$\Gamma$-convergence and stochastic homogenisation of singularly-perturbed elliptic functionals

Abstract

We study the limit behaviour of singularly-perturbed elliptic functionals of the form

$$\begin{aligned} {\mathscr {F}}_k(u,v)=\int _A v^2\,f_k(x,\nabla u)\, \textrm{d}x+\frac{1}{\varepsilon _k}\int _A g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x, \end{aligned}$$

where u is a vector-valued Sobolev function, $v \in [0,1]$ a phase-field variable, and $\varepsilon _k>0$ a singular-perturbation parameter; i.e., $\varepsilon _k \rightarrow 0$, as $k\rightarrow +\infty$. Under mild assumptions on the integrands $f_k$ and $g_k$, we show that if $f_k$ grows superlinearly in the gradient-variable, then the functionals ${\mathscr {F}}_k$ $\Gamma$-converge (up to subsequences) to a brittle energy-functional; i.e., to a free-discontinuity functional whose surface integrand does not depend on the jump-amplitude of u. This result is achieved by providing explicit asymptotic formulas for the bulk and surface integrands which show, in particular, that volume and surface term in ${\mathscr {F}}_k$ decouple in the limit. The abstract $\Gamma$-convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.

1 Introduction

Since the seminal work of Modica and Mortola [48, 49] and of Ambrosio and Tortorelli [7, 8], singularly-perturbed elliptic functionals have proven to be an effective tool to approximate free-discontinuity problems in a variety of situations. Elliptic-functionals based approximations have been successfully used, e.g., for numerical simulations in imaging or brittle fracture (see, e.g., [18,19,20]), in cavitation problems [43, 44], and to define a notion of regular evolution in fracture mechanics [10, 41], just to mention few examples. Besides being an approximation tool, these kinds of functionals are also commonly employed to model a range of phenomena where “diffuse” interfaces appear (see, e.g., [15, 16, 30, 31, 37, 38, 51, 52, 55]), or as instances of gradient damage models (see, e.g., [34, 45, 53]).

In this paper we study the $\Gamma$-convergence, as $k \rightarrow +\infty$, of general elliptic functionals of the form

$$\begin{aligned} {\mathscr {F}}_k(u,v)=\int _A \psi (v)\,f_k(x,\nabla u)\, \textrm{d}x+\frac{1}{\varepsilon _k}\int _A g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x, \end{aligned}$$

(1.1)

where $\varepsilon _k \searrow 0$ is a singular-perturbation parameter. The set $A\subset {\mathbb {R}}^n$ is open bounded and with Lipschitz boundary, $u :A \rightarrow {\mathbb {R}}^m$ is a vectorial function, $v:A \rightarrow [0,1]$ is a phase-field variable, and $\psi :[0,1] \rightarrow [0,1]$ is an increasing and continuous function satisfying $\psi (0)=0$, $\psi (1)=1$, and $\psi (s)>0$ for $s>0$. For every $k\in {\mathbb {N}}$, the integrands $f_k :{\mathbb {R}}^n \times {\mathbb {R}}^{m\times n} \rightarrow [0,+\infty )$ and $g_k :{\mathbb {R}}^n \times [0,1] \times {\mathbb {R}}^n \rightarrow [0,+\infty )$ belong to suitable classes of functions denoted by ${\mathcal {F}}$ and ${\mathcal {G}}$, respectively (see Sect. 2.2 for their definition). The requirement $(f_k) \subset {\mathcal {F}}$ and $(g_k) \subset {\mathcal {G}}$ in particular ensures the existence of an exponent $p>1$ such that for every $k\in {\mathbb {N}}$ and every $x\in {\mathbb {R}}^n$:

$$\begin{aligned} c_1 |\xi |^p \le f_k(x,\xi )\le c_2 |\xi |^p, \end{aligned}$$

(1.2)

for every $\xi \in {\mathbb {R}}^{m\times n}$ and for some $0<c_1 \le c_2<+\infty$, and

$$\begin{aligned} c_3 \big ( |1-v|^p+|w|^p\big ) \le g_k(x,v,w)\le c_4 \big (|1-v|^p +|w|^p\big ), \end{aligned}$$

(1.3)

for every $v\in [0,1]$ and $w\in {\mathbb {R}}^n$, and for some $0<c_3 \le c_4<+\infty$. As a consequence, the functionals ${\mathscr {F}}_k$ are finite in $W^{1,p}(A;{\mathbb {R}}^m)\times W^{1,p}(A)$ and are bounded both from below and from above by Ambrosio-Tortorelli functionals of the form

$$\begin{aligned} AT_{k}(u,v)=\int _A \psi (v)\,|\nabla u|^p\, \textrm{d}x+\int _A\bigg (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\bigg )\, \textrm{d}x. \end{aligned}$$

(1.4)

Therefore if $(u_k,v_k) \subset W^{1,p}(A;{\mathbb {R}}^m)\times W^{1,p}(A)$ is a pair satisfying $\sup _{k}{\mathscr {F}}_k(u_k,v_k)<+\infty$, the lower bound on ${\mathscr {F}}_k$ immediately yields $v_k \rightarrow 1$ in $L^p(A)$, as $k \rightarrow +\infty$. On the other hand, $|\nabla u_k|$ can blow up in the regions where $v_k$ is asymptotically small, so that one expects a limit u which may develop discontinuities. In [7, 8] Ambrosio and Tortorelli showed that functionals of type (1.4) provide a variational approximation, in the sense of $\Gamma$-convergence, of the free-discontinuity functional of Mumford-Shah type given by

$$\begin{aligned} \int _A |\nabla u|^p\, \textrm{d}x+c_p{\mathcal {H}}^{n-1}(S_u), \end{aligned}$$

(1.5)

where now the variable u belongs to the space of generalised special functions of bounded variation $GSBV^p(A;{\mathbb {R}}^m)$. As in the case of the Modica-Mortola approximation of the perimeter-functional [48, 49], the effect of the singular perturbation, $\varepsilon _k^{p-1}|\nabla v|^p$, in (1.4) is that of producing a transition layer around the discontinuity set of u, denoted by $S_u$. Similarly, the pre-factor $c_p>0$ is related to the cost of an optimal transition of the phase-field variable, now taking place between the value zero, where $\psi$ vanishes, and the value one. Another similarity shared by the Modica-Mortola and the Ambrosio-Tortorelli approximation is that they are essentially one-dimensional, that is, in both cases the n-dimensional analysis can be carried out by resorting to an integral-geometric argument, the slicing procedure, which allows to reduce the general situation to the one-dimensional case.

A relevant feature of the Ambrosio-Tortorelli approximation is that the “regularised” bulk and surface terms in (1.4) separately converge to their sharp-interface counterparts in (1.5). This kind of volume-surface decoupling can be also observed in a number of variants of (1.4). Indeed, this is the case, e.g., of the anisotropic functionals analysed in [36], of the phase-field approximation of brittle fracture in linearised elasticity in [29], of the second-order variants proposed in [11, 26], and of the finite-difference discretisation of the Ambrosio-Tortorelli functional on periodic [12] and on stochastic grids [13]. More specifically, in [11, 26, 29, 36] the volume-surface decoupling is obtained by means of general integral-geometric arguments which can be employed thanks to the specific form of the approximating functionals. On the other hand, in [12, 13] the interplay between the singular perturbation and the discretisation parameters makes for a subtle problem for which an ad hoc proof is needed. In [12] this proof relies on an explicit geometric construction which, however, is feasible only in dimension $n=2$. In fact, more refined arguments are necessary to deal with the case $n\ge 3$, as shown in [13]. Namely, in [13] the limit volume-surface decoupling is achieved by resorting to a weighted co-area formula, which is reminiscent of a technique introduced by Ambrosio [5] (see also [24, 27, 42, 54]). This procedure allows to identify an asymptotically small region where the phase-field variable $v_k$ can be modified and set equal to zero while the corresponding $u_k$ makes a steep transition between two constant values. In this way, a pair $(u_k,v_k)$ is obtained whose bulk energy vanishes while the surface energy does not essentially increase.

In the present paper we show that the volume-surface decoupling illustrated above takes place also for the general functionals ${\mathscr {F}}_k$, whose integrands $f_k$ and $g_k$ combine both a k and an x dependence, and satisfy (1.2) and (1.3). Moreover, being the dependence on x only measurable, the case of homogenisation is covered by our analysis as well, as shown in this paper. We remark that the generality of the functionals does not allow us to use either the slicing or the blow-up method to establish a $\Gamma$-convergence result for ${\mathscr {F}}_k$, as it is instead customary for phase-field functionals of Ambrosio-Tortorelli type. Our approach is close in spirit to that of [27] and combines the localisation method of $\Gamma$-convergence [23, 32] with a careful local analysis which eventually allows us to completely characterise the integrands of the $\Gamma$-limit thus proving, in particular, that volume and surface term do not interact in the limit.

The volume-surface decoupling has been extensively analysed in the case of free-discontinuity functionals, starting with the seminal work [5]. It has then been proven that a decoupling takes place in the case of free-discontinuity functionals with periodically oscillating integrands [24], for scale-dependent scalar brittle energies both in the continuous [42] and in the discrete case [54], for general vectorial scale-dependent free-discontinuity functionals [27, 28], and, more recently, also in the setting of linearised elasticity [40]. The clear advantage of having such a decoupling is that limit volume and surface integrands can be determined independently from one another, by means of asymptotic formulas which are then easier to handle, e.g., computationally. Moreover in [42] it is shown that the noninteraction between volume and surface is crucial to prove the stability of unilateral minimality properties in the study of crack-propagation in composite materials. The same applies to the case of the evolution considered in [41], where this feature plays a central role in proving that the regular quasistatic evolution for the Ambrosio-Tortorelli functional converges to a quasistatic evolution for brittle fracture in the sense of [39]. These considerations also motivate the analysis carried out in the present paper.

The main result of this paper is contained in Theorem 3.1 and is a $\Gamma$-convergence and an integral representation result for the $\Gamma$-limit. Namely, in Theorem 3.1 we show that if $f_k$ and $g_k$ satisfy rather mild assumptions, (see Sect. 2.2 for the complete list of hypotheses) together with (1.2) and (1.3), then (up to subsequences) the functionals ${\mathscr {F}}_k$ $\Gamma$-converge to a free-discontinuity functional of the form

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

(1.6)

where now $u\in GSBV^p(A;{\mathbb {R}}^m)$ and $\nu _u$ denotes the generalised normal to $S_u$. We observe that the surface term in ${\mathscr {F}}$ is both inhomogeneous and anisotropic, however it does not depend on the jump-opening $[u]=u^+-u^-$; in other words, ${\mathscr {F}}$ is a so-called brittle energy. The form of the surface term in (1.6) is one of the effects of the volume-surface limit decoupling mentioned above, which is apparent from the asymptotic formulas defining $f_\infty$ and $g_\infty$. In fact, in Theorem 3.1 we also provide formulas for $f_\infty$ and $g_\infty$. Namely, we prove that

$$\begin{aligned} f_{\infty }(x,\xi )=\limsup _{\rho \rightarrow 0} \lim _{k \rightarrow +\infty } \frac{1}{\rho ^n} \inf \int _{Q_\rho (x)} f_k(y,\nabla u)\, \textrm{d}y\end{aligned}$$

(1.7)

where the infimum in (1.7) is taken over all functions $u\in W^{1,p}(Q_\rho (x);{\mathbb {R}}^m)$ with $u(y)=\xi y$ near $\partial Q_\rho (x)$. The surface energy density is given instead by

$$\begin{aligned} g_\infty (x,\nu )= \limsup _{\rho \rightarrow 0} \lim _{k \rightarrow +\infty } \frac{1}{\rho ^{n-1}} \inf \frac{1}{\varepsilon _k}\int _{Q^\nu _\rho (x)} g_k(y,v,\varepsilon _k \nabla v)\, \textrm{d}y\end{aligned}$$

(1.8)

where the cube $Q^\nu _\rho (x)$ is a suitable rotation of $Q_\rho (x)$ and the infimum in v is taken in a u-dependent class of functions. More precisely, the infimum in (1.8) is taken among all $v \in W^{1,p}(Q^\nu _\rho (x))$, with $0\le v \le 1$, for which there exists $u \in W^{1,p}(Q^\nu _\rho (x);{\mathbb {R}}^m)$ such that $v \nabla u$ = 0 a.e. in $Q^\nu _\rho (x)$ and $(u,v)=(u^\nu _x,1)$ in $U \cap \{|(y-x)\cdot \nu |>\varepsilon _k\}$ where U is a neighbourhood of $\partial Q^\nu _\rho (x)$, and $u^\nu _x$ is the jump function given by

$$\begin{aligned} u^{\nu }_{x}(y)={\left\{ \begin{array}{ll} e_1 &{} \text {if }\; (y-x) \cdot \nu \ge 0, \\ 0 &{} \text {if }\; (y-x) \cdot \nu < 0. \end{array}\right. } \end{aligned}$$

In (1.8) the boundary datum $(u^\nu _x,1)$ cannot be prescribed in the vicinity of $\{y \in {\mathbb {R}}^n :(y-x) \cdot \nu =0\}$ due to the discontinuity of $u^{\nu }_{x}$ and to the fact that v must be equal to zero (and not to one) where u jumps. However, this mixed Dirichlet-Neumann boundary condition can be replaced by a Dirichlet boundary condition prescribed on the whole boundary of $Q^\nu _{\rho }(x)$, up to replacing $u^{\nu }_{x}$ with a regularised counterpart defined, e.g., as in (l), Sect. 2.1.

