Integral representation and Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence for free-discontinuity problems with p(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot )$$\end{document}-growth

An integral representation result for free-discontinuity energies defined on the space GSBVp(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$GSBV^{p(\cdot )}$$\end{document} of generalized special functions of bounded variation with variable exponent is proved, under the assumption of log-Hölder continuity for the variable exponent p(x). Our analysis is based on a variable exponent version of the global method for relaxation devised in Bouchitté et al. (Arch Ration Mech Anal 165:187–242, 2002) for a constant exponent. We prove Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence of sequences of energies of the same type, we identify the limit integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions.


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 f (x, u(x), ∇u(x)) dx (1.1) (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 T V model and the isotropic diffusion away from the edges (see also [40] for a related model, [44] for simulations, and [41] for a -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-Shahtype functionals. 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 S BV p(·) of S BV functions with p(·)-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 -convergence (with respect to the convergence in measure) for functionals F j : G S BV p(·) ( ; R m ) → [0, +∞) of the form The variable exponent p(·) is assumed to be log-Hölder continuous, with p(x) ≥ 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 0 < α ≤ g j (x, ζ, ν) ≤ β.
Under a fairly general set of assumptions, devised in [16], we are able to show that the -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) ≥ 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 ∞ is completely determined by taking the -limit of the functionals (1.1) in the Sobolev space W 1, p(·) , while the surface limit density g ∞ can be recovered from the sole g j 's via an asymptotic cell formula on piecewise constant functions, that is G S BV 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 (here B( ) denote the Borel subsets of ) 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 < α < β such that for any u ∈ G S BV p(·) ( ; R m ) and B ∈ B( ) we have 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(·) 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 S BV 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 This can be done by first establishing the integral representation in the S BV p(·) case for functionals which satisfy 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 S BV p (·) . Then, by letting the perturbation parameter vanish and by truncating functions suitably, the representation can be extended to G S BV p (·) . 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 G S BV p(·) , 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 G S BV p (·) . 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 -convergence result for sequences of free-discontinuity functionals defined on G S BV p (·) . The identification of the -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.

Basic notation and preliminaries
We start with some basic notation. Let ⊂ R d be open, bounded with Lipschitz boundary. Let A( ) be the family of open subsets of , and denote by B( ) the family of Borel sets contained in . For every x ∈ R d and ε > 0 we indicate by B ε (x) ⊂ R d the open ball with center x and radius ε. If x = 0, we will often use the shorthand B ε . For x, y ∈ R d , we use the notation x · y for the scalar product and |x| for the Euclidean norm. Moreover, we let S d−1 := {x ∈ R d : |x| = 1}, we denote by R m×d the set of m × d matrices and by R d 0 the set R d \{0}. The m-dimensional Lebesgue measure of the unit ball in R m is indicated by γ m for every m ∈ N. We denote by L d and H k the d-dimensional Lebesgue measure and the k-dimensional Hausdorff measure, respectively. For A ⊂ R d , ε > 0, and x 0 ∈ R d we set (2.1) The closure of A is denoted by A. The diameter of A is indicated by diam(A). Given two sets A 1 , A 2 ⊂ R d , we denote their symmetric difference by A 1 A 2 . We write χ A for the characteristic function of any A ⊂ R d , which is 1 on A and 0 otherwise. If A is a set of finite perimeter, we denote its essential boundary by ∂ * A, see [6,Definition 3.60]. The notation L 0 (E; R m ) will be used for the space of Lebesgue measurable function from some measurable set E ⊂ R n to R m , endowed with the convergence in measure.

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 : → [1, +∞) will be called a variable exponent. Correspondingly, for every A ⊂ we define while p + and p − will be denoted by p + and p − , respectively. For a measurable function u : → R m we define the modular as The variable exponent Lebesgue space L p(·) ( ) is defined as the set of measurable functions u such that p(·) (u/λ) < +∞ for some λ > 0. In the case p + < +∞, L p(·) ( ) coincides with the set of functions such that p(·) (u) is finite. It can be checked that · L p(·) ( ) is a norm on L p(·) ( ). Moreover, if p + < +∞, it holds that if u L p(·) ( ) > 1, while an analogous inequality holds by exchanging the role of p − and p + if 0 ≤ u L p(·) ( ) ≤ 1. Another useful property of the modular, in the case p + < +∞, is the following one: for all λ > 0. We say that a function p : → R is log-Hölder continuous on if We recall the following geometric meaning of the p log-Hölder continuity (see, e.g., [ The following lemma provides an extension to the variable exponent setting of the wellknown embedding property of classical Lebesgue spaces (see, e.g., [29,Corollary 3.3.4]).

Lemma 2.2 Let p, q be measurable variable exponents on , and assume that
The embedding constant is less or equal to the minimum between 2(1 + L d ( )) and The following result generalizes the concept of Lebesgue points to the variable exponent Lebesgue spaces (see, e.g., [39,Theorem 3.1]).

