Abstract
An integral representation result for free-discontinuity energies defined on the space \(GSBV^{p(\cdot )}\) 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 \(\Gamma \)-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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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
where \(\Omega \) is a reference configuration, and a “surface" energy of the form
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
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
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
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
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
(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
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
This can be done by first establishing the integral representation in the \(SBV^{p(\cdot )}\) 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 \(SBV^{p(\cdot )}\). 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
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
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
and the (Luxembourg) norm
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
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:
for all \(\lambda >0\).
We say that a function \(p:\Omega \rightarrow \mathbb {R}\) is log-Hölder continuous on \(\Omega \) if
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:
-
(i)
p is log-Hölder continuous;
-
(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 p, q 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
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
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
where
for \(i=1,\dots ,m\).
For every \(u\in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) such that
we define
and the truncation operator
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
If \(1\le p < d\) then
and
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
Let \(B\subset \subset \Omega \) and \(u\in GSBV^{p(\cdot )}(B;\mathbb {R}^m)\), 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. \(\square \)
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
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
Proof
For every \(j\in \mathbb {N}\), we set
Correspondingly, we define
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
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 \(u_0\in GSBV^{{p^-_\eta }}(B;\mathbb {R}^m)\) and a subsequence (not relabeled) \({u}_j\) such that \(\bar{u}_j \rightarrow u_0\) in measure and
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
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
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
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
-
(i)
\(\displaystyle \sum _{l=1}^\infty \mathcal {H}^{d-1}((\partial ^*P_l\cap D^1)\backslash J_z)\le \theta \);
-
(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
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
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
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
for all \(u \in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(B \in \mathcal {B}(\Omega )\), where f is given by
for all \(x_0 \in \Omega \), \(u_0 \in \mathbb {R}^m\), \(\xi \in \mathbb {R}^{m \times d}\), and g is given by
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
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
and the remainder term is
where we used the notation introduced in (2.1).
Proof
We choose \(k \in \mathbb {N}\) such that
and for \(i=1,\ldots ,k\), we set
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 (\({H}_{3}\))
For the last term, we compute using (\({H}_{4}\))
Consequently, recalling (3.8) and using (\({H}_{1}\)) we find \(i_0 \in \lbrace 1, \ldots , k \rbrace \) such that
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
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
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
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
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
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
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
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
where \(\mu \) is defined in (3.10). As \(\textbf{m}^\delta _{\mathcal {F}}(u,A) \) is decreasing in \(\delta \), we can also introduce
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
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
We define
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
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
whence \(v^\delta \in GSBV^{p{(\cdot )}}(\Omega ;\mathbb {R}^m)\). We have
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
We now claim that
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 \(M>0\)
from the classical Poincaré inequality we get
whence
where \(|Dw^{\delta ,M}|(A)\) is bounded in view of (3.21) and the fact that \(u\in GSBV(A;\mathbb {R}^m)\), since
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
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
If, in addition, \(u \in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\), then \(u_\varepsilon \) also satisfies
Proof
It will suffice to treat the scalar case \(m=1\). Let \(x_0\in \Omega \) be such that
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
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
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. \(\sigma \in (0,1)\),
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
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,
We have
so, recalling that \(u_\varepsilon =v_\varepsilon \) in \(B_{\sigma \varepsilon }(x_0)\), (3.23)(ii) can be rephrased as
as \(\varepsilon \rightarrow 0\), where we have set
From (3.24)(a)-(b) we infer
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
The assertion (3.28) will then follow once we prove that
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
and also property
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
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
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. \(x_0\in \Omega \)
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
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
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
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
From this and (3.36) we infer that there exists a non-negative sequence \((\varrho _\varepsilon )_\varepsilon \), vanishing as \(\varepsilon \rightarrow 0\), such that
We set for brevity
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
On the other hand, by (\(H_4\)) we also obtain
Now, taking into account (3.23)(iii) we get
Since \(|\nabla u_\varepsilon |\le |\nabla u|\) \(\mathcal {L}^d\)-a.e., we have, with (3.24)(a),
Combining (3.41) with (3.42) we finally get
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
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
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
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
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
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
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
Now, with (3.43) and (3.45) and the fact that \(s\varepsilon \le (1-3\theta )\varepsilon \) we get
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
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
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
and
If, in addition, \(u\in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\), we also have
and
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