In view of the growth conditions (1.2) satisfied by $f_k$ and the properties of $\psi$, the constraint $v \nabla u=0$ satisfied a.e. in $Q^\nu _\rho (x)$ is equivalent to

$$\begin{aligned} \int _{Q^\nu _\rho (x)} \psi (v)f_k(y,\nabla u)\, \textrm{d}y=0, \end{aligned}$$

which makes apparent why the bulk term in ${\mathscr {F}}_k$ does not contribute to $g_\infty$. We notice, however, that due to the nature of the problem, the variable u must enter in the definition of $g_\infty$, so that in this case a decoupling is not to be intended as in the case of free-discontinuity functionals [27].

To derive the formula for $f_\infty$ we follow a similar strategy as in [21] and use the co-area formula in the Modica-Mortola term in (1.1) to show that, in the set where v is bounded away from zero, ${\mathscr {F}}_k$ behaves like a sequence of free-discontinuity functionals whose volume integrand is $f_k$. Then, we conclude by invoking the decoupling result for free-discontinuity functionals proven in [27]. In fact, we notice that (1.7) coincides with the asymptotic formula for the limit volume integrand provided in [27]. The proof of (1.8) is more subtle and is substantially different, e.g., from that in [13]. Namely, to prove (1.8) we need to modify a sequence $(u_k, v_k)$ with bounded energy in the cube $Q^\nu _\rho (x)$ to get a new sequence with zero volume energy which can be used as a test in (1.8), hence, in particular, the modification to $(u_k, v_k)$ shall not increase the surface energy. In the case of the discretised Ambrosio-Tortorelli functional considered in [13], the discrete nature of the problem allows for a construction which is not feasible in a continuous setting. In our case, instead, we follow an argument which is close in spirit to a construction in [41]. This argument amounts to partition the set where $\nabla u_k \ne 0$ and to use the bound on the energy to single out a set of the partition with small measure and small volume energy. Then in this set the function $v_k$ is modified by suitably interpolating between the value zero and two functions explicitly depending on $u_k$. The advantage of this interpolation is that it allows to easily estimate the increment in surface energy in terms of the volume energy and at the same time to define a test pair for (1.8). Eventually, to prove that the increment in surface energy is asymptotically negligible we need to use that $p>1$. We notice that the assumption $p>1$ is optimal in the sense that if $f_k$ is linear in the gradient variable; i.e., (1.2) holds with $p=1$, then it is well known [2, 4] that the corresponding Ambrosio-Tortorelli functional $\Gamma$-converges to a free-discontinuity functional whose surface energy explicitly depends on [u], this dependence being the result of a nontrivial limit volume-surface interaction.

Our general analysis is then applied to study the homogenisation of damage models; i.e., to deal with the case of integrands $f_k$ and $g_k$ of type

$$\begin{aligned} f_k(x,\xi )= f\Big (\frac{x}{\varepsilon _k},\xi \Big )\; \text { and } \; g_{k}(x,v,w)=g\Big (\frac{x}{\varepsilon _k},v, w\Big ), \end{aligned}$$

(1.9)

for some $f \in {\mathcal {F}}$ and $g\in {\mathcal {G}}$. More specifically, in Theorem 3.5 we prove a homogenisation result for ${\mathscr {F}}_k$, with $f_k$ and $g_k$ as in (1.9), without requiring any spatial periodicity of the integrands, but rather assuming the existence and spatial homogeneity of the limit of certain scaled minimisation problems (cf. (3.9) and (3.10)). Eventually, we show that the assumptions of Theorem 3.5 are satisfied, almost surely, in the case where the integrands f and g are stationary random variables and derive the corresponding stochastic homogenisation result, Theorem 8.4. Thanks to the decoupling result, Theorem 3.1, the stochastic homogenisation of the bulk term readily follows from [35]. On the other hand, the treatment of the regularised surface term requires a new ad hoc analysis which shares some similarities with that developed for random surface functionals [3, 25, 28]. We also mention here the recent paper [46, 50] where the stochastic homogenisation of Modica-Mortola functionals with a stationary and ergodic gradient-perturbation is studied.

To conclude we notice that our analysis also allows to deduce a $\Gamma$-convergence result for functionals with oscillating integrands of type (1.9) when the heterogeneity scale does not necessarily coincide with the Ambrosio-Tortorelli parameter $\varepsilon _k$, but is rather given by a different infinitesimal scale $\delta _k>0$. In this case, though, the asymptotic formulas provided by Theorem 3.1 would fully characterise the homogenised volume energy but not the surface energy. In fact, in this case a full characterisation of the homogenised surface integrand requires a further investigation which, in particular, shall distinguish between the regimes $\varepsilon _k \ll \delta _k$ and $\varepsilon _k \gg \delta _k$. A complete analysis of this type, in the spirit of [9], goes beyond the purpose of the present paper and is instead the object of the ongoing work [14].

Outline of the paper This paper is organised as follows. In Sect. 2 we collect some notation used throughout, introduce the mathematical setting and the functionals we are going to analyse, moreover we prove some preliminary results. In Sect. 3 we state the main results of the paper, namely, the $\Gamma$-convergence and integral representation result (Theorem 3.1), a convergence result for some associated minimisation problems (Theorem 3.4), and a homogenisation result without periodicity assumptions (Theorem 3.5). In Sect. 4 we prove some properties satisfied by the limit volume and surface integrands (Proposition 4.1 and Proposition 4.5). In Sect. 5 we implement the localisation method of $\Gamma$-convergence proving, in particular, a fundamental estimate for the functionals ${\mathscr {F}}_k$ (Proposition 5.1) and a compactness and integral representation result for the $\Gamma$-limit of ${\mathscr {F}}_k$ (Theorem 5.2). In Sect. 6 we characterise the volume integrand of the $\Gamma$-limit (Proposition 6.1) and in Sect. 7 the surface integrand (Proposition 7.4), thus fully achieving the proof of the main result, Theorem 3.1. In Sect. 8 we prove a stochastic homogenisation result for stationary random integrands (Theorem 8.4). Eventually in the Appendix we prove two technical lemmas which are used in Sect. 8.

2 Setting of the problem and preliminaries

In this section we collect some notation, introduce the functionals we are going to study, and prove some preliminary results.

2.1 Notation

The present subsection is devoted to the notation we employ throughout.

• (a) $m\ge 1$ and $n\ge 2$ are fixed positive integers; we set ${\mathbb {R}}^m_0:={\mathbb {R}}^m\setminus \{0\}$;

• (b) ${\mathbb {S}}^{n-1}{:}{=}\{\nu =(\nu _1,\ldots ,\nu _n)\in {\mathbb {R}}^n :\nu _1^2+\cdots +\nu _n^2=1\}$ and ${\widehat{{\mathbb {S}}}}^{n-1}_\pm {:}{=}\{\nu \in {\mathbb {S}}^{n-1}:\pm \nu _{i(\nu )}>0\}$, where $i(\nu ){:}{=}\max \{i\in \{1,\ldots ,n\}:\nu _i\ne 0\}$;

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

• (d) ${\mathcal {A}}$ denotes the collection of all open and bounded subsets of ${\mathbb {R}}^n$ with Lipschitz boundary. If $A,B \in {\mathcal {A}}$ by $A \subset \subset B$ we mean that A is relatively compact in B;

• (e) Q denotes the open unit cube in ${\mathbb {R}}^n$ with sides parallel to the coordinate axis, centred at the origin; for $x\in {\mathbb {R}}^n$ and $r>0$ we set $Q_r(x):= rQ+x$. Moreover, $Q'$ denotes the open unit cube in ${\mathbb {R}}^{n-1}$ with sides parallel to the coordinate axis, centred at the origin, for every $r>0$ we set $Q_r'{:}{=}r Q'$;

• (f) for every $\nu \in {{\mathbb {S}}}^{n-1}$ let $R_\nu$ denote an orthogonal $(n\times n)$-matrix such that $R_\nu e_n=\nu$; we also assume that $R_{-\nu }Q=R_\nu Q$ for every $\nu \in {\mathbb {S}}^{n-1}$, $R_\nu \in {\mathbb {Q}}^{n\times n}$ if $\nu \in {\mathbb {S}}^{n-1}\cap {\mathbb {Q}}^n$, and that the restrictions of the map $\nu \mapsto R_\nu$ to ${\widehat{{{\mathbb {S}}}}}_{\pm }^{n-1}$ are continuous. For an explicit example of a map $\nu \mapsto R_\nu$ satisfying all these properties we refer the reader, e.g., to [27, Example A.1];

• (g) for $x\in {\mathbb {R}}^n$, $r>0$, and $\nu \in {\mathbb {S}}^{n-1}$, we define $Q^\nu _r(x):=R_\nu Q_r(0)+x$.

• (h) for $\xi \in {\mathbb {R}}^{m \times n}$ we let $u_\xi$ be the linear function whose gradient is equal to $\xi$ ; i.e., $u_\xi (x):=\xi x$, for every $x\in {\mathbb {R}}^n$;

• (i) for $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}^m_0$, and $\nu \in {{\mathbb {S}}}^{n-1}$ we denote with $u_{x,\zeta }^{\nu }$ the piecewise constant function taking values $0,\zeta$ and jumping across the hyperplane $\Pi ^\nu (x):=\{y\in {\mathbb {R}}^n :(y-x) \cdot \nu =0\}$ ; i.e.,

  $$\begin{aligned} u^{\nu }_{x,\zeta }(y):={\left\{ \begin{array}{ll} \zeta &{} \text {if }\; (y-x) \cdot \nu \ge 0, \\ 0 &{} \text {if }\; (y-x) \cdot \nu < 0, \end{array}\right. } \end{aligned}$$

  when $\zeta =e_1$ we simply write $u_{x}^\nu$ in place of $u^{\nu }_{x,e_1}$;

• (j) let $\textrm{u} \in C^1({\mathbb {R}})$, $\textrm{v}\in C^1({\mathbb {R}})$, with $0\le \textrm{v}\le 1$, be one-dimensional functions satisfying the following two properties:

  • i. $\textrm{v}\textrm{u}'\equiv 0$ in ${\mathbb {R}}$;

  • ii. $(\textrm{u}(t),\textrm{v}(t))=(\chi _{(0,+\infty )}(t),1)$ for $|t|>1$.

• (k) for $x\in {\mathbb {R}}^n$ and $\nu \in {{\mathbb {S}}}^{n-1}$ we set

  $$\begin{aligned} {\bar{u}}^\nu _{x} (y):= \textrm{u}((y-x) \cdot \nu ) e_1, \quad {\bar{v}}^\nu _x(y):= \textrm{v}((y-x) \cdot \nu ); \end{aligned}$$

• (l) for $x\in {\mathbb {R}}^n$, $\nu \in {\mathbb {S}}^{n-1}$, $\zeta \in {\mathbb {R}}^m_0$ and $\varepsilon >0$ we set

  $$\begin{aligned} {\bar{u}}^\nu _{x,\zeta ,\varepsilon }(y){:}{=}\textrm{u}\big (\tfrac{1}{\varepsilon }(y-x)\cdot \nu )\zeta ,\quad {\bar{v}}^\nu _{x,\varepsilon }(y){:}{=}\textrm{v}\big (\tfrac{1}{\varepsilon }(y-x)\cdot \nu ). \end{aligned}$$

  When $\zeta =e_1$ we simply write $\bar{u}_{x,\varepsilon }^\nu$ in place of $\bar{u}_{x,e_1,\varepsilon }^\nu$. We notice that in particular, $\bar{u}_{x,1}^\nu =\bar{u}_x^\nu$, $\bar{v}_{x,1}^\nu =\bar{v}_x^\nu$;

• (m) for a given topological space X, ${\mathcal {B}}(X)$ denotes the Borel $\sigma$- algebra on X. If $X={\mathbb {R}}^d$, with $d\in {\mathbb {N}}$, $d\ge 1$ we simply write ${\mathcal {B}}^d$ in place of ${\mathcal {B}}({\mathbb {R}}^d)$. For $d=1$ we write ${\mathcal {B}}$.

For every ${\mathcal {L}}^n$-measurable set $A\subset {\mathbb {R}}^n$ we define $L^0(A;{\mathbb {R}}^m)$ as the space of all ${\mathbb {R}}^m$-valued Lebesgue measurable functions. We endow $L^0(A;{\mathbb {R}}^m)$ with the topology of convergence in measure on bounded subsets of A and recall that this topology is both metrisable and separable.