The space GSBV p(·) : Poincaré-type inequality
We denote by S BV p(·) ( ; R m ) the set of functions u ∈ S BV ( ; R m ) with ∇u ∈ L p(·) ( ; R m×d ) and H d−1 (J u ) < +∞. Here, ∇u denotes the approximate gradient, while J u stands for the (approximate) jump set with corresponding normal ν u and one-sided limits u + and u − . We say that u ∈ G S BV p(·) ( ; R m ) if for every φ ∈ C 1 (R m ) with the support of ∇φ compact, the composition φ • u belongs to S BV p(·) loc ( ; R m ). From the inclusion L p(·) ( ) ⊂ L p − ( ) and [4], one can also deduce that for u ∈ G S BV p(·) ( ) the approximate gradient ∇u exists L d -a.e. in .
In order to state a Poincaré-Wirtinger inequality in G S BV p(·) , we first fix some notation, following [11,17]. With given a = (a 1 , . . . , a m ) and the truncation operator where γ iso is the dimensional constant in the relative isoperimetric inequality. We recall the following Poincaré-Wirtinger inequality for S BV 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 ≥ d is discussed in [6,Remark 4.15]. and

Remark 2.6
More generally, Theorem 2.5 holds for functions in G S BV ( ; R m ) and for balls B ⊂⊂ , by applying the scalar result in S BV to truncated functions u M i := M ∧ u i ∨ −M for every i = 1, . . . , m, up to understand ∇u and J u in a weaker sense.
The analogous result in G S BV p(·) is as follows.
Theorem 2.7 Let p : → (1, +∞) be measurable and such that Let B ⊂⊂ and u ∈ G S BV p(·) (B; R m ), and assume that (2.6) holds. Then for some constant c depending on p − , d, and 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.
A first consequence of Theorem 2.7 is the following compactness result, which can be seen as the G S BV p(·) 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 ⊂
be a ball, ( p j ) j∈N be a sequence of variable exponents p j : B → (1, +∞) complying uniformly with (2.9) and converging uniformly to somep : Then there exist a function u 0 ∈ W 1,p(·) (B; R m ) and a subsequence (not relabeled) of (2.13) Proof For every j ∈ N, we set Correspondingly, we define We set for brevityū j := T B u j −med(u j ; B). Let η > 0 be fixed such that p − η := p − −η > 1 and p + η := p + + η < (p − η ) * . Note that, for j large enough, we have By virtue of (2.9), (2.10), the definition of T B u j and (2.12) we have This implies, by [4, Theorem 2.2] that there exists u 0 ∈ G S BV p − η (B; R m ) and a subsequence (not relabeled) u j such thatū j → u 0 in measure and (2.14) With (2.7), since p + η < (p − η ) * , we get that |ū j | p + η is equiintegrable, henceū j strongly converges to u 0 in L p + η (B; R m ). With Lemma 2.2, and the definition ofū j we then get the first assertion in (2.13). With (2.14), we have by the uniform convergence of p j . With the weak-L 1 convergence of ∇ū j to ∇u 0 and Ioffe's Theorem (see [42]), we get B |∇u 0 |p (y)−η dy ≤ C with a bound independent of η. Applying the Monotone Convergence Theorem in the set {|∇u 0 | ≥ 1} we get u 0 ∈ W 1,p(·) (B; R m ). The second assertion in (2.13) follows from (2.11) and (2.12).
To conclude this section, we recall the following result on the approximation of G S BV 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]).
be a Borel set with finite perimeter. Let θ > 0 be fixed. Then there exists a partition (P l ) ∞ l=1 of D, made of sets of finite perimeter, and a piecewise constant function z pc := ∞ l=1 b l χ P l such that

The integral representation result
In this section we will establish an integral representation result in the space G S BV p(·) ( ; R m ) for m ∈ N, where the variable exponent p : → (1, +∞) complies with the following assumptions (P 1 ) p − > 1 and p + < +∞; (P 2 ) p is log-Hölder continuous on (see (2.4)).
We consider functionals F : G S BV p(·) ( ; R m ) × B( ) → [0, +∞) with the following general assumptions: is lower semicontinuous with respect to convergence in measure on for any For every u ∈ G S BV p(·) ( ; R m ) and A ∈ A( ) we define Moreover, for x 0 ∈ , u 0 ∈ R m , and ξ ∈ R m×d we introduce the affine functions and, for a, b ∈ R m , ν ∈ S d−1 we define u x 0 ,a,b,ν : R d → R m by The main result of this section is the following integral representation theorem. → (1, +∞) be a variable exponent complying with (P 1 )-(P 2 ), and suppose that F : for all u ∈ G S BV p(·) ( ; R m ) and B ∈ B( ), where f is given by for all x 0 ∈ , u 0 ∈ R m , ξ ∈ R m×d , and g is given by for all x 0 ∈ , a, b ∈ R m , and ν ∈ S d−1 .

Fundamental estimate
In this section we prove an important tool in the proof of the integral representation, namely a fundamental estimate in G S BV p(·) for functionals F .

7)
and the remainder term is where we used the notation introduced in (2.1).

The global method
This section is entirely devoted to the proof of Theorem 3.1. As a first step, we show that F and m F , defined by (3.1), have the same Radon-Nikodym derivative with respect to the measure 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 ε → 0, the minimization problems m F (u, B ε (x 0 )) and m F (ū bulk x 0 , B ε (x 0 )) coincide for L d -a.e. x 0 ∈ , where we writeū bulk x 0 := x 0 ,u(x 0 ),∇u(x 0 ) for brevity, see (3.2).