(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
We now define the function \(w\in GSBV^{p(\cdot )}(B_{\varepsilon _0}(y);\mathbb {R}^m)\) by setting
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:
and for fixed \(\eta >0\) (small enough) there exists (a smaller, if necessary) \(\varepsilon _0>0\) such that
holds, for all \(\varepsilon <\varepsilon _0\), for \(q=p^-_\Omega \). Moreover, if we set for every \(\varepsilon >0\)
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
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
for \(\varepsilon \) 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 \(T_\varepsilon z\) defined as the truncation \(T_{B_\varepsilon (x_0)}z\) of the function z. We then set
and we have
Arguing exactly as in [11, Lemma 3], we also have
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. \(\sigma \in (0,1)\),
We then define
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
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
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
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
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
If we now set
with (3.55), (3.56), (3.61), and since truncations are 1-Lipschitz, we get
Now, as shown in [11, Remark 2, Formula (39)], one has componentwise
and the same properties also hold for z. With this, one has, for all \(E\subset B_1(x_0)\),
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
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
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
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
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
Plugging in (3.65) we find that, for a suitable non-negative vanishing sequence \(\varrho _\varepsilon \), it holds that
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
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
With this, using (\(H_4\)) and (3.50)(ii) we infer
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
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
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
as \(\varepsilon \rightarrow 0\). Then for each \(\varepsilon >0\) we can find \(\sigma \in (1-4\theta ,1-3\theta )\) such that
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
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
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
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
Now, with (3.70), (3.72) and \(\sigma \le (1-3\theta )\) we then obtain
and, as already proven in (3.68),
Collecting the estimates (3.74), (3.76), (3.77) and using \(\varrho _\varepsilon \rightarrow 0\) we infer
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
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
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:
-
(f1)
(measurability) f is Borel measurable on \(\mathbb {R}^d{\times } \mathbb {R}^{m{\times }d}\);
-
(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:
-
(g1)
(measurability) g is Borel measurable on \(\mathbb {R}^d{\times }\mathbb {R}^m_0{\times } {\mathbb {S}}^{d-1}\);
-
(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 \(c|\zeta _1|\le |\zeta _2|\);
-
(g3)
(lower and upper bound) for every \(x\in \mathbb {R}^d\), \(\zeta \in \mathbb {R}^m_0\), and \(\nu \in {\mathbb {S}}^{d-1}\)
$$\begin{aligned} \alpha \le g(x,\zeta ,\nu ) \le \beta ; \end{aligned}$$ -
(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
-
(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
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
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
and the densities \(f_\infty , g_\infty \) can be computed as
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
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
-
(i)
\(|\hat{u}|\le \mu \) on \(\mathbb {R}^d\);
-
(ii)
\(\hat{u}=u\) \(\mathcal {L}^d\)-a.e. in \(\{|u|\le \lambda \}\);
-
(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
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,
Then we have
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
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
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 \(\Omega \), with \(V\subset \subset W \subset \subset U\), 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 \(GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) converging in measure to u on W and \(U\backslash \overline{V}\), respectively, such that
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
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
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
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
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 \(k\rightarrow +\infty \), 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
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
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
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 \(\xi \in \mathbb {R}^{m{\times }d}\),
-
(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
-
(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
We set, for every \(\xi \in \mathbb {R}^{m\times d}\),
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
We consider the functionals \({F}_j: L^1(\Omega ;\mathbb {R}^m)\times \mathcal {A}(\Omega )\rightarrow [0,+\infty ]\) defined as
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
and
Moreover, \(f_\textrm{sob}\) is a Carathéodory function satisfying f2 and it holds that
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
Proof
We show the two inequalities in (5.7). We first prove
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
while by (5.5) and (5.1) we find
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
First, from the Radon-Nikodým Theorem we have that
holds for \(\mathcal {L}^d\)-a.e. \(x\in \Omega \). Let \((u_j)\) be a sequence of measurable functions such that \(u_j\in GSBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\)
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
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
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
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
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
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
We define, accordingly,
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
Moreover, with (5.15) and the fact that \(|\bar{\ell }_{\nabla u(x)}|\le |\nabla u(x)|<\lambda \) in \(B_1\), we get
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
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
for k large enough, and
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
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
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 \(\frac{2|\nabla \hat{v}_k(y)|}{t}\ge 1\) on \(S_k^t\). Now, taking into account (5.20), we get
Choosing
we have \(t_k\ge 1\) for k large enough and, taking into account (5.21), from (5.23) we obtain
By virtue of Lemma 2.1, and since \(p^-_k-1\ge p^--1>0\), it holds now
for k large. We then conclude that
whence, in particular, since \(\varepsilon _k<1\) for k large enough, we get
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
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
Finally, with (5.26) and (5.27), by a simple inequality we obtain
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
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 \(y\in B_1\),
Then, for every fixed \(E\subseteq B_1\), arguing as for (5.28) and taking into account (5.30) we obtain
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
as \(k\rightarrow +\infty \).
