# Non-local approximation of free-discontinuity problems in linear elasticity
and application to stochastic homogenisation

Roberta Marziani  Faculty of Mathematics, TU Dortmund University,
Vogelpothsweg 87, 44227 Dortmund, Germany<EMAIL_ADDRESS>and  Francesco Solombrino  Dipartimento di Matematica e Applicazioni “R.
Caccioppoli”, Università di Napoli Federico II, via Cintia, 80126 Napoli,
Italy<EMAIL_ADDRESS>
###### Abstract.

We analyse the $\Gamma$-convergence of general non-local convolution type
functionals with varying densities depending on the space variable and on the
symmetrized gradient. The limit is a local free-discontinuity functional,
where the bulk term can be completely characterized in terms of an asymptotic
cell formula. From that, we can deduce an homogenisation result in the
stochastic setting.

###### Key words and phrases:

Keywords: Non-local approximation, variational fracture, free discontinuity
problems, functions of bounded deformations, $\Gamma$-convergence,
deterministic and stochastic homogenisation.

###### 1991 Mathematics Subject Classification:

MSC 2010: 49J45, 49Q20, 74Q05, 74R10, 70G75.

##  1\. Introduction

This paper is focused on the approximation of brittle fracture energies for
linearly elastic materials, by means of non-local functionals defined on
Sobolev spaces. The asymptotic behavior of these functionals will
simultaneously show the emergence both of effective energies for the elastic
deformation (which may be, e.g., the output of homogenization), and of
Griffith-type surface energies accounting for crack formation. In turn, this
result can be further generalized to the setting of stochastic homogenization
with fracture.

Precisely our results will extend the range of application of the recent
papers [18, 23] while also providing some relevant technical improvement. We
briefly comment on these previous contributions, in order to introduce our
results. There, an approach originally devised by Braides and Dal Maso [7] for
the approximation of the Mumford-Shah functional has been generalized to the
linearly elastic setting. Namely, it was shown that, for a given bounded
increasing function $f\colon\mathbb{R}^{+}\to\mathbb{R}^{+}$ the energies

$\frac{1}{\varepsilon_{k}}\int_{U}f\Big{(}\varepsilon_{k}W(e(u))*\rho_{k}(x)\Big{)}\,{\rm
d}x\,,$ (1.1)

$\Gamma$-converge, in the $L^{1}(U)$-topology, to the functional

$\alpha\int_{U}W(e(u))\,\mathrm{d}x+\beta\int_{J_{u}}\phi_{\rho}(\nu_{u})\,{\rm
d}\mathcal{H}^{n-1}\,,$ (1.2)

with $\alpha=f^{\prime}(0)$ and $\beta=\lim_{t\to+\infty}f(t)$. Above,
$\rho_{k}$ are rescaled convolution kernels with unit mass and compact convex
symmetrical support $S$, $\phi_{\rho}$ is (twice) the support function of $S$
(see (2.4) for its precise definition), $W(e(u))$ is a convex elastic energy
with superlinear $p$ growth depending on the symmetrized gradient $e(u)$ of a
vector-valued displacement $u$, whose jump set is denoted by $J_{u}$. Notice
that the effective domains of the approximations and of the limit are
different. Actually (1.1) is finite on the Sobolev space
$W^{1,p}(U;\mathbb{R}^{n})$, while the energy space of (1.2) is the one of
generalized functions with bounded deformation $GSBD^{p}(U)$, introduced in
[14].

We stress that the above results allowed one for a general (convex) bulk
energy. The proof strategy cannot rely, at least when estimating the bulk
part, on any slicing procedure. The latter is instead successful in the
particular case $W(\xi)=|\xi|^{p}$, considered for instance in [21]. We also
remark that the results of [18, 23] were obtained under an additional
structural assumption on the kernels $\rho_{k}$, which have to be radial with
respect to the norm induced by $S$. In the particular case considered in [21],
this restriction was instead not needed.

A natural extension of the aforementioned models allows one to include an
explicit dependence on $k$ and on the space variable of the energy density.
This amounts to consider functionals of the form

$\frac{1}{\varepsilon_{k}}\int_{U}f\Big{(}\varepsilon_{k}W_{k}(\cdot,e(u))*\rho_{k}(x)\Big{)}\,{\rm
d}x\,,$ (1.3)

whose limit behaviour is the object of the present paper. This more general
setting is indeed suitable for further applications, if one thinks to the
mechanical counterpart of the model. Indeed energy densities of type
$W_{k}(y,M)$, where $y$ is the position in the reference configuration, are
customary when dealing with heterogeneous material with some microstructures.
The prototypical example is the case of homogenisation, that is, when
$W_{k}(y,M)=W(\frac{y}{\delta_{k}},M)$ with $\delta_{k}\searrow 0$.

The main result of our paper is contained in Theorem 2.1. There we show that
the functionals in (1.3) $\Gamma$-converge to a limit energy of the form

$\alpha\int_{U}W(x,e(u))\,\mathrm{d}x+\beta\int_{J_{u}}\phi_{\rho}(\nu_{u})\,{\rm
d}\mathcal{H}^{n-1}\,.$ (1.4)

Above, the limit bulk density $W$ can be characterised in terms of cell
formula (see (2.7)-(2.8)). Remarkably, that coincides exactly with the
asymptotic formula that one would obtain by considering the limit behaviour of
the local energies $\int_{U}W_{k}(x,e(u))\,{\rm d}x$ in the Sobolev space
$W^{1,p}(U)$. Hence, a decoupling effect between bulk and surface contribution
occurs, since the volume energy only depends on $f$ trough its derivative at
the origin. A similar effect has been observed in [11] where the analogue of
(1.3) for energies depending on the full deformation gradient was taken into
account. On the one hand, the possibility of using smooth truncations (a tool
which is not available in $GSBD$) allowed the author there to replace $f$ by a
sequence $f_{k}$ and to derive more general surface energies in the limit. On
the other hand, the precise characterisation of the volume energy density was
obtained at the expense of an additional technical condition on the $W_{k}$’s
(the so-called stable $\gamma$-convergence). It actually turns out, as an
output of our proof strategy, that this extra assumption can be dropped out
(see Appendix A). Thus, our results also permit some improvement in the
previous literature about non-local approximation of free-discontinuity
problems.

We now come to the description of our proof technique. The most difficult
point is the lower bound for the bulk contribution. This is done in
Proposition 5.1, by means of a localisation and blow-up procedure which
contains some elements of novelty in the non-local setting. More precisely we
consider the blow-up of sequences with equi-bounded energies at a Lebesgue
point for the limit energy. A crucial task is to gain a uniform control on the
$L^{p}$ norm of the symmetrized gradients of the blow-up functions up to sets
with vanishing perimeter. This allows us to apply [19, Lemma 5.1] (which
relies on the Korn-type inequality of [8]): we can substitute, with almost no
change in the energy, the above mentioned sequence with a more regular one
bounded in $W^{1,p}$. Exploiting the properties of $f$ we are then reconducted
to analyse the limit behaviour on small squares of a local energy in
$W^{1,p}$, which can be estimated from below via a cell formula.

An optimal estimate from below for the surface term can be obtained by means
of a slicing procedure (Proposition 5.3). As for the $\Gamma$-limsup
inequality it can be achieved by a direct construction for a class of
competitors with regular jump set, which are dense in energy. Here we use the
classical approximation result of [9, 12].

We underline that even in the case of (1.1) (i.e., with $W$ not depending on
$k$) we have some technical improvement in comparison with the result of [18,
23]. First of all we do not need anymore to assume the kernels $\rho_{k}$ to
be radially symmetric. Secondly our $\Gamma$-convergence argument is carried
out with respect to the convergence in measure instead of the $L^{1}$
convergence. This is (almost) the natural one for sequences with equibounded
energy (see Theorem 2.1-$(ii)$). It can be indeed shown that such sequences
are compact in the measure convergence up to an exceptional set $U^{\infty}$,
where their modulus diverges. However, this set can be easily made empty by
adding a penalisation term in the energy (see the statement of Theorem 3.5 and
Remark 2.2).

Eventually we complement our analysis with a stochastic homogenisation result
Theorem 7.4. Namely we consider functionals of type (1.3) with stationary
random integrands

$W_{k}(\omega,y,M)=W\Big{(}\omega,\frac{y}{\delta_{k}},M\Big{)}\,,$ (1.5)

where $\omega$ belongs to the sample space $\Omega$ of a probability space
$(\Omega,\mathcal{T},P)$ and $\delta_{k}\searrow 0$. Following the approach
proposed by [15] (which relies on the Subadditive Ergodic Theorem in [1]) we
show that, almost surely, such functionals $\Gamma$-converge to a free-
discontinuity functional of the form (1.4) where the bulk energy density is
independent of the space variable. A similar result was obtained in [3] in the
context of elliptic approximation of free-discontinuity functionals.

Plan of the paper. The paper is structured as follows. After fixing the
notation, in Section 2, we introduce the problem, discuss the assumptions and
state our main results. Section 3 is devoted to recalling preliminary results
which are useful for the analysis. The proof of Theorem 2.1 is carried out
trough the Sections 4–6, dealing with compactness, lower, and upper bound,
respectively. In Section 7 we prove a stochastic homogenisation result Theorem
7.4. Eventually in the Appendix we briefly comment on the result of [11,
Theorem 3.2], highlighting that the assumptions made there can actually be
weakened. A complete statement is given for the readers convenience in Theorem
A.1.

##  2\. Setting of the problem and main results

###  2.1. Notation

We start by collecting the notation adopted throughout the paper.

  1. (a)

$n\geq 2$ is a fixed integer and $p>1$ is a fixed real number;

  2. (b)

$\mathbb{M}^{n\times n}$ denotes the space of $n\times n$ real matrices;
$\mathbb{M}^{n\times n}_{\rm sym}$ and $\mathbb{M}^{n\times n}_{\rm skew}$
denote the spaces of symmetric and skew-symmetric matrices respectively;

  3. (c)

for a subset $A\subset\mathbb{R}^{n}$ $\partial^{*}A$ denotes the essential
boundary of $A$;

  4. (d)

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

  5. (e)

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

  6. (f)

$U$ denotes an open bounded subset of $\mathbb{R}^{n}$ with Lipschitz
boundary;

  7. (g)

we denote by $\mathcal{A}(U)$ and $\mathcal{A}$ the collection of all open and
bounded subsets of $U$ and $\mathbb{R}^{n}$ respectively;

  8. (h)

If $A,B\in\mathcal{A}(U)$ (or $\mathcal{A}$) by $A\subset\subset B$ we mean
that $A$ is relatively compact in $B$;

  9. (i)

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

  10. (j)

for every $\xi\in{\mathbb{S}}^{n-1}$ let $R_{\xi}$ denote an orthogonal
$(n\times n)$-matrix such that $R_{\xi}e_{n}=\xi$;

  11. (k)

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

  12. (l)

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 $\mathcal{B}$;

  13. (m)

we denote by $L^{0}(U;\mathbb{R}^{n})$ the space of measurable functions;

  14. (n)

for $a,b\in\mathbb{R}^{n}$ the symbol $a\otimes b$ denotes the tensor product
between $a$ and $b$, while $a\odot b:=\frac{1}{2}(a\otimes b+b\otimes a)$.

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

###  2.2. (G)SBV and (G)SBD functions

We will work with the functional spaces $(G)SBV^{p}(U;\mathbb{R}^{n})$ and
$(G)SBD^{p}(U)$ for which we will recall the main properties and refer the
reader to [2, 14] for a complete exposition of the subject. We say that $u\in
L^{1}(U;\mathbb{R}^{n})$ belongs to the space of special functions with
bounded variation, i.e., $u\in SBV(U;\mathbb{R}^{n})$, if its distributional
gradient is a finite $\mathbb{M}^{n\times n}$-valued Radon measure without
Cantor part, that is,

$Du=\nabla
u\mathcal{L}^{n}+[u]\otimes\nu_{u}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits J_{u}\,,$

where $\nabla u$ is the approximate gradient, $J_{u}$ is the approximate jump
set, $[u]=u^{+}-u^{-}$ the jump opening and $\nu_{u}$ the unit normal to
$J_{u}$. A function $u\in L^{0}(U;\mathbb{R}^{n})$ belongs to the space of
generalised special functions with bounded variation, i.e., $u\in
GSBV(U;\mathbb{R}^{n})$, if for any $\varphi\in
C^{1}(\mathbb{R}^{n};\mathbb{R}^{n})$ with support of $\nabla\varphi$ compact
it holds $\varphi\circ u\in SBV_{\rm loc}(U;\mathbb{R}^{n})$.

We say that $u\in L^{1}(U;\mathbb{R}^{n})$ belongs to the space of special
functions with bounded deformation, and we write $u\in SBD(U)$, if its
symmetrized distributional gradient is a finite $\mathbb{M}^{n\times n}_{\rm
sym}$-valued Radon measure without Cantor part, that is,

$Eu=\frac{Du+(Du)^{T}}{2}=e(u)\mathcal{L}^{n}+[u]\odot\nu_{u}\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits J_{u}\,,$

with $e(u)$ is the approximate symmetric gradient with respect to the Lebesgue
measure. On the contrary the space of generalised special functions with
bounded deformation, $GSBD(U)$, cannot be defined analogously to the space
$GSBV(U;\mathbb{R}^{n})$ as if $u\in SBD(U)$ and $\varphi$ is as above, then
in general $\varphi\circ u\notin SBD(U)$. To overcome this issue, Dal Maso in
[14] proposed a definition of this space by relying on a slicing argument
which we describe in the following.
For $\xi\in\mathbb{R}^{n}\setminus\\{0\\}$ we let
$\Pi^{\xi}:=\\{y\in\mathbb{R}^{n}\colon\langle\xi,y\rangle=0\\}$; for any
$y\in\Pi^{\xi}$ and $A\in\mathcal{B}(U)$ we set

$A_{\xi,y}:=\\{t\in\mathbb{R}\colon y+t\xi\in A\\}\,.$

Given $u\colon U\to\mathbb{R}^{n}$ we define $u^{\xi,y}\colon
U_{\xi,y}\to\mathbb{R}$ by

$u^{\xi,y}(t):=\langle u(y+t\xi),\xi\rangle\,.$

If $u^{\xi,y}\in SBV(U_{\xi,y};\mathbb{R})$ we set

$J_{u^{\xi,y}}^{1}:=\\{t\in J_{u^{\xi,y}}\colon|[J_{u^{\xi,y}}](t)|\geq
1\\}\,.$

We then say that $u\in L^{0}(U;\mathbb{R}^{n})$ belongs to the space of
generalised special functions with bounded deformation, and we write $u\in
GSBD(U)$, if there exists a bounded Radon measure $\lambda$ on $U$ such that
$u^{\xi,y}\in SBV_{\rm loc}(U_{\xi,y})$ for all $\nu\in\mathbb{S}^{n-1}$ and
all $y\in\Pi^{\xi}$ and

$\int_{\Pi^{\xi}}\Big{(}|Du^{\xi,y}|(A_{\xi,y}\setminus
J_{u^{\xi,y}}^{1})+\mathcal{H}^{0}(A_{\xi,y}\cap
J_{u^{\xi,y}}^{1})\Big{)}\,{\rm d}\mathcal{H}^{n-1}(t)\leq\lambda(A)\,,$

for all $A\in\mathcal{B}(U)$. Eventually we set

$GSBV^{p}(U):=\\{u\in(G)SBV(U;\mathbb{R}^{n})\colon\nabla u\in
L^{p}(U;\mathbb{M}^{n\times n})\ \text{and}\
\mathcal{H}^{n-1}(J_{u})<+\infty\\}\,;$

and

$GSBD^{p}(U):=\\{u\in(G)SBD(U)\colon e(u)\in L^{p}(U;\mathbb{M}^{n\times
n}_{\rm sym})\ \text{and}\ \mathcal{H}^{n-1}(J_{u})<+\infty\\}\,,$

where $\nabla u$ and $e(u)$ are well defined also in $GSBV(U;\mathbb{R}^{n})$
and $GSBD(U)$ respectively.