Lemma 3.4 Let p :
→ (1, +∞) be a Riemann-integrable variable exponent satisfying (P 1 ). Suppose that F satisfies (H 1 ) and (H 3 )-(H 4 ) and let u ∈ G S BV p(·) ( ; R m ). Then for L d -a.e. x 0 ∈ we have The final step is that, asymptotically as ε → 0, the minimization problems m F (u, B ε (x 0 )) and m F (ū surf 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.
In the remaining part of the section we prove Lemma 3.3. For this, we need to fix some notation. For δ > 0 and A ∈ A( ), we define where μ is defined in (3.10). As m δ F (u, A) is decreasing in δ, we can also introduce In the following lemma, we prove that F and m * F coincide under our assumptions. Lemma 3.6 Let p : → (1, +∞) be complying with (P 1 ). Suppose that F satisfies (H 1 )- Proof We can follow the argument of [11,Lemma 4] (see also [12,Lemma 3.3]). We start by proving the inequality m * for all δ > 0, whence the assertion follows taking into account (3.15).
We now address the reverse inequality. We fix A ∈ A( ) and by (3.16)-(3.17) and (H 4 ). Moreover, v δ,n → v δ pointwise a.e. in , and then in measure on . Then, [4, Theorem 2.2] combined with the compactness in L 0 of (v δ,n ) yields v δ ∈ G S BV p − ( ; R m ) and ∇v δ,n ∇v δ weakly in L p − ( ; R m×d ). Now, by Ioffe's Theorem (see [42]) and (3.19) we get where we also used the fact that μ(N δ . For later purpose, we also note by (H 4 ) that this implies We now claim that With this, using (H 2 ), (3.15), and (3.20) we will get the required inequality m * F (u, A) ≥ F (u, A) in the limit as δ → 0. To prove (3.22), we first note that w δ from the classical Poincaré inequality we get where |Dw δ,M |(A) is bounded in view of (3.21) and the fact that This implies w δ,M → 0 in L 1 (A; R m ), and then in measure on A, as δ → 0 for every M > 0. Now, with fixed M = 1 and ε ∈ (0, 1) we have 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.
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.

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 L d -a.e. point in 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 L d -a.e. x ∈ 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.
If, in addition, u ∈ S BV p(·) ( ; R m ), then u ε also satisfies Proof It will suffice to treat the scalar case m = 1. Let x 0 ∈ be such that Properties (a) and (c) hold for L d -a.e. x 0 ∈ by Theorem 2.3 since |∇u| ∈ L p(·) ( ; R m×d ) and by Lemma 2.4, respectively, while (b) follows from the fact that J u is countably H d−1rectifiable (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 with a fixed neighborhood of Note that We notice that (3.24)(b) implies also (2.6) for ε small enough, which combined with (2.11) gives as ε → 0. Therefore, for every sequence ε → 0 one can find a subsequence (not relabeled) such that, for L 1 -a.e. σ ∈ (0, 1), Now, we fix a sequence ε → 0 and consider a subsequence (not relabeled) and σ ∈ (0, 1) for which (3.27) holds. We then define From the definition of u ε and the argument of (3.26) we get the assertions in (3.23)(i). We now prove (3.23)(ii). We set u ε (y) :=ū ε (x 0 + εy).
We have as ε → 0, where we have set Then, by virtue of Theorem 2.8 there exist a function The assertion (3.28) will then follow once we prove that For this, notice that (3.24)(c) implies u 0 (y) = ∇u(x 0 ) · y for L d -a.e. y ∈ B σ . The a.e. convergence in measure of u ε − med( u ε ; B σ ) to ∇u(x 0 ) · y is now enough to reproduce the proof of [11, eq. (21)], and obtain (3.29). We therefore omit the details.
is satisfied. Then as a consequence of Fubini's Theorem, we can fix σ ∈ (0, 1) such that Now, we can define the sequence u ε as above and prove (i), (ii) and (iii). Assertion (i) will follow from (iii), (3.31), (c) and Hölder's inequality, since We are now in a position to prove Lemma 3.4, which will follow as a consequence of Lemmas 3.8 and 3.9.
Proof We will prove the assertion for those points x 0 ∈ for which the statement of Lemma 3.7 holds and lim ε→0 ( Let (u ε ) ε be the sequence of Lemma 3.7 and we fix σ ∈ (0, 1) such that (3.23)(ii) holds.
. Now, we apply Lemma 3.2 with u and v replaced by z ε and u ε , respectively, and (3.35) where, to enlighten the notation, we denote by (3.37) From this and (3.36) we infer that there exists a non-negative sequence ( ε ) ε , vanishing as ε → 0, such that (3.38) We set for brevity Then, by using that z ε =ū bulk On the other hand, by (H 4 ) we also obtain Recall that w ε = u in a neighborhood of ∂ B ε (x 0 ). This along with (3.38), (3.40), (3.43) and (3.32) follows up to passing to η, θ → 0. The proof is concluded.

