1 Introduction

Originally introduced in the setting of image restoration [46], free-discontinuity functionals are by now ubiquitous in the mathematical modeling of elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. If u is the variable of the problem (representing, e.g., the output image or the deformation of the solid), these problems are characterized by the competition between a “bulk” energy, usually taking the form of a variational integral

$$\begin{aligned} \int _\Omega f(x, u(x), \nabla u(x))\,\textrm{d}x \end{aligned}$$
(1.1)

where \(\Omega \) is a reference configuration, and a “surface" energy of the form

$$\begin{aligned} \int _{J_u} g\big (x, u^+(x), u^-(x), \nu _u(x)\big ) \, \textrm{d} \mathcal {H}^{d-1} (x) \end{aligned}$$
(1.2)

where \(J_u\) is the set of discontinuities of u with normal \(\nu _u\). This latter term is, for instance, accounting for the “cost” of an interface in the image (enforcing optimal segmentation), or for the energy spent to produce a crack [34, 38]. If one imposes a p-growth assumption of the form

$$\begin{aligned} c|\xi |^p \le f(x,u, \xi )\le C(1+|\xi |^p) \end{aligned}$$

with \(p>1\) on the bulk integrand f, and \(g \ge \alpha >0\), then the existence of minimizers is guaranteed in the space of Generalized Special functions of Bounded Variation (GSBV) whenever f is quasiconvex and g BV-elliptic, see [6]. In particular, compactness of minimizing sequences with respect to the convergence in measure can be recovered by Ambrosio’s results [3, 4] if some lower order fidelity terms are included in the problem, or some boundary data are considered (see [35]).

A wide attention has been also paid, over the last two decades, to the theory of variational limits of free-discontinuity functionals, with applications in various contexts, such as homogenization, dimension reduction, or atomistic-to-continuum approximations. Starting from the first results on the subspace SBV of special functions of bounded variation [11, 13, 14] and on piecewise constant functions [5], this analysis has been further improved to deal with functionals and variational limits on \(GSBV^p\) (generalized special functions of bounded variation with p-integrable bulk density), see, e.g., [7,8,9,10, 16, 31, 35]. Most of these results are based on the so-called global method for relaxation, which has been developed by Bouchitté, Fonseca, Leoni, and Mascarenhas in [11, 12]. This very powerful method is essentially based on comparing asymptotic Dirichlet problems on small balls with different boundary data depending on the local properties of the functions and allows one to characterize limit energy densities in terms of cell formulas. Recently, it has also been used for analyzing the limit behavior of free-discontinuity problems in the space GSBD of generalized functions of bounded deformations, involving the symmetric gradient, see for example [15, 18, 20, 23], with applications to crack energies in linear elastic materials.

The topic of the present paper are, instead, free discontinuity problems in variable exponent spaces. These spaces were originally considered by the Russian school, see [49] and the Czech one [43]. Subsequently, motivated by models for the behavior of composite materials, Zhikov initiated the so-called theory of variational integrals with non standard growth in the mid 80’s. Since then, the subject of variable exponent spaces has undergone a large interest, both from the standpoint of regularity theory (see [50] for the scalar case and [2, 21] for the vectorial one) and in view of applications ranging from electrorheological fluids right up to homogenization, see [47, 48, 51, 52], and the references in [24, 29]. Motivated by the aforementioned applications, in [22] Coscia and Mucci analyzed the \(\Gamma \)-convergence of variational integrals of the form (1.1) with a p(x)-growth condition

$$\begin{aligned} c|\xi |^{p(x)} \le f(x,u, \xi )\le C(1+|\xi |^{p(x)}), \end{aligned}$$
(1.3)

where \(p(x)\ge 1+\delta >1\) is a variable exponent, in the Sobolev space \(W^{1, p(\cdot )}(\Omega ; \mathbb {R}^m)\). They proved that the \(\Gamma \)-limit of these energies is still an integral functional of the same type and growth, under a key assumption on the modulus of continuity of the variable exponent, the so-called log-Hölder continuity, see (2.4) below. In some sense, this condition says that we can freeze the exponent on small balls around a point, as pointed out in [28, Lemma 3.2] (see also Lemma 2.1 below). As such, it is particularly suitable for blow-up methods: for instance, in [1] it allows the authors to prove the singular part of the measure representation of relaxed functionals with growth (1.3) disappears. More in general, log-Hölder continuity plays a central role in the theory of functionals with p(x)-growth, as Zhikov proved in [50] that such functionals exhibit the Lavrentiev phenomenon if it is violated.

In recent years, variational problems in spaces of functions of bounded variation with variable integrability exponent on the gradient have been proposed, especially in the setting of image restoration. In the pioneering paper [19] Chen, Levine and Rao proposed for the first time a model considering a kind of intermediate regime between the TV model and the isotropic diffusion away from the edges (see also [40] for a related model, [44] for simulations, and [41] for a \(\Gamma \)-convergence result). Observe that in these models, the value \(p(x)=1\) is allowed. A related, but different, point of view takes instead into account the coupling of a strictly superlinear bulk energy (1.1) under the growth conditions (1.3) with a surface energy (1.2), which can be seen as a variable-exponent version of Mumford-Shah-type functionals.Footnote 1 This kind of functionals will constitute the object of the present paper. From an analytical point of view, they were considered in [26]. There, provided the bulk integrand is quasiconvex and the exponent is log-Hölder continuous, a lower semicontinuity result for sequences with bounded energy has been proved, which entails well-posedness of such variational problems in the subspace \(SBV^{p(\cdot )}\) of SBV functions with \(p(\cdot )\)-integrable gradients (again, if some lower order terms are added to the problem in order to apply Ambrosio’s compactness Theorem).

Description of our results This leads us to the purpose of the present paper. Our focus is to study the \(\Gamma \)-convergence (with respect to the convergence in measure) for functionals \(\mathcal {F}_j:GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m) \rightarrow [0,+\infty )\) of the form

$$\begin{aligned} \mathcal {F}_j(u) = \int _{\Omega } f_j\big (x, \nabla u(x) \big ) \, \textrm{d}x + \int _{J_u} g_j\big (x, [u](x), \nu _u(x)\big ) \, \textrm{d} \mathcal {H}^{d-1} (x) \end{aligned}$$
(1.4)

for each \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\), where \([u](x):=u^+(x)-u^-(x)\). The variable exponent \(p(\cdot )\) is assumed to be log-Hölder continuous, with \(p(x) \ge p^- > 1\) for all x (see (\({P}_{1}\))). We assume that the bulk integrands \(f_j\) satisfy (1.3) uniformly in j, while the surface integrands \(g_j\) satisfy

$$\begin{aligned} 0<\alpha \le g_j(x, \zeta , \nu )\le \beta . \end{aligned}$$

Under a fairly general set of assumptions, devised in [16], we are able to show that the \(\Gamma \)-limit is again an integral functional of the same form (Theorem 4.1). Furthermore, as shown in Sect. 5, due to the assumption \(p(x) \ge p^- > 1\) a separation of scales effect takes place, exactly as in the case of a constant exponent: bulk and surface effects decouple in the limit. Namely, the bulk limit density \(f_\infty \) is completely determined by taking the \(\Gamma \)-limit of the functionals (1.1) in the Sobolev space \(W^{1,p(\cdot )}\), while the surface limit density \(g_\infty \) can be recovered from the sole \(g_j\)’s via an asymptotic cell formula on piecewise constant functions, that is GSBV functions whose gradient is a.e. equal to 0.

As we mentioned, for the proof of Theorem 4.1 we follow quite closely the global method for relaxation of [11]. The main point is recovering an integral representation for functionals

$$\begin{aligned} \mathcal {F}:GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m) \times \mathcal {B}(\Omega ) \rightarrow [0,+\infty ) \end{aligned}$$

(here \(\mathcal {B}(\Omega )\) denote the Borel subsets of \(\Omega \)) that satisfy the standard abstract conditions to be Borel measure in the second argument, lower semicontinuity with respect to the convergence in measure, and local in the first argument. In addition, we require a coercivity and control condition of variable exponent type: there exist \(0< \alpha < \beta \) such that for any \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(B \in \mathcal {B}(\Omega )\) we have

$$\begin{aligned} \alpha \bigg (\int _{ B } |\nabla u|^{p(x)} \, \textrm{d} x+ \mathcal {H}^{d-1}(J_u \cap B)\bigg ) \le \mathcal {F}(u,B) \le \beta \bigg (\int _{ B } (1 + |\nabla u|^{p(x)}) \, \textrm{d} x+ \mathcal {H}^{d-1}(J_u \cap B)\bigg ). \end{aligned}$$

The result is proved in Theorem 3.1. The proof strategy recovers the integral bulk and surface densities as blow-up limits of cell minimization formulas, as a consequence of the estimates in Lemmas 3.7 and 3.10. In particular, in this latter the interplay between the asymptotic estimates and the variable exponent setting causes some nontrivial difficulties, which are overcome by means of assumption (2.4). It allows us to estimate the asymptotic distance between a suitable modification of u and its blow-up at jump points in some variable exponent space, keeping bounded some constants which depend on the oscillation of \(p(\cdot )\) in a small ball around the blow-up point (see equation (3.57)). The log-Hölder continuity assumption plays also a crucial role in Theorem 5.2, where separation of scales for the bulk energy is shown. There, a Lusin-type approximation for SBV functions is used to reduce the asymptotic minimization problems defining the cell formula for the bulk energy to the (variable exponent) Sobolev setting. Again, via (2.4) it is possible to estimate the rest term coming from this approximation (see Eqs. (5.26)–(5.28)).

Our results can be also adapted to the case where the surface integrands \(g_j\)’s satisfy a more general growth condition, as in [16], namely

$$\begin{aligned} \alpha \le g_j(x,\zeta , \nu )\le \beta (1+|\zeta |). \end{aligned}$$

This can be done by first establishing the integral representation in the \(SBV^{p(\cdot )}\) case for functionals which satisfy

$$\begin{aligned} \begin{aligned} \alpha \bigg (\int _{ B } |\nabla u|^{p(x)} \, \textrm{d} x&+ \int _{J_u \cap B}(1+|[u]|)\,\textrm{d}\mathcal {H}^{d-1}\bigg ) \le \mathcal {F}(u,B)\\&\le \beta \bigg (\int _{ B } (1 + |\nabla u|^{p(x)}) \, \textrm{d} x+ \int _{J_u \cap B}(1+|[u]|)\,\textrm{d}\mathcal {H}^{d-1}\bigg ). \end{aligned} \end{aligned}$$

The analysis can be reconducted to this setting by a perturbation trick: one considers a small perturbation of the functional, depending on the jump opening, to represent functionals on \(SBV^{p(\cdot )}\). Then, by letting the perturbation parameter vanish and by truncating functions suitably, the representation can be extended to \(GSBV^{p(\cdot )}\). In order to do this, one can follow quite closely the arguments in [16], with some minor changes due to the variable exponent setting: for the sake of completeness and self-containedness, statements and proofs are given in “Appendix A”.

Outline of the paper The paper is structured as follows. In Sect. 2 we fix the basic notation and recall some basic facts about Lebesgue spaces with variable exponent (Sect. 2.1). Then, in Sect. 2.2 we introduce the space \(GSBV^{p(\cdot )}\), and prove some regularity and compactness properties useful in the sequel. Section 3 is entirely devoted to the proof of the integral representation result in \(GSBV^{p(\cdot )}\). Specifically, in Sect. 3.1 we prove a fundamental estimate, which is a key tool for the global method, Sect. 3.2. The proofs of the necessary blow-up properties are postponed to Sects. 3.3 and 3.4. In Sect. 4 we prove a \(\Gamma \)-convergence result for sequences of free-discontinuity functionals defined on \(GSBV^{p(\cdot )}\). The identification of the \(\Gamma \)-limit is contained in Sect. 5. Eventually, in “Appendix A”, we develop the analysis of Sects. 3 and 4 for free-discontinuity energies with a weaker growth condition from above in the surface term.

2 Basic notation and preliminaries

We start with some basic notation. Let \(\Omega \subset \mathbb {R}^d\) be open, bounded with Lipschitz boundary. Let \(\mathcal {A}(\Omega )\) be the family of open subsets of \(\Omega \), and denote by \(\mathcal {B}(\Omega )\) the family of Borel sets contained in \(\Omega \). For every \(x\in {\mathbb {R}}^d\) and \(\varepsilon >0\) we indicate by \(B_\varepsilon (x) \subset {\mathbb {R}}^d\) the open ball with center x and radius \(\varepsilon \). If \(x=0\), we will often use the shorthand \(B_\varepsilon \). For x, \(y\in {\mathbb {R}}^d\), we use the notation \(x\cdot y\) for the scalar product and |x| for the Euclidean norm. Moreover, we let \({\mathbb {S}}^{d-1}:=\{x \in {\mathbb {R}}^d:|x|=1\}\), we denote by \(\mathbb {R}^{m \times d}\) the set of \(m\times d\) matrices and by \(\mathbb {R}^d_0\) the set \(\mathbb {R}^d\backslash \{0\}\). The m-dimensional Lebesgue measure of the unit ball in \(\mathbb {R}^m\) is indicated by \(\gamma _m\) for every \(m \in \mathbb {N}\). We denote by \({\mathcal {L}}^d\) and \(\mathcal {H}^k\) the d-dimensional Lebesgue measure and the k-dimensional Hausdorff measure, respectively. For \(A \subset \mathbb {R}^d\), \(\varepsilon >0\), and \(x_0 \in \mathbb {R}^d\) we set

$$\begin{aligned} A_{\varepsilon ,x_0} := x_0 + \varepsilon (A - x_0). \end{aligned}$$
(2.1)

The closure of A is denoted by \(\overline{A}\). The diameter of A is indicated by \(\textrm{diam}(A)\). Given two sets \(A_1,A_2 \subset \mathbb {R}^d\), we denote their symmetric difference by \(A_1 \triangle A_2\). We write \(\chi _A\) for the characteristic function of any \(A\subset \mathbb {R}^d\), which is 1 on A and 0 otherwise. If A is a set of finite perimeter, we denote its essential boundary by \(\partial ^* A\), see [6, Definition 3.60]. The notation \(L^0(E; \mathbb {R}^m)\) will be used for the space of Lebesgue measurable function from some measurable set \(E\subset \mathbb {R}^n\) to \(\mathbb {R}^m\), endowed with the convergence in measure.

2.1 Variable exponent Lebesgue spaces

We briefly recall the notions of variable exponents and variable exponent Lebesgue spaces. We refer the reader to [29] for a comprehensive treatment of the topic.

A measurable function \(p:\Omega \rightarrow [1,+\infty )\) will be called a variable exponent. Correspondingly, for every \(A\subset \Omega \) we define

$$\begin{aligned} p^+_A:=\mathop {\textrm{ess}\,\sup }_{x\in A} p(x)\,\hbox {\,\, and \,\,}p^-_A:=\mathop {\textrm{ess}\,\inf }_{x\in A} p(x), \end{aligned}$$

while \(p^+_\Omega \) and \(p^-_\Omega \) will be denoted by \(p^+\) and \(p^-\), respectively.

For a measurable function \(u:\Omega \rightarrow \mathbb {R}^m\) we define the modular as

$$\begin{aligned} \varrho _{p(\cdot )}(u):=\int _\Omega |u(x)|^{p(x)}\,\textrm{d}x \end{aligned}$$

and the (Luxembourg) norm

$$\begin{aligned} \Vert u\Vert _{L^{p(\cdot )}(\Omega )}:=\inf \{\lambda >0:\,\, \varrho _{p(\cdot )}(u/\lambda )\le 1\}. \end{aligned}$$

The variable exponent Lebesgue space \(L^{p(\cdot )}(\Omega )\) is defined as the set of measurable functions u such that \(\varrho _{p(\cdot )}(u/\lambda )<+\infty \) for some \(\lambda >0\). In the case \(p^+<+\infty \), \(L^{p(\cdot )}(\Omega )\) coincides with the set of functions such that \(\varrho _{p(\cdot )}(u)\) is finite. It can be checked that \(\Vert \cdot \Vert _{L^{p(\cdot )}(\Omega )}\) is a norm on \(L^{p(\cdot )}(\Omega )\). Moreover, if \(p^+<+\infty \), it holds that

$$\begin{aligned} \varrho _{p(\cdot )}(u)^\frac{1}{p^+} \le \Vert u\Vert _{L^{p(\cdot )}(\Omega )} \le \varrho _{p(\cdot )}(u)^\frac{1}{p^-} \end{aligned}$$
(2.2)

if \(\Vert u\Vert _{L^{p(\cdot )}(\Omega )}>1\), while an analogous inequality holds by exchanging the role of \(p^-\) and \(p^+\) if \(0\le \Vert u\Vert _{L^{p(\cdot )}(\Omega )}\le 1\). Another useful property of the modular, in the case \(p^+<+\infty \), is the following one:

$$\begin{aligned} \min \{\lambda ^{p^+}, \lambda ^{p^-}\}\varrho _{p(\cdot )}(u) \le \varrho _{p(\cdot )}(\lambda u) \le \max \{\lambda ^{p^+}, \lambda ^{p^-}\}\varrho _{p(\cdot )}(u) \end{aligned}$$
(2.3)

for all \(\lambda >0\).

We say that a function \(p:\Omega \rightarrow \mathbb {R}\) is log-Hölder continuous on \(\Omega \) if

$$\begin{aligned} \exists C>0 \hbox { \,\, such that \,\, } |p(x)-p(y)|\le \frac{C}{-\log |x-y|},\quad \forall x,y\in \Omega ,\, |x-y|\le \frac{1}{2}. \end{aligned}$$
(2.4)

We recall the following geometric meaning of the p log-Hölder continuity (see, e.g., [28, Lemma 3.2]).

Lemma 2.1

Let \(p:\Omega \rightarrow [1,+\infty )\) be a bounded, continuous variable exponent. The following conditions are equivalent:

  1. (i)

    p is log-Hölder continuous;

  2. (ii)

    for all open balls B, we have

    $$\begin{aligned} \mathcal {L}^d(B)^{(p^-_B-p^+_B)}\le C_1. \end{aligned}$$

The following lemma provides an extension to the variable exponent setting of the well-known embedding property of classical Lebesgue spaces (see, e.g., [29, Corollary 3.3.4]).

Lemma 2.2

Let pq be measurable variable exponents on \(\Omega \), and assume that \(\mathcal {L}^d(\Omega )<+\infty \). Then \(L^{p(\cdot )}(\Omega )\hookrightarrow L^{q(\cdot )}(\Omega )\) if and only if \(q(x)\le p(x)\) for \(\mathcal {L}^d\)-a.e. x in \(\Omega \). The embedding constant is less or equal to the minimum between \(2(1+ \mathcal {L}^d(\Omega ))\) and \(2\max \{\mathcal {L}^d(\Omega )^{(\frac{1}{q}- \frac{1}{p})^+}, \mathcal {L}^d(\Omega )^{(\frac{1}{q}- \frac{1}{p})^-}\}\).

The following result generalizes the concept of Lebesgue points to the variable exponent Lebesgue spaces (see, e.g., [39, Theorem 3.1]).

Theorem 2.3

Let \(\displaystyle p^+:=\mathop \mathrm{ess\,sup}_{x\in \mathbb {R}^d}p(x)<+\infty \). If \(u\in L^{p(\cdot )}(\mathbb {R}^d)\) then

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \frac{1}{\varepsilon ^d}\int _{B_\varepsilon (x)}|u(y)-u(x)|^{p(y)}\,\textrm{d}y=0 \end{aligned}$$

for a.e. \(x\in \mathbb {R}^d\).

2.2 The space \(GSBV^{p(\cdot )}\): Poincaré-type inequality

We denote by \(SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) the set of functions \(u\in SBV(\Omega ;\mathbb {R}^m)\) with \(\nabla u\in L^{p(\cdot )}(\Omega ;\mathbb {R}^{m \times d})\) and \(\mathcal {H}^{d-1}(J_u)<+\infty \). Here, \(\nabla u\) denotes the approximate gradient, while \(J_u\) stands for the (approximate) jump set with corresponding normal \(\nu _u\) and one-sided limits \(u^+\) and \(u^-\). We say that \(u\in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) if for every \(\phi \in C^1(\mathbb {R}^m)\) with the support of \(\nabla \phi \) compact, the composition \(\phi \circ u\) belongs to \(SBV^{p(\cdot )}_\textrm{loc}(\Omega ;\mathbb {R}^m)\).

From the inclusion \(L^{p(\cdot )}(\Omega )\subset L^{p^-}(\Omega )\) and [4], one can also deduce that for \(u\in GSBV^{p(\cdot )}(\Omega )\) the approximate gradient \(\nabla u\) exists \(\mathcal {L}^d\)-a.e. in \(\Omega \).

Lemma 2.4

(Approximate gradient) Let \(\Omega \subset \mathbb {R}^d\) be open, bounded (with Lipschitz boundary), let \(p:\Omega \rightarrow [1,+\infty ]\) be a variable exponent, and \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then for \(\mathcal {L}^d\)-a.e. \(x_0 \in \Omega \) there exists a matrix in \(\mathbb {R}^{m\times d}\), denoted by \(\nabla u(x_0)\), such that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \ \varepsilon ^{-d} \mathcal {L}^d\Big (\Big \{x \in B_\varepsilon (x_0) :\, \frac{|u(x) - u(x_0) - \nabla u(x_0)(x-x_0)|}{|x - x_0|}> \varrho \Big \} \Big ) = 0 \text { for all } \varrho >0. \end{aligned}$$

In order to state a Poincaré-Wirtinger inequality in \(GSBV^{p(\cdot )}\), we first fix some notation, following [11, 17]. With given \(a=(a_1,\dots , a_m)\), \(b=(b_1,\dots ,b_m)\in \mathbb {R}^m\), we denote \(a\wedge b:=(\min (a_1,b_1),\dots ,\min (a_m,b_m))\) 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 \(u:B\rightarrow \mathbb {R}^m\), with \(u=(u_1,\dots ,u_m)\), we set

$$\begin{aligned} u_*(s;B):= ((u_1)_*(s;B),\dots , (u_m)_*(s;B)),\quad \textrm{med}(u;B):=u_*\left( \frac{1}{2}\mathcal {L}^d(B);B\right) , \end{aligned}$$

where

$$\begin{aligned} (u_i)_*(s;B):=\inf \{t\in \mathbb {R}:\,\, |\{u_i<t\}\cap B|\ge s\} \qquad \hbox { for } 0\le s \le \mathcal {L}^d(B), \end{aligned}$$

for \(i=1,\dots ,m\).

For every \(u\in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) such that

$$\begin{aligned} \left( 2\gamma _\textrm{iso}\mathcal {H}^{d-1}(J_u\cap B)\right) ^{\frac{d}{d-1}} \le \frac{1}{2}\mathcal {L}^d(B), \end{aligned}$$

we define

$$\begin{aligned} \begin{aligned} \tau '(u;B)&:= u_*\left( \left( 2\gamma _\textrm{iso}\mathcal {H}^{d-1}(J_u\cap B)\right) ^{\frac{d}{d-1}};B\right) , \\ \tau ''(u;B)&:= u_*\left( \mathcal {L}^d(B)-\left( 2\gamma _\textrm{iso}\mathcal {H}^{d-1}(J_u\cap B)\right) ^{\frac{d}{d-1}};B\right) , \end{aligned} \end{aligned}$$

and the truncation operator

$$\begin{aligned} T_Bu(x):= (u(x)\wedge \tau ''(u;B)) \vee \tau '(u;B), \end{aligned}$$
(2.5)

where \(\gamma _\textrm{iso}\) is the dimensional constant in the relative isoperimetric inequality.

We recall the following Poincaré-Wirtinger inequality for SBV functions with small jump set in a ball, which was first proven in the scalar setting in [27, Theorem 3.1], and then extended to vector-valued functions in [17, Theorem 2.5]. In the statement below, the case \(p\ge d\) is discussed in [6, Remark 4.15].

Theorem 2.5

Let \(u\in SBV(B;\mathbb {R}^m)\) and assume that

$$\begin{aligned} \left( 2\gamma _\textrm{iso}\mathcal {H}^{d-1}(J_u\cap B)\right) ^{\frac{d}{d-1}} \le \frac{1}{2}\mathcal {L}^d(B). \end{aligned}$$
(2.6)

If \(1\le p < d\) then

$$\begin{aligned} \left( \int _B |T_Bu-\textrm{med}(u;B)|^{p^*}\,\textrm{d}x\right) ^\frac{1}{p^*} \le \frac{2\gamma _\textrm{iso}p(d-1)}{d-p} \left( \int _B|\nabla u|^p\,\textrm{d}x\right) ^\frac{1}{p} \end{aligned}$$
(2.7)

and

$$\begin{aligned} \mathcal {L}^d(\{T_Bu\ne u\}\cap B) \le 2 \left( 2\gamma _\textrm{iso}\mathcal {H}^{d-1}(J_u\cap B)\right) ^{\frac{d}{d-1}}, \end{aligned}$$
(2.8)

where \(p^*:=\frac{dp}{d-p}\).

If \(p\ge d\), inequality (2.7) holds with \(p^*\) replaced by an arbitrary \(q\in [1, +\infty )\).

Remark 2.6

More generally, Theorem 2.5 holds for functions in \(GSBV(\Omega ;\mathbb {R}^m)\) and for balls \(B\subset \subset \Omega \), by applying the scalar result in SBV to truncated functions \(u_i^M:=M\wedge u_i \vee -M\) for every \(i=1,\dots ,m\), up to understand \(\nabla u\) and \(J_u\) in a weaker sense.

The analogous result in \(GSBV^{p(\cdot )}\) is as follows.

Theorem 2.7

Let \(p:\Omega \rightarrow (1,+\infty )\) be measurable and such that

$$\begin{aligned} {\textrm{either}\,p^-\ge d \quad \textrm{ or }\quad 1<p^-<d,\quad p^+< (p^-)^*.} \end{aligned}$$
(2.9)

Let \(B\subset \subset \Omega \) and \(u\in GSBV^{p(\cdot )}(B;\mathbb {R}^m)\), and assume that (2.6) holds. Then

$$\begin{aligned} \Vert T_Bu-\textrm{med}(u;B)\Vert _{L^{p(\cdot )}(B;\mathbb {R}^m)} \le c(1+\mathcal {L}^d(B))^2\mathcal {L}^d(B)^{\frac{1}{d}+\frac{1}{p^+}-\frac{1}{p^-}} \Vert \nabla {u}\Vert _{L^{p(\cdot )}(B;\mathbb {R}^{m \times d})} \nonumber \\ \end{aligned}$$
(2.10)

for some constant c depending on \(p^-,d\), and

$$\begin{aligned} \mathcal {L}^d(\{T_Bu\ne u\}\cap B) \le 2 \left( 2\gamma _\textrm{iso}\mathcal {H}^{d-1}(J_u\cap B)\right) ^{\frac{d}{d-1}}. \end{aligned}$$
(2.11)

Proof

In view of Remark 2.6, we are reduced to prove the validity of (2.10). For this, it will suffice to write (2.7) for \(p=p^-\), and then the desired inequality will be a consequence of (2.9) and Lemma 2.2. \(\square \)

A first consequence of Theorem 2.7 is the following compactness result, which can be seen as the \(GSBV^{p(\cdot )}\) counterpart of [27, Theorem 3.5]. Motivated by the blow-up analysis of Lemma 3.7, we will prove the result for a fixed ball and a uniformly convergent sequence of continuous variable exponents satisfying (2.9) (see also [26, Theorem 4.1] for a related result under the additional stronger assumption (2.4)).