Let $A\in {\mathcal {A}}$; in this paper we deal with the functional space $SBV(A;{\mathbb {R}}^m)$ (resp. $GSBV(A;{\mathbb {R}}^m)$) of special functions of bounded variation (resp. of generalised special functions of bounded variation) on A, for which we refer the reader to the monograph [6]. Here we only recall that if $u\in SBV(A;{\mathbb {R}}^m)$ then its distributional derivative can be represented as

$$\begin{aligned} Du(B)=\int _B \nabla u \, \textrm{d}x+\int _{B\cap S_u}[u]\otimes \nu _u \, \textrm{d}{\mathcal {H}}^{n-1}, \end{aligned}$$

(2.1)

for every $B \in {\mathcal {B}}^n$. In (2.1) $\nabla u$ denotes the approximate gradient of u (which makes sense also for $u\in GSBV$), $S_u$ the set of approximate discontinuity points of u, $[u]:=u^+-u^-$ where $u^\pm$ are the one-sided approximate limit points of u at $S_u$, and $\nu _u$ is the measure theoretic normal to $S_u$.

Let $p>1$; we also consider

$$\begin{aligned} SBV^{p}(A;{\mathbb {R}}^m):= \{u \in SBV(A;{\mathbb {R}}^m):\nabla u\in L^{p}(A;{\mathbb {R}}^{m\times n}) \text { and } {\mathcal {H}}^{n-1}(S_{u})<+\infty \}, \end{aligned}$$

and

$$\begin{aligned} GSBV^{p}(A;{\mathbb {R}}^m):= \{u \in GSBV(A;{\mathbb {R}}^m):\nabla u\in L^{p}(A;{\mathbb {R}}^{m\times n}) \text { and } {\mathcal {H}}^{n-1}(S_{u})<+\infty \}. \end{aligned}$$

We recall that $GSBV^p(A;{\mathbb {R}}^m)$ is a vector space; moreover, if $u\in GSBV^p(A;{\mathbb {R}}^m)$ then we have that $\phi (u) \in SBV^p(A;{\mathbb {R}}^m)\cap L^\infty (A;{\mathbb {R}}^m)$, for every $\phi \in C^1_c({\mathbb {R}}^m;{\mathbb {R}}^m)$ (see [33]).

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

2.2 Setting of the problem

Let $p \in (1,+\infty )$; let $c_1,c_2, c_3, c_4, L_1, L_2$ be given constants such that $0<c_1\le c_2<+\infty$, $0<c_3\le c_4<+\infty$, $0<L_1,L_2<+\infty$. Let ${\mathcal {F}}:={\mathcal {F}}(p, c_1,c_2,L_1)$ denote the collection of all functions $f:{\mathbb {R}}^n\times {\mathbb {R}}^{m\times n}\rightarrow [0,+\infty )$ satisfying the following conditions:

(f1):

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

(f2):

(lower bound) for every $x \in {\mathbb {R}}^n$ and every $\xi \in {\mathbb {R}}^{m\times n}$

$$\begin{aligned} c_1 |\xi |^p \le f(x,\xi )\,; \end{aligned}$$

(f3):

(upper bound) for every $x \in {\mathbb {R}}^n$ and every $\xi \in {\mathbb {R}}^{m\times n}$

$$\begin{aligned} f(x,\xi ) \le c_2|\xi |^p\,; \end{aligned}$$

(f4):

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

$$\begin{aligned} |f(x,\xi _1)-f(x,\xi _2)|\le & {} L_1\big (1+|\xi _1|^{p-1}+|\xi _2|^{p-1}\big )|\xi _1-\xi _2|, \end{aligned}$$

for every $\xi _1$, $\xi _2 \in {\mathbb {R}}^{m\times n}$;

Moreover, ${\mathcal {G}}:={\mathcal {G}}(p, c_3,c_4,L_2)$ denotes the collection of all functions $g:{\mathbb {R}}^n\times {\mathbb {R}}\times {\mathbb {R}}^n\rightarrow [0,+\infty )$ satisfying the following conditions:

(g1):

(measurability) g is Borel measurable on ${\mathbb {R}}^n\times {\mathbb {R}}\times {\mathbb {R}}^n$;

(g2):

(lower bound) for every $x \in {\mathbb {R}}^n$, every $v\in {\mathbb {R}}$, and every $w \in {\mathbb {R}}^{n}$

$$\begin{aligned} c_3\big ( |1-v|^p +|w|^p\big ) \le g(x,v,w)\,; \end{aligned}$$

(g3):

(upper bound) for every $x \in {\mathbb {R}}^n$, every $v\in {\mathbb {R}}$, and every $w \in {\mathbb {R}}^{n}$

$$\begin{aligned} g(x,v,w) \le c_4( |1-v|^p +|w|^p)\,. \end{aligned}$$

(g4):

(continuity in v and w) for every $x \in {\mathbb {R}}^n$ we have

$$\begin{aligned} |g(x,v_1,w_1)-g(x,v_2,w_2)|\le & {} L_2\Big ( \big (1+|v_1|^{p-1}+|v_2|^{p-1}\big )|v_1-v_2|\\{} & {} +\big (1+|w_1|^{p-1}+|w_2|^{p-1}\big )|w_1-w_2|\Big ) \end{aligned}$$

for every $v_1$, $v_2\in {\mathbb {R}}$ and every $w_1$, $w_2 \in {\mathbb {R}}^{n}$;

(g5):

(monotonicity in v) for every $x\in {\mathbb {R}}^n$ and every $w\in {\mathbb {R}}^n$, $g(x,\cdot \,,w)$ is decreasing on $(-\infty , 1)$ and increasing on $[1,+\infty )$;

(g6):

(minimum in w) for every $x\in {\mathbb {R}}^n$ and every $v\in {\mathbb {R}}$ it holds

$$\begin{aligned} g(x,v,0)\le g(x,v,w) \end{aligned}$$

for every $w\in {\mathbb {R}}^n$.

Remark 2.1

Let $x \in {\mathbb {R}}^n$; we notice that gathering (f2) and (f3) readily implies that

$$\begin{aligned} f(x,\xi ) = 0\; \text { if and only if }\; \xi =0. \end{aligned}$$

(2.2)

Moreover, from (g2) and (g3) we deduce that

$$\begin{aligned} g(x,v,w)=0\; \text { if and only if }\; (v,w)=(1,0). \end{aligned}$$

(2.3)

Let $\psi :{\mathbb {R}}\rightarrow [0,+\infty )$ be continuous, decreasing on $(-\infty ,0]$, increasing on $(0,+\infty )$, such that $\psi (1)=1$, and $\psi (v)=0$ iff $v=0$.

For $k\in {\mathbb {N}}$ let $(f_k) \subset {\mathcal {F}}$ and $(g_k)\subset {\mathcal {G}}$ and let $(\varepsilon _k)$ be a decreasing sequence of strictly positive real numbers converging to zero, as $k \rightarrow +\infty$.

We consider the sequence of elliptic functionals ${\mathscr {F}}_k :L^0({\mathbb {R}}^n;{\mathbb {R}}^m) \times L^0({\mathbb {R}}^n) \times {\mathcal {A}}\longrightarrow [0,+\infty ]$ defined by

$$\begin{aligned} {\mathscr {F}}_k(u,v, A){:}{=}{\left\{ \begin{array}{ll} \displaystyle \int _A \psi (v)\,f_k(x,\nabla u)\, \textrm{d}x&{} \\ \quad +\frac{1}{\varepsilon _k}\int _A g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x&{}\quad \text {if}\ (u,v)\in W^{1,p}(A;{\mathbb {R}}^m)\times W^{1,p}(A)\, ,\\ {} &{}\quad 0\le v\le 1 \, ,\\ +\infty &{}\quad \text {otherwise}\,. \end{array}\right. } \end{aligned}$$

(2.4)

Remark 2.2

Assumptions (f2)–(f3) and (g2)–(g3) imply that for every $A\in {\mathcal {A}}$ and every $(u,v)\in W^{1,p}(A;{\mathbb {R}}^m)\times W^{1,p}(A)$, $0\le v\le 1$ it holds

$$\begin{aligned} \begin{aligned}&c_1\int _A \psi (v)|\nabla u|^p\, \textrm{d}x+c_2\int _A\bigg (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\bigg )\, \textrm{d}x\le {\mathscr {F}}_k(u,v,A)\\&\quad \le c_3\int _A \psi (v)|\nabla u|^p\, \textrm{d}x+c_4\int _A \bigg (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\bigg )\, \textrm{d}x; \end{aligned} \end{aligned}$$

(2.5)

that is, up to a multiplicative constant, the functionals ${\mathscr {F}}_k$ are bounded from below and from above by the Ambrosio-Tortorelli functionals

$$\begin{aligned} AT_{k}(u,v){:}{=}\int _A \psi (v)\,|\nabla u|^p\, \textrm{d}x+\int _A\bigg (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\bigg )\, \textrm{d}x. \end{aligned}$$

Remark 2.3

For later use, we notice that the assumptions on $\psi$, $f_k$ and $g_k$ ensure that for every $A\in {\mathcal {A}}$ the functionals ${\mathscr {F}}_k(\,\cdot ,\,\cdot ,A)$ are continuous in the strong $W^{1,p}(A;{\mathbb {R}}^m)\times W^{1,p}(A)$ topology.

For every $A\in {\mathcal {A}}$, $u\in L^0({\mathbb {R}}^n;{\mathbb {R}}^m)$ and $v \in L^0({\mathbb {R}}^n)$ it is also convenient to write

$$\begin{aligned} {\mathscr {F}}_k(u,v,A)={\mathscr {F}}_k^b(u,v,A)+{\mathscr {F}}_k^s(v,A), \end{aligned}$$

where ${\mathscr {F}}_k^b :L^0({\mathbb {R}}^n;{\mathbb {R}}^m) \times L^0({\mathbb {R}}^n) \times {\mathcal {A}}\longrightarrow [0,+\infty ]$ and ${\mathscr {F}}_k^s :L^0({\mathbb {R}}^n) \times {\mathcal {A}}\longrightarrow [0,+\infty ]$ denote the bulk and the surface part of ${\mathscr {F}}_k$, respectively; i.e.,

$$\begin{aligned} {\mathscr {F}}_k^b(u,v,A){:}{=}{\left\{ \begin{array}{ll} \displaystyle \int _A \psi (v)f_k(x,\nabla u)\, \textrm{d}x&{}\text {if}\ (u,v)\in W^{1,p}(A;{\mathbb {R}}^m)\times W^{1,p}(A)\,,\ 0\le v\le 1\,,\\ +\infty &{}\text {otherwise}\, \end{array}\right. } \end{aligned}$$

(2.6)

and

$$\begin{aligned} {\mathscr {F}}_k^s(v,A){:}{=}{\left\{ \begin{array}{ll} \displaystyle \frac{1}{\varepsilon _k}\int _A g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x&{}\text {if}\ v\in W^{1,p}(A)\,,\ 0\le v\le 1\,,\\ +\infty &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

(2.7)

For $\rho > 2\varepsilon _k$, $x\in {\mathbb {R}}^n$, $\xi \in {\mathbb {R}}^{m \times n}$, and $\nu \in {{\mathbb {S}}}^{n-1}$ we consider the following two minimisation problems

$$\begin{aligned} {\textbf{m}}_{k}^b(u_\xi ,Q_\rho (x)){:}{=}\inf \{{\mathscr {F}}_{k}^b(u,1,Q_\rho (x)):u\in W^{1,p}(Q_\rho (x);{\mathbb {R}}^m)\, ,\ u=u_\xi \ \text {near}\ \partial Q_\rho (x)\} \end{aligned}$$

(2.8)

and

$$\begin{aligned} {\textbf{m}}_{k, \textrm{N}}^s(u_{x}^\nu ,Q_\rho ^\nu (x)){:}{=}\inf \{{\mathscr {F}}_k^s(v,Q_\rho ^\nu (x)):v\in {\mathscr {A}}_{\varepsilon _k,\rho }(x,\nu )\}, \end{aligned}$$

(2.9)

where

$$\begin{aligned} {\mathscr {A}}_{\varepsilon _k,\rho }(x,\nu ):= & {} \big \{v\in W^{1,p}(Q_\rho ^\nu (x)),\; 0\le v\le 1 :\, \exists \, u\in W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)\nonumber \\ \text { with } v \,\nabla u= & {} 0\ \text {a.e. in}\ Q_\rho ^\nu (x)\nonumber \\ \text {and}\ (u,v)= & {} (u_{x}^\nu ,1)\ \text {near}\ \partial Q_\rho ^\nu (x)\ \text {in}\ \{|(y-x)\cdot \nu |>\varepsilon _k\} \big \}. \end{aligned}$$

(2.10)

We observe that in (2.8) by “$u=u_\xi \ \text {near}\ \partial Q_\rho (x)$” we mean that the boundary datum is attained in a neighbourhood of $\partial Q_\rho (x)$. Whereas in (2.10) the boundary datum is prescribed only in

$$\begin{aligned} U\cap \{y\in {\mathbb {R}}^n :|(y-x)\cdot \nu |>\varepsilon _k\}, \end{aligned}$$

for some neighbourhood U of $\partial Q_\rho ^\nu (x)$.