In order to prove (5.22), we notice that
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
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
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
Then, with (5.17), (5.22) and recalling the definition of \(\mathcal {I}_k\), we obtain
Setting
we have \(\widetilde{\textrm{w}}_k\in W^{1,p(\cdot )}(B_{\varepsilon _k}(x);\mathbb {R}^m)\) and
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
whence passing to the limsup as \(k\rightarrow +\infty \), recalling (5.35), and then letting \(\eta \rightarrow 0^+\), we get
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
and, correspondingly, we define the sequence of minimum problems
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
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
where
Proof
For every \(x \in \mathbb {R}^d\), \(\zeta \in \mathbb {R}_0^m\), and \(\nu \in \mathbb {S}^{d-1}\) we define
Step 1. We start with the proof of the inequality
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
Now, given \(\lambda > |\zeta |\), by virtue of Lemma 4.2, for every j there exists \(\hat{u}_j\) such that
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
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
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
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
Taking into account the upper bounds in , (g3), and (5.44)–(5.45) we obtain
where \(C_d:=2+\beta \gamma _{d-1}(2^{d-1}-1)\). This inequality, together with (5.48), gives
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
whence by taking the limit as \(\eta \rightarrow 0^+\) we get (5.43).
Step 2. We now prove
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
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
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.,
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
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
Hence there exists \(j_0(\varepsilon )>0\) such that whenever \(j \ge j_0(\varepsilon )\)
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
and the blow-up variable exponent at x as
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
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
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
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
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
Since \(v_j \rightarrow u\) in \(L^{p(\cdot )}(B_{\eta \varepsilon }(x),\mathbb {R}^m)\), it follows that
Therefore, from (5.57) and (5.59) we have
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
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
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
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
for every \(j \ge j_{2}(\varepsilon )\). Finally, collecting (5.61), (5.62), and (5.64) we conclude that
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,
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 \(j \ge j_{ 2 }(\varepsilon )\)
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
Moreover, with the upper bound in and (5.64), the volume integral in the right hand side of (5.71) can be estimated as
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
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
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
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
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
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
Finally, dividing by \(\gamma _{d-1}\), taking the limsup as \(\varepsilon \rightarrow 0^+\) and using (5.42), (5.51), and (5.52), we obtain
with \(C:= (2\beta \gamma _d + c(d))/\gamma _{d-1}\). Recalling the definition of \(\zeta \) and \(\nu \), we obtain that
holds true for \({\mathcal {H}}^{n-1}\)-a.e. \(x\in J_u \cap A\). Taking the limit as \(\eta \rightarrow 0^+\) we get
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. \(\square \)
Notes
Indeed, for such functionals, also in the case of a constant integrability exponent it is customary to assume \(p>1\), in order to get the scale separation effect we describe later on.
References
Acerbi, E., Bouchitté, G., Fonseca, I.: Relaxation of convex functionals: the gap phenomenon. Ann. Inst. Henri Poincare (C) Anal. Non Lineaire 20, 359–390 (2003)
Acerbi, E., Mingione, G.: Regularity results for a class of quasiconvex functionals with nonstandard growth. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30(2), 311–339 (2001)
Ambrosio, L.: A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital. B (7) 3(4), 857–881 (1989)
Ambrosio, L.: Existence theory for a new class of variational problems. Arch. Rat. Mech. 111, 291–322 (1990)
Ambrosio, L., Braides, A.: Functionals defined on partitions of sets of finite perimeter, I: integral representation and \(\Gamma \)-convergence. J. Math. Pures Appl. 69, 285–305 (1990)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, New York (2000)
Bach, A., Braides, A., Zeppieri, C.I.: Quantitative analysis of finite-difference approximations of free-discontinuity problems. Interfaces Free Bound. 22, 317–381 (2020)
Bach, A., Cicalese, M., Ruf, M.: Random finite-difference discretizations of the Ambrosio–Tortorelli functional with optimal mesh size. SIAM J. Math. Anal. 53(2), 2275–2318 (2021)
Barchiesi, M., Focardi, M.: Homogenization of the Neumann problem in perforated domains: an alternative approach. Calc. Var. PDEs 42, 257–288 (2011)
Barchiesi, M., Lazzaroni, G., Zeppieri, C.I.: A bridging mechanism in the homogenisation of brittle composites with soft inclusions. SIAM J. Math. Anal. 48, 1178–1209 (2016)
Bouchitté, G., Fonseca, I., Leoni, G., Mascarenhas, L.: A global method for relaxation in \(W^{1, p}\) and in \({SBV}_p\). Arch. Ration. Mech. Anal. 165, 187–242 (2002)