###  2.3. Setting of the problem

Let $1<p<+\infty$; let $c_{1},c_{2}$ be given positive constants such that
$0<c_{1}\leq c_{2}<+\infty$. Let $\mathcal{W}:=\mathcal{W}(p,c_{1},c_{2})$ be
the collection of all functions
$W\colon\mathbb{R}^{n}\times\mathbb{M}^{n\times n}\to\mathbb{R}$ satisfying
the following conditions:

  1. ($W1$)

$W$ is a Carathéodory function on $\mathbb{R}^{n}\times\mathbb{M}^{n\times
n}$;

  2. ($W2$)

$W(x,0)=0$ for every $x\in\mathbb{R}^{n}$;

  3. ($W3$)

for every $x\in\mathbb{R}^{n}$, $M\in\mathbb{M}^{n\times n}$ and
$S\in\mathbb{M}^{n\times n}_{\rm skew}$

$W(x,M+S)=W(x,M)\,;$

  4. ($W4$)

for every $x\in\mathbb{R}^{n}$ and every $M\in\mathbb{M}^{n\times n}$

$c_{1}|M+M^{T}|^{p}\leq W(x,M)\leq c_{2}(|M+M^{T}|^{p}+1)\,.$

Let $f\colon[0,+\infty)\to[0,+\infty)$ be a lower semi-continuous increasing
function such that there exist $\alpha,\beta>0$ with

$\lim_{t\to
0^{+}}\frac{f(t)}{t}=\alpha\,,\quad\lim_{t\to+\infty}f(t)=\beta\,.$ (2.1)

Note that for such $f$ it holds

$f(t)\leq\hat{\alpha}t\quad\forall\hat{\alpha}>\alpha\,;$ (2.2)

moreover by [23, Lemma 2.10] there exist $(\alpha_{i})_{i\in\mathbb{N}}$,
$(\beta_{i})_{i\in\mathbb{N}}$ sequences of positive numbers with
$\sup_{i}\alpha_{i}=\alpha$, $\sup_{i}\beta_{i}=\beta$ such that

$f(t)\geq f_{i}(t):=\alpha_{i}t\wedge\beta_{i}\quad\forall i\in\mathbb{N}\,,\
t\in\mathbb{R}\,.$ (2.3)

Let $\rho\in L^{\infty}(\mathbb{R}^{n};[0,+\infty))$ be a lower semi-
continuous convolution kernel with $\int_{\mathbb{R}^{n}}\rho\,{\rm d}x=1$ and
$S:={\rm supp}(\rho)$ bounded, convex, symmetrical and that $0\in S$. We
denote by $|\cdot|_{S}$ the norm induced by $S$, namely,

$|x|_{S}:=\inf\\{\lambda>0\colon x\in\lambda S\\}\,,$

so that $S=\\{|x|_{S}<1\\}$. Then for any bounded set $K\subset\mathbb{R}^{n}$
and $x\in\mathbb{R}^{n}$ we let

$\,\mathrm{d}_{S}(x,K):=\inf_{y\in K}|x-y|_{S}\,.$

For any Borel set $E$ and any $r>0$ we denote by $E_{r}$ and $E_{-r}$
respectively the sets

$E_{r}:=\\{x\in\mathbb{R}^{n}\colon\,\mathrm{d}_{S}(x,E)<r\\}\,,\quad
E_{-r}:=\\{x\in\mathbb{R}^{n}\colon\,\mathrm{d}_{S}(x,E^{c})>r\\}\,.$

Finally we let $\phi_{\rho}\colon\mathbb{R}^{n}\to[0,+\infty)$ be given by

$\phi_{\rho}(\nu):=2\sup_{y\in S}|y\cdot\nu|\,.$ (2.4)

For $\delta>0$ we set
$\rho_{\delta}(x):=\frac{1}{\delta^{n}}\rho(\frac{x}{\delta})$,
$S_{\delta}(x):=x+\delta S$.

For $k\in\mathbb{N}$ let $(W_{k})\subset\mathcal{W}$, let $(\varepsilon_{k})$
be a decreasing sequence of strictly positive real numbers converging to zero,
as $k\to+\infty$ and let $\rho_{k}:=\rho_{\varepsilon_{k}}$. We consider the
family of functionals $F_{k}\colon L^{0}(U;\mathbb{R}^{n})\to[0,+\infty]$
defined as

$F_{k}(u):=\begin{cases}\displaystyle\frac{1}{\varepsilon_{k}}\int_{U}f\Big{(}\varepsilon_{k}W_{k}(\cdot,e(u))*\rho_{k}(x)\Big{)}\,{\rm
d}x&\text{ if }u\in W^{1,p}(U;\mathbb{R}^{n})\,,\\\ +\infty&\text{ otherwise
}.\end{cases}$ (2.5)

Let $x\in\mathbb{R}^{n}$, $M\in\mathbb{M}^{n\times n}$, $A\in\mathcal{A}$ and
$u\in W^{1,p}(A;\mathbb{R}^{n})$ be fixed. Set $u_{M}(y):=My$. We then define
the minimisation problem

$\textbf{m}_{k}(u_{M},A):=\inf\left\\{\int_{A}W_{k}(x,e(v))\,{\rm d}x\colon
v\in W^{1,p}(A;\mathbb{R}^{n}),\ v=u_{M}\ \text{near}\ \partial A\right\\}\,,$
(2.6)

and the cell formulas

$\begin{split}W^{\prime}(x,M):=\limsup_{r\searrow
0^{+}}\liminf_{k\to+\infty}\frac{\textbf{m}_{k}(u_{M},Q_{r}(x))}{r^{n}}\,,\end{split}$
(2.7) $\begin{split}W^{\prime\prime}(x,M):=\limsup_{r\searrow
0^{+}}\limsup_{k\to+\infty}\frac{\textbf{m}_{k}(u_{M},Q_{r}(x))}{r^{n}}\,.\end{split}$
(2.8)

###  2.4. Main Results

In this Section we state our main results. The first one is a
$\Gamma$-convergence theorem for the energies $F_{k}$.

######  Theorem 2.1 ($\Gamma$-convergence of $F_{k}$).

Let $F_{k}$ be as in (2.5). Then the following hold:

  1. $(i)$

There exists a subsequence, not relabelled, such that $F_{k}$
$\Gamma$-converges with respect to the convergence in measure to the
functional $F\colon L^{0}(U;\mathbb{R}^{n})\to[0,+\infty]$ given by

$F(u):=\begin{cases}\displaystyle\ \alpha\int_{U}W(x,e(u))\,{\rm
d}x+\beta\int_{J_{u}}\phi_{\rho}(\nu_{u})\,{\rm d}\mathcal{H}^{n-1}&\text{if
}u\in GSBD^{p}(U)\,,\\\ +\infty&\text{otherwise}\,,\end{cases}$ (2.9)

with $\phi_{\rho}$ as in (2.4) and for every $x\in U$ and every
$M\in\mathbb{M}^{n\times n}$

$\begin{split}W(x,M)=W(x,{\rm
sym}(M))=W^{\prime}(x,M)=W^{\prime\prime}(x,M)\,,\end{split}$

with $W^{\prime\prime}$, $W^{\prime\prime}$ as in (2.7) and (2.8);

  2. $(ii)$

Let $(u_{k})\subset L^{0}(U;\mathbb{R}^{n})$ be such that
$\sup_{k}F_{k}(u_{k})<+\infty$. Then there exists $u\in GSBD^{p}(U)$ such
that, up to subsequence, it holds

$u_{k}\to u\quad\text{in measure on }U\setminus U^{\infty}\,,$
$e(u_{k})\rightharpoonup e(u)\quad\text{in }L^{p}(U\setminus
U^{\infty};\mathbb{M}^{n\times n}_{\rm sym})\,,$
$\liminf_{k\to+\infty}\mathcal{H}^{n-1}(J_{u_{k}})\geq\mathcal{H}^{n-1}(J_{u}\cup(\partial^{*}U^{\infty}\cap
U))\,,$

where $U^{\infty}:=\\{x\in U\colon|u_{k}(x)|\to+\infty\\}$. If in addition

$\sup_{k\in\mathbb{N}}\int_{U}\psi(|u_{k}|)\,{\rm d}x<+\infty\,,$

for some $\psi\colon[0,+\infty)\to[0,+\infty)$, continuous, increasing with
$\lim_{s\to+\infty}\psi(s)=+\infty$, then $U^{\infty}=\emptyset$, so that
$|u|$ is finite a.e., and $u_{k}\to u$ in measure on $U$.

######  Remark 2.2.

The addition of a penalty term of the form $\int_{U}\psi(|u|)\,{\rm d}x$ to
the energy enforces then compactness in measure, while causing no troubles in
the $\Gamma$-convergence analysis. Indeed, such a term is clearly lower
semicontinuous, hence the corresponding lower bound follows immediately. As
for the upper bound, if one takes $\psi$ as in Theorem 3.4, the argument of
Proposition 6.1 can be readily adapted also in presence of such an additional
term. As this is not the core of the argument, we will neglect lower order
terms in our statements and proofs, directly assuming that convergence in
measure holds everywhere. The technical details left to prove the upper bound
are summarized in Remark 6.2 for the readers convenience.

The proof of Theorem 2.1 is divided into three main steps contained
respectively in sections 4, 5 and 6. As a consequence of Theorem 2.1 and the
Urysohn property of $\Gamma$-convergence [13, Proposition 8.3] we deduce the
following corollary.

######  Corollary 2.3.

Let $(W_{k})\subset\mathcal{W}$ and let $F_{k}$ be the functionals as in
(2.5). Let $W^{\prime}$, $W^{\prime\prime}$ be as in (2.7) and (2.8),
respectively. Assume that

$W^{\prime}(x,M)=W^{\prime\prime}(x,M)=:W(x,M)\,,\quad\text{for a.e.
}x\in\mathbb{R}^{n}\text{ and for every }M\in\mathbb{M}^{n\times n}\,,$

for some Borel function $W\colon\mathbb{R}^{n}\times\mathbb{M}^{n\times
n}\to[0,+\infty)$. Let $F$ defined as in (2.9) accordingly. Then the
functionals $F_{k}$ $\Gamma$-converge with respect to the convergence in
measure to $F$. Moreover

$\begin{split}W(x,M)=W(x,{\rm
sym}(M))=W^{\prime}(x,M)=W^{\prime\prime}(x,M)\,,\end{split}$

for every $x\in U$ and every $M\in\mathbb{M}^{n\times n}$.

We now state a homogenisation theorem without assuming any spatial periodicity
of the energy densities $W_{k}$. We start by introducing some notation. We fix
$W\in\mathcal{W}$ and set

$\mathbf{m}(u_{M},A):=\inf\left\\{\int_{A}W(x,e(v))\,{\rm d}x\colon v\in
W^{1,p}(A;\mathbb{R}^{n}),\ v=u\ \text{near}\ \partial A\right\\}\,,$ (2.10)

for all $A\in\mathcal{A}$ and all $M\in\mathbb{M}^{n\times n}$ Let also
$(W_{k})\subset\mathcal{W}$ be given by

$W_{k}(x,M):=W\Big{(}\frac{x}{\delta_{k}},M\Big{)}\,,$ (2.11)

with $\delta_{k}\searrow 0$ when $k\to+\infty$.

######  Theorem 2.4 (Deterministic homogenisation).

Let Let $W\in\mathcal{W}$ and let $\mathbf{m}(u_{M},Q_{t}(tx))$ be as in
(2.10) with $A=Q_{t}(tx)$. Assume that for every $x\in\mathbb{R}^{n}$,
$M\in\mathbb{M}^{n\times n}$ the following limit

$\lim_{t\to+\infty}\frac{\mathbf{m}(u_{M},Q_{t}(tx))}{t^{n}}=:W_{\rm
hom}(M)\,,$ (2.12)

exists and is independent of $x$. Then the functionals $F_{k}$ defined in
(2.5) with $W_{k}$ as in (2.11) $\Gamma$-converge with respect to the
convergence in measure to the functional $F_{\rm hom}\colon
L^{0}(U;\mathbb{R}^{n})\to[0,+\infty]$ given by

$F_{\rm hom}(u):=\begin{cases}\displaystyle\ \alpha\int_{U}W_{\rm
hom}(e(u))\,{\rm d}x+\beta\int_{J_{u}}\phi_{\rho}(\nu_{u})\,{\rm
d}\mathcal{H}^{n-1}&\text{if }u\in GSBD^{p}(U)\,,\\\
+\infty&\text{otherwise}\,,\end{cases}$ (2.13)

with $\phi_{\rho}$ as in (2.4). Moreover $W_{\rm hom}(M)=W_{\rm hom}({\rm
sym}(M))$ for all $M\in\mathbb{M}^{n\times n}$.

###### Proof.

Let $W^{\prime}$, $W^{\prime\prime}$ be respectively as in (2.7) and (2.8). By
Corollary 2.3 it is sufficient to show that

$W_{\rm hom}(M)=W^{\prime}(x,M)=W^{\prime\prime}(x,M)\,,$ (2.14)

for all $x\in\mathbb{R}^{n}$ and $M\in\mathbb{M}^{n\times n}$. We fix
$x\in\mathbb{R}^{n}$, $M\in\mathbb{M}^{n\times n}$, $r>0$ and
$k\in\mathbb{N}$. For any $u\in W^{1,p}(Q_{r}(x);\mathbb{R}^{n})$ with
$u=u_{M}$ near $\partial Q_{r}(x)$ we let $u_{k}\in
W^{1,p}(Q_{\frac{r}{\delta_{k}}}(\frac{x}{\delta_{k}});\mathbb{R}^{n})$ be
given by $u_{k}(y):=\frac{1}{\delta_{k}}u(\delta_{k}y)$. Then clearly
$u_{k}=u_{M}$ near $\partial Q_{\frac{r}{\delta_{k}}}(\frac{x}{\delta_{k}})$.
Moreover by performing the change of variable $\hat{y}=\frac{y}{\delta_{k}}$
we find

$\int_{Q_{r}(x)}W\Big{(}\frac{y}{\delta_{k}},e(u)\Big{)}\,{\rm
d}y=\delta_{k}^{n}\int_{Q_{\frac{r}{\delta_{k}}}(\frac{x}{\delta_{k}})}W(y,e(u_{k}))\,{\rm
d}y\,.$

Hence in particular

$\mathbf{m}_{k}(u_{M},Q_{r}(x))=\delta_{k}^{n}\mathbf{m}(u_{M},Q_{\frac{r}{\delta_{k}}}(\tfrac{x}{\delta_{k}}))=\frac{r^{n}}{t_{k}^{n}}\mathbf{m}(u_{M},Q_{t_{k}}(t_{k}\tfrac{x}{r}))\,,$

with $t_{k}:=\frac{r}{\delta_{k}}$. Eventually passing to the limit as
$k\to+\infty$ by (2.12) we deduce

$\lim_{k\to+\infty}\frac{\mathbf{m}_{k}(u_{M},Q_{r}(x))}{r^{n}}=\lim_{k\to+\infty}\frac{\mathbf{m}(u_{M},Q_{t_{k}}(t_{k}\tfrac{x}{r}))}{t_{k}^{n}}=W_{\rm
hom}(M)\,.$

∎

##  3\. Some preliminary results

In this section we collect some useful results that will be employed
throughout the paper. We start by recalling a $\Gamma$-convergence result for
the bulk energies defined in (3.1) (Theorem 3.1) and a $\Gamma$-convergence
result for one-dimensional non-local energies (Theorem 3.3). To follow we
recall a density and a compactness result (cf., Theorem 3.4 and Theorem 3.5).
We conclude this section with a series of technical lemmas (cf. Lemmas 3.6,
3.7, 3.8 and Corollary 3.9).

We consider the family of functionals $E_{k}\colon
L^{0}(\mathbb{R}^{n};\mathbb{R}^{n})\times\mathcal{A}\to[0,+\infty]$ given by

$E_{k}(u,A):=\begin{cases}\displaystyle\int_{A}W_{k}(x,e(u))\,{\rm d}x&\text{
if }u\in W^{1,p}(A;\mathbb{R}^{n})\,,\\\ +\infty&\text{ otherwise
}.\end{cases}$ (3.1)

######  Theorem 3.1 ($\Gamma$-convergence of $E_{k}$).

Let $E_{k}$ be as in (3.1). Then there exists a subsequence, not relabelled,
such that for every $A\in\mathcal{A}$ the functionals $E_{k}(\cdot,A)$
$\Gamma$-converge, with respect to the convergence in measure, to the
functional $E(\cdot,A)$ with $E\colon
L^{0}(\mathbb{R}^{n};\mathbb{R}^{n})\times\mathcal{A}\to[0,+\infty]$ given by

$E(u,A)=\begin{cases}\displaystyle\int_{A}W(x,e(u))\,{\rm d}x&\text{ if }u\in
W^{1,p}(A;\mathbb{R}^{n})\,,\\\ +\infty&\text{ otherwise }\,,\end{cases}$
(3.2)

where for every $x\in\mathbb{R}^{n}$ and every $M\in\mathbb{M}^{n\times n}$

$\begin{split}W(x,M)=W(x,{\rm
sym}(M))=W^{\prime}(x,M)=W^{\prime\prime}(x,M)\,,\end{split}$ (3.3)

with $W^{\prime}$, $W^{\prime\prime}$ as in (2.7) and (2.8) respectively. The
same $\Gamma$-convergence holds with respect to the $L^{p}_{\rm
loc}(\mathbb{R}^{n};\mathbb{R}^{n})$ convergence.