Theorem 2.8

Let \(B\subset \Omega \) be a ball, \((p_j)_{j\in \mathbb {N}}\) be a sequence of variable exponents \(p_j:B\rightarrow (1,+\infty )\) complying uniformly with (2.9) and converging uniformly to some \(\bar{p}:B\rightarrow (1,+\infty )\) in B. Let \(\{u_j\}_{j\in \mathbb {N}}\subset GSBV^{p_j(\cdot )}(B;\mathbb {R}^m)\) be such that

$$\begin{aligned} \sup _{j\in \mathbb {N}}\int _B |\nabla u_j|^{p_j(y)}\,\textrm{d}y < +\infty ,\quad \lim _{j\rightarrow +\infty } \mathcal {H}^{d-1}(J_{u_j}\cap B)=0. \end{aligned}$$
(2.12)

Then there exist a function \(u_0\in W^{1,\bar{p}(\cdot )}(B;\mathbb {R}^m)\) and a subsequence (not relabeled) of \(\{u_j\}\) such that

$$\begin{aligned} \begin{aligned} \int _B |T_B{u}_j-\textrm{med}(u_j;B)-u_0|^{p_j(y)}\,\textrm{d}y\rightarrow 0,\quad {u}_j-\textrm{med}(u_j;B) \rightarrow u_0 \quad&\mathcal {L}^d-\hbox {a.e. in } B. \end{aligned} \nonumber \\ \end{aligned}$$
(2.13)

Proof

For every \(j\in \mathbb {N}\), we set

$$\begin{aligned} p^-_j:= \inf _{y\in B} p_j(y),\quad p^+_j:= \sup _{y\in B} p_j(y). \end{aligned}$$

Correspondingly, we define

$$\begin{aligned} p^-:=\mathop {\lim \inf }_{j\rightarrow +\infty } p^-_j,\quad p^+:=\mathop {\lim \sup }_{j\rightarrow +\infty } p^+_j. \end{aligned}$$

We set for brevity \(\bar{u}_j:=T_B{u}_j-\textrm{med}(u_j;B)\). Let \(\eta >0\) be fixed such that \(p^-_\eta :=p^--\eta >1\) and \(p^+_\eta :=p^++\eta <(p^-_\eta )^*\). Note that, for j large enough, we have

$$\begin{aligned} p^-_\eta<p^-_j\le p_j(\cdot ) \le p^+_j<p^+_\eta \quad \hbox { on } B. \end{aligned}$$

By virtue of (2.9), (2.10), the definition of \(T_B{u}_j\) and (2.12) we have

$$\begin{aligned} \sup _{j\in \mathbb {N}} \left( \Vert \bar{u}_j\Vert _{L^{p^-_\eta }(B;\mathbb {R}^m)} + \Vert \nabla \bar{u}_j\Vert _{L^{p^-_\eta }(B;\mathbb {R}^{m\times d})} + \mathcal {H}^{d-1}(J_{u_j}\cap B)\right) <+\infty . \end{aligned}$$

This implies, by [4, Theorem 2.2] that there exists \(u_0\in GSBV^{{p^-_\eta }}(B;\mathbb {R}^m)\) and a subsequence (not relabeled) \({u}_j\) such that \(\bar{u}_j \rightarrow u_0\) in measure and

$$\begin{aligned} \mathcal {H}^{d-1}(J_{u_0}\cap B) \le \mathop {\lim \inf }_{j\rightarrow +\infty } \mathcal {H}^{d-1}(J_{T_B{u}_j}\cap B)=0. \end{aligned}$$
(2.14)

With (2.7), since \(p^+_\eta <(p^-_\eta )^*\), we get that \(|\bar{u}_j|^{p^+_\eta }\) is equiintegrable, hence \(\bar{u}_j\) strongly converges to \(u_0\) in \(L^{p^+_\eta }(B;\mathbb {R}^m)\). With Lemma 2.2, and the definition of \(\bar{u}_j\) we then get the first assertion in (2.13). With (2.14), we have \(u_0\in W^{1,{p^-_\eta }}(B;\mathbb {R}^m)\). Now, for each \(\eta >0\) we further have

$$\begin{aligned} \sup _{j\in \mathbb {N}}\int _B |\nabla u_j|^{\bar{p}(y)-\eta }\,\textrm{d}y \le C< +\infty \end{aligned}$$

by the uniform convergence of \(p_j\). With the weak-\(L^1\) convergence of \(\nabla \bar{u}_j\) to \(\nabla u_0\) and Ioffe’s Theorem (see [42]), we get

$$\begin{aligned} \int _B |\nabla u_0|^{\bar{p}(y)-\eta }\,\textrm{d}y \le C \end{aligned}$$

with a bound independent of \(\eta \). Applying the Monotone Convergence Theorem in the set \(\{|\nabla u_0|\ge 1\}\) we get \(u_0 \in W^{1,\bar{p}(\cdot )}(B;\mathbb {R}^m)\). The second assertion in (2.13) follows from (2.11) and (2.12). \(\square \)

To conclude this section, we recall the following result on the approximation of GSBV functions with piecewise constant functions (see [37, Theorem 4.9]), which can be seen as a piecewise Poincaré inequality and essentially relies on the BV coarea formula. We refer the reader for a proof to [36, Theorem 2.3], although the argument can be retrieved in previous literature (see, e.g., [4, 14]).

Theorem 2.9

Let \(d\ge 1\) and \(z\in GSBV(\Omega ;\mathbb {R}^m)\) with

$$\begin{aligned} \Vert \nabla z\Vert _{L^1(\Omega ;\mathbb {R}^{m\times d})} + \mathcal {H}^{d-1}(J_z) < +\infty . \end{aligned}$$

Let \(D\subset \Omega \) be a Borel set with finite perimeter. Let \(\theta >0\) be fixed. Then there exists a partition \((P_l)_{l=1}^\infty \) of D, made of sets of finite perimeter, and a piecewise constant function \(z_\textrm{pc}:=\sum _{l=1}^\infty b_l \chi _{P_l}\) such that

  1. (i)

    \(\displaystyle \sum _{l=1}^\infty \mathcal {H}^{d-1}((\partial ^*P_l\cap D^1)\backslash J_z)\le \theta \);

  2. (ii)

    \(\Vert z-z_\textrm{pc}\Vert _{L^\infty (D;\mathbb {R}^m)}\le c \theta ^{-1}\Vert \nabla z\Vert _{L^1(D;\mathbb {R}^{m\times d})}\),

for a dimensional constant \(c=c(d)>0\), where \(D^1\) denotes the set of points with density one. If, in addition, the i-th component \(z^i\) satisfies the bound \(\Vert z^i\Vert _{L^\infty (D;\mathbb {R})}\le M\), then also \(\Vert z^i_\textrm{pc}\Vert _{L^\infty (D;\mathbb {R})}\le M\) holds.

3 The integral representation result

In this section we will establish an integral representation result in the space \(GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) for \(m \in \mathbb {N}\), where the variable exponent \(p:\Omega \rightarrow (1,+\infty )\) complies with the following assumptions

(\(P_1\)):

\(p^->1\) and \(p^+<+\infty \);

(\(P_2\)):

p is log-Hölder continuous on \(\Omega \) (see (2.4)).

We consider functionals \(\mathcal {F}:GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m) \times \mathcal {B}(\Omega ) \rightarrow [0,+\infty )\) with the following general assumptions:

(\(H_1\)):

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

(\(H_2\)):

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

(\(H_3\)):

\(\mathcal {F}(\cdot , A)\) is local for any \(A \in \mathcal {A}(\Omega )\), in the sense that if \(u,v \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) satisfy \(u=v\) a.e. in A, then \(\mathcal {F}(u,A) = \mathcal {F}(v,A)\);

(\(H_4\)):

there exist \(0< \alpha < \beta \) such that for any \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(B \in \mathcal {B}(\Omega )\) we have

$$\begin{aligned}{} & {} \alpha \bigg (\int _{ B } |\nabla u|^{p(x)} \, \textrm{d} x+ \mathcal {H}^{d-1}(J_u \cap B)\bigg ) \le \mathcal {F}(u,B) \\ {}{} & {} \qquad \quad \le \beta \bigg (\int _{ B } (1 + |\nabla u|^{p(x)}) \, \textrm{d} x+ \mathcal {H}^{d-1}(J_u \cap B)\bigg ). \end{aligned}$$

For every \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(A \in \mathcal {A}(\Omega )\) we define

$$\begin{aligned} \textbf{m}_{\mathcal {F}}(u,A) = \inf _{v \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)} \ \lbrace \mathcal {F}(v,A): \ v = u \ \text { in a neighborhood of } \partial A \rbrace . \end{aligned}$$
(3.1)

Moreover, for \(x_0 \in \Omega \), \(u_0 \in \mathbb {R}^m\), and \(\xi \in \mathbb {R}^{m \times d} \) we introduce the affine functions \(\ell _{x_0,u_0,\xi }:\mathbb {R}^d \rightarrow \mathbb {R}^m\) by

$$\begin{aligned} \ell _{x_0,u_0,\xi }(x) = u_0 + \xi (x-x_0), \end{aligned}$$
(3.2)

and, for \(a,b \in \mathbb {R}^m\), \(\nu \in \mathbb {S}^{d-1}\) we define \(u_{x_0,a,b,\nu } :\mathbb {R}^d \rightarrow \mathbb {R}^m\) by

$$\begin{aligned} u_{x_0,a,b,\nu }(x) = {\left\{ \begin{array}{ll} a &{} \text {if } (x-x_0) \cdot \nu > 0,\\ b &{} \text {if } (x-x_0) \cdot \nu < 0. \end{array}\right. } \end{aligned}$$
(3.3)

The main result of this section is the following integral representation theorem.

Theorem 3.1

(Integral representation in \(GSBV^{p(\cdot )}\)) Let \(\Omega \subset \mathbb {R}^d\) be open, bounded with Lipschitz boundary, let \(m \in \mathbb {N}\). Let \(p:\Omega \rightarrow (1,+\infty )\) be a variable exponent complying with (\({P}_{1}\))-(\({P}_{2}\)), and suppose that \(\mathcal {F}:GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m) \times \mathcal {B}(\Omega ) \rightarrow [0,+\infty )\) satisfies (\(\hbox {H}_{1}\))–(\(\hbox {H}_{4}\)). Then

$$\begin{aligned} \mathcal {F}(u,B) = \int _B f\big (x,u(x),\nabla u(x)\big ) \, \textrm{d}x + \int _{J_u\cap B} g\big (x,u^+(x),u^-(x),\nu _u(x)\big )\, \textrm{d} \mathcal {H}^{d-1}(x) \end{aligned}$$

for all \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(B \in \mathcal {B}(\Omega )\), where f is given by

$$\begin{aligned} f(x_0,u_0,\xi ) = \limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\ell _{x_0,u_0,\xi },B_\varepsilon (x_0))}{\gamma _d\varepsilon ^{d}} \end{aligned}$$
(3.4)

for all \(x_0 \in \Omega \), \(u_0 \in \mathbb {R}^m\), \(\xi \in \mathbb {R}^{m \times d}\), and g is given by

$$\begin{aligned} g(x_0,a,b,\nu ) = \limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(u_{x_0,a,b,\nu },B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \end{aligned}$$
(3.5)

for all \( x_0 \in \Omega \), \(a,b \in \mathbb {R}^m\), and \(\nu \in \mathbb {S}^{d-1}\).

3.1 Fundamental estimate

In this section we prove an important tool in the proof of the integral representation, namely a fundamental estimate in \(GSBV^{p(\cdot )}\) for functionals \(\mathcal {F}\).

Lemma 3.2

(Fundamental estimate in \(GSBV^{p(\cdot )}\)) Let \(\Omega \subset \mathbb {R}^d\) be open and bounded, and let \(p:\Omega \rightarrow (1,+\infty )\) be a variable exponent in \(\Omega \) satisfying (\(\hbox {P}_{1}\)). Let \(\eta >0\) and let \(D', D'', E \in \mathcal {A}(\Omega )\) with \(D' \subset \subset D''\), and set \(\delta :=\frac{1}{2}\textrm{dist}(D',\partial D'')\). For every functional \(\mathcal {F}\) satisfying (\(\hbox {H}_{1}\)), (\(\hbox {H}_{3}\)), and (\(\hbox {H}_{4}\)) and for every \(u \in GSBV^{p(\cdot )}(D';\mathbb {R}^m)\), \(v \in GSBV^{p(\cdot )}(E;\mathbb {R}^m)\) there exists a function \(\varphi \in C^\infty (\mathbb {R}^d;[0,1])\) such that \(w:= \varphi u + (1- \varphi )v \in GSBV^{p(\cdot )}(D'\cup E;\mathbb {R}^m)\) satisfies

$$\begin{aligned} \mathrm{(i)}&\ \ \mathcal {F} ( w, D' \cup E) \le (1+ \eta )\big (\mathcal {F}(u,D'') + \mathcal {F}(v, E) \big ) + M \int _{F}\left( \frac{|u-v|}{\delta }\right) ^{p(x)}\,\textrm{d}x\nonumber \\&\qquad \qquad \qquad \qquad +\eta \mathcal {L}^d(D' \cup E), \nonumber \\ \mathrm{(ii)}&\ \ w = u \text { on } D' \text { and } w = v \text { on } E {\setminus } D'', \end{aligned}$$
(3.6)

where \(F:= (D'' {\setminus } D') \cap E\) and \(M=M(D',D'',E,p^+,\eta )>0\) depends only on \(D',D'',E,p^+,\eta \), but is independent of u and v. Moreover, if for \(\varepsilon >0\) and \(x_0 \in \mathbb {R}^d\) we have \(D'_{\varepsilon ,x_0}, D''_{\varepsilon ,x_0}, E_{\varepsilon ,x_0} \subset \Omega \), then

$$\begin{aligned} M(D'_{\varepsilon ,x_0}, D''_{\varepsilon ,x_0},E_{\varepsilon ,x_0},p^+,\eta ) = M(D',D'',E,p^+,\eta ), \end{aligned}$$
(3.7)

and the remainder term is

$$\begin{aligned} M \int _{F_{\varepsilon ,x_0}}\left( \frac{|u-v|}{\delta \varepsilon }\right) ^{p(x)}\,\textrm{d}x, \end{aligned}$$

where we used the notation introduced in (2.1).

Proof

We choose \(k \in \mathbb {N}\) such that

$$\begin{aligned} k \ge \max \Big \{\frac{ 3^{p^+-1}\beta }{\eta \alpha }, \frac{\beta }{\eta } \Big \}, \end{aligned}$$
(3.8)

and for \(i=1,\ldots ,k\), we set

$$\begin{aligned} D_{i+1}:=\left\{ x\in D'':\,\, \textrm{dist}(x,D')<\frac{\delta i}{k}\right\} . \end{aligned}$$

We then have \(D_1:=D' \subset \subset D_2 \subset \subset \cdots \subset \subset D_{k+1} \subset \subset D''\). Correspondingly, let \(\varphi _i\in C_0^\infty (D_{i+1})\) with \(0\le \varphi _i\le 1\) and \(\varphi _i=1\) in a neighborhood \(U_i\) of \(\overline{D_i}\). Note that \(\Vert \nabla \varphi _i\Vert _\infty \le \frac{2k}{\delta }\).

Let \(u \in GSBV^{p(\cdot )}(D'';\mathbb {R}^m)\) and \(v \in GSBV^{p(\cdot )}(E;\mathbb {R}^m)\) be such that \(u-v \in L^{p(\cdot )}((D'' {\setminus } D') \cap E;\mathbb {R}^m)\), as otherwise the result is trivial. We define the function \(w_i = \varphi _i u + (1-\varphi _i)v \in GSBV^{p(\cdot )}(D' \cup E;\mathbb {R}^m)\) (this can be easily proved as in [22, Lemma 2.11]), where u and v are extended arbitrarily outside \(D''\) and E, respectively. Letting \(I_i = D'' \cap (D_{i+1} {\setminus } \overline{D_i})\) we get by (\({H}_{1}\)) and (\({H}_{3}\))

$$\begin{aligned} \mathcal {F}(w_i,D' \cup E)&\le \mathcal {F}(u, (D' \cup E) \cap U_i) + \mathcal {F}(v, E {\setminus } \textrm{supp}\,\varphi _i) + \mathcal {F}(w_i,I_i) \nonumber \\&\le \mathcal {F}(u,D'') + \mathcal {F}(v,E) + \mathcal {F}(w_i,I_i). \end{aligned}$$
(3.9)

For the last term, we compute using (\({H}_{4}\))

$$\begin{aligned} \mathcal {F}(w_i,I_i)&\le \beta \int _{I_i} (1+|\nabla w_i|^{p(x)})\, \textrm{d}x + \beta \mathcal {H}^{d-1}(J_{w_i} \cap J_i)\\&\le \beta \int _{I_i} (1+|\varphi _i \nabla u + (1-\varphi _i)\nabla v + \nabla \varphi _i (u-v) |^{p(x)})\,\textrm{d}x + \beta \mathcal {H}^{d-1}( (J_{u} \cup J_v) \cap I_i) \\&\le \beta \mathcal {L}^d(I_i) + 3^{p^+-1}\beta \int _{I_i} \big (|\nabla u|^{p(x)} + |\nabla v|^{p(x)} + |\nabla \varphi _i|^{p(x)} |u-v|^{p(x)} \big )\,\textrm{d}x \\&\quad + \beta \mathcal {H}^{d-1}( J_{u} \cap I_i) + \beta \mathcal {H}^{d-1}( J_v \cap I_i) \\&\le 3^{p^+-1} \frac{\beta }{\alpha } \big ( \mathcal {F}(u,I_i) + \mathcal {F}(v,I_i)\big ) + (2k)^{p^+}\cdot 3^{p^+-1}\beta \int _{I_i}\left( \frac{|u-v|}{\delta }\right) ^{p(x)}\,\textrm{d}x\\&\quad + \beta \mathcal {L}^d(I_i). \end{aligned}$$

Consequently, recalling (3.8) and using (\({H}_{1}\)) we find \(i_0 \in \lbrace 1, \ldots , k \rbrace \) such that

$$\begin{aligned} \begin{aligned} \mathcal {F}(w_{i_0},I_{i_0})&\le \frac{1}{k}\sum _{i=1}^k \mathcal {F}(w_{i},I_i) \\&\le \eta \big ( \mathcal {F}(u,D'') + \mathcal {F}(v,E)\big ) + M \int _{F}\left( \frac{|u-v|}{\delta }\right) ^{p(x)}\,\textrm{d}x + \eta \mathcal {L}^d(F), \end{aligned} \end{aligned}$$

where \(M:= (2k)^{p^+}\cdot 3^{p^+-1} \beta k^{-1}\). This along with (3.9) concludes the proof of (3.6) by setting \(w = w_{i_0}\). To see the scaling property (3.7), it suffices to use the cut-off functions \(\varphi ^\varepsilon _i\in C_0^\infty ((D_{i+1})_{\varepsilon ,x_0};[0,1])\) \(i=1,\ldots ,k\), defined by \(\varphi _i^\varepsilon (x) = \varphi _i( x_0 + \frac{(x-x_0)}{\varepsilon }) \) for \(x \in (D_{i+1})_{\varepsilon ,x_0}\). This concludes the proof. \(\square \)

3.2 The global method

This section is entirely devoted to the proof of Theorem 3.1. As a first step, we show that \(\mathcal {F}\) and \(\textbf{m}_{\mathcal {F}}\), defined by (3.1), have the same Radon-Nikodym derivative with respect to the measure

$$\begin{aligned} \mu := \mathcal {L}^d\lfloor _{\Omega }\,\, +\,\, \mathcal {H}^{d-1}\lfloor _{J_u \cap \Omega }. \end{aligned}$$
(3.10)

Lemma 3.3

Let \(p:\Omega \rightarrow (1,+\infty )\) be a variable exponent satisfying (\(\hbox {P}_{1}\)). Suppose that \(\mathcal {F}\) satisfies (\(\hbox {H}_{1}\))–(\(\hbox {H}_{4}\)). Let \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(\mu \) as in (3.10). Then for \(\mu \)-a.e. \(x_0 \in \Omega \) we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(u,B_\varepsilon (x_0))}{\mu (B_\varepsilon (x_0))} = \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\mu (B_\varepsilon (x_0))}. \end{aligned}$$

The proof of this lemma is postponed to the end of this section. The second step in the proof of Theorem 3.1 is that, asymptotically as \(\varepsilon \rightarrow 0\), the minimization problems \(\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))\) and \(\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{bulk}_{x_0},B_\varepsilon (x_0))\) coincide for \(\mathcal {L}^{d}\)-a.e. \(x_0 \in \Omega \), where we write \(\bar{u}^\textrm{bulk}_{x_0}:= \ell _{x_0,u(x_0),\nabla u(x_0)} \) for brevity, see (3.2).

Lemma 3.4

Let \(p:\Omega \rightarrow (1,+\infty )\) be a Riemann-integrable variable exponent satisfying (\(\hbox {P}_{1}\)). Suppose that \(\mathcal {F}\) satisfies (\(\hbox {H}_{1}\)) and (\(\hbox {H}_{3}\))–(\(\hbox {H}_{4}\)) and let \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then for \(\mathcal {L}^{d}\)-a.e. \(x_0 \in \Omega \) we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _{d}\varepsilon ^{d}} = \limsup _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{bulk}_{x_0},B_\varepsilon (x_0))}{\gamma _{d}\varepsilon ^{d}}. \end{aligned}$$
(3.11)

The final step is that, asymptotically as \(\varepsilon \rightarrow 0\), the minimization problems \(\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))\) and \(\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{surf}_{x_0},B_\varepsilon (x_0))\) coincide for \(\mathcal {H}^{d-1}\)-a.e. \(x_0 \in J_u\), where we write \(\bar{u}^\textrm{surf}_{x_0}:= u_{x_0,u^+(x_0),u^-(x_0),\nu _u(x_0)}\) for brevity, see (3.3).

Lemma 3.5

Let \(p:\Omega \rightarrow (1,+\infty )\) be a variable exponent satisfying (\(\hbox {P}_{1}\))–(\(\hbox {P}_{2}\)). Suppose that \(\mathcal {F}\) satisfies (\(\hbox {H}_{1}\)) and (\(\hbox {H}_{3}\))–(\(\hbox {H}_{4}\)) and let \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then for \(\mathcal {H}^{d-1}\)-a.e. \(x_0 \in J_u\) we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} = \limsup _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{surf}_{x_0},B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}. \end{aligned}$$
(3.12)

We defer the proof of Lemma 3.4 and Lemma 3.5 to Sect. 3.3 and Sect. 3.4, respectively. Now, we proceed to prove Theorem 3.1.

Proof of Theorem 3.1

In view of the assumption (\({H}_{4}\)) on \(\mathcal {F}\) and of the Besicovitch derivation theorem (cf. [6, Theorem 2.22]), we need to show that

$$\begin{aligned} \frac{\textrm{d}\mathcal {F}(u,\cdot )}{\textrm{d}\mathcal {L}^{d}}(x_0)&= f\big (x_0,u(x_0),\nabla u(x_0)\big ), \quad \hbox { for } \mathcal {L}^{d}-\hbox {a.e.}\ x_0 \in \Omega , \end{aligned}$$
(3.13)
$$\begin{aligned} \frac{\textrm{d}\mathcal {F}(u,\cdot )}{\textrm{d}\mathcal {H}^{d-1}\lfloor _{J_u}}(x_0)&= g\big (x_0,u^+(x_0),u^-(x_0),\nu _u(x_0)\big ), \quad \hbox { for } \mathcal {H}^{d-1}-\hbox {a.e.}\ x_0 \in J_u, \end{aligned}$$
(3.14)

where f and g were defined in (3.4) and (3.5), respectively.

By Lemma 3.3 and the fact that \(\lim _{\varepsilon \rightarrow 0} (\gamma _{d}\varepsilon ^{d})^{-1}\mu (B_\varepsilon (x_0))=1\) for \(\mathcal {L}^{d}\)-a.e. \(x_0 \in \Omega \) we deduce

$$\begin{aligned} \frac{\textrm{d}\mathcal {F}(u,\cdot )}{\textrm{d}\mathcal {L}^{d}}(x_0) = \lim _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(u,B_\varepsilon (x_0))}{\mu (B_\varepsilon (x_0))} = \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\mu (B_\varepsilon (x_0))} = \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _{d}\varepsilon ^{d}} < \infty \end{aligned}$$

for \(\mathcal {L}^{d}\)-a.e. \(x_0 \in \Omega \). Then, (3.13) follows from (3.4) and Lemma 3.4. By Lemma 3.3 and the fact that \(\lim _{\varepsilon \rightarrow 0} (\gamma _{d-1}\varepsilon ^{d-1})^{-1}\mu (B_\varepsilon (x_0))=1\) for \(\mathcal {H}^{d-1}\)-a.e. \(x_0 \in J_u\) we deduce

$$\begin{aligned} \frac{\textrm{d}\mathcal {F}(u,\cdot )}{\textrm{d}\mathcal {H}^{d-1}\lfloor _{J_u}}(x_0) = \lim _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(u,B_\varepsilon (x_0))}{\mu (B_\varepsilon (x_0))} = \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\mu (B_\varepsilon (x_0))} = \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} < \infty \end{aligned}$$

for \(\mathcal {H}^{d-1}\)-a.e. \(x_0 \in J_u\). Now, (3.14) follows from (3.5) and Lemma 3.5. \(\square \)

In the remaining part of the section we prove Lemma 3.3. For this, we need to fix some notation. For \(\delta >0\) and \(A \in \mathcal {A}(\Omega )\), we define

$$\begin{aligned} \textbf{m}^\delta _{\mathcal {F}}(u,A)&= \inf \Bigg \{ \sum \limits _{i=1}^\infty \textbf{m}_{\mathcal {F}}(u,B_i):\ B_i \subset A \text { pairwise disjoint balls}, \, \textrm{diam}(B_i) \le \delta ,\\&\quad \qquad \qquad \mu \Big ( A {\setminus } \bigcup \limits _{i=1}^\infty B_i\Big ) = 0\Bigg \}, \end{aligned}$$

where \(\mu \) is defined in (3.10). As \(\textbf{m}^\delta _{\mathcal {F}}(u,A) \) is decreasing in \(\delta \), we can also introduce

$$\begin{aligned} \textbf{m}^*_{\mathcal {F}}(u,A) := \lim _{\delta \rightarrow 0 } \textbf{m}^\delta _{\mathcal {F}}(u,A). \end{aligned}$$
(3.15)

In the following lemma, we prove that \(\mathcal {F}\) and \(\textbf{m}^*_{\mathcal {F}}\) coincide under our assumptions.

Lemma 3.6

Let \(p:\Omega \rightarrow (1,+\infty )\) be complying with (\({P}_{1}\)). Suppose that \(\mathcal {F}\) satisfies (\({H_1}\))–(\({H_4}\)) and let \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then, for all \(A \in \mathcal {A}(\Omega )\) there holds \(\mathcal {F}(u,A) = \textbf{m}^*_{\mathcal {F}}(u,A)\).

Proof

We can follow the argument of [11, Lemma 4] (see also [12, Lemma 3.3]). We start by proving the inequality \(\textbf{m}_{\mathcal {F}}^*(u,A) \le \mathcal {F}(u,A)\). For each ball \(B \subset A\) we have \(\textbf{m}_{\mathcal {F}}(u,B) \le \mathcal {F}(u,B)\) by definition. By (\({H}_{1}\)) we get \(\textbf{m}_{\mathcal {F}}^\delta (u,A) \le \mathcal {F}(u,A)\) for all \(\delta >0\), whence the assertion follows taking into account (3.15).