Bouchitté, G., Fonseca, I., Mascarenhas, L.: A global method for relaxation. Arch. Ration. Mech. Anal. 145, 51–98 (1998)
Braides, A., Chiadò Piat, V.: Integral representation results for functionals defined on SBV(\(\Omega {\mathbb{R} }^{m})\). J. Math. Pures Appl. (9) 75, 595–626 (1996)
Braides, A., Defranceschi, A., Vitali, E.: Homogenization of free discontinuity problems. Arch. Ration. Mech. Anal. 135, 297–356 (1996)
Cagnetti, F., Chambolle, A., Scardia, L.: Korn and Poincaré-Korn inequalities for functions with a small jump set. Math. Ann. 383, 1179–1216 (2022). https://doi.org/10.1007/s00208-021-02210-w
Cagnetti, F., Dal Maso, G., Scardia, L., Zeppieri, C.I.: \(\Gamma \text{- }\)convergence of free discontinuity problems. Ann. Inst. Henri Poincare (C) Anal. Non Lineaire 36, 1035–1079 (2019)
Carriero, M., Leaci, A.: \(S^k\)-valued maps minimizing the \(L^p\) norm of the gradient with free discontinuities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18(3), 321–352 (1991)
Chambolle, A., Crismale, V.: A density result in \(GSBD^p\) with applications to the approximation of brittle fracture energies. Arch. Ration. Mech. Anal. 232, 1329–1378 (2019)
Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006)
Conti, S., Focardi, M., Iurlano, F.: Integral representation for functionals defined on \(SBD^p\) in dimension two. Arch. Ration. Mech. Anal. 223, 1337–1374 (2017)
Coscia, A., Mingione, G.: Hölder continuity of the gradient of \(p(x)\)-harmonic mappings. C. R. Acad. Sci. Paris Sér. I Math. 328(4), 363–368 (1999)
Coscia, A., Mucci, D.: Integral representation and \(\Gamma \)-convergence of variational integrals with \({p(x)}\)-growth. ESAIM: Control Optim. Calc. Var. 7, 495–519 (2002)
Crismale, V., Friedrich, M., Solombrino, F.: Integral representation for energies in linear elasticity with surface discontinuities. Adv. Calc. Var. 15(4), 705–733 (2022). https://doi.org/10.1515/acv-2020-0047
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Heidelberg (2013)
Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Birkhäuser, Boston (1993)
De Cicco, V., Leone, C., Verde, A.: Lower semicontinuity in \(SBV\) for integrals with variable growth. SIAM J. Math. Anal. 42(6), 3112–3128 (2010)
De Giorgi, E., Carriero, M., Leaci, A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. Anal. 108, 195–218 (1989)
Diening, L.: Maximal function on generalized Lebesgue spaces \(L^{p(\cdot )}\). Math. Inequal. Appl. 7(2), 245–253 (2004)
Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics. Springer, New York (2010)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Math. CRC Press, Boca Raton (1992)
Focardi, M., Gelli, M.S., Ponsiglione, M.: Fracture mechanics in perforated domains: a variational model for brittle porous media. Math. Models Methods Appl. Sci. 19, 2065–2100 (2009)
Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: \(L^p\) Spaces. Springer, New York (2007)
Fonseca, I., Müller, S., Pedregal, P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29, 736–756 (1998)
Francfort, G.A., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46(8), 1319–1342 (1998)
Friedrich, M.: A compactness result in \(GSBV^p\) and applications to \(\Gamma \)-convergence for free discontinuity problems. Calc. Var. 58, 86 (2019)
Friedrich, M.: A piecewise Korn inequality in \(SBD\) and applications to embedding and density results. SIAM J. Math. Anal. 50, 3842–3918 (2018)
Friedrich, M., Solombrino, F.: Functionals defined on piecewise rigid functions: integral representation and \(\Gamma \)-convergence. Arch. Ration. Mech. Anal. 236(3), 1325–1387 (2020)
Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. 221, 163–198 (1921)
Harjulehto, P., Hästö, P.: Lebesgue points in variable exponent spaces. Ann. Acad. Sci. Fenn. 29, 295–306 (2004)
Harjulehto, P., Hästö, P., Latvala, V.: Minimizers of the variable exponent, non-uniformly convex Dirichlet energy. J. Math. Pures Appl. 89(2), 174–197 (2008)
Harjulehto, P., Hästö, P., Latvala, V., Toivanen, O.: Critical variable exponent functionals in image restoration. Appl. Math. Lett. 26(1), 56–60 (2013)
Ioffe, A.D.: On lower semicontinuity of integral functionals. I. SIAM J. Control Optim. 15, 521–538 (1977)
Kovácik, O., Rákosnák, J.: On spaces \(L^{p(x)}\) and \(W^{1, p(x)}\). Czechoslovak Math. J. 41(116), 592–618 (1991)
Li, F., Li, Z., Pi, L.: Variable exponent functionals in image restoration. Appl. Math. Comput. 216, 870–882 (2010)
Morse, A.P.: Perfect blankets. Trans. Am. Math. Soc. 61, 418–442 (1947)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)
Rajagopal, K.R., R\(\mathring{\rm u}\)žička, M.: Mathematical modelling of electrorheological fluids. Contin. Mech. Thermodyn. 13, 59–78 (2001)
R\(\mathring{\rm u}\)žička, M.: Electrorheological fluids: mathematical modeling and mathematical theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000)
Sharapudinov, I.I.: Approximation of functions in the metric of the space \(L^{p(t)}([a, b])\) and quadrature formulas, (Russian). In: Constructive Function Theory’81 (Varna, 1981), pp. 189–193. Publ. House Bulgar. Acad. Sci., Sofia (1983)
Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5, 105–116 (1997)
Zhikov, V.V.: Meyers type estimates for solving the non linear Stokes system. Differ. Equ. 33, 107–114 (1997)
Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)
Acknowledgements
The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of INdAM. The authors have been supported by the project STAR PLUS 2020—Linea 1 (21-UNINA-EPIG-172) “New perspectives in the Variational modeling of Continuum Mechanics”. The work of FS is part of the project “Variational methods for stationary and evolution problems with singularities and interfaces” PRIN 2017 financed by the Italian Ministry of Education, University, and Research.