The proof of Theorem (3.1) is rather standard and follows by the localisation
method (see e.g., [13, Sections 18,19]) and by suitably adapting the integral
representation result in [4, Theorem 2] to our setting with the help of Korn-
Poincaré inequality. For this reason we omit the proof here and we refer the
reader to [19, Proposition 3.13] for more details. We only highlight that the
result holds also for non regular open bounded subsets of $\mathbb{R}^{n}$.
Since this may not be immediately clear from the statement given in [19,
Proposition 3.13], we discuss this point in the remark below.

######  Remark 3.2.

Let $A$ be any open bounded subset of $\mathbb{R}^{n}$ and $u\in
W^{1,p}(A;\mathbb{R}^{n})$. We show that there exists a sequence
$(u_{k})\subset W^{1,p}(A;\mathbb{R}^{n})$ such that $u_{k}\to u$ in
$L^{p}(A;\mathbb{R}^{n})$ and $E_{k}(u_{k},A)\to E(u,A)$. With the use of
Korn-Poincaré inequality, it is clear that this can be done if $A$ is an
extension domain. In the general case, consider smooth relatively compact
subsets $A^{\prime}\subset\subset A^{\prime\prime}\subset\subset A$, and fix
$\eta>0$. We find a sequence $(v_{k})\subset
W^{1,p}(A^{\prime\prime};\mathbb{R}^{n})$ such that
$E_{k}(v_{k},A^{\prime\prime})\to E(u,A^{\prime\prime})$. By the liminf
inequality, this also gives $E_{k}(v_{k},A^{\prime\prime}\setminus
A^{\prime})\to E(u,A^{\prime\prime}\setminus A^{\prime})$. With ($W4$) we have

$\limsup_{k\to+\infty}\int_{A^{\prime\prime}\setminus
A^{\prime}}|e(v_{k})|^{p}\,{\rm
d}x\leq\frac{c_{2}}{c_{1}}\int_{A^{\prime\prime}\setminus
A^{\prime}}(1+|e(u)|^{p})\,{\rm d}x$

Then, considering a cut-off $\varphi$ between $A^{\prime}$ and
$A^{\prime\prime}$ we set $u_{k}:=\varphi v_{k}+(1-\varphi)u$. Clearly
$u_{k}\to u$ in $L^{p}(A;\mathbb{R}^{n})$. Furthermore, by ($W4$) one has

$\begin{split}\limsup_{k\to+\infty}E_{k}(u_{k},A)&=\limsup_{k\to+\infty}E_{k}(v_{k},A^{\prime})+E_{k}(u_{k},A\setminus
A^{\prime})\\\
&\leq\limsup_{k\to+\infty}E_{k}(v_{k},A^{\prime\prime})+c_{2}\left[\int_{A\setminus
A^{\prime}}(1+|e(u)|^{p})\,{\rm d}x\right.\\\
&\left.+\int_{A^{\prime\prime}\setminus
A^{\prime}}(1+|\nabla\phi\odot(v_{k}-u)|^{p}+|e(v_{k})|^{p})\,{\rm
d}x\right]\\\ &\leq
E(u,A^{\prime\prime})+c_{2}\left(1+\frac{c_{2}}{c_{1}}\right)\int_{A\setminus
A^{\prime}}(1+|e(u)|^{p})\,{\rm d}x\leq E(u,A)+\eta\,,\end{split}$

provided $\mathcal{L}^{n}(A\setminus A^{\prime})$ is sufficiently small. The
limsup inequality, which is the only relevant one, follows by a diagonal
argument.

We recall now the following one-dimensional result for non-local energies
given in [5, Theorem 3.30].

######  Theorem 3.3 ($\Gamma$-convergence in 1d).

Let $I\subset\mathbb{R}$ be a bounded interval. Let
$f\colon[0,+\infty)\mapsto[0,+\infty)$ be a lower semi-continuos function
satisfying (2.1) for some $\alpha,\beta>0$. Consider the family of functionals
$G_{k}\colon L^{0}(I)\to[0,+\infty]$ defined by

$G_{k}(w):=\frac{1}{\varepsilon_{k}}\int_{I}f\Big{(}\frac{1}{2}\int_{x-\varepsilon_{k}}^{x+\varepsilon_{k}}|\dot{w}(y)|^{p}\,{\rm
d}y\Big{)}\,{\rm d}x\,,$

if $u\in W^{1,p}(I)$ and $+\infty$ otherwise. Then $G_{k}$ $\Gamma$-converge
with respect to the convergence in measure, to the functional $G\colon
L^{0}(I)\to[0,+\infty]$ given by

$G(w):=\alpha\int_{I}|\dot{w}|^{p}\,{\rm d}x+2\beta\\#(J_{w})\,,$

if $w\in SBV(I)$ and $+\infty$ otherwise.

We next recall an approximation result [9, Theorem 1.1] and a compactness
result in $GSBD^{p}$ in [10] (which generalises [14, Theorem 11.3]). To this
aim we denote by $\mathcal{W}^{\infty}_{\rm pw}(U;\mathbb{R}^{n})\subset
GSBD^{p}(U)$ the space of “piecewise smooth” $SBV$-functions, that is,

$\begin{split}\mathcal{W}^{\infty}_{\rm pw}(U;\mathbb{R}^{n}):=\Big{\\{}u\in
GSBD^{p}(U)\colon&u\in SBV(U;\mathbb{R}^{n})\cap W^{m,\infty}(U\setminus
J_{u};\mathbb{R}^{n}),\ \forall m\in\mathbb{N},\\\
&\mathcal{H}^{n-1}(\overline{J}_{u}\setminus J_{u})=0,\
\overline{J}_{u}=\cup_{i=1}^{k}K_{i}\subset\subset U\\\ &\text{with $K_{i}$
connected (n-1)-rectifiable set},\forall\,1\leq i\leq k\Big{\\}}\end{split}$
(3.4)

######  Theorem 3.4 (Density in $GSBD^{p}$).

Let $\phi$ be a norm on $\mathbb{R}^{n}$. Let $u\in GSBD^{p}(U)$. Then there
exists a sequence $(u_{j})\subset\mathcal{W}^{\infty}_{\rm
pw}(U;\mathbb{R}^{n})$ such that

  1. $(i)$

$u_{j}\to u\ \text{in measure on }U$ ;

  2. $(ii)$

$e(u_{j})\to e(u)\ \text{in }L^{p}(U;\mathbb{M}^{n\times n}_{\rm sym})$ ;

  3. $(iii)$

$\lim_{j\to\infty}\int_{J_{u_{j}}}\phi(\nu_{u_{j}})\,{\rm
d}\mathcal{H}^{n-1}=\int_{J_{u}}\phi(\nu_{u})\,{\rm d}\mathcal{H}^{n-1}$ .

Moreover, if

$\int_{U}\psi(|u|)\,{\rm d}x<+\infty\,,$

for some $\psi\colon[0,+\infty)\to[0,+\infty)$, continuous, increasing with

$\psi(0)=0\,,\ \psi(s+t)\leq C(\psi(s)+\psi(t))\,,\ \psi(s)\leq
C(1+s^{p})\,,\lim_{s\to+\infty}\psi(s)=+\infty\,;$

then

$\lim_{j\to+\infty}\int_{U}\psi(|u_{j}-u|)\,{\rm d}x=0\,.$

We notice that the approximating class considered above fulfils the additional
requirement of having a jump set compactly contained in $U$. This is possible,
as shown in [16, Theorem C].

######  Theorem 3.5 (Compactness in $GSBD^{p}$).

Let $(u_{j})\subset GSBD^{p}(U)$ be a sequence satisfying

$\sup_{j\in\mathbb{N}}\Big{(}\|e(u_{j})\|_{L^{p}(U)}+\mathcal{H}^{n-1}(J_{u_{j}})\Big{)}<+\infty\,.$

Then there exist a subsequence, still denoted by $(u_{j})$, and $u\in
GSBD^{p}(U)$ with the following properties:

  1. $(i)$

the set $U^{\infty}:=\\{x\in U\colon|u_{j}|\to+\infty\\}$ has finite
perimeter;

  2. $(ii)$

$u_{j}\to u$ in measure on $U\setminus U^{\infty}$ ;

  3. $(iii)$

$e(u_{j})\rightharpoonup e(u)$ in $L^{p}(U\setminus
U^{\infty};\mathbb{M}^{n\times n}_{\rm sym})$ ;

  4. $(iv)$

$\liminf_{j\to+\infty}\mathcal{H}^{n-1}(J_{u_{j}})\geq\mathcal{H}^{n-1}(J_{u}\cup(U\cap\partial^{*}U^{\infty}))$
.

If in addition

$\sup_{j\in\mathbb{N}}\int_{U}\psi(|u_{j}|)\,{\rm d}x<+\infty\,,$

for some $\psi\colon[0,+\infty)\to[0,+\infty)$, continuous, increasing with
$\lim_{s\to+\infty}\psi(s)=+\infty$, then $U^{\infty}=\emptyset$ so that $|u|$
is finite a.e. and $(ii)$ holds in $U$.

In the rest of this section we give some technical Lemmas.

######  Lemma 3.6.

For $j\in\mathbb{N}$ let $g_{j}\colon\mathbb{R}^{n}\to[0,+\infty)$ be a family
of equi-integrable functions. Let $E_{j}\subset\mathbb{R}^{n}$ be such that
$\mathcal{L}^{n}(E_{j})\to 0$ and let $\delta_{j}\searrow 0$ as $j\to+\infty$.
Then $(g_{j}\chi_{E_{j}})*\rho_{\delta_{j}}$ converges to 0 strongly in
$L^{1}(\mathbb{R}^{n})$.

###### Proof.

By properties of convolution it holds that

$\|g_{j}\chi_{E_{j}}*\rho_{\delta_{j}}\|_{L^{1}(\mathbb{R}^{n})}\leq\|g_{j}\chi_{E_{j}}\|_{L^{1}(\mathbb{R}^{n})}.$

By equi-integrability we have that for every $\varepsilon>0$ there is
$J\in\mathbb{N}$ such that for every $j\geq J$

$\|g_{j}\chi_{E_{j}}\|_{L^{1}(\mathbb{R}^{n})}=\int_{E_{j}}g_{j}\,{\rm
d}x\leq\varepsilon,$

from which the thesis follows. ∎

######  Lemma 3.7.

Let $A^{\prime}$ be an open bounded subset of $\mathbb{R}^{n}$. For
$j\in\mathbb{N}$ let $g_{j}\colon A^{\prime}\to[0,+\infty)$ be a family of
equi-integrable functions. Let $\delta_{j}\searrow 0$ as $j\to+\infty$. Then
for every $A\subset\subset A^{\prime}$ there holds

$\liminf_{j\to+\infty}\int_{A}g_{j}*\rho_{\delta_{j}}\,{\rm
d}x\geq\liminf_{j\to+\infty}\int_{A}g_{j}\,{\rm d}x\,.$

###### Proof.

We consider the sequence of positive measures
$\nu_{j}:=g_{j}*\rho_{\delta_{j}}\mathcal{L}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits A$. Since $A^{\prime}$ is
bounded $g_{j}$ turn out to be equi-bounded in $L^{1}(A^{\prime})$, hence we
get

$\nu_{j}(A)=\int_{A}g_{j}*\rho_{\delta_{j}}\,{\rm
d}x\leq\int_{A^{\prime}}g_{j}\,{\rm d}x\leq C\,.$

Therefore there exist a positive measure $\nu\in\mathcal{M}_{b}(A)$, a
function $g\in L^{1}(A^{\prime})$, and a not-relabelled subsequence such that
$\nu_{j}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\nu$ weakly $*$ in
$\mathcal{M}_{b}(A)$ and $g_{j}\to g$ weakly in $L^{1}(A^{\prime})$. It
remains to show that $\nu=g\mathcal{L}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits A$, indeed this would imply

$\liminf_{j\to+\infty}\int_{A}g_{j}*\rho_{\delta_{j}}\,{\rm
d}x=\liminf_{j\to+\infty}\nu_{j}(A)\geq\nu(A)=\int_{A}g\,{\rm
d}x=\liminf_{j\to+\infty}\int_{A}g_{j}\,{\rm d}x\,,$

and we could conclude. Let $\varphi\in C^{\infty}_{c}(A)$ and let
$A\subset\subset A^{\prime\prime}\subset\subset A^{\prime}$ be fixed. By
Fubini’s theorem we have

$\begin{split}\int_{A}\varphi\,\mathrm{d}\nu=\lim_{j\to+\infty}\int_{A}\varphi\,\mathrm{d}\nu_{j}&=\lim_{j\to+\infty}\int_{A}\varphi(g_{j}*\rho_{\delta_{j}})\,{\rm
d}x\\\
&=\lim_{j\to+\infty}\int_{A^{\prime\prime}}(\varphi*\hat{\rho}_{\delta_{j}})g_{j}\,{\rm
d}x=\int_{A^{\prime\prime}}\varphi g\,{\rm d}x=\int_{A}\varphi g\,{\rm
d}x,\end{split}$

where $\hat{\rho}_{\delta_{j}}(x):=\rho_{\delta_{j}}(-x)$ and the last
inequality follows since $g_{j}\rightharpoonup g$ weakly in
$L^{1}(A^{\prime})$ and $\varphi*\hat{\rho}_{\delta_{j}}(x)\to\varphi$
strongly in $L^{1}(A^{\prime})$. Thus we deduce
$\nu=g\mathcal{L}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits A$ and the proof is concluded.
∎

######  Lemma 3.8.

Let $A\subset\mathbb{R}^{n-1}$. Let $(u_{k})\subset L^{1}(A)$ be a sequence
converging to $u$ in $L^{1}(A)$. Let $A^{\prime}\subset\subset A$ and let
$w_{k}\colon A^{\prime}\times Q^{\prime}\to\mathbb{R}$ be given by
$w_{k}(x,y):=u_{k}(x+\varepsilon_{k}y)$. Then $w_{k}$ converges to $u$ in
$L^{1}(A^{\prime}\times Q^{\prime})$.

###### Proof.

By Frechet-Kolmogoroff’s Theorem for every $\eta>0$, $y\in Q^{\prime}$ there
is $h\in\mathbb{N}$ such that for all $k\geq h$ there holds

$\int_{A^{\prime}}|u_{k}(x+\varepsilon_{k}y)-u(x)|\,{\rm d}x\leq\eta\,.$

This together with Fubini’s theorem yield

$\begin{split}\int_{A^{\prime}\times Q^{\prime}}|w_{k}(x,y)-u(x)|\,{\rm
d}x\,{\rm
d}y\leq\int_{Q^{\prime}}\int_{A^{\prime}}|u_{k}(x+\varepsilon_{k}y)-u(x)|\,{\rm
d}x\,{\rm d}y\leq\eta\,,\end{split}$

for all $k\geq h$. Eventually by letting $\eta\to 0$ we conclude. ∎

######  Corollary 3.9.

Let $A\subset\mathbb{R}^{n-1}$. Let $(u_{k})\subset L^{0}(A)$ be a sequence
converging to $u$ in measure. Let $A^{\prime}\subset\subset A$ and let
$w_{k}\colon A^{\prime}\times Q^{\prime}\to\mathbb{R}$ be given by
$w_{k}(x,y):=u_{k}(x+\varepsilon_{k}y)$. Then $w_{k}$ converges to $u$ in
measure.

###### Proof.

Since ${\rm arctan}(u_{k})$ converges to ${\rm arctan}(u)$ in $L^{1}(A)$ by
Lemma 3.8 we have that ${\rm arctan}(w_{k})$ converges to ${\rm arctan}(u)$ in
$L^{1}(A^{\prime}\times Q^{\prime})$. Hence $w_{k}$ converges to $u$ in
measure. ∎

##  4\. Compactness

In this section we prove point $(ii)$ of Theorem 2.1.

