# Lower semicontinuity and relaxation for free discontinuity functionals with
non-standard growth

Stefano Almi  Dipartimento di Matematica e Applicazioni “R. Caccioppoli”,
Università di Napoli Federico II, via Cintia, 80126 Napoli, Italy.
<EMAIL_ADDRESS>,  Dario Reggiani  Scuola Superiore Meridionale,
Universitá di Napoli Federico II Largo San Marcellino 10, 80138 Napoli, Italy.
<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.

A lower semicontinuity result and a relaxation formula for free discontinuity
functionals with non-standard growth in the bulk energy are provided. Our
analysis is based on a non-trivial adaptation of the blow-up [8] and of the
global method for relaxation [17] to the setting of generalized special
function of bounded variation with Orlicz growth. Key tools developed in this
paper are an integral representation result and a Poincaré inequality under
non-standard growth.

###### Key words and phrases:

Free discontinuity functionals, non-standard growth, integral representation,
lower semicontinuity, relaxation

###### 2020 Mathematics Subject Classification:

49J45, 46E30, 49M20

##  1\. Introduction

Integral functionals with non-standard growth first appeared in the works of
Zhikov [62, 61] for modeling composite materials characterized by a strongly
anisotropic behavior. The non-standard character of such functionals is
typically expressed in terms of a point-dependent integrability of the
deformation gradient, which may be captured, in a functional setting, in terms
of variable exponents spaces [51] or, more in general, in Orlicz type of
spaces (see, e.g., [45]). As relevant examples of bulk energies undergoing
non-standard growth we report here the so-called variable exponent and the
double-phase case

$|\xi|^{p(x)}\qquad\text{and}\qquad|\xi|^{p}+a(x)|\xi|^{q}\qquad\text{for
$(x,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d\times m}$,}$ (1.1)

for suitable choices of the exponent function
$p\colon\mathbb{R}^{d}\to(1,+\infty)$, of the exponents $1<p<q<+\infty$, and
of the weight function $a\colon\mathbb{R}^{d}\to[0,+\infty)$. We refer to
Section 4.3 for a list of relevant examples in the literature.

In a Sobolev/Orlicz setting, the study of integral functionals with non-
standard growth has attracted an increasing attention in the last decades.
Lower semicontinuity, relaxation, and integral representation have been
tackled in a number of papers: we mention the works [1, 18, 42, 43], dealing
with the role of convexity and quasiconvexity in the gap problem for
functionals characterized by a $(p,q)$-growth, i.e., different grow rates from
above and below. Further results on relaxation in the gap problem have been
presented in [56], where some explicit examples of concentration effects for
bulk densities of double-phase type (1.1) are discussed, pointing out the
importance of the Hölder regularity of $a(\cdot)$ in the relaxation procedure.
The variable exponent setting has been dealt with in [30] under the so called
log-Hölder continuity assumption on the exponent function $p(\cdot)$. Further
instances of non-standard growth in partial differential equations may be
found in [15, 16, 22, 31, 38, 55, 58], while applications ranging from optimal
design to electro-rheological fluids and homogeneization appeared in [13, 14,
59, 64, 65]. Let us also mention the somehow related topic of regularity of
minimizers of functionals with non-standard growth, which have been thoroughly
discussed in [2, 4, 29, 39, 40] for the variable exponent, in, e.g., [11, 12,
27, 28, 35, 36, 54] for the double phase, and in [49] in a unified and
generalized framework.

When dealing with composite materials, it is rather natural to account for
failure phenomena, such as fracture, which can not be captured by a mere bulk
energy defined on Sobolev or Orlicz spaces. This leads us to the extension of
the above non-standard growth functionals to a Free Discontinuity setting,
where singularities may appear in the form of jump discontinuities. Besides
Materials Science, applications of a variable exponent in the setting of
functions of bounded variation already appeared in image reconstruction [23]
(see also [46, 47]), where an intermediate regime between Total Variation and
the isotropic diffusion away from the edges was proposed.

The aim of this paper it to provide a unified framework for lower
semicontinuity and relaxation of functionals of the form

$\mathcal{G}(u):=\int_{\Omega}f(x,\nabla
u)\,dx+\int_{J_{u}}g(x,[u](x),\nu_{u})\,d\mathcal{H}^{d-1}\,,$ (1.2)

focusing on the role played by the non-standard growth condition of volume
integrand $f(x,\xi)$. In formula (1.2), $\Omega\subseteq\mathbb{R}^{d}$ is an
open bounded subset of $\mathbb{R}^{d}$, $u\in GSBV(\Omega;\mathbb{R}^{m})$ is
a function of generalized special bounded variation (see [10, Section 4.5]),
$J_{u}$ denotes the jump set of $u$, $\nu_{u}$ stands for the approximate unit
normal to $u$, and $[u]:=u^{+}-u^{-}$ represents the jump of $u$, that is, the
difference between the traces $u^{+}$ and $u^{-}$ of $u$ on $J_{u}$, defined
according to the orientation of $\nu_{u}$. For the sole variable exponent
case, lower semicontinuity results for $\mathcal{G}$ have been obtained in
[34], while $\Gamma$-convergence and relaxation issues have been recently
considered in [60]. The key assumptions in the mentioned papers are
superlinearity of $p(\cdot)$ (meaning $\min_{\Omega}p(\cdot)>1$), which allows
for a separation of scales in the $\Gamma$-convergence and relaxation
processes, and a log-Hölder continuity of the exponent, necessary to avoid the
Lavrentiev’s phenomenon, as originally observed in [63].

Our primary interest is in providing suitable conditions which either imply
the lower semicontinuity of the functional $\mathcal{G}$ in (1.2) or guarantee
an explicit formula for its lower semicontinuous envelope, while assuming
$BV$-ellipticity [10, Chapter 5] of $g$ and non-standard growth for $f$. In
this regard, we will suppose that there exists a superlinear generalized
$\Phi$-function (see also Definitions 4.3 and 4.13)
$\psi\colon\Omega\times[0,+\infty)\to[0,+\infty)$ such that

$a\psi(x,|\xi|)\leq f(x,\xi)\leq b(1+\psi(x,|\xi|))\qquad\text{for
$(x,\xi)\in\Omega\times\mathbb{R}^{d\times m}$}$ (1.3)

for some $0<a<b<+\infty$. The superlinearity of $\psi$ (and thus of $f$) is
expressed by the conditions $\textnormal{(Inc)}_{p}$ and
$\textnormal{(Dec)}_{q}$ for some $p,q\in(1,+\infty)$ (see also Definition
4.12), meaning that for a.e. $x\in\Omega$ the maps
$t\mapsto\frac{\psi(x,t)}{t^{p}}$ and $t\mapsto\frac{\psi(x,t)}{t^{q}}$ are
monotone increasing and monotone decreasing, respectively. The third basic
hypothesis on $\psi$, common to all our results, is (A0). Loosely speaking,
such condition does not allow for a too degenerate behavior on small balls
contained in $\Omega$ of the functions

$\psi^{+}_{B}(t):=\sup_{x\in B}\,\psi(x,t)\qquad\psi^{-}_{B}(t):=\inf_{x\in
B}\,\psi(x,t)\qquad\text{for $t\in[0,+\infty)$ and $B\subseteq\Omega$}.$ (1.4)

We remark that $\textnormal{(Inc)}_{p}$, $\textnormal{(Dec)}_{q}$, and (A0)
are well-suited for a blow-up argument, which is at the core of our proofs
strategy, and are rather standard in the theory of generalized Orlicz spaces
[45].

Let us discuss our results in more details. In Theorem 3.3 we prove the lower
semicontinuity of the functional $\mathcal{G}$ in the space
$GSBV^{\psi}(\Omega;\mathbb{R}^{m})$, the space of functions $u\in
GSBV(\Omega;\mathbb{R}^{m})$ with $L^{\psi}$-integrable approximate gradient
$\nabla u$. We refer to Definition 4.28 and Section 4.2 for more details on
such space. Besides $\textnormal{(Inc)}_{p}$, $\textnormal{(Dec)}_{q}$, and
(A0), in Theorem 3.3 we ask for the quasiconvexity of $f$ and for a mild
continuity property of $\psi(x,t)$ with respect to $x\in\Omega$ (see (3.6) for
a precise statement). We notice that the last condition allows for
improvements in the existing theory of generalized Orlicz spaces. For
instance, in the variable exponent case the log-Hölder continuity of
$p(\cdot)$ is not necessary for (3.6), as pointed out in Section 4.3. The
proof of Theorem 3.3 is based on the approximation strategy of [53] and on a
localization technique which leads us to study the asymptotic behavior of
$\mathcal{G}$ around Lebesgue points of the limit energy, as first done in [8]
in the $GSBV$-setting (see also [10, Theorem 5.29]). Our assumptions are
designed in such a way that the maximal operator for $SBV^{p}$-functions,
exploited in [8] for the construction of more regular approximating sequences,
can be replaced by the maximal operator in Orlicz spaces, for which continuity
estimates has been obtained in [50] (see also Theorem 4.35).

In Theorem 3.4 we show a relaxation formula of the functional $\mathcal{G}$ of
the form

$\overline{\mathcal{G}}(u):=\int_{\Omega}\mathcal{Q}f(x,\nabla
u)\,dx+\int_{J_{u}}\mathcal{R}g(x,[u],\nu_{u})d\mathcal{H}^{d-1},$ (1.5)

which therefore maintains the original structure (1.2). In particular, in
(1.5) the functions $\mathcal{Q}f\colon\Omega\times\mathbb{R}^{m\times
d}\to[0,+\infty)$ and
$\mathcal{R}g\colon\Omega\times\mathbb{R}^{m}\times\mathbb{S}^{d-1}\to[0,+\infty)$
denote the quasiconvex and the BV-elliptic envelope of $f$ and $g$,
respectively. When dealing with relaxation, we have to strengthen condition
(3.6). Thus, in Theorem 3.4 we replace the latter with the stronger assumption
(adA1), which allows to estimate $\psi^{+}_{B}$ with $\psi^{-}_{B}$ on small
balls $B\subseteq\Omega$ with a fixed control rate. We remark that condition
(adA1) is weaker than the more traditional (A1), introduced in the framework
of Orlicz spaces (see, e.g., [45]). In the variable exponent case, (adA1)
still requires log-Hölder continuity of $p(\cdot)$, while in the double-phase
case it calls for a Hölder continuity of the weight $a(x)$, which may be
however weaker than the one considered in regularity theory in Sobolev/Orlicz
setting (see, for instance, [12, 28]). Further examples are discussed in
Section 4.3.

The proof of Theorem 3.4 follows the well-established strategy of the global
method of relaxation [17, 19], which need to be adapted to the non-standard
growth framework (1.4). The crucial step towards the relaxation formula (1.5)
is the integral representation of the lower semicontinuous envelope
$\overline{\mathcal{G}}$ of $\mathcal{G}$ in
$GSBV^{\psi}(\Omega;\mathbb{R}^{m})$ (see Theorem 3.1 and Corollary 3.2). This
ensures that $\overline{\mathcal{G}}$ can still be written as the sum of a
bulk energy depending on the approximate gradient $\nabla u\in
L^{\psi}(\Omega;\mathbb{R}^{m\times d})$ and of a surface term obtained by
integrating over the jump set $J_{u}$ of $u$ a suitable function depending on
$x$, $[u]$, and $\nu_{u}$. The fundamental tool to characterize the behavior
of the blow-up energy on jump points and on approximate differentiability
points of $u$ is a new Poincaré inequality for special functions of bounded
variation with $L^{\varphi}$-integrable approximate gradient, where $\varphi$
denotes a (generalized) $\Phi$-function in the sense of Definition 4.3. Such
inequality, obtained in Theorems 5.1 and 5.9, is of independent mathematical
interest, as it can not be recovered by the classical approach of [37] or of
[60] for the variable exponent, while it builds upon fine properties of
rearrangments of $BV$-functions, adapting and combining the techniques of [5,
24, 25, 26]. The possible non-homogeneity of $\varphi$ in space implies that a
suitable truncation of $u$ can not be controlled only in term of $\nabla u$,
as some remainder term appears, that can be estimated along the blow-up
procedure thanks to (adA1). The crucial role played by the Poincaré inequality
is evident in Lemmas 6.6 and 6.9 and enables us to recover the densities of
the lower semicontinuous envelope $\overline{\mathcal{G}}$ as blow-up limits
of cell minimization formulas. In both lemmas, the Poincaré inequality is
exploited to replace the optimal blow-up sequences with more regular
functions, without excessively increasing the energy.

### Outlook and open problems.

Our paper presents a general framework for lower semicontinuity and relaxation
of free discontinuity functionals under non-standard growth of the bulk
functional. The generalized $\Phi$-functions we consider cover most of the
examples appeared in the literature, and in particular among them the variable
exponent and the double phase (1.1). Because of the superlinear assumption
$\textnormal{(Inc)}_{p}$ for $p>1$, however, we can not handle an
$\ell\log\ell$-kind of behavior, which has been studied in [52] for a lower-
semicontinuity problem in $SBV$ without $x$-dependence on the integrand
function in (1.2). Hence, an extension of our results in the above direction
will be considered in a forthcoming research. Furthermore, regularity issues
for minimizers of the functional $\mathcal{G}$ may be investigated, in the
spirit of [10, Chapter 6], and could lead to stronger assumptions on the
growth function $\psi$ (see, e.g., [2, 3, 4] for the variable exponent). Also
the investigation of lower semicontinuity and relaxation issues for free
discontinuity functionals with bulk energies having mixed $(p,q)$-growth
condition, relevant for the modeling of determinant constraints, is not fully
covered by our theory and deserves further analysis.

### Plan of the paper.

In Section 2 we introduce the basic notation of the paper. In Section 3 we
state the main results of our paper: the integral representation in Theorem
3.1 and Corollary 3.2, the lower semicontinuity result in Theorem 3.3, and the
relaxation formula in Theorem 3.4. Section 4 is devoted to some preliminaries
on (generalized) $\Phi$-function and on $(G)SBV$-spaces, as well as to the
discussion of the main assumptions of the above theorems. In particular, in
Section 4.3 we report a number of generalized $\Phi$-function known in the
literature, and discuss how they fall into our theory. In Section 5 we state
and prove the Poincaré inequality for $SBV$-functions with
$L^{\varphi}$-integrable approximate gradient (see Theorems 5.1 and 5.9).
Finally, in Sections 6–8 we proceed with the proof of Theorems 3.1, 3.3, and
3.4, respectively.

##  2\. Notations

Through the paper we assume that $\Omega\subset\mathbb{R}^{d}$ is a bounded
open set with Lipschitz boundary and that $d\geq 2$. We denote by
$\mathcal{A}(\Omega)$ and $\mathcal{B}(\Omega)$ the family of open sets and
the family of Borel measurable sets contained in $\Omega$, respectively. For
every $x\in\mathbb{R}^{d}$ and $\varepsilon>0$ we indicate with
$B_{\varepsilon}(x)$ the ball centered in $x$ with radius $\varepsilon$, if
$x=0$ we write $B_{\varepsilon}$. Given $x\in\mathbb{R}^{d}$ we indicate with
$|x|$ its Euclidean norm. The set $\mathbb{R}^{m\times d}$ is the set of
$m\times d$ matrices with real coefficients,
$\mathbb{S}^{d-1}:=\\{x\in\mathbb{R}^{d}\colon|x|=1\\}$, and
$\mathbb{R}^{d}_{0}:=\mathbb{R}^{d}\setminus\\{0\\}$. The Lebesgue measure of
the $d$-dimensional unit ball is indicated $\omega_{d}$. We denote by
$\mathcal{L}^{d}$ and $\mathcal{H}^{k}$ the $d$-dimensional Lebesgue measure
and the $k$-dimensional Hausdorff measure, respectively. The space
$L^{0}(\Omega)$ stands for the space of all measurable functions in $\Omega$.
Given $x_{0}\in\mathbb{R}^{d}$ and $\varepsilon>0$, for any set
$A\subset\mathbb{R}^{d}$ we set

$A_{\varepsilon,x_{0}}:=x_{0}+\varepsilon(A-x_{0}).$ (2.1)

The closure of a set $A$ is indicated with $\overline{A}$, the diameter with
$\textnormal{diam}(A)$. Given two sets $A_{1},A_{2}\subset\mathbb{R}^{d}$ we
denote their symmetric difference with $A_{1}\Delta A_{2}$. We write
$\chi_{A}$ for the characteristic function of any set
$A\subset\mathbb{R}^{d}$. If $A$ is a set of finite perimeter we indicate with
$\partial^{M}A$ its essential boundary (the points of the boundary which do
not have density zero nor one) and with $\partial^{*}A$ its reduced boundary
(the points for which a "normal" can be defined). For a Carathéodory function
$\varphi\colon\Omega\times\mathbb{R}^{+}\to\mathbb{R}$ and a ball
$B\subset\mathbb{R}^{d}$ we define

$\varphi^{-}_{B}(t):=\operatorname*{ess\,inf}_{x\in B\cap\Omega}\varphi(x,t)\
\ \ \mbox{and}\ \ \ \varphi^{-}_{B}(t):=\operatorname*{ess\,sup}_{x\in
B\cap\Omega}\varphi(x,t).$ (2.2)

for every $t\geq 0$. For a monotone function $\varphi$ on the real line, the
customary notations $\varphi(t-)$ and $\varphi(t+)$ are used to denote left
and right limits in $t$, respectively. For an increasing coercive function
$\varphi$, setting for simplicity $\varphi(+\infty):=+\infty$, we denote by
$\varphi^{-1}\colon[0,+\infty]\to[0,+\infty]$ the left inverse of $\varphi$
defined as

$\varphi^{-1}(s):=\inf\\{t\geq 0\colon\varphi(t)\geq s\\}.$

##  3\. Main Results

The paper will be concerned with integral representation, lower
semicontinuity, and relaxation of functionals defined on the space
$GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ of generalized special functions of
bounded variation with $\psi$-growth on the gradient. Here $\psi$ is a
suitable (generalized) Orlicz function $\psi$. We will consider functionals

$\mathcal{F}:GSBV^{\psi}(\Omega,\mathbb{R}^{m})\times\mathcal{A}(\Omega)\to[0,+\infty)$

satisfying the following assumptions:

  1. (H1)

$\mathcal{F}(u,\cdot)$ is a Borel measure for any $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$;

  2. (H2)

$\mathcal{F}(\cdot,A)$ is lower semicontinuous with respect to convergence in
measure in $\Omega$ for any $A\in\mathcal{A}(\Omega)$;

  3. (H3)

$\mathcal{F}(\cdot,A)$ is local for every $A\in\mathcal{A}(\Omega)$, that is,
if $u,v\in GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ satisfy $u=v$
$\mathcal{L}^{d}$-a.e. in $A$ then $\mathcal{F}(u,A)=\mathcal{F}(v,A)$;

  4. (H4)

there exist $0<a<b$ such that for every $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ and every $B\in\mathcal{B}(\Omega)$ it
holds

$a\left(\int_{B}\psi(x,|\nabla u|)dx+\mathcal{H}^{d-1}(J_{u}\cap
B)\right)\leq\mathcal{F}(u,B)\leq b\left(\int_{B}(1+\psi(x,|\nabla
u|))dx+\mathcal{H}^{d-1}(J_{u}\cap B)\right).$

In (H4), as we said, we consider a generalized Orlicz function $\psi$. The
assumptions we make on $\psi$ will be detailed in Section 4 (see (aInc),
(aDec) and the so-called weight condition (A0)). These properties are standard
in the theory of generalized Orlicz functions. We also assume that $\psi$
satisfies

  * (adA1)

For every ball $B\subset\Omega$ with $\textnormal{diam}(B)\leq 1$ there exists
$\beta\in(0,1)$ such that

$\psi_{B}^{+}(\beta t)\leq\psi_{B}^{-}(t)\ \ \ \mbox{for all
$t\in\left[\sigma,(\psi_{B}^{-})^{-1}\left(\frac{1}{\textnormal{diam}(B)}\right)\right]$}\,,$

where $\sigma\geq 1$ is the constant in the weight condition (A0). We remark
that this condition is a weaker variant of the standard assumption (A1) (see
Definition 4.21), called local continuity condition and usually considered for
generalized Orlicz spaces. All the main results of the paper will hold, if one
requires that (A0), (adA1), (aInc) and (aDec) are satisfied on $\Omega$. They
encompass a broad range of applications, with some relevant examples that will
be discussed in Section 4.3.

Our first result concerns the integral representation, and requires some
notation to be fixed. For every $u\in GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ and
every $A\in\mathcal{A}(\Omega)$ we define

$\mathbf{m}_{\mathcal{F}}(u,A):=\inf_{v\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})}\\{\mathcal{F}(v,A)\colon v=u\ \mbox{in a
neighborhood of $\partial A$}\\}.$ (3.1)

Moreover, given $x_{0}\in\Omega$, $u_{0}\in\mathbb{R}^{m}$ and
$\xi\in\mathbb{R}^{m\times d}$, we define the affine function
$\ell_{x_{0},u_{0},\xi}\colon\mathbb{R}^{d}\to\mathbb{R}^{m}$ as

$\ell_{x_{0},u_{0},\xi}:=u_{0}+\xi(x-x_{0}).$

Given $x_{0}\in\Omega$, $\mathfrak{a},\mathfrak{b}\in\mathbb{R}^{m}$ and
$\nu\in\mathbb{S}^{d-1}$ we also introduce
$u_{x_{0},\mathfrak{a},\mathfrak{b},\nu}\colon\mathbb{R}^{d}\to\mathbb{R}^{m}$
as

$u_{x_{0},\mathfrak{a},\mathfrak{b},\nu}=\begin{cases*}\mathfrak{a}&if
$(x-x_{0})\cdot\nu>0$,\\\ \mathfrak{b}&if $(x-x_{0})\cdot\nu<0$.\end{cases*}$

Our main result concerning the integral representation is the following.
Below, $\Phi_{w}(\Omega)$ denotes the class of weak generalized
$\Phi$-functions, whose definition is recalled in Definition 4.13.