Funding
Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Mondino.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A: A \(\Gamma \)-convergence result with weaker growth conditions from above
Appendix A: A \(\Gamma \)-convergence result with weaker growth conditions from above
In this section we will prove a \(\Gamma \)-convergence result for energies whose surface densities satisfy a weaker assumption than (g3) of Sect. 4. To do this, we will also take advantage an integral representation result on \(SBV^{p(\cdot )}\) (see Theorem A.1) via a perturbation argument.
Let \((f_j)_{j\in \mathbb {N}}\) and \((g_j)_{j\in \mathbb {N}}\) be sequences of functions satisfying f1–f2 and (g1), (g2), (g4), respectively. In place of (g3), we require each \(g_j\) to comply with the additional property
- (\(g3^\prime \)):
-
(lower and upper bound) for every \(x\in \mathbb {R}^d\), \(\zeta \in \mathbb {R}^d_0\), and \(\nu \in {\mathbb {S}}^{d-1}\)
$$\begin{aligned} \alpha \le g(x,\zeta ,\nu ) \le \beta (1+|\zeta |), \end{aligned}$$
together with (g5).
Correspondingly, we define the functionals \(\mathcal {E}_j: L^0(\Omega ;\mathbb {R}^m)\times \mathcal {A}(\Omega )\rightarrow [0,+\infty ]\) as
1.1 A.1 Integral representation: the \(SBV^{p(\cdot )}\) case
In this section we discuss the minor modifications needed in order to obtain an integral representation result for functionals \(\mathcal {F}:SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m) \times \mathcal {B}(\Omega ) \rightarrow [0,+\infty )\), satisfying assumptions (\(H_1\))–(\(H_3\)) and the following
- (\(\textrm{H}_4'\)):
-
there exist \(0< \alpha < \beta \) such that for any \(u \in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(B \in \mathcal {B}(\Omega )\) we have
$$\begin{aligned} \begin{aligned} \alpha \bigg (\int _{ B } |\nabla u|^{p(x)} \, \textrm{d} x&+ \int _{J_u \cap B}(1+|[u]|)\,\textrm{d}\mathcal {H}^{d-1}\bigg ) \le \mathcal {F}(u,B) \\&\le \beta \bigg (\int _{ B } (1 + |\nabla u|^{p(x)}) \, \textrm{d} x+ \int _{J_u \cap B}(1+|[u]|)\,\textrm{d}\mathcal {H}^{d-1}\bigg ). \end{aligned} \end{aligned}$$(A.2)
For every \(u \in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(A \in \mathcal {A}(\Omega )\) we define
The main result of this section is the following integral representation theorem.
Theorem A.1
(Integral representation in \(SBV^{p(\cdot )}\)) 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 (\(\hbox {P}_{1}\))-(\(\hbox {P}_{2}\)), and suppose that \(\mathcal {F}:SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m) \times \mathcal {B}(\Omega ) \rightarrow [0,+\infty )\) satisfies (\(\hbox {H}_{1}\))–(\(\hbox {H}_{3}\)) and (\(\hbox {H}_4'\)). Then
for all \(u \in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(B \in \mathcal {B}(\Omega )\), where f is given by
for all \(x_0 \in \Omega \), \(u_0 \in \mathbb {R}^m\), \(\xi \in \mathbb {R}^{m \times d}\) and \(\ell _{x_0,u_0,\xi }\) as in (3.2), g is given by
for all \( x_0 \in \Omega \), \(a,b \in \mathbb {R}^m\), \(\nu \in \mathbb {S}^{d-1}\) and \(u_{x_0,a,b,\nu }\) as in (3.3), and \(\textbf{m}_{\mathcal {F}}\) 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 \(GSBV^{p(\cdot )}\) functions. For this, Lemmas 3.3, 3.4 and 3.5 are replaced by the corresponding \(SBV^{p(\cdot )}\) 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 \(p:\Omega \rightarrow (1,+\infty )\) be a variable exponent satisfying (\(\hbox {P}_{1}\))-(\(\hbox {P}_{2}\)). Suppose that \(\mathcal {F}\) satisfies (\(\hbox {H}_{1}\))–(\(\hbox {H}_{3}\)) and (\(\hbox {H}_4'\)). Let \(u \in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(\mu \) be defined as
Then for \(\mu \)-a.e. \(x_0 \in \Omega \) we have
Proof
The only needed modification concerns the proof of Lemma 3.6. Indeed, under assumption (\({H}_4'\)), for \(u\in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) the same construction provides a sequence \(v^{\delta ,n}\) in \(SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) such that
Then, an analogous compactness argument, based on [4, Theorem 2.1] yields \(v^\delta \in SBV^{p^-}(\Omega ;\mathbb {R}^m)\), which can be improved to \(v^\delta \in SBV^{p{(\cdot )}}(\Omega ;\mathbb {R}^m)\) 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). \(\square \)
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 \(p:\Omega \rightarrow (1,+\infty )\) 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 \(u \in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then for \(\mathcal {L}^{d}\)-a.e. \(x_0 \in \Omega \) we have
Proof
The proof of “\(\le \)” inequality in (A.8) can be obtained with the same construction of Lemma 3.8 applied to the sequence \((u_\varepsilon )\) 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
whence by Lemma 3.7(iii), \((iii)'\), (3.42) we obtain the analogous of (3.43), and this concludes the proof of the first inequality in (A.8).