######  Proposition 4.1 (Compactness).

Let $F_{k}$ be as in (2.5). Let $(u_{k})\subset L^{0}(U;\mathbb{R}^{n})$ be
such that $\sup_{k}F_{k}(u_{k})<+\infty$. Then there exists $u\in GSBD^{p}(U)$
such that, up to subsequence, it holds

$u_{k}\to u\quad\text{in measure on }U\setminus U^{\infty}\,,$
$e(u_{k})\rightharpoonup e(u)\quad\text{in }L^{p}(U\setminus
U^{\infty};\mathbb{M}^{n\times n}_{\rm sym})\,,$
$\liminf_{k\to+\infty}\mathcal{H}^{n-1}(J_{u_{k}})\geq\mathcal{H}^{n-1}(J_{u}\cup(\partial^{*}U^{\infty}\cap
U))\,,$

where $U^{\infty}:=\\{x\in U\colon|u_{k}(x)|\to+\infty\\}$. If in addition

$\sup_{k\in\mathbb{N}}\int_{U}\psi(|u_{k}|)\,{\rm d}x<+\infty\,,$

for some $\psi\colon[0,+\infty)\to[0,+\infty)$, continuous, increasing with
$\lim_{s\to+\infty}\psi(s)=+\infty$, then $U^{\infty}=\emptyset$, and all
implications hold on $U$.

###### Proof.

The proof follows by suitably adaptations of proof of [18, Proposition 4.1].
We recall it here for completeness. Let $(u_{k})$ be as in the statement and
let $U^{\prime}\subset\subset U$ be fixed. Then it is sufficient to prove the
following claim: there exist ($\bar{u}_{k})\subset GSBV^{p}(U;\mathbb{R}^{n})$
and $c_{0}>0$ (independent of $k$) such that

$\bar{u}_{k}-u_{k}\to 0\ \text{in measure on }U\,,$ (4.1) $F_{k}(u_{k})\geq
c_{0}\bigg{(}\int_{U^{\prime}}|e(\bar{u}_{k})|^{p}\,{\rm
d}x+\mathcal{H}^{n-1}(J_{\bar{u}_{k}})\bigg{)}\,.$ (4.2)

Indeed by (4.2) and Theorem 3.5 we deduce the existence of $u\in
GSBD^{p}(U^{\prime})$ such that, up to subsequence,

$\bar{u}_{k}\to u\ \text{in measure on
}\Omega^{\prime}\setminus{\Omega^{\prime}}^{\infty}\,,$
$e(\bar{u}_{k})\rightharpoonup e(u)\ \text{in}\
L^{p}(U^{\prime}\setminus{U^{\prime}}^{\infty};\mathbb{M}^{n\times n}_{\rm
sym})\,,$

and

$\liminf_{k\to+\infty}\mathcal{H}^{n-1}(J_{\bar{u}_{k}}\cap
U^{\prime})\geq\mathcal{H}^{n-1}(J_{u}\cup({U^{\prime}}\cap\partial^{*}{U^{\prime}}^{\infty}))\,.$

Eventually by (4.1) and the arbitrariness of $U^{\prime}$ we would conclude
(observing indeed that also the remaining part of the statement follows
directly from Theorem 3.5). Thus we are only left to prove the claim.
For fixed $i\in\mathbb{N}$ let $f_{i}(t)=\alpha_{i}\wedge\beta_{i}$ be as in
(2.3). Choose $\eta\in(0,1)$ such that $Q_{\eta}(0)\subset\subset S$ and let

$m_{\eta}:=\min_{x\in\overline{Q}_{\eta}(0)}\rho(x)>0\quad\text{and}\quad
f_{i}^{\eta}(t):=f_{i}(m_{\eta}\eta^{n}t)=\alpha_{i}m_{\eta}\eta^{n}t\wedge\beta_{i}\,.$
(4.3)

Then we have

$\begin{split}F_{k}(u_{k})&\geq\frac{1}{\varepsilon_{k}}\int_{U}f_{i}\Big{(}\varepsilon_{k}W_{k}(\cdot,e(u_{k}))*\rho_{k}(x)\Big{)}\,{\rm
d}x\\\
&\geq\frac{1}{\varepsilon_{k}}\int_{U}f_{i}^{\eta}\Big{(}\varepsilon_{k}\fint_{Q_{\eta\varepsilon_{k}}(x)}W_{k}(y,e(u_{k}))\,{\rm
d}y\Big{)}\,{\rm d}x\,.\end{split}$ (4.4)

We set

$A_{k}^{1}:=\Biggl{\\{}x\in
U\colon\varepsilon_{k}\fint_{Q_{\eta\varepsilon_{k}(x)}}W_{k}(y,e(u_{k}))\,{\rm
d}y\geq\frac{\beta_{i}}{\alpha_{i}m_{\eta}\eta^{2n}}\Biggr{\\}}\,,$
$A_{k}^{2}:=\Biggl{\\{}x\in
U\colon\operatorname{dist}(x,A_{k}^{1})\leq(1-\eta)\varepsilon_{k}\Biggr{\\}}\,.$

Note that

$A_{k}^{1}\subset A_{k}^{2}\subset\Biggl{\\{}x\in
U\colon\varepsilon_{k}\fint_{Q_{\varepsilon_{k}}(x)}W_{k}(y,e(u_{k}))\,{\rm
d}y\geq\frac{\beta_{i}}{\alpha_{i}m_{\eta}\eta^{n}}\Biggr{\\}}\,.$ (4.5)

Indeed if $x\in A_{k}^{2}$ there is $z\in A_{k}^{1}$ with
$Q_{\eta\varepsilon_{k}}(z)\subset Q_{\varepsilon_{k}}(x)$ and therefore

$\varepsilon_{k}\fint_{Q_{\varepsilon_{k}}(x)}W_{k}(y,e(u_{k}))\,{\rm
d}y\geq\eta^{n}\varepsilon_{k}\fint_{Q_{\eta\varepsilon_{k}}(z)}W_{k}(y,e(u_{k}))\,{\rm
d}y\geq\frac{\beta_{i}}{\alpha_{i}m_{\eta}\eta^{n}}\,.$

By combining together (4.4) and (4.5) we find

$F_{k}(u_{k})\geq\frac{\beta_{i}}{\varepsilon_{k}}\mathcal{L}^{n}(A_{k}^{2})\,.$
(4.6)

By the coarea formula (see e.g., [17, Theorem 3.14]) and the mean value
theorem there exists $t_{k}\in(0,(1-\eta)\varepsilon_{k})$ such that the set
$A_{k}^{3}:=\\{\operatorname{dist}(\cdot,A_{k}^{1})\leq t_{k}\\}\subset
A_{k}^{2}$ satisfies

$\mathcal{L}^{n}(A_{k}^{2})\geq(1-\eta)\varepsilon_{k}\mathcal{H}^{n-1}(\partial
A_{k}^{3})\,.$ (4.7)

Let now

$\bar{u}_{k}(x):=\begin{cases}0&\text{ if }x\in A_{k}^{3}\,,\\\ u_{k}&\text{
otherwise in }U\,.\end{cases}$

By (4.6) and the fact that $A_{k}^{3}\subset A_{k}^{2}$ we have
$\mathcal{L}^{n}(A_{k}^{3})\to 0$ as $k\to+\infty$ from which (4.1) follows.
On the other hand as $J_{\bar{u}_{k}}=\partial A_{k}^{3}$ (4.6) and (4.7)
yield

$F_{k}(u_{k})\geq(1-\eta)\beta_{i}\mathcal{H}^{n-1}(J_{\bar{u}_{k}})\,.$ (4.8)

We next show that there exists $K(n)\geq 1$ such that for every
$x\in\overline{Q}$

$c_{1}\varepsilon_{k}\fint_{Q_{\eta\varepsilon_{k}}(x)}|e(\bar{u}_{k}(y))|^{p}\,{\rm
d}y\leq K\frac{\beta_{i}}{\alpha_{i}m_{\eta}\eta^{n}}\,.$ (4.9)

By ($W4$) we have

$W_{k}(x,e(u_{k}(x)))\geq c_{1}|e(u_{k}(x))|^{p}\geq
c_{1}|e(\bar{u}_{k}(x))|^{p}\quad\text{for a.e. }x\in U\,.$ (4.10)

Now if $x\in U\setminus A_{k}^{3}$, then $x\notin A_{k}^{1}$ and

$c_{1}\varepsilon_{k}\fint_{Q_{\eta\varepsilon_{k}}(x)}|e(\bar{u}_{k}(y))|^{p}\,{\rm
d}y\leq\varepsilon_{k}\fint_{Q_{\eta\varepsilon_{k}}(x)}W_{k}(y,e(u_{k}(y)))\,{\rm
d}y\leq\frac{\beta_{i}}{\alpha_{i}m_{\eta}\eta^{n}}\,.$

Assume instead that $x\in A_{k}^{3}$. Observe that $\bar{u}_{k}=0$ in
$Q_{\eta\varepsilon_{k}}(x)\cap A_{k}^{3}$, so that

$\int_{Q_{\eta\varepsilon_{k}}(x)}|e(\bar{u}_{k}(y))|^{p}\,{\rm
d}y=\int_{Q_{\eta\varepsilon_{k}}(x)\setminus
A_{k}^{3}}|e(\bar{u}_{k}(y))|^{p}\,{\rm d}y\,.$

Furthemore, we can cover $Q_{\eta\varepsilon_{k}}(x)\cap(Q\setminus
A_{k}^{3})$ with a finite number $K(n)\geq 1$ of balls of radius
$\eta\varepsilon_{k}$ and centres $x_{1},...,x_{K}\in U\setminus A_{k}^{3}$
(see e.g. [23, Remark 2.8]). Hence, we find

$\begin{split}c_{1}\varepsilon_{k}\fint_{Q_{\eta\varepsilon_{k}}(x)}|e(\bar{u}_{k}(y))|^{p}\,{\rm
d}y&\leq
c_{1}\varepsilon_{k}\sum_{i=1}^{K}\fint_{Q_{\eta\varepsilon_{k}}(x_{i})}|e(\bar{u}_{k}(y))|^{p}\,{\rm
d}y\\\
&\leq\varepsilon_{k}\sum_{i=1}^{K}\fint_{Q_{\eta\varepsilon_{k}}(x_{i})}W_{k}(y,e(u_{k}(y)))\,{\rm
d}y\leq K\frac{\beta_{i}}{\alpha_{i}m_{\eta}\eta^{n}}\,,\end{split}$

and (4.9) follows. Finally by (4.4), the monotonicity of $f_{i}^{\eta}$,
(4.10) we infer

$\begin{split}F_{k}(u_{k})&\geq\frac{1}{\varepsilon_{k}}\int_{U}f_{i}^{\eta}\Big{(}\varepsilon_{k}\fint_{Q_{\eta\varepsilon_{k}}(x)}c_{1}|e(\bar{u}_{k}(y))|^{p}\,{\rm
d}y\Big{)}\,{\rm d}x\\\ &\geq
c_{1}\frac{\alpha_{i}m_{\eta}\eta^{n}}{K}\int_{U}\fint_{Q_{\eta\varepsilon_{k}}(x)}|e(\bar{u}_{k}(y))|^{p}\,{\rm
d}y\,{\rm d}x\,,\end{split}$ (4.11)

where the last inequality follows from (4.9) and the fact that
$f_{i}^{\eta}(t)\geq\frac{\alpha_{i}m_{\eta}\eta^{n}}{K}t$ when $t\leq
K\frac{\beta_{i}}{m_{\eta}\eta^{n}}$. Moreover by using in order the change of
variable $y=x-\eta\varepsilon_{k}z$, Fubini’s theorem, and the change of
variable $\hat{x}=x-\eta\varepsilon_{k}z$ (for $k$ large enough), we find

$\begin{split}\int_{U}\fint_{Q_{\eta\varepsilon_{k}}(x)}|e(\bar{u}_{k}(y))|^{p}\,{\rm
d}y\,{\rm
d}x&=\int_{Q}\int_{U}|e(\bar{u}_{k}(x-\eta\varepsilon_{k}z))|^{p}\,{\rm
d}x\,{\rm d}z\\\ &\geq\int_{U^{\prime}}|e(\bar{u}_{k}(x))|^{p}\,{\rm
d}x\,.\end{split}$ (4.12)

Eventually gathering together (4.7), (4.11), and (4.12), we deduce (4.2) with

$c_{0}:=\frac{1}{2}\Big{(}\frac{c_{1}\alpha_{i}m_{\eta}\eta^{n}}{K}\wedge(1-\eta)\beta_{i}\Big{)}\,,$

and in particular, by arbitrariness of $U^{\prime}$, $\bar{u}_{k}\in
GSBV^{p}(U;\mathbb{R}^{n})$. ∎

##  5\. Lower bound

In this section we prove the lower bound. To this purpose it is convenient to
localise the functionals $F_{k}$, namely we set

$F_{k}(u,A):=\frac{1}{\varepsilon_{k}}\int_{A}f\Big{(}\varepsilon_{k}W_{k}(\cdot,e(u))*\rho_{k}(x)\Big{)}\,{\rm
d}x\,,\quad\text{for }u\in W^{1,p}(U)\,,\ A\subset U\,.$ (5.1)

######  Proposition 5.1 (Lower bound: bulk contribution).

Let $(u_{k})\subset L^{0}(U;\mathbb{R}^{n})$ be a sequence that converges to
in measure to $u\in L^{0}(U;\mathbb{R}^{n})$. Assume moreover that
$F_{k}(u_{k})\leq C$ . Then there exists a subsequence, not relabelled, such
that

$\liminf_{k\to+\infty}F_{k}(u_{k},A)\geq\alpha\int_{A}W(x,e(u))\,{\rm
d}x\quad\forall A\in\mathcal{A}(U)\,,$

with $W$ given by (3.3).

###### Proof.

Let $(u_{k})$ and $u$ be as in the statement. By Proposition 4.1 $u\in
GSBD^{p}(U)$. Let $A\in\mathcal{A}(U)$ be fixed. Up to extracting a
subsequence we may assume, by Theorem 3.1, that

$W^{\prime}(x,M)=W^{\prime\prime}(x,M)=W(x,M)\,,$

with $W^{\prime}$, $W^{\prime\prime}$ defined as in (2.7) and (2.8). For every
$k\in\mathbb{N}$ let $\mu_{k}$ be the Radon measure on $(U,\mathcal{B}(U))$
given by

$\mu_{k}(A):=F_{k}(u,A)\,,\quad\forall A\in\mathcal{B}(U)\,.$ (5.2)

As $\mu_{k}(A)\leq C$ by [2, Theorem 1.59] we deduce the existence of a
subsequence, not relabelled, and of a Radon measure $\mu$ on
$(A,\mathcal{B}(A))$ such that

$\mu_{k}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu\quad\text{ and
}\quad\liminf_{k\to+\infty}\mu_{k}(A)\geq\mu(A)\,.$ (5.3)

By Radon-Nikodym’s Theorem there exist two measures $\mu^{a},\mu^{s}$ with
$\mu^{a}\ll\mathcal{L}^{n}$ and $\mu^{s}\perp\mathcal{L}^{n}$, and a function
$h\in L^{1}(A)$ such that $\mu=\mu^{a}+\mu^{s}$ and
$\mu^{a}=h\mathcal{L}^{n}$. This together with (5.3) imply that

$\liminf_{k\to+\infty}F_{k}(u_{k},A)\geq\int_{A}h(x)\,{\rm d}x\,.$

Hence to conclude we need to show that

$h(x)\geq\alpha W(x,e(u(x)))\quad\text{ for a.e. }x\in U\,.$ (5.4)

with $W$ as in (3.3) For $i\in\mathbb{N}$ fixed let
$f_{i}(t)=\alpha_{i}t\wedge\beta_{i}$ be as in (2.3). Then it is enough to
show that

$h(x)\geq\alpha_{i}W(x,e(u(x)))\quad\text{ for a.e. }x\in U\,,$ (5.5)

We divide the proof of (5.5) into four steps.