Lemma 3.9 Under the assumptions of Lemma 3.8, for
Proof We can restrict the proof to those points x 0 ∈ considered in Lemma 3.8. Let η > 0, σ = 1 − θ fixed as in Lemma 3.8, and let (u ε ) ε 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 ε > 0 we can find s ∈ (1 − 4θ, 1 − 3θ) such that From now on, the argument of the proof closely follows that of Lemma 3.8. We choose a sequence . Now, we apply Lemma 3.2 with u and v replaced by z ε andū bulk x 0 , respectively, and the same choice for the sets D ε,x 0 , D ε,x 0 and E ε,x 0 as in Lemma 3.8, see (3.35).
By virtue of Lemma 3.2, we then find 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 ( ε ) ε , vanishing as ε → 0, such that We now proceed to the estimate of the terms in (3.47).
where |∇u(x 0 )| p is defined as in (3.39). The analogous of the estimate for F (ū bulk Finally, letting η and θ to 0, and recalling that w ε =ū bulk and this concludes the proof of (3.44).

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(·). This requires a careful analysis of the asymptotic behavior of some constants, where the assumption of log-Hölder continuity of the variable exponent p(·), see (P 2 ), plays a crucial role. We state and prove the announced blow-up properties for u ∈ G S BV p(·) around each jump point x 0 ∈ J u . Lemma 3.10 Assume that p : → (1, +∞) be continuous and complying with (P 2 ). Let u ∈ G S BV p(·) ( ; R m ). Then for H d−1 -a.e. x 0 ∈ J u , for L 1 -a.e. σ ∈ (0, 1) and for every If, in addition, u ∈ S BV p(·) ( ; R m ), we also have

53)
where 0 is the hyperplane passing through x 0 with normal ν u (x 0 ).
Proof We first note that since |∇u| p(·) ∈ L 1 ( ), the points x 0 ∈ J u can be fixed such that  (N ) = 0, and each K j is a subset of a C 1 hypersurface. Then, in a neighborhood B ε 0 (y) ⊂ of each point y ∈ K j , up to a rotation, we may find a C 1 function j : We now define the function w ∈ G S BV p(·) (B ε 0 (y); R m ) by setting Notice indeed that by construction we have |∇w| ≤ C|∇u| a.e., hence w ∈ G S BV p(·) (B ε 0 (y); R m ). Furthemore, J w ∩ B ε 0 (y) ⊂ B ε 0 (y)\K j . Now, following the argument of [11, Lemma 3], we can fix x 0 ∈ B ε 0 (y) ∩ K j with the following properties: and for fixed η > 0 (small enough) there exists (a smaller, if necessary) ε 0 > 0 such that holds, for all ε < ε 0 , for q = p − . Moreover, if we set for every ε > 0 combining with (3.54) we have that (3.56) is indeed satisfied for q = p − ε . Now, fix q such that (3.56) holds. Define T ε w(x) := T B ε (x 0 ) w(x) as in (2.5) with u = w and B = B ε (x 0 ). From the Poincaré inequality (2.7), (3.56) and for any q ≤ r < q * we Since arguing as for the proof of [11, eq.
(34)] we can prove that for ε small enough, collecting the previous estimates we finally obtain If we define the function z as 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 ε z defined as the truncation T B ε (x 0 ) z of the function z. We then set and we have Arguing exactly as in [11,Lemma 3], we also have (3.59) Now, an analogous argument as for (3.26) shows that for every sequence ε → 0 one can find a subsequence (not relabeled) such that, for L 1 -a.e. σ ∈ (0, 1), We then defineū 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 η > 0, the estimate (3.58) with q = p − ε and r = p + ε implies that for ε small enough. Observe that, by its definition, the constant C(d, p − ε , p + ε ) is a bounded function of ε. Now, since and with p + ε ≤ p + , assertion (iii) in (3.50) will follow sending ε → 0 first and then η → 0, once we note that lim sup for some constant c 1 by virtue of (P 2 ). Assertion (3.52) for a function u ∈ S BV ( ; 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 − , and the property We omit further details. Concerning (3.53), we begin by observing that, if u ∈ S BV ( ; If we now set with (3.55), (3.56), (3.61), and since truncations are 1-Lipschitz, we get and the same properties also hold for z. With this, one has, for all E ⊂ B 1 (x 0 ), since the last property is satisfied at H d−1 -a.e. x 0 ∈ J u by the definition of measure-theoretic normal to a rectifiable set. This is clearly equivalent to (3.53).
We now prove Lemma 3.5. The two inequalities in (3.12) will be shown with Lemmas 3.11 and 3.12 below.
We will prove the compactness of -convergence via the localization technique forconvergence (see [25, for the general method), where the main ingredient is the fundamental estimate in G S BV p(·) , proven with Lemma 3.2. The representation (4.2) in terms of the densities f ∞ and g ∞ then will follow by the integral representation result of Theorem 3.1. Indeed, since each F j is invariant under translations of u, then also F ∞ , as -limit, satisfies the same property. Thus, from Theorem 3.1, in particular (3.4)-(3.5), we and the densities f ∞ , g ∞ can be computed as for all x 0 ∈ , ξ ∈ R m×d , ζ ∈ R m and ν ∈ S d−1 . For our purposes, it will be useful to consider functionals I : We recall a result concerning the existence of suitable truncations of a measurable function u by which functionals F as above almost decrease (see [16,Lemma 4.1]). For our purposes, the statement below is formulated in the p(·)-setting, and since the adaptation of the original proof requires only minor changes, we omit the details. (4.1), where we assume that f satisfies (f1)-(f2) and g satisfies (g1),(g2), (g4) and (g5). Let I be as in (4.6). Let η, λ > 0. Then there exists μ > λ depending on η, λ, α, β, c such that the following holds: for every open set A ⊂ and for every u

Lemma 4.2 Let F be as in
Moreover, there existsv with the same properties ofû such that (iii) holds for the functional I withv in place ofû.
Let (F j ) j be a sequence of functionals of the form (4.1). We start by proving some properties of the -liminf and -limsup with respect to the topology of the convergence in measure. To this end, we define for all u ∈ G S BV p(·) ( ; R m ) and A ∈ A( ).