We now address the reverse inequality. We fix \(A\in \mathcal {A}(\Omega )\) and \(\delta >0\). Let \((B^\delta _i)_i\) be balls as in the definition of \(\textbf{m}_{\mathcal {F}}^\delta (u,A)\) such that

$$\begin{aligned} \sum \limits _{i=1}^\infty \textbf{m}_{\mathcal {F}}(u,B^\delta _i) \le \textbf{m}_{\mathcal {F}}^\delta (u,A) + \delta . \end{aligned}$$
(3.16)

By the definition of \(\textbf{m}_{\mathcal {F}}\), we find \(v_i^\delta \in GSBV^{p(\cdot )}(B_i^\delta ;\mathbb {R}^m)\) such that \(v_i^\delta = u\) in a neighborhood of \(\partial B_i^\delta \) and

$$\begin{aligned} \mathcal {F}(v_i^\delta ,B_i^\delta ) \le \textbf{m}_{\mathcal {F}}(u,B_i^\delta ) + \delta \mathcal {L}^d(B_i^\delta ). \end{aligned}$$
(3.17)

We define

$$\begin{aligned} v^{\delta ,n} := \sum \limits _{i=1}^n v^\delta _i \chi _{B^\delta _i} + u \chi _{N_0^{\delta ,n}} \quad \text {for } n\in \mathbb {N},\quad \quad v^\delta := \sum \limits _{i=1}^\infty v^\delta _i \chi _{B^\delta _i} + u \chi _{N_0^\delta }, \end{aligned}$$
(3.18)

where \(N_0^{\delta ,n}:= \Omega {\setminus } \bigcup _{i=1}^n B^\delta _i\) and \(N_0^\delta := \Omega {\setminus } \bigcup _{i=1}^\infty B^\delta _i\). By construction, we have that each \( v^{\delta ,n} \) lies in \(GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and that

$$\begin{aligned} \sup _{n \in \mathbb {N}} \left( \int _\Omega |\nabla v^{\delta ,n}(x)|^{p(x)}\,\textrm{d}x + \mathcal {H}^{d-1}(J_{ v^{\delta ,n} })\right) <+\infty \end{aligned}$$
(3.19)

by (3.16)–(3.17) and (\({H}_{4}\)). Moreover, \( v^{\delta ,n} \rightarrow v^\delta \) pointwise a.e. in \(\Omega \), and then in measure on \(\Omega \). Then, [4, Theorem 2.2] combined with the compactness in \(L^0\) of \((v^{\delta ,n})\) yields \(v^\delta \in GSBV^{p^{-}}(\Omega ;\mathbb {R}^m)\) and \(\nabla v^{\delta ,n}\rightharpoonup \nabla v^{\delta }\) weakly in \(L^{p^-}(\Omega ;\mathbb {R}^{m \times d})\). Now, by Ioffe’s Theorem (see [42]) and (3.19) we get

$$\begin{aligned} \int _\Omega |\nabla v^{\delta }(x)|^{p(x)}\,\textrm{d}x \le \displaystyle \mathop {\lim \inf }_{n\rightarrow +\infty } \int _\Omega |\nabla v^{\delta ,n}(x)|^{p(x)}\,\textrm{d}x <+\infty , \end{aligned}$$

whence \(v^\delta \in GSBV^{p{(\cdot )}}(\Omega ;\mathbb {R}^m)\). We have

$$\begin{aligned} \mathcal {F}(v^\delta ,A)&= \sum \limits _{i=1}^\infty \mathcal {F}(v_i^\delta ,B_i^\delta ) + \mathcal {F}(u,N_0^\delta \cap A) \le \sum \limits _{i=1}^\infty \big (\textbf{m}_{\mathcal {F}}(u,B_i^\delta ) + \delta \mathcal {L}^d(B_i^\delta )\big ) \nonumber \\&\le \textbf{m}_{\mathcal {F}}^\delta (u,A) + \delta (1+\mathcal {L}^d(A)), \end{aligned}$$
(3.20)

where we also used the fact that \(\mu (N_0^\delta \cap A) = \mathcal {F}(u,N_0^\delta \cap A) = 0\) by the definition of \((B^\delta _i)_i\) and (\({H}_{4}\)). For later purpose, we also note by (\({H}_{4}\)) that this implies

$$\begin{aligned} \Vert \nabla v^\delta \Vert _{L^{p^-}(A;\mathbb {R}^{m\times d})} + \mathcal {H}^{d-1}(J_{v^\delta } \cap A) \le c\alpha ^{-1} \big (\textbf{m}_{\mathcal {F}}^\delta (u,A) + \delta (1+\mathcal {L}^d(A)) \big ). \end{aligned}$$
(3.21)

We now claim that

$$\begin{aligned} w^\delta := u-v^\delta \rightarrow 0 \quad \hbox { in measure on } A. \end{aligned}$$
(3.22)

With this, using (\({H}_{2}\)), (3.15), and (3.20) we will get the required inequality \(\textbf{m}_{\mathcal {F}}^*(u,A) \ge \mathcal {F}(u,A)\) in the limit as \(\delta \rightarrow 0\). To prove (3.22), we first note that \(w^\delta \lfloor _{B^\delta _i} \in GSBV^{p^-}(B^\delta _i;\mathbb {R}^m)\) has trace zero on \(\partial B^\delta _i\). Then, setting for every \(M>0\)

$$\begin{aligned} w^{\delta ,M}:= (-M \vee w^\delta )\wedge M, \end{aligned}$$

from the classical Poincaré inequality we get

$$\begin{aligned} \begin{aligned} \Vert w^{\delta ,M}\Vert _{L^1(B^\delta _i;\mathbb {R}^m)}&\le C\delta |Dw^{\delta ,M}|(B^\delta _i), \end{aligned} \end{aligned}$$

whence

$$\begin{aligned} \Vert w^{\delta ,M}\Vert _{L^1(A;\mathbb {R}^m)} \le C\delta |Dw^{\delta ,M}|(\cup _{i=1}^\infty B^\delta _i) \le C\delta |Dw^{\delta ,M}|(A), \end{aligned}$$

where \(|Dw^{\delta ,M}|(A)\) is bounded in view of (3.21) and the fact that \(u\in GSBV(A;\mathbb {R}^m)\), since

$$\begin{aligned} |Dw^{\delta ,M}|(A)\le & {} \int _A|\nabla v^\delta |\,\textrm{d}x + \int _A|\nabla u|\,\textrm{d}x + 2M \left( \mathcal {H}^{d-1}(J_{v^\delta }\cap A)+\mathcal {H}^{d-1}(J_{u}\cap A)\right) \\{} & {} <+\infty . \end{aligned}$$

This implies \(w^{\delta ,M}\rightarrow 0\) in \(L^1(A;\mathbb {R}^m)\), and then in measure on A, as \(\delta \rightarrow 0\) for every \(M>0\). Now, with fixed \(M=1\) and \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} E_\varepsilon ^\delta :=\{x\in A:\,\,|w^\delta (x)|>\varepsilon \} \subseteq \{x\in A:\,\,|w^{\delta ,1}(x)|>\varepsilon \}, \end{aligned}$$

whence \(\mathcal {L}^d(E_\varepsilon ^\delta )\rightarrow 0\) as \(\delta \rightarrow 0\), thus proving (3.22). The proof is concluded. \(\square \)

Proof of Lemma 3.3

We may follow the same argument as in [11, Proofs of Lemma 5 and Lemma 6], by exploiting also Lemma 3.6. We then omit the details. \(\square \)

To conclude the proof of Theorem 3.1, it remains to prove Lemmas 3.4 and 3.5. This is the subject of the following two sections.

3.3 The bulk density

This section is devoted to the proof of Lemma 3.4. With the following lemma, we analyze the blow-up at points with approximate gradient, which exists for \(\mathcal {L}^d\)-a.e. point in \(\Omega \) by Lemma 2.4. It is noteworthy that in order to develop the blow-up arguments of this section, it will suffice to consider a Riemann integrable exponent p satisfying (\({P}_{1}\)), as \(\mathcal {L}^d\)-a.e. \(x\in \Omega \) is a continuity point for p. On the contrary, the stronger assumption (\({P}_2\)) will be crucial in Sect. 3.4 when dealing with the surface scaling.

Lemma 3.7

Let \(p:\Omega \rightarrow (1,+\infty )\) be a Riemann integrable variable exponent complying with (\(\hbox {P}_{1}\)). Let \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then for \(\mathcal {L}^{d}\)-a.e. \(x_0 \in \Omega \) and \(\mathcal {L}^{1}\)-a.e. \(\sigma \in (0,1)\) there exists a sequence \(u_\varepsilon \in GSBV^{p(\cdot )}(B_\varepsilon (x_0);\mathbb {R}^m)\) such that

$$\begin{aligned} \begin{aligned}&(i)\,\, u_\varepsilon =u \,\, \hbox {in } B_\varepsilon (x_0)\backslash \overline{B_{\sigma \varepsilon }(x_0)},\quad \displaystyle \lim _{\varepsilon \rightarrow 0}\,{\varepsilon ^{-(d+1)}} \mathcal {L}^d(\{u_\varepsilon \ne u\}\cap B_{\varepsilon }(x_0))=0; \\&(ii)\,\, \displaystyle \lim _{\varepsilon \rightarrow 0} \ \varepsilon ^{-d} \int _{B_{\sigma \varepsilon }(x_0)} \left( \frac{|u_\varepsilon (x) - u(x_0)-\nabla u(x_0)(x-x_0)|}{\varepsilon } \right) ^{p(x)} \, \textrm{d}x = 0;\\&(iii)\,\, \displaystyle \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d} \mathcal {H}^{d-1}(J_{u_\varepsilon })=0. \end{aligned} \end{aligned}$$
(3.23)

If, in addition, \(u \in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\), then \(u_\varepsilon \) also satisfies

$$\begin{aligned} \begin{aligned}&(i)'\,\, \lim _{\varepsilon \rightarrow 0}\varepsilon ^{-(d+1)}\int _{B_\varepsilon (x_0)}|u_\varepsilon -u|\,\textrm{d}x=0, \\&(iii)'\,\, \lim _{\varepsilon \rightarrow 0}\varepsilon ^{-d}\int _{J_{u_\varepsilon }}|[u_\varepsilon ]|\,\textrm{d}\mathcal {H}^{d-1}=0. \end{aligned} \end{aligned}$$

Proof

It will suffice to treat the scalar case \(m=1\). Let \(x_0\in \Omega \) be such that

$$\begin{aligned} {(a)}&\ \ \lim _{\varepsilon \rightarrow 0} \ \varepsilon ^{-d} \int _{B_{\varepsilon }(x_0)} \big |\nabla u(x) - \nabla u(x_0)\big |^{p(x)} \, \textrm{d}x = 0\,;\nonumber \\ {(b)}&\ \ \lim _{\varepsilon \rightarrow 0} \ \varepsilon ^{-d}\,\mathcal {H}^{d-1}(J_{u} \cap B_\varepsilon (x_0))= 0\,; \\ {(c)}&\ \ {\lim _{\varepsilon \rightarrow 0} \ \varepsilon ^{-d} \mathcal {L}^d\Big (\Big \{x \in B_\varepsilon (x_0) :\, \frac{|u(x) - u(x_0) - \nabla u(x_0)(x-x_0)|}{\varepsilon }> \varrho \Big \} \Big ) = 0 \text { for all } \varrho >0 }.\nonumber \end{aligned}$$
(3.24)

Properties (a) and (c) hold for \(\mathcal {L}^d\)-a.e. \(x_0 \in \Omega \) by Theorem 2.3 since \(|\nabla u| \in L^{p(\cdot )}(\Omega ;\mathbb {R}^{m\times d})\) and by Lemma 2.4, respectively, while (b) follows from the fact that \(J_u\) is countably \(\mathcal {H}^{d-1}\)-rectifiable (see, e.g., [6]). We can also assume that \(x_0\) is a continuity point for p(x); hence, it is not restrictive to assume that (2.9) holds, up to replacing \(\Omega \) with a fixed neighborhood of \(x_0\) where it is satisfied.

We set \(\bar{u}_\varepsilon (x):=\frac{u(x)-u(x_0)}{\varepsilon }\), define the truncated functions \(T_\varepsilon \bar{u}_\varepsilon := T_{B_{\varepsilon }(x_0)}\bar{u}_\varepsilon \) as in (2.5), and \(v_\varepsilon (x):= u(x_0)+\varepsilon T_\varepsilon \bar{u}_\varepsilon (x)\).

Note that

$$\begin{aligned} |\nabla v_\varepsilon |\le |\nabla u| \quad \mathcal {L}^d-\hbox { a.e.} \end{aligned}$$
(3.25)

and \(J_{v_\varepsilon }\subseteq J_u\), \(\mathcal {H}^{d-1}(J_{v_\varepsilon }\backslash J_u)=0\). This along with (3.24)(b) implies (3.23)(iii).

We notice that (3.24)(b) implies also (2.6) for \(\varepsilon \) small enough, which combined with (2.11) gives

$$\begin{aligned} \begin{aligned} 0\le \varepsilon ^{-d} \int _0^1 \mathcal {H}^{d-1}(\{v_\varepsilon \ne u\}\cap \partial B_{\sigma \varepsilon }(x_0))\,\textrm{d}\sigma&= \frac{2}{\varepsilon ^{d+1}} \mathcal {L}^d(\{v_\varepsilon \ne u\}\cap B_{\varepsilon }(x_0)) \\&\le \frac{4 (2\gamma _\textrm{iso})^{\frac{d}{d-1}}}{\varepsilon ^{d+1}}\left( \mathcal {H}^{d-1}(J_{u}\cap B_\varepsilon (x_0))\right) ^{\frac{d}{d-1}} \\&\le 4 (2\gamma _\textrm{iso}\gamma _d)^{\frac{d}{d-1}}C^{\frac{d}{d-1}}\varepsilon ^\frac{1}{d-1} \rightarrow 0 \end{aligned} \nonumber \\ \end{aligned}$$
(3.26)

as \(\varepsilon \rightarrow 0\).

Therefore, for every sequence \(\varepsilon \rightarrow 0\) one can find a subsequence (not relabeled) such that, for \(\mathcal {L}^1\)-a.e. \(\sigma \in (0,1)\),

$$\begin{aligned} \begin{aligned} \mu (\partial B_{\sigma \varepsilon }(x_0)) = \mathcal {H}^{d-1}(\partial B_{\sigma \varepsilon }(x_0)\cap J_{v_\varepsilon })=0, \\ \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d} \mathcal {H}^{d-1}(\{v_\varepsilon \ne u\}\cap \partial B_{\sigma \varepsilon }(x_0)) = 0. \end{aligned} \end{aligned}$$
(3.27)

Now, we fix a sequence \(\varepsilon \rightarrow 0\) and consider a subsequence (not relabeled) and \(\sigma \in (0,1)\) for which (3.27) holds. We then define

$$\begin{aligned} u_\varepsilon (x) = {\left\{ \begin{array}{ll} v_\varepsilon (x) &{} \hbox { in } B_{\sigma \varepsilon }(x_0),\\ u(x) &{} \hbox { in } B_\varepsilon (x_0)\backslash \overline{B_{\sigma \varepsilon }(x_0)}. \end{array}\right. } \end{aligned}$$

From the definition of \(u_\varepsilon \) and the argument of (3.26) we get the assertions in (3.23)(i). We now prove (3.23)(ii). We set \(\widetilde{u}_\varepsilon (y):=\bar{u}_\varepsilon (x_0+\varepsilon y)\). Then, for \(s\in [0,\varepsilon ^d]\) we have \((\bar{u}_\varepsilon )_*(s;B_{\sigma \varepsilon }(x_0)) = (\widetilde{u}_\varepsilon )_*(s/\varepsilon ^d;B_\sigma )\) and, in turn,

$$\begin{aligned}{} & {} \tau '(\widetilde{u}_\varepsilon ;B_{\sigma }) = \tau '(\bar{u}_\varepsilon ;B_{\sigma \varepsilon }(x_0)),\,\,\tau ''(\widetilde{u}_\varepsilon ;B_{\sigma }) = \tau ''(\bar{u}_\varepsilon ;B_{\sigma \varepsilon }(x_0)),\,\, \\{} & {} \textrm{med}(\widetilde{u}_\varepsilon ;B_\sigma ) =\textrm{med}(\bar{u}_\varepsilon ;B_{\sigma \varepsilon }(x_0)). \end{aligned}$$

We have

$$\begin{aligned} T_\varepsilon \bar{u}_\varepsilon (x_0+\varepsilon y)=T_\sigma \widetilde{u}_\varepsilon (y), \end{aligned}$$

so, recalling that \(u_\varepsilon =v_\varepsilon \) in \(B_{\sigma \varepsilon }(x_0)\), (3.23)(ii) can be rephrased as

$$\begin{aligned} \int _{B_\sigma }|T_\sigma \widetilde{u}_\varepsilon (y)-\nabla u(x_0)\cdot y|^{p_\varepsilon (y)}\,\textrm{d}y\rightarrow 0, \end{aligned}$$
(3.28)

as \(\varepsilon \rightarrow 0\), where we have set

$$\begin{aligned} p_\varepsilon (y):=p(x_0+\varepsilon y),\quad y\in B_\sigma . \end{aligned}$$

From (3.24)(a)-(b) we infer

$$\begin{aligned} \int _{B_\sigma } |\nabla \widetilde{u}_\varepsilon |^{p_\varepsilon (y)}\,\textrm{d}y\le C,\quad \lim _{\varepsilon \rightarrow 0} \mathcal {H}^{d-1}(J_{\widetilde{u}_\varepsilon }\cap B_\sigma )=0. \end{aligned}$$

Then, by virtue of Theorem 2.8 there exist a function \(\widetilde{u}_0\in W^{1,p(x_0)}(B_\sigma ;\mathbb {R})\) and a subsequence (not relabeled) of \(\{\widetilde{u}_\varepsilon \}\) such that

$$\begin{aligned} \begin{aligned} \int _{B_\sigma }|T_\sigma \widetilde{u}_\varepsilon (y)-\textrm{med}(\widetilde{u}_\varepsilon ;B_\sigma )-\widetilde{u}_0|^{p_\varepsilon (y)}\,\textrm{d}y\rightarrow 0,\quad \widetilde{u}_\varepsilon -\textrm{med}(\widetilde{u}_\varepsilon ;B_\sigma ) \rightarrow \widetilde{u}_0 \\ {}&\mathcal {L}^d-\text{ a.e. } \text{ in } B_\sigma . \end{aligned} \end{aligned}$$

The assertion (3.28) will then follow once we prove that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \textrm{med}(\widetilde{u}_\varepsilon ;B_\sigma )=0. \end{aligned}$$
(3.29)

For this, notice that (3.24)(c) implies \(\widetilde{u}_0(y)=\nabla u(x_0)\cdot y\) for \(\mathcal {L}^d\)-a.e. \(y\in B_\sigma \). The a.e. convergence in measure of \(\widetilde{u}_\varepsilon -\textrm{med}(\widetilde{u}_\varepsilon ;B_\sigma )\) to \(\nabla u(x_0)\cdot y\) is now enough to reproduce the proof of [11, eq. (21)], and obtain (3.29). We therefore omit the details.

If \(u\in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\), we may fix \(x_0\in \Omega \) such that, in addition to (3.24)(a), (3.24)(c) holds in the stronger form

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \ \varepsilon ^{-(d+1)}\int _{B_\varepsilon (x_0)}{|u(x) - u(x_0) - \nabla u(x_0)(x-x_0)|}\,\textrm{d}x =0, \end{aligned}$$

and also property

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d}\int _{J_u\cap B_\varepsilon (x_0)}|[u]|\,\textrm{d}\mathcal {H}^{d-1}=0 \end{aligned}$$
(3.30)

is satisfied. Then as a consequence of Fubini’s Theorem, we can fix \(\sigma \in (0,1)\) such that \(\mathcal {H}^{d-1}(J_u\cap \partial B_{\sigma \varepsilon }(x_0))=0\) and

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d}\int _{\partial B_{\sigma \varepsilon }(x_0)}{|u(x) - u(x_0) - \nabla u(x_0)(x-x_0)|}\,\textrm{d}\mathcal {H}^{d-1} =0. \end{aligned}$$
(3.31)

Now, we can define the sequence \(u_\varepsilon \) as above and prove (i), (ii) and (iii). Assertion \((i)'\) will follow from (iii), (3.31), (c) and Hölder’s inequality, since \(u_\varepsilon =u\) in \(B_\varepsilon (x_0)\backslash B_{\sigma \varepsilon }(x_0)\).

Finally, since by construction it holds that \(|[u_\varepsilon ]|\le |[u]|\) \(\mathcal {H}^{d-1}\)-a.e., property \((iii)'\) is a consequence of (3.30). \(\square \)

We are now in a position to prove Lemma 3.4, which will follow as a consequence of Lemmas 3.8 and 3.9.

Lemma 3.8

Let \(p:\Omega \rightarrow (1,+\infty )\) be a Riemann integrable variable exponent satisfying (\(\hbox {P}_{1}\)). Suppose that \(\mathcal {F}\) satisfies (\(\hbox {H}_{1}\)) and (\(\hbox {H}_{3}\))–(\(\hbox {H}_{4}\)) and let \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then for \(\mathcal {L}^{d}\)-a.e. \(x_0 \in \Omega \) we have

$$\begin{aligned} \displaystyle \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _d\varepsilon ^{d}} \le \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\bar{u}_{x_0}^\textrm{bulk},B_{\varepsilon }(x_0))}{\gamma _d\varepsilon ^{d}}. \end{aligned}$$
(3.32)

Proof

We will prove the assertion for those points \(x_0\in \Omega \) for which the statement of Lemma 3.7 holds and \(\lim _{\varepsilon \rightarrow 0} (\gamma _{d}\varepsilon ^{d})^{-1}\mu (B_\varepsilon (x_0))=1\). This holds for \(\mathcal {L}^d\)-a.e. \(x_0\in \Omega \). Also, by Lemma 3.3, we know that for \(\mathcal {L}^d\)-a.e. \(x_0\in \Omega \)

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(u,B_\varepsilon (x_0))}{\gamma _d \varepsilon ^d} = \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _d \varepsilon ^d}<+\infty . \end{aligned}$$
(3.33)

Let \((u_\varepsilon )_\varepsilon \) be the sequence of Lemma 3.7 and we fix \(\sigma \in (0,1)\) such that (3.23)(ii) holds. We write \(\sigma =1-\theta \) for some \(\theta \in (0,1)\).

Given \(z_\varepsilon \in GSBV^{p(\cdot )}(B_{(1-3\theta )\varepsilon }(x_0);\mathbb {R}^m)\) such that \(z_\varepsilon =\bar{u}_{x_0}^\textrm{bulk}\) in a neighborhood of \(\partial B_{(1-3\theta )\varepsilon }(x_0)\) and

$$\begin{aligned} \mathcal {F}(z_\varepsilon ,B_{(1-3\theta )\varepsilon }(x_0)) \le \textbf{m}_{\mathcal {F}}(\bar{u}_{x_0}^\textrm{bulk},B_{(1-3\theta )\varepsilon }(x_0)) + \gamma _d\varepsilon ^{d+1}, \end{aligned}$$
(3.34)

we extend it to \(z_\varepsilon \in GSBV^{p(\cdot )}(B_{\varepsilon }(x_0);\mathbb {R}^m)\) by setting \(z_\varepsilon =\bar{u}_{x_0}^\textrm{bulk}\) outside \(B_{(1-3\theta )\varepsilon }(x_0)\). Now, we apply Lemma 3.2 with u and v replaced by \(z_\varepsilon \) and \(u_\varepsilon \), respectively, and

$$\begin{aligned} D'_{\varepsilon ,x_0}:=B_{(1-2\theta )\varepsilon }(x_0),\,\, D''_{\varepsilon ,x_0}:=B_{(1-\theta )\varepsilon }(x_0),\,\, E_{\varepsilon ,x_0}:=C_{\varepsilon ,\theta }(x_0), \end{aligned}$$
(3.35)

where, to enlighten the notation, we denote by \(C_{\varepsilon ,\theta }(x_0)\) the annulus \(B_\varepsilon (x_0)\backslash \overline{B_{(1-4\theta )\varepsilon }(x_0)}\). Note that \(C_{\varepsilon ,\theta }(x_0)=(C_{1,\theta }(x_0))_{\varepsilon ,x_0}\) according to notation (2.1), where \(C_{1,\theta }(x_0):=B_1(x_0)\backslash \overline{B_{(1-4\theta )}(x_0)}\). Also, \(\mathcal {L}^d(C_{1,\theta }(x_0))=\gamma _d(1-(1-4\theta )^d)\rightarrow 0\) as \(\theta \rightarrow 0\).

With fixed \(\eta >0\), we then find \({w}_\varepsilon \in GSBV^{p(\cdot )}(B_\varepsilon (x_0);\mathbb {R}^m)\) such that \({w}_\varepsilon =u_\varepsilon \) on \(B_\varepsilon (x_0)\backslash B_{(1-\theta )\varepsilon }(x_0)\) and

$$\begin{aligned} \begin{aligned} \mathcal {F}({w}_\varepsilon , B_\varepsilon (x_0))&\le (1+\eta ) \left( \mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0)) + \mathcal {F}(u_\varepsilon , C_{\varepsilon ,\theta }(x_0))\right) \\&+ M \int _{(B_{(1-\theta )\varepsilon }(x_0) {\setminus } B_{(1-2\theta )\varepsilon }(x_0))}\left( \frac{|z_\varepsilon -u_\varepsilon |}{\varepsilon }\right) ^{p(x)}\,\textrm{d}x +\eta \mathcal {L}^d(B_\varepsilon (x_0)). \end{aligned} \end{aligned}$$
(3.36)

Recalling the definition of \(u_\varepsilon \), we have \(w_\varepsilon =u_\varepsilon =u\) in a neighborhood of \(\partial B_\varepsilon (x_0)\). Moreover, since \(z_\varepsilon =\bar{u}_{x_0}^\textrm{bulk}\) outside \(B_{(1-3\theta )\varepsilon }(x_0)\), by virtue of (3.23)(ii) we conclude that

$$\begin{aligned} \begin{aligned}&\lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d} \int _{(B_{(1-\theta )\varepsilon }(x_0) {\setminus } B_{(1-2\theta )\varepsilon }(x_0))}\left( \frac{|z_\varepsilon -u_\varepsilon |}{\varepsilon }\right) ^{p(x)}\,\textrm{d}x \\&\quad = \lim _{\varepsilon \rightarrow 0} \ \varepsilon ^{-d} \int _{B_{(1-\theta )\varepsilon }(x_0)} \left( \frac{|u_\varepsilon - \bar{u}_{x_0}^\textrm{bulk}|}{\varepsilon } \right) ^{p(x)} \, \textrm{d}x = 0. \end{aligned} \end{aligned}$$
(3.37)

From this and (3.36) we infer that there exists a non-negative sequence \((\varrho _\varepsilon )_\varepsilon \), vanishing as \(\varepsilon \rightarrow 0\), such that

$$\begin{aligned} \mathcal {F}&(w_\varepsilon , B_\varepsilon (x_0)) \le (1+\eta )\left( \mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0)) + \mathcal {F}(u_\varepsilon , C_{\varepsilon ,\theta }(x_0))\right) + \varepsilon ^d\varrho _\varepsilon +\gamma _d\varepsilon ^d\eta . \end{aligned}$$
(3.38)

We set for brevity

$$\begin{aligned} |\nabla u(x_0)|^{\widetilde{p}}:=\max \{|\nabla u(x_0)|^{p^-}, |\nabla u(x_0)|^{p^+}\}. \end{aligned}$$
(3.39)

Then, by using that \(z_\varepsilon = \bar{u}^\textrm{bulk}_{x_0}\) on \(B_{\varepsilon }(x_0) {\setminus } B_{(1-3\theta )\varepsilon }(x_0) \subset C_{\varepsilon ,\theta }(x_0)\),  (\(H_1\)),  (\(H_4\)), and (3.34) we compute

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0))}{\varepsilon ^{d}}&\le \limsup _{\varepsilon \rightarrow 0} \frac{\mathcal {F}(z_\varepsilon ,B_{(1- 3 \theta )\varepsilon }(x_0))}{\varepsilon ^{d}}+ \limsup _{\varepsilon \rightarrow 0} \frac{\mathcal {F}(\bar{u}^\textrm{bulk}_{x_0}, C_{\varepsilon ,\theta }(x_0))}{\varepsilon ^{d}}\nonumber \\&\le \limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{bulk}_{x_0},B_{(1-3\theta )\varepsilon }(x_0))}{\varepsilon ^{d}} \nonumber \\&\quad + \beta \, \mathcal {L}^d(C_{1,\theta }(x_0)) (1+|\nabla u(x_0)|^{\widetilde{p}}) \nonumber \\&\le (1-3\theta )^{d}\limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{bulk}_{x_0},B_{(1-3\theta )\varepsilon }(x_0))}{(1-3\theta )^{d}\varepsilon ^{d}} \\&+\beta \, \mathcal {L}^d(C_{1,\theta }(x_0)) (1+|\nabla u(x_0)|^{\widetilde{p}}). \nonumber \end{aligned}$$
(3.40)

On the other hand, by (\(H_4\)) we also obtain

$$\begin{aligned} \mathcal {F}(u_\varepsilon , C_{\varepsilon ,\theta }(x_0))&\le \beta \int _{C_{\varepsilon ,\theta }(x_0)} (1+ |\nabla u_\varepsilon |^{p(x)})\,\textrm{d}x + \beta \,\mathcal {H}^{d-1}(J_{u_\varepsilon } \cap C_{\varepsilon ,\theta }(x_0))) \\&\le \beta \mathcal {L}^d(C_{\varepsilon ,\theta }(x_0)) + \beta \int _{ C_{\varepsilon ,\theta }(x_0)} |\nabla u_\varepsilon |^{p(x)}\,\textrm{d}x +\beta \mathcal {H}^{d-1}(J_{u_\varepsilon }\cap C_{\varepsilon ,\theta }(x_0)). \end{aligned}$$

Now, taking into account (3.23)(iii) we get

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0} \frac{\mathcal {F}(u_\varepsilon , C_{\varepsilon ,\theta }(x_0))}{\varepsilon ^{d}} \le \beta \, \mathcal {L}^d( C_{1,\theta }(x_0)) + \limsup _{\varepsilon \rightarrow 0} \frac{\beta }{\varepsilon ^d} \int _{ C_{\varepsilon ,\theta }(x_0)} |\nabla u_\varepsilon |^{p(x)}\,\textrm{d}x. \end{aligned}$$
(3.41)

Since \(|\nabla u_\varepsilon |\le |\nabla u|\) \(\mathcal {L}^d\)-a.e., we have, with (3.24)(a),

$$\begin{aligned} \begin{aligned}&\limsup _{\varepsilon \rightarrow 0} \frac{\beta }{\varepsilon ^d} \int _{ C_{\varepsilon ,\theta }(x_0)} |\nabla u_\varepsilon |^{p(x)}\,\textrm{d}x \\&\quad \le \limsup _{\varepsilon \rightarrow 0} \frac{\beta }{\varepsilon ^d} \int _{ C_{\varepsilon ,\theta }(x_0)} |\nabla u|^{p(x)}\,\textrm{d}x \\&\quad \le \limsup _{\varepsilon \rightarrow 0} 2^{p^+-1}\left( \frac{\beta }{\varepsilon ^d}\int _{B_\varepsilon (x_0)} |\nabla u -\nabla u(x_0)|^{p(x)}\,\textrm{d}x + \beta |\nabla u(x_0)|^{\widetilde{p}}\mathcal {L}^d(C_{1,\theta }(x_0))\right) \\&\quad \le 2^{p^+-1} \beta |\nabla u(x_0)|^{\widetilde{p}} \mathcal {L}^d(C_{1,\theta }(x_0)). \end{aligned} \end{aligned}$$
(3.42)

Combining (3.41) with (3.42) we finally get

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0} \frac{\mathcal {F}(u_\varepsilon , C_{\varepsilon ,\theta }(x_0))}{\varepsilon ^{d}} \le \beta \,\mathcal {L}^d(C_{1,\theta }(x_0)) (1+2^{p^+-1} |\nabla u(x_0)|^{\widetilde{p}}). \end{aligned}$$
(3.43)

Recall that \(w_\varepsilon = u\) in a neighborhood of \(\partial B_\varepsilon (x_0)\). This along with (3.38), (3.40), (3.43) and \(\varrho _\varepsilon \rightarrow 0\) yields

$$\begin{aligned} \displaystyle \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _{d}\varepsilon ^{d}}&\le \limsup _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(w_\varepsilon , B_\varepsilon (x_0))}{\gamma _{d}\varepsilon ^{d}} \\&\le (1+\eta ) \, (1-3\theta )^{d} \limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{bulk}_{x_0},B_\varepsilon (x_0))}{\gamma _{d}\varepsilon ^{d}} \\ {}&\ \ \ + (1+\eta ) \beta \gamma _d^{-1}\mathcal {L}^d(C_{1,\theta }(x_0)) (1+2^{p^+}|\nabla u(x_0)|^{\widetilde{p}}) + \eta , \end{aligned}$$

whence (3.32) follows up to passing to \(\eta ,\theta \rightarrow 0\). The proof is concluded. \(\square \)

Lemma 3.9

Under the assumptions of Lemma 3.8, for \(\mathcal {L}^{d}\)-a.e. \(x_0 \in \Omega \) we have

$$\begin{aligned} \displaystyle \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _d\varepsilon ^{d}} \ge \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\bar{u}_{x_0}^\textrm{bulk},B_{\varepsilon }(x_0))}{\gamma _d\varepsilon ^{d}}. \end{aligned}$$
(3.44)

Proof

We can restrict the proof to those points \(x_0\in \Omega \) considered in Lemma 3.8. Let \(\eta >0\), \(\sigma =1-\theta \) fixed as in Lemma 3.8, and let \((u_\varepsilon )_\varepsilon \) be the sequence provided by Lemma 3.7. An argument based on Fubini’s Theorem (see (3.26) and (3.27)) shows that for each \(\varepsilon >0\) we can find \(s\in (1-4\theta , 1-3\theta )\) such that

$$\begin{aligned} \begin{aligned}&\mathcal {H}^{d-1}(\partial B_{s\varepsilon }(x_0)\cap (J_{u_\varepsilon }\cup J_u))=0,\quad \hbox { for all } \varepsilon >0, \\&\lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d} \mathcal {H}^{d-1}(\{u\ne u_\varepsilon \}\cap \partial B_{s\varepsilon }(x_0)) =0. \end{aligned} \end{aligned}$$
(3.45)

From now on, the argument of the proof closely follows that of Lemma 3.8. We choose a sequence \(z_\varepsilon \in GSBV^{p(\cdot )}(B_{s\varepsilon }(x_0);\mathbb {R}^m)\) such that \(z_\varepsilon =u\) in a neighborhood of \(\partial B_{s\varepsilon }(x_0)\) and

$$\begin{aligned} \mathcal {F}(z_\varepsilon ,B_{s\varepsilon }(x_0)) \le \textbf{m}_{\mathcal {F}}(u,B_{s\varepsilon }(x_0)) + \gamma _d\varepsilon ^{d+1}. \end{aligned}$$
(3.46)

Setting \(z_\varepsilon =u_\varepsilon \) outside \(B_{s\varepsilon }(x_0)\), we extend it to \(z_\varepsilon \in GSBV^{p(\cdot )}(B_{\varepsilon }(x_0);\mathbb {R}^m)\). Now, we apply Lemma 3.2 with u and v replaced by \(z_\varepsilon \) and \(\bar{u}_{x_0}^\textrm{bulk}\), respectively, and the same choice for the sets \(D'_{\varepsilon ,x_0}\), \(D''_{\varepsilon ,x_0}\) and \(E_{\varepsilon ,x_0}\) as in Lemma 3.8, see (3.35).