Step 1: In this step we show that for a.e. $x_{0}\in U$ there exists a
sequence $(k_{j},r_{j})\to(+\infty,0)$ as $j\to+\infty$ such that setting
$\delta_{j}:=\frac{\varepsilon_{k_{j}}}{r_{j}}$,

$u_{k}^{r}(v):=\frac{u_{k}(x_{0}+rv)-u_{k}(x_{0})}{r}\quad\text{and}\quad
W_{k}^{r}(x,M):=W(x_{0}+rx,M)\,,$ (5.6)

there hold

$h(x_{0})\geq\lim_{j\to+\infty}\frac{1}{r_{j}}\frac{1}{\delta_{j}}\int_{Q}f_{i}\Big{(}r_{j}{\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(u_{k_{j}}^{r_{j}}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x\,,$ (5.7)

and

$u_{k_{j}}^{r_{j}}\to\nabla u(x_{0})(\cdot)\quad\text{in measure on }Q\,.$
(5.8)

By Besicovitch differentiation theorem and [8, Corollary 5.2] we have that for
a.e. $x_{0}\in U$ the following hold:

$h(x_{0})=\lim_{r\searrow
0^{+}}\frac{\mu(\overline{Q}_{r}(x_{0}))}{|Q_{r}(x_{0})|}\,,$ (5.9)
$\lim_{r\searrow 0^{+}}\frac{1}{r^{n}}\mathcal{L}^{n}\left(\left\\{y\in
Q_{r}(x_{0})\colon\frac{|u(y)-u(x_{0})-\nabla
u(x_{0})(y-x_{0})|}{|y-x_{0}|}>\delta\right\\}\right)=0\quad\forall\delta>0\,.$
(5.10)

We fix $x_{0}\in\Omega$ for which (5.9) and (5.10) hold. By [2, Proposition
1.62] we have

$\mu(\overline{Q}_{r}(x_{0}))\geq\limsup_{k\to+\infty}\mu_{k}(\overline{Q}_{r}(x_{0}))\,,$

for every $r>0$, which together with (5.9) yield

$h(x_{0})\geq\lim_{r\searrow
0^{+}}\limsup_{k\to+\infty}\frac{\mu_{k}(\overline{Q}_{r}(x_{0}))}{|Q_{r}(x_{0})|}\,.$
(5.11)

Moreover from (5.2) and the change of variable $x=x_{0}+rx^{\prime}$ we get

$\begin{split}\mu_{k}(\overline{Q}_{r}(x_{0}))&=\frac{1}{\varepsilon_{k}}\int_{\overline{Q}_{r}(x_{0})}f\Big{(}\varepsilon_{k}W_{k}(\cdot,e(u_{k}))*\rho_{k}(x)\Big{)}\,{\rm
d}x\\\
&=\frac{r^{n}}{\varepsilon_{k}}\int_{\overline{Q}}f\Big{(}\varepsilon_{k}W_{k}(\cdot,e(u_{k}))*\rho_{k}(x_{0}+rx)\Big{)}\,{\rm
d}x\,.\end{split}$ (5.12)

From (5.6) and the change of variable $y=ry^{\prime}$ we may deduce that

$W_{k}(\cdot,e(u_{k}))*\rho_{k}(x_{0}+rx)=W_{k}^{r}(\cdot,e(u_{k}^{r}))*\rho_{\frac{\varepsilon_{k}}{r}}(x)\,,$
(5.13)

as $k\to+\infty$. Gathering together (5.11), (5.12), (5.13) and using (2.3) we
obtain

$h(x_{0})\geq\lim_{r\searrow
0^{+}}\limsup_{k\to+\infty}\frac{1}{r}\frac{r}{\varepsilon_{k}}\int_{\overline{Q}}f_{i}\Big{(}r{\frac{\varepsilon_{k}}{r}}W_{k}^{r}(\cdot,e(u_{k}^{r}))*\rho_{\frac{\varepsilon_{k}}{r}}(x)\Big{)}\,{\rm
d}x\,.$

Furthermore from (5.10) and the fact that $u_{k}$ converges to $u$ in measure
we can deduce that

$\lim_{r\to 0}\lim_{k\to+\infty}\mathcal{L}^{n}\left(\left\\{v\in
Q\colon|u_{k}^{r}(v)-\nabla
u(x_{0})(v)|>\delta\right\\}\right)=0\quad\forall\delta>0\,.$

We now can find a subsequence $(k_{j},r_{j})\to(+\infty,0)$ as $j\to+\infty$
for which (5.7) and (5.8) hold with
$\delta_{j}:=\tfrac{\varepsilon_{k_{j}}}{r_{j}}$ and the proof of step 1 is
conluded.
Step 2: In this step we show that for any $0<\zeta<1$ and a.e. $x_{0}\in U$
there exist $(\bar{u}_{j})\subset GSBV^{p}(Q;\mathbb{R}^{n})$ and $c_{0}>0$
independent of $j$ such that

$\lim_{j\to+\infty}\mathcal{L}^{n}\\{\bar{u}_{j}\neq u_{k_{j}}^{r_{j}}\\}=0;$
(5.14) $\bar{u}_{j}\to\nabla u(x_{0})(\cdot)\quad\text{ in measure on }Q\,;$
(5.15) $\mathcal{H}^{n-1}(J_{\bar{u}_{j}}\cap Q)\to 0\,;$ (5.16)
$\int_{Q_{1-\zeta}(0)}|e(\bar{u}_{j})|^{p}\,{\rm d}x\leq c_{0}\,.$ (5.17)

By step 1 we have that $u_{k_{j}}^{r_{j}}$ converges in measure to $\nabla
u(x_{0})(\cdot)$ in $Q$ as $j\to+\infty$ and for $j$ large enough it satisfies

$\frac{1}{r_{j}}\frac{1}{\delta_{j}}\int_{Q}f_{i}\Big{(}r_{j}{\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(u_{k_{j}}^{r_{j}}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x\leq C\,.$ (5.18)

Next we fix $\eta\in(0,1)$ such that $Q_{\eta}(0)\subset\subset S$ and let
$m_{\eta}$ and $f_{i}^{\eta}$ be as in (4.3). Then we get

$\int_{Q}f_{i}\Big{(}{r_{j}\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(u_{k_{j}}^{r_{j}}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x\geq\int_{Q}f_{i}^{\eta}\Big{(}{r_{j}\delta_{j}}\fint_{Q_{\eta\delta_{j}}(x)}W_{k_{j}}^{r_{j}}(y,e(u_{k_{j}}^{r_{j}}))\,{\rm
d}y\Big{)}\,{\rm d}x\,.$ (5.19)

We define the sets

$A_{j}^{1}:=\Biggl{\\{}x\in Q\colon
r_{j}\delta_{j}\fint_{Q_{\eta\delta_{j}}(x)}W_{k_{j}}^{r_{j}}(y,e(u_{k_{j}}^{r_{j}}))\,{\rm
d}y\geq\frac{\beta_{i}}{\alpha_{i}m_{\eta}\eta^{2n}}\Biggr{\\}}\,,$
$A_{j}^{2}:=\Biggl{\\{}x\in
Q\colon\operatorname{dist}(x,A_{j}^{1})\leq(1-\eta)\delta_{j}\Biggr{\\}}\,.$

Then arguing as in the proof of Proposition 4.1 we find that

$A_{j}^{1}\subset A_{j}^{2}\subset\Biggl{\\{}x\in Q\colon
r_{j}\delta_{j}\fint_{Q_{\delta_{j}}(x)}W_{k_{j}}^{r_{j}}(y,e(u_{k_{j}}^{r_{j}}))\,{\rm
d}y\geq\frac{\beta_{i}}{\alpha_{i}m_{\eta}\eta^{n}}\Biggr{\\}}\,.$ (5.20)

(5.20) together with (5.18) and (5.19) imply that (for $j$ large enough)

$C\geq\frac{\beta_{i}}{r_{j}\delta_{j}}\mathcal{L}^{n}(A_{j}^{2})=\frac{\beta_{i}}{\varepsilon_{j}}\mathcal{L}^{n}(A_{j}^{2})\,.$
(5.21)

By the coarea formula and the mean value theorem we can find
$t_{j}\in(0,(1-\eta)\delta_{j})$ such that setting
$A_{j}^{3}:=\\{\operatorname{dist}(\cdot,A_{j}^{1})\leq t_{j}\\}\subset
A_{j}^{2}$

$\mathcal{L}^{n}(A_{j}^{2})\geq(1-\eta)\delta_{j}\mathcal{H}^{n-1}(\partial
A_{j}^{3})\,.$ (5.22)

We finally define

$\bar{u}_{j}(x):=\begin{cases}0&\text{ if }x\in A_{j}^{3},\\\
u_{k_{j}}^{r_{j}}&\text{ otherwise in }Q.\end{cases}$

Recall that, by definition, $\frac{\varepsilon_{j}}{\delta_{j}}\to 0$. With
this, as a consequence of (5.21) and (5.22) we have that both
$\mathcal{L}^{n}(A_{j}^{3})$ and $\mathcal{H}^{n-1}(\partial
A_{j}^{3})=\mathcal{H}^{n-1}(J_{\bar{u}_{j}})$ converge to $0$ as
$j\to+\infty$. Hence $\bar{u}_{j}\subset GSBV^{p}(B_{1}(0);\mathbb{R}^{n})$
and $\bar{u}_{j}-u_{k_{j}}^{r_{j}}\to 0$ in measure on $Q$ which combined with
(5.8) yield $\bar{u}_{j}\to\nabla u(x_{0})(\cdot)$ in measure on $Q$. It
remains to show (5.17). To this aim notice that arguing exactly as in the
proof of Proposition 4.1 one can find $K(n)\geq 1$ such that for every
$x\in\overline{Q}$

$c_{1}{r_{j}\delta_{j}}\fint_{Q_{\eta\delta_{j}}(x)}|e(\bar{u}_{j}(y))|^{p}\,{\rm
d}y\leq K\frac{\beta_{i}}{\alpha_{i}m_{\eta}\eta^{n}}\,.$ (5.23)

Next from (5.19), the monotonicity of $f_{i}^{\eta}$ we infer

$\begin{split}\frac{1}{r_{j}\delta_{j}}\int_{Q}f_{i}\Big{(}{r_{j}\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(u_{k_{j}}^{r_{j}}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x&\geq\frac{1}{r_{j}\delta_{j}}\int_{Q}f_{i}^{\eta}\Big{(}c_{1}{r_{j}\delta_{j}}\fint_{Q_{\eta\delta_{j}}(x)}|e(\bar{u}_{j}(y))|^{p}\,{\rm
d}y\Big{)}\,{\rm d}x\\\ &\geq
c_{1}\frac{\alpha_{i}m_{\eta}\eta^{n}}{K}\int_{Q}\fint_{Q_{\eta\delta_{j}}(x)}|e(\bar{u}_{j}(y))|^{p}\,{\rm
d}y\,{\rm d}x\,\end{split}$ (5.24)

where the last inequality follows from (5.23) and fact that
$f_{i}^{\eta}(t)\geq\frac{\alpha_{i}m_{\eta}\eta^{n}}{K}t$ when $t\leq
K\frac{\beta_{i}}{m_{\eta}\eta^{n}}$. Finally, for a fixed $0<\zeta<1$,
arguing exactly as for (4.12) we get

$\begin{split}\int_{Q}\fint_{Q_{\eta\delta_{j}}(x)}|e(\bar{u}_{j}(y))|^{p}\,{\rm
d}y\,{\rm d}x\geq\int_{Q_{1-\zeta}(0)}|e(\bar{u}_{j}(x))|^{p}\,{\rm
d}x\,\end{split}$ (5.25)

when $j$ is sufficiently large. Eventually gathering together (5.18), (5.24),
and (5.25), we deduce (5.17) with
$c_{0}:=\frac{CK}{c_{1}\alpha_{i}m_{\eta}\eta^{n}}$.
Step 3: In this step show that for a.e. $x_{0}\in U$ there exists a sequence
$(w_{j})\subset W^{1,p}(Q;\mathbb{R}^{n})$ such that:

$(|\nabla w_{j}|^{p})\quad\text{ is equi-integrable; }$ (5.26)
$\lim_{j\to+\infty}\mathcal{L}^{n}(\left\\{w_{j}\neq\bar{u}_{j}\right\\}=0\,;$
(5.27) $\lim_{j\to+\infty}\|w_{j}-\nabla u(x_{0})(\cdot)\|_{L^{p}(Q)}=0\,;$
(5.28)
$h(x_{0})\geq\liminf_{j\to+\infty}\frac{1}{r_{j}\delta_{j}}\int_{Q_{1-\zeta}(0)}f_{i}\Big{(}{r_{j}\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x\quad\forall i\in\mathbb{N}\,.$ (5.29)

From step 2 we can apply [19, Lemma 5.1] to the sequence $\bar{u}_{j}$ and get
the existence of $(w_{j})\subset W^{1,p}(Q;\mathbb{R}^{n})$ that satisfies
(5.26)–(5.28). Moreover recalling ($W4$) and the equi-integrability of
$(|\nabla w_{j}|^{p})$ we have that $W_{k_{j}}^{r_{j}}(x,e(w_{j}))$ is equi-
integrable as well, while from the inclusion

$E_{j}:=\left\\{e(w_{j})\neq e(u_{k_{j}}^{r_{j}})\right\\}\subset\\{w_{j}\neq
u_{k_{j}}^{r_{j}}\\}\subset\left\\{w_{j}\neq\bar{u}_{j}\right\\}\cup\\{\bar{u}_{j}\neq
u^{r_{j}}_{k_{j}}\\}\,,$

it follows that $\mathcal{L}^{n}\left(\left\\{e(w_{j})\neq
e(u_{k_{j}}^{r_{j}})\right\\}\right)\to 0$. Thus, we can apply Lemma 3.6 with
$g_{j}=W_{k_{j}}^{r_{j}}(x,e(w_{j}))$, and $E_{j}=\left\\{e(w_{j})\neq
e(u_{k_{j}}^{r_{j}})\right\\}$, and deduce that

$\int_{E_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\,{\rm
d}x\to 0\,.$ (5.30)

Using that $f_{i}(t)\leq\alpha_{i}t$ we obtain the following estimate

$\begin{split}\frac{1}{r_{j}\delta_{j}}&\int_{Q_{1-\zeta}(0)}f_{i}\Big{(}{r_{j}}\delta_{j}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x\\\
&\leq\frac{1}{r_{j}\delta_{j}}\int_{Q_{1-\zeta}(0)}f_{i}\Big{(}{r_{j}\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(u_{k_{j}}^{r_{j}}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x+\alpha_{i}\int_{E_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\,{\rm
d}x\,.\end{split}$

Passing to the limit as $j\to+\infty$ in the above inequality and using (5.7)
and (5.30) we infer (5.29).
Step 4: In this step we show that for a.e. $x_{0}\in U$

$\liminf_{j\to+\infty}\frac{1}{r_{j}\delta_{j}}\int_{Q_{1-\zeta}(0)}f_{i}\Big{(}{r_{j}\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x\geq\alpha_{i}W(x_{0},e(u(x_{0})))\quad\forall i\in\mathbb{N}\,.$ (5.31)

We define the following partition

$B_{j}^{1}:=\left\\{x\in
Q_{1-\zeta}(0)\colon{r_{j}\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\geq\frac{\beta_{i}}{\alpha_{i}}\right\\}\,,\quad
B_{j}^{2}:=Q_{1-\eta}(0)\setminus B_{j}^{1}\,.$

Then for $j$ large enough there holds

$\mathcal{L}^{n}(B_{j}^{1})\leq Cr_{j}\delta_{j}\to 0\,,$

which, together with Lemma 3.6, implies

$\int_{B_{j}^{1}}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\,{\rm
d}x\to 0\,.$ (5.32)

Since $f_{i}(t)=\alpha_{i}t$ when $t\leq\frac{\beta_{i}}{\alpha_{i}}$, we then
have

$\frac{1}{r_{j}\delta_{j}}\int_{Q_{1-\zeta}(0)}f_{i}\Big{(}{r_{j}\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x\geq{\alpha_{i}}\int_{B_{j}^{2}}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\,{\rm
d}x\,.$ (5.33)

Now, taking the liminf as $j\to+\infty$ in (5.33), and using (5.32), we get

$\begin{split}\liminf_{j\to+\infty}\frac{1}{r_{j}\delta_{j}}\int_{Q_{1-\zeta}(0)}f_{i}\Big{(}{r_{j}\delta_{j}}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\Big{)}\,{\rm
d}x\geq\liminf_{j\to+\infty}{\alpha_{i}}\int_{Q_{1-\zeta}(0)}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\,{\rm
d}x\,.\end{split}$ (5.34)

From this, applying Lemma 3.7 with $g_{j}=W_{k_{j}}^{r_{j}}(x,e(w_{j}))$ we
have

$\liminf_{j\to+\infty}{\alpha_{i}}\int_{Q_{1-\zeta}(0)}W_{k_{j}}^{r_{j}}(\cdot,e(w_{j}))*\rho_{\delta_{j}}(x)\,{\rm
d}x\geq{\alpha_{i}}\liminf_{j\to+\infty}\int_{Q_{1-\zeta}(0)}W_{k_{j}}(x_{0}+r_{j}x,e(w_{j}))\,{\rm
d}x\,.$ (5.35)