######  Theorem 3.1 (Integral representation in $GSBV^{\psi}$).

Let $\psi\in\Phi_{w}(\Omega)$ satisfy (A0), (adA1), (aInc) and (aDec) on
$\Omega$. Assume that
$\mathcal{F}:GSBV^{\psi}(\Omega,\mathbb{R}^{m})\times\mathcal{A}(\Omega)\to[0,+\infty)$
satisfies assumptions (H1)–(H4). Then, for all $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ and all $A\in\mathcal{A}(\Omega)$

$\mathcal{F}(u,A)=\int_{A}f(x,u(x),\nabla u(x))\,dx+\int_{J_{u}\cap
A}g(x,u^{+}(x),u^{-}(x),\nu_{u}(x))\,d\mathcal{H}^{d-1},$

where

$\displaystyle f(x_{0},u_{0},\xi):=\limsup_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(\ell_{x_{0},u_{0},\xi},B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}\
\ \ \mbox{for
$(x_{0},u_{0},\xi_{0})\in\Omega\times\mathbb{R}^{m}\times\mathbb{R}^{m\times
d}$},$ (3.2) $\displaystyle
g(x_{0},\mathfrak{a},\mathfrak{b},\nu):=\limsup_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u_{x_{0},\mathfrak{a},\mathfrak{b},\nu},B_{\varepsilon}(x_{0}))}{\omega_{d-1}\varepsilon^{d-1}}\
\ \ \mbox{for all $x_{0}\in\Omega$,
$\mathfrak{a},\mathfrak{b}\in\mathbb{R}^{m}$ and $\nu\in\mathbb{S}^{d-1}$}.$
(3.3)

If translation invariance with respect to $u$ is assumed on $\mathcal{F}$,
namely

  1. (H5)

$\mathcal{F}(u+c,A)=\mathcal{F}(u,A)$ for every $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$, every $A\in\mathcal{B}(\Omega)$ and every
$c\in\mathbb{R}^{m}$ ,

we have the following result.

######  Corollary 3.2.

Let $\psi\in\Phi_{w}(\Omega)$ be as in Theorem 3.1 and suppose that
$\mathcal{F}:GSBV^{\psi}(\Omega,\mathbb{R}^{m})\times\mathcal{A}(\Omega)\to[0,+\infty)$
satisfies assumptions (H1)–(H5). Then, for all $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ and all $A\in\mathcal{A}(\Omega)$

$\mathcal{F}(u,A)=\int_{A}f(x,\nabla u(x))\,dx+\int_{J_{u}\cap
A}g(x,[u](x),\nu_{u}(x))\,d\mathcal{H}^{d-1},$

where $f$ and $g$ are as in (3.2) and (3.3), respectively.

Our next result concerns the lower semicontinuity in
$GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ of variational functionals
$\mathcal{G}:L^{0}(\Omega,\mathbb{R}^{m})\times\mathcal{A}(\Omega)\to[0,+\infty]$
of the form

$\mathcal{G}(u,A)=\int_{A}f(x,\nabla
u(x))+\int_{J_{u}}g(x,[u](x),\nu_{u}(x))\,d\mathcal{H}^{d-1},$ (3.4)

In (3.4) the function $f:\Omega\times\mathbb{R}^{m\times d}\to[0,+\infty)$
satisfies the following assumptions:

  1. (f1)

$f$ is a Carathéodory function;

  2. (f2)

there exist two constants $a,b>0$ such that

$a\psi(x,|\xi|)\leq f(x,\xi)\leq b(1+\psi(x,|\xi|))$ (3.5)

for every $x\in\Omega$ and $\xi\in\mathbb{R}^{m\times d}$.

On the other hand, the function
$g\colon\Omega\times\mathbb{R}^{m}_{0}\times\mathbb{S}^{d-1}\to[0,+\infty)$
satisfies the following assumptions:

  1. (g1)

$g$ is a Borel measurable function lower semicontinuous in $x$ and continuous
in the remaining variables;

  2. (g2)

there exist $\alpha_{1},\alpha_{2}>0$ such that for every $x\in\Omega$,
$\zeta\in\mathbb{R}^{m}_{0}$ and $\nu\in\mathbb{S}^{d-1}$

$\alpha_{1}\leq g(x,\zeta,\nu)\leq\alpha_{2}.$

For the result below, we can further weaken (adA1). We will namely assume that
$\psi$ complies with the following property: for $\sigma\geq 1$ being the
constant in the weight condition (A0), and for $\mathcal{L}^{d}$-a.e.
$x_{0}\in\Omega$ there exists $C=C(x_{0})>0$ such that

$\displaystyle\begin{split}\mbox{given $\theta>\sigma$, we can find
$\varepsilon_{0}>0$}&\mbox{ such that for every
$\varepsilon\leq\varepsilon_{0}$ and every $t\in[\sigma,\theta],$}\\\
&\psi^{+}_{B_{\varepsilon}(x_{0})}(t)\leq
C\psi^{-}_{B_{\varepsilon}(x_{0})}(t).\end{split}$ (3.6)

As we will discuss in Section 4, if $\psi\in\Phi_{w}(\Omega)$ satisfies (A0),
(aDec) and (adA1) on $\Omega$, then it also satisfies (3.6). Hence, the
assumptions for the lower semicontinuity Theorem below are weaker than those
in Theorem 3.1.

######  Theorem 3.3.

Let $\psi\in\Phi_{w}(\Omega)$ satisfying (A0), (aInc) and (aDec) on $\Omega$.
Assume also that $\psi$ is satisfies (3.6). Consider a functional
$\mathcal{G}\colon
L^{0}(\Omega,\mathbb{R}^{m})\times\mathcal{A}(\Omega)\to[0,+\infty]$ as in
(3.4). Let $f\colon\Omega\times\mathbb{R}^{m\times d}\to[0,+\infty)$ be a
function satisfying (f1)-(f2) and such that $z\mapsto f(x,z)$ is quasiconvex
in $\mathbb{R}^{m\times d}$ for every $x\in\Omega$. Let
$g\colon\Omega\times\mathbb{R}^{m}_{0}\times\mathbb{S}^{d-1}\to[0,+\infty)$ be
a function satisfying (g1)–(g2) and such that $(\zeta,\nu)\mapsto
g(x,\zeta,\nu)$ is BV-elliptic for every $x\in\Omega$. Then, for every
$A\in\mathcal{A}(\Omega)$, we have

$\mathcal{G}(u,A)\leq\liminf_{k\to+\infty}\mathcal{G}(u_{k},A)$

for every sequence $\\{u_{k}\\}_{k}\subset GSBV^{\psi}(A,\mathbb{R}^{m})$
converging to a function $u\in GSBV^{\psi}(A,\mathbb{R}^{m})$ in measure.

The third main result is a relaxation result, which requires both the use of
Theorems 3.1 and 3.3. Given $\mathcal{G}$ as in (3.4), for every $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ and every $A\in\mathcal{A}(\Omega)$ we
denote the lower semicontinuous envelope of the functional $\mathcal{G}$ as

$\overline{\mathcal{G}}(u,A):=\inf\left\\{\liminf_{k\to+\infty}\mathcal{G}(u_{k},A)\colon\\{u_{k}\\}_{k}\subset
GSBV^{\psi}(A,\mathbb{R}^{m})\mbox{ and }u_{k}\to u\mbox{ in measure on
$A$}\right\\}.$

We assume that
$g\colon\Omega\times\mathbb{R}^{m}_{0}\times\mathbb{S}^{d-1}\to[0,+\infty)$
satisfies also the following additional properties:

  1. (g3)

there exists $c>0$ such that for every $x\in\Omega$ and every
$\nu\in\mathbb{S}^{d-1}$ it holds

$g(x,\zeta_{1},\nu)\leq g(x,\zeta_{2},\nu)\ \ \ \mbox{for every
$\zeta_{1},\zeta_{2}\in\mathbb{R}^{m}_{0}$ with
$c|\zeta_{1}|\leq|\zeta_{2}|$;}$

  1. (g4)

for every $x\in\Omega$, $\zeta\in\mathbb{R}^{m}_{0}$ and
$\nu\in\mathbb{S}^{d-1}$

$g(x,\zeta,\nu)=g(x,-\zeta,-\nu).$

######  Theorem 3.4.

Let $\psi\in\Phi_{w}(\Omega)$ satisfying (A0), (adA1), (aInc) and (aDec) on
$\Omega$. Let $\mathcal{G}$ be as in (3.4) and
$f\colon\Omega\times\mathbb{R}^{m\times d}\to[0,+\infty)$ satisfying (f1) and
(f2). Assume also that
$g\colon\Omega\times\mathbb{R}^{m}_{0}\times\mathbb{S}^{d-1}\to[0,+\infty)$ is
a continuous function satisfying (g1)–(g4). Then, for every $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ and every $A\in\mathcal{A}(\Omega)$,

$\overline{\mathcal{G}}(u,A)=\int_{A}\mathcal{Q}f(x,\nabla
u(x))\,dx+\int_{J_{u}\cap
A}\mathcal{R}g(x,[u](x),\nu_{u}(x))\,d\mathcal{H}^{d-1},$

where $\mathcal{Q}f$ is the quasiconvex envelope of $f$ and $\mathcal{R}g$ is
the BV-elliptic envelope of $g$.

##  4\. Preliminaries

In this preliminary section we recall some basic facts about (generalized)
$\Phi$-functions and Orlicz spaces, and on $GSBV$ functions. Finally, in
Subsection 4.3 we give a list of non-standard growth functions $\psi$ which
fit into the scope of our results.

###  4.1. (Generalized) $\Phi$-functions and Orlicz spaces

We begin by collecting some basic definitions and useful facts about
$\Phi$-functions and generalized Orlicz spaces. For a complete treatment of
the topic (and the proofs of the statements below which are left without
proof) we refer to [45].

######  Definition 4.1.

A function $g\colon(0,+\infty)\to\mathbb{R}$ is almost increasing (resp.
almost decreasing) if there exists a constant $a\geq 1$ such that $g(s)\leq
ag(t)$ (resp $ag(s)\geq g(t)$) for every $0<s<t$.

Increasing and decreasing functions are included in the above definition if
$a=1$.

######  Definition 4.2.

Let $f\colon(0,\infty)\to\mathbb{R}$ and $p,q>0$. We say that $f$ satisfies

  * $\textnormal{(Inc)}_{p}$

if $\frac{f(t)}{t^{p}}$ is increasing;

  * $\textnormal{(aInc)}_{p}$

if $\frac{f(t)}{t^{p}}$ is almost increasing;

  * $\textnormal{(Dec)}_{q}$

if $\frac{f(t)}{t^{q}}$ is decreasing;

  * $\textnormal{(aDec)}_{q}$

if $\frac{f(t)}{t^{q}}$ is almost decreasing;

We say that $f$ satisfies (aInc), (Inc), (aDec) or (Dec) if there exist $p>1$
or $q<\infty$ such that $f$ satisfies $\textnormal{(Inc)}_{p}$,
$\textnormal{(aInc)}_{p}$, $\textnormal{(Dec)}_{q}$ or
$\textnormal{(aDec)}_{q}$, respectively.

######  Definition 4.3.

Let $\varphi:[0,+\infty)\to[0,+\infty]$ be increasing with
$\varphi(0)=0=\lim_{t\to 0^{+}}\varphi(t)$ and
$\lim_{t\to+\infty}\varphi(t)=+\infty$. Such $\varphi$ is called a
$\Phi$-prefunction. We say that a $\Phi$-prefunction $\varphi$ is a

  * -

weak $\Phi$-function if it satisfies $\textnormal{(aInc)}_{1}$ on
$(0,+\infty)$;

  * -

convex $\Phi$-function if it is left-continuous and convex;

  * -

strong $\Phi$-function if it is continuous in the topology of $[0,+\infty]$
and convex.

The set of weak, convex and strong $\Phi$-functions are denoted by $\Phi_{w}$,
$\Phi_{c}$ and $\Phi_{s}$, respectively.

It follows by definition that $\Phi_{s}\subset\Phi_{c}\subset\Phi_{w}$. If a
function $\varphi\in\Phi_{c}$ satisfies (aDec) then $\varphi\in\Phi_{s}$.

######  Definition 4.4.

Two function $\varphi$ and $\psi$ are called equivalent, $\varphi\simeq\psi$,
if there exists $L\geq 1$ such that $\varphi(t/L)\leq\psi(t)\leq\varphi(Lt)$
for all $t\geq 0$. The notation "$\approx$" is instead used with the following
meaning: $\varphi\approx\psi$ if and only if there exist two constants
$c_{1},c_{2}>0$ such that $c_{1}\varphi\leq\psi\leq c_{2}\varphi$.

######  Lemma 4.5.

Let $\varphi,\psi\colon[0,+\infty)\to[0,+\infty]$ be increasing with
$\varphi\simeq\psi$. Then, the following facts hold:

  1. (a)

if $\varphi$ is a $\Phi$-prefunction, then $\psi$ is a $\Phi$-prefunction;

  2. (b)

if $\varphi$ satisfies $\textnormal{(aInc)}_{p}$, then $\psi$ satisfies
$\textnormal{(aInc)}_{p}$;

  3. (c)

if $\varphi$ satisfies $\textnormal{(aDec)}_{q}$, then $\psi$ satisfies
$\textnormal{(aDec)}_{q}$.

######  Lemma 4.6.

If $\varphi\in\Phi_{w}$ satisfies $\textnormal{(aInc)}_{p}$ with $p\geq 1$,
then there exists $\psi\in\Phi_{c}$ equivalent to $\varphi$ such that
$\psi^{1/p}$ is convex. In particular, $\psi$ satisfies
$\textnormal{(Inc)}_{p}$.

######  Theorem 4.7.

Every weak $\Phi$-function is equivalent to a strong $\Phi$-function.
Moreover, if $\varphi\in\Phi_{w}$ is finite valued and satisfies
$\textnormal{(Inc)}_{1}$, then we can find $\psi\in\Phi_{s}$ such that

$\psi(t)\leq\varphi(t)\leq\psi(2t)\ \ \ t\geq 0.$

Now, we recall the concept of doubling functions and its equivalence with
(aDec).

######  Definition 4.8.

We say that a function $\varphi\colon[0,+\infty)\to[0,+\infty]$ satisfies
$\Delta_{2}$, or that it is doubling, if there exists a constant $K\geq 2$
such that

$\varphi(2t)\leq K\varphi(t)\ \ \ \mbox{for all $t\geq 0$}.$

######  Lemma 4.9.

The following statements hold.

  1. (a)

If $\varphi\in\Phi_{w}$, then $\Delta_{2}$ is equivalent to (aDec).

  2. (b)

If $\varphi\in\Phi_{c}$, then $\Delta_{2}$ is equivalent to (Dec).

######  Remark 4.10.

By the previous Lemma it follows that if $\varphi\in\Phi_{c}$ satisfies
$\textnormal{(aDec)}_{q}$, then it satisfies $\textnormal{(Dec)}_{q_{2}}$ for
some possibly larger $q_{2}$. This and Lemma 4.5(c) imply that if $\varphi$
satisfies $\Delta_{2}$ and $\psi\simeq\varphi$, then $\psi$ satisfies
$\Delta_{2}$.

######  Theorem 4.11.

Let $\psi,\varphi\in\Phi_{w}$. Then $\varphi\simeq\psi$ if and only if
$\varphi^{-1}\approx\psi^{-1}$. In particular, if
$\psi(t/L)\leq\varphi(t)\leq\psi(Lt)$ for some $L\geq 1$ then,

$\psi^{-1}(t)/L\leq\varphi^{-1}(t)\leq L\psi^{-1}(t).$

We now come to the generalized setting, where explicit dependence on the space
variable $x$ is allowed.

######  Definition 4.12.

Let $(A,\Sigma,\mu)$ be a complete, $\sigma$-finite measure space. Let
$\varphi\colon A\times[0,+\infty)\to[0,+\infty]$ and $p,q>0$. We say that
$\varphi$ satisfies $\textnormal{(aInc)}_{p}$ or $\textnormal{(aDec)}_{q}$ if
there exists a constant $a\geq 1$ such that $t\mapsto\varphi(x,t)$ satisfies
$\textnormal{(aInc)}_{p}$ or $\textnormal{(aDec)}_{q}$ respectively, for
$\mu$-a.e. $x\in A$. When $a=1$ we use the notation $\textnormal{(Inc)}_{p}$
and $\textnormal{(Dec)}_{q}$. For (Inc) and (Dec) the definition is analogous.

######  Definition 4.13.

Let $(A,\Sigma,\mu)$ be a complete, $\sigma$-finite measure space. A function
$\varphi\colon A\times[0,+\infty)\to[0,+\infty]$ is said to be a generalized
$\Phi$-prefunction on $(A,\Sigma,\mu)$ if $x\mapsto\varphi(x,|f(x)|)$ is
measurable for any $f\in L^{0}(A,\mu)$ and $\varphi(x,\cdot)$ is a
$\Phi$-prefunction for $\mu$-a.e. $x\in A$. We say that the generalized
$\Phi$-prefunction $\varphi$ is

  1. -

a weak generalized $\Phi$-function if it satisfies $\textnormal{(Inc)}_{1}$ on
$(0,+\infty)$;

  2. -

a convex generalized $\Phi$-function if $\varphi(x,\cdot)\in\Phi_{c}$ for
$\mu$-a.e. $x\in A$;

  3. -

a strong generalized $\Phi$-function if $\varphi(x,\cdot)\in\Phi_{s}$ for
$\mu$-a.e. $x\in A$.

If $\varphi$ is a generalized weak $\Phi$-function on $(A,\Sigma,\mu)$, we
write $\varphi\in\Phi_{w}(A,\mu)$. Similarly we define
$\varphi\in\Phi_{c}(A,\mu)$ and $\varphi\in\Phi_{s}(A,\mu)$. If
$A\subset\mathbb{R}^{d}$ is an open set and $\mu=\mathcal{L}^{d}$, we omit the
measure dependence and simply write $\Phi_{w}(A)$, $\Phi_{c}(A)$ or
$\Phi_{s}(A)$ and we say that $\varphi$ is a generalized $\Phi$-function on
$A$.

######  Proposition 4.14.

Let $\varphi\colon A\times[0,+\infty)\to[0,+\infty]$, $x\mapsto\varphi(x,t)$
be measurable for every $t\geq 0$ and $t\mapsto\varphi(x,t)$ be increasing and
left-continuous for $\mu$-a.e. $x\in A$. If $f\in L^{0}(A,\mu)$. Then
$x\mapsto\varphi(x,|f(x)|)$ is measurable.

Properties of $\Phi$-functions are generalized point-wise uniformly to the
generalized $\Phi$-function case.

######  Definition 4.15.

Two function $\varphi,\psi\colon A\times[0,+\infty)\to[0,+\infty]$ are called
equivalent, $\varphi\simeq\psi$, if there exists $L\geq 1$ such that

$\varphi(x,t/L)\leq\psi(x,t)\leq\varphi(x,Lt)\ \ \ \mbox{for $\mu$-a.e. $x\in
A$ and for all $t\geq 0$.}$

######  Lemma 4.16.

Let $\varphi,\psi\colon A\times[0,+\infty)\to[0,+\infty]$, be increasing with
respect to the second variable, such that $\varphi\simeq\psi$, and
$x\mapsto\varphi(x,|f(x)|)$ and $x\mapsto\psi(x,|f(x)|)$ be measurable for any
$f\in L^{0}(A,\mu)$. Then

  1. (a)

if $\varphi$ is a generalized $\Phi$-prefunction, then $\psi$ is a generalized
$\Phi$-prefunction;

  2. (b)

if $\varphi$ satisfies $\textnormal{(aInc)}_{p}$, then $\psi$ satisfies
$\textnormal{(aInc)}_{p}$;

  3. (c)

if $\varphi$ satisfies $\textnormal{(aDec)}_{q}$, then $\psi$ satisfies
$\textnormal{(aDec)}_{q}$.

######  Lemma 4.17.

If $\varphi\in\Phi_{w}(A,\mu)$ satisfies $\textnormal{(aInc)}_{p}$ with $p\geq
1$, then there exists $\psi\in\Phi_{c}(A,\mu)$ equivalent to $\varphi$ such
that $\psi^{1/p}$ is convex. In particular, $\psi$ satisfies
$\textnormal{(Inc)}_{p}$.

We can now discuss our main assumptions on the growth function $\psi$,
starting from the weight condition (A0) and the local continuity condition
(A1).

######  Definition 4.18.

Let $\varphi\in\Phi_{w}(\Omega)$. We say that $\varphi$ satisfies (A0) if
there exists a constant $\sigma\geq 1$ such that

$\varphi\left(x,\frac{1}{\sigma}\right)\leq 1\leq\varphi(x,\sigma)\ \ \
\mbox{for $\mathcal{L}^{d}$-a.e. $x\in\Omega$}.$

######  Proposition 4.19.

Given $\varphi,\psi\in\Phi_{w}(\Omega)$, if $\varphi\simeq\psi$ and $\varphi$
satisfies (A0), then $\psi$ satisfies (A0).

If $\varphi\in\Phi_{w}(\Omega)$ satisfies (A0), then
$\varphi_{B}^{-},\varphi_{B}^{+}\in\Phi_{w}$ for every $B\Subset\Omega$ (see
[45, Lemma 2.5.16]).

######  Definition 4.20.

Let $\varphi\in\Phi_{w}(\Omega)$ satisfy (A0). We say that $\varphi$ satisfies
(A1) if for every ball $B\subset\Omega$ with $\mathcal{L}^{d}(B)\leq 1$ there
exists $\beta\in(0,1)$ such that

$\varphi_{B}^{+}(\beta t)\leq\varphi_{B}^{-}(t)\ \ \ \mbox{for all
$t\in\left[\sigma,(\varphi_{B}^{-})^{-1}\left(\frac{1}{\mathcal{L}^{d}(B)}\right)\right]$}.$
(4.1)

Notice that the above definition explicitly depends on the space dimension
$d$. To our purposes, an adimensional (and weaker) version of (A1) will be
sufficient. We state it below.

######  Definition 4.21.

Let $\varphi\in\Phi_{w}(\Omega)$ satisfy (A0). We say that $\varphi$ satisfies
(adA1) if for every ball $B\subset\Omega$ with $\textnormal{diam}(B)\leq 1$
there exists $\beta\in(0,1)$ such that

$\varphi_{B}^{+}(\beta t)\leq\varphi_{B}^{-}(t)\ \ \ \mbox{for all
$t\in\left[\sigma,(\varphi_{B}^{-})^{-1}\left(\frac{1}{\textnormal{diam}(B)}\right)\right]$}.$
(4.2)

######  Lemma 4.22.

Let $\varphi,\psi\in\Phi_{w}(\Omega)$ and $\varphi\simeq\psi$. If $\varphi$
satisfies (A0) and (A1) (resp. (adA1)), then $\psi$ does as well.