By virtue of Lemma 3.2, we then find \({w}_\varepsilon \in GSBV^{p(\cdot )}(B_\varepsilon (x_0);\mathbb {R}^m)\) such that \({w}_\varepsilon =\bar{u}_{x_0}^\textrm{bulk}\) on \(B_\varepsilon (x_0)\backslash B_{(1-\theta )\varepsilon }(x_0)\) and

$$\begin{aligned} \begin{aligned} \mathcal {F}({w}_\varepsilon , B_\varepsilon (x_0))&\le (1+\eta ) \left( \mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0)) + \mathcal {F}(\bar{u}_{x_0}^\textrm{bulk}, C_{\varepsilon ,\theta }(x_0))\right) \\&\quad + M \int _{(B_{(1-\theta )\varepsilon }(x_0) {\setminus } B_{(1-2\theta )\varepsilon }(x_0))}\left( \frac{|z_\varepsilon -\bar{u}_{x_0}^\textrm{bulk}|}{\varepsilon }\right) ^{p(x)}\,\textrm{d}x +\eta \mathcal {L}^d(B_\varepsilon (x_0)). \end{aligned} \end{aligned}$$

Since \(z_\varepsilon =u_\varepsilon \) outside \(B_{(1-3\theta )\varepsilon }(x_0)\) from the choice of s, by arguing as in Lemma 3.8, see in particular (3.37) and (3.38), we find a non-negative sequence \((\varrho _\varepsilon )_\varepsilon \), vanishing as \(\varepsilon \rightarrow 0\), such that

$$\begin{aligned} \mathcal {F}&(w_\varepsilon , B_\varepsilon (x_0)) \le (1+\eta )\left( \mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0)) + \mathcal {F}(\bar{u}_{x_0}^\textrm{bulk}, C_{\varepsilon ,\theta }(x_0))\right) + \varepsilon ^d\varrho _\varepsilon +\gamma _d\varepsilon ^d\eta . \end{aligned}$$
(3.47)

We now proceed to the estimate of the terms in (3.47). Using that \(z_\varepsilon = u_\varepsilon \) on \(B_{\varepsilon }(x_0) {\setminus } B_{s\varepsilon }(x_0) \subset C_{\varepsilon ,\theta }(x_0)\),  (\(H_1\)),  (\(H_4\)), and (3.46) we obtain

$$\begin{aligned} \mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0))\le & {} \textbf{m}_{\mathcal {F}}(u,B_{s\varepsilon }(x_0)) + \gamma _d\varepsilon ^{d+1} + \beta \mathcal {H}^{d-1}(\partial B_{s\varepsilon }(x_0)\cap (J_{u_\varepsilon }\cup J_u))\nonumber \\{} & {} + \mathcal {F}(u_\varepsilon , C_{\varepsilon ,\theta }(x_0)). \end{aligned}$$
(3.48)

Now, with (3.43) and (3.45) and the fact that \(s\varepsilon \le (1-3\theta )\varepsilon \) we get

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0))}{\varepsilon ^{d}}&\le s^d \limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(u,B_{s\varepsilon }(x_0))}{(s\varepsilon )^{d}} + \beta \, \mathcal {L}^d(C_{1,\theta }) (1+{|\nabla u(x_0)|^{\widetilde{p}}}) \nonumber \\&\le (1-3\theta )^{d}\limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(u,B_{\varepsilon }(x_0))}{\varepsilon ^{d}} \nonumber \\&\quad +\beta \, \mathcal {L}^d(C_{1,\theta }(x_0)) (1+{|\nabla u(x_0)|^{\widetilde{p}}}), \end{aligned}$$
(3.49)

where \(|\nabla u(x_0)|^{\widetilde{p}}\) is defined as in (3.39).

The analogous of the estimate for \(\mathcal {F}(\bar{u}_{x_0}^\textrm{bulk}, C_{\varepsilon ,\theta }(x_0))\) in (3.40), the estimates (3.47), (3.48), (3.49) and \(\varrho _\varepsilon \rightarrow 0\) give

$$\begin{aligned} \begin{aligned} \limsup _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(w_\varepsilon , B_{\varepsilon }(x_0))}{\varepsilon ^{d}}&\le (1+\eta ) (1-3\theta )^{d}\limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(u,B_{\varepsilon }(x_0))}{\varepsilon ^{d}} \\&\quad +2(1+\eta )\beta \, \mathcal {L}^d(C_{1,\theta }(x_0)) (1+{|\nabla u(x_0)|^{\widetilde{p}}})+\gamma _d\eta . \end{aligned} \end{aligned}$$

Finally, letting \(\eta \) and \(\theta \) to 0, and recalling that \(w_\varepsilon =\bar{u}_{x_0}^\textrm{bulk}\) in a neighborhood of \(\partial B_\varepsilon (x_0)\), we can write

$$\begin{aligned} \begin{aligned} \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\bar{u}_{x_0}^\textrm{bulk},B_{\varepsilon }(x_0))}{\gamma _d\varepsilon ^{d}}&\le \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{\mathcal {F}(w_\varepsilon ,B_{\varepsilon }(x_0))}{\gamma _d\varepsilon ^{d}} \\&\le \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(u,B_{\varepsilon }(x_0))}{\gamma _d\varepsilon ^{d}} = \lim _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(u,B_{\varepsilon }(x_0))}{\gamma _d\varepsilon ^{d}}, \end{aligned} \end{aligned}$$

and this concludes the proof of (3.44). \(\square \)

3.4 The surface density

The proof of Lemma 3.5 requires the analysis of the blow-up at the jump points of function u. To this aim, we need a refinement of the results of [11, Lemma 3] to the case of a variable exponent \(p(\cdot )\). This requires a careful analysis of the asymptotic behavior of some constants, where the assumption of log-Hölder continuity of the variable exponent \(p(\cdot )\), see (\(P_2\)), plays a crucial role.

We state and prove the announced blow-up properties for \(u\in GSBV^{p(\cdot )}\) around each jump point \(x_0\in J_u\).

Lemma 3.10

Assume that \(p:\Omega \rightarrow (1,+\infty )\) be continuous and complying with (\(\hbox {P}_{2}\)). Let \(u\in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then for \(\mathcal {H}^{d-1}\)-a.e. \(x_0\in J_u\), for \(\mathcal {L}^1\)-a.e. \(\sigma \in (0,1)\) and for every \(\varepsilon >0\) there exists a function \(\bar{u}_\varepsilon \in GSBV^{p(\cdot )}(B_\varepsilon (x_0);\mathbb {R}^m)\) with \(\nu =\nu _u(x_0)\) such that

$$\begin{aligned} \begin{aligned}&(i) \,\, \mathcal {H}^{d-1}((J_{\bar{u}_\varepsilon }\backslash J_u)\cap B_\varepsilon (x_0))=0,\\&(ii) \,\, \displaystyle \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-(d-1)} \int _{B_\varepsilon (x_0)}|\nabla \bar{u}_\varepsilon |^{p(x)}\,\textrm{d}x=0, \\&(iii) \,\,\displaystyle \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-(d-1)} \int _{B_{\sigma \varepsilon }(x_0)}\left( \frac{|\bar{u}_\varepsilon -\bar{u}^\textrm{surf}_{x_0}|}{\varepsilon }\right) ^{p(x)}\,\textrm{d}x=0,\\&(iv) \,\, \bar{u}_\varepsilon =u \hbox {\,\,on } B_\varepsilon (x_0)\backslash \overline{B_{\sigma \varepsilon }(x_0)}\, \end{aligned} \end{aligned}$$
(3.50)

and

$$\begin{aligned} \begin{aligned}&\lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d} \mathcal {L}^d(\{x\in B_\varepsilon (x_0):\,\, \bar{u}_\varepsilon \ne u\})=0. \end{aligned} \end{aligned}$$
(3.51)

If, in addition, \(u\in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\), we also have

$$\begin{aligned} \displaystyle \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d}\int _{B_\varepsilon (x_0)}|\bar{u}_\varepsilon (x)-u(x)|\,\textrm{d}x=0, \end{aligned}$$
(3.52)

and

$$\begin{aligned} \displaystyle \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-(d-1)}\int _{J_{\bar{u}_\varepsilon }\cap E_{\varepsilon ,x_0}}|[\bar{u}_\varepsilon ]|\,\textrm{d}\mathcal {H}^{d-1}=|[\bar{u}^\textrm{surf}_{x_0}]|\mathcal {H}^{d-1}(\Pi _0\cap E) \quad \hbox { for all Borel sets } E\subset B_1(x_0), \nonumber \\ \end{aligned}$$
(3.53)

where \(\Pi _0\) is the hyperplane passing through \(x_0\) with normal \(\nu _u(x_0)\).

Proof

We first note that since \(|\nabla u|^{p(\cdot )}\in L^1(\Omega )\), the points \(x_0\in J_u\) can be fixed such that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-(d-1)}\int _{B_\varepsilon (x_0)}|\nabla u|^{p(x)}\,\textrm{d}x=0 \end{aligned}$$
(3.54)

(see, e.g., [30, Section 2.4.3, Theorem 3]). Further, since \(J_u\) is \((d-1)\)-rectifiable, there exists a sequence of compact sets \(K_j\) such that \(J_u=\bigcup _{j=1}^\infty K_j\cup N\), for some N such that \(\mathcal {H}^{d-1}(N)=0\), and each \(K_j\) is a subset of a \(C^1\) hypersurface. Then, in a neighborhood \(B_{\varepsilon _0}(y)\subset \Omega \) of each point \(y\in K_j\), up to a rotation, we may find a \(C^1\) function \(\Gamma _j:\mathbb {R}^{d-1}\rightarrow \mathbb {R}\) such that

$$\begin{aligned} K_j\cap B_{\varepsilon _0}(y) \subset \{x=(x',x_d)\in B_{\varepsilon _0}(y):\,\,x_d=\Gamma _j(x')\}. \end{aligned}$$

We now define the function \(w\in GSBV^{p(\cdot )}(B_{\varepsilon _0}(y);\mathbb {R}^m)\) by setting

$$\begin{aligned} w(x):= {\left\{ \begin{array}{ll} u(x',x_d) &{} \hbox { if } x_d>\Gamma _j(x'),\\ u(x', -x_d+2\Gamma _j(x')) &{} \hbox { if } x_d<\Gamma _j(x'). \end{array}\right. } \end{aligned}$$

Notice indeed that by construction we have \(|\nabla w|\le C |\nabla u|\) a.e., hence \(w\in GSBV^{p(\cdot )}(B_{\varepsilon _0}(y);\mathbb {R}^m)\). Furthemore, \(J_w\cap B_{\varepsilon _0}(y)\subset B_{\varepsilon _0}(y)\backslash K_j\). Now, following the argument of [11, Lemma 3], we can fix \(x_0\in B_{\varepsilon _0}(y)\cap K_j\) with the following properties:

$$\begin{aligned} w(x_0)=u^+(x_0),\quad \lim _{\varepsilon \rightarrow 0^+}\frac{1}{\varepsilon ^{d-1}}\int _{B_\varepsilon (x_0)\cap K_j} |w-u^+(x_0)|\,\textrm{d}x=0, \end{aligned}$$
(3.55)

and for fixed \(\eta >0\) (small enough) there exists (a smaller, if necessary) \(\varepsilon _0>0\) such that

$$\begin{aligned} \int _{B_\varepsilon (x_0)} |\nabla w|^{q}\,\textrm{d}x + \mathcal {H}^{d-1}(J_w\cap B_\varepsilon (x_0)) < \eta \varepsilon ^{d-1} \end{aligned}$$
(3.56)

holds, for all \(\varepsilon <\varepsilon _0\), for \(q=p^-_\Omega \). Moreover, if we set for every \(\varepsilon >0\)

$$\begin{aligned} p^-_\varepsilon := \inf _{x\in B_\varepsilon (x_0)} p(x),\quad p^+_\varepsilon := \sup _{x\in B_\varepsilon (x_0)} p(x), \end{aligned}$$

combining with (3.54) we have that (3.56) is indeed satisfied for \(q=p^-_\varepsilon \).

Now, fix q such that (3.56) holds. Define \(T_\varepsilon w(x):= T_{B_\varepsilon (x_0)}w(x)\) as in (2.5) with \(u=w\) and \(B=B_\varepsilon (x_0)\). From the Poincaré inequality (2.7), (3.56) and for any \(q\le r < q^*\) we have

$$\begin{aligned} \begin{aligned}&\int _{B_\varepsilon (x_0)}{|T_\varepsilon w-\textrm{med}(w;B_\varepsilon (x_0))|}^r\,\textrm{d}x \\&\quad \le \left( \int _{B_\varepsilon (x_0)}{|T_\varepsilon w-\textrm{med}(w;B_\varepsilon (x_0))|}^{q^*}\,\textrm{d}x\right) ^{\frac{r}{q^*}}[\mathcal {L}^d(B_\varepsilon )]^{1-\frac{r}{q^*}} \\&\quad \le \left( {2\gamma _\textrm{iso}q^*(d-1)}\right) ^r\left( \int _{B_\varepsilon (x_0)}{|\nabla w|}^{q}\,\textrm{d}x\right) ^{\frac{r}{q}}[\mathcal {L}^d(B_\varepsilon )]^{1-\frac{r}{q^*}} \\&\quad \le C(d,q,r)\eta ^{\frac{r}{q}} \varepsilon ^{\frac{r(d-1)}{q}+d-\frac{r(d-q)}{q}} \\&\quad = C(d,q,r)\eta ^{\frac{r}{q}} \varepsilon ^{\frac{r(q-1)}{q}+d}, \end{aligned} \end{aligned}$$

where \(C(d,q,r):=\left( {2\gamma _\textrm{iso}q^*(d-1)}\gamma _d^{\frac{1}{r}-\frac{1}{q^*}}\right) ^r\). Since arguing as for the proof of [11, eq. (34)] we can prove that

$$\begin{aligned} \begin{aligned}&\int _{B_\varepsilon (x_0)}{|\textrm{med}(w,B_\varepsilon (x_0)) - u^+(x_0)|}^r\,\textrm{d}x \le \eta ^r \varepsilon ^{\frac{r(q-1)}{q}+d} \end{aligned} \end{aligned}$$

for \(\varepsilon \) small enough, collecting the previous estimates we finally obtain

$$\begin{aligned} \frac{1}{\varepsilon ^{d-1}}\int _{B_\varepsilon (x_0)}\left( {\frac{|T_\varepsilon w-u^+(x_0)|}{\varepsilon }}\right) ^r\,\textrm{d}x \le 2\max \{C(d,q,r)\eta ^{\frac{r}{q}},\eta ^r\}\varepsilon ^{-\frac{(r-q)}{q}}. \end{aligned}$$
(3.57)

If we define the function z as

$$\begin{aligned} z(x):= {\left\{ \begin{array}{ll} u(x',x_d) &{} \hbox { if } x_d<\Gamma _j(x'),\\ u(x', -x_d+2\Gamma _j(x')) &{} \hbox { if } x_d>\Gamma _j(x'), \end{array}\right. } \end{aligned}$$

then z complies with the (3.55)–(3.56), up to replacing w with z and \(u^+(x_0)\) with \(u^-(x_0)\). Hence, an analogous estimate as in (3.57) can be inferred for the sequence \(T_\varepsilon z\) defined as the truncation \(T_{B_\varepsilon (x_0)}z\) of the function z. We then set

$$\begin{aligned} u_\varepsilon (x):= {\left\{ \begin{array}{ll} T_\varepsilon w(x) &{} \hbox { if } x_d>\Gamma _j(x'),\\ T_\varepsilon z(x) &{} \hbox { if } x_d<\Gamma _j(x'), \end{array}\right. } \end{aligned}$$

and we have

$$\begin{aligned} \frac{1}{\varepsilon ^{d-1}}\int _{B_{\sigma \varepsilon }(x_0)}\left( {\frac{|{u}_\varepsilon -\bar{u}^\textrm{surf}_{x_0}|}{\varepsilon }}\right) ^{q}\,\textrm{d}x \le 2\max \{C(d,q,r)\eta ^{\frac{r}{q}},\eta ^{r}\}\varepsilon ^{-\frac{(r-q)}{q}},\quad \forall r\in [q,q^*), \nonumber \\ \end{aligned}$$
(3.58)

Arguing exactly as in [11, Lemma 3], we also have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d} \mathcal {L}^d(\{x\in B_\varepsilon (x_0):\,\, u_\varepsilon \ne u\})=0. \end{aligned}$$
(3.59)

Now, an analogous argument as for (3.26) shows that for every sequence \(\varepsilon \rightarrow 0\) one can find a subsequence (not relabeled) such that, for \(\mathcal {L}^1\)-a.e. \(\sigma \in (0,1)\),

$$\begin{aligned} \begin{aligned} \mu (\partial B_{\sigma \varepsilon }(x_0)) = \mathcal {H}^{d-1}(\partial B_{\sigma \varepsilon }(x_0)\cap J_{u_\varepsilon })=0, \\ \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d} \mathcal {H}^{d-1}(\{u_\varepsilon \ne u\}\cap \partial B_{\sigma \varepsilon }(x_0)) = 0. \end{aligned} \end{aligned}$$

We then define

$$\begin{aligned} \bar{u}_\varepsilon (x) = {\left\{ \begin{array}{ll} u_\varepsilon (x) &{} \hbox { in } B_{\sigma \varepsilon }(x_0),\\ u(x) &{} \hbox { in } B_\varepsilon (x_0)\backslash \overline{B_{\sigma \varepsilon }(x_0)}. \end{array}\right. } \end{aligned}$$

Now, property (3.50) (i), (ii) and (iv) follow from the definition and (3.54), while (3.51) is immediate from (3.59). As for (3.50) (iii), with fixed \(\eta >0\), the estimate (3.58) with \(q=p^-_\varepsilon \) and \(r=p^+_\varepsilon \) implies that

$$\begin{aligned} \frac{1}{\varepsilon ^{d-1}}\int _{B_{\sigma \varepsilon }(x_0)}\left( {\frac{|\bar{u}_\varepsilon -\bar{u}^\textrm{surf}_{x_0}|}{\varepsilon }}\right) ^{p^+_\varepsilon }\,\textrm{d}x \le 2 \max \{C(d,p^-_\varepsilon , p^+_\varepsilon )\eta ^{\frac{p^+_\varepsilon }{p^-_\varepsilon }},\eta ^{p^+_\varepsilon }\}\varepsilon ^{-\frac{(p^+_\varepsilon -p^-_\varepsilon )}{p^-_\varepsilon }} \nonumber \\ \end{aligned}$$
(3.60)

for \(\varepsilon \) small enough. Observe that, by its definition, the constant \(C(d,p^-_\varepsilon , p^+_\varepsilon )\) is a bounded function of \(\varepsilon \). Now, since

$$\begin{aligned} \begin{aligned} \frac{1}{\varepsilon ^{d-1}}\int _{B_{\sigma \varepsilon }(x_0)}\left( {\frac{|\bar{u}_\varepsilon -\bar{u}^\textrm{surf}_{x_0}|}{\varepsilon }}\right) ^{p(\cdot )}\,\textrm{d}x&\le \varepsilon + \frac{1}{\varepsilon ^{d-1}}\int _{B_{\sigma \varepsilon }(x_0)}\left( {\frac{|\bar{u}_\varepsilon -\bar{u}^\textrm{surf}_{x_0}|}{\varepsilon }}\right) ^{p^+_\varepsilon }\,\textrm{d}x \\&\le \varepsilon + 2\max \{C(d,p^-_\varepsilon , p^+_\varepsilon )\eta ^{\frac{p^+_\varepsilon }{p^-_\varepsilon }},\eta ^{p^+_\varepsilon }\}\varepsilon ^{-\frac{(p^+_\varepsilon -p^-_\varepsilon )}{p^-_\varepsilon }}, \end{aligned} \end{aligned}$$

and with \(p_\varepsilon ^+\le p^+_\Omega \), assertion (iii) in (3.50) will follow sending \(\varepsilon \rightarrow 0\) first and then \(\eta \rightarrow 0\), once we note that

$$\begin{aligned} \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \varepsilon ^{-\frac{(p^+_\varepsilon -p^-_\varepsilon )}{p^-_\varepsilon }}\le c_1 \end{aligned}$$

for some constant \(c_1\) by virtue of (\(P_2\)).

Assertion (3.52) for a function \(u\in SBV(\Omega ;\mathbb {R}^m)\) can be obtained exactly as in [11, Lemma 3] as a consequence of Hölder’s inequality, combining (3.58), written for \(r=q=p^-_\Omega \), and the property

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d}\int _{B_\varepsilon (x_0)}|u(x)-\bar{u}^\textrm{surf}_{x_0}(x)|\,\textrm{d}x=0. \end{aligned}$$

We omit further details.

Concerning (3.53), we begin by observing that, if \(u\in {SBV}(\Omega ; \mathbb {R}^m)\) for \(\mathcal {H}^{d-1}\)-a.e. \(x_0\in J_u\) we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{1}{\varepsilon ^{d-1}}\int _{(J_u\cap B_\varepsilon (x_0)){\setminus } K_j}(1+|[u]|)\,\textrm{d}\mathcal {H}^{d-1}=0. \end{aligned}$$
(3.61)

If we now set

$$\begin{aligned} \bar{u}^\textrm{surf}_{\varepsilon , x_0}(x):= {\left\{ \begin{array}{ll} \tau ' (w, B_\varepsilon (x_0))\wedge u^+(x_0)\vee \tau {''} (w, B_\varepsilon (x_0)) &{} \hbox { if } x_d>\Gamma _j(x'),\\ \tau ' (z, B_\varepsilon (x_0))\wedge u^-(x_0)\vee \tau {''} (z, B_\varepsilon (x_0)) &{} \hbox { if } x_d<\Gamma _j(x'), \end{array}\right. } \end{aligned}$$

with (3.55), (3.56), (3.61), and since truncations are 1-Lipschitz, we get

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{1}{\varepsilon ^{d-1}}\int _{J_{u_\varepsilon }}|u_\varepsilon -\bar{u}^\textrm{surf}_{\varepsilon , x_0}|\,\textrm{d}\mathcal {H}^{d-1}=0. \end{aligned}$$

Now, as shown in [11, Remark 2, Formula (39)], one has componentwise

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0}\tau '' (w_i, B_\varepsilon (x_0))\ge u^+_i(x_0),\quad \liminf _{\varepsilon \rightarrow 0}\tau ' (w_i, B_\varepsilon (x_0))\le u^-_i(x_0) \end{aligned}$$

and the same properties also hold for z. With this, one has, for all \(E\subset B_1(x_0)\),

$$\begin{aligned} \begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{1}{\varepsilon ^{d-1}}\int _{J_{u_\varepsilon }\cap E_{\varepsilon , x_0}}|[u_\varepsilon ]|\,\textrm{d}\mathcal {H}^{d-1}&=|[u]|(x_0)\lim _{\varepsilon \rightarrow 0}\frac{1}{\varepsilon ^{d-1}}\mathcal {H}^{d-1}(J_{u_\varepsilon }\cap E_{\varepsilon , x_0}) \\&=|[u]|(x_0)\mathcal {H}^{d-1}(\Pi _0\cap E), \end{aligned} \end{aligned}$$

since the last property is satisfied at \(\mathcal {H}^{d-1}\)-a.e. \(x_0\in J_u\) by the definition of measure-theoretic normal to a rectifiable set. This is clearly equivalent to (3.53). \(\square \)

We now prove Lemma 3.5. The two inequalities in (3.12) will be shown with Lemmas 3.11 and 3.12 below.