Next we modify $w_{j}$ so that it coincides with $\nabla u(x_{0})(\cdot)$ on
$\partial Q_{1-\zeta}(0)$ without essentially increasing the energy. This can
be achieved by relying on the following Fundamental Estimate than can be
proved with standard arguments: there exist $C(\gamma)$ and a sequence
$(\overline{w}_{j})\subset W^{1,p}(Q_{1-\zeta}(0);\mathbb{R}^{d})$ with
$\overline{w}_{i}=\nabla u(x_{0})(\cdot)$ in a neighbourhood of $\partial
Q_{1-\zeta}(0)$ such that

$\begin{split}\int_{Q_{1-\zeta}(0)}W_{k_{j}}(x_{0}+r_{j}x,e(\overline{w}_{j}))\,{\rm
d}x&\leq(1+\gamma)\int_{Q_{1-\zeta}(0)}W_{k_{j}}(x_{0}+r_{j}x,e(w_{j}))\,{\rm
d}x\\\ &+(1+\gamma)\int_{Q_{1-\zeta}(0)\setminus
Q_{1-\zeta-\gamma}(0)}W_{k_{j}}(x_{0}+r_{j}x,e(u(x_{0})))\,{\rm d}x\\\
&+C(\gamma)\|w_{j}-\nabla
u(x_{0})(\cdot)\|^{p}_{L^{p}(Q_{1-\zeta}(0))}+\gamma\,.\end{split}$ (5.36)

By (5.28) we know that $w_{j}$ converges to $\nabla u(x_{0})(\cdot)$ in
$L^{p}(Q)$, moreover from (($W4$)) there holds

$\begin{split}\int_{Q_{1-\zeta-\gamma}(0)}W_{k_{j}}(x_{0}+r_{j}x,e(u(x_{0})))\,{\rm
d}x&\leq c_{2}(|e(u(x_{0}))|^{p}+1)\mathcal{L}^{n}({Q_{1-\zeta}(0)\setminus
Q_{1-\zeta-\gamma}(0)})\\\ &\leq
c_{2}(|e(u(x_{0}))|^{p}+1)n\gamma\,.\end{split}$

This fact and (5.36) imply that

$\begin{split}\liminf_{j\to+\infty}&\int_{Q_{1-\zeta}(0)}W_{k_{j}}(x_{0}+r_{j}x,e(w_{j}))\,{\rm
d}x\\\
&\geq\frac{1}{1+\gamma}\liminf_{j\to+\infty}\int_{Q_{1-\zeta}(0)}W_{k_{j}}(x_{0}+r_{j}x,e(\overline{w}_{j}))\,{\rm
d}x-c_{2}(|e(u(x_{0}))|^{p}+1)n\gamma-\frac{\gamma}{1+\gamma}\,.\end{split}$
(5.37)

We now set $\widetilde{w}_{j}(x):=r_{j}\overline{w}_{j}((x-x_{0})/r_{j})$,
which is admissible for
$\mathbf{m}_{k}(u_{e(u(x_{0}))},Q_{(1-\zeta)r_{j}}(0))$ in (2.6). Hence, by a
change of variable in (5.37) we obtain

$\begin{split}\liminf_{j\to+\infty}{\alpha_{i}}\int_{Q_{1-\zeta}(0)}W_{k_{j}}(x_{0}+r_{j}x,e(\overline{w}_{j}))\,{\rm
d}x&\geq\liminf_{j\to+\infty}\frac{\alpha_{i}}{r_{j}^{n}}\int_{Q_{(1-\zeta)r_{j}}(x_{0})}W_{k_{j}}(x,e(\widetilde{w}_{j}))\,{\rm
d}x\\\
&\geq\liminf_{j\to+\infty}\alpha_{i}\frac{\mathbf{m}_{k_{j}}(u_{e(u(x_{0}))},Q_{(1-\zeta)r_{j}}(x_{0}))}{r_{j}^{n}}\\\
&=(1-\zeta)\alpha_{i}W(x_{0},e(u(x_{0})))\,.\end{split}$ (5.38)

Gathering together (5.37) and (5.38) we deduce

$\liminf_{j\to+\infty}{\alpha_{i}}\int_{Q_{1-\zeta}(0)}W_{k_{j}}(x_{0}+r_{j}x,e(w_{j}))\,{\rm
d}x\geq\frac{(1-\zeta)^{n}}{1+\gamma}\alpha_{i}W(x_{0},e(u(x_{0})))-C\left(\gamma+\frac{\gamma}{1+\gamma}\right)\,.$

With this, (5.34), and (5.35), we eventually deduce (5.31) by arbitrariness of
$\zeta$ and $\gamma$.
Conclusion: from step 3 and step 4 we deduce the validity of (5.5) and the
proof is concluded. ∎

######  Remark 5.2.

We observe en passant that Proposition 5.1 indeed holds also for a sequence of
functionals

$F_{k}(u,A):=\frac{1}{\varepsilon_{k}}\int_{A}f_{k}\Big{(}\varepsilon_{k}W_{k}(\cdot,e(u))*\rho_{k}(x)\Big{)}\,{\rm
d}x\,,\quad\text{for }u\in W^{1,p}(U)\,,\ A\subset U\,,$

provided the functions $f_{k}$ satisfy an estimate of the form

$f_{k}(t)\geq\alpha_{k}t\wedge\beta$

for all $t\in[0,+\infty)$, where $\beta$ is a uniform constant and
$\alpha=\lim_{k\to+\infty}\alpha_{k}$.

######  Proposition 5.3 (Lower bound: surface contribution).

Let $(u_{k})\subset L^{0}(U;\mathbb{R}^{n})$ be a sequence that converges to
in measure to $u\in L^{0}(U;\mathbb{R}^{n})$. Assume moreover that
$F_{k}(u_{k})\leq C$. Then there holds

$\liminf_{k\to+\infty}F_{k}(u_{k},A)\geq\beta\int_{J_{u}\cap
A}\phi_{\rho}(\nu_{u})\,{\rm d}\mathcal{H}^{n-1}\quad\forall
A\in\mathcal{A}(U)\,.$

###### Proof.

Let $(u_{k})$ and $u$ be as in the statement, so that by Proposition 4.1 $u\in
GSBD^{p}(U)$. Let $A\in\mathcal{A}(U)$ be fixed. We claim that it suffices to
show that for any $\xi\in\mathbb{S}^{n-1}$ fixed there holds

$\liminf_{k\to+\infty}F_{k}^{\xi}(u_{k},A)\geq\beta\int_{J_{u}^{\xi}\cap
A}\mu_{\xi}|\langle\nu_{u},\xi\rangle|\,{\rm d}\mathcal{H}^{n-1}\,,$ (5.39)

with

$F_{k}^{\xi}(u_{k},A):=\frac{1}{\varepsilon_{k}}\int_{A}f\Big{(}c_{1}\varepsilon_{k}|\langle
e(u_{k})\xi,\xi\rangle|^{p}*\rho_{k}(x)\Big{)}\,{\rm d}x\,,$
$J_{u}^{\xi}:=\\{x\in J_{u}\colon\langle u^{+}(x)-u^{-}(x),\xi\rangle\neq
0\\}\quad\text{and}\quad\mu_{\xi}:=\mathcal{H}^{1}(\\{x\in S\colon x=t\xi\
\text{ for }\ t\in\mathbb{R}\\})\,.$

Indeed, assume for the moment (5.39) holds true. Then (($W4$)) gives

$W_{k}(x,e(u_{k}))\geq c_{1}|e(u_{k})|^{p}\geq
c_{1}|\langle(e(u_{k}))\xi,\xi\rangle|^{p}\,.$

Since $f$ is nondecreasing, the above implies

$\liminf_{k\to+\infty}F_{k}(u_{k},A)\geq\liminf_{k\to+\infty}F_{k}^{\xi}(u_{k},A)\geq\beta\int_{J_{u}^{\xi}\cap
A}\mu_{\xi}|\langle\nu_{u},\xi\rangle|\,{\rm
d}\mathcal{H}^{n-1}=\beta\int_{J_{u}\cap A}\varphi_{\xi}\,{\rm
d}\mathcal{H}^{n-1}\,,$

with $\varphi_{\xi}\colon J_{u}\to[0,+\infty]$ given by

$\varphi_{\xi}(x):=\begin{cases}\mu_{\xi}|\langle\nu_{u}(x),\xi\rangle|&\text{if
}x\in J_{u}^{\xi}\,,\\\ 0&\text{otherwise}\,.\end{cases}$

Now let $(\xi_{h})\subset\mathbb{S}^{n-1}$ be a dense subset, in this way by
[6, Proposition 1.16] it holds

$\liminf_{k\to+\infty}F_{k}(u_{k},A)\geq\beta\int_{J_{u}\cap
A}\sup_{h}\varphi_{\xi_{h}}\,{\rm d}\mathcal{H}^{n-1}\,.$

On the other hand by [18, Lemma 4.5] we have

$\phi_{\rho}(\nu)=\sup_{\xi\in\mathbb{S}^{n-1}}\mu_{\xi}|\langle\nu,\xi\rangle|\,,$
(5.40)

which in turn implies $\phi_{\rho}(\nu_{u}(x))=\sup_{h}\varphi_{\xi_{h}}(x)$
and the thesis follows. It remains to show (5.39) for which we will argue by
slicing. As the set $S$ is convex for $\delta\in(0,1)$ fixed we can find
$r=r(\delta,S)$ such that the cylinder

$C_{\xi}^{r,\delta}:=R_{\xi}(Q^{\prime}_{r}(0)\times\big{(}-{\mu_{\xi}\delta}/2,{\mu_{\xi}\delta}/2\big{)})\subset\subset
S\,,$

where $R_{\xi}\in SO(n)$ is such that $R_{\xi}e_{n}=\xi$ (see (j)). Let now
$m_{\xi}:=\min_{x\in\overline{C}_{\xi}^{r,\delta}}\rho(x)$ and

$C_{\xi,k}^{r,\delta}(x):=\varepsilon_{k}C_{\xi}^{r,\delta}+x\,.$

Next for any $x\in A$ we denote by $\hat{x}_{\xi}$ the projection of $x$ onto
$\Pi^{\xi}:=\\{y\in\mathbb{R}^{n}\colon\langle y,\xi\rangle=0\\}$ and

$I_{\xi}:=\\{\tau\in\mathbb{R}\colon\hat{x}_{\xi}+\tau\xi\in A\\}\,.$

Thus we have

$\begin{split}F_{k}^{\xi}(u_{k},A)&\geq\frac{1}{\varepsilon_{k}}\int_{A}f\biggl{(}\frac{c_{1}m_{\xi}}{\varepsilon_{k}^{n-1}}\int_{C_{\xi,k}^{r,\delta}(x)}|\langle
e(u_{k}(z))\xi,\xi\rangle|^{p}\,{\rm d}z\biggr{)}\,{\rm d}x\\\
&=\frac{1}{\varepsilon_{k}}\int_{\Pi^{\xi}}\int_{I_{\xi}}f\biggl{(}\frac{c_{1}m_{\xi}}{\varepsilon_{k}^{n-1}}\int_{C_{\xi,k}^{r,\delta}(\hat{x}_{\xi}+\tau\xi)}|\langle
e(u_{k}(z))\xi,\xi\rangle|^{p}\,{\rm d}z\biggr{)}{\rm d}\tau\,{\rm
d}\mathcal{H}^{n-1}(\hat{x}_{\xi})\ \,,\end{split}$ (5.41)

where the last equality follows by Fubini’s Theorem. Noticing that $f$ is
concave and using the change of variable
$z=\hat{x}_{\xi}+s\xi+\varepsilon_{k}rz^{\prime}$ with

$z^{\prime}\in
Q^{\prime}_{\xi}:=R_{\xi}\big{(}Q^{\prime}\times\\{0\\}\big{)}\quad\text{and}\quad
s\in\bigg{(}\tau-\frac{\mu_{\xi}\delta\varepsilon_{k}}{2},\tau+\frac{\mu_{\xi}\delta\varepsilon_{k}}{2}\bigg{)}\,,$

together with Jensen’s inequality yield

$\begin{split}f\biggl{(}\frac{c_{1}m_{\xi}}{\varepsilon_{k}^{n-1}}\int_{C_{\xi,k}^{r,\delta}(\hat{x}_{\xi}+\tau\xi)}|\langle
e(u_{k}(z))\xi,\xi\rangle|^{p}\,{\rm
d}z\biggr{)}&\geq\fint_{Q^{\prime}_{\xi}}\widetilde{f}\biggl{(}\int_{\tau-\frac{\mu_{\xi}\delta\varepsilon_{k}}{2}}^{\tau+\frac{\mu_{\xi}\delta\varepsilon_{k}}{2}}|\langle
e(u_{k}(\hat{x}_{\xi}+\varepsilon_{k}rz^{\prime}+s\xi))\xi,\xi\rangle|^{p}\,{\rm
d}s\biggr{)}\,{\rm d}z^{\prime}\\\
&=\fint_{Q^{\prime}_{\xi}}\widetilde{f}\biggl{(}\int_{\tau-\frac{\mu_{\xi}\delta\varepsilon_{k}}{2}}^{\tau+\frac{\mu_{\xi}\delta\varepsilon_{k}}{2}}\Big{|}\frac{\partial}{\partial
s}w_{\xi,k}(\hat{x}_{\xi},z^{\prime},s)\Big{|}^{p}\,{\rm d}s\biggr{)}\,{\rm
d}z^{\prime}\end{split}$ (5.42)

with
$\widetilde{f}(t):=f\big{(}\frac{c_{1}m_{\xi}}{\varepsilon_{k}^{n-1}}t\big{)}$
and $w_{\xi,k}(\hat{x}_{\xi},z^{\prime},s):=\langle
u_{k}(\hat{x}_{\xi}+\varepsilon_{k}rz^{\prime}+s\xi),\xi\rangle$. Observe now
that applying Corollary 3.9 and Fubini’s Theorem to the functions
$w_{\xi,k}(\hat{x}_{\xi},z^{\prime},s)$ we have that, for a.e.
$(\hat{x}_{\xi},z^{\prime})\in\Pi_{\xi}\times Q^{\prime}_{\xi}$ the functions
$s\mapsto w_{\xi,k}(\hat{x}_{\xi},z^{\prime},s)$ converge to the section
$u^{\hat{x}_{\xi}}(s):=u(\hat{x}_{\xi}+s\xi)\cdot\xi$ in measure on $I_{\xi}$.
Further, gathering together (5.41) and (5.42), and exchanging the order of
integration it holds

$F_{k}^{\xi}(u_{k},A)\geq\int_{\Pi^{\xi}}\fint_{Q^{\prime}_{\xi}}\bigg{(}\frac{1}{\varepsilon_{k}}\int_{I_{\xi}}\widetilde{f}\biggl{(}\int_{\tau-\frac{\mu_{\xi}\delta\varepsilon_{k}}{2}}^{\tau+\frac{\mu_{\xi}\delta\varepsilon_{k}}{2}}|\dot{w}^{\hat{x}_{\xi},z^{\prime}}_{\xi,k}(s)|^{p}\,{\rm
d}s\biggr{)}{\rm d}\tau\bigg{)}\,{\rm d}z^{\prime}\,{\rm
d}\mathcal{H}^{n-1}(\hat{x}_{\xi})\,,$ (5.43)

where the shortcut $w^{\hat{x}_{\xi},z^{\prime}}_{\xi,k}(s)$ denotes the
function $s\mapsto w_{\xi,k}(\hat{x}_{\xi},z^{\prime},s)$ for fixed
$(\hat{x}_{\xi},z^{\prime})$. By Theorem 3.3 we get

$\liminf_{k\to+\infty}\frac{1}{\varepsilon_{k}}\int_{I_{\xi}}\widetilde{f}\biggl{(}\int_{\tau-\frac{\mu_{\xi}\delta\varepsilon_{k}}{2}}^{\tau+\frac{\mu_{\xi}\delta\varepsilon_{k}}{2}}|\dot{w}^{\hat{x}_{\xi},z^{\prime}}_{\xi,k}(s)|^{p}\,{\rm
d}s\biggr{)}{\rm d}\tau\geq\beta\delta\mu_{\xi}\\#(J_{u^{\hat{x}_{\xi}}}\cap
I_{\xi})\,.$ (5.44)

Combining (5.43) with (5.44) we finally obtain

$\liminf_{k\to+\infty}F_{k}^{\xi}(u_{k},A)\geq\delta\beta\int_{\Pi^{\xi}}\mu_{\xi}\\#(J_{u^{\hat{x}_{\xi}}}\cap
I_{\xi})\,{\rm
d}\mathcal{H}^{n-1}(\hat{x}_{\xi})=\delta\beta\int_{J_{u}^{\xi}\cap
A}\mu_{\xi}|\langle\nu_{u},\xi\rangle|\,{\rm d}\mathcal{H}^{n-1}\,.$

Eventually by the arbitrariness of $\delta$ we deduce (5.39). ∎

With the help of Propositions 5.1 and 5.3 we can now prove the following lower
bound.