Remark 2.4

Clearly, the class of competitors ${\mathscr {A}}_{\varepsilon _k,\rho }(x,\nu )$ is nonempty. Indeed, the pair $(\bar{u}_{x,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu )$ defined as in (l), with $\varepsilon =\varepsilon _k$, satisfies both

$$\begin{aligned} (\bar{u}_{x,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu )=(u_x^\nu ,1)\; \text { in }\; \{|(y-x)\cdot \nu |>\varepsilon _k\}\quad \text {and}\quad {\bar{v}}_{x,\varepsilon _k}^\nu \nabla \bar{u}_{x,\varepsilon _k}^\nu \equiv 0. \end{aligned}$$

Thus the restriction of ${\bar{v}}_{x,\varepsilon _k}^\nu$ to $Q_\rho ^\nu (x)$ belongs to ${\mathscr {A}}_{\varepsilon _k,\rho }(x,\nu )$.

In view of (2.2) and of the properties satisfied by $\psi$, we also observe that in (2.10) the constraint

$$\begin{aligned} v\, \nabla u=0 \; \text { a.e. in }\; Q_\rho ^\nu (x) \end{aligned}$$

can be equivalently replaced by

$$\begin{aligned} {\mathscr {F}}_k^b(u,v,Q_\rho ^\nu (x))=0. \end{aligned}$$

Hence, in particular, for every $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}^m_0$, $\nu \in {\mathbb {S}}^{n-1}$, $\rho >2\varepsilon _k$, and $k\in {\mathbb {N}}$ we have

$$\begin{aligned} {\mathscr {F}}_k(\bar{u}_{x,\zeta ,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))={\mathscr {F}}_k^s(\bar{v}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)). \end{aligned}$$

(2.11)

Finally, for every $x \in {\mathbb {R}}^{n}$ and every $\xi \in {\mathbb {R}}^{m\times n}$ we define

$$\begin{aligned} f'(x,\xi ):= & {} \limsup _{\rho \rightarrow 0}\frac{1}{\rho ^n}\liminf _{k\rightarrow +\infty }{\textbf{m}}_{k}^b(u_\xi ,Q_\rho (x))\,, \end{aligned}$$

(2.12)

$$\begin{aligned} f''(x,\xi ):= & {} \limsup _{\rho \rightarrow 0}\frac{1}{\rho ^n}\limsup _{k\rightarrow +\infty }{\textbf{m}}_{k}^b(u_\xi ,Q_\rho (x)), \end{aligned}$$

(2.13)

while for every $x \in {\mathbb {R}}^n$ and every $\nu \in {\mathbb {S}}^{n-1}$ we set

$$\begin{aligned} g'(x,\nu ):= & {} \limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\liminf _{k\rightarrow +\infty }{\textbf{m}}_{k, \textrm{N}}^s(u_{x}^\nu ,Q^\nu _\rho (x))\,, \end{aligned}$$

(2.14)

$$\begin{aligned} g''(x,\nu ):= & {} \limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\limsup _{k\rightarrow +\infty }{\textbf{m}}_{k, \textrm{N}}^s(u_{x}^\nu ,Q^\nu _\rho (x)). \end{aligned}$$

(2.15)

2.3 Equivalent formulas for $g'$ and $g''$

For later use, in Proposition 2.6 below, we prove that $g'$ and $g''$ can be equivalently defined by replacing the boundary conditions in (2.14)–(2.15) with suitable Dirichlet boundary conditions on the whole boundary of $Q^\nu _\rho (x)$. More precisely, we consider the minimum values defined as follows: For every $x\in {\mathbb {R}}^n$, $\nu \in {{\mathbb {S}}}^{n-1}$, and $A\in {\mathcal {A}}$ we set

$$\begin{aligned} {\textbf{m}}_k^s({\bar{u}}_{x,\varepsilon _k}^\nu ,A){:}{=}\inf \{{\mathscr {F}}_k^s(v,A):v\in {\mathscr {A}}(\bar{u}_{x,\varepsilon _k}^\nu ,A)\}, \end{aligned}$$

(2.16)

where

$$\begin{aligned} {\mathscr {A}}(\bar{u}_{x,\varepsilon _k}^\nu ,A):= & {} \{v\in W^{1,p}(A),\, 0\le v\le 1:\exists \, u\in W^{1,p}(A;{\mathbb {R}}^m)\ \nonumber \\ \text {with}\ v\nabla u= & {} 0\ \text {a.e. in}\ A\nonumber \\ \text {and}\ (u,v)= & {} (\bar{u}_{x,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu )\ \text {near}\ \partial A\}, \end{aligned}$$

(2.17)

with $(\bar{u}_{x,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu )$ as in (l).

Remark 2.5

Let $A\in {\mathcal {A}}$ be such that $A=A'\times I$ with $A'\subset {\mathbb {R}}^{n-1}$ open and bounded and $I\subset {\mathbb {R}}$ open interval. Let $\nu \in {\mathbb {S}}^{n-1}$ and set $A_{\nu }:=R_\nu A$, with $R_\nu$ as in (f). For every $k \in {\mathbb {N}}$ we have

$$\begin{aligned} \int _{A_\nu }\Bigg (\frac{(1-\bar{v}_{x,\varepsilon _k}^\nu (y))^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla \bar{v}_{x,\varepsilon _k}^\nu (y)|^p\Bigg )\, \textrm{d}y\le & {} \int _{A'}\int _ {\mathbb {R}}\big ((1- \textrm{v}(t))^p+|\textrm{v}'(t)|^p\big )\, \textrm{d}t\, \textrm{d}y'\nonumber \\\le & {} C_{\textrm{v}}{\mathcal {L}}^{n-1}(A'), \end{aligned}$$

(2.18)

where

$$\begin{aligned} C_{\textrm{v}}:=\int _{\mathbb {R}}\big ((1- \textrm{v}(t))^p+|\textrm{v}'(t)|^p\big )\, \textrm{d}t=\int _{-1}^1\big ((1- \textrm{v}(t))^p+|\textrm{v}'(t)|^p\big )\, \textrm{d}t<+\infty . \end{aligned}$$

In particular from (g3), Remark 2.4, and (2.18) we infer

$$\begin{aligned} {\textbf{m}}_k^s({\bar{u}}_{x,\varepsilon _k}^\nu ,A_\nu )\le {\mathscr {F}}_k^s(\bar{v}_{x,\varepsilon _k}^\nu ,A_\nu )\le c_4C_{\textrm{v}}{\mathcal {L}}^{n-1}(A'). \end{aligned}$$

(2.19)

We are now in a position to prove the following equivalent formulation for $g'$ and $g''$. We observe that the most delicate part in the proof of this result is to show that a suitable Dirichlet boundary datum can be prescribed on the whole $\partial Q_\rho ^\nu (x)$ while keeping the nonconvex constraint $v\,\nabla u=0$ a.e. in $Q_\rho ^\nu (x)$.

Proposition 2.6

For every $\rho >2\varepsilon _k$, $x\in {\mathbb {R}}^n$, and $\nu \in {\mathbb {S}}^{n-1}$ let ${\textbf{m}}_k^s(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$ be as in (2.16) with $A=Q_\rho ^\nu (x)$. Then we have

$$\begin{aligned} g'(x,\nu )&=\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\liminf _{k\rightarrow +\infty }{\textbf{m}}_k^s(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\,,\\ g''(x,\nu )&=\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\limsup _{k\rightarrow +\infty }{\textbf{m}}_k^s(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\,, \end{aligned}$$

where $g'$, $g''$ are as in (2.14) and (2.15), respectively.

Proof

We only prove the equality for $g'$, the proof of the equality for $g''$ being analogous. Let $x\in {\mathbb {R}}^n$ and $\nu \in {\mathbb {S}}^{n-1}$ be fixed and set

$$\begin{aligned} \underline{g}'(x,\nu ){:}{=}\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\liminf _{k\rightarrow +\infty }{\textbf{m}}_k^s(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)). \end{aligned}$$

In view of Remark 2.4 we readily have

$$\begin{aligned} g'(x,\nu )\le \underline{g}'(x,\nu ), \end{aligned}$$

(2.20)

for every $x\in {\mathbb {R}}^n$ and every $\nu \in {\mathbb {S}}^{n-1}$. To prove the opposite inequality, let $\rho >0$ and $\alpha \in (0,1)$ be fixed and let ${\bar{k}}\in {\mathbb {N}}$ be such that $\varepsilon _k<\frac{\alpha \rho }{2}$, for every $k\ge {\bar{k}}$. Let moreover $v_k\in {\mathscr {A}}_{\varepsilon _k,\rho }(x,\nu )$ be such that

$$\begin{aligned} {\mathscr {F}}_k^s(v_k,Q_{\rho }^\nu (x))\le {\textbf{m}}^s_{k,\mathrm N}(u_{x}^\nu ,Q_\rho ^\nu (x))+\rho ^n. \end{aligned}$$

(2.21)

Then, there exists $u_k\in W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)$ satisfying $v_k \nabla u_k=0$ a.e. in $Q_{\rho }^\nu (x)$ and

$$\begin{aligned} (u_k,v_k)=(u_{x}^\nu ,1)=({\bar{u}}_{x,\varepsilon _k}^\nu ,{\bar{v}}_{x,\varepsilon _k}^\nu )\quad \text {in}\quad U_k\cap \{|(y-x)\cdot \nu |>\varepsilon _k\}, \end{aligned}$$

(2.22)

where $U_k$ is a neighbourhood of $\partial Q_\rho ^\nu (x)$.

Starting from $(u_k,v_k) \in W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m) \times W^{1,p}(Q_\rho ^\nu (x))$ we are now going to define a new pair $({\widetilde{u}}_k, {\widetilde{v}}_k) \in W^{1,p}(Q_{(1+\alpha )\rho }^\nu (x);{\mathbb {R}}^m) \times W^{1,p}(Q_{(1+\alpha )\rho }^\nu (x))$ with ${\widetilde{v}}_k\in {\mathscr {A}}(\bar{u}_{x,\varepsilon _k}^\nu ,Q_{(1+\alpha )\rho }^\nu (x))$. Moreover, we will do this in a way such that ${\mathscr {F}}_k^s({\widetilde{v}}_k,Q_{(1+\alpha )\rho }^\nu (x))$ will be bounded from above by ${\mathscr {F}}_k^s(v_k,Q_\rho ^\nu (x))$ and hence, thanks to (2.21), by ${\textbf{m}}^s_{k,\mathrm N}(u_{x}^\nu ,Q_\rho ^\nu (x))$.

Fig. 1

The cubes $Q_{\rho -\beta _k}^\nu (x)$, $Q_\rho ^\nu (x)$, $Q_{(1+\alpha )\rho }^\nu (x)$ and in gray the sets $R_k$ (dark gray) and $\{d_k<1\}$ (light gray)

To this end, let $\beta _k\in (0,2\varepsilon _k)\subset (0,\rho )$ be such that $Q_\rho ^\nu (x)\setminus {\overline{Q}}_{\rho -\beta _k}^\nu (x)\subset U_k$ and set

$$\begin{aligned} R_k{:}{=}R_\nu \Big (\big (Q_{\rho }' \setminus \overline{Q'}_{\!\!\rho -\beta _k}\big )\times (-\varepsilon _k,\varepsilon _k)\Big )+x, \end{aligned}$$

where $R_\nu$ is as in (f). By construction we have that

$$\begin{aligned} \big (Q_{\rho }^\nu (x)\setminus {\overline{Q}}_{\rho -\beta _k}^\nu (x)\big )\setminus {\overline{R}}_k\subset U_k\cap \{|(y-x)\cdot \nu |>\varepsilon _k\}\, \end{aligned}$$

(2.23)

(see Fig. 1). Now let $\varphi _k \in C_c^1(Q_{\rho }')$ be a cut-off function between $Q'_{\rho -\beta _k}$ and $Q'_{\rho }$ ; i.e., $0\le \varphi _k \le 1$, and $\varphi _k\equiv 1$ on $Q_{\rho -\beta _k}'$. Eventually, for $y=(y',y_n)\in Q_{(1+\alpha )\rho }^\nu (x)$ we define the pair $({\widetilde{u}}_k,{\widetilde{v}}_k)$ by setting

$$\begin{aligned} {\widetilde{u}}_k(y):=\varphi _k\big ((R_\nu ^T(y-x))'\big )u_k(y)+\Big (1-\varphi _k\big ((R_\nu ^T(y-x))'\big )\Big ){\bar{u}}_{x,\varepsilon _k}^\nu (y) \end{aligned}$$

and

$$\begin{aligned} {\widetilde{v}}_k(y):={\left\{ \begin{array}{ll} \min \{v_k(y),d_k(y)\}&{}\text {in }\; Q_{\rho }^\nu (x),\\ \min \{{\bar{v}}_{x,\varepsilon _k}^\nu (y),d_k(y)\}&{}\text {in }\; Q_{(1+\alpha )\rho }^\nu (x)\setminus {\overline{Q}}_{\rho }^\nu (x), \end{array}\right. } \end{aligned}$$

where $d_k(y){:}{=}\frac{1}{\varepsilon _k}{{\,\textrm{dist}\,}}(y,R_k)$.