######  Remark 4.23.

Let $\varphi\in\Phi_{s}(B_{\varepsilon})$. Define
$\varphi_{\varepsilon}(x,\cdot):=\varphi(\varepsilon x,\cdot)$ for every $x\in
B_{1}$. If $\varphi$ satisfies (A0) with a constant $\sigma\geq 1$ then $\psi$
satisfies (A0) with the same constant $\sigma$ of $\varphi$. If $\varphi$
satisfies $\textnormal{(Inc)}_{p}$ with $p\in[1,\infty)$ (resp.
$\textnormal{(Dec)}_{q}$ with $q\in[1,\infty)$), then $\psi$ satisfies
$\textnormal{(Inc)}_{p}$ with the same $p$ (resp. $\textnormal{(Dec)}_{q}$
with the same $q$).

The next two results show that the validity of our main assumptions is
unchanged by passing to equivalent $\Phi$-functions. will be used to simplify
our proofs while remaining in full generality.

######  Lemma 4.24.

Let $\varphi\in\Phi_{w}(\Omega)$ satisfy (A0), (adA1), (aInc) and (aDec) on
$\Omega$. Then, there exist $\psi\in\Phi_{s}(\Omega)$ which satisfies (A0),
(adA1), (Inc) and (Dec) on $\Omega$ and such that $\psi\simeq\varphi$ and
$\psi\approx\varphi$.

###### Proof.

By assumption we have that $\varphi$ satisfies (aDec) on $\Omega$ and thus is
finite valued. Therefore, using Lemma 4.17, there exists
$\psi\in\Phi_{c}(\Omega)$ and hence in $\Phi_{s}(\Omega)$ such that
$\psi\simeq\varphi$ and satisfying (Inc) on $\Omega$. Moreover, by Lemma 4.16
and Lemma 4.9, we have that $\psi$ satisfies (Dec) on $\Omega$. Finally,
properties (A0) and (adA1) come from Lemma 4.19 and 4.22, respectively, while
the fact that $\psi\approx\varphi$ is a consequence of $\psi\simeq\varphi$ and
the fact that both functions are doubling. ∎

######  Lemma 4.25.

Let $\varphi\in\Phi_{w}(\Omega)$ satisfy (A0), and (aDec) on $\Omega$ and
(3.6) $\mathcal{L}^{d}$-a.e. in $\Omega$. Given $\psi\in\Phi_{w}(\Omega)$ such
that $\varphi\simeq\psi$ then $\psi$ satisfies (3.6) $\mathcal{L}^{d}$-a.e. in
$\Omega$.

###### Proof.

We have that $\psi$ satisfies (A0) with some $\sigma\geq 1$, and it is
doubling on $\Omega$ with a constant $K\geq 2$. Let $x_{0}\in\Omega$ such that
(3.6) holds for $\varphi$, we want to prove that (3.6) holds for $\psi$ in
$x_{0}$ as well. We have that there exists $C_{\varphi}=C_{\varphi}(x_{0})>0$
such that given $\theta>\sigma$, we can find $\varepsilon_{0}>0$ such that for
every $\varepsilon\leq\varepsilon_{0}$ and every $t\in[\sigma,\theta]$

$\varphi_{B_{\varepsilon}(x_{0})}^{+}(t)\leq
C_{\varphi}\varphi_{B_{\varepsilon}(x_{0})}^{-}(t).$

There exists $L\geq 1$ with $\varphi(t/L)\leq\psi(t)\leq\varphi(Lt)$ for every
$t\geq 0$. Observe that $\psi$ satisfies (A0) with $L\sigma$. Then, for every
$\theta\geq L\sigma$ we deduce that there exists
$\varepsilon_{0}=\varepsilon_{0}(L\theta)>0$ such that for every
$\varepsilon\leq\varepsilon_{0}$

$\psi_{B_{\varepsilon}(x_{0})}^{+}(t)\leq\varphi_{B_{\varepsilon}(x_{0})}^{+}(Lt)\leq
C_{\varphi}\varphi_{B_{\varepsilon}(x_{0})}^{-}(Lt)\leq
C_{\varphi}\psi_{B_{\varepsilon}(x_{0})}^{-}(L^{2}t)\leq
C_{\varphi}K^{2\log_{2}(L)+1}\psi_{B_{\varepsilon}(x_{0})}^{-}(t),\ \ \
t\in[L\sigma,\theta],$

where we have used property (3.6) of $\varphi$ in $x_{0}$ and the fact that
$\psi$ is doubling. Therefore $\psi$ satisfies property (3.6) in $x_{0}$ with
the constant $C_{\varphi}K^{2\log_{2}(L)+1}$. ∎

Finally, in the next elementary Lemma we observe that (adA1), together with
(A0), (aDec), is stronger than condition (3.6) on $\Omega$.

######  Lemma 4.26.

Let $\varphi\in\Phi_{w}(\Omega)$ satisfy (A0), (aDec) and (adA1). Then
$\varphi$ satisfies (3.6) on every $x_{0}\in\Omega$.

###### Proof.

Let $x_{0}\in\Omega$, let $\sigma$ be the constant of (A0) and let $K$ be the
doubling constant of $\psi$. Fix $\theta>\sigma$. We take $\varepsilon_{0}>0$
such that
$\theta\leq\varphi_{B_{\varepsilon_{0}}(x_{0})}^{-}\left(\frac{1}{2\varepsilon_{0}}\right)$.
By (adA1) there exists $\beta\in(0,1)$ such that for every
$\varepsilon\leq\varepsilon_{0}$

$\varphi_{B_{\varepsilon}(x_{0})}^{+}(\beta
t)\leq\varphi_{B_{\varepsilon}(x_{0})}^{-}(t),\ \ \ \mbox{ for every
$t\in[\sigma,\theta]\subseteq\left[\sigma,\varphi_{B_{\varepsilon}(x_{0})}^{-}\left(\frac{1}{2\varepsilon}\right)\right]$}.$

This together with the doubling property of $\varphi$ implies that (3.6) holds
for every $x_{0}\in\Omega$ with the uniform constant $K^{1-\log_{2}(\beta)}$.
∎

We recall the concept of generalized Orlicz space and report a brief list of
useful definitions and properties of such spaces. Given
$\varphi\in\Phi_{w}(\Omega)$ and $f\in L^{0}(\Omega,\mathbb{R}^{m})$ we define
the modular $\rho_{\varphi}(f)$ as

$\rho_{\varphi}(f):=\int_{\Omega}\varphi(x,|f(x)|)\,dx.$

The set

$L^{\varphi}(\Omega,\mathbb{R}^{m}):=\\{f\in
L^{0}(\Omega)\colon\rho_{\varphi}(\lambda f)<+\infty\ \ \ \mbox{for some
$\lambda>0$}\\},$

is called a generalized Orlicz space.

######  Lemma 4.27.

Let $\varphi\in\Phi_{w}(\Omega)$. Then

  1. (a)

$L^{\varphi}(\Omega,\mathbb{R}^{m})=\\{f\in
L^{0}(\Omega,\mathbb{R}^{m})\colon\lim_{\lambda\to
0^{+}}\rho_{\varphi}(\lambda f)=0\\}$;

  2. (b)

if $\varphi$ additionally satisfies (aDec) we have

$L^{\varphi}(\Omega,\mathbb{R}^{m})=\\{f\in
L^{0}(\Omega,\mathbb{R}^{m})\colon\rho_{\varphi}(f)<+\infty\\}.$

Next we define the (quasi)-norm associated to generalized Orlicz spaces.

######  Definition 4.28.

Let $\varphi\in\Phi_{w}(\Omega)$. For $f\in L^{0}(\Omega,\mathbb{R}^{m})$ we
define

$\|f\|_{L^{\varphi}(\Omega,\mathbb{R}^{m})}:=\inf\left\\{\lambda>0\colon\rho_{\varphi}\left(\frac{f}{\lambda}\right)\leq
1\right\\}.$

Most of the time we will abbreviate the notation writing $\|f\|_{\varphi}$.
Now we present some properties linking the modular with the Orlicz norm.

######  Lemma 4.29.

If $\varphi\in\Phi_{w}(\Omega)$ then $\|\cdot\|_{\varphi}$ is a quasi-norm. If
$\varphi\in\Phi_{c}(\Omega)$ then $\|\cdot\|_{\varphi}$ is a norm.

######  Proposition 4.30.

Let $\varphi,\psi\in\Phi_{w}(\Omega)$. If $\varphi\simeq\psi$, then
$L^{\varphi}(\Omega,\mathbb{R}^{m})=L^{\psi}(\Omega,\mathbb{R}^{m})$ and the
two norms are equivalent.

######  Remark 4.31.

Lemmas 4.24, 4.25 and 4.30 entail, in particular, that it suffices to prove
all our results under the stronger assumption that $\psi\in\Phi_{s}(\Omega)$
satisfying (A0), (Inc), (Dec) and (adA1) (or (3.6), depending on the need) on
$\Omega$. The extension to a $\psi\in\Phi_{w}(\Omega)$ is indeed immediate,
upon noticing that, if we take equivalent Orlicz functions, then the
generalized Orlicz space remains the same.

######  Lemma 4.32 (Unit ball property).

Let $\varphi\in\Phi_{w}(\Omega)$. Given $f\in L^{0}(\Omega,\mathbb{R}^{m})$ we
have

$\|f\|_{\varphi}<1\ \ \ \Rightarrow\ \ \ \rho_{\varphi}(f)\leq 1\ \ \
\Rightarrow\ \ \ \|f\|_{\varphi}\leq 1.$

If in addition $\varphi$ is left continuous then $\rho_{\varphi}(f)\leq
1\Leftrightarrow\|f\|_{\varphi}\leq 1$. Moreover, the following properties
hold:

  1. (a)

if $\|f\|_{\varphi}<1$, then $\rho_{\varphi}(f)\leq\|f\|_{\varphi}$;

  2. (b)

if $\|f\|_{\varphi}>1$, then $\|f\|_{\varphi}\leq\rho_{\varphi}(f)$;

  3. (c)

in any case, $\|f\|_{\varphi}\leq\rho_{\varphi}(f)+1$.

In generalized Orlicz spaces we also have a generalization of the concept of
Lebesgue points.

######  Proposition 4.33.

Let $\varphi\in\Phi_{w}(\Omega)$ satisfying (A0) and (aDec). Then, for every
$f\in L^{0}(\Omega,\mathbb{R}^{m})$ such that $\rho_{\varphi}(f)<+\infty$, we
have

$\lim_{\varepsilon\to
0^{+}}\fint_{B_{\varepsilon}(x_{0})}\varphi(x,|f(x)-f(x_{0})|)\,dx=0\ \ \
\mbox{for $\mathcal{L}^{d}$-a.e. $x_{0}\in\Omega$}.$

###### Proof.

The proof can be carried out exactly as the proof of Theorem 3.1 in [44] with
some minor changes, so we omit it. ∎

We present some properties of the maximal operator in Orlicz spaces (see [45,
Chapter 4] or [50]). Since the proof of Proposition 7.3 only needs the
estimate for $\Phi$-functions independent of $x$, we give the statement only
for non generalized $\Phi$-functions.

######  Definition 4.34.

Given $f\in L^{1}_{\textnormal{loc}}(\Omega,\mathbb{R}^{m})$ we define the
restricted maximal operator to $\Omega$ as

$Mf(x):=\sup_{r>0}\frac{1}{\omega_{d}r^{d}}\int_{B_{r}(x)\cap\Omega}|f(y)|\,dy.$

Analogously, for a non-negative, finite Radon measure $\mu$ on $\Omega$ one
can define

$M\mu(x):=\sup_{r>0}\frac{\mu(B_{r}(x)\cap\Omega)}{\omega_{d}r^{d}}.$

######  Theorem 4.35.

Let $\varphi\in\Phi_{w}$ finite valued satisfy $\textnormal{(Inc)}_{\gamma}$
with $\gamma>1$. Then, the restricted maximal operator $M\colon
L^{\varphi}(\Omega,\mathbb{R}^{m})\to L^{\varphi}(\Omega,\mathbb{R}^{m})$ is
bounded. In particular, we have that if $\|f\|_{\varphi}\neq 0$, setting
$\varepsilon:=\frac{1}{2\|f\|_{\varphi}}$, there exists
$c=c(\varphi^{-1}(1),d,\gamma)$ such that $\rho_{\varphi}(c\varepsilon Mf)\leq
1$.

###### Proof.

The proof builds upon the key estimate [45, Theorem 4.3.3]. In particular, let
$c_{1}=c_{1}(d,\gamma)$ be the constant such that
$\|Mf\|_{L^{\gamma}(\Omega)}\leq c_{1}\|f\|_{L^{\gamma}(\Omega)}$. Then, the
constant $c$ is given by

$c:=\frac{(\varphi^{-1}(1))^{3}}{16c_{1}}.$

∎

######  Corollary 4.36.

Let $\varphi\in\Phi_{w}$ finite valued satisfy $\textnormal{(Inc)}_{\gamma}$
with $\gamma>1$. Assume in addition that it satisfies $\textnormal{(Dec)}_{q}$
with $1<q<\infty$. Then, there exists $C=C(\varphi^{-1}(1),d,\gamma,q)>0$ such
that for every $f\in L^{\varphi}(\Omega,\mathbb{R}^{m})$

$\rho_{\varphi}(Mf)\leq C(\rho_{\varphi}(f)+1)^{q}.$ (4.3)

###### Proof.

Using Theorem 4.7, we can find $\psi\in\Phi_{s}$ such that
$\psi(t)\leq\varphi(t)\leq\psi(2t)$ for every $t\geq 0$. Moreover, by Lemma
4.5, $\psi$ satisfies $\textnormal{(Inc)}_{\gamma}$ and
$\textnormal{(Dec)}_{q}$. Let us prove (4.3) for $\psi$. If $f\equiv 0$ then
there is nothing to prove. Assume that $f$ is not identically zero on
$\Omega$. Since $\|\cdot\|_{\psi}$ is a norm by Lemma 4.29, we have that
$\|f\|_{\psi}\neq 0$. Let $\varepsilon:=\frac{1}{2\|f\|_{\psi}}$ and
$c=c(\psi^{-1}(1),d,\gamma)=c(\varphi^{-1}(1),d,\gamma)>0$ be the constant
from Theorem 4.35. If $c\varepsilon\leq 1$ by $\textnormal{(Dec)}_{q}$ and
Theorem 4.35 we have

$1\geq\int_{\Omega}\psi(c\varepsilon
Mf(x))\,dx\geq(c\varepsilon)^{q}\int_{\Omega}\psi(x,Mf(x))\,dx.$

Therefore, using also Lemma 4.32(c) we get

$\int_{\Omega}\psi(Mf(x))\,dx\leq c^{-q}(2\|f\|_{\psi})^{q}\leq
2^{q}c^{-q}\left(1+\int_{\Omega}\psi(|f(x)|)\,dx\right)^{q},$

If $c\varepsilon>1$, then it is enough to observe that since $\psi$ is
increasing

$\int_{\Omega}\psi(Mf(x))\,dx\leq\int_{\Omega}\psi(c\varepsilon Mf(x))\,dx\leq
1\leq 1+\|f\|_{\psi}\leq 2+2\rho_{\psi}(f)\leq 2(1+\rho_{\psi}(f))^{q}.$

Setting $C:=\max\\{2^{q}c^{-q},2\\}$ we conclude (4.3) for $\psi$. Finally,
for $\varphi$ we have

$\rho_{\varphi}(Mf/2)\leq\rho_{\psi}(Mf)\leq C(\rho_{\psi}(f)+1)^{q}\leq
C(\rho_{\varphi}(f)+1)^{q},$

using again $\textnormal{(Dec)}_{q}$, we conclude (4.3) for $\varphi$ as well
with $C=C(\varphi^{-1}(1),d,\gamma,q)$. ∎

###  4.2. (Generalized) Special functions of Bounded Variation

We now discuss the definition and some useful properties of $GSBV$ and $SBV$
function. For a complete treatment of the topic we refer to [10].

Let $A\in\mathcal{A}(\mathbb{R}^{d})$ and $x\in A$ such that

$\limsup_{r\searrow 0}\frac{\mathcal{L}^{d}(A\cap B(x,r))}{r^{d}}>0,$

Given $u\in L^{0}(A,\mathbb{R}^{m})$, we say that $a\in\mathbb{R}^{m}$ is the
approximate limit of $u$ at $x$ if

$\lim_{r\searrow 0}\frac{\mathcal{L}^{d}(\Omega\cap
B(x,r)\cap\left\\{|a-v(x)|>\varepsilon\right\\})}{r^{d}}=0\ \ \ \mbox{for
every $\varepsilon>0$}.$

In that case we write

$\operatorname*{ap-\,lim}_{y\to x}u(y)=a.$

We say that $x\in A$ is an approximate jump point of $u$, and we write $x\in
J_{u}$, if there exists $a,b\in\mathbb{R}^{m}$ with $a\neq b$ and
$\nu\in\mathbb{S}^{d-1}$ such that

$\underset{\underset{(y-x)\cdot\nu>0}{y\to x}}{\textnormal{ap-}\lim}u(y)=a\ \
\ \ \mbox{and}\ \ \ \ \underset{\underset{(y-x)\cdot\nu<0}{y\to
x}}{\textnormal{ap-}\lim}u(y)=b.$

In particular, for every $x\in J_{u}$ the triple $(a,b,\nu)$ is uniquely
determined up to a change of sign of $\nu$ and a permutation of $a$ and $b$.
We indicate such triple by $(u^{+}(x),u^{-}(x),\nu_{u}(x))$. The jump of $u$
at $x\in J_{u}$ is defined as
$[u](x):=u^{+}(x)-u^{-}(x)\in\mathbb{R}^{m}_{0}$. The set $J_{u}$ is countably
rectifiable and has $\nu$ as an approximate unit normal vector at
$\mathcal{H}^{d-1}$-every point (see [57]).

The space $\textnormal{BV}(A,\mathbb{R}^{m})$ of functions of bounded
variation is the set of $u\in L^{1}(A;\mathbb{R}^{m})$ whose distributional
gradient $Du$ is a bounded Radon measure on $A$ with values in
$\mathbb{R}^{m\times d}$. Given $u\in\textnormal{BV}(A,\mathbb{R}^{m})$ we can
write $Du=D^{a}u+D^{s}u$, where $D^{a}u$ is absolutely continuous and $D^{s}u$
is singular w.r.t. $\mathcal{L}^{d}$. The density $\nabla u\in
L^{1}(A,\mathbb{R}^{m\times d})$ of $D^{a}u$ w.r.t. $\mathcal{L}^{d}$
coincides a.e. in $A$ with the approximate gradient of $u$. That is, for a.e.
$x\in A$ it holds

$\underset{y\to x}{\textnormal{ap-}\lim}\frac{u(y)-u(x)-\nabla
u(x)\cdot(y-x)}{|x-y|}=0.$

The space $SBV(A,\mathbb{R}^{m})$ of special functions of bounded variation is
defined as the set of all $u\in\textnormal{BV}(A,\mathbb{R}^{m})$ such that
$|D^{s}u|(A\setminus J_{u})=0$. Moreover, we denote by
$SBV_{\textnormal{loc}}(\Omega,\mathbb{R}^{m})$ the space of functions
belonging to $SBV(U,\mathbb{R}^{m})$ for every $U\Subset A$ open. For
$p\in[1,+\infty)$, $SBV^{p}(A,\mathbb{R}^{m})$ stands for the set of functions
$u\in SBV(A,\mathbb{R}^{m})$, with approximate gradient $\nabla u\in
L^{p}(A,\mathbb{R}^{m\times d})$ and $\mathcal{H}^{d-1}(J_{u})<+\infty$, that
is

$SBV^{p}(A,\mathbb{R}^{m}):=\\{u\in SBV(A,\mathbb{R}^{m})\colon\nabla u\in
L^{p}(\Omega,\mathbb{R}^{m}),\mathcal{H}^{d-1}(J_{u})<+\infty\\}.$

We say that $u\in GSBV(A,\mathbb{R}^{m})$ if $X(u)\in
SBV_{\textnormal{loc}}(A,\mathbb{R}^{m})$ for every $X\in
C_{c}^{1}(\mathbb{R}^{m},\mathbb{R}^{m})$. Also for $u\in
GSBV(A,\mathbb{R}^{m})$ the approximate gradient $\nabla u$ exists
$\mathcal{L}^{d}$-a.e. in $A$ and the jump set $J_{u}$ is countably
$\mathcal{H}^{d-1}$-rectifiable with approximate unit normal vector $\nu_{u}$.
Finally, for $p\in[1,+\infty)$, we define as before

$GSBV^{p}(A,\mathbb{R}^{m}):=\\{u\in GSBV(A,\mathbb{R}^{m})\colon\nabla u\in
L^{p}(\Omega,\mathbb{R}^{m}),\mathcal{H}^{d-1}(J_{u})<+\infty\\};$

it is known that $GSBV^{p}(A,\mathbb{R}^{m})$ is a vector space (see e.g. [33,
pg. 172]).

Moreover, given a generalized Orlicz function $\varphi\in\Phi_{w}(\Omega)$ we
denote with $SBV^{\varphi}(\Omega,\mathbb{R}^{m})$ the space of functions
$u\in SBV(\Omega,\mathbb{R}^{m})$ with $\mathcal{H}^{d-1}(J_{u})<+\infty$ and
$\nabla u\in L^{\varphi}(\Omega,\mathbb{R}^{m\times d})$. The definition of
$GSBV^{\varphi}(\Omega,\mathbb{R}^{m})$ is analogous.

The following three theorems can be found in [37] for the scalar case and in
[21] for vector valued $SBV$ functions.