######  Proposition 5.4 (Lower-bound).

Let $F_{k}$ and $F$ be as in (2.5) and (2.9) respectively. Let $(u_{k})\subset
L^{0}(U;\mathbb{R}^{n})$, $u\in GSBD^{p}(U;\mathbb{R}^{n})$ be such that
$u_{k}$ converges to $u$ in measure. Then there exists a subsequence, not
relabelled, such that

$\liminf_{k\to+\infty}F_{k}(u_{k})\geq F(u)\,.$

###### Proof.

Let $(u_{k})$ and $u$ be as in the statement. We assume without loss of
generality that

$\liminf_{k\to+\infty}F_{k}(u_{k})=\lim_{k\to+\infty}F_{k}(u_{k})<+\infty\,.$

Thus in particular there exists $C>0$ such that $F_{k}(u_{k})\leq C$ and by
Proposition 4.1 it follows that $u\in GSBD^{p}(U)$. We define
$\varphi_{1},\varphi_{2}\colon U\to[0,+\infty)$ as follows

$\varphi_{1}(x):=\begin{cases}\alpha W\big{(}x,e(u(x))\big{)}&\text{if }x\in
U\setminus J_{u}\,,\\\ 0&\text{otherwise}\,,\end{cases}$
$\varphi_{2}(x):=\begin{cases}\beta\phi_{\rho}(\nu_{u}(x))&\text{if }x\in
J_{u}\,,\\\ 0&\text{otherwise}\,.\end{cases}$

Moreover set

$\lambda:=\mathcal{L}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega+\mathcal{H}^{n-1}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits J_{u}\,.$

Now by invoking Propositions 5.1 and 5.3 we have that

$\liminf_{k\to+\infty}F_{k}(u_{k},A)\geq\int_{A}\varphi_{1}\,\mathrm{d}\lambda\,,$

and

$\liminf_{k\to+\infty}F_{k}(u_{k},A)\geq\int_{A}\varphi_{2}\,\mathrm{d}\lambda\,,$

for all $A\in\mathcal{A}(U)$. Then by [6, Proposition 1.16] we find that

$\liminf_{k\to+\infty}F_{k}(u_{k},A)\geq\int_{A}\varphi_{1}\vee\varphi_{2}\,\mathrm{d}\lambda=F(u,A)\,,$

from which in particular

$\liminf_{k\to+\infty}F_{k}(u_{k})\geq\int_{A}\varphi_{1}\vee\varphi_{2}\,\mathrm{d}\lambda=F(u)\,.$

∎

##  6\. Upper bound

In this section we prove the upper bound.

######  Proposition 6.1.

Let $F_{k}$ and $F$ be as in (2.5) and (2.9) respectively. Then for each $u\in
L^{0}(U;\mathbb{R}^{n})$ there is $(u_{k})\subset L^{0}(U;\mathbb{R}^{n})$
that converges in measure to $u$ and such that

$\limsup_{k\to\infty}F_{k}(u_{k})\leq F(u)\,.$

###### Proof.

Without loss of generality we assume $F(u)<C$ so that $u\in GSBD^{p}(U)$.
Moreover by Theorem 3.4 we can assume that $u\in\mathcal{W}^{\infty}_{\rm
pw}(U;\mathbb{R}^{n})$ and that $J_{u}$ is a connected $(n-1)$-rectifiable set
compactly contained in $U$. We fix $U^{\prime}\in\mathcal{A}$ with
$U\subset\subset U^{\prime}$ and consider an extension of $u$ on $U^{\prime}$,
not relabelled, such that $u\in\mathcal{W}^{\infty}_{\rm
pw}(U^{\prime};\mathbb{R}^{n})$. Then by Theorem 3.1 and Remark 3.2 we can
find $(v_{k})\subset W^{1,p}(U^{\prime}\setminus J_{u};\mathbb{R}^{n})$ such
that $v_{k}$ converges strongly to $u$ in $L^{p}(U^{\prime}\setminus
J_{u};\mathbb{R}^{n})$ and

$\lim_{k\to\infty}E_{k}(v_{k},U^{\prime}\setminus
J_{u})=E(u,U^{\prime}\setminus J_{u})=\int_{U^{\prime}}W(x,e(u))\,{\rm d}x\,.$
(6.1)

where the last equality clearly holds as $J_{u}$ is a null set. For every
$h>0$ we set

$(J_{u})_{h}:=\\{x\in U\colon{\rm d}_{S}(x,J_{u})<h\\}\,,$

so that for $h$ small enough $(J_{u})_{h}\subset\subset U$. Fix now
$0<\delta_{k}<<\varepsilon_{k}$ and take $\varphi_{k}\in
C^{\infty}_{c}(U^{\prime})$ cutoff between $(J_{u})_{\delta_{k}}$ and
$(J_{u})_{2\delta_{k}}$. Next define $(u_{k})\subset
W^{1,p}(U^{\prime};\mathbb{R}^{n})$ as

$u_{k}:=v_{k}(1-\varphi_{k})\to u\quad\text{strongly in
}L^{p}(U^{\prime}\setminus J_{u};\mathbb{R}^{n})\,,$

and in particular $u_{k}\to u$ in measure on $U^{\prime}$. Then using that
$u_{k}=v_{k}$ in $U^{\prime}\setminus(J_{u})_{2\delta_{k}}$ we have

$\begin{split}F_{k}(u_{k})\leq F_{k}(v_{k},U\setminus
J_{u})+\beta\frac{\mathcal{L}^{n}((J_{u})_{2\delta_{k}+\varepsilon_{k}})}{\varepsilon_{k}}\,.\end{split}$
(6.2)

Now invoking [20, Theorem 3.7] we have

$\lim_{k\to\infty}\frac{\mathcal{L}^{n}((J_{u})_{2\delta_{k}+\varepsilon_{k}})}{\varepsilon_{k}}=\int_{J_{u}}\phi_{\rho}(\nu_{u})\,{\rm
d}\mathcal{H}^{n-1}\,.$ (6.3)

Moreover as $f$ is increasing and satisfies (2.1) then $f(t)\leq\hat{\alpha}t$
for all $\hat{\alpha}>\alpha$. This together with the change of variable
$y=x-\varepsilon_{k}z$, Fubini’s theorem, and the change of variable
$\hat{x}=x-\varepsilon_{k}z$ yield

$\begin{split}F_{k}(v_{k},U\setminus J_{u})&\leq\hat{\alpha}\int_{U\setminus
J_{u}}\int_{\mathbb{R}^{n}}W_{k}(y,e(v_{k}))\rho_{k}(x-y)\,{\rm d}y\,{\rm
d}x\\\
&=\hat{\alpha}\int_{U}\int_{\mathbb{R}^{n}}W_{k}\big{(}x-\varepsilon_{k}z,e(v_{k}(x-\varepsilon_{k}\cdot))\big{)}\rho(z)\,{\rm
d}z\,{\rm d}x\\\
&=\hat{\alpha}\int_{\mathbb{R}^{n}}\rho(z)\int_{U}W_{k}\big{(}x-\varepsilon_{k}z,e(v_{k}(x-\varepsilon_{k}\cdot))\big{)}\,{\rm
d}x\,{\rm d}z\\\ &\leq\hat{\alpha}\int_{U^{\prime}}W_{k}(x,e(v_{k}))\,{\rm
d}x=E_{k}(v_{k},U^{\prime}\setminus J_{u})\,.\end{split}$

Hence passing to the limit in $k$ in the above inequality and using (6.1) we
get

$\limsup_{k\to\infty}F_{k}(v_{k},{U\setminus
J_{u}})\leq\hat{\alpha}E(u,U^{\prime}\setminus
J_{u})=\hat{\alpha}\int_{U^{\prime}}W(x,e(u))\,{\rm d}x\,,$ (6.4)

for all $\hat{\alpha}>\alpha$. Finally gathering together (6.2)-(6.4) we
obtain

$\limsup_{k\to\infty}F_{k}(u_{k})\leq\hat{\alpha}\int_{U^{\prime}}W(x,e(u))\,{\rm
d}x+\beta\int_{J_{u}}\phi_{\rho}(\nu_{u})\,{\rm d}\mathcal{H}^{n-1}\,.$

Eventually by the arbitrariness of $U^{\prime}$ and $\hat{\alpha}$ we
conclude. ∎

######  Remark 6.2.

If a lower order term $\int_{U}\psi(|u|)\,{\rm d}x$, is added to the energy,
the density argument above can still be applied if $\psi$ complies with the
assumptions of Theorem 3.4. Also observe that within the same assumptions,
$\int_{U}\psi(|u_{k}|)\,{\rm d}x$ is equiintegrable whenever $(u_{k})$ is
converging in $L^{p}$. For $u_{k}$ and $u$ as in the proof above, this entails
the convergence $\int_{U}\psi(|u_{k}|)\,{\rm d}x\to\int_{U}\psi(|u|)\,{\rm
d}x$.

We are now in a position to prove Theorem 2.1.

###### Proof of Theorem 2.1.

Point $(i)$ follows by combining Propositions 5.4 and 6.1, while $(ii)$ is a
consequence of Proposition 4.1. ∎

##  7\. Stochastic homogenisation

In this section we are concerned with the $\Gamma$-convergence analysis of the
functionals $F_{k}$ when $W_{k}$ are random integrands of type

$W_{k}(\omega,y,M)=W\Big{(}\omega,\frac{y}{\delta_{k}},M\Big{)}\,,$

with $\omega$ belonging to the sample space $\Omega$ of a complete probability
space $(\Omega,\mathcal{T},P)$ and $\delta_{k}\searrow 0$. In order to do that
we first give some definitions.

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

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

  1. $(a)$

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

  2. $(b)$

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

  3. $(c)$

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

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

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

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

and we set

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

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

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

  1. $(a)$

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

  2. $(b)$

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

  3. $(c)$

(subadditivity) for every $A\in\mathcal{I}_{n}$ and for every finite family
$(A_{i})_{i\in I}\subset\mathcal{I}_{n}$ 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. $(d)$

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

######  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
$W\colon\Omega\times\mathbb{R}^{n}\times\mathbb{M}^{n\times n}\to[0,+\infty)$
is stationary with respect to $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ if

$W(\omega,x+z,M)=W(\tau_{z}(\omega),x,M)$

for every $\omega\in\Omega$, $x\in\mathbb{R}^{n}$, $z\in\mathbb{Z}^{n}$ and
$M\in\mathbb{M}^{n\times n}$. Moreover we say that a stationary random
integrand $W$ is ergodic if $(\tau_{z})_{z\in\mathbb{Z}^{n}}$ is ergodic.

For our purposes we consider random integrands
$W\colon\Omega\times\mathbb{R}^{n}\times\mathbb{M}^{n\times n}\to[0,+\infty)$
satisfying the following assumptions:

  1. $(w1)$

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

  2. $(w2)$

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

  3. $(w3)$

the map $M\mapsto W(\omega,x,M)$ is lower semicontinuous for every
$\omega\in\Omega$ and every $x\in\mathbb{R}^{n}$.

Let $W$ be a random integrand satisfying $(w1)$-$(w3)$ and $\delta_{k}\searrow
0$. We consider the family of functionals $F_{k}(\omega)\colon
L^{0}(U;\mathbb{R}^{n})\to[0,+\infty]$ defined as

$F_{k}(\omega)(u):=\frac{1}{\varepsilon_{k}}\int_{U}f\bigg{(}\varepsilon_{k}W\Big{(}\omega,\frac{\cdot}{\delta_{k}},e(u)\Big{)}*\rho_{k}(x)\bigg{)}\,{\rm
d}x\,,$ (7.1)

if $u\in W^{1,p}(U;\mathbb{R}^{n})$, and extended to $+\infty$ otherwise. Let
also for $\omega\in\Omega$ and $A\in\mathcal{A}$

$\mathbf{m}_{\omega}(u_{M},A):=\inf\left\\{\int_{A}W(\omega,x,e(v))\,{\rm
d}x\colon v\in W^{1,p}(A;\mathbb{R}^{n}),\ v=u\ \text{near}\ \partial
A\right\\}\,.$ (7.2)

We now state the main theorem of this section.

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

Let $W$ be a random integrand satisfying $(w1)$-$(w3)$. Assume moreover $W$ 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 $F_{k}(\omega)$ 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 $\omega\in\Omega^{\prime}$,
$x\in\mathbb{R}^{n}$, $M\in\mathbb{M}^{n\times n}$ the limit

$\lim_{t\to+\infty}\frac{\mathbf{m}_{\omega}(u_{M},Q_{t}(tx))}{t^{n}}=\lim_{t\to+\infty}\frac{\mathbf{m}_{\omega}(u_{M},Q_{t}(0))}{t^{n}}=:W_{\rm
hom}(\omega,M)$ (7.3)

exists and is independent of $x$. The function $W_{\rm
hom}\colon\Omega\times\mathbb{M}^{n\times n}\to[0,+\infty)$ is
$(\mathcal{T}\otimes\mathcal{B}^{n\times n})$-measurable. Moreover, for every
$\omega\in\Omega^{\prime}$ the functionals $F_{k}(\omega)$ $\Gamma$-converge
in measure to the functional $F_{\rm hom}(\omega)\colon
L^{0}(U;\mathbb{R}^{n})\to[0,+\infty]$ given by

$F_{\rm hom}(\omega)(u):=\begin{cases}\displaystyle\ \alpha\int_{U}W_{\rm
hom}(\omega,e(u))\,{\rm d}x+\beta\int_{J_{u}}\phi_{\rho}(\nu_{u})\,{\rm
d}\mathcal{H}^{n-1}&\text{if }u\in GSBD^{p}(U)\,,\\\
+\infty&\text{otherwise}\,.\end{cases}$

If, in addition, $W$ is ergodic, then $W_{\rm hom}$ is independent of $\omega$
and

$W_{\rm
hom}(M)=\lim_{t\to+\infty}\frac{1}{t^{n}}\int_{\Omega}\mathbf{m}_{\omega}(u_{M},Q_{t}(0))\,{\rm
d}P(\omega)\,,$ (7.4)

and thus $F_{\rm hom}$ is deterministic.

The rest of this section is devoted to prove Theorem 7.4. This will be done in
a number of steps.

######  Proposition 7.5.

Let $W$ be a stationary random integrand satisfying $(w1)$-$(w3)$ and let
$\mathbf{m}_{\omega}$ be as in (7.2). Then for every $M\in\mathbb{M}^{n\times
n}$ the function $\mu_{M}\colon\Omega\times\mathcal{I}_{n}\to\mathbb{R}$ given
by $\mu_{M}(\omega,A):=\mathbf{m}_{\omega}(u_{M},A)$ defines a subadditive
process on $(\Omega,\mathcal{T},P)$.