Lemma 4.3 (Properties of -liminf and -limsup) Let ⊂ R d be an open set, and
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 ∈ N. Define F ∞ and F ∞ as in (4.7), and write, for brevity, Then we have where α, β have been introduced in (f2) and (g3).

Proof
The monotonicity property (i) follows from the fact that Now, since v j → u in measure on A, by arguing as in the proof of Lemma 3.6 and exploiting the lower semicontinuity inequalities we easily obtain the lower bound. In order to prove (iii) and (iv), we preliminary show that for every U , V and W open subsets of , with V ⊂⊂ W ⊂⊂ U , we have 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 G S BV p(·) ( ; R m ) converging in measure to u on W and U \V , respectively, such that We may assume, up to passing to a not relabeled subsequence, that each liminf above is a limit. We fix η ∈ (0, 1) and λ > 0 such that (4.11) By virtue of Lemma 4.2, there exists μ > λ such that, for every k ≥ 1 we can find We apply Lemma 3.2 with η above, D : (4.13) Note that, by the dominated convergence in measure,û j k −v j k → 0 in L p(·) (W \D ; R m ) as k → +∞. Moreover, recalling (3.6)(ii), we have thatŵ j k → u in measure on U as k → +∞. By a diagonal argument this implies, in particular, that (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 for k large enough. Then, combining (4.12) with (4.13), (4.10) and passing to the limit as k → +∞, and then letting η → 0 + , assertion (4.9) follows.
We now prove the inner regularity of F ∞ , the first property in (iii). Combining (4.8)(ii) and (4.9) we find Now, we can choose V ⊂⊂ U and U in such a way that L d (U \V ) and G(u, U \V ) be arbitrarily small, and recalling that F ∞ (u, ·) is an increasing set function by (4.8)(i), we obtain (4.8)(iii) for F ∞ . The proof of the analogous property for F ∞ is similar.
We conclude by showing property (iv) for F ∞ . First, we note that it is not restrictive to assume that A ∩ B = ∅, otherwise the inequalities in (iv) are straightforward. It is well known (see, e.g., [5, Proof of Lemma 5.2]) that given η > 0, one can choose in open sets Then using, (4.8)(i),(ii) and (4.9) we get Since η was arbitrary, the statement follows.
We can now prove Theorem 4.1.

Proof of Theorem 4.1
First, we prove the existence of the -limit by applying an abstract compactness result for¯ -convergence, see [25,Theorem 16.9]. This implies the existence of an increasing sequence of integers ( j k ) k such that F ∞ and F ∞ defined in (4.7) with respect to ( j k ) k satisfy for all u ∈ G S BV p(·) ( ; R m ) and A ∈ A( ), where (F ∞ ) − and (F ∞ ) − denote the inner regular envelopes of F ∞ and F ∞ , respectively. By (4.8)(iii) we know that F ∞ and F ∞ are inner regular, and thus they both coincide with their respective inner regular envelopes. This shows that the -limit, denoted by F ∞ := F ∞ = F ∞ , exists for all u ∈ G S BV p(·) ( ; R m ) and all A ∈ A( ). We now check that F ∞ 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 F j show that F ∞ (·, A) is local according to (H 3 ) for any A ∈ A( ). Moreover, F ∞ (·, A) complies with (H 2 ) for any A ∈ A( ) in view of [25,Remark 16.3]. Now, since F ∞ 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 F ∞ (u, ·) 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 F ∞ admits a representation of the form (4.2).

Identification of the 0-limit
In this section we identify the structure of the -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 ∞ is only determined by ( f j ) j and g ∞ is only determined by (g j ) j .