Clearly ${\widetilde{u}}_k \in W^{1,p}(Q_{(1+\alpha )\rho }^\nu (x);{\mathbb {R}}^m)$, moreover, thanks to (2.22) the function $\widetilde{v}_k$ belongs to $W^{1,p}(Q_{(1+\alpha )\rho }^\nu (x))$. Furthermore, it holds $\widetilde{v}_k\,\nabla \widetilde{u}_k=0$ a.e. in $Q_{(1+\alpha )\rho }^\nu (x)$. Indeed, $\widetilde{v}_k|\nabla \widetilde{u}_k|\le v_k|\nabla u_k|=0$ a.e. in $Q_{\rho -\beta _k}^\nu (x)$; similarly in $Q_{(1+\alpha )\rho }^\nu (x)\setminus {\overline{Q}}_{\rho }^\nu (x)$ there holds $\widetilde{v}_k|\nabla \widetilde{u}_k|\le {\bar{v}}_{x,\varepsilon _k}^\nu |\nabla {\bar{u}}_{x,\varepsilon _k}^\nu |=0$. Finally, thanks to (2.22) and (2.23), in $Q_{\rho }^\nu (x)\setminus {\overline{Q}}_{\rho -\beta _k}^\nu (x)\setminus {\overline{R}}_k$ we have $\nabla \widetilde{u}_k=\nabla \bar{u}_{x,\varepsilon _k}^\nu =0$, while, by definition, $\widetilde{v}_k=0$ in $R_k$.

Since moreover $d_k>1$ in $Q_{(1+\alpha )\rho }^\nu (x)\setminus {\overline{Q}}_{\rho +2\varepsilon _k}^\nu (x)$, we also have $({\widetilde{u}}_k,{\widetilde{v}}_k)=({\bar{u}}_{x,\varepsilon _k}^\nu ,{\bar{v}}_{x,\varepsilon _k}^\nu )$ in $Q_{(1+\alpha )\rho }^\nu (x)\setminus {\overline{Q}}_{\rho +2\varepsilon _k}^\nu (x)$, and hence $\widetilde{v}_k\in {\mathscr {A}}(\bar{u}_{x,\varepsilon _k}^\nu ,Q_{(1+\alpha )\rho }^\nu (x))$ for $k\ge \bar{k}$.

Then, to conclude it only remains to estimate ${\mathscr {F}}_k^s({\widetilde{v}}_k,Q_{(1+\alpha )\rho }^\nu (x))$. To this end, we start noticing that $\widetilde{v}_k=v_k$ in $Q_\rho ^\nu (x)\cap \{d_k>1\}$, $\widetilde{v}_k=\bar{v}_{x,\varepsilon _k}^\nu$ in $(Q_{(1+\alpha )\rho }^\nu (x)\setminus {\overline{Q}}_\rho ^\nu (x))\cap \{d_k>1\}$, while in $Q_\rho ^\nu (x)\cap \{d_k<1\}$ we have

$$\begin{aligned} \frac{1}{\varepsilon _k}g_k(y,\widetilde{v}_k,\varepsilon _k\nabla \widetilde{v}_k)&\le \frac{1}{\varepsilon _k}g_k(y,v_k,\varepsilon _k\nabla v_k)+\frac{1}{\varepsilon _k}g_k(x,d_k,\varepsilon _k\nabla d_k)\\&\le \frac{1}{\varepsilon _k}g_k(y,v_k,\varepsilon _k\nabla v_k)+\frac{2c_4}{\varepsilon _k}\,, \end{aligned}$$

where the last estimate follows from (g3). Similarly, in $(Q_{(1+\alpha )\rho }^\nu (x)\setminus {\overline{Q}}_\rho ^\nu (x))\cap \{d_k<1\}$ we have

$$\begin{aligned} \frac{1}{\varepsilon _k}g_k(y,\widetilde{v}_k,\varepsilon _k\nabla \widetilde{v}_k)\le \frac{1}{\varepsilon _k}g_k(y,\bar{v}_{x,\varepsilon _k}^\nu ,\varepsilon _k\nabla \bar{v}_{x,\varepsilon _k}^\nu )+\frac{2c_4}{\varepsilon _k}. \end{aligned}$$

Thus, from (2.3) and (2.19) we infer

$$\begin{aligned} \begin{aligned} {\mathscr {F}}_k^s({\widetilde{v}}_k,Q_{(1+\alpha )\rho }^\nu (x))&\le {\mathscr {F}}_k^s(v_k,Q_\rho ^\nu (x)) + {\mathscr {F}}_k^s({\bar{v}}_{x,\varepsilon _k}^\nu ,Q_{(1+\alpha )\rho }^\nu (x)\setminus {\overline{Q}}_{\rho }^\nu (x))\\&\quad + \frac{2c_4}{\varepsilon _k}{\mathcal {L}}^n(\{d_k<1\})\\&\le {\mathscr {F}}_k^s(v_k,Q_\rho ^\nu (x))+c_4C_\textrm{v}{\mathcal {L}}^{n-1}(Q'_{(1+\alpha )\rho }\setminus \overline{Q'_{\rho }})+C(\beta _k+\varepsilon _k)\rho ^{n-2}. \end{aligned} \end{aligned}$$

(2.24)

Then, thanks to (2.21), dividing both sides of (2.24) by $((1+\alpha )\rho )^{n-1}$, since $\beta _k<2\varepsilon _k$ we obtain

$$\begin{aligned} \frac{{\textbf{m}}_k^s({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_{(1+\alpha )\rho }^\nu (x))}{((1+\alpha )\rho )^{n-1}}\le & {} \frac{1}{(1+\alpha )^{n-1}}\bigg (\frac{ {\textbf{m}}_{k,\mathrm N}^s(u_{x}^\nu ,Q_{\rho }^\nu (x))}{\rho ^{n-1}}\nonumber \\{} & {} +\rho + c_4C_{\textrm{v}}((1+\alpha )^{n-1}-1)+\frac{C\varepsilon _k}{\rho }\bigg ). \end{aligned}$$

(2.25)

Eventually, from (2.25) passing first to the liminf as $k \rightarrow +\infty$ and then to the limsup as $\rho \rightarrow 0$ we deduce

$$\begin{aligned} (1+\alpha )^{n-1}\underline{g}'(x,\nu )\le g'(x,\nu )+c_4C_\textrm{v}((1+\alpha )^{n-1}-1), \end{aligned}$$

which, together with (2.20), thanks to the arbitrariness of $\alpha >0$ yields $\underline{g}'(x,\nu )= g'(x,\nu )$. $\square$

3 Statements of the main results

In this section we state the main results of this paper, namely, a $\Gamma$-convergence and integral representation result (Theorem 3.1), a converge result for some associated minimisation problems (Theorem 3.4), and a homogenisation result without periodicity assumptions (Theorem 3.5).

3.1 $\Gamma$-convergence

The following result asserts that, up to subsequences, the functionals ${\mathscr {F}}_k$ $\Gamma$-converge to an integral functional of free-discontinuity type. Furthermore, it provides asymptotic cell formulas for the volume and surface limit integrands. These asymptotic cell formulas show, in particular, that volume and surface term decouple in the limit.

Theorem 3.1

($\Gamma$-convergence) Let $(f_k) \subset {\mathcal {F}}$, $(g_k) \subset {\mathcal {G}}$ and let ${\mathscr {F}}_k$ be the functionals as in (2.4). Then there exists a subsequence, not relabelled, such that for every $A \in {\mathcal {A}}$ the functionals ${\mathscr {F}}_{k}(\cdot , \cdot , A)$ $\Gamma$-converge in $L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ to ${\mathscr {F}}(\cdot ,\cdot ,A)$ with ${\mathscr {F}}:L^0({\mathbb {R}}^n;{\mathbb {R}}^m) \times L^0({\mathbb {R}}^n) \times {\mathcal {A}}\longrightarrow [0,+\infty ]$ given by

$$\begin{aligned} {\mathscr {F}}(u, v, A){:}{=}{\left\{ \begin{array}{ll} \displaystyle \int _A f_\infty (x,\nabla u)\, \textrm{d}x+\int _{S_u \cap A} g_\infty (x,\nu _u)\, \textrm{d}{\mathcal {H}}^{n-1}&{}\text {if}\ u \in GSBV^p(A;{\mathbb {R}}^m)\,,\\ &{} v= 1\, \text {a.e. in } A\,,\\ +\infty &{}\text {otherwise}, \end{array}\right. } \end{aligned}$$

where $f_\infty :{\mathbb {R}}^n \times {\mathbb {R}}^{m \times n} \rightarrow [0,+\infty )$ and $g_\infty :{\mathbb {R}}^n \times {{\mathbb {S}}}^{n-1} \rightarrow [0,+\infty )$ are Borel functions. Moreover, it holds:

• i. for a.e. $x \in {\mathbb {R}}^n$ and for every $\xi \in {\mathbb {R}}^{m\times n}$

  $$\begin{aligned} f_\infty (x,\xi )= f'(x,\xi ) = f''(x,\xi )\,, \end{aligned}$$

  with $f'$, $f''$ as in (2.12) and (2.13), respectively;

• ii. for every $x\in {\mathbb {R}}^n$ and every $\nu \in {{\mathbb {S}}}^{n-1}$

  $$\begin{aligned} g_\infty (x,\nu )= g'(x,\nu ) = g''(x,\nu )\,, \end{aligned}$$

  with $g'$, $g''$ as in (2.14) and (2.15), respectively.

Remark 3.2

The choice of considering functionals ${\mathscr {F}}_{k}$ which are finite when the variable v satisfies the bounds $0\le v \le 1$ is a choice of convenience and it is not restrictive. Indeed, in view of (g5) and (g6) and of the properties of $\psi$, the functionals ${\mathscr {F}}_{k}$ decrease under the transformation $v \rightarrow \min \{\max \{v,0\},1\}$. Hence a $\Gamma$-convergence result for functionals ${\mathscr {F}}_k$ defined on functions v with values in ${\mathbb {R}}$ can be easily deduced from Theorem 3.1.

The proof of Theorem 3.1 will be achieved in four main steps which are addressed in Sects. 4, 5, 6, and 7, respectively. Firstly, we show that the functions $f', f'', g'$, and $g''$ satisfy a number of properties and, in particular, they are Borel measurable (see Proposition 4.1 and Proposition 4.5). In the second step, we prove the existence of a sequence $(k_j)$, with $k_j \rightarrow +\infty$ as $j\rightarrow +\infty$, such that for every $A\in {\mathcal {A}}$ the corresponding functionals ${\mathscr {F}}_{k_j}(\,\cdot \,,\, \cdot \,, A)$ $\Gamma$-converge to a free-discontinuity functional which is finite in $GSBV^p(A;{\mathbb {R}}^m)\times \{1\}$ and is of the form

$$\begin{aligned} \int _A {\hat{f}}(x,\nabla u)\, \textrm{d}x+\int _{S_u\cap A}{\hat{g}}(x,[u],\nu _u)\, \textrm{d}{\mathcal {H}}^{n-1}, \end{aligned}$$

(3.1)

for some Borel functions ${\hat{f}}$ and ${\hat{g}}$ (see Theorem 5.2).

In the third step we identify ${\hat{f}}$ by showing that it is equal both to $f'$ and $f''$ (see Proposition 6.1). Eventually, in the final step we identify ${\hat{g}}$ by proving that it coincides with both $g'$ and $g''$ (see Proposition 7.4). The representation result for ${\hat{g}}$ implies, in particular, that the surface term in (3.1) does not depend on [u].

The following result is an immediate consequence of Theorem 3.1 and of the Urysohn property of $\Gamma$-convergence [32, Proposition 8.3].

Corollary 3.3

Let $(f_k) \subset {\mathcal {F}}$, $(g_k) \subset {\mathcal {G}}$ and let ${\mathscr {F}}_k$ be the functionals as in (2.4). Let $f'$, $f''$ be as in (2.12) and (2.13), respectively, and $g'$, $g''$ be as in (2.14) and (2.15), respectively. Assume that

$$\begin{aligned} f'(x,\xi ) = f''(x,\xi )=:f_\infty (x,\xi ),\; \text { for a.e. }x \in {\mathbb {R}}^n\text { and for every }\xi \in {\mathbb {R}}^{m\times n} \end{aligned}$$

and

$$\begin{aligned} g'(x,\nu ) = g''(x,\nu )=:g_\infty (x,\nu ),\; \text { for every }x\in {\mathbb {R}}^n\text { and every }\nu \in {{\mathbb {S}}}^{n-1}, \end{aligned}$$

for some Borel functions $f_\infty :{\mathbb {R}}^n \times {\mathbb {R}}^{m\times n} \rightarrow [0,+\infty )$ and $g_\infty :{\mathbb {R}}^n \times {{\mathbb {S}}}^{n-1} \rightarrow [0,+\infty )$. Then, for every $A \in {\mathcal {A}}$ the functionals ${\mathscr {F}}_{k}(\cdot , \cdot , A)$ $\Gamma$-converge in $L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ to ${\mathscr {F}}(\cdot ,\cdot ,A)$ with ${\mathscr {F}}:L^0({\mathbb {R}}^n;{\mathbb {R}}^m) \times L^0({\mathbb {R}}^n) \times {\mathcal {A}}\longrightarrow [0,+\infty ]$ given by

$$\begin{aligned} {\mathscr {F}}(u, v, A){:}{=}{\left\{ \begin{array}{ll} \displaystyle \int _A f_\infty (x,\nabla u)\, \textrm{d}x+\int _{S_u \cap A} g_\infty (x,\nu _u)\, \textrm{d}{\mathcal {H}}^{n-1}&{}\text {if}\ u \in GSBV^p(A;{\mathbb {R}}^m)\,,\\ &{} v= 1\, \text { a.e. in } A\,,\\ +\infty &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

3.2 Convergence of minimisation problems

In view of Theorem 3.1 and Corollary 3.3 we are in a position to prove the following result on the convergence of certain minimisation problems associated with ${\mathscr {F}}_k$. Other minimisation problems can be treated similarly.