The reverse inequality in (A.8) can be proved following the argument of Lemma 3.9. For this, we first notice that since \(u_\varepsilon \) satisfies Lemma 3.7\((i)'\), in addition to (3.45) we may require that
where \(u^-_\varepsilon \) and \(u^+\) denote the inner and outer traces at \(\partial B_{s\varepsilon }(x_0)\) of \(u_\varepsilon \) and u, respectively. Then, estimates (3.43) and (3.48) (with the additional term \(\beta \int _{\partial B_{s\varepsilon }(x_0)} |u^+-u^-_\varepsilon |\,\textrm{d}\mathcal {H}^{d-1}\) in the left hand side) can be established. Finally, combining (3.43), (3.45), (A.9) and the fact that \(s\varepsilon \le (1-3\theta )\varepsilon \), we obtain also (3.49). This will suffice to conclude the argument of Lemma 3.9 and then the proof of the inequality “\(\ge \)” in (A.8). \(\square \)
Lemma A.4
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 \(u \in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\). Then for \(\mathcal {H}^{d-1}\)-a.e. \(x_0 \in J_u\) we have
Proof
The construction of Lemma 3.11 can be performed using the sequence \((\bar{u}_\varepsilon )\) which complies with Lemma 3.10(i)-(iv), (3.52) and (3.53), thus obtaining the analogous of estimates (3.67) and (3.68) where the constant \(\beta \) is replaced by \(\beta (1+|[\bar{u}^\textrm{surf}_{x_0}]|)\). Now, taking into account (\({H}_4'\)), (3.50)(ii), (3.69) and (3.53), we obtain the analogous of (3.70); i.e.,
With this, we can easily infer the upper inequality in (A.10).
As for the reverse inequality, given \((\bar{u}_\varepsilon )\) as above, by (3.52) we may require, in addition to (3.72), also the property
where \(u^+\) and \(\bar{u}_\varepsilon ^-\) 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, as a consequence of (3.72), (5.21), (A.11) and the fact that \(\sigma \le (1-3\theta )\) we then obtain
which corresponds to (3.75). The estimate (3.77) now reads
whence the conclusion follows exactly in the same way as in Lemma 3.12. We omit further details. \(\square \)
1.2 A.2 \(\Gamma \)-convergence
Let \(\sigma >0\). We define the family of perturbed functionals \(\mathcal {E}_j^\sigma : L^0(\Omega ;\mathbb {R}^m)\times \mathcal {A}(\Omega )\rightarrow [0,+\infty ]\), \(j\in \mathbb {N}\), as
where
First, we prove a \(\Gamma \)-convergence result for the perturbed functionals \(\mathcal {E}_j^\sigma \).
Theorem A.5
(\(\Gamma \)-convergence of perturbed functionals) 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), (g2), (g3), (g4), (g5), respectively. Let \(\sigma >0\) and \(\mathcal {E}_j^\sigma :SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m) \times \mathcal {A}(\Omega ) \rightarrow [0,+\infty )\) be the sequence of functionals given in (A.12). Then, there exists a functional \(\mathcal {E}^\sigma :SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\times \mathcal {A}(\Omega ) \rightarrow [0,+\infty )\) and a subsequence (not relabeled) such that
for all \(A \in \mathcal {A}(\Omega ) \). Let \(f_\infty ^\sigma \) and \(g_\infty ^\sigma \) be defined as
for all \(x_0 \in \Omega \), \(u_0 \in \mathbb {R}^m\), \(\xi \in \mathbb {R}^{m \times d}\), and
for all \( x_0 \in \Omega \), \(\zeta \in \mathbb {R}^m\), and \(\nu \in \mathbb {S}^{d-1}\), where \(\textbf{m}_{\mathcal {E}^\sigma }\) is as in (A.3) with \(\mathcal {F}=\mathcal {E}^\sigma \).