Lemma 3.11

Let \(p:\Omega \rightarrow (1,+\infty )\) be a variable exponent satisfying (\(\hbox {P}_{1}\))-(\(\hbox {P}_{2}\)). Suppose that \(\mathcal {F}\) satisfies (\(\hbox {H}_{1}\)) and (\(\hbox {H}_{3}\))–(\(\hbox {H}_{4}\)) and let \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then for \(\mathcal {H}^{d-1}\)-a.e. \(x_0 \in J_u\) we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \le \limsup _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{surf}_{x_0},B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}. \end{aligned}$$
(3.62)

Proof

Let \(\bar{u}_\varepsilon \) be the sequence of Lemma 3.10, let \(x_0\) be such that Lemma 3.10 holds, and set \(\nu :=\nu _u(x_0)\). By Lemma 3.3, for \(\mathcal {H}^{d-1}\)-a.e. \(x_0\in J_u\cap \Omega \) we have

$$\begin{aligned} \frac{\textrm{d}\mathcal {F}(u,\cdot )}{\textrm{d}\mathcal {H}^{d-1}\lfloor _{J_u}}(x_0) = \lim _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(u,B_\varepsilon (x_0))}{\mu (B_\varepsilon (x_0))}= \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} < \infty . \end{aligned}$$
(3.63)

Let \(\eta >0\) and \(\sigma \in (0,1)\) be fixed such that Lemma 3.10 holds, and set \(\sigma =1-\theta \) for some \(\theta \in (0,1)\). We consider a sequence \(\bar{z}_\varepsilon \in GSBV^{p(\cdot )}(B_{(1-3\theta )\varepsilon }(x_0);\mathbb {R}^m)\) with \(\bar{z}_\varepsilon = \bar{u}^\textrm{surf}_{x_0}\) in a neighborhood of \(\partial B_{(1-3\theta )\varepsilon }(x_0)\) and

$$\begin{aligned} \mathcal {F}\big (\bar{z}_\varepsilon ,B_{(1-3\theta )\varepsilon }(x_0)\big ) \le \textbf{m}_{\mathcal {F}}\big (\bar{u}^\textrm{surf}_{x_0},B_{(1-3\theta )\varepsilon }(x_0)\big ) + \gamma _{d-1} \varepsilon ^{d}. \end{aligned}$$
(3.64)

We extend \(\bar{z}_\varepsilon \) to a function in \(GSBV^{p(\cdot )}(B_\varepsilon (x_0);\mathbb {R}^m)\) by setting \(\bar{z}_\varepsilon = \bar{u}^\textrm{surf}_{x_0}\) outside \(B_{(1-3\theta )\varepsilon }(x_0)\). Now, we apply Lemma 3.2 with u and v replaced by \(\bar{z}_\varepsilon \) and \(\bar{u}_\varepsilon \), respectively, and \(D'_{\varepsilon ,x_0}:=B_{(1-2\theta )\varepsilon }(x_0)\), \(D''_{\varepsilon ,x_0}:=B_{(1-\theta )\varepsilon }(x_0)\) and \(E_{\varepsilon ,x_0}:=C_{\varepsilon ,\theta }(x_0)\), where \(C_{\varepsilon ,\theta }(x_0)\) still denotes the annulus \(B_\varepsilon (x_0)\backslash \overline{B_{(1-4\theta )\varepsilon }(x_0)}\) (see (3.35)). We then find \(\bar{w}_\varepsilon \in GSBV^{p(\cdot )}(B_\varepsilon (x_0);\mathbb {R}^m)\) such that \(\bar{w}_\varepsilon =\bar{u}_\varepsilon \) on \(B_\varepsilon (x_0)\backslash B_{(1-\theta )\varepsilon }(x_0)\) and

$$\begin{aligned} \begin{aligned} \mathcal {F}(\bar{w}_\varepsilon , B_\varepsilon (x_0))&\le (1+\eta ) \left( \mathcal {F}(\bar{z}_\varepsilon , B_{(1-\theta )\varepsilon }(x_0)) + \mathcal {F}(\bar{u}_\varepsilon , C_{\varepsilon ,\theta }(x_0))\right) \\&\quad + M \int _{(B_{(1-\theta )\varepsilon }(x_0) {\setminus } B_{(1-2\theta )\varepsilon }(x_0))}\left( \frac{|\bar{z}_\varepsilon -\bar{u}_\varepsilon |}{\varepsilon }\right) ^{p(x)}\,\textrm{d}x +\eta \mathcal {L}^d(B_\varepsilon (x_0)). \end{aligned} \nonumber \\ \end{aligned}$$
(3.65)

In particular, by (3.50)(iv) we have that \(\bar{w}_\varepsilon =\bar{z}_\varepsilon =u\) in a neighborhood of \(\partial B_\varepsilon (x_0)\). By (3.50)(iii) and the fact that \(\bar{z}_\varepsilon =\bar{u}^\textrm{surf}_{x_0}\) outside \(B_{(1-3\theta )\varepsilon }(x_0)\) we get

$$\begin{aligned} \begin{aligned}&\lim _{\varepsilon \rightarrow 0} \varepsilon ^{-(d+1)} \int _{(B_{(1-\theta )\varepsilon }(x_0) {\setminus } B_{(1-2\theta )\varepsilon }(x_0))}\left( \frac{|\bar{z}_\varepsilon -\bar{u}_\varepsilon |}{\varepsilon }\right) ^{p(x)}\,\textrm{d}x \\&\quad = \lim _{\varepsilon \rightarrow 0} \ \varepsilon ^{-(d+1)} \int _{B_{(1-\theta )\varepsilon }(x_0)} \left( \frac{|\bar{u}_\varepsilon - \bar{u}^\textrm{surf}_{x_0}|}{\varepsilon } \right) ^{p(x)} \, \textrm{d}x = 0. \end{aligned} \end{aligned}$$
(3.66)

Plugging in (3.65) we find that, for a suitable non-negative vanishing sequence \(\varrho _\varepsilon \), it holds that

$$\begin{aligned} \begin{aligned} \mathcal {F}(\bar{w}_\varepsilon , B_\varepsilon (x_0))\le (1+\eta ) \left( \mathcal {F}(\bar{z}_\varepsilon , B_{(1-\theta )\varepsilon }(x_0)) + \mathcal {F}(\bar{u}_\varepsilon , C_{\varepsilon ,\theta }(x_0))\right) + \varepsilon ^{d-1}\varrho _\varepsilon +\gamma _d\varepsilon ^d\eta . \end{aligned}\nonumber \\ \end{aligned}$$
(3.67)

In order to estimate the terms in (3.67), using that \(\bar{z}_\varepsilon = \bar{u}^\textrm{surf}_{x_0}\) on \(B_{\varepsilon }(x_0) {\setminus } B_{(1-3\theta )\varepsilon }(x_0) \subset C_{\varepsilon ,\theta }(x_0)\),  (\(H_1\)),  (\(H_4\)), and (3.64) we compute

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(\bar{z}_\varepsilon , B_{(1-\theta )\varepsilon }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}&\le \limsup _{\varepsilon \rightarrow 0} \frac{\mathcal {F}(\bar{z}_\varepsilon ,B_{(1- 3 \theta )\varepsilon }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}+ \limsup _{\varepsilon \rightarrow 0} \frac{\mathcal {F}(\bar{u}^\textrm{surf}_{x_0}, C_{\varepsilon ,\theta }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}\nonumber \\&\le \limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{surf}_{x_0},B_{(1-3\theta )\varepsilon }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \nonumber \\&\quad + \frac{\beta }{\gamma _{d-1}} \, \mathcal {H}^{d-1}(C_{1,\theta }(x_0)\cap \Pi _0) \nonumber \\&\le (1-3\theta )^{d-1}\limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{surf}_{x_0},B_{\varepsilon }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \\&\quad +\beta \,(1-(1-4\theta )^{d-1}), \nonumber \end{aligned}$$
(3.68)

where we denote by \(\Pi _0\) the hyperplane passing through \(x_0\) with normal \(\nu _u(x_0)\).

To estimate the remaining term, observe that by rectifiability of \(J_u\) and (3.50) (i) it holds

$$\begin{aligned} \begin{aligned} \mathop {\lim \sup }_{\varepsilon \rightarrow 0}\frac{\beta \mathcal {H}^{d-1}(J_{\bar{u}_\varepsilon } \cap C_{\varepsilon ,\theta }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}&\le \lim _{\varepsilon \rightarrow 0}\frac{\beta \mathcal {H}^{d-1}(J_u \cap C_{\varepsilon ,\theta }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}\\&=\frac{\beta }{\gamma _{d-1}} \, \mathcal {H}^{d-1}(C_{1,\theta }(x_0)\cap \Pi _0)=\beta \,(1-(1-4\theta )^{d-1}). \end{aligned} \end{aligned}$$
(3.69)

With this, using  (\(H_4\)) and (3.50)(ii) we infer

$$\begin{aligned} \begin{aligned} \mathop {\lim \sup }_{\varepsilon \rightarrow 0}\frac{\mathcal {F}(\bar{u}_\varepsilon , C_{\varepsilon ,\theta }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}&\le \mathop {\lim \sup }_{\varepsilon \rightarrow 0}\frac{\beta \int _{C_{\varepsilon ,\theta }(x_0)} (1+ |\nabla \bar{u}_\varepsilon |^{p(x)})\,\textrm{d}x}{\gamma _{d-1}\varepsilon ^{d-1}}\\&\quad + \mathop {\lim \sup }_{\varepsilon \rightarrow 0}\frac{\beta \mathcal {H}^{d-1}(J_{\bar{u}_\varepsilon } \cap C_{\varepsilon ,\theta }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \\&= \beta (1-(1-4\theta )^{d-1}). \end{aligned} \end{aligned}$$
(3.70)

Finally, collecting the estimates in (3.67), (3.68) and (3.70), recalling that \(\varrho _\varepsilon \rightarrow 0\) and that \(\bar{w}_\varepsilon =u\) in a neighborhood of \(\partial B_\varepsilon (x_0)\), we obtain

$$\begin{aligned} \displaystyle \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}&\le \limsup _{\varepsilon \rightarrow 0}\frac{\mathcal {F}(\bar{w}_\varepsilon , B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \\&\le (1+\eta ) \, (1-3\theta )^{d-1} \limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{surf}_{x_0},B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \\ {}&\ \ \ + 2\beta (1+\eta )(1-(1-4\theta )^{d-1}), \end{aligned}$$

whence (3.62) follows up to passing to \(\eta ,\theta \rightarrow 0\). The proof is concluded. \(\square \)

Lemma 3.12

Under the assumptions of Lemma 3.11, for \(\mathcal {H}^{d-1}\)-a.e. \(x_0 \in J_u\) we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(u,B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \ge \limsup _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{surf}_{x_0},B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}. \end{aligned}$$
(3.71)

Proof

Let \(\bar{u}_\varepsilon \) be the sequence of Lemma 3.10, and let \(\theta \in (0,1)\), \(\eta >0\) be fixed. From (3.51) it follows that

$$\begin{aligned} \begin{aligned} 0\le \varepsilon ^{-(d-1)} \int _0^1 \mathcal {H}^{d-1}(\{u\ne \bar{u}_\varepsilon \}\cap \partial B_{\sigma \varepsilon }(x_0))\,\textrm{d}\sigma&= \frac{2}{\varepsilon ^{d}} \mathcal {L}^d(\{u\ne \bar{u}_\varepsilon \}\cap B_{\varepsilon }(x_0))\rightarrow 0 \end{aligned} \end{aligned}$$

as \(\varepsilon \rightarrow 0\). Then for each \(\varepsilon >0\) we can find \(\sigma \in (1-4\theta ,1-3\theta )\) such that

$$\begin{aligned} \begin{aligned}&\mu (\partial B_{\sigma \varepsilon }(x_0)) = \mathcal {H}^{d-1}(\partial B_{\sigma \varepsilon }(x_0)\cap (J_{\bar{u}_\varepsilon }\cup J_u))=0,\quad \hbox { for all } \varepsilon >0, \\&\lim _{\varepsilon \rightarrow 0} \varepsilon ^{-d} \mathcal {H}^{d-1}(\{u\ne \bar{u}_\varepsilon \}\cap \partial B_{\sigma \varepsilon }(x_0)) =0. \end{aligned} \end{aligned}$$
(3.72)

We consider a sequence \(z_\varepsilon \in GSBV^{p(\cdot )}(B_{\sigma \varepsilon }(x_0);\mathbb {R}^m)\) with \(z_\varepsilon = u\) in a neighborhood of \(\partial B_{\sigma \varepsilon }(x_0)\) and

$$\begin{aligned} \mathcal {F}\big (z_\varepsilon ,B_{\sigma \varepsilon }(x_0)\big ) \le \textbf{m}_{\mathcal {F}}\big (u,B_{\sigma \varepsilon }(x_0)\big ) + \gamma _{d-1}\varepsilon ^{d}. \end{aligned}$$
(3.73)

We extend \(z_\varepsilon \) to a function in \(GSBV^{p(\cdot )}(B_\varepsilon (x_0);\mathbb {R}^m)\) by setting \(z_\varepsilon = \bar{u}_\varepsilon \) outside \(B_{\sigma \varepsilon }(x_0)\). By applying Lemma 3.2 with u and v replaced by \(z_\varepsilon \) and \(\bar{u}^\textrm{surf}_{x_0}\), respectively, and the same choice for the sets \(D'_{\varepsilon ,x_0}\), \(D''_{\varepsilon ,x_0}\) and \(E_{\varepsilon ,x_0}\) as in Lemma 3.11, we find \(\bar{w}_\varepsilon \in GSBV^{p(\cdot )}(B_\varepsilon (x_0);\mathbb {R}^m)\) such that \(\bar{w}_\varepsilon =\bar{u}^\textrm{surf}_{x_0}\) on \(B_\varepsilon (x_0)\backslash B_{(1-\theta )\varepsilon }\) and

$$\begin{aligned} \begin{aligned} \mathcal {F}(\bar{w}_\varepsilon , B_\varepsilon (x_0))&\le (1+\eta ) \left( \mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0)) + \mathcal {F}(\bar{u}^\textrm{surf}_{x_0}, C_{\varepsilon ,\theta }(x_0))\right) \\&\quad + M \int _{(B_{(1-\theta )\varepsilon }(x_0) {\setminus } B_{(1-2\theta )\varepsilon }(x_0))}\left( \frac{|z_\varepsilon -\bar{u}^\textrm{surf}_{x_0}|}{\varepsilon }\right) ^{p(x)}\,\textrm{d}x +\eta \mathcal {L}^d(B_\varepsilon (x_0)). \end{aligned} \end{aligned}$$

We notice that, as a consequence of the choice of \(\sigma \), \(z_\varepsilon = \bar{u}_\varepsilon \) outside \(B_{(1-3\theta )}(x_0)\). Then, by virtue of (3.50)\({}_3\), we can find a non-negative sequence \(\varrho _\varepsilon \), vanishing as \(\varepsilon \rightarrow 0\), such that

$$\begin{aligned} \begin{aligned} \mathcal {F}(\bar{w}_\varepsilon , B_\varepsilon (x_0))&\le (1+\eta ) \left( \mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0)) + \mathcal {F}(\bar{u}^\textrm{surf}_{x_0}, C_{\varepsilon ,\theta }(x_0))\right) \\&\quad + \varepsilon ^{d-1}\varrho _\varepsilon +\eta \gamma _d\varepsilon ^d. \end{aligned} \end{aligned}$$
(3.74)

We now estimate each term in the right hand side of (3.74). Taking into account (\(H_1\)), (\(H_4\)), (3.73), the fact that \(z_\varepsilon = \bar{u}_\varepsilon \) on \(B_\varepsilon (x_0)\backslash B_{\sigma \varepsilon }(x_0)\) and the choice of \(\sigma \), we get

$$\begin{aligned} \begin{aligned}&\mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0))\\&\le \textbf{m}_{\mathcal {F}}\big (u,B_{\sigma _\varepsilon }(x_0)\big ) + \gamma _{d-1}\varepsilon ^{d} + \beta \mathcal {H}^{d-1}\left( (\{\bar{u}_\varepsilon \ne u\}\cup J_u\cup J_{\bar{u}_\varepsilon })\cap \partial B_{\sigma _\varepsilon }(x_0)\right) \\&\quad + \mathcal {F}(\bar{u}_\varepsilon , C_{\varepsilon ,\theta }(x_0)). \end{aligned}\nonumber \\ \end{aligned}$$
(3.75)

Now, with (3.70), (3.72) and \(\sigma \le (1-3\theta )\) we then obtain

$$\begin{aligned} \begin{aligned} \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{\mathcal {F}(z_\varepsilon , B_{(1-\theta )\varepsilon }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}&\le \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{ \textbf{m}_{\mathcal {F}}\big (u,B_{\sigma \varepsilon }(x_0)\big )}{\gamma _{d-1}\varepsilon ^{d-1}}+\beta (1-(1-4\theta )^{d-1}) \\&\le (1-3\theta )^{d-1}\mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{ \textbf{m}_{\mathcal {F}}\big (u,B_{\varepsilon }(x_0)\big )}{\gamma _{d-1}\varepsilon ^{d-1}}+\beta (1-(1-4\theta )^{d-1}), \end{aligned} \nonumber \\ \end{aligned}$$
(3.76)

and, as already proven in (3.68),

$$\begin{aligned} \mathop {\lim \sup }_{\varepsilon \rightarrow 0}\frac{\mathcal {F}(\bar{u}^\textrm{surf}_{x_0}, C_{\varepsilon ,\theta }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \le \beta (1-(1-4\theta )^{d-1}). \end{aligned}$$
(3.77)

Collecting the estimates (3.74), (3.76), (3.77) and using \(\varrho _\varepsilon \rightarrow 0\) we infer

$$\begin{aligned} \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{\mathcal {F}(\bar{w}_\varepsilon , B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \le (1+\eta ) \left( (1-3\theta )^{d-1}\mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{ \textbf{m}_{\mathcal {F}}\big (u,B_{\varepsilon }(x_0)\big )}{\gamma _{d-1}\varepsilon ^{d-1}}+2\beta (1-(1-4\theta )^{d-1})\right) . \end{aligned}$$

Finally, since \(\bar{w}_\varepsilon =\bar{u}^\textrm{surf}_{x_0}\) in a neighborhood of \(\partial B_\varepsilon (x_0)\), and using the arbitrariness of \(\eta \) and \(\theta \), we derive

$$\begin{aligned} \begin{aligned} \limsup _{\varepsilon \rightarrow 0}\frac{\textbf{m}_{\mathcal {F}}(\bar{u}^\textrm{surf}_{x_0},B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}&\le \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{\mathcal {F}(\bar{w}_\varepsilon , B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \\&\le \mathop {\lim \sup }_{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}({u},B_{\varepsilon }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}} \\&= \lim _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}({u},B_{\varepsilon }(x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}. \end{aligned} \end{aligned}$$

The proof of (3.71) is concluded. \(\square \)

4 \(\Gamma \)-convergence

In this section, we present a general \(\Gamma \)-convergence result for functionals \(\mathcal {F}:GSBV^{p(\cdot )}(\Omega ; \mathbb {R}^m) \times \mathcal {A}(\Omega ) \rightarrow [0,+\infty )\) of the form

$$\begin{aligned} \mathcal {F}(u,A) = \int _{A} f\big (x, \nabla u(x) \big ) \, \textrm{d}x + \int _{J_u \cap A} g\big (x, [u](x), \nu _u(x)\big ) \, \textrm{d} \mathcal {H}^{d-1} (x) \end{aligned}$$
(4.1)

for each \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and each \(A \in \mathcal {A}(\Omega )\), where \([u](x):=u^+(x)-u^-(x)\) (we refer the reader to [25] for an exhaustive treatment of the topic). To formulate the result, we adopt the notation of Sect. 3 and define the minimization problems \(\textbf{m}_{ \mathcal {F}}(u,A)\) and the functions \(\ell _{x_0,u_0,\xi }\) and \(u_{x_0,a,b,\nu }\) as in (3.1), (3.2) and (3.3), respectively.

Let \(0<\alpha \le \beta <+\infty \) and \(1\le c<+\infty \) be fixed constants. We assume that \(f:\mathbb {R}^d{\times } \mathbb {R}^{m{\times }d}\rightarrow [0,+\infty )\) satisfies the following assumptions:

  1. (f1)

    (measurability) f is Borel measurable on \(\mathbb {R}^d{\times } \mathbb {R}^{m{\times }d}\);

  2. (f2)

    (lower and upper bound) for every \(x \in \mathbb {R}^d\) and every \(\xi \in \mathbb {R}^{m{\times }d}\),

    $$\begin{aligned} \alpha |\xi |^{p(\cdot )} \le f(x,\xi )\le \beta (1+|\xi |^{p(\cdot )}), \end{aligned}$$

and that \(g:\mathbb {R}^d{\times }\mathbb {R}^m_0{\times } {\mathbb {S}}^{d-1} \rightarrow [0,+\infty )\) complies with the following assumptions:

  1. (g1)

    (measurability) g is Borel measurable on \(\mathbb {R}^d{\times }\mathbb {R}^m_0{\times } {\mathbb {S}}^{d-1}\);

  2. (g2)

    (estimate for \(c|\zeta _1|\le |\zeta _2|\)) for every \(x\in \mathbb {R}^d\) and every \(\nu \in {\mathbb {S}}^{d-1}\) we have

    $$\begin{aligned} g(x,\zeta _1,\nu ) \le \,g(x,\zeta _2,\nu ) \end{aligned}$$

    for every \(\zeta _1\), \(\zeta _2\in \mathbb {R}^m_0\) with \(c|\zeta _1|\le |\zeta _2|\);

  3. (g3)

    (lower and upper bound) for every \(x\in \mathbb {R}^d\), \(\zeta \in \mathbb {R}^m_0\), and \(\nu \in {\mathbb {S}}^{d-1}\)

    $$\begin{aligned} \alpha \le g(x,\zeta ,\nu ) \le \beta ; \end{aligned}$$
  4. (g4)

    (symmetry) for every \(x\in \mathbb {R}^d\), \(\zeta \in \mathbb {R}^m_0\), and \(\nu \in {\mathbb {S}}^{d-1}\)

    $$\begin{aligned} g(x,\zeta ,\nu ) = g(x,-\zeta ,-\nu ). \end{aligned}$$

    For future reference, we also introduce the property

  5. (g5)

    (estimate for \(|\zeta _1|\le |\zeta _2|\)) for every \(x\in \mathbb {R}^d\) and every \(\nu \in {\mathbb {S}}^{d-1}\) we have

    $$\begin{aligned} g(x,\zeta _1,\nu ) \le c \,g(x,\zeta _2,\nu ) \end{aligned}$$

    for every \(\zeta _1\), \(\zeta _2 \in \mathbb {R}^m_0\) with \(|\zeta _1|\le |\zeta _2|\).

Notice that assumption  implies (g5) with \(c:=\frac{\beta }{\alpha }\).

The first main result is the following.

Theorem 4.1

(\(\Gamma \)-convergence) Let \(\Omega \subset \mathbb {R}^d\) be open. Let \((f_j)_j\) and \((g_j)_j\) be sequences of functions satisfying f1–f2 and (g1)–(g4), respectively. Let \(\mathcal {F}_j :GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m) \times \mathcal {A}(\Omega ) \rightarrow [0,+\infty )\) be the corresponding sequence of functionals given in (4.1). Then, there exists a functional \(\mathcal {F}_\infty :GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\times \mathcal {A}(\Omega ) \rightarrow [0,+\infty )\) and a subsequence (not relabeled) such that

$$\begin{aligned} \mathcal {F}_\infty (\cdot ,A) =\Gamma \text {-}\lim _{j \rightarrow \infty } \mathcal {F}_j(\cdot ,A) \ \ \ \ \text {with respect to convergence in measure on } A \end{aligned}$$

for all \(A \in \mathcal {A}(\Omega ) \). Moreover, for every \(u\in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(A\in \mathcal {A}(\Omega )\) we have that

$$\begin{aligned} \mathcal {F}_\infty (u,A)= \int _A f_{\infty }\big (x,u(x), \nabla u(x) \big )\, \textrm{d}x +\int _{J_u\cap A}g_{\infty }(x,u^+(x), u^-(x),\nu _u(x))\, \textrm{d}\mathcal {H}^{d-1}(x), \end{aligned}$$
(4.2)

where \(f_\infty =f_{\infty }(x_0,u_0,\xi )\) is given by (3.4) for all \(x_0 \in \Omega \), \(u_0 \in \mathbb {R}^m\), \(\xi \in \mathbb {R}^{m\times d}\), and \(g_\infty =g_{\infty }(x_0,a,b,\nu )\) is given by (3.5) for all \( x_0 \in \Omega \), \(a,b \in \mathbb {R}^m\), and \(\nu \in \mathbb {S}^{d-1}\).

We will prove the compactness of \(\Gamma \)-convergence via the localization technique for \(\Gamma \)-convergence (see [25, Ch. 14–20] for the general method), where the main ingredient is the fundamental estimate in \(GSBV^{p(\cdot )}\), proven with Lemma 3.2. The representation (4.2) in terms of the densities \(f_\infty \) and \(g_\infty \) then will follow by the integral representation result of Theorem 3.1. Indeed, since each \(\mathcal {F}_j\) is invariant under translations of u, then also \(\mathcal {F}_\infty \), as \(\Gamma \)-limit, satisfies the same property. Thus, from Theorem 3.1, in particular (3.4)–(3.5), we infer that \(f_\infty =f_{\infty }(x_0,\xi )\), \(g_\infty =g_{\infty }(x_0,a-b,\nu )\) so that \(\mathcal {F}_\infty \) has the form

$$\begin{aligned} \mathcal {F}_\infty (u,A)= \int _A f_{\infty }\big (x, \nabla u(x) \big )\, \textrm{d}x +\int _{J_u\cap A}g_{\infty }(x,[u](x),\nu _u(x))\, \textrm{d}\mathcal {H}^{d-1}(x), \end{aligned}$$
(4.3)

and the densities \(f_\infty , g_\infty \) can be computed as

$$\begin{aligned} f_\infty (x_0,\xi )&= \limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(\ell _{0,0,\xi },B_\varepsilon (x_0))}{\gamma _d\varepsilon ^{d}}, \end{aligned}$$
(4.4)
$$\begin{aligned} g_\infty (x_0,\zeta ,\nu )&= \limsup _{\varepsilon \rightarrow 0} \frac{\textbf{m}_{\mathcal {F}}(u_{x_0,\zeta ,0,\nu },B_\varepsilon (x_0))}{\gamma _{d-1}\varepsilon ^{d-1}}, \end{aligned}$$
(4.5)

for all \(x_0 \in \Omega \), \(\xi \in \mathbb {R}^{m \times d}\), \(\zeta \in \mathbb {R}^m\) and \(\nu \in \mathbb {S}^{d-1}\).

For our purposes, it will be useful to consider functionals \(\mathcal {I}: L^0(\Omega ;\mathbb {R}^m)\times \mathcal {A}(\Omega )\rightarrow [0,+\infty ]\) defined as

$$\begin{aligned} \mathcal {I}(u,A):= {\left\{ \begin{array}{ll} \int _A f(x,\nabla u(x))\,\textrm{d}x, &{} u\in GSBV^{p(\cdot )}(A;\mathbb {R}^m), \\ +\infty &{} \hbox { otherwise. } \end{array}\right. } \end{aligned}$$
(4.6)

We recall a result concerning the existence of suitable truncations of a measurable function u by which functionals \(\mathcal {F}\) as above almost decrease (see [16, Lemma 4.1]). For our purposes, the statement below is formulated in the \(p(\cdot )\)-setting, and since the adaptation of the original proof requires only minor changes, we omit the details.

Lemma 4.2

Let \(\mathcal {F}\) be as in (4.1), where we assume that f satisfies (f1)–(f2) and g satisfies (g1),(g2), (g4) and (g5). Let \(\mathcal {I}\) be as in (4.6). Let \(\eta ,\lambda >0\). Then there exists \(\mu >\lambda \) depending on \(\eta \), \(\lambda \), \(\alpha \), \(\beta \), c such that the following holds: for every open set \(A\subset \Omega \) and for every \(u\in L^0(\mathbb {R}^d, \mathbb {R}^m)\) such that \({u}|_A\in GSBV^{p(\cdot )}(A,\mathbb {R}^m)\), there exists \(\hat{u}\in L^\infty (\mathbb {R}^d,\mathbb {R}^m)\) such that \(\hat{u}|_A\in SBV^{p(\cdot )}(A,\mathbb {R}^m)\) and

  1. (i)

    \(|\hat{u}|\le \mu \) on \(\mathbb {R}^d\);

  2. (ii)

    \(\hat{u}=u\) \(\mathcal {L}^d\)-a.e. in \(\{|u|\le \lambda \}\);

  3. (iii)

    \(\mathcal {F}(\hat{u},A) \le (1+\eta ) \mathcal {F}(u,A) + \beta \mathcal {L}^d (A\cap \{|u|\ge \lambda \}).\)

Moreover, there exists \(\hat{v}\) with the same properties of \(\hat{u}\) such that (iii) holds for the functional \(\mathcal {I}\) with \(\hat{v}\) in place of \(\hat{u}\).

Let \((\mathcal {F}_j)_j\) be a sequence of functionals of the form (4.1). We start by proving some properties of the \(\Gamma \)-liminf and \(\Gamma \)-limsup with respect to the topology of the convergence in measure. To this end, we define

$$\begin{aligned} \mathcal {F}_\infty '(u,A)&:=\Gamma -\liminf _{n \rightarrow \infty } \mathcal {F}_j(u,A) = \inf \big \{ \liminf _{j \rightarrow \infty } \mathcal {F}_j(u_j,A): \ u_j \rightarrow u \hbox { in measure on } A \big \}, \nonumber \\ \mathcal {F}_\infty ''(u,A)&:= \Gamma -\limsup _{n \rightarrow \infty } \mathcal {F}_j(u,A) = \inf \big \{ \limsup _{j \rightarrow \infty } \mathcal {F}_j(u_j,A): \ u_j \rightarrow u \hbox { in measure on } A \big \} \end{aligned}$$
(4.7)

for all \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(A \in \mathcal {A}(\Omega )\).

Lemma 4.3

(Properties of \(\Gamma \)-liminf and \(\Gamma \)-limsup) Let \(\Omega \subset \mathbb {R}^d\) be an open set, and

\(\mathcal {F}_j:GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\times \mathcal {A}(\Omega ) \rightarrow [0,\infty )\) be a sequence of functionals as in (4.1), where we assume that \(f_j\) and \(g_j\) comply with f1–f2 and (g1)–(g4), respectively, for all \(j\in \mathbb {N}\). Define \(\mathcal {F}_\infty '\) and \(\mathcal {F}_\infty ''\) as in (4.7), and write, for brevity,

$$\begin{aligned} \mathcal {G}(u,A):= \int _A |\nabla u|^{p(\cdot )}\,\textrm{d}x+ \mathcal {H}^{d-1}(J_u \cap A). \end{aligned}$$

Then we have

$$\begin{aligned} \mathrm{(i)}&\ \ \mathcal {F}_\infty '(u,A) \le \mathcal {F}_\infty '(u,B), \ \ \ \ \ \ \mathcal {F}_\infty ''(u,A) \le \mathcal {F}_\infty ''(u,B) \ \ \ \text { whenever } A \subset B, \nonumber \\ \mathrm{(ii)}&\ \ \alpha \mathcal {G}(u,A) \le \mathcal {F}_\infty '(u,A) \le \mathcal {F}_\infty ''(u,A) \le \beta \mathcal {G}(u,A) + \beta \mathcal {L}^d(A), \nonumber \\ \mathrm{(iii)}&\ \ \mathcal {F}_\infty '(u,A) = \sup \nolimits _{B \subset \subset A} \mathcal {F}_\infty '(u,B), \ \ \ \ \mathcal {F}_\infty ''(u,A) = \sup \nolimits _{B \subset \subset A} \mathcal {F}_\infty ''(u,B) \ \ \ \nonumber \\&\text { whenever } A \in \mathcal {A}(\Omega ), \nonumber \\ \mathrm{(iv)}&\ \ \mathcal {F}_\infty '(u,A\cup B) \le \mathcal {F}_\infty '(u,A) + \mathcal {F}_\infty '(u,B), \nonumber \\&\ \ \mathcal {F}_\infty ''(u,A\cup B) \le \mathcal {F}_\infty ''(u,A) + \mathcal {F}_\infty ''(u,B) \ \ \ \text { whenever } A,B \in \mathcal {A}(\Omega ), \end{aligned}$$
(4.8)

where \(\alpha , \beta \) have been introduced in (f2) and (g3).