Identification of the bulk density
We start with the identification of the bulk density. To do this, we restrict functionals F as in (4.1) to Sobolev functions W 1, p(·) ( ; R m ). Indeed, since every Sobolev function has a H d−1 -negligible jump set we have We set, for every ξ ∈ R m×d ,¯ ξ := 0,0,ξ , where x 0 ,u 0 ,ξ is defined as in (3.2). In analogy to (3.1), for every u ∈ W 1, p(·) ( ; R m ) and A ∈ A( ) we define We consider the functionals F j : where f j satisfies (f1),(f2) and (f3) for every j ∈ N. We then have the followingconvergence result. Moreover, f sob is a Carathéodory function satisfying (f2) and it holds that We can now proceed with the announced identification of the bulk density.
Proof We show the two inequalities in (5.7). We first prove First, in view of (3.1) and (5. while by (5.5) and (5.1) we find Thus, since both f ∞ and f sob are continuous with respect to ξ by (f3), combining (5.9)-(5.10) we obtain (5.8).
We now prove the reverse inequality First, from the Radon-Nikodým Theorem we have that holds for L d -a.e. x ∈ . Let (u j ) be a sequence of measurable functions such that u j ∈ G S BV p(·) ( ; R m ) u j → u in measure on and lim j→+∞ F j (u j , ) = F (u, ).
Since u ∈ G S BV p(·) ( ; R d ), by virtue of Lemma 2.4 the approximate gradient ∇u(x) exists for L d -a.e. x ∈ . Then, since (5.11) needs to hold for L d -a.e. x ∈ , we may assume that (5.12) holds at x and that ∇u(x) exists. Since F (u, ·) is a Radon measure, there exists a subsequence (ε k ) ⊂ (0, +∞) with ε k 0 as k → +∞ such that F (u, ∂ B ε k (x)) = 0 for every k ∈ N and such that (5.6) holds along (ε k ), namely (5.13) Moreover, with fixed η ∈ (0, 1), since (u j ) is a recovery sequence and F (u, ·) is a Radon measure, for every k ∈ N we can find j k ∈ N (depending also on η) such that, for every j ≥ j k it holds that Now, we have to modify the sequence (u j ) to construct a competitor for the minimization problem m B ε (x)) which defines f sob . We introduce the functions Then, since u j → u in measure on , we have that u ε k j → u ε k in measure on B 1 as j → +∞. In addition, by a diagonal argument and up to passing to a larger j k ∈ N, we also havê By virtue of (5.13), we may choose ( j k ) k such that also holds. Finally, taking into account (4.1), (5.12), (5.14) and with a change of variables we find lim sup (5.17) Let I k be defined as in (4.6), with f j k (x + ε k y, ·) in place of f (x, ·), and set We define, accordingly, Let λ > |∇u(x)|. Then, by virtue of Lemma 4.2 there exists μ > λ such that, for every k, we can find a functionv k ∈ S BV p k (·) ( Moreover, with (5.15) and the fact that |¯ ∇u(x) | ≤ |∇u(x)| < λ in B 1 , we get v k →¯ ∇u(x) in measure on B 1 as k → +∞ (5.19) and L d (B 1 ∩ {|û k | ≥ λ}) ≤ ε k for k large enough. Taking into account (f2), (g3), (5.18) and a change of variables we get for k large enough. Then, with (5.12) and (5.14), we can find a constant M > 0 independent of k and η such that for k large enough, and Now, we regularize the sequence (v k ) in order to obtain a sequenceŵ k ∈ W 1, p k (·) (B 1 ; R m ) such that 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 R t k := y ∈ B 1 : ≤ t for every r > 0 with B r (y) ⊂ B 1 , We claim that 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 2|∇v k (y)| t ≥ 1 on S t k . Now, taking into account (5.20), we get Choosing we have t k ≥ 1 for k large enough and, taking into account (5.21), from (5.23) we obtain ≤ C. (5.27) Finally, with (5.26) and (5.27), by a simple inequality we obtain (5.28) Then, by applying [33, Lemma 1.2] to (ẑ k ), we find a sequence of Lipschitz functions (ŵ k ) which satisfyŵ k ∈ W 1,p (B 1 ; R m ), |∇ŵ k |p equi-integrable uniformly with respect to k, and L d ({ẑ k =ŵ k }) → 0 as k → +∞. Since |ẑ k | ≤ μ in B 1 , we may assume also that |ŵ k | ≤ μ L d -a.e. in B 1 . An inspection to the proof of [33,Lemma 1.2] shows that (ŵ k ) can be chosen in such a way that holds. We claim that (|∇ŵ k | p k (·) ) is equi-integrable on B 1 uniformly with respect to k. Indeed, arguing as for (5.27) we first get, for every y ∈ B 1 , Then, for every fixed E ⊆ B 1 , arguing as for (5.28) and taking into account (5.30) we obtain This and the equi-integrability of |∇ŵ k |p imply the claim. Moreover, from (5.25), and since by (5.19) the equibounded sequence (ŵ k −¯ ∇u(x) ) tends to 0 in measure on B 1 , we have as k → +∞.
In order to prove (5.22), we notice that Now, taking into account the equi-integrability of (|∇ŵ k | p k (·) ), the upper bound (f2) and (5.32), for ε k small enough we get whence (5.22) follows. Finally, we have to modify the sequence (ŵ k ) in such a way that it attains the boundary datum¯ ∇u(x) in a neighborhood of ∂ B 1 . We know that the functionals I k (u, A) above for u ∈ W 1, p k (·) (A; R m ) and A ∈ A( ) satisfy uniformly the Fundamental Estimate proved in Lemma 3.2. Namely, corresponding to the fixed η above, there exist a constant C η > 0 and a sequence (ŵ k ) in W 1, p k (·) (B 1 ; R m ) withŵ k =¯ ∇u(x) in a neighborhood of ∂ B 1 for all k ∈ N such that Now, taking into account (f2), (5.32)