Then, for every \(u\in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(A\in \mathcal {A}(\Omega )\) we have that
Proof
The proof of Theorem A.5 can be obtained along the lines of the argument of Theorem 4.1. We then briefly sketch the proof, referring the reader to Theorem 4.1 for further details.
We start by observing that some properties of the \(\Gamma \)-liminf and \(\Gamma \)-limsup with respect to the topology of the convergence in measure, established in Lemma 4.3 for functionals \(\mathcal {F}_j\), still hold true for \(\mathcal {E}_j^\sigma \). To this end, we define
for all \(u \in SBV^{p(\cdot )}(\Omega ;\mathbb {R}^m)\) and \(A \in \mathcal {A}(\Omega )\).
Then the analogous of assertions (i), (iii) and (iv) of Lemma 4.3 still hold true for \((\mathcal {E}^\sigma )'\) and \((\mathcal {E}^\sigma )''\), since the arguments are based on Lemmas 4.2 and 3.2. Setting
we only have to check that
The upper bound for \((\mathcal {E}^\sigma )''\) in (A.18) can be inferred choosing the constant sequence \(u_j=u\) in (A.17) and taking into account f2, the definition of \(g_j^\sigma \) (equation (A.13)) together with (\(g3^\prime \)). For what concerns the lower bound in (A.18), we consider an (almost) optimal sequence \((v_j)_j\) in (A.17). Then, with f2, (A.13) and (\(g3^\prime \)) we get
Now, since \(v_j\rightarrow u\) in measure on A, we may appeal to the closure property of SBV (see, e.g., [6, Theorem 4.7]). Then, by arguing as in the proof of Lemma 3.6 and exploiting the lower semicontinuity inequalities
for any concave function \(\theta :(0,+\infty )\rightarrow (0,+\infty )\), we easily obtain the lower bound.
The existence of the \(\Gamma \)-limit is still a consequence of the abstract result [25, Theorem 16.9], in view of the inner regularity of both \((\mathcal {E}^\sigma )'\) and \((\mathcal {E}^\sigma )''\). Since (A.18) implies (\({H}_4'\)), the functional \(\mathcal {E}^\sigma =(\mathcal {E}^\sigma )'=(\mathcal {E}^\sigma )''\) satisfies all the assumptions of Theorem A.1. This concludes the proof. \(\square \)
Now, we are in position to deduce the \(\Gamma \)-convergence result for the family of functionals \(\mathcal {E}_j\), defined in (A.1). The argument of the proof is analogous to that of [16, Theorem 5.1], but with some simplifications due to the fact that, by virtue of Theorem A.5, we do not need to use the \(\Gamma \)-convergence of the restrictions to \(L^{p(\cdot )}\) of our functionals.
Theorem A.6
Let \(\Sigma \) be a countable subset of \((0,+\infty )\), with \(0\in \overline{\Sigma }\). Assume that for every \(\sigma \in \Sigma \) there exists a functional \(\mathcal {E}^\sigma : L^0(\Omega ;\mathbb {R}^m)\times \mathcal {A}(\Omega )\rightarrow [0,+\infty ]\) such that for every \(A\in \mathcal {A}(\Omega )\) the sequence \(\mathcal {E}_j^\sigma (\cdot ,A)\) defined in (A.12) \(\Gamma \)-converges to \(\mathcal {E}^\sigma (\cdot ,A)\) in \(L^0(\mathbb {R}^d;\mathbb {R}^m)\). Let \(f_\infty ^\sigma \) and \(g_\infty ^\sigma \) be the functions defined in (A.4) and (A.5), respectively. Let \(f_\infty ^0:\mathbb {R}^d\times \mathbb {R}^{m\times d}\rightarrow [0,+\infty ]\) and \(g_\infty ^0:\mathbb {R}^d\times \mathbb {R}^{m}_0\times \mathbb {S}^{d-1}\rightarrow [0,+\infty ]\) be the functions defined as
Then, the functionals \(\mathcal {E}_j(\cdot ,A)\) defined in (A.1) \(\Gamma \)-converge in \(L^0(\mathbb {R}^d;\mathbb {R}^m)\) to the functional \(\mathcal {E}^0(\cdot ,A)\) given by
for every \(A\in \mathcal {A}(\Omega )\) and \(u\in GSBV^{p(\cdot )}(A;\mathbb {R}^m)\).
Proof
It follows from (A.4) and (A.5) that \(f^{\sigma _1}_\infty \le f^{\sigma _2}_\infty \) and \(g^{\sigma _1}_\infty \le g^{\sigma _2}_\infty \) for \(0<\sigma _1<\sigma _2\). Then, by the Monotone Convergence Theorem we have
for every \(A\in \mathcal {A}(\Omega )\) and every \(u\in L^0(\mathbb {R}^d,\mathbb {R}^m)\) with \(u|_A\in SBV^{p(\cdot )}(A,\mathbb {R}^m)\).