######  Theorem 4.37 ([37, Lemma 2.6]).

Let $p\geq 1$ and $u\in SBV(\Omega,\mathbb{R}^{m})$ be such that

$\int_{K}|\nabla u|^{p}\,dx+\mathcal{H}^{d-1}(J_{u}\cap K)<+\infty$

for every compact $K\Subset\Omega$. Then

$\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d-1}}\left(\int_{B_{\varepsilon}(x_{0})}|\nabla
u(x)|^{p}\,dx+\mathcal{H}^{d-1}(B_{\varepsilon}(x_{0})\cap J_{u})\right)=0\ \
\ \mbox{for $\mathcal{H}^{d-1}$-a.e. $x_{0}\in\Omega\setminus J_{u}$.}$

For the other two theorems we need first to fix some notations. Given
$a=(a_{1},\dots,a_{m})$ and $b=(b_{1},\dots,b_{m})$ vectors in
$\mathbb{R}^{m}$, we set

$a\wedge b:=(\min(a_{1},b_{1}),\dots,\min(a_{m},b_{m}))\ \ \ \mbox{and}\ \ \
a\vee b:=(\max(a_{1},b_{1}),\dots,\max(a_{m},b_{m})).$

Let $B$ be a ball in $\mathbb{R}^{d}$. For every measurable function $u\colon
B\to\mathbb{R}^{m}$, given $0\leq s\leq\mathcal{L}^{d}(B)$, we define

$u_{*}(s;B):=((u_{1})_{*}(s;B),\dots,(u_{m})_{*}(s;B)),$

where, for every $i=1,\dots,m$,

$(u_{i})_{*}(s;B):=\inf\left\\{t\in\mathbb{R}\colon\mathcal{L}^{d}(\left\\{u_{i}<t\right\\})\geq
s\right\\}.$

Moreover, we define
$\displaystyle\textnormal{med}(u;B):=u_{*}\left(\frac{\mathcal{L}^{d}(B)}{2};B\right)$.

Let $\gamma_{\textnormal{iso}}$ be the dimensional constant in the relative
isoperimetric inequality in balls. For every $u\in
GSBV(\Omega,\mathbb{R}^{m})$ such that

$\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}\cap
B)\right)^{\frac{d}{d-1}}\leq\frac{1}{2}\mathcal{L}^{d}(B),$

we define

$\displaystyle\begin{split}&\tau^{\prime}(u;B):=u_{*}\left(\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}\cap
B)\right)^{\frac{d}{d-1}};B\right),\\\
&\tau^{\prime\prime}(u;B):=u_{*}\left(\mathcal{L}^{d}(B)-\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}\cap
B)\right)^{\frac{d}{d-1}};B\right)\end{split}$ (4.4)

and the truncation operator in $B$

$T_{B}u(x):=(\tau^{\prime}(u;B)\vee u(x))\wedge\tau^{\prime\prime}(u;B).$
(4.5)

Notice that it holds

$\mathcal{L}^{d}(\\{T_{B}u\neq u\\}\cap B)\leq
2\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}\cap
B)\right)^{\frac{d}{d-1}}.$ (4.6)

######  Theorem 4.38 ([37, Theorem 3.1 and Remark 3.2]).

Let $B\subset\mathbb{R}^{d}$ be a ball and $u\in SBV(B,\mathbb{R}^{m})$
satisfy

$\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}\cap
B)\right)^{\frac{d}{d-1}}\leq\frac{1}{2}\mathcal{L}^{d}(B).$ (4.7)

If $1\leq p<d$ then,

$\left(\int_{B}|T_{B}u-\textnormal{med}(u,B)|^{p^{*}}\,dx\right)^{1/p^{*}}\leq\frac{2\gamma_{\textnormal{iso}}p(d-1)}{d-p}\left(\int_{B}|\nabla
u|^{p}\,dx\right)^{1/p},$ (4.8)

and

$\mathcal{L}^{d}(\\{T_{B}u\neq u\\}\cap B)\leq
2\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}\cap
B)\right)^{\frac{d}{d-1}}.$ (4.9)

If $p\geq d$ instead, for every $q\geq 1$ we have

$\|T_{B}u-\textnormal{med}(u,B)\|_{L^{q}(B,\mathbb{R}^{m})}\leq\frac{2\gamma_{\textnormal{iso}}q(d-1)}{d}\mathcal{L}^{d}(B)^{\frac{1}{d}+\frac{1}{q}-\frac{1}{p}}\|\nabla
u\|_{L^{p}(B,\mathbb{R}^{m})}.$ (4.10)

######  Theorem 4.39 ([37, Theorem 3.6]).

Let $u\in SBV_{\textnormal{loc}}(\Omega,\mathbb{R}^{m})$ and $p>1$ and let
$x_{0}\in\Omega$. If

$\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d-1}}\left(\int_{B_{\varepsilon}(x_{0})}|\nabla
u(x)|^{p}\,dx+\mathcal{H}^{d-1}(B_{\varepsilon}(x_{0})\cap J_{u})\right)=0,$

then $x_{0}\neq J_{u}$ and

$\lim_{\varepsilon\to
0^{+}}\textnormal{med}(u,B_{\varepsilon}(x_{0}))=\operatorname*{ap-\,lim}_{x\to
x_{0}}u(x)\in\mathbb{R}^{m}.$

Moreover, there exists $\varepsilon_{0}>0$ such that for $\delta\in(3/4,1)$
and $\varepsilon,\rho>0$ with
$\delta\varepsilon<\rho<\varepsilon<\varepsilon_{0}$,

$\displaystyle\tau^{\prime}(u,B_{\rho}(x_{0}))\leq\textnormal{med}(u,B_{\varepsilon}(x_{0}))\leq\tau^{\prime\prime}(u,B_{\rho}(x_{0})),$
$\displaystyle\tau^{\prime}(u,B_{\varepsilon}(x_{0}))\leq\textnormal{med}(u,B_{\rho}(x_{0}))\leq\tau^{\prime\prime}(u,B_{\varepsilon}(x_{0})).$

###  4.3. Examples

In this last subsection of preliminaries, we present some examples of
generalized Orlicz functions to which our theory applies. We consider the
following Orlicz functions:

  * (I)

Variable exponent: $\psi(x,t)=t^{p(x)}$. For the variable exponent case, the
semicontinuity was already addressed in [34], while integral representation
and $\Gamma$-convergence are studied in [60].

  * (II)

Perturbed Orlicz: $\psi(x,t)=a(x)\varphi(t)$.

  * (III)

Double phase: $\psi(x,t)=t^{p}+a(x)t^{q}$.

  * (IV)

Degenerate double phase: $\psi(x,t)=t^{p}+a(x)t^{p}\log(\mathrm{e}+t)$.

  * (V)

Triple phase: $\psi(x,t)=t^{p}+a(x)t^{q}+b(x)t^{r}$.

  * (VI)

Variable exponent double phase: $\psi(x,t)=t^{p(x)}+a(x)t^{q(x)}$. This type
of generalized Orlicz function was studied recently in [31].

We say that a function $p\colon\Omega\to[1,+\infty)$ is log-Hölder continuous
on $\Omega$ if

$\exists\ C>0\ \ \ \mbox{such that}\ \ \ |p(x)-p(y)|\leq\frac{C}{-\log|x-y|}\
\ \ \mbox{for every $x,y\in\Omega$ with $|x-y|$}\leq\frac{1}{2}.$

We use the usual shortcut $p\in C^{\mathrm{log}}(\Omega)$ to denote a log-
Hölder continuous variable exponent on $\Omega$. In the Table below we collect
conditions that are sufficient (sometimes necessary) for properties (aInc),
(aDec), (A0), (3.6) and (adA1) in the special cases above. The usual notations
$C^{0}$ and $C^{0,\beta}$ are used for continuous and Hölder continuous
functions with exponent $\beta$, respectively. In each line, the checkmark is
used to denote that a property needs no new assumption to be satisfied than
the previously considered ones.

| (aInc) | (aDec) | (A0) | (3.6) |  (adA1)
---|---|---|---|---|---
(I) | $\operatorname*{ess\,inf}_{\Omega}p>1$ | $\operatorname*{ess\,sup}_{\Omega}p<+\infty$ | $\checkmark$ | $p\in C^{0}$ |  $p\in C^{\log}$
(II) |  $\varphi$ is (aInc) |  $\varphi$ is (aDec) | $c_{1}\leq a\leq c_{2}$ | $\checkmark$ |  $\checkmark$
(III) |  $a\geq 0$, $p>1$ | $q<+\infty$ | $a\in L^{\infty}$ | $a\in C^{0}$ |  $a\in C^{0,q/p-1}$
(IV) |  $a\geq 0$, $p>1$ | $p<+\infty$ | $a\in L^{\infty}$ | $a\in C^{0}$ |  $a\in C^{\log}$
(V) |  $a,b\geq 0$, $p>1$ | $q\leq r<+\infty$ | $a,b\in L^{\infty}$ | $a,b\in C^{0}$ |  $a\in C^{0,q/p-1}$, $b\in C^{0,r/p-1}$
(VI) | $\operatorname*{ess\,inf}_{\Omega}p>1$ | $\operatorname*{ess\,sup}_{\Omega}q<+\infty$ | $a\in L^{\infty}$ | $a,p,q\in C^{0}$ |  $p\in C^{\log}$, $a\in C^{0,\alpha}$,
$q\in C^{0,\alpha/q^{-}}$, $\frac{q}{p}\leq 1+\alpha$

We remark that in the case of variable exponent, the sole continuity
assumption will be enough to obtain (3.6) and thus Theorem 3.3. This
represents an improvement with respect to the result of [34], where log–Hölder
continuity has been assumed.

We briefly discuss how our conditions can be checked in all the aforementioned
cases. Checking the validity of (aInc) and (aDec) is straightforward, while we
refer to Table 1 in [48] for condition (A0). Concerning (3.6), it follows from
the following general claim: given $\varphi\in\Phi_{w}(\Omega)$ satisfying
(A0) on $\Omega$, we have that if $\varphi\in C^{0}(\Omega\times[0,+\infty))$,
then it satisfies property (3.6). Now, assume that $\varphi$ satisfies (A0)
with $\sigma\geq 1$ and let for simplicity $x_{0}=0$. Let $\zeta>\sigma$
fixed. Since $\varphi$ is uniformly continuous on
$B_{\delta}\times[\sigma,\zeta]$ for some $\delta>0$ small enough, we observe
that there exists a uniform modulus of continuity
$\omega\colon[0,+\infty)\to[0,+\infty)$ such that

$|\varphi(x,t)-\varphi(y,t)|\leq\omega(|x-y|)\ \ \ \mbox{for every $x,y\in
B_{\delta}$ and every $t\in[\sigma,\zeta]$.}$

Let $\varepsilon_{0}\in(0,\delta)$ such that $\omega(2\varepsilon_{0})\leq 1$.
Thanks to (A0) and the fact that $\varphi(y,\cdot)$ is increasing, we estimate

$\varphi(x,t)\leq\varphi(y,t)+\omega(|x-y|)\leq\varphi(y,t)+1\leq
2\varphi(y,t),\ \ \ \mbox{for every $x,y\in B_{\varepsilon}$ and every
$t\in[\sigma,\zeta]$}\,.$

Thus, property (3.6) is satisfied for every $x_{0}\in\Omega$ with $C=2$.
As for condition (adA1), we discuss it here for example in the double phase
case. Namely, we prove that if $\psi(x,t)=|t|^{p}+a(x)|t|^{q}$ with $a\in
C^{0,\alpha}(\overline{\Omega})$, $a(x)\geq 0$ and $1<p<q$ with
$\frac{q}{p}\leq 1+\alpha$ for some $\alpha\in(0,1]$, then $\psi$ satisfies
(adA1).

Note that $\psi(x,t)\approx\max\\{t^{p},a(x)t^{q}\\}$. Let $B\Subset\Omega$ a
ball with $\textnormal{diam}(B)\leq 1$, denote $a^{-}_{B}:=\min_{B}a(\cdot)$
and $a^{+}_{B}:=\max_{B}a(\cdot)$. We need to show that
$\max\\{t^{p},a_{B}^{+}t^{q}\\}\lesssim\max\\{t^{p},a^{-}_{B}t^{q}\\}$
whenever $\psi^{-}_{B}(t)\leq\frac{1}{\textnormal{diam}(B)}$. In view of the
fact that $a\in C^{0,\alpha}(\overline{\Omega})$ and $\frac{q}{p}\leq
1+\alpha$, we have

$a_{B}^{+}\lesssim\max\\{\textnormal{diam}(B)^{\frac{q-p}{p}},a_{B}^{-}\\}.$

Now, when $t^{p}\leq\frac{1}{\textnormal{diam}(B)}$, one has in particular
$\psi^{-}_{B}(t)\leq\frac{1}{\textnormal{diam}(B)}$, so let us assume that
$t^{p}\leq\frac{1}{\textnormal{diam}(B)}$. Hence,
$a^{+}_{B}\lesssim\max\\{t^{p-q},a^{-}_{B}\\}$, so that
$a^{+}_{B}t^{q}\lesssim\max\\{t^{p},a^{-}_{B}t^{q}\\}$. From this, we
conclude.

##  5\. Poincaré inequality in $SBV^{\varphi}$

In this section we present a fundamental ingredient we need in the proofs of
our main results and which is in our opinion of independent interest. Namely,
a Poincaré inequality for functions belonging to $SBV^{\varphi}$ with small
jump set a là De Giorgi-Carriero-Leaci. We start with the case of
$\Phi$-functions, while the extension to the generalized setting will be given
in Subsection 5.3.

######  Theorem 5.1 (Poincaré inequality in $SBV^{\varphi}$).

Let $\varphi\in\Phi_{w}$ be a finite valued Orlicz function satisfying
$\textnormal{(Inc)}_{1}$ and let $B_{r}(x)$ be a ball in $\mathbb{R}^{d}$ with
radius $r$ and centred in $x\in\mathbb{R}^{d}$. Set
$\phi:=\varphi^{\frac{d}{d-1}}t^{-\frac{1}{d-1}}$ and let $u\in
SBV^{\varphi}(B_{r}(x))$ be such that

$\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}\cap
B_{r}(x))\right)^{\frac{d}{d-1}}\leq\frac{1}{2}\mathcal{L}^{d}(B_{r}).$ (5.1)

Then, there exists a dimensional constant $C=C(d)$ such that the following
inequality holds

$\phi^{-1}\left(\fint_{B_{r}(x)}\phi\left(\frac{|T_{B_{r}(x)}u(y)-\textnormal{med}(u,B_{r}(x))|}{r}\right)\,dy\right)\leq
C\varphi^{-1}\left(\fint_{B_{r}(x)}\varphi(C|\nabla u(y)|)\,dy\right),$ (5.2)

where $T_{B_{r}(x)}u$ is defined in (4.5).

Notice that since we have a truncation of a scalar $SBV$ function, the result
above applies also to real valued $GSBV$ functions. Also observe that if
$\varphi$ has linear growth, then the inequality in (5.2) reduces to the
standard homogeneous Poincaré inequality with $p=1$ for $SBV$ functions.

######  Remark 5.2.

By translation invariance and rescaling, it is enough to verify inequality
(5.2) only on $B_{1}$. Indeed, let $u\colon B_{r}(x)\to\mathbb{R}^{m}$, define
$v\colon B_{1}\to\mathbb{R}^{m}$ as $v(y):=u(x+ry)/r$. We have
$u_{*}(s;B_{r})=rv_{*}(s/r^{d};B_{1})$ for every
$s\in[0,\mathcal{L}^{d}(B_{r})]$. Thus,

$\tau^{\prime}(v,B_{1})=\frac{\tau^{\prime}(u,B_{r})}{r},\ \ \
\tau^{\prime\prime}(v,B_{1})=\frac{\tau^{\prime\prime}(u,B_{r})}{r},\ \ \
\textnormal{med}(v,B_{1})=\frac{\textnormal{med}(u,B_{r})}{r},$

and $T_{B_{1}}v(y):=T_{B_{r}(x)}u(x+ry)/r$ for every $y\in B_{1}$.

Then, if the Poincaré inequality (5.2) holds on $B_{1}$ for $v$, we deduce

$\displaystyle\phi^{-1}\left(\fint_{B_{r}(x)}\phi\left(\frac{|T_{B_{r}(x)}u(y)-\textnormal{med}(u,B_{r}(x))|}{r}\right)\,dy\right)$
$\displaystyle=\phi^{-1}\left(\fint_{B_{1}}\phi\left(|T_{B_{1}}v(z)-\textnormal{med}(v,B_{1})|\right)\,dz\right)$
$\displaystyle\leq C\varphi^{-1}\left(\fint_{B_{1}}\varphi(C|\nabla
v(z)|)\,dz\right)$
$\displaystyle=C\varphi^{-1}\left(\fint_{B_{r}}\varphi(C|\nabla
u(y)|)\,dx\right).$

The proof of Theorem 5.1 will be given in Subsection 5.2, after recalling and
proving some preliminary material.

###  5.1. Rearrangements of $BV$ functions

We are going to make use of the theory of rearrangements for $BV$ functions.
Since they are used only in this section, we report here their main
definitions and properties. For a complete treatment of the topic see [26]. We
say that two functions $u$ and $v$ are equi-measurable if
$\mathcal{L}^{d}(\\{u>t\\})=\mathcal{L}^{d}(\\{v>t\\})$ for every
$t\in\mathbb{R}$.

For simplicity, let us denote by $B$ the unit ball in $\mathbb{R}^{d}$ with
$\mathcal{L}^{d}(B)=\omega_{d}$. Let $u\colon B\to\mathbb{R}$ be a measurable
function. We define the distribution function
$\nu_{u}\colon\mathbb{R}\to[0,\omega_{d}]$ of $u$ as

$\nu_{u}(t):=\mathcal{L}^{d}(\\{x\in B\colon u(x)>t\\})\ \ \ t\in\mathbb{R}.$
(5.3)

We have that $\nu_{u}$ is right-continuous and non-increasing. By definition,
$\nu_{u}(\operatorname*{ess\,sup}u)=0$. Moreover,
$\nu_{u}(t-)=\mathcal{L}^{d}(\\{u\geq t\\})$ and
$\nu_{u}(t-)-\nu_{u}(t)=\mathcal{L}^{d}(\\{u=t\\})$. Thus, $\nu_{u}$ is
continuous in $t$ if and only if $\mathcal{L}^{d}(\\{u=t\\})=0$. We can now
define also the signed decreasing rearrangement of $u$ as a function
$u^{0}\colon[0,\omega_{d}]\to\mathbb{R}\cup\\{\pm\infty\\}$ given by

$u^{0}(s):=\sup\\{t\in\mathbb{R}\colon\nu_{u}(t)>s\\}\ \ \
s\in[0,\omega_{d}].$ (5.4)

We have that $u^{0}$ is right-continuous, non-increasing and it is equi-
measurable with $u$. By definition, $u^{0}(0)=\operatorname*{ess\,sup}u$.
Moreover, notice that

$\\{s\in[0,\omega_{d}]\colon u^{0}(s)>t\\}=[0,\nu_{u}(t))\ \ \ \mbox{for every
$t\in\mathbb{R}$}.$

Therefore, $u^{0}$ and $u$ are equi-distributed, that is
$\nu_{u}=\nu_{u^{0}}$. We also have

$u^{0}(\nu_{u}(t))\leq t\ \ \mbox{for every $t\in\mathbb{R}$}\ \ \ \mbox{and}\
\ \ t\leq u^{0}(\nu_{u}(t)-)\ \ \mbox{for every
$t\in(-\infty,\operatorname*{ess\,sup}u)$}$ (5.5)

and

$\nu_{u}(u^{0}(s))\leq s\ \ \mbox{for every $s\in[0,\omega_{d}]$}\ \ \
\mbox{and}\ \ \ s\leq\nu_{u}(u^{0}(s)-)\ \ \mbox{for every
$s\in[0,\omega_{d})$}.$ (5.6)

To summarize, $u^{0}$ is a monotone real valued function on $(0,\omega_{d})$
and thus $u^{0}\in BV_{\textnormal{loc}}((0,\omega_{d}))$. Moreover, $u^{0}$
is constant in the interval $(s_{1},s_{2})$ if and only if there exists
$t\in\mathbb{R}$ where $\nu_{u}$ jumps and
$(s_{1},s_{2})\subset(\nu_{u}(t),\nu_{u}(t-))$. Vice versa, $u^{0}$ jumps at
some point $s_{0}\in(0,\omega_{d})$ and
$(t_{1},t_{2})\subset(u^{0}(s_{0}),u^{0}(s_{0}-))$ if and only if $\nu_{u}$ is
constant on $(t_{1},t_{2})$. Therefore, a countable family of left-open
intervals $(\alpha_{i},\beta_{i}]$ exists such that

$(\operatorname*{ess\,inf}u,\operatorname*{ess\,sup}u)\setminus\bigcup_{i\in
I}(\alpha_{i},\beta_{i}]\subseteq
u^{0}((0,\omega_{d}))\subseteq[\operatorname*{ess\,inf}u,\operatorname*{ess\,sup}u]\setminus\bigcup_{i\in
I}(\alpha_{i},\beta_{i}].$ (5.7)

Finally, if $u$ is a non-negative function on $B$, we indicate the
distribution function of $u$ as

$\mu_{u}(t):=\mathcal{L}^{d}(\\{x\in B\colon u(x)>t\\})\,,\ \
t\in[0,+\infty),$ (5.8)

and the decreasing rearrangement as $u^{*}\colon[0,\omega_{d}]\to[0,+\infty]$
is given by

$u^{*}(s):=\sup\\{t\in[0,+\infty)\colon\mu_{u}(t)>s\\}\,,\ \
s\in[0,\omega_{d}].$ (5.9)

We start by proving the following fact.

######  Proposition 5.3.

Let $\psi\in\Phi_{s}$ be a finite valued Orlicz function satisfying
$\textnormal{(Inc)}_{1}$. Then, for every $u\in SBV(B)$ such that (5.1) holds,
we have that $(T_{B}u)^{0}$ is absolutely continuous in $(0,\omega_{d})$ and

$\int_{0}^{\omega_{d}}\psi\left(c(d)\min\\{s,1-s\\}^{\frac{d-1}{d}}\left(-\frac{d(T_{B}u)^{0}}{ds}(s)\right)\right)\,ds\leq\int_{B}\psi\left(2|\nabla
u(x)|\right)\,dx,$ (5.10)

where $c(d)$ is a positive constant depending only on the dimension.

In order to prove Proposition 5.3, we need the following two results, the
first one is [26, Equation (3.21)] and the second is contained in [5,
Proposition 2.1].