Theorem 3.4

(Convergence of minimisation problems) Assume that the hypotheses of Corollary 3.3 are satisfied. Let $q\ge 1$, let $h\in L^q(A;{\mathbb {R}}^m)$, and set

$$\begin{aligned} M_k:=\inf \bigg \{{\mathscr {F}}_{k}(u,v,A)+\int _A |u-h|^q\, \textrm{d}x:(u,v)\in L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)\bigg \}. \end{aligned}$$

Then $M_k \rightarrow M$ as $k \rightarrow +\infty$ where

$$\begin{aligned} M:=\min \bigg \{{\mathscr {F}}(u,1,A)+\int _A |u-h|^q\, \textrm{d}x:u\in GSBV^p(A;{\mathbb {R}}^m) \cap L^q(A;{\mathbb {R}}^m)\bigg \}. \end{aligned}$$

(3.2)

Moreover if $(u_k,v_k) \subset L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ is such that

$$\begin{aligned} \lim _{k \rightarrow +\infty } \bigg ({\mathscr {F}}_{k}(u_k,v_k,A)+\int _A |u_k-h|^q\, \textrm{d}x-M_k\bigg )=0, \end{aligned}$$

(3.3)

then $v_k \rightarrow 1$ in $L^p(A)$ and there exists a subsequence of $(u_k)$ which converges in $L^q(A;{\mathbb {R}}^m)$ to a solution of (3.2).

Proof

Let $(u_k,v_k) \subset L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ be as in (3.3). Then the convergence $v_k \rightarrow 1$ in $L^p(A)$ readily follows by (g2), whereas the lower bound in (2.5) on the functionals ${\mathscr {F}}_k$ together with [36, Lemma 4.1] give the existence of a subsequence $(u_{k_j})\subset (u_k)$ with $u_{k_j} \rightarrow u$ in $L^0(A;{\mathbb {R}}^m)$, for some $u\in GSBV^p(A;{\mathbb {R}}^m)$. Eventually, the convergence $M_k \rightarrow M$, the improved convergence $u_{k_j} \rightarrow u$ in $L^q(A;{\mathbb {R}}^m)$, and the fact that u is a solution to (3.2) follow arguing as in [34, Theorem 7.1], now invoking Corollary 3.3. $\square$

3.3 Homogenisation

In this subsection we prove a general homogenisation theorem without assuming any spatial periodicity of the integrands $f_k$ and $g_k$. This theorem will be crucial to prove the stochastic homogenisation result Theorem 8.4.

In order to state the homogenisation result, we need to introduce some further notation.

For $f\in {\mathcal {F}}$, $g\in {\mathcal {G}}$, $A\in {\mathcal {A}}$, and $u\in W^{1,p}(A;{\mathbb {R}}^m)$ set

$$\begin{aligned} {\mathscr {F}}^b(u,A):=\int _A f (x,\nabla u)\, \textrm{d}x, \end{aligned}$$

(3.4)

while for $v\in W^{1,p}(A)$ with $0\le v \le 1$ set

$$\begin{aligned} {\mathscr {F}}^s(v,A):=\int _A g (x,v,\nabla v)\, \textrm{d}x. \end{aligned}$$

(3.5)

Let $A\in {\mathcal {A}}$; for $x\in {\mathbb {R}}^n$ and $\xi \in {\mathbb {R}}^{m \times n}$ we define

$$\begin{aligned} {\textbf{m}}^b(u_\xi ,A){:}{=}\inf \{{\mathscr {F}}^b(u,A):u\in W^{1,p}(A;{\mathbb {R}}^m)\,,\ u=u_\xi \ \text {near}\ \partial A\}, \end{aligned}$$

(3.6)

while for $z\in {\mathbb {R}}^n$, $\nu \in {{\mathbb {S}}}^{n-1}$, and ${\bar{u}}^\nu _z$ as in (k) (with $x=z$) we set

$$\begin{aligned} {\textbf{m}}^s({\bar{u}}^\nu _z,A):=\inf \{{\mathscr {F}}^s(v,A):v\in {\mathscr {A}}({\bar{u}}^\nu _z,A)\}, \end{aligned}$$

(3.7)

where ${\mathscr {A}}({\bar{u}}^\nu _z,A)$ is as in (2.17), with ${\bar{u}}^\nu _{x,\varepsilon _k}$ and ${\bar{v}}^\nu _{x,\varepsilon _k}$ replaced, respectively, by ${\bar{u}}^\nu _z$ and ${\bar{v}}^\nu _z$ (that is, $x=z$ and $\varepsilon _k=1$).

We are now ready to state the homogenisation result; the latter corresponds to the choice

$$\begin{aligned} f_k(x,\xi ):= f\Big (\frac{x}{\varepsilon _k},\xi \Big )\quad \text {and} \quad g_{k}(x,v,w):=g\Big (\frac{x}{\varepsilon _k},v, w\Big ), \end{aligned}$$

(3.8)

with $f\in {\mathcal {F}}$ and $g\in {\mathcal {G}}$. We stress again that we will not assume any spatial periodicity of f and g.

Theorem 3.5

(Deterministic homogenisation) Let $f\in {\mathcal {F}}$ and $g\in {\mathcal {G}}$. Let also ${\textbf{m}}^b(u_\xi ,Q_r(rx))$ be as in (3.6) with $A=Q_r(rx)$ and ${\textbf{m}}^s({\bar{u}}_{rx}^\nu ,Q^\nu _r(rx))$ be as in (3.7) with $z=rx$ and $A=Q_r^\nu (rx)$. Assume that for every $x\in {\mathbb {R}}^n$, $\xi \in {\mathbb {R}}^{m\times n}$, $\nu \in {\mathbb {S}}^{n-1}$ the two following limits

$$\begin{aligned}{} & {} \lim _{r\rightarrow +\infty } \frac{{\textbf{m}}^b(u_\xi ,Q_r(rx))}{r^n}=:f_{\textrm{hom}}(\xi ), \end{aligned}$$

(3.9)

$$\begin{aligned}{} & {} \lim _{r\rightarrow +\infty } \frac{{\textbf{m}}^s(\bar{u}^\nu _{rx},Q^\nu _r(rx))}{r^{n-1}}=:g_{\textrm{hom}}(\nu ), \end{aligned}$$

(3.10)

exist and are independent of x. Then, for every $A\in {\mathcal {A}}$ the functionals ${\mathscr {F}}_k(\cdot ,\cdot , A)$ defined in (2.4) with $f_k$ and $g_k$ as in (3.8) $\Gamma$-converge in $L^0({\mathbb {R}}^n;{\mathbb {R}}^m) \times L^0({\mathbb {R}}^n)$ to the functional ${\mathscr {F}}_{\textrm{hom}}(\cdot ,\cdot ,A)$, with ${\mathscr {F}}_{\textrm{hom}}:L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)\times {\mathcal {A}}\longrightarrow [0,+\infty ]$ given by

$$\begin{aligned} {\mathscr {F}}_{\textrm{hom}}(u,v,A){:}{=}{\left\{ \begin{array}{ll} \displaystyle \int _A f_{\textrm{hom}}(\nabla u)\, \textrm{d}x+\int _{S_u \cap A} g_{\textrm{hom}}(\nu _u)\, \textrm{d}{\mathcal {H}}^{n-1}&{}\text {if}\ u \in GSBV^p(A;{\mathbb {R}}^m)\,,\\ &{} v= 1\, \text {a.e. in } A\,,\\ +\infty &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

Proof

Let $f'$, $f''$ be as in (2.12), (2.13), respectively, and $g'$, $g''$ be as in (2.14), (2.15), respectively. By virtue of Corollary 3.3 it suffices to show that

$$\begin{aligned} f_{\textrm{hom}}(\xi )=f'(x,\xi )=f''(x,\xi ),\quad g_{\textrm{hom}}(\nu )=g'(x,\nu )=g''(x,\nu ), \end{aligned}$$

(3.11)

for every $x\in {\mathbb {R}}^n$, $\xi \in {\mathbb {R}}^{m\times n}$, and $\nu \in {{\mathbb {S}}}^{n-1}$.

We start by proving the first two equalities in (3.11). To this end, fix $x\in {\mathbb {R}}^n$, $\xi \in {\mathbb {R}}^{m\times n}$, $\rho >0$ and $k\in {\mathbb {N}}$. Take $u\in W^{1,p}(Q_\rho (x);{\mathbb {R}}^m)$ and let $u_k\in W^{1,p}(Q_{\frac{\rho }{\varepsilon _k}}(\frac{x}{\varepsilon _k});{\mathbb {R}}^m)$ be defined as $u_k(y):=\frac{1}{\varepsilon _k}u(\varepsilon _ky)$. We notice that $u=u_\xi$ near $\partial Q_\rho (x)$ if and only if $u_k=u_\xi$ near $\partial Q_{\frac{\rho }{\varepsilon _k}}(\frac{x}{\varepsilon _k})$. Moreover, a change of variables gives ${\mathscr {F}}^b_{k}(u,1,Q_\rho (x))=\varepsilon _k^n{\mathscr {F}}^b\big (u_k,Q_{\frac{\rho }{\varepsilon _k}}\big (\tfrac{x}{\varepsilon _k}\big )\big )\,,$ from which, setting $r_k:=\frac{\rho }{\varepsilon _k}$ we immediately deduce

$$\begin{aligned} {\textbf{m}}^b_{k}(u_\xi ,Q_\rho (x))=\varepsilon _k^n{\textbf{m}}^b\big (u_\xi ,Q_{\frac{\rho }{\varepsilon _k}}\big (\tfrac{x}{\varepsilon _k}\big )\big )= \frac{\rho ^n}{r_k^n}{\textbf{m}}^b\big (u_\xi ,Q_{r_k}\big (r_k\tfrac{x}{\rho }\big )\big ), \end{aligned}$$

and thus (3.9) applied with x replaced by $x/\rho$ yields

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{{\textbf{m}}^b_{\varepsilon _k}(u_\xi ,Q_\rho (x))}{\rho ^n}=\lim _{k\rightarrow +\infty }\frac{{\textbf{m}}^b\big (u_\xi ,Q_{r_k}\big (r_k\tfrac{x}{\rho }\big )\big )}{r_k^n}=f_{\textrm{hom}}(\xi ). \end{aligned}$$

Eventually, by (2.12) and (2.13) we get $f'(x,\xi )=f''(x,\xi )=f_{\textrm{hom}}(\xi )$, for every $x\in {\mathbb {R}}^n$, $\xi \in {\mathbb {R}}^{m\times n}$.