###### Proof.

Let $M\in\mathbb{M}^{n\times s}$ be fixed. Then we need to show that $\mu_{M}$
satisfies properties $(a)$–$(d)$.
Step 1: measurability. Let $A\in\mathcal{I}_{n}$ be fixed. For
$N\in\mathbb{N}$ let

$W^{N}(\omega,x,M):=\inf_{\xi\in\mathbb{M}^{n\times
n}}\\{W(\omega,x,\xi)+N|\xi-M|\\}$

be the Moreau-Yosida regularisation of $M\mapsto W(\omega,x,M)$ which is
$N$-Lipschitz. Let also $F^{N}(\omega)\colon W^{1,p}(A)\to[0,+\infty)$ be
defined as

$F^{N}(\omega)(u):=\int_{A}W^{N}(\omega,x,e(u))\,{\rm d}x\,.$

Arguing as in the proof of [22, Lemma C.1.] it can be shown that
$(\omega,u)\mapsto F^{N}(\omega)(u)$ is
$\mathcal{T}\otimes\mathcal{B}(W^{1,p}(A))$-measurable. By $(w3)$
$W^{N}\nearrow W$ pointwise, and in particular $F^{N}(\omega)(u)$ converges to
$\int_{A}W(\omega,x,e(u))\,{\rm d}x$ pointwise. As a consequence
$(\omega,u)\mapsto\int_{A}W(\omega,x,e(u))\,{\rm d}x$ is also
$\mathcal{T}\otimes\mathcal{B}(W^{1,p}(A))$-measurable. Now we note that
$F(\omega)(u_{M})<+\infty$. This together with $(w3)$ and [22, Lemma C.2.]
imply that $\omega\mapsto\mu_{M}(u_{M},A)$ is $\mathcal{T}$-measurable.
Step 2: covariance. Let $\omega\in\Omega$, $A\in\mathcal{I}_{n}$ and
$z\in\mathbb{Z}^{n}$ be fixed. Let $v\in W^{1,p}(A+z)$. Then $v=u_{M}$ near
$\partial(A+z)$ if and only if $v(\cdot+z)=u_{M}$ near $\partial A$; moreover
the stationarity of $W$ yields

$\int_{A+z}W(\omega,x,e(v))\,{\rm
d}x=\int_{A}W(\omega,x+z,e(v(\cdot+z)))\,{\rm
d}x=\int_{A}W(\tau_{z}(\omega),x,e(v(\cdot+z)))\,{\rm d}x\,.$

Hence we have that

$\mu_{M}(\omega,A+z)=\mu_{M}(\tau_{z}(\omega),A)\,.$

Step 3: subadditivity. Let $\omega\in\Omega$ and $A\in\mathcal{I}_{n}$ be
fixed. Let $\\{A_{1},...,A_{N}\\}\subset\mathcal{I}_{n}$ be pairwise disjoint
such that $A=\cup_{i=1}^{N}A_{i}$. For each $i\in\\{1,...,N\\}$ we take
$v_{i}\in W^{1,p}(A_{i})$ with $v_{i}=u_{M}$ near $\partial A_{i}$. Let $v\in
W^{1,p}(A)$ be given by $v:=v_{i}$ in $A_{i}$. Then clearly $v=u_{M}$ near
$\partial A$ and

$\mu_{M}(\omega,A)\leq\int_{A}W(\omega,x,e(v))\,{\rm
d}x=\sum_{i=1}^{N}\int_{A_{i}}W(\omega,x,e(v_{i}))\,{\rm d}x\,.$

By the arbitrariness of $v_{i}$ we then conclude

$\mu_{M}(\omega,A)\leq\sum_{i=1}^{N}\mu_{M}(\omega,A_{i})\,.$

Step 4: boundedness. Let $\omega\in\Omega$ and $A\in\mathcal{I}_{n}$ be fixed.
Then by ($W4$) we get

$\mu_{M}(\omega,A)\leq\int_{A}W(\omega,x,e(u_{M}))\,{\rm d}x\leq
c_{2}(|M+M^{T}|^{p}+1)\mathcal{L}^{n}(A)\,.$

∎

With the help of Proposition 7.5 we can now prove the main result of this
section.

###### Proof of Theorem 7.4.

By Theorem 2.4 the $\Gamma$-convergence of $F_{k}(\omega)$ follows if we show
that there exists a set $\Omega^{\prime}\in\mathcal{T}$ of full probability
such that (7.3) holds true for all $\omega\in\Omega^{\prime}$.
Let $M\in\mathbb{M}^{n\times n}$ be fixed. By Proposition 7.5 and the
Subadditive Ergodic Theorem [1, Theorem 2.4] we deduce the existence of
$\Omega_{M}\in\mathcal{T}$ with $P(\Omega_{M})=1$ and of a
$\mathcal{T}$-measurable function $\phi_{M}\colon\Omega\to[0,+\infty)$ such
that

$\phi_{M}(\omega)=\lim_{t\to+\infty}\frac{\mathbf{m}_{\omega}(u_{M},t\widetilde{Q})}{|t\widetilde{Q}|}\,,$

for all $\omega\in\Omega_{M}$ and all cubes
$\widetilde{Q}\subset\mathbb{R}^{n}$. We set

$\Omega^{\prime}:=\bigcap_{M\in\mathbb{Q}^{n\times n}}\Omega_{M}\,,$ (7.5)

which satisfies $P(\Omega^{\prime})=1$, and let $W_{\rm
hom}\colon\Omega\times\mathbb{M}^{n\times n}\to[0,+\infty)$ be given by

$W_{\rm
hom}(\omega,M):=\limsup_{t\to+\infty}\frac{\mathbf{m}_{\omega}(u_{M},Q_{t}(0))}{t^{n}}\,.$

Clearly $W_{\rm hom}(\omega,M)=\phi_{M}(\omega)$ for all
$\omega\in\Omega^{\prime}$, $M\in\mathbb{Q}^{n\times n}$. Let now
$\omega\in\Omega^{\prime}$, $t>0$, $\eta\in(0,1)$ and $\widetilde{Q}$ (cube
centred at $x$) be fixed. Choose $M_{1}\in\mathbb{Q}^{n\times n}$ and
$M_{2}\in\mathbb{M}^{n\times n}$. Choose also $u\in
W^{1,p}(t\widetilde{Q};\mathbb{R}^{n})$ with $u=u_{M_{2}}$ near
$\partial(t\widetilde{Q})$ and

$\int_{t\widetilde{Q}}W(\omega,x,e(u))\,{\rm
d}x\leq\mathbf{m}_{\omega}(u_{M_{2}},t\widetilde{Q})+\eta\,.$

We extend $u$ with $u_{M_{2}}$ in $\mathbb{R}^{n}\setminus A$ without
relabelling it. Next we take $\varphi_{\eta}\in
C^{\infty}_{c}(\mathbb{R}^{n},[0,1])$ with $\varphi_{\eta}\equiv 1$ on
$t\widetilde{Q}$, $\varphi_{\eta}\equiv 0$ on
$\mathbb{R}^{n}\setminus(1+\eta)t\widetilde{Q}$ and
$\|\nabla\varphi_{\eta}\|_{L^{\infty}(\mathbb{R}^{n})}\leq C/\eta$, and set

$v:=u\varphi_{\eta}+u_{M_{1}}(1-\varphi_{\eta})\,,$

which is admissible in the definition of
$\mathbf{m}_{\omega}(u_{M_{1}},(1+\eta)t\widetilde{Q})$. By ($W4$) it holds

$\begin{split}\int_{(1+\eta)t\widetilde{Q}}W(\omega,x,e(v))\,{\rm
d}x\leq\int_{t\widetilde{Q}}W(\omega,x,e(u))\,{\rm
d}x+{c_{2}}\int_{(1+\eta)t\widetilde{Q}\setminus t\widetilde{Q}}(|\nabla
v+\nabla v^{T}|^{p}+1)\,{\rm d}x\,.\end{split}$ (7.6)

As $v=u_{M_{2}}\varphi_{\eta}+u_{M_{1}}(1-\varphi_{\eta})$ in
$(1+\eta)t\widetilde{Q}\setminus t\widetilde{Q}$ we get

$\begin{split}|\nabla v+\nabla v^{T}|^{p}&\leq
C\Big{(}|M_{1}+M_{1}^{T}|^{p}+|M_{2}+M_{2}^{T}|^{p}+\frac{1}{\eta^{p}}|{M_{1}}-{M_{2}}|^{p}|x|^{p}\Big{)}\\\
&\leq
C\Big{(}|M_{1}+M_{1}^{T}|^{p}+|M_{2}+M_{2}^{T}|^{p}+\frac{1}{\eta^{p}}|{M_{1}}-{M_{2}}|^{p}(1+\eta)^{p}\Big{)}\,.\end{split}$
(7.7)

Combining (7.6) with (7.7) we obtain

$\begin{split}\mathbf{m}_{\omega}(u_{M_{1}},&(1+\eta)t\widetilde{Q})\leq\mathbf{m}_{\omega}(u_{M_{2}},t\widetilde{Q})\\\
&+C\Big{(}|M_{1}+M_{1}^{T}|^{p}+|M_{2}+M_{2}^{T}|^{p}+\frac{1}{\eta^{p}}|{M_{1}}-{M_{2}}|^{p}+1\Big{)}((1+\eta)^{n}-1)t^{n}|\widetilde{Q}|\,.\end{split}$

Rescaling by $(1+\eta)^{n}t^{n}$ and passing to the limit as $t\to+\infty$ in
the above inequality we find

$\begin{split}W_{\rm
hom}(\omega,M_{1})&\leq\frac{1}{(1+\eta)^{n}}\liminf_{t\to+\infty}\frac{\mathbf{m}_{\omega}(u_{M_{2}}t\widetilde{Q})}{t^{n}|\widetilde{Q}|}\\\
&+C\Big{(}|M_{1}+M_{1}^{T}|^{p}+|M_{2}+M_{2}^{T}|^{p}+\frac{1}{\eta^{p}}|{M_{1}}-{M_{2}}|^{p}+1\Big{)}\frac{((1+\eta)^{n}-1)}{(1+\eta)^{n}}\,.\end{split}$
(7.8)

By inverting the role of $M_{1}$ and $M_{2}$ we can similarly find

$\begin{split}\frac{1}{(1+\eta)^{n}}\limsup_{t\to+\infty}&\frac{\mathbf{m}_{\omega}(u_{M_{2}},t\widetilde{Q})}{t^{n}|\widetilde{Q}|}\leq
W_{\rm hom}(\omega,M_{1})\\\
&+C\Big{(}|M_{1}+M_{1}^{T}|^{p}+|M_{2}+M_{2}^{T}|^{p}+\frac{1}{\eta^{p}}|{M_{1}}-{M_{2}}|^{p}+1\Big{)}\frac{((1+\eta)^{n}-1)}{(1+\eta)^{n}}\,.\end{split}$
(7.9)

Thus by the arbitrariness of $M_{1}$ we can choose a sequence
$(M^{j}_{1})\subset\mathbb{Q}^{n\times n}$ converging to $M_{2}$ such that by
combining (7.8) and (7.9) we obtain

$\limsup_{t\to+\infty}\frac{\mathbf{m}_{\omega}(u_{M_{2}},t\widetilde{Q})}{t^{n}|\widetilde{Q}|}\leq\liminf_{t\to+\infty}\frac{\mathbf{m}_{\omega}(u_{M_{2}},t\widetilde{Q})}{t^{n}|\widetilde{Q}|}+C\Big{(}|M_{2}+M_{2}^{T}|^{p}+1\Big{)}((1+\eta)^{n}-1)\,.$

Eventually by letting $\eta\to 0$ we find that for all
$\omega\in\Omega^{\prime}$ there exists the limit

$\lim_{t\to+\infty}\frac{\mathbf{m}_{\omega}(u_{M_{2}},t\widetilde{Q})}{t^{n}|\widetilde{Q}|}=\limsup_{t\to+\infty}\frac{\mathbf{m}_{\omega}(u_{M_{2}},Q_{t}(0))}{t^{n}}=W_{\rm
hom}(\omega,M_{2})\,,$ (7.10)

and is independent of $x$. Eventually if $W$ is ergodic then [1, Theorem 2.4]
ensures that $W_{\rm hom}$ does not depend on $\omega$. Moreover by Birkhoff
ergodic Theorem we deduce (7.4). ∎

## Acknowledgments

The work of FS was partially supported by the project _Variational methods for
stationary and evolution problems with singularities and interfaces_ PRIN 2017
(2017BTM7SN) financed by the Italian Ministry of Education, University, and
Research and by the project Starplus 2020 Unina Linea 1 _New challenges in the
variational modeling of continuum mechanics_ from the University of Naples
“Federico II” and Compagnia di San Paolo (CUP: E65F20001630003). He is also
member of the GNAMPA group of INdAM.

##  Appendix A A remark on the non-local approximation of free-discontinuity
problems in $GSBV$

This Appendix is devoted to the statement of a $\Gamma$-convergence Theorem
for non-local functionals depending on the full deformation gradient $\nabla
u$. The result we are going to state has actually been proved in [11, Theorem
3.2], under an additional technical assumption, the so called stable
$\gamma$-convergence of the functionals

$\widetilde{E}_{k}(u,A):=\begin{cases}\displaystyle\int_{A}W_{k}(x,\nabla
u)\,{\rm d}x&\text{ if }u\in W^{1,p}(A;\mathbb{R}^{n})\,,\\\ +\infty&\text{
otherwise }\,.\end{cases}$ (A.1)

This assumption, stated in [11, Definition 7.2] is stronger than simple
$\Gamma$-convergence, and introduces a limitation to the class of functionals
to which the theorem applies, although relevant examples fulfilling this
condition can be readily provided (see [11, Examples 7.3-7.5]). Actually, the
inspection of the proof of Proposition 5.1, which can be clearly adapted to
the $GSBV$ setting, shows that it is not needed. For the reader’s convenience
we give a precise statement of the result, after recalling the structural
assumptions on the non-local approximation energies under which it is
formulated.

The functions $W_{k}$ are assumed to satisfy ($W1$)–($W2$), together with

  1. $(W4^{\prime})$

for every $x\in\mathbb{R}^{n}$ and every $M\in\mathbb{M}^{n\times n}$

$c_{1}|M|^{p}\leq W_{k}(x,M)\leq c_{2}(|M|^{p}+1)\,.$

We will denote with $\widetilde{E}$ the $\Gamma$-limit with respect to the
convergence in measure of the functionals $\widetilde{E}_{k}$ in (A.1), given
by (see [13, Theorem 20.4])

$\widetilde{E}(u,A):=\begin{cases}\displaystyle\int_{A}W(x,\nabla u)\,{\rm
d}x&\text{ if }u\in W^{1,p}(A;\mathbb{R}^{n})\,,\\\ +\infty&\text{ otherwise
}\,.\end{cases}$

where, for every $x\in u$ and every $M\in\mathbb{M}^{n\times n}$

$W(x,M)=W^{\prime}(x,M)=W^{\prime\prime}(x,M)\,.$ (A.2)

Above, $W^{\prime}$ and $W^{\prime\prime}$ are defined in (2.7), and (2.8),
respectively, provided that $E_{k}$ is replaced by $\widetilde{E}_{k}$. We
then consider the non-local functionals

$\widetilde{F}_{k}(u):=\begin{cases}\displaystyle\frac{1}{\varepsilon_{k}}\int_{U}f_{k}\Big{(}\varepsilon_{k}W_{k}(\cdot,\nabla
u)*\rho_{k}(x)\Big{)}\,{\rm d}x&\text{ if }u\in
W^{1,p}(U;\mathbb{R}^{n})\,,\\\ +\infty&\text{ otherwise }.\end{cases}$ (A.3)

where $\rho_{k}$ are as in Section 2.3, while
$f_{k}\colon[0,+\infty)\to[0,+\infty)$ are concave and satisfy

$a_{1}t\wedge b_{1}\leq f_{k}(t)\leq b_{2}$ (A.4)

for suitable uniform constants $a_{1}$, $b_{1}$, $b_{2}>0$. We then have the
following theorem.

######  Theorem A.1.

Assume ($W1$), ($W2$), and $(W4^{\prime})$. Consider a sequence of concave
functions $f_{k}$ as in (A.4) and convolution kernels $\rho_{k}$ as in Section
2.3. Let the functionals $\widetilde{F}_{k}$ be given by (A.3). Finally,
assume that

$\alpha_{k}t\wedge b_{1}\leq f_{k}(t)\leq
b_{2}\quad\mbox{with}\quad\lim_{k\to+\infty}\alpha_{k}-f^{\prime}_{k}(0)=0\,.$
(A.5)

Then $\widetilde{F}_{k}$ $\Gamma$-converge, with respect to the convergence in
measure, to a functional of the form

$\alpha\int_{U}W(x,\nabla u)\,{\rm
d}x+\int_{J_{u}}\varphi(x,[u],\nu_{u})\,{\rm d}\mathcal{H}^{n-1}$

where $W$ is given by (A.2), $\alpha=\liminf f_{k}^{\prime}(0)$, and $\varphi$
is a suitable Carathéodory integrand.

###### Proof.

By [11, Theorem 3.1] we have that the $\Gamma$-limit of $\widetilde{F}_{k}$ is
an integral functional of the form

$\int_{U}W_{\infty}(x,\nabla u)\,{\rm
d}x+\int_{J_{u}}\varphi(x,[u],\nu_{u})\,{\rm d}\mathcal{H}^{n-1}\,.$

For $W^{\prime}$ and $W^{\prime\prime}$ as in (2.7), and (2.8), respectively,
one has only to show that $W_{\infty}\leq\alpha W^{\prime\prime}$ and
$W_{\infty}\geq\alpha W^{\prime}$. The first inequality is actually already
proved in [11, Proposition 7.1]. As for the second, notice under assumption
(A.5) and taking into account Remark 5.2, it can be recovered by exactly
following the argument of Proposition 5.1, provided one is willing to replace
each occurrence of $e(u)$ with $\nabla u$. ∎

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