Proof

The monotonicity property (i) follows from the fact that \(\mathcal {F}_j(u,\cdot )\) are measures. The upper bound in (ii) can be inferred choosing the constant sequence \(u_j=u\) in (4.7) and taking into account the upper bounds in (f2) and (g3). For what concerns the lower bound in (ii), we consider an (almost) optimal sequence \((v_j)_j\) in (4.7). Then, with the lower bounds in (f2) and (g3) we get

$$\begin{aligned} \sup _{j \in \mathbb {N}} \alpha \mathcal {G}(v_j, A)<+\infty . \end{aligned}$$

Now, since \(v_j\rightarrow u\) in measure on A, by arguing as in the proof of Lemma 3.6 and exploiting the lower semicontinuity inequalities

$$\begin{aligned} \begin{aligned} \int _A |\nabla u(x)|^{p(x)}\,\textrm{d}x&\le \displaystyle \mathop {\lim \inf }_{j\rightarrow +\infty } \int _A|\nabla v_j(x)|^{p(x)}\,\textrm{d}x <+\infty , \\ \mathcal {H}^{d-1}(J_u\cap A)&\le \displaystyle \mathop {\lim \inf }_{j\rightarrow +\infty } \mathcal {H}^{d-1}(J_{v_j}\cap A), \end{aligned} \end{aligned}$$

we easily obtain the lower bound.

In order to prove (iii) and (iv), we preliminary show that for every UV and W open subsets of \(\Omega \), with \(V\subset \subset W \subset \subset U\), we have

$$\begin{aligned} \mathcal {F}_\infty '(u,U) \le \mathcal {F}_\infty '(u,W) + \mathcal {F}_\infty '(u,U\backslash \overline{V}),\quad \mathcal {F}_\infty ''(u,U) \le \mathcal {F}_\infty ''(u,W) + \mathcal {F}_\infty ''(u,U\backslash \overline{V}). \nonumber \\ \end{aligned}$$
(4.9)

We confine ourselves to the proof of the first assertion in (4.9), the other one being similar. Let \((u_j)_j\) and \((v_j)_j\) be sequences in \(GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) converging in measure to u on W and \(U\backslash \overline{V}\), respectively, such that

$$\begin{aligned} \mathcal {F}_\infty '(u,W) = \mathop {\lim \inf }_{j\rightarrow +\infty } \mathcal {F}_j(u_j,W), \quad \mathcal {F}_\infty '(u,U\backslash \overline{V}) = \mathop {\lim \inf }_{j\rightarrow +\infty } \mathcal {F}_j(v_j,U\backslash \overline{V}). \end{aligned}$$
(4.10)

We may assume, up to passing to a not relabeled subsequence, that each liminf above is a limit. We fix \(\eta \in (0,1)\) and \(\lambda >0\) such that

$$\begin{aligned} \beta \mathcal {L}^d (U\cap \{|u|\ge \lambda \})<\eta . \end{aligned}$$
(4.11)

By virtue of Lemma 4.2, there exists \(\mu >\lambda \) such that, for every \(k\ge 1\) we can find \(\hat{u}_{j_k}\in SBV^{p(\cdot )}(W;\mathbb {R}^m)\cap L^\infty (W;\mathbb {R}^m)\), with \(|\hat{u}_{j_k}|\le \mu \), \(\hat{v}_{j_k}\in SBV^{p(\cdot )}(U\backslash \overline{V};\mathbb {R}^m)\cap L^\infty (U\backslash \overline{V};\mathbb {R}^m)\), with \(|\hat{v}_{j_k}|\le \mu \), such that \(\hat{u}_{j_k}=u_j\) \(\mathcal {L}^d\)-a.e. in \(W\cap \{|u_j|\le \lambda \}\), \(\hat{v}_{j_k}=v_j\) \(\mathcal {L}^d\)-a.e. in \((U\backslash \overline{V})\cap \{|v_j|\le \lambda \}\) and

$$\begin{aligned} \begin{aligned} \mathcal {F}_j(\hat{u}_{j_k},W)&\le \left( 1+\eta \right) \mathcal {F}_j(u_j,W) + \beta \mathcal {L}^d (W\cap \{|u_j|\ge \lambda \}), \\ \mathcal {F}_j(\hat{v}_{j_k},U\backslash \overline{V})&\le \left( 1+\eta \right) \mathcal {F}_j(v_j,U\backslash \overline{V}) + \beta \mathcal {L}^d ((U\backslash \overline{V})\cap \{|v_j|\ge \lambda \}). \end{aligned} \end{aligned}$$
(4.12)

We apply Lemma 3.2 with \(\eta \) above, \(D'':=W\), \(E:=U\backslash \overline{V}\), \(u=\hat{u}_{j_k}\), \(v=\hat{v}_{j_k}\), for some \(D'\) with \(V\subset \subset D'\subset \subset W\). Note that \(W\backslash D'\subset U\backslash \overline{V}\). We then find a function \(\hat{w}_{j_k}\in SBV^{p(\cdot )}(U;\mathbb {R}^m)\cap L^\infty (U;\mathbb {R}^m)\) such that

$$\begin{aligned} \begin{aligned} \mathcal {F}_j (\hat{w}_{j_k}, U) \le&\; {\left( 1+\eta \right) }\big (\mathcal {F}_j(\hat{u}_{j_k},W) + \mathcal {F}_j(\hat{v}_{j_k},U\backslash \overline{V}) \big ) \\&+ M \int _{W\backslash D'}\left( \frac{|\hat{u}_{j_k}-\hat{v}_{j_k}|}{\delta }\right) ^{p(x)}\,\textrm{d}x + {\eta } \mathcal {L}^d(D' \cup E). \end{aligned} \end{aligned}$$
(4.13)

Note that, by the dominated convergence in measure, \(\hat{u}_{j_k}-\hat{v}_{j_k}\rightarrow 0\) in \(L^{p(\cdot )}(W\backslash D';\mathbb {R}^m)\) as \(k\rightarrow +\infty \). Moreover, recalling (3.6)(ii), we have that \(\hat{w}_{j_k}\rightarrow u\) in measure on U as \(k\rightarrow +\infty \). By a diagonal argument this implies, in particular, that

$$\begin{aligned} \mathcal {F}_\infty '(u,U) \le \mathop {\lim \inf }_{k\rightarrow +\infty } \mathcal {F}_{j_k} (\hat{w}_{j_k}, U). \end{aligned}$$
(4.14)

Note also that, from (4.11) and the convergence in measure of both \(u_{j_k}\) and \(v_{j_k}\) to u, we have

$$\begin{aligned} \mathcal {L}^d (W\cap \{|u_{j_k}|\ge \lambda \})<\eta \quad \hbox { and } \quad \mathcal {L}^d ((U\backslash \overline{V})\cap \{|v_{j_k}|\ge \lambda \})<\eta \end{aligned}$$

for k large enough. Then, combining (4.12) with (4.13), (4.10) and passing to the limit as \(k\rightarrow +\infty \), and then letting \(\eta \rightarrow 0^+\), assertion (4.9) follows.

We now prove the inner regularity of \(\mathcal {F}_\infty '\), the first property in (iii). Combining (4.8)(ii) and (4.9) we find

$$\begin{aligned} \mathcal {F}_\infty '(u,U) \le \mathcal {F}_\infty '(u,W)+ \beta \mathcal {G}(u,U\backslash \overline{V}) + \beta \mathcal {L}^d(U\backslash \overline{V}). \end{aligned}$$

Now, we can choose \(V\subset \subset U\) and U in such a way that \(\mathcal {L}^d(U\backslash \overline{V})\) and \(\mathcal {G}(u,U\backslash \overline{V})\) be arbitrarily small, and recalling that \(\mathcal {F}_\infty '(u,\cdot )\) is an increasing set function by (4.8)(i), we obtain (4.8)(iii) for \(\mathcal {F}_\infty '\). The proof of the analogous property for \(\mathcal {F}_\infty ''\) is similar.

We conclude by showing property (iv) for \(\mathcal {F}_\infty '\). First, we note that it is not restrictive to assume that \(A \cap B \ne \emptyset \), otherwise the inequalities in (iv) are straightforward. It is well known (see, e.g., [5, Proof of Lemma 5.2]) that given \(\eta >0\), one can choose in \(\Omega \) open sets \(U \subset \subset U' \subset \subset A\) and \(V \subset \subset V' \subset \subset B\) such that \(U' \cap V' = \emptyset \), and \(\mathcal {G}(u,(A\cup B) {\setminus } (\overline{U \cup V} )) + \mathcal {L}^d((A\cup B) {\setminus } (\overline{U \cup V} )) \le \eta \). Then using, (4.8)(i),(ii) and (4.9) we get

$$\begin{aligned} \mathcal {F}_\infty '(u,A \cup B)&\le \mathcal {F}_\infty '(u,U' \cup V') + \mathcal {F}_\infty '(u, (A\cup B) {\setminus } \overline{U \cup V} ) \\&\le \mathcal {F}_\infty '(u,U')+ \mathcal {F}_\infty '(u,V') + \beta \eta \\ {}&\le \mathcal {F}_\infty '(u,A) + \mathcal {F}_\infty '(u,B) + \beta \eta , \end{aligned}$$

where we also used \(\mathcal {F}_\infty '(u,U' \cup V') \le \mathcal {F}_\infty '(u,U') + \mathcal {F}_\infty '(u,V')\) which holds due to \(U' \cap V' = \emptyset \). Since \(\eta \) was arbitrary, the statement follows. \(\square \)

We can now prove Theorem 4.1.

Proof of Theorem 4.1

First, we prove the existence of the \(\Gamma \)-limit by applying an abstract compactness result for \(\bar{\Gamma }\)-convergence, see [25, Theorem 16.9]. This implies the existence of an increasing sequence of integers \((j_k)_k\) such that \(\mathcal {F}_\infty '\) and \(\mathcal {F}_\infty ''\) defined in (4.7) with respect to \((j_k)_k\) satisfy

$$\begin{aligned} (\mathcal {F}_\infty ')_-(u,A) = (\mathcal {F}_\infty '')_-(u,A) \end{aligned}$$

for all \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(A \in \mathcal {A}(\Omega )\), where \((\mathcal {F}_\infty ')_-\) and \((\mathcal {F}_\infty '')_-\) denote the inner regular envelopes of \(\mathcal {F}_\infty '\) and \(\mathcal {F}_\infty ''\), respectively. By (4.8)(iii) we know that \(\mathcal {F}_\infty '\) and \(\mathcal {F}_\infty ''\) are inner regular, and thus they both coincide with their respective inner regular envelopes. This shows that the \(\Gamma \)-limit, denoted by \(\mathcal {F}_\infty := \mathcal {F}_\infty ' = \mathcal {F}_\infty ''\), exists for all \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and all \(A \in \mathcal {A}(\Omega )\).

We now check that \(\mathcal {F}_\infty \) satisfies assumptions (\(H_1\))–(\(H_4\)) of the integral representation result, Theorem 3.1. First, the definition in (4.7) and the locality of each \(\mathcal {F}_j\) show that \(\mathcal {F}_\infty (\cdot , A)\) is local according to (\(H_3\)) for any \(A \in \mathcal {A}(\Omega )\). Moreover, \(\mathcal {F}_\infty (\cdot ,A)\) complies with (\(H_2\)) for any \(A \in \mathcal {A}(\Omega )\) in view of [25, Remark 16.3]. Now, since \(\mathcal {F}_\infty \) is increasing, superadditive, inner regular (see [25, Proposition 16.12 and Remark 16.3]) and subadditive by (4.8)(iv), the De Giorgi-Letta criterion (see [25, Theorem 14.23]) ensures that \(\mathcal {F}_\infty (u,\cdot )\) can be extended to a Borel measure. Thus, also (\(H_1\)) is satisfied. Eventually, by (4.8)(ii) we get (\(H_4\)). Therefore, we can conclude that \(\mathcal {F}_\infty \) admits a representation of the form (4.2). \(\square \)

5 Identification of the \(\Gamma \)-limit

In this section we identify the structure of the \(\Gamma \)-limit provided by Theorem 4.1, by showing a separation of scales effect; i.e., that there is no interaction between the bulk and surface densities, as \(f_\infty \) is only determined by \((f_j)_j\) and \(g_\infty \) is only determined by \((g_j)_j\).

We assume that \(f:\mathbb {R}^d{\times } \mathbb {R}^{m{\times }d}\rightarrow [0,+\infty )\) satisfies f1,f2 and the following: for every \(x \in \mathbb {R}^d\) and every \(\xi \in \mathbb {R}^{m{\times }d}\),

  1. (f3)

    (continuity in \(\xi \)) for every \(x \in \mathbb {R}^d\) we have

    $$\begin{aligned} |f(x,\xi _1)-f(x,\xi _2)| \le \omega _1(|\xi _1-\xi _2|)\big (1+f(x,\xi _1)+f(x,\xi _2)\big ) \end{aligned}$$

    for every \(\xi _1\), \(\xi _2 \in \mathbb {R}^{m{\times }d}\);

and that \(g:\mathbb {R}^d{\times }\mathbb {R}^m_0{\times } {\mathbb {S}}^{d-1} \rightarrow [0,+\infty )\) satisfies (g1), (g2), (g3), (g4) and complies with

  1. (g6)

    (continuity in \(\zeta \)) for every \(x\in \mathbb {R}^d\) and every \(\nu \in {\mathbb {S}}^{d-1}\) we have

    $$\begin{aligned} |g(x,\zeta _2,\nu )-g(x,\zeta _1,\nu )|\le \omega _2(|\zeta _1-\zeta _2|)\big (g(x,\zeta _1,\nu )+g(x,\zeta _2,\nu )\big ) \end{aligned}$$

    for every \(\zeta _1\), \(\zeta _2\in \mathbb {R}^m_0\), where \(\omega _2:[0,+\infty ) \rightarrow [0,+\infty )\) is a nondecreasing continuous function such that \(\omega _2(0)=0\).

5.1 Identification of the bulk density

We start with the identification of the bulk density. To do this, we restrict functionals \(\mathcal {F}\) as in (4.1) to Sobolev functions \(W^{1,p(\cdot )}(\Omega ;\mathbb {R}^m)\). Indeed, since every Sobolev function has a \(\mathcal {H}^{d-1}\)-negligible jump set we have

$$\begin{aligned} \mathcal {F}(u,A)= \int _\Omega f(x,\nabla u)\,\textrm{d}x,\quad \hbox { for all } u\in W^{1,p(\cdot )}(\Omega ;\mathbb {R}^m). \end{aligned}$$
(5.1)

We set, for every \(\xi \in \mathbb {R}^{m\times d}\),

$$\begin{aligned} \bar{\ell }_\xi :=\ell _{0,0,\xi }, \end{aligned}$$
(5.2)

where \(\ell _{x_0,u_0,\xi }\) is defined as in (3.2). In analogy to (3.1), for every \(u\in W^{1,p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(A\in \mathcal {A}(\Omega )\) we define

$$\begin{aligned} \textbf{m}_{\mathcal {F}}^{1,p(\cdot )}(u,A) = \inf _{v \in W^{1,p(\cdot )}(\Omega ;\mathbb {R}^m)} \ \lbrace \mathcal {F}(v,A): \ v = u \ \text { in a neighborhood of } \partial A \rbrace . \end{aligned}$$
(5.3)

We consider the functionals \({F}_j: L^1(\Omega ;\mathbb {R}^m)\times \mathcal {A}(\Omega )\rightarrow [0,+\infty ]\) defined as

$$\begin{aligned} {F}_j(u,A):= {\left\{ \begin{array}{ll} \int _A f_j(x,\nabla u(x))\,\textrm{d}x, &{} u\in W^{1,p(\cdot )}(\Omega ;\mathbb {R}^m), \\ +\infty &{} \hbox { otherwise. } \end{array}\right. } \end{aligned}$$

where \(f_j\) satisfies (f1),(f2) and (f3) for every \(j\in \mathbb {N}\). We then have the following \(\Gamma \)-convergence result.

Proposition 5.1

The functionals \({F}_j(\cdot ,A)\) \(\Gamma \)-converge (up to a not relabeled subsequence) as \(j\rightarrow +\infty \) in the strong topology of \(L^1(\Omega ;\mathbb {R}^m)\) to the functional \({F}(\cdot ,A)\) for every \(A\in \mathcal {A}(\Omega )\), where

$$\begin{aligned} {F}(u,A)=\int _A f_\textrm{sob}(x,\nabla u(x))\,\textrm{d}x \end{aligned}$$
(5.4)

and

$$\begin{aligned} f_\textrm{sob}(x,\xi ):= \mathop {\lim \sup }_{\varepsilon \rightarrow 0^+} \frac{\textbf{m}_{{F}}^{1,p(\cdot )}(\bar{\ell }_\xi , B_\varepsilon (x))}{\gamma _d \varepsilon ^d},\quad \hbox { for all } x\in \Omega \text { and } \xi \in \mathbb {R}^{m\times d}. \end{aligned}$$
(5.5)

Moreover, \(f_\textrm{sob}\) is a Carathéodory function satisfying f2 and it holds that

$$\begin{aligned} f_\textrm{sob}(x,\xi ) = \mathop {\lim \sup }_{\varepsilon \rightarrow 0^+} \mathop {\lim \inf }_{j\rightarrow +\infty } \frac{\textbf{m}_{{F}_j}^{1,p(\cdot )}(\bar{\ell }_\xi , B_\varepsilon (x))}{\gamma _d \varepsilon ^d} = \mathop {\lim \sup }_{\varepsilon \rightarrow 0^+} \mathop {\lim \sup }_{j\rightarrow +\infty } \frac{\textbf{m}_{{F}_j}^{1,p(\cdot )}(\bar{\ell }_\xi , B_\varepsilon (x))}{\gamma _d \varepsilon ^d}. \nonumber \\ \end{aligned}$$
(5.6)

Proof

The proof of the \(\Gamma \)-convergence result and the integral representation (5.4) can be obtained as in [22, Theorem 4.1 and 4.2]. The characterization (5.5) follows by adapting the global method of Sect. 3.2 to the variable exponent Sobolev setting, while (5.6) is a standard consequence of the \(\Gamma \)-convergence. We omit the details. \(\square \)

We can now proceed with the announced identification of the bulk density.

Theorem 5.2

Under the assumptions of Theorem 4.1 and assumption (f3) on the sequence \((f_j)\), let \(f_\textrm{sob}\) and \(f_\infty \) be defined as in (5.5) and (4.4), respectively. Then, for all \(u\in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) we have that

$$\begin{aligned} f_\infty (x,\nabla u(x)) = f_\textrm{sob}(x,\nabla u(x))\quad \text { for } \mathcal {L}^d-\text {a.e.}\ x\in \Omega . \end{aligned}$$
(5.7)

Proof

We show the two inequalities in (5.7). We first prove

$$\begin{aligned} f_\infty (x,\nabla u(x)) \le f_\textrm{sob}(x,\nabla u(x)) \quad \hbox { for } \mathcal {L}^d-\text {a.e.}\ x\in \Omega . \end{aligned}$$
(5.8)

First, in view of (3.1) and (5.3), we get \(\textbf{m}_{\mathcal {F}}(\bar{\ell }_{ \xi },B_{\varepsilon }(x)) \le \textbf{m}^{1,p(\cdot )}_{\mathcal {F}}(\bar{\ell }_{ \xi },B_{\varepsilon }(x))\) for all \(\xi \in \mathbb {R}^{m\times d}\), where we recall the notation \(\bar{\ell }_{\xi }\) introduced in (5.2). Then (3.4) implies

$$\begin{aligned} f_\infty (x,\xi )&=\limsup _{\varepsilon \rightarrow 0^+}\frac{\textbf{m}_{\mathcal {F}}(\bar{\ell }_{\xi }, B_{\varepsilon }(x))}{\gamma _d\varepsilon ^d} \le \limsup _{\varepsilon \rightarrow 0^+}\frac{\textbf{m}^{1,p(\cdot )}_{\mathcal {F}}(\bar{\ell }_{\xi },B_{\varepsilon }(x))}{\gamma _d\varepsilon ^d}, \end{aligned}$$
(5.9)

while by (5.5) and (5.1) we find

$$\begin{aligned} f_\textrm{sob}(x,\xi ) = \limsup _{\varepsilon \rightarrow 0^+} \frac{\textbf{m}^{1,p(\cdot )}_{\mathcal {F}}(\bar{\ell }_{\xi },B_{\varepsilon }(x))}{\gamma _d\varepsilon ^d}. \end{aligned}$$
(5.10)

Thus, since both \(f_\infty \) and \(f_\textrm{sob}\) are continuous with respect to \(\xi \) by (f3), combining (5.9)–(5.10) we obtain (5.8).

We now prove the reverse inequality

$$\begin{aligned} f_\textrm{sob}(x,\nabla u(x)) \le f_\infty (x,\nabla u(x)) \quad \hbox { for } \mathcal {L}^d-\text {a.e.}\ x\in \Omega . \end{aligned}$$
(5.11)

First, from the Radon-Nikodým Theorem we have that

$$\begin{aligned} f_\infty (x,\nabla u(x))= \lim _{\varepsilon \rightarrow 0^+} \frac{\mathcal {F}(u,B_\varepsilon (x))}{\gamma _d\varepsilon ^d}<+\infty \end{aligned}$$
(5.12)

holds for \(\mathcal {L}^d\)-a.e. \(x\in \Omega \). Let \((u_j)\) be a sequence of measurable functions such that \(u_j\in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\)

$$\begin{aligned} u_j\rightarrow u \hbox { in measure on } \Omega \quad \hbox { and } \quad \lim _{j\rightarrow +\infty } \mathcal {F}_j(u_j,\Omega )=\mathcal {F}(u,\Omega ). \end{aligned}$$

Since \(u\in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^d)\), by virtue of Lemma 2.4 the approximate gradient \(\nabla u(x)\) exists for \(\mathcal {L}^d\)-a.e. \(x\in \Omega \). Then, since (5.11) needs to hold for \(\mathcal {L}^d\)-a.e. \(x\in \Omega \), we may assume that (5.12) holds at x and that \(\nabla u(x)\) exists. Since \(\mathcal {F}(u,\cdot )\) is a Radon measure, there exists a subsequence \((\varepsilon _k)\subset (0,+\infty )\) with \(\varepsilon _k\searrow 0\) as \(k\rightarrow +\infty \) such that \(\mathcal {F}(u,\partial B_{\varepsilon _k}(x))=0\) for every \(k\in \mathbb {N}\) and such that (5.6) holds along \((\varepsilon _k)\), namely

$$\begin{aligned} f_\textrm{sob}(x,\nabla u(x)) = \mathop {\lim }_{k\rightarrow +\infty } \mathop {\lim \sup }_{j\rightarrow +\infty } \frac{\textbf{m}_{\mathcal {F}_j}^{1,p(\cdot )}(\bar{\ell }_{\nabla u(x)}, B_{\varepsilon _k}(x))}{\gamma _d \varepsilon _k^d}. \end{aligned}$$
(5.13)

Moreover, with fixed \(\eta \in (0,1)\), since \((u_j)\) is a recovery sequence and \(\mathcal {F}(u,\cdot )\) is a Radon measure, for every \(k\in \mathbb {N}\) we can find \(j_k\in \mathbb {N}\) (depending also on \(\eta \)) such that, for every \(j\ge j_k\) it holds that

$$\begin{aligned} \frac{\mathcal {F}_j(u_j, B_{\varepsilon _k}(x))}{\gamma _d \varepsilon _k^d} \le \frac{\mathcal {F}(u, B_{\varepsilon _k}(x))}{\gamma _d \varepsilon _k^d} + {\eta }. \end{aligned}$$
(5.14)

Now, we have to modify the sequence \((u_j)\) to construct a competitor for the minimization problem \(\textbf{m}_{\mathcal {F}_j}^{1,p(\cdot )}(\bar{\ell }_{\nabla u(x)}, B_{\varepsilon }(x))\) which defines \(f_\textrm{sob}\).

We introduce the functions

$$\begin{aligned} u_j^{\varepsilon _k}(y):= \frac{u_j(x+\varepsilon _ky)-u_j(x)}{\varepsilon _k} \quad \hbox { and } \quad u^{\varepsilon _k}(y):= \frac{u(x+\varepsilon _ky)-u(x)}{\varepsilon _k} \quad \hbox { for } y\in B_1. \end{aligned}$$

Then, since \(u_j\rightarrow u\) in measure on \(\Omega \), we have that \(u_j^{\varepsilon _k}\rightarrow u^{\varepsilon _k}\) in measure on \(B_1\) as \(j\rightarrow +\infty \). In addition, by a diagonal argument and up to passing to a larger \(j_k\in \mathbb {N}\), we also have

$$\begin{aligned} \hat{u}_k:=u_{j_k}^{\varepsilon _k} \rightarrow \bar{\ell }_{\nabla u(x)} \quad \hbox { in measure on } B_1 \text { as } k\rightarrow +\infty . \end{aligned}$$
(5.15)

By virtue of (5.13), we may choose \((j_k)_k\) such that also

$$\begin{aligned} f_\textrm{sob}(x,\nabla u(x)) = \mathop {\lim }_{k\rightarrow +\infty } \frac{\textbf{m}_{\mathcal {F}_{j_k}}^{1,p(\cdot )}(\bar{\ell }_{\nabla u(x)}, B_{\varepsilon _k}(x))}{\gamma _d \varepsilon _k^d} \end{aligned}$$
(5.16)

holds. Finally, taking into account (4.1), (5.12), (5.14) and with a change of variables we find

$$\begin{aligned} \mathop {\lim \sup }_{k\rightarrow +\infty } \int _{B_1} f_{j_k}(x+\varepsilon _ky,\nabla \hat{u}_k(y))\,\textrm{d}y \le \lim _{k\rightarrow +\infty } \frac{\mathcal {F}(u,B_{\varepsilon _k}(x))}{\gamma _d\varepsilon _k^d}{+\eta } = f_\infty (x,\nabla u(x)) {+\eta }. \nonumber \\ \end{aligned}$$
(5.17)

Let \(\mathcal {I}_k\) be defined as in (4.6), with \(f_{j_k}(x+\varepsilon _ky,\cdot )\) in place of \(f(x,\cdot )\), and set

$$\begin{aligned} p_k(y):=p(x+\varepsilon _k y),\quad y\in B_1. \end{aligned}$$

We define, accordingly,

$$\begin{aligned} p_k^+:=\sup _{y\in B_1} p_k(y),\quad p_k^-:=\inf _{y\in B_1} p_k(y). \end{aligned}$$

Let \(\lambda >|\nabla u(x)|\). Then, by virtue of Lemma 4.2 there exists \(\mu >\lambda \) such that, for every k, we can find a function \(\hat{v}_k\in SBV^{p_k(\cdot )}(B_1;\mathbb {R}^d)\cap L^\infty (B_1;\mathbb {R}^d)\) such that \(\hat{v}_k= \hat{u}_k\) \(\mathcal {L}^d\)-a.e. in \(B_1\cap \{|\hat{u}_k|\le \lambda \}\), \(|\hat{v}_k|\le \mu \) and

$$\begin{aligned} \mathcal {I}_k(\hat{v}_k, B_1) \le \left( 1+\eta \right) \mathcal {I}_k(\hat{u}_k, B_1) + \beta \mathcal {L}^d (B_1\cap \{|\hat{u}_k|\ge \lambda \}). \end{aligned}$$
(5.18)

Moreover, with (5.15) and the fact that \(|\bar{\ell }_{\nabla u(x)}|\le |\nabla u(x)|<\lambda \) in \(B_1\), we get

$$\begin{aligned} \hat{v}_k\rightarrow \bar{\ell }_{\nabla u(x)} \quad \hbox { in measure on } B_1 \text { as } k\rightarrow +\infty \end{aligned}$$
(5.19)

and \(\mathcal {L}^d (B_1\cap \{|\hat{u}_k|\ge \lambda \})\le \varepsilon _k\) for k large enough. Taking into account (f2), (g3), (5.18) and a change of variables we get

$$\begin{aligned} \begin{aligned}&\alpha \int _{B_1} |\nabla \hat{v}_k|^{p_k(y)}\,\textrm{d}y \le \frac{1+\eta }{\varepsilon _k^d}\int _{B_{\varepsilon _k}(x)}f_{j_k}(y,\nabla u_{j_k}(y))\,\textrm{d}y + \beta \varepsilon _k, \\&\frac{\alpha }{\varepsilon _k}\mathcal {H}^{d-1}(J_{\hat{v}_k}\cap B_1) \le \frac{\alpha }{\varepsilon _k^d}\mathcal {H}^{d-1}(J_{{u}_{j_k}}\cap B_{\varepsilon _k}(x)) \le \frac{1}{\varepsilon _k^d}\int _{J_{{u}_{j_k}}\cap B_{\varepsilon _k}(x)} g_{j_k}(y, [u_{j_k}], \nu _{u_{j_k}})\,\textrm{d}\mathcal {H}^{d-1}, \end{aligned} \end{aligned}$$

for k large enough. Then, with (5.12) and (5.14), we can find a constant \(M>0\) independent of k and \(\eta \) such that

$$\begin{aligned} \int _{B_1} |\nabla \hat{v}_k|^{p_k(\cdot )}\,\textrm{d}y \le M \quad \hbox { and } \quad \mathcal {H}^{d-1}(J_{\hat{v}_k}\cap B_1)\le M \varepsilon _k, \end{aligned}$$
(5.20)

for k large enough, and

$$\begin{aligned} |D^s \hat{v}_k|(B_1)\le 2\mu M\varepsilon _k. \end{aligned}$$
(5.21)

Now, we regularize the sequence \((\hat{v}_k)\) in order to obtain a sequence \(\hat{w}_k\in W^{1,p_k(\cdot )}(B_1;\mathbb {R}^m)\) such that

$$\begin{aligned} \int _{B_1} f_{j_k}(x+\varepsilon _ky,\nabla \hat{w}_k(y))\,\textrm{d}y \le \int _{B_1} f_{j_k}(x+\varepsilon _ky,\nabla \hat{v}_k(y))\,\textrm{d}y + {\eta }. \end{aligned}$$
(5.22)

For this, we may adapt to the variable exponent setting the argument for the proof of [16, Theorem 5.2(b), Step 1], devised for a constant exponent q. We just provide the main steps of this adaptation.