######  Lemma 5.4.

Let $u\in BV(B)$ and set $C_{u}\subseteq B$ the set of points in $B$ where $u$
is approximately continuous. We have that

$\partial^{M}\\{u>t\\}\cap C_{u}=\emptyset\ \ \mbox{for a.e.
$t\in(\operatorname*{ess\,inf}u,\operatorname*{ess\,sup}u)\setminus
u^{0}((0,\omega_{d}))$}.$

###### Proof.

The proof can be carried out exactly as the proof of [26, Equation (3.21)]. ∎

######  Lemma 5.5.

Let $f,g\in L^{1}(0,\omega_{d})$ such that $f(x)\geq 0$ and $g(x)\geq 0$ for
$\mathcal{L}^{1}$-a.e. $x\in(0,\omega_{d})$ and let
$F\colon[0,+\infty)\to[0,+\infty)$ be a convex function with $F(0)=0$. Assume
also that

$\int_{0}^{t}f^{*}(s)\,ds\leq\int_{0}^{t}g^{*}(s)\,ds\ \ \ \mbox{for every
$t\in[0,\omega_{d}]$,}\ \ \ \ \int_{0}^{\omega_{d}}F(g^{*}(s))\,ds<+\infty.$

Then,

$\int_{0}^{\omega_{d}}F(f(x))\,dx\leq\int_{0}^{\omega_{d}}F(g(x))\,dx.$ (5.11)

###### Proof.

The result is essentially proved in [5, Proposition 2.1], which actually
requires $F$ to be also Lipschitz. However, an inspection of the proof reveals
that we need $F$ to be Lipschitz only on the set $\\{f^{*}(s)\colon
s\in[0,\omega_{d}]\\}$. Since $F$ is convex and thus locally Lipschitz, fixing
$\eta>0$ arbitrarily small, we can apply [5, Proposition 2.1] to $f\wedge
f^{*}(\eta)$ and $g$ getting

$\int_{\eta}^{\omega_{d}}F(f^{*}(s))\,ds\leq\int_{0}^{\omega_{d}}F(g^{*}(s))\,ds.$

Letting $\eta\to 0^{+}$, by monotone convergence theorem and using the fact
that $F$ is strictly increasing, we deduce

$\int_{0}^{\omega_{d}}(F(f))^{*}(s)\,ds\leq\int_{0}^{\omega_{d}}(F(g))^{*}(s)\,ds.$

Finally, using the equi-measurability of rearrangements we conclude (5.11). ∎

We are now in a position to prove Proposition 5.3.

###### Proof of Proposition 5.3.

In order to ease the notation we set
$(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}))^{\frac{d}{d-1}}=:\Lambda$.
We split the proof into three steps.

Step 1: $u^{0}$ is continuous in $[\Lambda,\omega_{d}-\Lambda]$. We actually
prove that $u^{0}$ is continuous in

$\left(\left(\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u})\right)^{\frac{d}{d-1}},\omega_{d}-\left(\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u})\right)^{\frac{d}{d-1}}\right).$
(5.12)

Assume this is false. Then, we can find $s$ belonging to the interval in
(5.12) such that $u^{0}$ jumps in $s$ and thus
$(u^{0}(s),u^{0}(s-))\neq\emptyset$. Hence, owning Lemma 5.4 and the fact that
$u\in SBV(B)$, we can find $t\in(u^{0}(s),u^{0}(s-))$ such that
$\partial^{M}\\{u>t\\}\cap C_{u}=\emptyset$ and the set $\\{u>t\\}$ has finite
perimeter. This implies $\partial^{*}\\{u>t\\}\subset J_{u}$. Moreover, recall
that
$\mathcal{H}^{d-1}(\partial^{M}\\{u>t\\}\setminus\partial^{*}\\{u>t\\})=0$. By
definition of $u^{0}$ we have that $\mathcal{L}^{d}(\\{u>t\\})=s$. Thus, both
$\\{u>t\\}$ and $B\setminus\\{u>t\\}$ have measure strictly greater than
$(\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}))^{\frac{d}{d-1}}$.
Recalling the isoperimetric inequality this gives

$\left(\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(\partial^{*}\\{u>t\\})\right)^{\frac{d}{d-1}}=\left(\gamma_{\textnormal{iso}}P(\\{u>t\\},B)\right)^{\frac{d}{d-1}}>\left(\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u})\right)^{\frac{d}{d-1}}.$

This means that
$\mathcal{H}^{d-1}(J_{u})<\mathcal{H}^{d-1}(\partial^{*}\\{u>t\\})$, a
contradiction. Hence, $u^{0}$ is continuous in the interval in (5.12) and thus
in $[\Lambda,\omega_{d}-\Lambda]$. Since $u^{0}$ admits a continuous
representative in $[\Lambda,\omega_{d}-\Lambda]$, let us define

$\tau^{\prime}:=u^{0}\left(\omega_{d}-\Lambda\right)\ \ \ \mbox{and}\ \ \
\tau^{\prime\prime}:=u^{0}\left(\Lambda\right).$

Notice that $\tau^{\prime}$ and $\tau^{\prime\prime}$ by definition of $u^{0}$
and $\Lambda$ coincide with $\tau^{\prime}(u,B)$ and
$\tau^{\prime\prime}(u,B)$, respectively, defined in (LABEL:tau).

Step 2: $u^{0}$ is absolutely continuous in $(\Lambda,\omega_{d}-\Lambda)$. We
shall use some arguments from [24, Theorem 6.5 and Lemma 6.6]. Assume
$\mathcal{H}^{d-1}(J_{u})>0$, otherwise the absolute continuity of $u^{0}$
follows by [24, Lemma 6.6]. Let

$h_{B}(s):=\frac{1}{\gamma_{\textnormal{iso}}}\min\\{s,\omega_{d}-s\\}^{\frac{d-1}{d}}.$

Fix $\varepsilon>0$. Since $|\nabla u|\in L^{1}(B)$, by the standard theory of
rearrangements we have $|\nabla u|^{*}\in L^{1}((0,\omega_{d}))$. Thus, we can
find $\delta>0$ such that

$\int_{0}^{\delta}|\nabla u|^{*}(s)\,ds\leq\varepsilon.$ (5.13)

Let $\alpha,\beta\in u^{0}((\Lambda,\omega_{d}-\Lambda))$ with
$\alpha\leq\beta$. We have that
$\alpha,\beta\in(\tau^{\prime},\tau^{\prime\prime})$ by Step 1. Set
$u^{\beta}_{\alpha}\in SBV(B)$ as $u^{\beta}_{\alpha}:=\alpha\vee
u\wedge\beta$. Using the coarea formula we deduce

$\displaystyle\int_{\alpha}^{\beta}\\!P(\\{u>t\\},B)\,dt=\int_{-\infty}^{+\infty}\\!P(\\{u_{\alpha}^{\beta}>t\\},B)\,dt=|Du^{\beta}_{\alpha}|(B)\leq\int_{B}|\nabla
u^{\alpha}_{\beta}(x)|\,dx+(\beta-\alpha)\mathcal{H}^{d-1}(J_{u}).$ (5.14)

On the other hand, by definition of $h_{B}$ and the isoperimetric inequality
in $B$ we infer

$\displaystyle\int_{\alpha}^{\beta}P(\\{u>t\\},B)dt\geq\int_{\alpha}^{\beta}h_{B}(\nu_{u}(t))\,dt.$
(5.15)

Since $t\in(\alpha,\beta)\subset(\tau^{\prime},\tau^{\prime\prime})$, using
(5.6) we have that

$\lim_{t\to u^{0}(\Lambda)^{-}}\nu_{u}(t)=\nu_{u}(u^{0}(\Lambda)-)\geq\Lambda\
\ \ \mbox{and}\ \ \ \lim_{t\to
u^{0}(\omega_{d}-\Lambda)^{+}}\nu_{u}(t)=\nu_{u}(u^{0}(\omega_{d}-\Lambda))\leq\omega_{d}-\Lambda.$

This gives that $\nu_{u}(t)\in(\Lambda,\omega_{d}-\Lambda)$ for every
$t\in(\alpha,\beta)$. Hence, for every $t\in(\alpha,\beta)$,

$h_{B}(\nu_{u}(t))\geq
h_{B}(\Lambda)=h_{B}\left(\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u})\right)^{\frac{d}{d-1}}\right)=2\mathcal{H}^{d-1}(J_{u}).$
(5.16)

Combining (5.14)–(5.16) gives

$\int_{\alpha}^{\beta}h_{B}(\nu_{u}(t))\,dt\leq\int_{B}|\nabla
u^{\beta}_{\alpha}(x)|\,dx+\frac{1}{2}\int_{\alpha}^{\beta}h_{B}(\nu_{u}(t))\,dt,$

and finally,

$\int_{\alpha}^{\beta}h_{B}(\nu_{u}(t))\,dt\leq 2\int_{\\{\alpha\leq
u\leq\beta\\}}|\nabla u(x)|\,dx.$ (5.17)

Consider now a finite family of disjoint open intervals contained in
$(\Lambda,\omega_{d}-\Lambda)$, namely $(a_{i},b_{i})$ for $i=1,\dots,N$, such
that $\sum_{i=1}^{N}(b_{i}-a_{i})\leq\delta$ where $\delta>0$ is as in (5.13).
Now consider inequality (5.17) with $\alpha:=u^{0}(b_{i})\leq
u^{0}(a_{i})=:\beta$ for every $i=1,\dots,N$. If we add all these inequalities
we get

$\int_{\cup_{i=1}^{N}(u^{0}(b_{i}),u^{0}(a_{i}))}h_{B}(\nu_{u}(t))\,dt\leq
2\int_{\cup_{i=1}^{N}\\{u^{0}(b_{i})<u<u^{0}(a_{i})\\}}|\nabla u(x)|\,dx,$
(5.18)

where we have also used the fact that $\int_{\\{u=t\\}}|\nabla u|\,dx=0$ for
every $t\in\mathbb{R}$ (see [10, Proposition 3.73 (c)]). Recalling (5.6), for
every $i=1,\dots,N$ it holds

$\lim_{t\to u^{0}(a_{i})^{-}}\nu_{u}(t)=\nu_{u}(u^{0}(a_{i})-)\geq a_{i}\ \ \
\mbox{and}\ \ \ \lim_{t\to
u^{0}(b_{i})^{+}}\nu_{u}(t)=\nu_{u}(u^{0}(b_{i}))\leq b_{i}.$ (5.19)

Hence, by definition of $\nu_{u}$ we have that
$\mathcal{L}^{d}(\\{u^{0}(b_{i})<u<u^{0}(a_{i})\\})=\nu_{u}(u^{0}(b_{i}))-\nu_{u}(u^{0}(a_{i})-)\leq
b_{i}-a_{i}$ for every $i=1,\dots,N$. For every non-negative measurable
function $f$ and every Borel set $E$, by the Hardy-Littlewood inequality, it
holds

$\int_{E}f(x)\,dx\leq\int_{0}^{\mathcal{L}^{d}(E)}f^{*}(t)\,dt.$

Combining this result with inequality (5.18) gives

$\displaystyle\begin{split}\int_{\cup_{i=1}^{N}(u^{0}(b_{i}),u^{0}(a_{i}))}h_{B}(\nu_{u}(t))\,dt&\leq
2\int_{0}^{\sum_{i=1}^{N}[\nu_{u}(u^{0}(b_{i}))-\nu_{u}(u^{0}(a_{i}))]}|\nabla
u|^{*}\,dt\leq 2\int_{0}^{\sum_{i=1}^{N}(b_{i}-a_{i})}|\nabla u|^{*}\,dt\\\
&\leq 2\int_{0}^{\delta}|\nabla u|^{*}\,dt\leq 2\varepsilon.\end{split}$
(5.20)

Again by (5.19) we get that $\nu_{u}(t)\in(\Lambda,\omega_{d}-\Lambda)$ for
every $t\in\cup_{i=1}^{N}(a_{i},b_{i})$. Therefore, arguing as before,

$h_{B}(\nu_{u}(t))\geq 2\mathcal{H}^{d-1}(J_{u})$

for every $t\in(a_{i},b_{i})$, $i=1,\dots,N$. Thus,

$2\mathcal{H}^{d-1}(J_{u})\sum_{i=1}^{N}(u^{0}(a_{i})-u^{0}(b_{i}))\leq
2\varepsilon,$

and this gives the absolute continuity of $u^{0}$ on
$(\Lambda,\omega_{d}-\Lambda)$ since the family of sub-intervals and
$\varepsilon>0$ where arbitrary.

Step 3: Conclusion. By the previous steps we have that $u^{0}$ is uniformly
continuous and absolutely continuous in $[\Lambda,\omega_{d}-\Lambda]$ and
$(\Lambda,\omega_{d}-\Lambda)$, respectively. Hence, by definition of
$\tau^{\prime},\tau^{\prime\prime}$, the function $\tau^{\prime}\vee
u^{0}\wedge\tau^{\prime\prime}$ is absolutely continuous in $(0,\omega_{d})$.
Observe that by definition of $u^{0}$ and $T_{B}u$ (see (4.5)), we have that

$\tau^{\prime}\vee u^{0}\wedge\tau^{\prime\prime}=(\tau^{\prime}\vee
u\wedge\tau^{\prime\prime})^{0}=(T_{B}u)^{0}.$

Let $h_{B}$ be as in Step 2, and let us set $v:=T_{B}u$. Recalling
inequalities (5.18) and (5.20), since $v^{0}$ is constant in
$(0,\omega_{d})\setminus(\Lambda,\omega_{d}-\Lambda)$, without any additional
effort, we can deduce that for every family of disjoint intervals
$\\{(a_{i},b_{i})\\}_{i=1}^{N}$ in $(0,\omega_{d})$, it holds

$\int_{\cup_{i=1}^{N}(v^{0}(b_{i}),v^{0}(a_{i}))}h_{B}(\nu_{v}(t))\,dt\leq
2\int_{0}^{\sum_{i=1}^{N}(b_{i}-a_{i})}|\nabla u|^{*}\,dt.$

Moreover, since $v^{0}$ is absolutely continuous on $(0,\omega_{d})$, by a
change of variables we get

$\int_{\cup_{i=1}^{N}(a_{i},b_{i})}-\frac{dv^{0}}{dr}(r)h_{B}(r)\,dr\leq
2\int_{0}^{\sum_{i=1}^{N}(b_{i}-a_{i})}|\nabla u|^{*}\,dt,$

Observe that by approximation the inequality holds even if the union is
countable. Since every open subset of $(0,\omega_{d})$ is the union of
disjoint open intervals contained in $(0,\omega_{d})$, given
$\ell\in(0,\omega_{d})$, it holds

$\sup_{A\in\mathcal{A}((0,\omega_{d})),\ \mathcal{L}^{1}(A)=\ell\
}\int_{A}-\frac{dv^{0}}{dr}(r)h_{B}(r)\,dr\leq 2\int_{0}^{\ell}|\nabla
u|^{*}\,dt.$

In particular, taking $A=(0,\omega_{d})$ we deduce that
$-\frac{dv^{0}}{dr}(r)h_{B}(r)\in L^{1}((0,\omega_{d}))$. Thus, since every
Borel set $E$ can be approximated arbitrarily well in the sense of the
Lebesgue measure by open sets, by the absolute continuity of the integral we
conclude that for every $\ell\in(0,\omega_{d})$

$\sup_{E\in\mathcal{B}((0,\omega_{d})),\ \mathcal{L}^{1}(E)=\ell\
}\int_{E}-\frac{dv^{0}}{dr}(r)h_{B}(r)\,dr\leq 2\int_{0}^{\ell}|\nabla
u|^{*}\,dt.$

In particular, this implies that for every $\ell\in(0,\omega_{d})$

$\int_{0}^{\ell}\left(-\frac{dv^{0}}{dr}(r)h_{B}(r)\right)^{*}\,dr\leq\int_{0}^{\ell}|2\nabla
u|^{*}\,dt.$ (5.21)

Finally, since $\psi\in\Phi_{s}$ is convex and (5.21) holds for every
$\ell\in(0,\omega_{d})$, we can apply Lemma 5.5 to the functions
$-\frac{dv^{0}}{dr}h_{B}$ and $|2\nabla u|^{*}$ to infer

$\int_{0}^{\omega_{d}}\psi\left(-\frac{dv^{0}}{dr}(r)h_{B}(r)\right)\,dr\leq\int_{0}^{\omega_{d}}\psi(|2\nabla
u|^{*}(t))\,dt=\int_{0}^{\omega_{d}}(\psi(|2\nabla u|))^{*}(t)\,dt,$

where we have also used that $\psi$ is strictly increasing. Using the equi-
measurability of rearrangements and recalling the definition of $h_{B}$, we
get (5.10) with $c(d):=1/\gamma_{\textnormal{iso}}$. This concludes the proof.
∎

###  5.2. Proof of Theorem 5.1.

To prove Theorem 5.1 we need the following Hardy type inequality proven in
[25].

######  Lemma 5.6 ([25, Lemma 2.2]).

Let $\ell\in(0,+\infty)$. Let $\varphi\in\Phi_{w}$ be a finite valued Orlicz
function satisfying $\textnormal{(Inc)}_{1}$, set
$\phi:=\varphi^{\frac{d}{d-1}}t^{-\frac{1}{d-1}}$. Then, there exists a
constant $c_{1}:=c_{1}(d,\ell)>0$ such that for every measurable functions
$h\colon(0,\ell)\to[0,+\infty)$ it holds

$\phi^{-1}\left(\int_{0}^{\ell}\phi\left(\int_{s}^{\ell}h(t)\,dt\right)\,ds\right)\leq\varphi^{-1}\left(\int_{0}^{\ell}\varphi(c_{1}s^{\frac{d-1}{d}}h(s))\,ds\right).$

Using the definitions of $u^{0}$, $T_{B}u$ and $\textnormal{med}(u,B)$ we
have:

$\displaystyle\begin{split}&u^{0}\left(\frac{\omega_{d}}{2}\right)=\textnormal{med}(u,B)=:m(u),\\\
&((T_{B}u-m(u))_{+})^{*}(s)=((T_{B}u)^{0}(s)-m(u))_{+}\ \ \ \mbox{for a.e.
$s\in(0,\omega_{d})$},\\\
&((T_{B}u-m(u))_{-})^{*}(s)=((T_{B}u)^{0}(\omega_{d}-s)-m(u))_{-}=(m(u)-(T_{B}u)^{0}(\omega_{d}-s))_{+}\
\ \ \mbox{for a.e. $s\in(0,\omega_{d})$}.\end{split}$ (5.22)

Now we are in a position to prove Theorem 5.1. In this proof we proceed
similarly to the proof of [25, Theorem 1.2].

###### Proof of Theorem 5.1.

We begin by recalling that since $\varphi$ satisfies $\textnormal{(Inc)}_{1}$,

$\lambda\varphi(t)\leq\varphi(\lambda t)\ \ \mbox{and}\ \
\varphi^{-1}(s+t)\leq 2(\varphi^{-1}(s)+\varphi^{-1}(t))\ \ \ \mbox{for every
$\lambda\geq 1$, $s,t\geq 0$}.$ (5.23)

The same properties hold for $\phi$ by the very same definition of $\phi$.
Moreover, in virtue of Theorems 4.7 and 4.11, we have that there exists
$\psi\in\Phi_{s}$ such that

$\psi(t)\leq\varphi(t)\leq\psi(2t)\ \ \mbox{and}\ \ \psi^{-1}(t)\leq
2\varphi^{-1}(t)\leq 2\psi^{-1}(t)\ \ \ \mbox{for every $t\geq 0$}.$ (5.24)

We have the following chain of inequalities for the left hand side term in
(5.2)

$\displaystyle\phi^{-1}\left(\int_{B}\right.$
$\displaystyle\phi(|T_{B}u(x)-m(u)|)\left.\,\vphantom{\int}dx\right)=\phi^{-1}\left(\int_{B}\phi((T_{B}u(x)-m(u))_{+})\,dx+\int_{B}\phi((T_{B}u(x)-m(u))_{-})\,dx\right)$
$\displaystyle\leq
2\phi^{-1}\left(\int_{0}^{\omega_{d}/2}\phi((T_{B}u-m(u))^{*}_{+}(s))\,ds\right)+2\phi^{-1}\left(\int_{0}^{\omega_{d}/2}\phi((T_{B}u-m(u))^{*}_{-}(s))\,ds\right)$
$\displaystyle=2\phi^{-1}\left(\int_{0}^{\omega_{d}/2}\phi\left(\int_{s}^{\omega_{d}/2}-\frac{d(T_{B}u)^{0}}{dr}(r)\,dr\right)\,ds\right)$
(5.25) $\displaystyle\ \ \
+2\phi^{-1}\left(\int_{0}^{\omega_{d}/2}\phi\left(\int_{s}^{\omega_{d}/2}-\frac{d(T_{B}u)^{0}}{dr}(\omega_{d}-r)\,dr\right)\,ds\right),$

where in the first inequality we have used the equi-measurability of the
rearrangement, the fact that $\phi$ is strictly increasing and (5.23) for
$\phi$, in the last equality we have used (5.22).

Now we estimate the right hand side term. Let $c=c(d)$ be the constant from
Proposition 5.3 and $c_{1}=c_{1}(d,\omega_{d}/2)=c_{1}(d)$ be the constant
from Lemma 5.6. We have

$\displaystyle\begin{split}\varphi^{-1}&\left(\int_{B}\varphi\left(4\frac{c_{1}}{c}|\nabla
u(x)|\right)\,dx\right)\geq\frac{1}{2}\psi^{-1}\left(\int_{B}\psi\left(4\frac{c_{1}}{c}|\nabla
u(x)|\right)\,dx\right)\\\
&\geq\frac{1}{2}\psi^{-1}\left(\int_{0}^{\omega_{d}}\psi\left(2c_{1}\min\\{s,1-s\\}^{\frac{d-1}{d}}\left(-\frac{d(T_{B}u)^{0}}{ds}(s)\right)\right)\,ds\right)\\\
&\geq\frac{1}{4}\psi^{-1}\left(\int_{0}^{\omega_{d}/2}\psi\left(2c_{1}s^{\frac{d-1}{d}}\left(-\frac{d(T_{B}u)^{0}}{ds}(s)\right)\right)\,ds\right)\\\
&\ \ \
+\frac{1}{4}\psi^{-1}\left(\int_{0}^{\omega_{d}/2}\psi\left(2c_{1}s^{\frac{d-1}{d}}\left(-\frac{d(T_{B}u)^{0}}{ds}(\omega_{d}-s)\right)\right)\,ds\right)\\\
&\geq\frac{1}{4}\varphi^{-1}\left(\int_{0}^{\omega_{d}/2}\varphi\left(c_{1}s^{\frac{d-1}{d}}\left(-\frac{d(T_{B}u)^{0}}{ds}(s)\right)\right)\,ds\right)\\\
&\ \ \
+\frac{1}{4}\varphi^{-1}\left(\int_{0}^{\omega_{d}/2}\varphi\left(c_{1}s^{\frac{d-1}{d}}\left(-\frac{d(T_{B}u)^{0}}{ds}(\omega_{d}-s)\right)\right)\,ds\right),\end{split}$
(5.26)

where we have used in this order: (5.24), Proposition 5.3, the fact that
$\psi^{-1}$ is increasing and, for the last inequality, we used (5.24)
together with (5.23).