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 F j when restricted to the space S BV pc (A, R m ) of those functions u ∈ S BV (A, R m ) such that ∇u = 0 L d -a.e. in A and H d−1 (J u ) < +∞.
In order to do that, we consider the sequence of surface energies and, correspondingly, we define the sequence of minimum problems where u x,ζ,ν coincides with u x,ζ,0,ν defined in (3.3). Since, to the best of our knowledge, a -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 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 ⊂ R d be open and p : → (1, +∞) 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 ∞ be defined by (4.5). Then, for all u ∈ G S BV p(·) ( , R m ) we have that Step 1. We start with the proof of the inequality Now, given λ > |ζ |, by virtue of Lemma 4.2, for every j there existsû j such that Moreover,û j = u x,ζ,ν in a neighborhood of ∂ B ε (x), |û j | ≤ μ in R d and, from the chain rule, ∇û j = 0 L d -a.e. in B ε (x). Consequently, the functions v j defined for every j ∈ N as satisfy v j | A ∈ S BV pc (A, R m ) for every A ∈ A( ) and, from the definition, also the uniform bound Now, arguing as for the proof of [16, eq. (8.4)], with (g3), (g4) and (g5) (which holds with c = β α ) we find that for every j where M d := β α 2 (βγ d−1 + 1). Since v j ∈ S BV pc (B ε (x), 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 ∈ S BV pc (B ε (x), R m ) ∩ L ∞ (B ε (x), R m ) and a subsequence (not relabelled) converging in measure to v on B ε (x).
We extend v to R d by setting v = u x,ζ,ν in R d \B ε (x) and observe that v| A ∈ S BV pc (A, R m ) for every A ∈ A( ). Moreover, by the definitions of v j and v and by (5.46), the convergence in measure on B ε (x) implies that |v| ≤ μ L d -a.e. in R d .
Step 2. We now prove We will prove (5.49) for functions u which belong to S BV p(·) (A, R m ) ∩ L ∞ (A, R m ), while the general case of (unbounded) functions in G S BV p(·) (A, 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 We fix x ∈ J u such that, by setting ζ := [u](x) and ν := ν u (x), we have We extend u to R d by setting u = 0 on R d \A. By the -convergence of F j (·, A) to F (·, A) there exists a sequence (u j ) converging to u in L 0 (R d , R m ) such that Since F (u, ·) is a finite Radon measure, we have that F (u, ∂ B ηε (x)) = 0 for all ε > 0 such that B ηε (x) ⊂ A, except for a countable set. As a consequence (u j ) is a recovery sequence for F (u, ·) also in B ηε (x); i.e., for all ε > 0 except for a countable set.
Let ε be such that (5.53) holds. We now fix λ > max{ u L ∞ (R d ,R m ) , |ζ |} and μ as in Lemma 4.2. Then for every j there exists v j such that Hence there exists j 0 (ε) > 0 such that whenever j ≥ j 0 (ε) 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, ζ, ν).
Step 2.1. We first define the blow-up function v ε j at x as v ε j (y) := v j (x + εy) for y ∈ B η , and the blow-up variable exponent at x as p ε (y) := p(x + εy) for y ∈ B η . Now, we modify v ε j so that it agrees with the boundary datum u 0,ζ,ν in a neighbourhood of ∂ B η . To this end, we apply the Fundamental Estimate (Lemma 3.2) to the functionals F j,ε : S BV p ε (·) (B η  Let K η ⊂ B η be a compact set such that Then, the argument of the proof of Lemma 3.2 allows us to deduce the existence of a constant M η > 0 and a finite family of cut-off functions ϕ 1 , . . . , ϕ N ∈ C ∞ c (B η ) such that 0 ≤ ϕ i ≤ 1 in B η , ϕ i = 1 in a neighbourhood of K η , and andv ε j = u 0,ζ,ν in a neighborhood of ∂ B η . By the upper bounds in (f2) and (g3), and by (5.56), we deduce that Since Therefore, from (5.57) and (5.59) we have Step 2.2. We now show that ∇v ε j is small in L p − ε -norm for j large and ε small. By the definition ofv ε j we have where the constant C η > 0 is an upper bound for ∇ϕ i j L ∞ (B η ,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 (ε) ≥ j 0 (ε) such that, for j ≥ j 1 (ε) and from (5.51), where ω I (ε) is independent of j and ω I (ε) → 0 as ε → 0 + . As for the second term in (5.61), by the definition of v ε j , the lower bound in (f2), and the positivity of g j , for ε small enough we have that Now, by (5.52) there exists ε 0 > 0 such that for every 0 < ε < ε 0 satisfying (5.53) we can find j 2 (ε) ≥ j 1 (ε) such that, taking into account also (5.63), we have ≤ ω I I (ε) (5.65) for every 0 < ε < ε 0 satisfying (5.53) and every j ≥ j 2 (ε), where ω I I (ε) is independent of j and ω I I (ε) → 0 as ε → 0 + .
for all x 0 ∈ , u 0 ∈ R m , ξ ∈ R m×d and x 0 ,u 0 ,ξ as in (3.2), g is given by for all x 0 ∈ , a, b ∈ R m , ν ∈ S d−1 and u x 0 ,a,b,ν as in (3.3), and m F is defined in (A.3).
The proof of Theorem A.1 can be obtained by adapting the argument of Theorem 3.1, which concerns with G S BV p(·) functions. For this, Lemmas 3.3, 3.4 and 3.5 are replaced by the corresponding S BV p(·) versions, Lemma A.2, A.3 and A.4 below, respectively. We will briefly list the main changes in the proofs due to the different assumption (H 4 ).