For fixed \(t>0\), we first define the sets

$$\begin{aligned} \begin{aligned} R^t_k&:=\left\{ y\in B_1:\,\, \frac{|D^s \hat{v}_k|(\overline{B_r(y)})}{\mathcal {L}^d(B_r(y))}\le t \quad \hbox { for every } r>0 \hbox { with }\overline{B_r(y)}\subset B_1 \right\} , \\ S^t_k&:= J_{\hat{v}_k} \cup \left\{ y\in B_1:\,\, |\nabla \hat{v}_k(y)|\ge \frac{t}{2} \right\} . \end{aligned} \end{aligned}$$

We claim that

$$\begin{aligned} \begin{aligned} \mathcal {L}^d(B_1\backslash R^t_k)&\le \frac{2\cdot 5^d}{t} \left( |D^s \hat{v}_k|(B_1) + \int _{S_k^t} |\nabla \hat{v}_k(y)|\,\textrm{d}y\right) \\&\le \frac{2\cdot 5^d}{t} |D^s \hat{v}_k|(B_1) + 2^{p^+_k+1}\cdot 5^d \int _{S_k^t} \left( \frac{|\nabla \hat{v}_k(y)|}{t} \right) ^{p_k(y)}\,\textrm{d}y, \end{aligned} \end{aligned}$$

Indeed, the first inequality follows from the Vitali Covering Lemma, arguing exactly as in [16, Theorem 5.2(b), Step 1]. The second inequality follows from the first one, using that \(\frac{2|\nabla \hat{v}_k(y)|}{t}\ge 1\) on \(S_k^t\). Now, taking into account (5.20), we get

$$\begin{aligned} \begin{aligned} \mathcal {L}^d(B_1\backslash R^t_k)&\le \frac{2\cdot 5^d}{t} |D^s \hat{v}_k|(B_1) + \frac{2^{p^+_k+1}\cdot 5^d M}{\min \{t^{p^+_k}, t^{p^-_k}\}}. \end{aligned} \end{aligned}$$
(5.23)

Choosing

$$\begin{aligned} t_k:= (2\mu M \varepsilon _k)^{-\frac{1}{p^-_k-1}} \end{aligned}$$

we have \(t_k\ge 1\) for k large enough and, taking into account (5.21), from (5.23) we obtain

$$\begin{aligned} \begin{aligned} t_k^{p^-_k}\mathcal {L}^d(B_1\backslash R^{t_k}_k)&\le {2\cdot 5^d}+ {2^{p^+_k+1}\cdot 5^d M}. \end{aligned} \end{aligned}$$

By virtue of Lemma 2.1, and since \(p^-_k-1\ge p^--1>0\), it holds now

$$\begin{aligned} t_k^{p^+_k-p^-_k} \le {\gamma _d^{-(p^-_k-p^+_k)}}[\mathcal {L}^d(B_{\varepsilon _k})]^{\frac{p^-_k-p^+_k}{d(p^-_k-1)}} \le C \end{aligned}$$

for k large. We then conclude that

$$\begin{aligned} \begin{aligned} t_k^{p^+_k}\mathcal {L}^d(B_1\backslash R^{t_k}_k) \le C({2\cdot 5^d}+ {2^{p^++1}\cdot 5^d M})=:\widetilde{M}, \end{aligned} \end{aligned}$$
(5.24)

whence, in particular, since \(\varepsilon _k<1\) for k large enough, we get

$$\begin{aligned} \begin{aligned} \mathcal {L}^d(B_1\backslash R^{t_k}_k) \le \frac{\widetilde{M}}{t_k^{p^+_k}}= \widetilde{M}(2\mu M \varepsilon _k)^{\frac{{p^+_k}}{p^-_k-1}}\le \overline{M}_{p}\varepsilon _k^{\frac{{p^-}}{p^+-1}}. \end{aligned} \end{aligned}$$
(5.25)

Now, by a Lusin’s type approximation argument (see, e.g., [30]), one can construct a sequence of Lipschitz functions \((\hat{z}_k)\) on \(B_1\), with Lip\((\hat{z}_k) \le c_d t_k\) for some constant \(c_d\) depending only on the dimension, such that \(\hat{z}_k=\hat{v}_k\) \(\mathcal {L}^d\)-a.e. in \(R_k^{t_k}\). Setting \(\bar{p}:=p(x)\), we claim that \((\hat{z}_k)\) are bounded in \(W^{1,\bar{p}}(B_1;\mathbb {R}^d)\). Indeed, with (5.24) and (5.20) we first have

$$\begin{aligned} \begin{aligned} \int _{B_1} |\nabla \hat{z}_k|^{p_k(y)}\,\textrm{d}y&\le 2^{p^+_k-1}\left( \int _{R_k^{t_k}} |\nabla \hat{v}_k|^{p_k(y)}\,\textrm{d}y + \max \{c_d^{p^+_k},c_d^{p^-_1}\} t_k^{p^+_k}\mathcal {L}^d(B_1\backslash R^{t_k}_k)\right) \\&\le 2^{p^+-1} (M+ \max \{c_d^{p^+},c_d^{p^-}\}\widetilde{M})=:M_{d,p}. \end{aligned} \nonumber \\ \end{aligned}$$
(5.26)

Note that \(\bar{p}=p_k(0)\) for every \(k\in \mathbb {N}\). Then, since \((\bar{p}-p_k(y))^+\le p_k^+-p_k^-\) and \(t_k\ge 1\) for k large enough, with (5.25) for every \(y\in B_1\) we get

$$\begin{aligned} |\nabla \hat{z}_k|^{(\bar{p}-p_k(y))^+} \le c_d^{p_k^+-p_k^-}t_k^{p^+_k-p^-_k}\le C. \end{aligned}$$
(5.27)

Finally, with (5.26) and (5.27), by a simple inequality we obtain

$$\begin{aligned} \begin{aligned} \int _{B_1} |\nabla \hat{z}_k|^{\bar{p}}\,\textrm{d}y&\le \mathcal {L}^d(B_1) + \int _{B_1} |\nabla \hat{z}_k|^{(\bar{p}-p_k(y))^+}|\nabla \hat{z}_k|^{p_k(y)}\,\textrm{d}y \\&\le \mathcal {L}^d(B_1) + CM_{d,p}. \end{aligned} \end{aligned}$$
(5.28)

Then, by applying [33, Lemma 1.2] to \((\hat{z}_k)\), we find a sequence of Lipschitz functions \((\hat{w}_k)\) which satisfy \(\hat{w}_k\in W^{1,\bar{p}}(B_1;\mathbb {R}^m)\), \(|\nabla \hat{w}_k|^{\bar{p}}\) equi-integrable uniformly with respect to k, and \(\mathcal {L}^d(\{\hat{z}_k\ne \hat{w}_k\})\rightarrow 0\) as \(k\rightarrow +\infty \). Since \(|\hat{z}_k|\le \mu \) in \(B_1\), we may assume also that \(|\hat{w}_k|\le \mu \) \(\mathcal {L}^d\)-a.e. in \(B_1\). An inspection to the proof of [33, Lemma 1.2] shows that \((\hat{w}_k)\) can be chosen in such a way that

$$\begin{aligned} \textrm{Lip}(\hat{w}_k)\le c_d\,\textrm{Lip}(\hat{z}_k) \end{aligned}$$
(5.29)

holds.

We claim that \((|\nabla \hat{w}_k|^{p_k(\cdot )})\) is equi-integrable on \(B_1\) uniformly with respect to k. Indeed, arguing as for (5.27) we first get, for every \(y\in B_1\),

$$\begin{aligned} |\nabla \hat{w}_k|^{(p_k(y)-\bar{p})^+} \le \widetilde{c_d}^{p_k^+-p_k^-}t_k^{p^+_k-p^-_k}\le C. \end{aligned}$$
(5.30)

Then, for every fixed \(E\subseteq B_1\), arguing as for (5.28) and taking into account (5.30) we obtain

$$\begin{aligned} \begin{aligned} \int _{E} |\nabla \hat{w}_k|^{p_k(y)}\,\textrm{d}y&\le \mathcal {L}^d(E) + \int _{E} |\nabla \hat{w}_k|^{(p_k(y)-\bar{p})^+}|\nabla \hat{w}_k|^{\bar{p}}\,\textrm{d}y \\&\le \mathcal {L}^d(E) + C\int _{E} |\nabla \hat{w}_k|^{\bar{p}}\,\textrm{d}y. \end{aligned} \end{aligned}$$
(5.31)

This and the equi-integrability of \(|\nabla \hat{w}_k|^{\bar{p}}\) imply the claim.

Moreover, from (5.25), and since by (5.19) the equibounded sequence \((\hat{w}_k-\bar{\ell }_{\nabla u(x)})\) tends to 0 in measure on \(B_1\), we have

$$\begin{aligned} \mathcal {L}^d(\{\hat{w}_k\ne \hat{v}_k\})\le \overline{M}_{p}\varepsilon _k^{\frac{{p^-}}{p^+-1}}, \quad \hbox { and } \quad \int _{B_1} |\hat{w}_k-\bar{\ell }_{\nabla u(x)} |^{p_k(y)}\,\textrm{d}y\rightarrow 0 \end{aligned}$$
(5.32)

as \(k\rightarrow +\infty \).

In order to prove (5.22), we notice that

$$\begin{aligned} \int _{B_1} f_{j_k}(x+\varepsilon _ky,\nabla \hat{w}_k(y))\,\textrm{d}y\le & {} \int _{B_1} f_{j_k}(x+\varepsilon _ky,\nabla \hat{v}_k(y))\,\textrm{d}y \\{} & {} + \int _{\{\hat{w}_k\ne \hat{v}_k\}}f_{j_k}(x+\varepsilon _ky,\nabla \hat{w}_k(y))\,\textrm{d}y. \end{aligned}$$

Now, taking into account the equi-integrability of \((|\nabla \hat{w}_k|^{p_k(\cdot )})\), the upper bound f2 and (5.32), for \(\varepsilon _k\) small enough we get

$$\begin{aligned} \int _{\{\hat{w}_k\ne \hat{v}_k\}}f_{j_k}(x+\varepsilon _ky,\nabla \hat{w}_k(y))\,\textrm{d}y<{\eta }, \end{aligned}$$

whence (5.22) follows.

Finally, we have to modify the sequence \((\hat{w}_k)\) in such a way that it attains the boundary datum \(\bar{\ell }_{\nabla u(x)}\) in a neighborhood of \(\partial B_1\). We know that the functionals \(\mathcal {I}_k(u,A)\) above for \(u\in W^{1,p_k(\cdot )}(A;\mathbb {R}^m)\) and \(A\in \mathcal {A}(\Omega )\) satisfy uniformly the Fundamental Estimate proved in Lemma 3.2. Namely, corresponding to the fixed \(\eta \) above, there exist a constant \(C_\eta >0\) and a sequence \((\hat{\textrm{w}}_k)\) in \(W^{1,p_k(\cdot )}(B_1;\mathbb {R}^m)\) with \(\hat{\textrm{w}}_k=\bar{\ell }_{\nabla u(x)}\) in a neighborhood of \(\partial B_1\) for all \(k\in \mathbb {N}\) such that

$$\begin{aligned} \mathcal {I}_k(\hat{\textrm{w}}_k,B_1)\le & {} (1+\eta )\left( \mathcal {I}_k(\hat{w}_k,B_1) + \mathcal {I}_k(\bar{\ell }_{\nabla u(x)},B_1\backslash \overline{B_{1-\eta }})\right) \nonumber \\{} & {} + C_\eta \int _{B_1} |\hat{w}_k - \bar{\ell }_{\nabla u(x)}|^{p_k(y)}\,\textrm{d}y + \gamma _d\eta . \nonumber \\ \end{aligned}$$
(5.33)

Now, taking into account (f2), (5.32) and the fact that \(\mathcal {L}^d(B_1\backslash \overline{B_{1-\eta }})\le d\eta \), we get

$$\begin{aligned} \mathop {\lim \sup }_{k\rightarrow +\infty }\mathcal {I}_k(\hat{\textrm{w}}_k,B_1) \le (1+\eta ) \mathop {\lim \sup }_{k\rightarrow +\infty }\mathcal {I}_k(\hat{w}_k,B_1) + d\eta (1+\eta )\beta (1+|\nabla u(x)|^{\bar{p}})+\gamma _d\eta . \nonumber \\ \end{aligned}$$
(5.34)

Then, with (5.17), (5.22) and recalling the definition of \(\mathcal {I}_k\), we obtain

$$\begin{aligned} \mathop {\lim \sup }_{k\rightarrow +\infty }\mathcal {I}_k(\hat{\textrm{w}}_k,B_1) \le {(1+\eta )( f_\infty (x,\nabla u(x)) +\eta )} + d\eta (1+\eta )\beta (1+|\nabla u(x)|^{\bar{p}})+\gamma _d\eta . \nonumber \\ \end{aligned}$$
(5.35)

Setting

$$\begin{aligned} \widetilde{\textrm{w}}_k(y):= \varepsilon _k \hat{\textrm{w}}_k((y-x)/\varepsilon _k) + \bar{\ell }_{\nabla u(x)}x \quad \hbox { for } y\in B_{\varepsilon _k}(x), \end{aligned}$$

we have \(\widetilde{\textrm{w}}_k\in W^{1,p(\cdot )}(B_{\varepsilon _k}(x);\mathbb {R}^m)\) and

$$\begin{aligned} \mathcal {I}_k(\hat{\textrm{w}}_k,B_1) = \frac{1}{\varepsilon _k^d} \int _{B_{\varepsilon _k}(x)} f_{j_k}(y, \nabla \widetilde{\textrm{w}}_k(y))\,\textrm{d}y. \end{aligned}$$
(5.36)

Moreover, since \(\hat{\textrm{w}}_k=\bar{\ell }_{\nabla u(x)}\) in a neighborhood of \(\partial B_1\), it follows that \(\widetilde{\textrm{w}}_k=\bar{\ell }_{\nabla u(x)}\) in a neighborhood of \(\partial B_{\varepsilon _k}(x)\). Then, with (5.3), (5.1) and (5.36) we obtain

$$\begin{aligned} \frac{\textbf{m}_{\mathcal {F}_{j_k}}^{1,p(\cdot )}(\bar{\ell }_{\nabla u(x)}, B_{\varepsilon _k}(x))}{\gamma _d \varepsilon _k^d} \le \mathcal {I}_k(\hat{\textrm{w}}_k,B_1), \end{aligned}$$

whence passing to the limsup as \(k\rightarrow +\infty \), recalling (5.35), and then letting \(\eta \rightarrow 0^+\), we get

$$\begin{aligned} \mathop {\lim \sup }_{k\rightarrow +\infty } \frac{\textbf{m}_{\mathcal {F}_{j_k}}^{1,p(\cdot )}(\bar{\ell }_{\nabla u(x)}, B_{\varepsilon _k}(x))}{\gamma _d \varepsilon _k^d} \le f_\infty (x,\nabla u(x)). \end{aligned}$$

The assertion (5.11) then follows from (5.16). \(\square \)

5.2 Identification of the surface density

We conclude our analysis with the identification of the surface density. We will prove that it coincides with the asymptotic surface density of functionals \(\mathcal {F}_j\) when restricted to the space \(SBV_{\textrm{pc}}(A,\mathbb {R}^m)\) of those functions \(u\in SBV(A,\mathbb {R}^m)\) such that \(\nabla u=0\) \(\mathcal {L}^d\)-a.e. in A and \(\mathcal {H}^{d-1}(J_u)<+\infty \).

In order to do that, we consider the sequence of surface energies

$$\begin{aligned} G_j(u,A):= {\left\{ \begin{array}{ll} \displaystyle \int _{J_u\cap A}g_j(x,[u],\nu _u)d \mathcal {H}^{d-1} &{}\text {if} \; u|_A\in GSBV^{p(\cdot )}(A,\mathbb {R}^m),\\ +\infty \quad &{} \hbox {otherwise in}\; L^0(\mathbb {R}^d,\mathbb {R}^m), \end{array}\right. } \end{aligned}$$
(5.37)

and, correspondingly, we define the sequence of minimum problems

$$\begin{aligned} \textbf{m}^{{PC}}_{G_j} (u_{x,\zeta ,\nu },A):= & {} \inf \Big \{G_j(u,A): u\in L^0(\mathbb {R}^d,\mathbb {R}^m),\ u|_A\in SBV_{\textrm{pc}}(A,\mathbb {R}^m),\ \nonumber \\{} & {} \qquad \quad u=u_{x,\zeta ,\nu } \text { near }\partial A \Big \}, \end{aligned}$$
(5.38)

where \(u_{x,\zeta ,\nu }\) coincides with \(u_{x,\zeta ,0,\nu }\) defined in (3.3).

Since, to the best of our knowledge, a \(\Gamma \)-convergence result for functionals \(G_j\) whose densities \(g_j\) explicitly depend on the jump [u] is still missing in literature, with Theorem 5.3 below we will show directly that

$$\begin{aligned} g_{\infty }(x,[u](x),\nu _u) = \limsup _{\varepsilon \rightarrow 0^+} \lim _{j \rightarrow + \infty } \frac{ \textbf{m}_{G_j}^{PC}(u_{x,[u](x),\nu _u}, B_\varepsilon (x))}{\gamma _{d-1}\varepsilon ^{d-1}},\quad \hbox { for } \mathcal {H}^{d-1}-\text {a.e. } x\in J_u. \end{aligned}$$

We also remark that, in the proof below, Theorem 2.9 allows for a quick construction in Step 2.3 of an optimal sequence of piecewise constant functions (cfr. the more involved arguments in [16, Theorem 5.2, (c)-(d)], whose compliance with the present setting was not investigated).

Theorem 5.3

Let \(\Omega \subset \mathbb {R}^d\) be open and \(p:\Omega \rightarrow (1,+\infty )\) be a continuous variable exponent. Let \((f_j)_j\) and \((g_j)_j\) be sequences functions satisfying (f1)–(f3) and (g1), (g2), (g3), (g4) and (g6), respectively. Let \(g_\infty \) be defined by (4.5). Then, for all \(u\in GSBV^{p(\cdot )}(\Omega ,\mathbb {R}^m)\) we have that

$$\begin{aligned} g_\infty (x,[u](x),\nu _u(x)) = g_\textrm{pc}(x,[u](x),\nu _u(x)),\quad \hbox {for } \mathcal {H}^{d-1}-\text {a.e. } x\in J_u, \end{aligned}$$
(5.39)

where

$$\begin{aligned} g_\textrm{pc}(x,\zeta ,\nu ):= \limsup _{\varepsilon \rightarrow 0^+} \lim _{j \rightarrow + \infty } \frac{ \textbf{m}_{G_j}^{PC}(u_{x,\zeta ,\nu }, B_\varepsilon (x))}{\gamma _{d-1}\varepsilon ^{d-1}}. \end{aligned}$$
(5.40)

Proof

For every \(x \in \mathbb {R}^d\), \(\zeta \in \mathbb {R}_0^m\), and \(\nu \in \mathbb {S}^{d-1}\) we define

$$\begin{aligned} g' (x, \zeta , \nu )&: = \limsup _{\varepsilon \rightarrow 0^+} \liminf _{j \rightarrow + \infty } \frac{ \textbf{m}_{G_j}^{PC}(u_{x,\zeta ,\nu }, B_\varepsilon (x))}{\gamma _{d-1}\varepsilon ^{d-1}}, \end{aligned}$$
(5.41)
$$\begin{aligned} g'' (x, \zeta , \nu )&: = \limsup _{\varepsilon \rightarrow 0^+} \limsup _{j \rightarrow + \infty } \frac{ \textbf{m}_{G_j}^{PC}(u_{x,\zeta ,\nu }, B_\varepsilon (x))}{\gamma _{d-1}\varepsilon ^{d-1}}. \end{aligned}$$
(5.42)

Step 1. We start with the proof of the inequality

$$\begin{aligned} g_\infty (x,\zeta ,\nu ) \le g'(x,\zeta ,\nu ). \end{aligned}$$
(5.43)

For this, we fix a triple \((x,\zeta ,\nu )\in \mathbb {R}^d\times \mathbb {R}^m_0 \times {\mathbb {S}}^{d-1}\) and \(0<\eta <1\). By the definition of \(\textbf{m}^{PC}_{G_j}\) (see (5.38)), for every j there exists \(u_j \in L^0(\mathbb {R}^d,\mathbb {R}^m)\), with \(u_j|_{B_\varepsilon ( x)}\in SBV_{\textrm{pc}}(B_\varepsilon ( x), \mathbb {R}^m)\), such that \(u_j=u_{x,\zeta ,\nu }\) in a neighborhood of \(\partial B_\varepsilon ( x)\) and

$$\begin{aligned} G_j(u_j,B_\varepsilon ( x)) \le \textbf{m}^{PC}_{G_j}(u_{x,\zeta ,\nu },B_\varepsilon ( x))+ \eta \, \varepsilon ^{d-1}. \end{aligned}$$
(5.44)

Now, given \(\lambda > |\zeta |\), by virtue of Lemma 4.2, for every j there exists \(\hat{u}_j\) such that

$$\begin{aligned} \mathcal {F}_j(\hat{u}_j,B_\varepsilon ( x))\le (1+\eta ) \mathcal {F}_j(u_j,B_\varepsilon ( x))+\beta \mathcal {L}^d(B_\varepsilon ( x)\cap \{|u_j|\ge \lambda \}). \end{aligned}$$

Moreover, \(\hat{u}_j=u_{x,\zeta ,\nu }\) in a neighborhood of \(\partial B_\varepsilon ( x)\), \(|\hat{u}_j|\le \mu \) in \(\mathbb {R}^d\) and, from the chain rule, \(\nabla \hat{u}_j=0\) \(\mathcal {L}^d\)-a.e. in \(B_\varepsilon ( x)\). Consequently, the functions \(v_j\) defined for every \(j\in \mathbb {N}\) as

$$\begin{aligned} v_j:= {\left\{ \begin{array}{ll} \hat{u}_j &{} \text {in }\; B_\varepsilon ( x) \\ u_{x,\zeta ,\nu } &{} \text {in }\; \mathbb {R}^d{\setminus } B_\varepsilon ( x) \end{array}\right. } \end{aligned}$$
(5.45)

satisfy \(v_j|_A\in SBV_{\textrm{pc}}(A, \mathbb {R}^m)\) for every \(A\in \mathcal {A}(\Omega )\) and, from the definition, also the uniform bound

$$\begin{aligned} |v_j|\le \mu \quad \hbox {in }\mathbb {R}^d. \end{aligned}$$
(5.46)

Now, arguing as for the proof of [16, eq. (8.4)], with (g3), (g4) and (g5) (which holds with \(c=\frac{\beta }{\alpha }\)) we find that for every j

$$\begin{aligned} \mathcal {H}^{d-1}(J_{v_j}\cap B_\varepsilon (x)) \le M_d \varepsilon ^{d-1}, \end{aligned}$$
(5.47)

where \(M_d:= \frac{\beta }{\alpha ^2}(\beta \gamma _{d-1} + 1)\).

Since \(v_j \in SBV_{\textrm{pc}}(B_\varepsilon (x), \mathbb {R}^m)\) and (5.46)–(5.47) hold, we can apply the compactness result [6, Theorem 4.8] to deduce the existence of a function \(v\in SBV_{\textrm{pc}}(B_\varepsilon (x),\mathbb {R}^m)\cap L^\infty (B_\varepsilon (x),\mathbb {R}^m)\) and a subsequence (not relabelled) converging in measure to v on \(B_\varepsilon (x)\). We extend v to \(\mathbb {R}^d\) by setting \(v=u_{x,\zeta ,\nu }\) in \(\mathbb {R}^d{\setminus } B_\varepsilon (x)\) and observe that \(v|_A\in SBV_{\textrm{pc}}(A, \mathbb {R}^m)\) for every \(A\in \mathcal {A}(\Omega )\). Moreover, by the definitions of \(v_j\) and v and by (5.46), the convergence in measure on \(B_\varepsilon (x)\) implies that \(|v|\le \mu \ \;\mathcal {L}^d\text {-a.e.\ in }\; \mathbb {R}^d\).

In particular, for \(A=B_{(1+\eta )\varepsilon }(x)\) we have \(v|_{B_{(1+\eta )\varepsilon }(x)}\in SBV_{\textrm{pc}}(B_{(1+\eta )\varepsilon }(x),\mathbb {R}^m)\) and \(v=u_{x,\zeta ,\nu }\) in \(B_{(1+\eta )\varepsilon }(x){\setminus } B_\varepsilon (x)\), which combined with the \(\Gamma \)-convergence of \(\mathcal {F}_j(\cdot , B_{(1+\eta )\varepsilon }(x))\) to \(\mathcal {F}(\cdot ,B_{(1+\eta )\varepsilon }(x))\) with respect to the convergence in measure gives

$$\begin{aligned} m_{\mathcal {F}}(u_{x,\zeta ,\nu },B_{(1+\eta )\varepsilon }(x ))\le \mathcal {F}(v,B_{(1+\eta )\varepsilon }(x)) \le \liminf _{j\rightarrow +\infty } {} \mathcal {F}_j(v_j,B_{(1+\eta )\varepsilon }(x)). \end{aligned}$$
(5.48)

Taking into account the upper bounds in , (g3), and (5.44)–(5.45) we obtain

$$\begin{aligned} \mathcal {F}_j(v_j&,B_{(1+\eta )\varepsilon }(x )) \le \mathcal {F}_j(v_j,B_\varepsilon (x )) +\mathcal {F}_j(u_{x,\zeta ,\nu },B_{(1+\eta )\varepsilon }(x ){\setminus } {\overline{B}}_\varepsilon (x ))\\&\le (1+\eta ) \mathcal {F}_j(u_j,B_\varepsilon ( x)) + \beta \gamma _d(1+2^d)\varepsilon ^d + G_j(u_{x,\zeta ,\nu },B_{(1+\eta )\varepsilon }(x ){\setminus } {\overline{B}}_\varepsilon (x ))\\&\le (1+\eta )G_j(u_j,B_\varepsilon (x )) + \beta \gamma _d(3+2^d)\varepsilon ^d + \beta \gamma _{d-1}((1+\eta )^{d-1}-1)\varepsilon ^{d-1}\\&\le (1+\eta )\, \textbf{m}^{PC}_{G_j}(u_{x,\zeta ,\nu },B_\varepsilon (x)) + \beta \gamma _d(3+2^d)\varepsilon ^d +C_d\eta \varepsilon ^{d-1} \end{aligned}$$

where \(C_d:=2+\beta \gamma _{d-1}(2^{d-1}-1)\). This inequality, together with (5.48), gives

$$\begin{aligned} \textbf{m}_{\mathcal {F}}(u_{x,\zeta ,\nu },B_{(1+\eta )\varepsilon }(x))\le (1+\eta )\liminf _{k\rightarrow +\infty } \textbf{m}^{PC}_{G_j}(u_{x,\zeta ,\nu },B_\varepsilon (x)) + \beta \gamma _d(3+2^d)\varepsilon ^d +C_d\eta \varepsilon ^{d-1}. \end{aligned}$$

Now, dividing both the sides by \(\gamma _{d-1}\varepsilon ^{d-1}\), taking the limsup as \(\varepsilon \rightarrow 0^+\), and recalling (5.41) and (4.5), we obtain

$$\begin{aligned} (1+\eta )^{d-1} g_\infty (x,\zeta ,\nu ) \le (1+\eta )g'(x,\zeta ,\nu )+\frac{\eta }{\gamma _{d-1}} C_d, \end{aligned}$$

whence by taking the limit as \(\eta \rightarrow 0^+\) we get (5.43).

Step 2. We now prove

$$\begin{aligned} g''(x,[u](x),\nu _u(x))\le g_\infty (x,[u](x),\nu _u(x)) \end{aligned}$$
(5.49)

for \(\mathcal {H}^{d-1}\)-a.e. \(x\in J_u\cap A\).

We will prove (5.49) for functions u which belong to \(SBV^{p(\cdot )}(A,\mathbb {R}^m)\cap L^\infty (A,\mathbb {R}^m)\), while the general case of (unbounded) functions in \(GSBV^{p(\cdot )}(A,\mathbb {R}^m)\) can be obtained from the previous case by constructing a sequence of truncations of function u as in the Step 5 of [16, Proof of Theorem 5.2(d)].