Finally, combining (5.25) and (5.26) and using Lemma 5.6, we infer that

$8\varphi^{-1}\left(\int_{B}\varphi\left(\frac{4c_{1}}{c}|\nabla
u(x)|\right)\,dx\right)\geq\phi^{-1}\left(\int_{B}\phi(|T_{B}u(x)-m(u)|)\,dx\right),$

which is (5.2) up to taking $C=C(d)$ larger in order to obtain an averaged
integral in the expression above. In view of Remark 5.2 we thus conclude. ∎

We will also make use of the following weaker version of the Poincaré
inequality in (5.2) for functions which are also doubling.

######  Theorem 5.7.

Let $\varphi\in\Phi_{w}$ be an Orlicz function satisfying
$\textnormal{(Inc)}_{1}$ and (aDec). Then, there exists $C=C(d,K)>0$, where
$K$ is the doubling constant of $\varphi$, such that for every ball
$B_{r}(x)\subset\mathbb{R}^{d}$ and every $u\in SBV(B_{r}(x))$ satisfying
(5.1), the following inequality holds

$\int_{B_{r}(x)}\varphi\left(\frac{|T_{B_{r}(x)}u(y)-\textnormal{med}(u,B_{r}(x))|}{r}\right)\,dy\leq
C\int_{B_{r}(x)}\varphi(|\nabla u(y)|)\,dy.$ (5.27)

In order to prove Theorem 5.7 we need the following elementary Lemma.

######  Lemma 5.8.

Let $\varphi,\phi\in\Phi_{s}$ be two finite valued Orlicz functions such that

$\lim_{t\to 0^{+}}\phi(\varphi^{-1}(t))=0,\ \ \
t\mapsto\frac{\phi(t)}{\varphi(t)}\ \ \mbox{is increasing}.$ (5.28)

Then, for every ball $B_{r}\subset\mathbb{R}^{d}$ and every measurable
function $u\in L^{1}(B_{r})$ we have that

$\varphi^{-1}\left(\fint_{B_{r}}\varphi(|u(x)|)\,dx\right)\leq\phi^{-1}\left(\fint_{B_{r}}\phi(2|u(x)|)\,dx\right)$
(5.29)

###### Proof.

Let us set $\psi:=\phi(\varphi^{-1})$. We claim that $\psi\in\Phi_{w}$
satisfies $\textnormal{(Inc)}_{1}$. Indeed, since $\varphi$ is continuous,
convex and finite valued, then it is a bijection, thus, $\varphi^{-1}$ is its
inverse which is continuous. This gives that $\psi$ is continuous and finite
valued on $(0,+\infty)$. Moreover, by the assumptions in (5.28) and definition
of $\psi$ we have that $\psi(0)=0=\lim_{t\to 0^{+}}\psi(t)$ and
$t\mapsto\psi(t)/t$ is increasing. Therefore, by Theorem 4.7, there exists
$\tilde{\psi}\in\Phi_{s}$ such that

$\tilde{\psi}(t)\leq\psi(t)\leq\tilde{\psi}(2t)\ \ \ t\geq 0.$

Since $\tilde{\psi}$ is convex, by Jensen inequality we get for every $v\in
L^{1}(B_{r})$

$\psi\left(\fint_{B_{r}}|v(x)|\,dx\right)\leq\tilde{\psi}\left(2\fint_{B_{r}}|v(x)|\,dx\right)\leq\fint_{B_{r}}\tilde{\psi}(2|v(x)|)\,dx\leq\fint_{B_{r}}\psi(2|v(x)|)\,dx.$

By substituting back the expression of $\psi$ with $v(x)=\varphi(|u(x)|)$ and
recalling that $\varphi^{-1}(2t)\leq 2\varphi^{-1}(t)$ we get (5.29). ∎

###### Proof of Theorem 5.7.

By translation invariance it is not restrictive to consider $x=0$. Using
Proposition 4.7 we can find $\tilde{\varphi}\in\Phi_{s}$ such that
$\tilde{\varphi}(t)\leq\varphi(t)\leq\tilde{\varphi}(2t)$. Setting
$\tilde{\phi}:=\tilde{\varphi}^{\frac{d}{d-1}}t^{-\frac{1}{d-1}}$ we have that
$\tilde{\varphi}$ and $\tilde{\phi}$ satisfy all the assumptions of Lemma 5.8.
Using Lemma 5.8 we thus infer

$\tilde{\varphi}^{-1}\left(\fint_{B_{r}}\tilde{\varphi}\left(\frac{|T_{B_{r}}u(x)-\textnormal{med}(u,B_{r})|}{r}\right)\,dx\right)\leq\tilde{\phi}^{-1}\left(\fint_{B_{r}}\tilde{\phi}\left(2\frac{|T_{B_{r}}u(x)-\textnormal{med}(u,B_{r})|}{r}\right)\,dx\right);$

which gives

$\varphi^{-1}\left(\fint_{B_{r}}\varphi\left(\frac{|T_{B_{r}}u(x)-\textnormal{med}(u,B_{r})|}{r}\right)\,dx\right)\leq
2\phi^{-1}\left(\fint_{B_{r}}\phi\left(8\frac{|T_{B_{r}}u(x)-\textnormal{med}(u,B_{r})|}{r}\right)\,dx\right).$

By Theorem 5.1 this implies that for every $u\in SBV(B_{r})$ satisfying (5.1)
then

$\varphi^{-1}\left(\fint_{B_{r}}\varphi\left(\frac{|T_{B_{r}}u(x)-\textnormal{med}(u,B_{r})|}{r}\right)\,dx\right)\leq
C\varphi^{-1}\left(\fint_{B_{r}}\varphi(C|\nabla u(x)|)\,dx\right)$

with $C=C(d)$. Using the fact that $\varphi$ is doubling and $\varphi^{-1}$ is
increasing, we get (5.7) with $C=C(d,K)$ where $K\geq 2$ is the doubling
constant of $\varphi$. ∎

###  5.3. Poincaré inequality for generalized Orlicz spaces

We now extend the Poincaré inequality for $SBV(B_{r}(x))$ functions with small
jump set to the cases where $\varphi\in\Phi_{s}(B_{r}(x))$ is a generalized
Orlicz function satisfying (A0), (adA1), $\textnormal{(Inc)}_{1}$ and (Dec) on
$B_{r}(x)$. Notice that the statement below requires the introduction of a
further truncation, depending on $\varphi$ and on the ball $B_{r}(x)$. In
order to ease the notation, we only consider the case $x=0$.

######  Theorem 5.9.

Let $\varphi\in\Phi_{s}(B_{r})$ be a generalized Orlicz function satisfying
(A0), (adA1), $\textnormal{(Inc)}_{1}$ and (Dec) on $B_{r}$. Let
$\phi(x,t):=\varphi(x,t)^{\frac{d}{d-1}}t^{-\frac{1}{d-1}}\in\Phi_{s}(B_{r})$.
Moreover, let $K$ be the doubling constant of $\varphi$, $\sigma$ be the
constant from (A0) and $\beta$ be the constant appearing in (adA1). For $u\in
SBV^{\varphi}(B_{r})$, set

$\mathfrak{u}^{\varphi}_{r}(x):=\left(\textnormal{med}(u,B_{r})-r(\varphi^{-}_{B_{r}})^{-1}\left(\frac{1}{2r}\right)\right)\vee
T_{B_{r}}u(x)\wedge\left(\textnormal{med}(u,B_{r})+r(\varphi_{B_{r}}^{-})^{-1}\left(\frac{1}{2r}\right)\right).$
(5.30)

Then, there exists $C=C(d,K,\beta)$ such that for every $u\in
SBV^{\varphi}(B_{r})$ satisfying (5.1), the following inequalities hold

$\displaystyle(\phi^{-}_{B_{r}})^{-1}\left(\fint_{B_{r}}\phi_{B_{r}}^{+}\left(\left|\frac{\mathfrak{u}^{\varphi}_{r}-\textnormal{med}(u,B_{r})}{r}\right|\right)\,dx\right)\leq
C(\varphi^{-}_{B_{r}})^{-1}\left(\fint_{B_{r}}\varphi_{B_{r}}^{-}(|\nabla
u|)\,dx\right)+C\sigma,$ (5.31)
$\displaystyle\int_{B_{r}}\varphi_{B_{r}}^{+}\left(\left|\frac{\mathfrak{u}^{\varphi}_{r}-\textnormal{med}(u,B_{r})}{r}\right|\right)\,dx\leq
C\int_{B_{r}}\left(\varphi_{B_{r}}^{-}(|\nabla
u|)+\varphi^{+}_{B_{r}}\left(\left|\frac{u-\textnormal{med}(u,B_{r})}{r}\right|\wedge\sigma\right)\right)\,dx.$
(5.32)

###### Proof.

By assumption we have that $\varphi^{-}_{B_{r}}$ satisfies
$\textnormal{(Inc)}_{1}$. Therefore inequality (5.2) hold with $\varphi$
replaced by $\varphi^{-}_{B_{r}}$ and $\phi$ replaced by $\phi^{-}_{B_{r}}$.
By definition of $\mathfrak{u}^{\varphi}_{r}$, we have that

$\frac{|\mathfrak{u}^{\varphi}_{r}(x)-\textnormal{med}(u,B_{r})|}{r}\leq(\varphi^{-}_{B_{r}})^{-1}\left(\frac{1}{2r}\right).$
(5.33)

Hence, observing that $\phi_{B_{r}}^{-}$ is increasing and

$\phi_{B_{r}}^{+}(\beta t)=(\varphi^{+}_{B_{r}}(\beta
t))^{\frac{d}{d-1}}(\beta
t)^{-\frac{1}{d-1}}\leq\frac{1}{\beta}(\varphi^{-}_{B}(t))^{\frac{d}{d-1}}t^{-\frac{1}{d-1}}=\frac{1}{\beta}\phi_{B_{r}}^{-}(t)\
\ \ \mbox{for
}t\in\left(\sigma,(\varphi_{B_{r}}^{-})^{-1}\left(\frac{1}{2r}\right)\right),$

inequality (5.31) is a consequence of (adA1).

Analogously, we have that inequality (5.32) is a consequence of the fact that
(5.27) holds with $\varphi$ replaced by $\varphi_{B_{r}}^{-}$, (5.33) and
(adA1). ∎

##  6\. Integral Representation

This section is devoted to the proof of Theorem 3.1 and its Corollary 3.2.
Throughout this section, $\psi$ stands for a function in $\Phi_{s}(\Omega)$
satisfying (A0), (adA1), (Inc) and (Dec) on $\Omega$. Notice that the
additional requirements that $\psi\in\Phi_{s}(\Omega)$ instead of
$\Phi_{w}(\Omega)$ and that (Inc) and (Dec) replace (aInc) and (aDec),
respectively, cause no restriction, as we discussed in Remark 4.31.

###  6.1. Fundamental estimate in $GSBV^{\psi}$

We begin by proving an important tool for integral representation, that is the
fundamental estimate.

######  Lemma 6.1 (Fundamental estimate in $GSBV^{\psi}$).

Let $\eta>0$ and $B\Subset\Omega$ a ball, let
$D^{\prime},D^{\prime\prime},E\in\mathcal{A}(B)$ with $D^{\prime}\Subset
D^{\prime\prime}$. For every functional $\mathcal{F}$ satisfying (H1), (H3)
and (H4) with $\psi$, for every $u\in
GSBV^{\psi}(D^{\prime\prime},\mathbb{R}^{m})$ and every $v\in
GSBV^{\psi}(E,\mathbb{R}^{m})$ there exists a function $\varphi\in
C^{\infty}(\mathbb{R}^{d},[0,1])$ such that the function $w:=\varphi
u+(1-\varphi)v\in GSBV^{\psi}(E\cup D^{\prime},\mathbb{R}^{m})$ satisfies

$\displaystyle\begin{split}&\textnormal{(i)}\mbox{ $w=u$ on $D^{\prime}$ and
$w=v$ on $E\setminus D^{\prime\prime}$};\\\ &\textnormal{(ii)}\mbox{
$\mathcal{F}(w,D^{\prime}\cup
E)\leq(1+\eta)(\mathcal{F}(u,D^{\prime\prime})+\mathcal{F}(v,E))+M\displaystyle\int_{F}\psi\left(x,\displaystyle\frac{|u-v|}{\delta}\right)\,dx+\eta\mathcal{L}^{d}(D^{\prime}\cup
E)$,}\end{split}$

with $\delta:=\frac{1}{2}\textnormal{dist}(D^{\prime},\partial
D^{\prime\prime})$, where $F:=(D^{\prime\prime}\setminus D^{\prime})\cap E$
and $M=M(D^{\prime},D^{\prime\prime},E,\psi,\eta)>0$ is independent of $u$ and
$v$. Moreover, given $\varepsilon>0$ and $x_{0}\in\mathbb{R}^{d}$ such that
$D_{\varepsilon,x_{0}}^{\prime},D_{\varepsilon,x_{0}}^{\prime\prime},E_{\varepsilon,x_{0}}\Subset\Omega$,
then

$M(D_{\varepsilon,x_{0}}^{\prime},D_{\varepsilon,x_{0}}^{\prime\prime},E_{\varepsilon,x_{0}},\psi,\eta)=M(D^{\prime},D^{\prime\prime},E,\psi,\eta)$

and the integral term in (ii) becomes:
$M\displaystyle\int_{F_{\varepsilon,x_{0}}}\psi\left(x,\displaystyle\frac{|u-v|}{\varepsilon\delta}\right)\,dx$.

###### Proof.

Let $K$ be the doubling constant of $\psi$, take $k\in\mathbb{N}$ such that
$k\geq\max\left\\{\frac{b(K+1)^{2}}{a\eta},\frac{b}{\eta}\right\\}$. We set
$D_{1}:=D^{\prime}$ and, for $i=1,\dots,k$,

$D_{i+1}:=\left\\{x\in
D^{\prime\prime}\colon\textnormal{dist}(x,D^{\prime})<\frac{i\delta}{k}\right\\}.$

We thus have $D^{\prime}=:D_{1}\Subset D_{2}\Subset\dots\Subset D_{k+1}\Subset
D^{\prime\prime}$. Let $\varphi_{i}\in C^{\infty}_{0}(D_{i+1},[0,1])$ with
$\varphi_{i}=1$ in a neighborhood $U_{i}$ of $\overline{D_{i}}$ and such that
$\|\nabla\varphi_{i}\|_{L^{\infty}}\leq\frac{2k}{\delta}$.

Let $u\in GSBV^{\psi}(D^{\prime\prime},\mathbb{R}^{m})$ and $v\in
GSBV^{\psi}(E,\mathbb{R}^{m})$ such that $u-v\in L^{\psi}(F,\mathbb{R}^{m})$,
otherwise the thesis is trivial. We define the function
$w_{i}:=\varphi_{i}u+(1-\varphi_{i})v$. We notice that $w_{i}\in
GSBV^{\psi}(D^{\prime}\cup E,\mathbb{R}^{m})$, this follows from properties
(A0) and (Dec) of $\psi$, with $u$ and $v$ extended arbitrarily outside
$D^{\prime\prime}$ and $E$ respectively. Set
$I_{i}:=E\cap(D_{i+1}\setminus\overline{D_{i}})$. Using (H1) and (H3) we infer

$\displaystyle\begin{split}\mathcal{F}(w_{i},D^{\prime}\cup
E)&\leq\mathcal{F}(u,(D^{\prime}\cup E)\cap
U_{i})+\mathcal{F}(v,E\setminus\textnormal{Supp}\varphi_{i})+\mathcal{F}(w_{i},I_{i})\\\
&\leq\mathcal{F}(u,D^{\prime\prime})+\mathcal{F}(v,E)+\mathcal{F}(w_{i},I_{i}).\end{split}$
(6.1)

We now estimate the last term of (6.1) using assumption (H4) and properties
(A0), (Dec) (i.e. doubling) of $\psi$ and the fact that $\psi$ is convex since
it belongs to the class $\Phi_{s}$. We have

$\displaystyle\mathcal{F}(w_{i},I_{i})$ $\displaystyle\leq
b\mathcal{L}^{d}(I_{i})+b(K+1)\int_{I_{i}}\\!\psi(x,|\nabla u+\nabla
v|)dx+b(K+1)\int_{I_{i}}\psi(x,|(u-v)\otimes\nabla\varphi|)dx$ $\displaystyle\
\ \ \ +b\mathcal{H}^{d-1}(J_{u}\cap I_{i})+b\mathcal{H}^{d-1}(J_{v}\cap
I_{i})$ $\displaystyle\leq
b\mathcal{L}^{d}(I_{i})+b(K+1)^{2}\int_{I_{i}}\psi(x,|\nabla
u|)\,dx+b(K+1)^{2}\int_{I_{i}}\psi(x,|\nabla v|)\,dx$ $\displaystyle\ \ \ \
+b(K+1)K^{\log_{2}(k)+2}\int_{I_{i}}\psi\left(x,\frac{|u-v|}{\delta}\right)dx+b\mathcal{H}^{d-1}(J_{u}\cap
I_{i})+b\mathcal{H}^{d-1}(J_{v}\cap I_{i})$
$\displaystyle\leq\frac{b}{a}(K+1)^{2}(\mathcal{F}(u,I_{i})+\mathcal{F}(v,I_{i}))+b(K+1)K^{\log_{2}(k)+2}\int_{I_{i}}\psi\left(x,\frac{|u-v|}{\delta}\right)dx+b\mathcal{L}^{d}(I_{i}).$

By our initial choice of $k$ and using (H1) we can find
$i_{0}\in\\{1,\dots,k\\}$ such that

$\mathcal{F}(w_{i_{0}},I_{i_{0}})\leq\frac{1}{k}\sum_{i=1}^{k}\mathcal{F}(w_{i},I_{i})\leq\eta(\mathcal{F}(u,D^{\prime\prime})+\mathcal{F}(v,E))+\eta\mathcal{L}^{d}(D^{\prime}\cup
E)+M\int_{F}\psi\left(x,\displaystyle\frac{|u-v|}{\delta}\right)dx$

with $M:=b(K+1)K^{\log_{2}(k)+2}k^{-1}$. This combined with (6.1) concludes
the proof by setting $w:=w_{i_{0}}$. For the second statement it is enough to
consider cut-off functions $\varphi_{i}^{\varepsilon}\in
C^{\infty}_{0}((D_{i+1})_{\varepsilon,x_{0}},[0,1])$ defined as
$\varphi_{i}^{\varepsilon}(x):=\varphi_{i}(x_{0}+(x-x_{0})/\varepsilon)$, for
$i=1,\dots,k$ and proceed as before. ∎

###  6.2. Proof of the integral representation theorem

Now we proceed with the proof of Theorem 3.1. We begin by showing that the
Radon-Nikodym derivatives of $\mathcal{F}$ and $\textbf{m}_{\mathcal{F}}$ with
respect to the measure

$\mu:=\mathcal{L}^{d}|_{\Omega}+\mathcal{H}^{d-1}|_{J_{u}\cap\Omega}$ (6.2)

coincide. To this aim, we state the following Lemma.

######  Lemma 6.2.

Let $\mathcal{F}$ satisfy (H1)–(H4), let $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ and let $\mu$ be as in (6.2). Then, for
$\mu$-a.e. $x_{0}\in\Omega$ it holds

$\lim_{\varepsilon\to
0^{+}}\frac{\mathcal{F}(u,B_{\varepsilon}(x_{0}))}{\mu(B_{\varepsilon}(x_{0}))}=\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\mu(B_{\varepsilon}(x_{0}))}.$

We postpone the proof of this lemma at the end of this section. As a second
step we prove that asymptotically as $\varepsilon\to 0^{+}$ the two
minimization problems $\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))$ and
$\mathbf{m}_{\mathcal{F}}(\overline{u}^{\textnormal{bulk}}_{x_{0}},B_{\varepsilon}(x_{0}))$
coincide for $\mathcal{L}^{d}$-a.e. $x_{0}\in\Omega$, where
$\overline{u}^{\textnormal{bulk}}_{x_{0}}:=\ell_{x_{0},u(x_{0}),\nabla
u(x_{0})}$.

######  Lemma 6.3.

Let $\mathcal{F}$ satisfy (H1) and (H3)-(H4) and let $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$. Then, for $\mathcal{L}^{d}$-a.e.
$x_{0}\in\Omega$ we have

$\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}=\limsup_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(\overline{u}^{\textnormal{bulk}}_{x_{0}},B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}.$
(6.3)

We defer the proof of this lemma to Section 6.3. As a third and final step we
prove that asymptotically as $\varepsilon\to 0^{+}$ the two minimization
problems $\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))$ and
$\mathbf{m}_{\mathcal{F}}(\overline{u}^{\textnormal{surf}}_{x_{0}},B_{\varepsilon}(x_{0}))$
coincide for $\mathcal{H}^{d-1}$-a.e. $x_{0}\in J_{u}$, where
$\overline{u}^{\textnormal{surf}}_{x_{0}}:=u_{x_{0},u^{+}(x_{0}),u^{-}(x_{0}),\nu_{u}(x_{0})}$.

######  Lemma 6.4.

Let $\mathcal{F}$ satisfy (H1) and (H3)-(H4) and let $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$. Then for $\mathcal{L}^{d}$-a.e.
$x_{0}\in\Omega$ we have

$\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\omega_{d-1}\varepsilon^{d-1}}=\limsup_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(\overline{u}^{\textnormal{surf}}_{x_{0}},B_{\varepsilon}(x_{0}))}{\omega_{d-1}\varepsilon^{d-1}}$
(6.4)

We defer the proof of this lemma to Section 6.4. We are now ready to prove
Theorem 3.1 and Corollary 3.2.