Lemma A.2 Let p :
→ (1, +∞) be a variable exponent satisfying (P 1 )-(P 2 ). Suppose that F satisfies (H 1 )-(H 3 ) and (H 4 ). Let u ∈ S BV p(·) ( ; R m ) and μ be defined as Then for μ-a.e. x 0 ∈ we have Proof The only needed modification concerns the proof of Lemma 3.6. Indeed, under assumption (H 4 ), for u ∈ S BV p(·) ( ; R m ) the same construction provides a sequence v δ,n in Then, an analogous compactness argument, based on [4, Theorem 2.1] yields v δ ∈ S BV p − ( ; R m ), which can be improved to v δ ∈ S BV p(·) ( ; R m ) by using Ioffe's theorem and the weak convegence of the gradients, exactly as in Lemma 3.6. Finally, assumption (H 4 ) does not change (3.21).
Note that the Fundamental estimate (3.6), proven with Lemma 3.2, still holds if we replace (H 4 ) by (H 4 ).

Lemma A.3 Let p :
→ (1, +∞) be a Riemann-integrable variable exponent satisfying (P 1 ). Suppose that F satisfies (H 1 ) and (H 3 ), (H 4 ) and let u ∈ S BV p(·) ( ; R m ). Then for L d -a.e. x 0 ∈ we have Proof The proof of "≤" inequality in (A.8) can be obtained with the same construction of Lemma 3.8 applied to the sequence (u ε ) complying with Lemma 3.7(i)-(iii) and (i) , (iii) . Applying the Fundamental estimate with the same choice of sets as in (3.35) and by assumption (H 4 ), we get F (u ε , C ε,θ (x 0 )) ≤ β The reverse inequality in (A.8) can be proved following the argument of Lemma 3.9. For this, we first notice that since u ε satisfies Lemma 3.7(i) , in addition to (3.45) we may require that where u − ε and u + denote the inner and outer traces at ∂ B sε (x 0 ) of u ε and u, respectively. Then, estimates (3.43) and (3.48) (with the additional term β ∂ B sε (x 0 ) |u + − u − ε | dH d−1 in the left hand side) can be established. Finally, combining (3.43), (3.45), (A.9) and the fact that sε ≤ (1 − 3θ)ε, we obtain also (3.49). This will suffice to conclude the argument of Lemma 3.9 and then the proof of the inequality "≥" in (A.8).
With this, we can easily infer the upper inequality in (A.10).
As for the reverse inequality, given (ū ε ) as above, by (3.52) we may require, in addition to (3.72), also the property where u + andū − ε have the same meaning as in Lemma A.3. Then we repeat the argument of Lemma 3.12, where (3.75) is now replaced by Now, since v j → u in measure on A, we may appeal to the closure property of S BV (see, e.g., [6,Theorem 4.7]). Then, by arguing as in the proof of Lemma 3.6 and exploiting the lower semicontinuity inequalities Then, the functionals E j (·, A) defined in (A.1) -converge in L 0 (R d ; R m ) to the functional E 0 (·, A) given by for every A ∈ A( ) and u ∈ G S BV p(·) (A; R m ).
Proof It follows from (A.4) and (A.5) that f σ 1 ∞ ≤ f σ 2 ∞ and g σ 1 ∞ ≤ g σ 2 ∞ for 0 < σ 1 < σ 2 . Then, by the Monotone Convergence Theorem we have where we use the topology of L 0 (R d , R m ). We subdivide the rest of the proof into steps.
Step 1: First, for every A ∈ A( ), u ∈ L 0 (R d , R m ) with u| A ∈ S BV p(·) (A, R m ) and for every σ ∈ we have E (u, A) ≤ E σ (u, A), whence by (A.22) we immediately get Step 2: We claim that for every A ∈ A( ) and every u ∈ L ∞ (R d , R m ).
With fixed A and u as above, by -convergence there exists a sequence (u j ) converging to u in L 0 (R d , R m ) such that E (u, A) = lim inf k→+∞ E j (u j , A). (A.25) Let us fix λ > u L ∞ (R d , R m ) and σ > 0. By Lemma 4.2 there exist μ > λ, independent of j, and a sequence (v j ) ⊂ L ∞ (R d , R m ), converging to u in measure on bounded sets, such that for every j we have If E j (u j , A) < +∞, by the lower bounds in (f2), (g3 ), and (A.28) the function v j belongs to G S BV p(·) (A, R m ) and By (A.12) and (A.26) this implies that , which, in its turn, by (A.28) and (A.29), leads to This inequality trivially holds also when E j (u j , A) = +∞. Therefore, using (A.25) and the inequality u L ∞ (R d , R m ) < λ, by -convergence we get for every σ ∈ . By (A.22), passing to the limit as σ → 0 + we obtain (A.24) whenever u ∈ L ∞ (R d , R m ).
Step 3: We now prove that E (u, A) ≤ E 0 (u, A) for every u ∈ L 0 (R d , R m ) and every A ∈ A( ). (A.30) Let us fix u and A. It is enough to prove the inequality when u| A ∈ G S BV p(·) (A, R m ). By Lemma 4.2 for every σ > 0 and for every integer j ≥ 1 there exists u j ∈ L ∞ (R d , R m ), with u j | A ∈ S BV p(·) (A, R m ), such that u j = u L d -a.e. in {|u| ≤ j} and .
Since u j → u in measure on bounded sets, passing to the limit as j → +∞, by the lower semicontinuity of the -limsup we deduce Thus, letting σ → 0 + we obtain (A.30).
Step 4: We now prove that Letting j → +∞ we get The -convergence of E j (·, A) to E 0 (·, A) in L 0 (R d , R m ) follows from (A.30) and (A.31). This concludes the proof.