Let \(A\in \mathcal {A}(\Omega )\), \(u\in SBV^{p(\cdot )}(A,\mathbb {R}^m)\cap L^\infty (A,\mathbb {R}^m)\). Let \(\eta \in (0,1)\). We fix \(x\in J_u\) such that, by setting \(\zeta :=[u](x)\) and \(\nu :=\nu _u(x)\), we have

$$\begin{aligned}&\displaystyle \zeta \ne 0, \end{aligned}$$
(5.50)
$$\begin{aligned}&\displaystyle \lim _{\varepsilon \rightarrow 0^+} \frac{1}{(\eta \varepsilon )^d}\int _{B_{\eta \varepsilon }(x)}|u(y)-u_{x,\zeta ,\nu }(y)|^{p(y)}\,\textrm{d}y = 0, \end{aligned}$$
(5.51)
$$\begin{aligned}&\displaystyle g_\infty (x,\zeta ,\nu ) = \lim _{\varepsilon \rightarrow 0^+} \frac{\mathcal {F}(u, B_{\eta \varepsilon }(x))}{\gamma _{d-1}(\eta \varepsilon )^{d-1}}. \end{aligned}$$
(5.52)

Note that (5.50) and (5.51) are satisfied for \({\mathcal {H}}^{d-1}\)-a.e. \(x\in J_u\) for \(p(\cdot )\equiv 1\) (see, e.g., [6, Definition 3.67 and Theorem 3.78]). This, combined with the boundedness of both u and \(u_{x,\zeta ,\nu }\), implies the (5.51) for any variable exponent such that \(p^-\ge 1\) and \(p^+<+\infty \). Also (5.52) holds for \({\mathcal {H}}^{d-1}\)-a.e. \(x\in J_u\), thanks to a generalized version of the Besicovitch Differentiation Theorem (see [45] and [32, Sections 1.2.1\(-\)1.2.2]).

We extend u to \(\mathbb {R}^d\) by setting \(u=0\) on \(\mathbb {R}^d{\setminus } A\). By the \(\Gamma \)-convergence of \(\mathcal {F}_j (\cdot ,A)\) to \(\mathcal {F} (\cdot ,A)\) there exists a sequence \((u_j)\) converging to u in \(L^0(\mathbb {R}^d,\mathbb {R}^m)\) such that

$$\begin{aligned} \lim _{k\rightarrow +\infty }\mathcal {F}_j(u_j,A)=\mathcal {F}(u,A). \end{aligned}$$

Since \(\mathcal {F}(u,\cdot )\) is a finite Radon measure, we have that \(\mathcal {F}(u,\partial B_{\eta \varepsilon }(x))=0\) for all \(\varepsilon >0\) such that \(B_{\eta \varepsilon }(x)\subset A\), except for a countable set. As a consequence \((u_j)\) is a recovery sequence for \(\mathcal {F}(u,\cdot )\) also in \(B_{\eta \varepsilon }(x)\); i.e.,

$$\begin{aligned} \lim _{k\rightarrow +\infty }\mathcal {F}_j(u_j,B_{\eta \varepsilon }(x))=\mathcal {F}(u,B_{\eta \varepsilon }(x)), \end{aligned}$$
(5.53)

for all \(\varepsilon >0\) except for a countable set.

Let \(\varepsilon \) be such that (5.53) holds. We now fix \(\lambda > \max \{ \Vert u\Vert _{L^\infty (\mathbb {R}^d,\mathbb {R}^m)}, |\zeta | \}\) and \(\mu \) as in Lemma 4.2. Then for every j there exists \(v_j\) such that

$$\begin{aligned} \mathcal {F}_j(v_j,B_{\eta \varepsilon }(x))\le (1+\eta ) \mathcal {F}_j(u_j,B_{\eta \varepsilon }(x))+\beta \mathcal {L}^d(B_{\eta \varepsilon }(x)\cap \{|u_j|\ge \lambda \}), \end{aligned}$$

and \( |v_j|\le \mu \) in \(\mathbb {R}^d\). We deduce that \(v_j\rightarrow u\) in \(L^{p(\cdot )}_{\textrm{loc}}(\mathbb {R}^d,\mathbb {R}^m)\) as well as

$$\begin{aligned} \limsup _{j\rightarrow +\infty } \mathcal {F}_j(v_j,B_{\eta \varepsilon }(x))\le (1+\eta ) \mathcal {F} (u,B_{\eta \varepsilon }(x)). \end{aligned}$$

Hence there exists \(j_0(\varepsilon )>0\) such that whenever \(j \ge j_0(\varepsilon )\)

$$\begin{aligned} \mathcal {F}_j(v_j, B_{\eta \varepsilon }(x)) \le (1+\eta ) \mathcal {F} (u,B_{\eta \varepsilon }(x))+ (\eta \varepsilon )^d. \end{aligned}$$
(5.54)

We now modify each \(v_j\) in order to obtain a function \(z_j\) which is an admissible competitor in the j-th minimization problem defining \(g''(x,\zeta ,\nu )\).

Step 2.1. We first define the blow-up function \(v_j^\varepsilon \) at x as

$$\begin{aligned} v_j^\varepsilon (y):= v_j(x+\varepsilon y)\quad \hbox {for }y \in B_\eta , \end{aligned}$$

and the blow-up variable exponent at x as

$$\begin{aligned} p_\varepsilon (y):= p(x+\varepsilon y)\quad \hbox {for }y \in B_\eta . \end{aligned}$$

Now, we modify \(v_j^\varepsilon \) so that it agrees with the boundary datum \(u_{0,\zeta ,\nu }\) in a neighbourhood of \(\partial B_\eta \). To this end, we apply the Fundamental Estimate (Lemma 3.2) to the functionals \(\mathcal {F}_{j,\varepsilon }:\big (SBV^{p_\varepsilon (\cdot )}(B_\eta , \mathbb {R}^m)\cap L^\infty (B_\eta , \mathbb {R}^m)\big ){\times } \mathcal {A}(B_\eta )\rightarrow [0,+\infty )\) defined as

$$\begin{aligned} \mathcal {F}_{j,\varepsilon }(v,A):= \int _A f_j (x+\varepsilon y,\nabla v(y) ) \textrm{d}y + \int _{J_v\cap A} g_j (x+\varepsilon y,[v](y),\nu _v(y) ) \textrm{d}\mathcal {H}^{d-1}(y),\nonumber \\ \end{aligned}$$
(5.55)

where \(\mathcal {A}(B_\eta )\) denotes the class of open subsets in \(B_\eta \).

Let \(K_\eta \subset B_\eta \) be a compact set such that

$$\begin{aligned} \beta \left( \mathcal {L}^{d}(B_\eta {\setminus } K_\eta ) + \mathcal {H}^{d-1}(\Pi ^\nu _0\cap (B_\eta {\setminus } K_\eta ))\right) < \eta ^d. \end{aligned}$$
(5.56)

Then, the argument of the proof of Lemma 3.2 allows us to deduce the existence of a constant \(M_\eta >0\) and a finite family of cut-off functions \(\varphi _1,\dots , \varphi _N\in C_c^\infty (B_\eta )\) such that \(0\le \varphi _i\le 1\) in \(B_\eta \), \(\varphi _i=1\) in a neighbourhood of \(K_\eta \), and

$$\begin{aligned} \mathcal {F}_{j,\varepsilon }(\hat{v}^\varepsilon _j,B_\eta ) \le {}&(1+\eta )\big (\mathcal {F}_{j,\varepsilon }(v^\varepsilon _j, B_\eta ) + \mathcal {F}_{j,\varepsilon }(u_{0,\zeta ,\nu },B_\eta {\setminus } K_\eta )\big )\nonumber \\&\quad + M_\eta \int _{B_\eta }|v^\varepsilon _j(y) - u_{0,\zeta ,\nu }(y)|^{p_\varepsilon (y)}\,\textrm{d}y + \gamma _d\eta ^{d+1}, \end{aligned}$$
(5.57)

where \(\hat{v}^\varepsilon _j:= \varphi _{i_j}v^\varepsilon _j + (1-\varphi _{i_j})u_{0,\zeta ,\nu }\) for a suitable \(i_ j\in \{1,\dots ,N\}\). It is clear from the definition that

$$\begin{aligned} |\hat{v}^\varepsilon _j| \le \mu \quad \hbox {in }\,B_\eta \end{aligned}$$
(5.58)

and \(\hat{v}^\varepsilon _j=u_{0,\zeta ,\nu }\) in a neighborhood of \(\partial B_\eta \). By the upper bounds in  and (g3), and by (5.56), we deduce that

$$\begin{aligned} \mathcal {F}_{j,\varepsilon }(u_{0,\zeta ,\nu },B_\eta {\setminus } K_\eta ) <\eta ^d. \end{aligned}$$

Since \(v_j \rightarrow u\) in \(L^{p(\cdot )}(B_{\eta \varepsilon }(x),\mathbb {R}^m)\), it follows that

$$\begin{aligned} v^\varepsilon _j (\cdot ) = v_j(x+\varepsilon \,\cdot ) \rightarrow u(x+\varepsilon \,\cdot ) \quad \text {in } \,\,L^{p_\varepsilon (\cdot )}(B_\eta ,\mathbb {R}^m) \ \text { as } \,\, j \rightarrow +\infty . \end{aligned}$$
(5.59)

Therefore, from (5.57) and (5.59) we have

$$\begin{aligned} \limsup _{j\rightarrow +\infty }\mathcal {F}_{j,\varepsilon }(\hat{v}^\varepsilon _j,B_\eta ) \le {}&(1+\eta )\Big (\limsup _{j\rightarrow +\infty }\mathcal {F}_{j,\varepsilon }(v^\varepsilon _j, B_\eta ) + \eta ^d\Big )\nonumber \\&+ M_\eta \int _{B_\eta }|u(x+\varepsilon \,y) - u_{0,\zeta ,\nu }(y)|^{p_\varepsilon (y)}\,\textrm{d}y + \gamma _d\eta ^{d+1}. \end{aligned}$$
(5.60)

Step 2.2. We now show that \(\nabla \hat{v}_j^\varepsilon \) is small in \(L^{p^-_\varepsilon }\)-norm for j large and \(\varepsilon \) small. By the definition of \(\hat{v}^\varepsilon _j\) we have

$$\begin{aligned} \Vert \nabla \hat{v}_j^\varepsilon \Vert _{L^{p^-_\varepsilon }(B_\eta , \mathbb {R}^{m\times d})}&\le C_\eta \Vert v_j^\varepsilon - u_{0,\zeta ,\nu }\Vert _{L^{p^-_\varepsilon }(B_\eta , \mathbb {R}^m)} + \Vert \nabla v_j^\varepsilon \Vert _{L^{p^-_\varepsilon }(B_\eta , \mathbb {R}^{m\times d})}, \end{aligned}$$
(5.61)

where the constant \(C_\eta >0\) is an upper bound for \(\Vert \nabla \varphi _{i_j}\Vert _{L^\infty (B_\eta , \mathbb {R}^m)}\).

We now estimate separately the two terms in the right-hand side of (5.61). Concerning the first term, by (5.59) we can find \(j_1(\varepsilon ) \ge j_0(\varepsilon )\) such that, for \(j \ge j_1(\varepsilon )\) and from (5.51), we have

$$\begin{aligned} \Vert v^\varepsilon _j&- u_{0,\zeta ,\nu }\Vert _{L^{p^-_\varepsilon }(B_\eta , \mathbb {R}^m)} \nonumber \\&\le \Vert v^\varepsilon _j(\cdot ) - u(x+\varepsilon \,\cdot )\Vert _{L^{p^-_\varepsilon }(B_\eta ,\mathbb {R}^m)} + \Vert u(x+\varepsilon \,\cdot ) - u_{0,\zeta ,\nu }(\cdot )\Vert _{L^{p^-_\varepsilon }(B_\eta , \mathbb {R}^m)} \le \omega _I (\varepsilon ), \end{aligned}$$
(5.62)

where \(\omega _I(\varepsilon )\) is independent of j and \(\omega _I(\varepsilon ) \rightarrow 0\) as \(\varepsilon \rightarrow 0^+\).

As for the second term in (5.61), by the definition of \(v_j^\varepsilon \), the lower bound in , and the positivity of \(g_j\), for \(\varepsilon \) small enough we have that

$$\begin{aligned} \int _{B_\eta }|\nabla v_j^\varepsilon |^{p_\varepsilon (y)} \textrm{d}y&\le \varepsilon ^{p^-_\varepsilon -d} \int _{B_{\eta \varepsilon }(x)}|\nabla v_j |^{p(y)} \,\textrm{d}y \nonumber \\&\le \frac{\varepsilon ^{p^-_\varepsilon -d}}{\alpha } \int _{B_{\eta \varepsilon }(x)} f_j (y,\nabla v_j )\,\textrm{d}y \\&\le \frac{\varepsilon ^{p^-_\varepsilon -1}}{\alpha }\cdot \frac{\mathcal {F}_j(v_j, B_{\eta \varepsilon }(x))}{\varepsilon ^{d-1}}. \nonumber \end{aligned}$$
(5.63)

Now, by (5.52) there exists \(\varepsilon _0>0\) such that for every \(0<\varepsilon <\varepsilon _0\) satisfying (5.53) we can find \(j_2(\varepsilon )\ge j_1(\varepsilon )\) such that, taking into account also (5.63), we have

$$\begin{aligned} \int _{B_\eta }|\nabla v_j^\varepsilon |^{p_\varepsilon (y)} \textrm{d}y \le \frac{\gamma _{d-1}\eta ^{d-1}\varepsilon ^{p^-_\varepsilon -1}}{\alpha }(g_\infty (x,\zeta ,\nu )+1) \end{aligned}$$
(5.64)

for every \(j \ge j_{2}(\varepsilon )\). Finally, collecting (5.61), (5.62), and (5.64) we conclude that

$$\begin{aligned} \Vert \nabla \hat{v}_j^\varepsilon \Vert _{L^{p^-_\varepsilon }(B_\eta , \mathbb {R}^{m\times d})} \le \omega _{II}(\varepsilon ) \end{aligned}$$
(5.65)

for every \(0<\varepsilon <\varepsilon _0\) satisfying (5.53) and every \(j \ge j_{ 2 }(\varepsilon )\), where \(\omega _{II}(\varepsilon )\) is independent of j and \(\omega _{II}(\varepsilon ) \rightarrow 0\) as \(\varepsilon \rightarrow 0^+\).

Step 2.3. As a next step, we need to modify \(\hat{v}_{j}^\varepsilon \) to make it piecewise constant.

Let \(\zeta _1,\dots ,\zeta _d\) be the coordinates of \(\zeta \). By (5.50) for every \(0<\varepsilon <\varepsilon _0\) satisfying (5.53) there exists an integer \(N_{\varepsilon }>0\), with \(\frac{1}{N_\varepsilon }< \mu \) and \(\frac{1}{N_\varepsilon }< |\zeta _i|\) for every i with \(\zeta _i\ne 0\), such that,

$$\begin{aligned} N_{\varepsilon } \rightarrow +\infty \quad \text {and}\quad \omega _{II}(\varepsilon ) \, N_\varepsilon \rightarrow 0^+ \quad \text {as } \, \varepsilon \rightarrow {0^+}. \end{aligned}$$
(5.66)

Note that, by (5.58), we have \(|{\hat{v}}_j^\varepsilon |<2\mu - \frac{1}{N_\varepsilon }\) in \(B_\eta \).

Since by (5.65) the functions \(\hat{v}_j^\varepsilon \) are equibounded in \(L^1(B_\eta ;\mathbb {R}^m)\) for every fixed \(\varepsilon \), by virtue of Theorem 2.9 applied with \(\theta :=N_\varepsilon \Vert \nabla \hat{v}_j^\varepsilon \Vert _{L^{1}(B_\eta , \mathbb {R}^{m\times d})}\) we can find a partition \((P_l^{\varepsilon ,j})_{l=1}^\infty \) of \(B_\eta \) made of sets of finite perimeter and a piecewise constant function \(w_j^\varepsilon :=\sum _{l=1}^\infty b_l \chi _{P_l^{\varepsilon ,j}}\) such that the following properties hold: for every \(0<\varepsilon <\varepsilon _0\) satisfying (5.53) and for every \(j \ge j_{ 2 }(\varepsilon )\)

$$\begin{aligned}&w_j^\varepsilon = u_{0,\zeta ,\nu } \text { in a neighborhood of }\partial B_\eta , \end{aligned}$$
(5.67)
$$\begin{aligned}&\Vert w_j^\varepsilon - {\hat{v}}_j^\varepsilon \Vert _{L^\infty (B_\eta ,\mathbb {R}^m)} \le \frac{1}{N_\varepsilon } < \mu , \end{aligned}$$
(5.68)
$$\begin{aligned}&\Vert w_j^\varepsilon \Vert _{L^\infty (B_\eta ,\mathbb {R}^m)} \le 2\mu , \end{aligned}$$
(5.69)
$$\begin{aligned}&\mathcal {H}^{d-1}((J_{w_j^\varepsilon }{\setminus } J_{{\hat{v}}_j^\varepsilon })\cap B_\eta ) \le \omega _{III} (\varepsilon ), \end{aligned}$$
(5.70)

where \(\omega _{III}(\varepsilon ):=c(d,p)\omega _{II}(\varepsilon )N_\varepsilon \) is independent of j and \(\omega _{III}(\varepsilon )\rightarrow 0^+\) as \(\varepsilon \rightarrow 0^+\). Note that (5.68) and (5.70) follow from Theorem 2.9(ii) and (i), respectively.

Step 2.4. Recalling the definition of \(\mathcal {F}_{j,\varepsilon }({\hat{v}}_j^\varepsilon ,B_\eta )\) (see (5.55)) and taking into account (5.60), we have

$$\begin{aligned} \limsup _{j\rightarrow +\infty }&\int _{J_{{\hat{v}}_j^\varepsilon }\cap B_\eta } g_j (x+\varepsilon y,[{\hat{v}}_j^\varepsilon ] (y),\nu _{{\hat{v}}_j^\varepsilon } (y) ) \,\textrm{d}\mathcal {H}^{d-1} (y)\nonumber \\&\le (1+\eta )\Big (\limsup _{j\rightarrow +\infty }\mathcal {F}_{j,\varepsilon }(v^\varepsilon _j, B_\eta ) + \eta ^d\Big ) \nonumber \\&\quad + M_\eta \int _{B_\eta }|u(x+\varepsilon \,y) - u_{0,\zeta ,\nu }(y)|^{p_\varepsilon (y)}\,\textrm{d}y + \gamma _d\eta ^{d+1}. \end{aligned}$$
(5.71)

Moreover, with the upper bound in  and (5.64), the volume integral in the right hand side of (5.71) can be estimated as

$$\begin{aligned}{} & {} \int _{B_\eta }f_j (x+\varepsilon y, \nabla v_j^\varepsilon (y) ) \,\textrm{d}y \le \beta \int _{B_\eta } (1+ |\nabla v_j^\varepsilon |^{p_\varepsilon (\cdot )})\,\textrm{d}y\\{} & {} \quad \le \beta \Big (\gamma _d \eta ^d + \frac{\gamma _{d-1}\eta ^{d-1}\varepsilon ^{p^-_\varepsilon -1}}{\alpha }(g_\infty (x,\zeta ,\nu )+1)\Big ) \end{aligned}$$

for every \(0<\varepsilon <\varepsilon _0\) satisfying (5.53) and every \(j \ge j_{ 2 }(\varepsilon )\).

By (5.55) again, this inequality and (5.71) yield in particular that

$$\begin{aligned} \limsup _{j\rightarrow +\infty }&\int _{J_{{\hat{v}}_j^\varepsilon }\cap B_\eta } g_j (x+\varepsilon y,[{\hat{v}}_j^\varepsilon ] (y),\nu _{{\hat{v}}_j^\varepsilon } (y))\,\textrm{d}\mathcal {H}^{d-1} (y)\nonumber \\&\le (1+\eta ) \limsup _{j \rightarrow + \infty }\int _{J_{ v_j^\varepsilon }\cap B_\eta } g_j (x+\varepsilon y,[v_j^\varepsilon ] (y),\nu _{v_j^\varepsilon } (y))\,\textrm{d}\mathcal {H}^{d-1} (y) \nonumber \\&\quad + 2 \beta \Big (\gamma _d \eta ^d+ \frac{\gamma _{d-1}\eta ^{d-1}\varepsilon ^{p^-_\varepsilon -1}}{\alpha }(g_\infty (x,\zeta ,\nu )+1)\Big ) \\&\quad + M_\eta \int _{B_\eta }|u(x+\varepsilon \,y) - u_{0,\zeta ,\nu }(y)|^{p_\varepsilon (y)}\,\textrm{d}y + c(d)\eta ^d, \nonumber \end{aligned}$$
(5.72)

where \(c(d):= 2+\gamma _d\).

Now, rewriting in terms of \(v_j\) the surface integral in the right hand side and combining with (5.54) and (5.72) we obtain

$$\begin{aligned}&\limsup _{j \rightarrow +\infty } \int _{J_{{\hat{v}}_j^\varepsilon }\cap B_\eta } g_j (x+\varepsilon y,[{\hat{v}}_j^\varepsilon ] (y),\nu _{{\hat{v}}_j^\varepsilon } (y)) d\mathcal {H}^{d-1} (y) \nonumber \\&\quad \le (1+\eta )^2 \frac{1}{\varepsilon ^{d-1}}\,\mathcal {F}(u, B_{\eta \varepsilon }(x)) + 2 \eta ^{d}\varepsilon + 2\beta \Big (\gamma _d \eta ^d+ \frac{\gamma _{d-1}\eta ^{d-1}\varepsilon ^{p^-_\varepsilon -1}}{\alpha }(g_\infty (x,\zeta ,\nu )+1)\Big ) \\&\qquad + M_\eta \int _{B_\eta }|u(x+\varepsilon \,y) - u_{0,\zeta ,\nu }(y)|^{p_\varepsilon (y)}\,\textrm{d}y + c(d)\eta ^d. \nonumber \end{aligned}$$
(5.73)

We now estimate the left-hand side in (5.73). Exploiting the assumptions (g3), (g4), (g6), and the properties of \({\hat{v}}_j^\varepsilon \) and \(w_j^\varepsilon \) we claim that

$$\begin{aligned}&\int _{J_{w_j^\varepsilon }\cap B_\eta }g_j (x+\varepsilon y,[w_j^\varepsilon ](y),\nu _{w_j^\varepsilon }(y))\,\textrm{d}\mathcal {H}^{d-1}(y)\nonumber \\&\quad \le \int _{J_{{\hat{v}}_j^\varepsilon }\cap B_\eta } g_j (x+\varepsilon y,[{\hat{v}}_j^\varepsilon ](y),\nu _{{\hat{v}}_j^\varepsilon }(y) )\,\textrm{d}\mathcal {H}^{d-1}(y)+\omega _{IV}(\varepsilon )+\omega _{V}(\varepsilon ), \end{aligned}$$
(5.74)

where \(\omega _{IV}(\varepsilon )\) and \(\omega _{V}(\varepsilon )\) are independent of j and tend to \(0^+\) as \(\varepsilon \rightarrow 0^+\). There, the key estimate is

$$\begin{aligned}&|g_j (x+\varepsilon y,[{\hat{v}}_j^\varepsilon ](y),\nu _{{\hat{v}}_j^\varepsilon }(y)) - g_j (x+\varepsilon y,[w_j^\varepsilon ](y),\nu _{w_j^\varepsilon }(y))| \\&\quad \le \omega _2(|[{\hat{v}}_j^\varepsilon ] (y) -[w_j^\varepsilon ] (y)|) \big (g_j (x+\varepsilon y,[{\hat{v}}_j^\varepsilon ] (y),\nu _{{\hat{v}}_j^\varepsilon } (y)) + g_j (x+\varepsilon y,[w_j^\varepsilon ] (y),\nu _{w_j^\varepsilon } (y))\big ) \\&\quad \le 4 \beta ^2 \omega _2(2\Vert {\hat{v}}_j^\varepsilon - w_j^\varepsilon \Vert _{L^{\infty }(B_\eta \!,\mathbb {R}^m)}), \end{aligned}$$

for \({\mathcal {H}}^{n-1}\)-a.e. \(y \in J_{{\hat{v}}_j^\varepsilon }\cap J_{w_j^\varepsilon }\). The claim follows then from (5.68), (5.70) and the bounds on \(g_j\).

Now, (5.74) together with (5.73) gives

$$\begin{aligned} \limsup _{j\rightarrow +\infty }&\int _{J_{w_j^\varepsilon }\cap B_\eta } g_j (x+\varepsilon y,[w_j^\varepsilon ] (y),\nu _{w_j^\varepsilon } (y))\,\textrm{d}\mathcal {H}^{d-1} (y)\nonumber \\&\le (1+\eta )^2 \frac{1}{\varepsilon ^{d-1}}\,\mathcal {F}(u, B_{\eta \varepsilon }(x)) + 2 \eta ^d \varepsilon + \omega _{IV}(\varepsilon )+\omega _{V}(\varepsilon ) \\&\quad + 2\beta \Big (\gamma _d \eta ^d+ \frac{\gamma _{d-1}\eta ^{d-1}\varepsilon ^{p^-_\varepsilon -1}}{\alpha }(g_\infty (x,\zeta ,\nu )+1)\Big ) \nonumber \\&\quad + M_\eta \int _{B_\eta }|u(x+\varepsilon \,y) - u_{0,\zeta ,\nu }(y)|^{p_\varepsilon (y)}\,\textrm{d}y + c(d)\eta ^d. \nonumber \end{aligned}$$
(5.75)

Defining \(z_j^\varepsilon (y):= w_j^\varepsilon ((y-x)/\varepsilon )\) for every \(y \in B_{\eta \varepsilon } (x)\), we clearly have that \(z_j^\varepsilon \in SBV_{\textrm{pc}} (B_{\eta \varepsilon } (x),\mathbb {R}^m)\) and \(z_j^\varepsilon = u_{x, \zeta , \nu }\) in a neighborhood of \(\partial B_{\eta \varepsilon } (x)\). Then, rewriting (5.75) in terms of the functions \(z_j^\varepsilon \) we find

$$\begin{aligned} \limsup _{j \rightarrow +\infty } \frac{1}{(\eta \varepsilon )^{d-1}}&\textbf{m}^{PC}_{G_j}(u_{x,\zeta ,\nu }, B_{\eta \varepsilon }(x)) \le \limsup _{j \rightarrow +\infty } \frac{1}{(\eta \varepsilon )^{d-1}} \textbf{m}^{PC}_{G_j}(u_{x,\zeta ,\nu }, B_{\eta \varepsilon }(x)) \\&\le \limsup _{j\rightarrow +\infty } \frac{1}{(\eta \varepsilon )^{d-1}}\int _{J_{z_j^\varepsilon }\cap B_{\eta \varepsilon }(x)} g_j (y,[z_j^\varepsilon ] (y),\nu _{z_j^\varepsilon } (y))\,\textrm{d}\mathcal {H}^{d-1} (y)\\&\le (1+\eta )^2 \frac{1}{(\eta \varepsilon )^{d-1}}\,\mathcal {F}(u, B_{\eta \varepsilon }(x)) + 2 \eta \varepsilon + \frac{\omega _{IV}(\varepsilon )}{\eta ^{d-1}} + \frac{\omega _{V}(\varepsilon )}{\eta ^{d-1}} \\&\quad +2\beta \Big (\gamma _d \eta + \frac{\gamma _{d-1}\varepsilon ^{p^-_\varepsilon -1}}{\alpha }(g_\infty (x,\zeta ,\nu )+1)\Big ) \\&\quad + \frac{M_\eta }{\eta ^{d-1}} \int _{B_1}|u(x+\varepsilon \,y) - u_{0,\zeta ,\nu }(y)|^{p_\varepsilon (y)}\,\textrm{d}y + c(d)\eta . \end{aligned}$$

Finally, dividing by \(\gamma _{d-1}\), taking the limsup as \(\varepsilon \rightarrow 0^+\) and using (5.42), (5.51), and (5.52), we obtain

$$\begin{aligned} g'' (x,\zeta ,\nu ) \le (1+\eta )^2 g_\infty (x,\zeta ,\nu ) + C \eta , \end{aligned}$$

with \(C:= (2\beta \gamma _d + c(d))/\gamma _{d-1}\). Recalling the definition of \(\zeta \) and \(\nu \), we obtain that

$$\begin{aligned} g'' (x,[u](x),\nu _{u}(x)) \le (1+\eta )^2 g_\infty (x,[u](x),\nu _u(x)) + C \eta \end{aligned}$$

holds true for \({\mathcal {H}}^{n-1}\)-a.e. \(x\in J_u \cap A\). Taking the limit as \(\eta \rightarrow 0^+\) we get

$$\begin{aligned} g'' (x,[u](x),\nu _{u}(x)) \le g_\infty (x,[u](x),\nu _u(x)) \end{aligned}$$

for \({\mathcal {H}}^{n-1}\)-a.e. \(x\in J_u \cap A\), thus proving (5.49) for \(u\in SBV^{p(\cdot )}(A,\mathbb {R}^m)\cap L^{\infty }(A,\mathbb {R}^m)\).

Finally, since by definition \(g'\le g''\), combining (5.43) and (5.49) we get (5.39)–(5.40). This concludes the proof. \(\square \)