Let \(\mathcal {E}'\), \(\mathcal {E}'':L^0(\mathbb {R}^d,\mathbb {R}^m){\times } \mathcal {A}(\Omega ) \rightarrow [0,+\infty ]\) be defined by
where we use the topology of \(L^0(\mathbb {R}^d,\mathbb {R}^m)\). We subdivide the rest of the proof into steps.
Step 1: First, for every \(A\in \mathcal {A}(\Omega )\), \(u\in L^0(\mathbb {R}^d,\mathbb {R}^m)\) with \(u|_A\in SBV^{p(\cdot )}(A,\mathbb {R}^m)\) and for every \(\sigma \in \Sigma \) we have \(\mathcal {E}''(u,A)\le \mathcal {E}^{\sigma }(u,A)\), whence by (A.22) we immediately get
Step 2: We claim that
for every \(A \in \mathcal {A}(\Omega )\) and every \(u\in L^\infty (\mathbb {R}^d,\mathbb {R}^m)\).
With fixed A and u as above, by \(\Gamma \)-convergence there exists a sequence \((u_j)\) converging to u in \(L^0(\mathbb {R}^d,\mathbb {R}^m)\) such that
Let us fix \(\lambda >\Vert u\Vert _{L^\infty (\mathbb {R}^d\!,\,\mathbb {R}^m)}\) and \(\sigma >0\). By Lemma 4.2 there exist \(\mu >\lambda \), independent of j, and a sequence \((v_j)\subset L^\infty (\mathbb {R}^d,\mathbb {R}^m)\), converging to u in measure on bounded sets, such that for every j we have
If \(\mathcal {E}_j(u_j,A)<+\infty \), by the lower bounds in , (\(g3^\prime \)), and (A.28) the function \(v_j\) belongs to \(GSBV^{p(\cdot )}(A,\mathbb {R}^m)\) 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 \(\mathcal {E}_j(u_j,A)=+\infty \). Therefore, using (A.25) and the inequality \(\Vert u\Vert _{L^\infty (\mathbb {R}^d\!,\,\mathbb {R}^m)}<\lambda \), by \(\Gamma \)-convergence we get
for every \(\sigma \in \Sigma \). By (A.22), passing to the limit as \(\sigma \rightarrow 0^+\) we obtain (A.24) whenever \(u\in L^\infty (\mathbb {R}^d,\mathbb {R}^m)\).
Step 3: We now prove that
Let us fix u and A. It is enough to prove the inequality when \(u|_A\in GSBV^{p(\cdot )}(A,\mathbb {R}^m)\). By Lemma 4.2 for every \(\sigma >0\) and for every integer \(j\ge 1\) there exists \(u_j \in L^\infty (\mathbb {R}^d,\mathbb {R}^m)\), with \(u_j|_A\in SBV^{p(\cdot )}(A,\mathbb {R}^m)\), such that \(u_j=u\) \(\mathcal {L}^d\)-a.e. in \(\{|u|\le j\}\) and
By (A.23) we have \(\mathcal {E}''(u_j,A)\le \mathcal {E}^0(u_j,A)\), hence
Since \(u_j\rightarrow u\) in measure on bounded sets, passing to the limit as \(j\rightarrow +\infty \), by the lower semicontinuity of the \(\Gamma \)-limsup we deduce
Thus, letting \(\sigma \rightarrow 0^+\) we obtain (A.30).
Step 4: We now prove that
Given an open set A, it is enough to prove the inequality for a function u such that \(u|_A\in GSBV^{p(\cdot )}(A,\mathbb {R}^m)\), since otherwise \(\mathcal {E}'(u,A)=+\infty \) due to the lower bounds in and (\(g3^\prime \)). By Lemma 4.2 for every \(\sigma >0\) and every integer \(j\ge 1\) there exists \(u_j \in L^\infty (\mathbb {R}^d,\mathbb {R}^m)\), with \(u_j|_A\in SBV^{p(\cdot )}(A,\mathbb {R}^m)\), such that
By (A.24) we have \(\mathcal {E}^0(u_j,A)\le \mathcal {E}'(u_j,A)\), which combined with (A.32) gives
Letting \(j\rightarrow +\infty \) we get
and then sending \(\sigma \rightarrow 0^+\) we obtain (A.31).
The \(\Gamma \)-convergence of \(\mathcal {E}_j(\cdot ,A)\) to \(\mathcal {E}^0(\cdot ,A)\) in \(L^0(\mathbb {R}^d,\mathbb {R}^m)\) follows from (A.30) and (A.31). This concludes the proof. \(\square \)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Scilla, G., Solombrino, F. & Stroffolini, B. Integral representation and \(\Gamma \)-convergence for free-discontinuity problems with \(p(\cdot )\)-growth. Calc. Var. 62, 213 (2023). https://doi.org/10.1007/s00526-023-02549-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-023-02549-9