We now prove the second two equalities in (3.11). To this end, for fixed $x\in {\mathbb {R}}^n$, $\nu \in {\mathbb {S}}^{n-1}$, $\rho >0$ and $k\in {\mathbb {N}}$, let $v\in {\mathscr {A}}(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$; then there exists $u\in W^{1,p}(Q^\nu _\rho (x);{\mathbb {R}}^m)$ such that

$$\begin{aligned} v\,\nabla u=0\;\text { a.e. in }\; Q_\rho ^\nu (x)\; \text { and }\; (u,v)=(\bar{u}_{x,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu )\; \text { near }\; \partial Q^\nu _\rho (x). \end{aligned}$$

(3.12)

Define $(u_k,v_k)\in W^{1,p}(Q^\nu _{\frac{\rho }{\varepsilon _k}}(\frac{x}{\varepsilon _k});{\mathbb {R}}^m)\times W^{1,p}(Q^\nu _{\frac{\rho }{\varepsilon _k}}(\frac{x}{\varepsilon _k}))$ as $u_k(y):=u_k(\varepsilon _ky)$, $v_k(y):=v(\varepsilon _ky)$. Then (3.12) is equivalent to

$$\begin{aligned} v_k\,\nabla u_k=0\;\text { a.e. in }\; Q^\nu _{\frac{\rho }{\varepsilon _k}}(\tfrac{x}{\varepsilon _k})\; \text { and }\; (u_k,v_k)=(\bar{u}_{\frac{x}{\varepsilon _k}}^\nu ,\bar{v}_{\frac{x}{\varepsilon _k}}^\nu )\;\text { near }\; \partial Q^\nu _{\frac{\rho }{\varepsilon _k}}(\tfrac{x}{\varepsilon _k}), \end{aligned}$$

that is, $v_k \in {\mathscr {A}}(\bar{u}_{\frac{x}{\varepsilon _k}}^\nu ,Q^\nu _{\frac{\rho }{\varepsilon _k}}(\tfrac{x}{\varepsilon _k}))$. Further, by a change of variables we immediately obtain that ${\mathscr {F}}_{k}^s(v,Q^\nu _\rho (x))=\varepsilon _k^{n-1}{\mathscr {F}}^s\big (v_k,Q^\nu _{\frac{\rho }{\varepsilon _k}}\big (\tfrac{x}{\varepsilon _k}\big )\big )\,.$ Hence, again setting $r_k:=\frac{\rho }{\varepsilon _k}$, we infer

$$\begin{aligned} {\textbf{m}}^s_{k}(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))=\varepsilon _k^{n-1}{\textbf{m}}^s\big (\bar{u}_{\frac{x}{\varepsilon _k}}^\nu ,Q_{\frac{\rho }{\varepsilon _k}}^\nu \big (\tfrac{x}{\varepsilon _k}\big )\big )= \frac{\rho ^{n-1}}{r_k^{n-1}}{\textbf{m}}^s\big ({\bar{u}}^\nu _{r_k\frac{x}{\rho }},Q_{r_k}\big (r_k\tfrac{x}{\rho }\big )\big ). \end{aligned}$$

Hence invoking (3.10) applied with x replaced by $x/\rho$ we get

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{{\textbf{m}}^s_{k}({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))}{\rho ^{n-1}}=\lim _{k\rightarrow +\infty }\frac{{\textbf{m}}^s\big ({\bar{u}}^\nu _{r_k\frac{x}{\rho }},Q_{r_k}^\nu \big (r_k\tfrac{x}{\rho }\big )\big )}{r_k^{n-1}}=g_{\textrm{hom}}(\nu ). \end{aligned}$$

Eventually, Proposition 2.6 gives $g'(x,\nu )=g''(x,\nu )=g_{\textrm{hom}}(\nu )$, for every $x\in {\mathbb {R}}^n$, $\nu \in {{\mathbb {S}}}^{n-1}$ and thus the claim. $\square$

4 Properties of $f',f'',g',g''$

This section is devoted to prove some properties satisfied by the functions $f', f'', g'$, and $g''$ defined in (2.12), (2.13), (2.14), and (2.15), respectively.

Proposition 4.1

Let $(f_k) \subset {\mathcal {F}}$; then the functions $f'$ and $f''$ defined, respectively, as in (2.12) and (2.13) satisfy properties 2.2-(f3). Moreover they also satisfy (f4), albeit with a different constant ${\widetilde{L}}_1>0$.

Proof

The proof readily follows from [27, Lemma A.6]. $\square$

Remark 4.2

As shown in [27, Lemma A.6], hypothesis (f4) in the definition of the class ${\mathcal {F}}$ can be weaken to obtain a larger class of volume integrands $\widetilde{{\mathcal {F}}}$ which is closed, in the sense that if $(f_k) \subset \widetilde{{\mathcal {F}}}$ then $f', f'' \in \widetilde{{\mathcal {F}}}$.

Before proving the corresponding result for $g',g''$ we need to prove the two following technical lemmas.

Lemma 4.3

Let $\rho ,\delta ,\varepsilon _k>0$ with $\rho> \delta >2\varepsilon _k$. Set

$$\begin{aligned} {\textbf{m}}_{k}^{s,\delta }({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)){:}{=}\inf \{{\mathscr {F}}_k^s(v,Q_\rho ^\nu (x)):v\in {\mathscr {A}}^\delta ({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\}, \end{aligned}$$

where

$$\begin{aligned} {\mathscr {A}}^\delta ({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)):= & {} \{v\in W^{1,p}(Q_\rho ^\nu (x)),\; \\ 0\le & {} v\le 1 :\, \exists \, u\in W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)\ \text { with } v \,\nabla u=0\\ \text {a.e. in}\ Q_\rho ^\nu (x)\ \text {and}\ (u,v)= & {} ({\bar{u}}_{x,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu )\ \text {in}\ Q_\rho ^\nu (x) \setminus {{\overline{Q}}}_{\rho -\delta }^\nu (x)\}, \end{aligned}$$

and $(\bar{u}_{x,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu )$ are as in (l). Moreover, let $g'_\rho ,g''_\rho :{\mathbb {R}}^n\times {\mathbb {S}}^{n-1}\rightarrow [0,+\infty ]$ be the functions defined as

$$\begin{aligned} g'_\rho (x,\nu )&:= \inf _{\delta>0}\liminf _{k\rightarrow +\infty }{\textbf{m}}_{k}^{s,\delta }({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))= \lim _{\delta \rightarrow 0}\liminf _{k\rightarrow +\infty }{\textbf{m}}_{k}^{s,\delta }({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\,,\nonumber \\ g''_\rho (x,\nu )&:= \inf _{\delta >0}\limsup _{k\rightarrow +\infty }{\textbf{m}}_{k}^{s,\delta }({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))=\lim _{\delta \rightarrow 0}\limsup _{k\rightarrow +\infty }{\textbf{m}}_{k}^{s,\delta }({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\,. \end{aligned}$$

(4.1)

Then the restrictions of $g'_\rho$, $g''_\rho$ to the sets ${\mathbb {R}}^n\times {\widehat{{\mathbb {S}}}}^{n-1}_+$ and ${\mathbb {R}}^n\times {\widehat{{\mathbb {S}}}}^{n-1}_-$ are upper semicontinuous.

Proof

We only show that the restriction of the function $g'_\rho$ to the set ${\mathbb {R}}^n\times {\widehat{{\mathbb {S}}}}^{n-1}_+$ is upper semicontinuous, the other proofs being analogous.

Let $\rho >0$, $x\in {\mathbb {R}}^n$, and $\nu \in {\widehat{{\mathbb {S}}}}^{n-1}_+$ be fixed. Let $\delta '>\delta$, since ${\mathscr {A}}^{\delta '}(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\subset {\mathscr {A}}^\delta (\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$, we immediately obtain that in (4.1) the infimum in $\delta >0$ coincides with the limit as $\delta \rightarrow 0$, which in particular exists. Thus, for fixed $\eta >0$ there exists $\delta _\eta >0$ such that

$$\begin{aligned} \liminf _{k \rightarrow +\infty }{\textbf{m}}_k^{s,\delta }(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\le g_\rho '(x,\nu )+\eta , \end{aligned}$$

(4.2)

for every $\delta \in (0,\delta _\eta )$. Fix $\delta _0\in (0,\frac{\delta _\eta }{3})$, hence by (4.2) we get

$$\begin{aligned} \liminf _{k\rightarrow +\infty }{\textbf{m}}_{k}^{s,3\delta _0}({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\le g_\rho '(x,\nu )+\eta . \end{aligned}$$

(4.3)

Now let $v_k \in {\mathscr {A}}^{3\delta _0}(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$ be such that

$$\begin{aligned} {\mathscr {F}}_{k}^s(v_k,Q_\rho ^\nu (x))\le {\textbf{m}}_{k}^{s,3\delta _0}(u_x^\nu ,Q_\rho ^\nu (x))+\eta , \end{aligned}$$

(4.4)

hence, in particular, $v_k \nabla u_k =0$ a.e. in $Q_\rho ^\nu (x)$, for some $u_k \in W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)$; further

$$\begin{aligned} (u_k,v_k)=({\bar{u}}_{x,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu )\quad \text {in }\big (Q_\rho ^\nu (x)\setminus {\overline{Q}}^\nu _{\rho -3\delta _0}(x)\big ). \end{aligned}$$

(4.5)

Now let $(x_j, \nu _j)\subset {\mathbb {R}}^n \times {\widehat{{\mathbb {S}}}}^{n-1}_+$ be a sequence converging to $(x,\nu )$. Thanks to the continuity of the map $\nu \mapsto R_\nu$ there exists ${\hat{\jmath }}= {\hat{\jmath }}(\delta _0)\in {\mathbb {N}}$ such that for every $j\ge {\hat{\jmath }}$ we have

$$\begin{aligned} Q_{\rho -2\delta _0}^\nu (x)\subset Q_{\rho -\delta _0}^{\nu _j}(x_j)\quad \text {and}\quad Q_{\rho -5\delta _0}^{\nu _j}(x_j)\subset Q_{\rho -4\delta _0}^\nu (x). \end{aligned}$$

(4.6)

We now modify $(u_k,v_k)$ to obtain a new pair $(\widetilde{u}_k,\widetilde{v}_k)\in W^{1,p}(Q_\rho ^{\nu _j}(x_j);{\mathbb {R}}^m)\times W^{1,p}(Q_\rho ^{\nu _j}(x_j))$ with $\widetilde{v}_k \in {\mathscr {A}}^\delta (\bar{u}_{x_j,\varepsilon _k}^{\nu _j},Q_\rho ^{\nu _j}(x_j))$ for any $j\ge {\hat{\jmath }}$ and any $\delta \in (0,\delta _0)$. To this end, set

$$\begin{aligned} h_{k,j}{:}{=}\varepsilon _k+|x-x_j|+\frac{\sqrt{n}}{2}\rho |\nu -\nu _j| \end{aligned}$$

and

$$\begin{aligned} R_{k,j}{:}{=}R_\nu \Big (\big (Q'_{\rho -3\delta _0+2\varepsilon _k}\setminus {\overline{Q}}'_{\rho -3\delta _0}\big )\times \big (-h_{k,j},h_{k,j}\big )\Big )+x, \end{aligned}$$

(see Fig. 2).

Fig. 2

The cubes $Q_{\rho }^\nu (x)$, $Q_{\rho -3\delta _0}^\nu (x)$, $Q_\rho ^{\nu _j}(x_j)$, and in gray the sets $R_{k,j}$ (dark gray) and $\{d_k<1\}$ (light gray)

Upon possibly increasing ${\hat{\jmath }}$, we can assume that $R_{k,j}\subset Q_{\rho -3\delta _0+2\varepsilon _k}^\nu (x)\setminus {\overline{Q}}^\nu _{\rho -3\delta _0}(x)$, for every $j\ge {\hat{\jmath }}$. Thus, setting $d_k(y){:}{=}\frac{1}{\varepsilon _k}{{\,\textrm{dist}\,}}(y,R_{k,j})$ and using (4.6), for $\varepsilon _k<\frac{\delta _0}{4}$ we also have

$$\begin{aligned} \{d_k<1\}\subset Q_{\rho -2\delta _0}^\nu (x)\setminus {\overline{Q}}_{\rho -4\delta _0}^\nu (x)\subset Q_{\rho -\delta _0}^{\nu _j}(x_j)\setminus {\overline{Q}}_{\rho -5\delta _0}^{\nu _j}(x_j). \end{aligned}$$

(4.7)

Now let $\varphi _k\in C_c^\infty (Q'_{\rho -3\delta _0+2\varepsilon _k})$ be a cut-off between $Q'_{\rho -3\delta _0}$ and $Q'_{\rho -3\delta _0+2\varepsilon _k}$ ; i.e., $0\le \varphi _k \le 1$ and $\varphi _k =1$ in $Q'_{\rho -3\delta _0}$. For $y\in Q_\rho ^{\nu _j}(x_j)$ we set

$$\begin{aligned} \widetilde{u}_k(y) {:}{=}\varphi _k\big ((R_\nu ^T(y-x))'\big )u_k(y)+\Big (1-\varphi _k\big (R_\nu ^T(y-x))'\big )\Big ){\bar{u}}_{x_j,\varepsilon _k}^{\nu _j}(y) \end{aligned}$$

and

$$\begin{aligned} \widetilde{v}_k(y){:}{=}{\left\{ \begin{array}{ll} \min \big \{v_k(y),d_k(y)\big \} &{}\text {in}\ Q_{\rho -3\delta _0}^\nu (x),\\ \min \big \{{\bar{v}}_{x_j,\varepsilon _k}^{\nu _j}(y),d_k(y)\big \} &{}\text {in}\ Q_{\rho }^{\nu _j}(x_j)\setminus Q_{\rho -3\delta _0}^\nu (x). \end{array}\right. } \end{aligned}$$