###### Proof of Theorem 3.1.

In view of the Besicovitch derivation theorem, we need to prove that

$\displaystyle\frac{\textnormal{d}\mathcal{F}(u,\cdot)}{\textnormal{d}\mathcal{L}^{d}}(x_{0})=f(x_{0},u(x_{0}),\nabla
u(x_{0})),\ \ \ \mbox{for $\mathcal{L}^{d}$-a.e. $x_{0}\in\Omega$},$ (6.5)
$\displaystyle\frac{\textnormal{d}\mathcal{F}(u,\cdot)}{\textnormal{d}\mathcal{H}^{d-1}|_{J_{u}}}(x_{0})=g(x_{0},u^{+}(x_{0}),u^{-}(x_{0}),\nu_{u}(x_{0})),\
\ \ \mbox{for $\mathcal{H}^{d-1}$-a.e. $x_{0}\in J_{u}$},$ (6.6)

where $f$ and $g$ are defined in (3.2) and (3.3), respectively.

Using Lemma 6.2 and that $\lim_{\varepsilon\to
0^{+}}(\omega_{d}\varepsilon^{d})^{-1}\mu(B_{\varepsilon}(x_{0}))=1$ for
$\mathcal{L}^{d}$-a.e. $x_{0}\in\Omega$, we infer that for
$\mathcal{L}^{d}$-a.e. $x_{0}\in\Omega$

$\frac{\textnormal{d}\mathcal{F}(u,\cdot)}{\textnormal{d}\mathcal{L}^{d}}(x_{0})=\lim_{\varepsilon\to
0^{+}}\frac{\mathcal{F}(u,B_{\varepsilon}(x_{0}))}{\mu(B_{\varepsilon}(x_{0}))}=\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\mu(B_{\varepsilon}(x_{0}))}=\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}<+\infty.$

Then, (6.5) follows by definition of $f$ ((3.2)) and Lemma 6.3.

By Lemma 6.2 and the fact that $\lim_{\varepsilon\to
0^{+}}(\omega_{d-1}\varepsilon^{d-1})^{-1}\mu(B_{\varepsilon}(x_{0}))=1$ for
$\mathcal{H}^{d-1}$-a.e. $x_{0}\in J_{u}$ we infer that for
$\mathcal{H}^{d-1}$-a.e. $x_{0}\in J_{u}$

$\frac{\textnormal{d}\mathcal{F}(u,\cdot)}{\textnormal{d}\mathcal{H}^{d-1}|_{J_{u}}}(x_{0})=\lim_{\varepsilon\to
0^{+}}\frac{\mathcal{F}(u,B_{\varepsilon}(x_{0}))}{\mu(B_{\varepsilon}(x_{0}))}=\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\mu(B_{\varepsilon}(x_{0}))}=\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\omega_{d-1}\varepsilon^{d-1}}<+\infty,$

and (6.6) follows by definition of $g$ ((3.3)) and Lemma 6.4. ∎

###### Proof of Corollary 3.2.

In view of Theorem 3.1 we have that for all $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ and all $A\in\mathcal{A}(\Omega)$

$\mathcal{F}(u,A)=\int_{A}\tilde{f}(x,u(x),\nabla u(x))\,dx+\int_{J_{u}\cap
A}\tilde{g}(x,u^{+}(x),u^{-}(x),\nu_{u}(x))\,d\mathcal{H}^{d-1},$

where $\tilde{f}$ and $\tilde{g}$ are defined in (3.2) and (3.3),
respectively. Recalling the definition of $\mathbf{m}_{\mathcal{F}}$, since
$\mathcal{F}$ satisfies assumption (H5), by definition of $\tilde{f}$ and
$\tilde{g}$ we deduce that for every $c\in\mathbb{R}^{m}$

$\displaystyle\tilde{f}(x,u(x)+c,\nabla u(x))=\tilde{f}(x,0,\nabla u(x)),\ \ \
\ \
\tilde{g}(x,u^{+}(x)+c,u^{-}(x)+c,\nu_{u}(x))=\tilde{g}(x,[u](x),0,\nu_{u}(x)).$

Hence, $\tilde{f}(x,u,\nabla u)=:f(x,\nabla u)$ and
$\tilde{g}(x,u^{+}(x),u^{-}(x),\nu_{u}(x))=:g(x,[u](x),\nu_{u}(x))$. ∎

In the remaining part of the section we prove Lemma 6.2 following the lines of
[17]. We start by fixing some notations. Given $\delta>0$ and
$A\in\mathcal{A}(\Omega)$ we define

$\displaystyle\mathbf{m}_{\mathcal{F}}^{\delta}(u,A):=\inf\left\\{\sum_{i=1}^{\infty}\mathbf{m}_{\mathcal{F}}(u,B_{i})\colon
B_{i}\subset A\ \mbox{pairwise disjoint balls},\
\textnormal{diam}(B_{i})<\delta,\
\mu\left(A\setminus\cup_{i=1}^{\infty}B_{i}\right)=0\right\\}$ (6.7)

where $\mu$ is the measure introduced in (6.2). Since
$\mathbf{m}_{\mathcal{F}}^{\delta}(u,A)$ is decreasing in $\delta$ we can also
introduce

$\mathbf{m}_{\mathcal{F}}^{*}(u,A):=\lim_{\delta\to
0^{+}}\mathbf{m}_{\mathcal{F}}^{\delta}(u,A).$ (6.8)

In the next lemma we prove that under assumptions (H1)–(H4) $\mathcal{F}$ and
$\mathbf{m}_{\mathcal{F}}^{*}$ coincide.

######  Lemma 6.5.

Let $\mathcal{F}$ satisfy (H1)–(H4) and let $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$. Then, for all $A\in\mathcal{A}(\Omega)$
we have $\mathcal{F}(u,A)=\mathbf{m}_{\mathcal{F}}^{*}(u,A)$.

###### Proof.

We mainly follow the lines of [19, Lemma 3.3]. We start by proving the
inequality $\mathcal{F}(u,A)\geq\mathbf{m}_{\mathcal{F}}^{*}(u,A)$. For every
ball $B\subset A$ we have that
$\mathbf{m}_{\mathcal{F}}(u,B)\leq\mathcal{F}(u,B)$ by definition. Using (H1)
we infer $\mathcal{F}(u,A)\geq\mathbf{m}_{\mathcal{F}}^{\delta}(u,A)$ for all
$\delta>0$. Thus the desired inequality follows by definition of
$\mathbf{m}_{\mathcal{F}}^{*}$ in (6.8).

We now prove the reverse inequality. Fix $\delta>0$ and
$A\in\mathcal{A}(\Omega)$ and let $\\{B_{i}^{\delta}\\}_{i}$ be a family of
balls as in (6.7) such that

$\sum_{i=1}^{\infty}\mathbf{m}_{\mathcal{F}}(u,B^{\delta}_{i})\leq\mathbf{m}^{\delta}_{\mathcal{F}}(u,A)+\delta.$
(6.9)

By definition of $\mathbf{m}_{\mathcal{F}}$ we can find $v_{i}^{\delta}\in
GSBV^{\psi}(B_{i}^{\delta},\mathbb{R}^{m})$ such that $v_{i}^{\delta}=u$ in a
neighborhood of $\partial B_{i}^{\delta}$ and

$\mathcal{F}(v_{i}^{\delta},B_{i}^{\delta})\leq\mathbf{m}_{\mathcal{F}}(u,B_{i}^{\delta})+\delta\mathcal{L}^{d}(B_{i}^{\delta}).$
(6.10)

We define

$v^{\delta,n}:=\sum_{i=1}^{n}v_{i}^{\delta}\chi_{B_{i}^{\delta}}+u\chi_{N_{0}^{\delta,n}}\
\ n\in\mathbb{N},\ \ \ \
v^{\delta}:=\sum_{i=1}^{\infty}v_{i}^{\delta}\chi_{B_{i}^{\delta}}+u\chi_{N_{0}^{\delta}},$
(6.11)

where $N_{0}^{\delta,n}:=\Omega\setminus\cup_{i=1}^{n}B_{i}^{\delta}$ and
$N_{0}^{\delta}:=\Omega\setminus\cup_{i=1}^{\infty}B_{i}^{\delta}$. By
construction, we have that $v^{\delta,n}\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$ for every $n\in\mathbb{N}$ and, using
(6.9), (6.10) and (H4),

$\sup_{n\in N}\left(\int_{\Omega}\psi\left(x,|\nabla
v^{\delta,n}(x)|\right)\,dx+\mathcal{H}^{d-1}(J_{v^{\delta,n}})\right)<+\infty.$
(6.12)

Moreover, we have that $v^{\delta,n}\to v^{\delta}$ pointwise
$\mathcal{L}^{d}$-a.e. and in measure on $\Omega$ by construction. Recalling
that $\psi$ satisfies $\textnormal{(Inc)}_{\gamma}$ for some $\gamma>1$, using
[7, Theorem 2.2] (see also [6] and [9]) together with the compactness in
$L^{0}$ of $\\{v^{\delta,n}\\}_{n}$ gives that $v^{\delta}\in
GSBV^{\gamma}(\Omega,\mathbb{R}^{m})$ and $\nabla
v^{\delta,n}\rightharpoonup\nabla v^{\delta}$ weakly in $L^{\gamma}$. Now
using Ioffe’s theorem and (6.12), we infer that

$\int_{\Omega}\psi\left(x,|\nabla
v^{\delta}(x)|\right)\,dx\leq\liminf_{n\to+\infty}\int_{\Omega}\psi\left(x,|\nabla
v^{\delta,n}(x)|\right)\,dx<+\infty.$

Therefore, $v^{\delta}\in GSBV^{\psi}(\Omega,\mathbb{R}^{m})$. Observe that it
holds

$\displaystyle\begin{split}\mathcal{F}(v^{\delta},A)&=\sum_{i=1}^{\infty}\mathcal{F}(v^{\delta}_{i},B^{\delta}_{i})+\mathcal{F}(u,A\cap
N_{0}^{\delta})\leq\sum_{i=1}^{\infty}(\mathbf{m}_{\mathcal{F}}(u,B_{i}^{\delta})+\delta\mathcal{L}^{d}(B_{i}^{\delta}))\\\
&\leq\mathbf{m}_{\mathcal{F}}^{\delta}(u,A)+\delta(1+\mathcal{L}^{d}(A)),\end{split}$
(6.13)

where we have used that $\mu(A\cap N_{0}^{\delta})=\mathcal{F}(u,A\cap
N_{0}^{\delta})=0$ by definition of $\\{B_{i}^{\delta}\\}_{i}$ and (H4).
Thanks to $\textnormal{(Inc)}_{\gamma}$ we have
$\psi^{+}_{\Omega}(1)t^{\gamma}\leq\psi(x,t)$ for $\mathcal{L}^{d}$-a.e.
$x\in\Omega$ and every $t\geq 1$. Therefore, (6.13) together with (H4) implies
that

$\int_{A}|\nabla v^{\delta}|^{\gamma}\,dx+\mathcal{H}^{d-1}(J_{v^{\delta}}\cap
A)\leq\frac{\psi_{\Omega}^{+}(1)}{a}\left(\mathbf{m}_{\mathcal{F}}^{\delta}(u,A)+\delta(1+\mathcal{L}^{d}(A))\right)+\mathcal{L}^{d}(A).$
(6.14)

We now claim that

$w^{\delta}:=u-v^{\delta}\to 0\ \ \mbox{in measure on $A$}.$ (6.15)

Notice that if (6.15) holds, then (H2), (6.8) and (6.13) imply that
$\mathbf{m}^{*}_{\mathcal{F}}(u,A)\geq\mathcal{F}(u,A)$ in the limit
$\delta\to 0^{+}$.

In order to prove (6.15), we first notice that
$w^{\delta}|_{B_{i}^{\delta}}\in GSBV^{\gamma}(B_{i}^{\delta},\mathbb{R}^{m})$
has zero trace on $\partial B_{i}^{\delta}$. Setting $w^{\delta,M}:=-M\vee
w^{\delta}\wedge M$ with $M>0$, by the homogeneous Poincaré inequality in $BV$
we have

$\|w^{\delta,M}\|_{L^{1}(B_{i}^{\delta})}\leq
C\delta|Dw^{\delta,M}|(B_{i}^{\delta}).$

Therefore, by definition of $\\{B_{i}^{\delta}\\}_{i}$ we deduce that

$\|w^{\delta,M}\|_{L^{1}(A)}\leq
C\delta|Dw^{\delta,M}|(\cup_{i=1}^{\infty}B_{i}^{\delta})\leq
C\delta|Dw^{\delta,M}|(A).$

The quantity $|Dw^{\delta,M}|(A)$ is bounded in view of (6.14) since $u\in
GSBV(A,\mathbb{R}^{m})$. Indeed,

$|Dw^{\delta,M}|(A)\leq\int_{A}|\nabla v^{\delta}|\,dx+\int_{A}|\nabla
u|\,dx+2M\left(\mathcal{H}^{d-1}(J_{u}\cap
A)+\mathcal{H}^{d-1}(J_{v^{\delta}}\cap A)\right)<+\infty.$

This implies $w^{\delta,M}\to 0$ in $L^{1}(A,\mathbb{R}^{m})$ as $\delta\to
0^{+}$ and thus in measure on $A$ for every $M>0$. Observe that if we take
$M=1$ then we have that for every $\varepsilon\in(0,1)$

$\\{x\in A\colon|w^{\delta}(x)|>\varepsilon\\}\subseteq\\{x\in
A\colon|w^{\delta,1}(x)|>\varepsilon\\},$

and this gives (6.15). ∎

We finally prove Lemma 6.2.

###### Proof of Lemma 6.2.

The proof follows the same arguments of [17, Lemma 5 and Lemma 6]. It
essentially relies on Lemma 6.5 and on the assumptions on $\mathcal{F}$ but it
is not hinged on the growth conditions. Hence we omit it. ∎

It remains to prove Lemmas 6.3 and 6.4. This is the subject of the following
two sections.

###  6.3. Lebesgue points

This section is devoted to the proof of Lemma 6.3. We now proceed with the
bulk part of the energy, that is, we analyze the blow up at points where the
approximate gradient exists.

######  Lemma 6.6.

Let $u\in GSBV^{\psi}(\Omega,\mathbb{R}^{m})$. Then for $\mathcal{L}^{d}$-a.e.
$x_{0}\in\Omega$ and $\mathcal{L}^{1}$-a.e. $\lambda\in(0,1)$ there exists a
sequence ${u}_{\varepsilon}\in GSBV^{\psi}(B_{\varepsilon}(x_{0}))$ such that
the following properties hold:

$\displaystyle\begin{split}&(i)\ \ {u}_{\varepsilon}=u\mbox{ on
}B_{\varepsilon}(x_{0})\setminus\overline{B_{\lambda\varepsilon}(x_{0})},\ \ \
\ \lim_{\varepsilon\to
0^{+}}\varepsilon^{-(d+1)}\mathcal{L}^{d}(\\{{u}_{\varepsilon}\neq u\\}\cap
B_{\varepsilon}(x_{0}))=0,\\\ &(ii)\ \ \lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d}}\int_{B_{\lambda\varepsilon}(x_{0})}\psi\left(x,\frac{|{u}_{\varepsilon}(x)-u(x_{0})-\nabla
u(x_{0})(x-x_{0})|}{\varepsilon}\right)\,dx=0;\\\ &(iii)\ \
\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d}}\mathcal{H}^{d-1}(J_{{u}_{\varepsilon}}\cap
B_{\varepsilon}(x_{0}))=0.\end{split}$ (6.16)

###### Proof.

Without loss of generality we can assume $m=1$. Take $x_{0}\in\Omega$
satisfying

$\displaystyle\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d}}\int_{B_{\varepsilon}(x_{0})}\\!\psi(x,|\nabla
u(x)-\nabla u(x_{0})|)\,dx=0;$ (6.17a) $\displaystyle\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d}}\mathcal{H}^{d-1}(J_{u}\cap
B_{\varepsilon}(x_{0}))=0;$ (6.17b) $\displaystyle\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d}}\mathcal{L}^{d}\left(\left\\{x\in
B_{\varepsilon}(x_{0})\colon\frac{|u(x)-u(x_{0})-\nabla
u(x_{0})(x-x_{0})|}{|x-x_{0}|}>\rho\right\\}\right)=0,\ \ \ \mbox{$\rho>0$}.$
(6.17c)

Notice that (6.17a)–(6.17c) are satisfied by $\mathcal{L}^{d}$-a.e
$x_{0}\in\Omega$ by Proposition 4.33. Set for brevity
$T_{\varepsilon}:=T_{B_{\varepsilon}(x_{0})}$. Observe that for $\varepsilon$
small enough, (6.17b) implies (5.1). For every $x\in B_{\varepsilon}(x_{0})$
we can thus define the truncated functions

$\overline{u}_{\varepsilon}(x):=T_{\varepsilon}\left(u(x)-u(x_{0})-\nabla
u(x_{0})(x-x_{0})\right)$

and, similarly as (5.30),

$\mathfrak{u}_{\varepsilon}^{\psi}(x):=\left(\textnormal{med}(\overline{u}_{\varepsilon}(x),B_{\varepsilon}(x_{0}))-\varepsilon(\psi^{-}_{B_{\varepsilon}(x_{0})})^{-1}\left(\frac{1}{2\varepsilon}\right)\right)\vee\overline{u}_{\varepsilon}(x)\wedge\left(\textnormal{med}(\overline{u}_{\varepsilon},B_{\varepsilon}(x_{0}))+\varepsilon(\psi_{B_{\varepsilon}(x_{0})}^{-})^{-1}\left(\frac{1}{2\varepsilon}\right)\right).$

Finally, we set $v_{\varepsilon}(x):=u(x_{0})+\nabla
u(x_{0})(x-x_{0})+\mathfrak{u}_{\varepsilon}^{\psi}(x)$.

We now want to estimate the quantity $\mathcal{L}^{d}(\\{v_{\varepsilon}\neq
u\\}\cap B_{\varepsilon}(x_{0}))$. We have

$\displaystyle\begin{split}\mathcal{L}^{d}(\\{v_{\varepsilon}\neq u\\}\cap
B_{\varepsilon}(x_{0}))\leq&\
\mathcal{L}^{d}(\\{\mathfrak{u}^{\psi}_{\varepsilon}\neq\overline{u}_{\varepsilon}\\}\cap
B_{\varepsilon}(x_{0}))\\\ &+\mathcal{L}^{d}(\\{\overline{u}_{\varepsilon}\neq
u(x)-u(x_{0})-\nabla u(x_{0})(x-x_{0})\\}\cap
B_{\varepsilon}(x_{0})).\end{split}$ (6.18)

By definition of $T_{\varepsilon}$ and (4.6), we can estimate the second
quantity on the right hand side of (6.18) with

$\mathcal{L}^{d}(\\{\overline{u}_{\varepsilon}\neq u(x)-u(x_{0})-\nabla
u(x_{0})(x-x_{0})\\}\cap
B_{\varepsilon}(x_{0}))\leq\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}\cap
B_{\varepsilon}(x_{0}))\right)^{\frac{d}{d-1}}.$

For the first term of (6.18) recall that by Theorem 4.7 and Lemma 4.5 we have
that for every $t\geq 0$

$\psi^{-}_{B_{\varepsilon}(x_{0})}\left((\psi^{-}_{B_{\varepsilon}(x_{0})})^{-1}(t)\right)\geq\frac{t}{2K},$

where $K\geq 2$ is the doubling constant of $\psi$.

We set $m_{\varepsilon}:=\textnormal{med}(u-\nabla u(x_{0})(\cdot-
x_{0}),B_{\varepsilon}(x_{0}))$. Using the definition of
$\mathfrak{u}^{\psi}_{\varepsilon}$, Chebychev inequality and Theorem 5.7,

$\displaystyle\mathcal{L}^{d}\left(\left\\{\mathfrak{u}^{\psi}_{\varepsilon}\neq\overline{u}_{\varepsilon}\right\\}\cap
B_{\varepsilon}(x_{0})\right)$
$\displaystyle\leq\mathcal{L}^{d}\left(B_{\varepsilon}(x_{0})\cap\left\\{|\overline{u}_{\varepsilon}-\textnormal{med}(\overline{u}_{\varepsilon},B_{\varepsilon}(x_{0}))|\geq\varepsilon(\psi^{-}_{B_{\varepsilon}(x_{0})})^{-1}\left(\frac{1}{2\varepsilon}\right)\right\\}\right)$
$\displaystyle\leq\mathcal{L}^{d}\left(B_{\varepsilon}(x_{0})\cap\left\\{\psi^{-}_{B_{\varepsilon}(x_{0})}\left(\frac{|\overline{u}_{\varepsilon}-\textnormal{med}(\overline{u}_{\varepsilon},B_{\varepsilon}(x_{0}))|}{\varepsilon}\right)\geq\frac{1}{4K\varepsilon}\right\\}\right)$
$\displaystyle\leq
4K\varepsilon\int_{B_{\varepsilon}(x_{0})}\psi^{-}_{B_{\varepsilon}(x_{0})}\left(\frac{|\overline{u}_{\varepsilon}(x)-\textnormal{med}(\overline{u}_{\varepsilon},B_{\varepsilon}(x_{0}))|}{\varepsilon}\right)\,dx$
$\displaystyle=4K\varepsilon\int_{B_{\varepsilon}(x_{0})}\psi^{-}_{B_{\varepsilon}(x_{0})}\left(\frac{|T_{\varepsilon}(u(x)-\nabla
u(x_{0})(x-x_{0}))-m_{\varepsilon}|}{\varepsilon}\right)\,dx$
$\displaystyle\leq\varepsilon
C\int_{B_{\varepsilon}(x_{0})}\psi_{B_{\varepsilon}(x_{0})}^{-}(|\nabla
u(x)-\nabla u(x_{0})|)\,dx$ $\displaystyle\leq\varepsilon
C\int_{B_{\varepsilon}(x_{0})}\psi(x,|\nabla u(x)-\nabla u(x_{0})|)\,dx.$

where $C=C(d,K)$. Thus, using Fubini Theorem and (6.17a)–(6.17b), we deduce
that

$\displaystyle\begin{split}\limsup_{\varepsilon\to
0^{+}}&\frac{1}{\varepsilon^{d}}\int_{0}^{1}\\!\mathcal{H}^{d-1}(\\{v_{\varepsilon}\neq
u\\}\cap\partial
B_{\lambda\varepsilon}(x_{0}))d\lambda=\limsup_{\varepsilon\to
0^{+}}\frac{2}{\varepsilon^{d+1}}\mathcal{L}^{d}(\\{v_{\varepsilon}\neq
u\\}\cap B_{\varepsilon}(x_{0}))\\\ &\leq\limsup_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d+1}}\left(\varepsilon
C\int_{B_{\varepsilon}(x_{0})}\psi(x,|\nabla u(x)-\nabla
u(x_{0})|)\,dx+\left(2\gamma_{\textnormal{iso}}\mathcal{H}^{d-1}(J_{u}\cap
B_{\varepsilon}(x_{0}))\right)^{\frac{d}{d-1}}\right)\\\ &=0.\end{split}$
(6.19)

Therefore, since $d\geq 2$, for every sequence $\varepsilon\to 0$, one can
find a subsequence such that for $\mathcal{L}^{1}$-a.e. $\lambda\in(0,1)$ it
holds

$\displaystyle\begin{split}&\mathcal{H}^{d-1}(\partial
B_{\lambda\varepsilon}(x_{0})\cap J_{v_{\varepsilon}})=0,\\\
&\lim_{\varepsilon\to
0^{+}}\varepsilon^{-d}\mathcal{H}^{d-1}(\\{v_{\varepsilon}\neq
u\\}\cap\partial B_{\lambda\varepsilon}(x_{0}))=0.\end{split}$ (6.20)

Now let us fix a subsequence $\varepsilon\to 0$ (not relabelled) and
$\lambda\in(0,1)$ such that (6.20) holds. We set

$u_{\varepsilon}(x):=\begin{cases*}v_{\varepsilon}(x)&if $x\in
B_{\lambda\varepsilon}(x_{0})$,\\\ u(x)&if $x\in
B_{\varepsilon}(x_{0})\setminus\overline{B_{\lambda\varepsilon}(x_{0})}$.\end{cases*}$

From the definition of $u_{\varepsilon}$ and the estimates in (6.19), we
deduce (6.16)(i) and (6.16)(iii). We are left to prove (6.16)(ii).

