_{r}^{\nu}(rx)\big{)},$
(A.9)

for some $c_{\alpha}>0$, with $c_{\alpha}\to 0$, as $\alpha\to 0$; as the
proof of the other inequality is analogous. Let $x\in\mathbb{R}^{n}$, $r>2$,
and set $r_{\alpha}^{\pm}:=(1\pm\alpha)r$. From (A.7) we might deduce that

$Q_{r_{\alpha}^{-}}^{\widetilde{\nu}}(rx)\subset\subset
Q_{r}^{\nu}(rx)\subset\subset Q_{r_{\alpha}^{+}}^{\widetilde{\nu}}(rx).$

Let $\eta>0$ be fixed and let $u\in W^{1,p}(Q^{\nu}_{r}(rx))$ be a test
function for $\mathbf{m}(\bar{u}_{rx}^{\nu},Q^{\nu}_{r}(rx))$ satisfying

$\mathscr{F}(u,Q^{\nu}_{r}(rx))\leq\mathbf{m}(\bar{u}_{rx}^{\nu},Q^{\nu}_{r}(rx))+\eta.$
(A.10)

Choose $\beta\in(0,1)$ such that
$Q_{r_{\alpha}^{-}}^{\widetilde{\nu}}(rx)\subset\subset Q^{\nu}_{r-\beta}(rx)$
and $u=\bar{u}^{\nu}_{rx}$ a.e. in
$Q_{r}^{\nu}(rx)\setminus\overline{Q}_{r-\beta}^{\nu}(rx)$ and set

$R:=R_{\nu}\Big{(}Q^{\prime}_{r}\setminus\overline{Q^{\prime}}_{r-\beta}\times(-1-\alpha
r,1+\alpha r)\Big{)}+rx\,,$

where $R_{\nu}$ is as in (g). Next define

$\widetilde{u}:=\varphi u+(1-\varphi)\bar{u}^{\widetilde{\nu}}_{rx}\,,$

where $\varphi$ is a smooth cutoff between $Q^{\nu}_{r-\beta}(rx)$ and
$Q^{\nu}_{r}(rx)$. Then
$\widetilde{u}\in\mathscr{A}(\bar{u}^{\widetilde{\nu}}_{rx},Q_{r_{\alpha}^{+}}^{\widetilde{\nu}}(rx))$
and it also satisfies

$\mathscr{F}(\widetilde{u},Q^{\widetilde{\nu}}_{r_{\alpha}^{+}}(rx))\leq\mathscr{F}(u,Q_{r}^{\nu}(rx))+\mathscr{F}(\widetilde{u},Q^{\nu}_{r}(rx)\setminus\overline{Q}_{r-\beta}^{\nu}(rx))+\mathscr{F}(\bar{u}^{\widetilde{\nu}}_{rx},Q^{\widetilde{\nu}}_{r_{\alpha}^{+}}(rx)\setminus\overline{Q}_{r_{\alpha}^{-}}^{\nu}(rx))\,.$
(A.11)

By definition we have

$\widetilde{u}=\varphi\bar{u}^{\nu}_{rx}+(1-\varphi)\bar{u}^{\widetilde{\nu}}_{rx}\quad\text{
in }\quad Q^{\nu}_{r}(rx)\setminus\overline{Q}_{r-\beta}^{\nu}(rx)\,,$

and in particular

$\widetilde{u}=u^{\nu}_{rx}={u}^{\widetilde{\nu}}_{rx}\in\\{0,1\\}\quad\text{
a.e. in
}\quad(Q^{\nu}_{r}(rx)\setminus\overline{Q}_{r-\beta}^{\nu}(rx))\setminus\overline{R}\,.$

Therefore from (2.6), (($f3$)) and (2.10) we deduce that

$\begin{split}\mathscr{F}(\widetilde{u},Q^{\nu}_{r}(rx)\setminus\overline{Q}_{r-\beta}^{\nu}(rx))&\leq
c_{2}\int_{R}\left(W(\widetilde{u})+|\nabla\widetilde{u}|^{p}\right)\,{\rm
d}y\\\ &\leq
C\mathcal{L}^{n}(R)+C\int_{R}\left(|\nabla\bar{u}^{\nu}_{rx}|^{p}+|\nabla\bar{u}^{\widetilde{\nu}}_{rx}|^{p}+|\nabla\varphi|^{p}|\bar{u}^{\nu}_{rx}-\bar{u}^{\widetilde{\nu}}_{rx}|^{p}\right)\,{\rm
d}y\\\ &\leq
C\mathcal{L}^{n}(R)+C\mathcal{L}^{n-1}(Q^{\prime}_{r}\setminus\overline{Q}^{\prime}_{r-\beta})\,.\end{split}$
(A.12)

On the other hand from (2.10) and (2.6) we infer

$\mathscr{F}(\bar{u}^{\widetilde{\nu}}_{rx}(y),Q^{\widetilde{\nu}}_{r_{\alpha}^{+}}(rx)\setminus
Q^{\widetilde{\nu}}_{r_{\alpha}^{-}}(rx))\leq c_{2}C_{\rm
u}((1+\alpha)^{n-1}-(1-\alpha)^{n-1})r^{n-1}\,.$ (A.13)

Using that

$\mathcal{L}^{n-1}(Q^{\prime}_{r}\setminus\overline{Q}^{\prime}_{r-\beta})\leq
Cr^{n-2}\,,\quad\text{ and }\quad\mathcal{L}^{n}(R)\leq C\beta r^{n-2}\alpha
r\leq C\alpha r^{n-1}\,.$

together with (A.11), (A.12), and (A.13) we obtain

$\mathscr{F}(\widetilde{u},Q^{\widetilde{\nu}}_{r_{\alpha}^{+}}(rx))\leq\mathscr{F}(u,Q_{r}^{\nu}(rx))+\mathscr{F}(\bar{u}^{\widetilde{\nu}}_{rx}(y),Q^{\widetilde{\nu}}_{r_{\alpha}^{+}}(rx)\setminus
Q^{\widetilde{\nu}}_{r_{\alpha}^{-}}(rx)+c_{\alpha}r^{n-1}\,,$

where $c_{\alpha}:=c_{4}C_{{\rm
u}}\big{(}(1+\alpha)^{n-1}-(1-\alpha)^{n-1}\big{)}+C\alpha$. Eventually as the
above inequality together with (A.10) imply

$\displaystyle\mathbf{m}(\bar{u}_{rx}^{\widetilde{\nu}},Q_{r_{\alpha}^{+}}^{\widetilde{\nu}}(rx))\leq\mathbf{m}(\bar{u}_{rx}^{\nu},Q_{r}^{\nu}(rx))+c_{\alpha}r^{n-1}+\eta\,,$

we may deduce (A.9) by the arbitrariness of $\eta>0$. ∎

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