We actually will prove a slightly stronger statement, namely

$\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d}}\int_{B_{\varepsilon}(x_{0})}\psi\left(x,\frac{|v_{\varepsilon}(x)-u(x_{0})-\nabla
u(x_{0})(x-x_{0})|}{\varepsilon}\right)\,dx=\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d}}\int_{B_{\varepsilon}(x_{0})}\psi\left(x,\frac{|\mathfrak{u}^{\psi}_{\varepsilon}(x)|}{\varepsilon}\right)\,dx=0.$
(6.21)

Let $\varepsilon_{0}>0$ be fixed. Set
$\psi_{\varepsilon}^{-}:=\psi^{-}_{B_{\varepsilon}(x_{0})}$ and
$\psi_{\varepsilon}^{+}:=\psi^{+}_{B_{\varepsilon}(x_{0})}$ for brevity. Using
property (adA1) and the fact that $\psi$ satisfies (aDec), by definition of
$\mathfrak{u}_{\varepsilon}^{\psi}$ and by inequality (5.32) we estimate, for
every $\varepsilon<\varepsilon_{0}$,

$\displaystyle\int_{B_{\varepsilon}(x_{0})}\psi\left(x,\frac{|\mathfrak{u}^{\psi}_{\varepsilon}(x)-\textnormal{med}(\overline{u}_{\varepsilon},B_{\varepsilon}(x_{0}))|}{\varepsilon}\right)\,dx$
$\displaystyle\leq
C\int_{B_{\varepsilon}(x_{0})}\psi^{-}_{\varepsilon}\left(|\nabla u(x)-\nabla
u(x_{0})|\right)\,dx+\int_{B_{\varepsilon}(x_{0})}\psi^{+}_{\varepsilon_{0}}\left(\frac{|\overline{u}_{\varepsilon}(x)-\textnormal{med}(\overline{u}_{\varepsilon},B_{\varepsilon}(x_{0}))|}{\varepsilon}\wedge\sigma\right)\,dx.$

with $C=C(d,K,\beta)$, where $\sigma$ and $\beta$ are the constants appearing
in (A0) and (adA1), respectively.

By (6.17a), in order to prove (6.21) we only need to show that

$\displaystyle\lim_{\varepsilon\to
0^{+}}\frac{\textnormal{med}(\overline{u}_{\varepsilon},B_{\varepsilon}(x_{0}))}{\varepsilon}=\lim_{\varepsilon\to
0^{+}}\textnormal{med}\left(\frac{u-u(x_{0})-\nabla u(x_{0})(\cdot-
x_{0})}{\varepsilon},B_{\varepsilon}(x_{0})\right)=0,$
$\displaystyle\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d}}\int_{B_{\varepsilon}(x_{0})}\psi^{+}_{\varepsilon_{0}}\left(\frac{|\overline{u}_{\varepsilon}(x)|}{\varepsilon}\wedge\sigma\right)\,dx=0.$

The first limit follows by (6.17c) The second limit follows again by (6.17c)
and dominate convergence theorem after rescaling. ∎

We can now prove Lemma 6.3 which will actually follow as a consequence of the
next two Lemmas.

######  Lemma 6.7.

Let $\mathcal{F}$ satisfy (H1) and (H3)-(H4) and let $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$. Then, for $\mathcal{L}^{d}$-a.e
$x_{0}\in\Omega$

$\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}\leq\limsup_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(\overline{u}^{\textnormal{bulk}}_{x_{0}},B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}.$
(6.22)

###### Proof.

We will prove the assertion for all the points $x_{0}\in\Omega$ for which the
statement of Lemma 6.6 holds, $\lim_{\varepsilon\to
0^{+}}(\omega_{d}\varepsilon^{d})^{-1}\mu(B_{\varepsilon}(x_{0}))=1$ and

$\lim_{\varepsilon\to
0^{+}}\frac{\mathcal{F}(u,B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}=\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}<+\infty.$
(6.23)

By Lemma 6.2, property (6.23) hold for $\mathcal{L}^{d}$-a.e.
$x_{0}\in\Omega$. Let $(u_{\varepsilon})_{\varepsilon}$ be the sequence given
by Lemma 6.6 and fix $\lambda\in(0,1)$ such that (6.16)(ii) holds. We write
$\lambda=1-\theta$ for some $\theta\in(0,1)$.

Given $z_{\varepsilon}\in
GSBV^{\psi}(B_{(1-3\theta)\varepsilon}(x_{0}),\mathbb{R}^{m})$ such that
$z_{\varepsilon}=\overline{u}_{x_{0}}^{\textnormal{bulk}}$ in a neighborhood
of $\partial B_{(1-3\theta)\varepsilon}(x_{0})$ and

$\mathcal{F}(z_{\varepsilon},B_{(1-3\theta)\varepsilon}(x_{0}))\leq\mathbf{m}_{\mathcal{F}}(\overline{u}_{x_{0}}^{\textnormal{bulk}},B_{(1-3\theta)\varepsilon}(x_{0}))+\omega_{d}\varepsilon^{d+1},$
(6.24)

we extend it to $z_{\varepsilon}\in
GSBV(B_{\varepsilon}(x_{0}),\mathbb{R}^{m})$ by setting
$z_{\varepsilon}=\overline{u}_{x_{0}}^{\textnormal{bulk}}$ outside of
$B_{(1-3\theta)\varepsilon}(x_{0})$. We further set
$C_{\varepsilon,\theta}(x_{0}):=B_{\varepsilon}(x_{0})\setminus\overline{B_{(1-4\theta)\varepsilon}(x_{0})}$.
Now we use Lemma 6.1 with $u$ and $v$ replaced by $z_{\varepsilon}$ and
$u_{\varepsilon}$, respectively, and with the sets

$D^{\prime}_{\varepsilon,x_{0}}:=B_{(1-2\theta)\varepsilon}(x_{0}),\ \ \
D^{\prime\prime}_{\varepsilon,x_{0}}:=B_{(1-\theta)\varepsilon}(x_{0}),\ \ \
E_{\varepsilon,x_{0}}:=C_{\varepsilon,\theta}(x_{0}).$ (6.25)

Notice that
$C_{\varepsilon,\theta}(x_{0})=(C_{1,\theta}(x_{0}))_{\varepsilon,x_{0}}$
according to the notation introduced in (2.1), where
$C_{1,\theta}(x_{0}):=B_{1}(x_{0})\setminus\overline{B_{(1-4\theta)}(x_{0})}$.
Moreover, we observe that
$\mathcal{L}^{d}(C_{1,\theta}(x_{0}))=\omega_{d}(1-(1-4\theta))\to 0$ as
$\theta\to 0$. For an arbitrarily fixed $\eta>0$, we find $w_{\varepsilon}\in
GSBV^{\psi}(B_{\varepsilon}(x_{0}),\mathbb{R}^{m})$ such that
$w_{\varepsilon}=u_{\varepsilon}$ on $B_{\varepsilon}(x_{0})\setminus
B_{(1-\theta)\varepsilon}(x_{0})$ and

$\displaystyle\begin{split}\mathcal{F}(w_{\varepsilon},B_{\varepsilon}(x_{0}))&\leq(1+\eta)(\mathcal{F}(z_{\varepsilon},B_{(1-\theta)\varepsilon}(x_{0}))+\mathcal{F}(u_{\varepsilon},C_{\varepsilon,\theta}(x_{0})))+\eta\mathcal{L}^{d}(B_{\varepsilon}(x_{0}))\\\
&+M\int_{B_{(1-\theta)\varepsilon}(x_{0})\setminus
B_{(1-2\theta)\varepsilon}(x_{0})}\psi\left(x,\frac{|z_{\varepsilon}-u_{\varepsilon}|}{\varepsilon}\right)\,dx\end{split}$
(6.26)

Recalling (6.16)(i) in Lemma 6.6, we have that
$w_{\varepsilon}=u_{\varepsilon}=u$ in a neighborhood of $\partial
B_{\varepsilon}(x_{0})$. Moreover, since
$z_{\varepsilon}=\overline{u}_{x_{0}}^{\textnormal{bulk}}$ outside
$B_{(1-3\theta)\varepsilon}(x_{0})$, using (6.16)(ii) we infer that

$\displaystyle\begin{split}\lim_{\varepsilon\to
0^{+}}&\frac{1}{\varepsilon^{d}}\int_{B_{(1-\theta)\varepsilon}(x_{0})\setminus
B_{(1-2\theta)\varepsilon}(x_{0})}\psi\left(x,\frac{|z_{\varepsilon}-u_{\varepsilon}|}{\varepsilon}\right)\,dx\\\
&\leq\lim_{\varepsilon\to
0^{+}}\frac{1}{\varepsilon^{d}}\int_{B_{(1-\theta)\varepsilon}(x_{0})}\psi\left(x,\frac{|\overline{u}_{x_{0}}^{\textnormal{bulk}}-u_{\varepsilon}|}{\varepsilon}\right)\,dx=0.\end{split}$
(6.27)

From (6.26) and (6.27) we deduce that there exists a non-negative
infinitesimal sequence $(\rho_{\varepsilon})_{\varepsilon}$ such that

$\mathcal{F}(w_{\varepsilon},B_{\varepsilon}(x_{0}))\leq(1+\eta)(\mathcal{F}(z_{\varepsilon},B_{(1-\theta)\varepsilon}(x_{0}))+\mathcal{F}(u_{\varepsilon},C_{\varepsilon,\theta}(x_{0})))+\varepsilon^{d}\rho_{\varepsilon}+\eta\omega_{d}\varepsilon^{d}.$
(6.28)

Then, using the fact that
$z_{\varepsilon}=\overline{u}_{x_{0}}^{\textnormal{bulk}}$ in
$B_{\varepsilon}(x_{0})\setminus B_{(1-3\theta)\varepsilon}(x_{0})\subset
C_{\varepsilon,\theta}(x_{0})$, (H1), (H4) and (6.24), setting
$\psi_{\varepsilon}^{+}(t):=\psi^{+}_{B_{\varepsilon}(x_{0})}(\cdot,t)$ for
$t>0$, we deduce that there exists $\varepsilon_{0}>0$ small enough such that
for every $\varepsilon<\varepsilon_{0}$

$\frac{\mathcal{F}(\overline{u}_{x_{0}}^{\textnormal{bulk}},C_{\varepsilon,\theta}(x_{0}))}{\varepsilon^{d}}\leq
b\mathcal{L}^{d}(C_{\varepsilon,\theta}(x_{0}))(1+\psi^{+}_{\varepsilon_{0}}(|\nabla
u(x_{0})|)).$ (6.29)

Thus,

$\displaystyle\limsup_{\varepsilon\to
0^{+}}\frac{\mathcal{F}(z_{\varepsilon},B_{(1-\theta)\varepsilon}(x_{0}))}{\varepsilon^{d}}$
$\displaystyle\leq\limsup_{\varepsilon\to
0^{+}}\left(\frac{\mathcal{F}(z_{\varepsilon},B_{(1-3\theta)\varepsilon}(x_{0}))}{\varepsilon^{d}}+\frac{\mathcal{F}(\overline{u}_{x_{0}}^{\textnormal{bulk}},C_{\varepsilon,\theta}(x_{0}))}{\varepsilon^{d}}\right)$
(6.30) $\displaystyle\leq\limsup_{\varepsilon\to
0^{+}}\left(\frac{\mathbf{m}_{\mathcal{F}}(\overline{u}_{x_{0}}^{\textnormal{bulk}},B_{(1-3\theta)\varepsilon}(x_{0}))}{\varepsilon^{d}}+b\mathcal{L}^{d}(C_{\varepsilon,\theta}(x_{0}))(1+\psi^{+}_{\varepsilon_{0}}(|\nabla
u(x_{0})|))\right)$ $\displaystyle\leq(1-3\theta)^{d}\limsup_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(\overline{u}_{x_{0}}^{\textnormal{bulk}},B_{(1-3\theta)\varepsilon}(x_{0}))}{(1-3\theta)^{d}\varepsilon^{d}}$
$\displaystyle\ \ \ \ +\limsup_{\varepsilon\to
0^{+}}b\mathcal{L}^{d}(C_{\varepsilon,\theta}(x_{0}))(1+\psi^{+}_{\varepsilon_{0}}(|\nabla
u(x_{0})|)).$

On the other hand, using again (H4) we also deduce

$\displaystyle\mathcal{F}(u_{\varepsilon},C_{\varepsilon,\theta}(x_{0}))\leq
b\mathcal{L}^{d}(C_{\varepsilon,\theta}(x_{0}))+b\int_{C_{\varepsilon,\theta}(x_{0})}\psi(x,|\nabla
u_{\varepsilon}(x)|)\,dx+b\mathcal{H}^{d-1}(J_{u_{\varepsilon}}\cap
C_{\varepsilon,\theta}(x_{0})).$

Now, using (6.16)(iii), we get

$\limsup_{\varepsilon\to
0^{+}}\frac{\mathcal{F}(u_{\varepsilon},C_{\varepsilon,\theta}(x_{0}))}{\varepsilon^{d}}\leq
b\mathcal{L}^{d}(C_{1,\theta}(x_{0}))+\limsup_{\varepsilon\to
0^{+}}\frac{b}{\varepsilon^{d}}\int_{C_{\varepsilon,\theta}(x_{0})}\psi(x,|\nabla
u_{\varepsilon}(x)|)\,dx.$ (6.31)

By exploiting the construction in Lemma 6.6, we notice that $|\nabla
u_{\varepsilon}|\leq|\nabla u|$ $\mathcal{L}^{d}$-a.e. in
$B_{\varepsilon}(x_{0})$ for any $\varepsilon>0$. Thus, using (6.17a) and the
fact that $\psi$ is doubling with constant $K\geq 2$, we have

$\displaystyle\limsup_{\varepsilon\to 0^{+}}\
\frac{b}{\varepsilon^{d}}\int_{C_{\varepsilon,\theta}(x_{0})}\psi(x,|\nabla
u_{\varepsilon}|)\,dx$ (6.32) $\displaystyle\ \leq\limsup_{\varepsilon\to
0^{+}}\
\frac{b}{\varepsilon^{d}}\int_{C_{\varepsilon,\theta}(x_{0})}\psi(x,|\nabla
u|)\,dx$ $\displaystyle\ \leq\limsup_{\varepsilon\to 0^{+}}\
(1+K)\left(\frac{b}{\varepsilon^{d}}\int_{B_{\varepsilon}(x_{0})}\psi(x,|\nabla
u(x)-\nabla
u(x_{0})|)\,dx+b\mathcal{L}^{d}(C_{\varepsilon,\theta}(x_{0}))\psi^{+}_{\varepsilon_{0}}(|\nabla
u(x_{0})|)\right)$ $\displaystyle\ \leq
b(1+K)\mathcal{L}^{d}(C_{1,\theta}(x_{0}))\psi^{+}_{\varepsilon_{0}}(|\nabla
u(x_{0})|).$

Combining (6.31) and (6.32) we infer

$\limsup_{\varepsilon\to
0^{+}}\frac{\mathcal{F}(u_{\varepsilon},C_{\varepsilon,\theta}(x_{0}))}{\varepsilon^{d}}\leq
b\mathcal{L}^{d}(C_{1,\theta}(x_{0}))\left(1+(1+K)\psi^{+}_{\varepsilon_{0}}(|\nabla
u(x_{0})|)\right).$ (6.33)

Recalling that $w_{\varepsilon}=u_{\varepsilon}$ in a neighborhood of
$\partial B_{\varepsilon}(x_{0})$, by (6.28), (6.30) and (6.33) we finally get

$\displaystyle\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}$
$\displaystyle\leq\limsup_{\varepsilon\to
0^{+}}\frac{\mathcal{F}(w_{\varepsilon},B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}$
$\displaystyle\leq(1+\eta)(1-3\theta)^{d}\limsup_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(\overline{u}_{x_{0}}^{\textnormal{bulk}},B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}$
$\displaystyle\ \ \
+(1+\eta)b\omega_{d}^{-1}\mathcal{L}^{d}(C_{1,\theta}(x_{0}))\left(1+(1+K)\psi^{+}_{\varepsilon_{0}}(|\nabla
u(x_{0})|)\right)+\eta.$

Letting $\eta\to 0$ and $\theta\to 0$ we get (6.22).

∎

######  Lemma 6.8.

Let $\mathcal{F}$ satisfy (H1) and (H3)-(H4) and let $u\in
GSBV^{\psi}(\Omega,\mathbb{R}^{m})$. Then, for $\mathcal{L}^{d}$-a.e
$x_{0}\in\Omega$ we have

$\lim_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(u,B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}\geq\limsup_{\varepsilon\to
0^{+}}\frac{\mathbf{m}_{\mathcal{F}}(\overline{u}^{\textnormal{bulk}}_{x_{0}},B_{\varepsilon}(x_{0}))}{\omega_{d}\varepsilon^{d}}.$
(6.34)

###### Proof.

We prove the statement for points $x_{0}\in\Omega$ considered in Lemma 6.7.
Let $\eta>0$ and $\lambda=1-\theta$ be fixed as in the proof of Lemma 6.7, and
let $(u_{\varepsilon})_{\varepsilon}$ be the sequence given by Lemma 6.6.
Using Fubini’s Theorem (as in (6.19) and (6.20)), we have that for every
$\varepsilon>0$ we can find $s\in(1-4\theta,1-3\theta)$ such that

$\displaystyle\begin{split}&\mathcal{H}^{d-1}(\partial
B_{s\varepsilon}(x_{0})\cap(J_{u}\cup J_{u_{\varepsilon}}))=0\ \ \ \mbox{for
every $\varepsilon>0$},\\\ &\lim_{\varepsilon\to
0^{+}}\varepsilon^{-d}\mathcal{H}^{d-1}(\\{u_{\varepsilon}\neq
u\\}\cap\partial B_{s\varepsilon}(x_{0}))=0.\end{split}$ (6.35)

From now on, we will follow mainly the arguments of the proof of Lemma 6.7. We
choose a sequence $z_{\varepsilon}\in
GSBV^{\psi}(B_{s\varepsilon}(x_{0}),\mathbb{R}^{m})$ such that
$z_{\varepsilon}=u$ in a neighborhood of $\partial B_{s\varepsilon}(x_{0})$
and

$\mathcal{F}(z_{\varepsilon},B_{s\varepsilon}(x_{0}))\leq\mathbf{m}_{\mathcal{F}}(u,B_{s\varepsilon}(x_{0}))+\omega_{d}\varepsilon^{d+1}.$
(6.36)

Setting $z_{\varepsilon}=u_{\varepsilon}$ outside $B_{s\varepsilon}(x_{0})$ we
extend it to $z_{\varepsilon}\in
GSBV^{\psi}(B_{\varepsilon}(x_{0}),\mathbb{R}^{m})$. We now use Lemma 6.1
applied with $u$ and $v$ replaced by $z_{\varepsilon}$ and
$\overline{u}_{x_{0}}^{\textnormal{bulk}}$, respectively, and the same choice
of sets
$D^{\prime}_{\varepsilon,x_{0}},D^{\prime\prime}_{\varepsilon,x_{0}},E_{\varepsilon,x_{0}}$
as in (6.25).

Hence, we find $w_{\varepsilon}\in
GSBV^{\psi}(B_{\varepsilon}(x_{0}),\mathbb{R}^{m})$ such that
$w_{\varepsilon}=\overline{u}_{x_{0}}^{\textnormal{bulk}}$ on
$B_{\varepsilon}(x_{0})\setminus B_{(1-\theta)\varepsilon}(x_{0})$ and

$\displaystyle\mathcal{F}(w_{\varepsilon},B_{\varepsilon}(x_{0}))\leq$
$\displaystyle(1+\eta)\left(\mathcal{F}(z_{\varepsilon},B_{(1-\theta)\varepsilon}(x_{0}))+\mathcal{F}(\overline{u}_{x_{0}}^{\textnormal{bulk}},C_{\varepsilon,\theta}(x_{0}))\right)+\eta\mathcal{L}^{d}(B_{\varepsilon}(x_{0}))+$
$\displaystyle+M\int_{B_{(1-\theta)\varepsilon}(x_{0})\setminus
B_{(1-2\theta)\varepsilon}(x_{0})}\psi\left(x,\frac{|z_{\varepsilon}-\overline{u}_{x_{0}}^{\textnormal{bulk}}|}{\varepsilon}\right)\,dx.$

Since from the initial choice of $s$ we have that