By construction, we have $\widetilde{u}_k=u_k$, $\widetilde{v}_k\le v_k$ in $Q_{\rho -3\delta _0}^\nu (x)$ and $\widetilde{u}_k={\bar{u}}_{x_j,\varepsilon _k}^{\nu _j}$, $\widetilde{v}_k\le {\bar{v}}_{x_j,\varepsilon _k}^{\nu _j}$ in $Q_\rho ^{\nu _j}(x_j)\setminus Q_{\rho -3\delta _0+2\varepsilon _k}^\nu (x)$; therefore, in particular, it holds

$$\begin{aligned} \begin{aligned} 0\le \widetilde{v}_k|\nabla \widetilde{u}_k|&\le v_k|\nabla u_k|=0\; \text { a.e. in }\; Q_{\rho -3\delta _0}^\nu (x)\\ 0\le \widetilde{v}_k|\nabla \widetilde{u}_k|&\le {\bar{v}}_{x_j,\varepsilon _k}^{\nu _j}|\nabla {\bar{u}}_{x_j,\varepsilon _k}^{\nu _j}|=0\; \text { a.e. in }\; Q_{\rho }^{\nu _j}(x_j)\setminus Q^\nu _{\rho -3\delta _0+2\varepsilon _k}(x). \end{aligned} \end{aligned}$$

(4.8)

Moreover, for $y\in \Big (Q_{\rho -3\delta _0+2\varepsilon _k}^\nu (x)\setminus {\overline{Q}}_{\rho -3\delta _0}^\nu (x)\Big )\setminus {\overline{R}}_{k,j}$ we have

$$\begin{aligned} |(y-x)\cdot \nu |=|R_\nu ^T(y-x)\cdot e_n|>h_{k,j}>\varepsilon _k. \end{aligned}$$

(4.9)

Then, by applying the triangular inequality twice, also noticing that $Q_\rho ^\nu (x)\subset B_{\frac{\sqrt{n}}{2}\rho }(x)$, we obtain

$$\begin{aligned} \begin{aligned} |(y-x_j)\cdot \nu _j|&> h_{k,j}-|x-x_j|-|y-x||\nu -\nu _j|\ge \varepsilon _k. \end{aligned} \end{aligned}$$

(4.10)

In view of (4.5), gathering (4.9) and (4.10) we infer that

$$\begin{aligned} u_k={\bar{u}}_{x,\varepsilon _k}^\nu ={\bar{u}}_{x_j,\varepsilon _k}^{\nu _j},\quad v_k={\bar{v}}_{x,\varepsilon _k}^\nu =v_{x_j,\varepsilon _k}^{\nu _j} \; \text { on }\; \Big (Q_{\rho -3\delta _0+2\varepsilon _k}^\nu (x)\setminus {\overline{Q}}_{\rho -3\delta _0}^\nu (x)\Big )\setminus {\overline{R}}_{k,j}. \end{aligned}$$

Hence $\widetilde{v}_k\in W^{1,p}(Q_\rho ^{\nu _j}(x_j))$ and $\nabla \widetilde{u}_k=0$ in $\big (Q_{\rho -3\delta _0+2\varepsilon _k}^\nu (x)\setminus {\overline{Q}}_{\rho -3\delta _0}^\nu (x)\big )\setminus {\overline{R}}_{k,j}$. Since moreover $\widetilde{v}_k=0$ in ${\overline{R}}_{k,j}$, we immediately obtain that $\widetilde{v}_k \nabla \widetilde{u}_k=0$ in $Q_{\rho -3\delta _0+2\varepsilon _k}^\nu (x)\setminus {\overline{Q}}_{\rho -3\delta _0}^\nu (x)$, which in view of (4.8) yields

$$\begin{aligned} \widetilde{v}_k\,\nabla \widetilde{u}_k=0 \; \text { a.e. in }\; Q_\rho ^{\nu _j}(x_j). \end{aligned}$$

Eventually, since in $Q_\rho ^{\nu _j}(x_j)\setminus Q_{\rho -3\delta _0}^\nu (x)$ we have $\widetilde{v}_k={\bar{v}}_{x_j,\varepsilon _k}^{\nu _j}$ if $d_k\ge 1$, from (4.7) we deduce that $(\widetilde{u}_k,\widetilde{v}_k)=({\bar{u}}_{x_j,\varepsilon _k}^{\nu _j},{\bar{v}}_{x_j,\varepsilon _k}^{\nu _j})$ in $Q_\rho ^{\nu _j}(x_j)\setminus {\overline{Q}}_{\rho -\delta _0}^{\nu _j}(x_j)$, hence also in $Q_\rho ^{\nu _j}(x_j)\setminus {\overline{Q}}_{\rho -\delta }^{\nu _j}(x_j)$ for every $\delta \in (0,\delta _0)$. In particular, $\widetilde{v}_k \in {\mathscr {A}}^\delta ({\bar{u}}_{x_j,\varepsilon _k}^{\nu _j},Q_\rho ^{\nu _j}(x_j))$ for any $\delta \in (0,\delta _0)$. Now it only remains to estimate ${\mathscr {F}}_{k}^s(\widetilde{v}_k,Q_\rho ^{\nu _j}(x_j))$. Arguing as in Proposition 2.6, we get

$$\begin{aligned} {\mathscr {F}}_{k}^s(\widetilde{v}_k,Q_\rho ^{\nu _j}(x_j))\le & {} {\mathscr {F}}_{k}^s(v_k,Q_\rho ^\nu (x))+ {\mathscr {F}}_{k}^s({\bar{v}}_{x_j,\varepsilon _k}^{\nu _j},Q_{\rho }^{\nu _j}(x_j)\setminus {\overline{Q}}_{\rho -3\delta _0}^\nu (x))\nonumber \\{} & {} +\frac{2c_4}{\varepsilon _k}{\mathcal {L}}^n(\{d_k<1\}). \end{aligned}$$

(4.11)

Moreover, the second inclusion in (4.6) together with (2.3) and (2.19) implies that

$$\begin{aligned} {\mathscr {F}}_{k}^s({\bar{v}}_{x_j,\varepsilon _k}^{\nu _j},Q_{\rho }^{\nu _j}(x_j)\setminus {\overline{Q}}_{\rho -3\delta _0}^\nu (x))&\le {\mathscr {F}}_{k}^s({\bar{v}}_{x_j,\varepsilon _k}^{\nu _j},Q_{\rho }^{\nu _j}(x_j)\setminus {\overline{Q}}_{\rho -5\delta _0}^{\nu _j}(x_j))\nonumber \\&\le c_4C_\textrm{v}{\mathcal {L}}^{n-1}\Big (Q'_{\rho }\setminus \overline{Q'}_{\!\!\rho -5\delta _0}\big )\Big )\le C\delta _0\rho ^{n-2}\,, \end{aligned}$$

(4.12)

also

$$\begin{aligned} \frac{2c_4}{\varepsilon _k}{\mathcal {L}}^n\big (\{d_k<1\}\big )&\le \frac{4c_4}{\varepsilon _k}{\mathcal {L}}^{n-1}\big (Q'_{\rho -3\delta _0+4\varepsilon _k}\setminus \overline{Q'}_{\!\!\rho -3\delta _0-2\varepsilon _k}\big ) (h_{k,j}+\varepsilon _k) \nonumber \\&\le Ch_{k,j}(\rho -3\delta _0)^{n-2}+{\mathcal {O}}(\varepsilon _k)\,, \end{aligned}$$

(4.13)

as $k\rightarrow +\infty$.

Since $\widetilde{v}_k \in {\mathscr {A}}^\delta ({\bar{u}}_{x_j,\varepsilon _k}^{\nu _j},Q_\rho ^{\nu _j}(x_j))$ for any $\delta \in (0,\delta _0)$, by combining (4.4) and (4.11)–(4.13), we get

$$\begin{aligned} {\textbf{m}}_{k}^{s,\delta }({\bar{u}}_{x_j,\varepsilon _k}^{\nu _j},Q_\rho ^{\nu _j}(x_j))\le {\textbf{m}}_{k}^{s,3\delta _0}({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))+\eta +C(\delta _0+h_{k,j})\rho ^{n-2}+{\mathcal {O}}(\varepsilon _k), \end{aligned}$$

(4.14)

as $k\rightarrow +\infty$. Using (4.3) and taking in (4.14) first the liminf as $k \rightarrow +\infty$, then the limit as $\delta \rightarrow 0$, and finally the limsup as $j \rightarrow +\infty$ gives

$$\begin{aligned} \limsup _{j\rightarrow +\infty }g_\rho '(x_j,\nu _j)\le g_\rho '(x,\nu )+2\eta +C\delta _0\rho ^{n-2}, \end{aligned}$$

since $\lim _j\lim _k h_{k,j}=0$. Therefore, letting $\delta _0\rightarrow 0$ the upper semicontinuity of $g_\rho '$ restricted to ${\widehat{{\mathbb {S}}}}_+^{n-1}\times {\mathbb {R}}^n$ follows by the arbitrariness of $\eta >0$. $\square$

Lemma 4.4

Let $g'$ and $g''$ be as in (2.14) and (2.15), respectively, and let $g_\rho '$, $g_\rho ''$ be as in (4.1). Then for every $x\in {\mathbb {R}}^n$ and every $\nu \in {\mathbb {S}}^{n-1}$ it holds

$$\begin{aligned} g'(x,\nu )=\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}g_\rho '(x,\nu )\;\text { and }\; g''(x,\nu )=\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}g_\rho ''(x,\nu ). \end{aligned}$$

Proof

We prove the statement only for $g'$, the proof for $g''$ being analogous. We notice that since ${\mathscr {A}}^\delta (\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\subset {\mathscr {A}}(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$ for every $\delta \in (0,\rho )$, we clearly have

$$\begin{aligned} g'(x,\nu )\le \limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}g_\rho '(x,\nu ), \end{aligned}$$

therefore to conclude we just need to prove the opposite inequality. This can be done by means of an easy extension argument as follows. For fixed $\rho > 0$, $x\in {\mathbb {R}}^n$, and $\nu \in {\mathbb {S}}^{n-1}$ and for every $k\in {\mathbb {N}}$ such that $\varepsilon _k\in (0,\frac{\rho }{2})$ let $v_k\in {\mathscr {A}}(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$ satisfy

$$\begin{aligned} {\mathscr {F}}_k^s(v_k,Q_\rho ^\nu (x))\le {\textbf{m}}_k^s(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))+\rho ^n, \end{aligned}$$

(4.15)

and let $u_k\in W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)$ be the corresponding u-variable. Let $\alpha >0$ be arbitrary; thanks to the boundary conditions satisfied by $(u_k,v_k)$ we can extend the pair $(u_k,v_k)$ to $Q_{(1+\alpha )\rho }^\nu (x)$ by setting $(u_k,v_k){:}{=}(\bar{u}_{x,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu )$ in $Q_{(1+\alpha )\rho }^\nu (x)\setminus {\overline{Q}}_\rho ^\nu (x)$. Then (2.3) and (2.19) yield

$$\begin{aligned} \begin{aligned} {\mathscr {F}}_k^s(v_k,Q_{(1+\alpha )\rho }^\nu (x))&\le {\mathscr {F}}_k^s(v_k,Q_\rho ^\nu (x))+{\mathscr {F}}_k^s\big (\bar{v}_{x,\varepsilon _k}^\nu ,(Q_{(1+\alpha )\rho }^\nu (x)\setminus {\overline{Q}}_\rho ^\nu (x)\big )\\&\le {\mathscr {F}}_k^s(v_k,Q_\rho ^\nu (x))+c_4C_\textrm{v}((1+\alpha )^{n-1}-1))\rho ^{n-1}. \end{aligned} \end{aligned}$$

(4.16)

Moreover, for $\delta \in (0,\alpha \rho )$ we have $v_k\in {\mathscr {A}}^\delta (\bar{u}_{x,\varepsilon _k}^\nu ,Q_{(1+\alpha )\rho }^\nu (x))$, thus (4.15) and (4.16) give

$$\begin{aligned} \inf _{\delta >0}\liminf _{k\rightarrow +\infty }{\textbf{m}}_k^{s,\delta }(\bar{u}_{x,\varepsilon _k}^\nu ,Q_{(1+\alpha )\rho }^\nu (x))\le {\textbf{m}}_k^s(\bar{u}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))+\rho ^n+c_4C_\textrm{v}((1+\alpha )^{n-1}-1))\rho ^{n-1}. \end{aligned}$$

Hence, dividing the above inequality by $((1+\alpha )\rho )^{n-1}$ and taking the limsup as $\rho \rightarrow 0$, thanks to Proposition 2.6 we obtain

$$\begin{aligned} (1+\alpha )^{n-1}\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}g_\rho '(x,\nu )\le g'(x,\nu )+c_4C_\textrm{v}((1+\alpha )^{n-1}-1)), \end{aligned}$$

thus we conclude by the arbitrariness of $\alpha >0$. $\square$

We are now ready to state and prove the following proposition which establishes the properties satisfied by $g'$ and $g''$.

Proposition 4.5

Let $(g_k) \subset {\mathcal {G}}$; then the functions $g'$ and $g''$ defined, respectively, as in (2.14) and (2.15) are Borel measurable and satisfy the following two properties:

• (1) (symmetry) for every $x\in {\mathbb {R}}^n$ and every $\nu \in {\mathbb {S}}^{n-1}$ it holds

  $$\begin{aligned} g'(x, \nu )=g'(x,-\nu ), \quad g''(x, \nu )=g''(x,-\nu ); \end{aligned}$$

  (4.17)

• (2) (boundedness) for every $x\in {\mathbb {R}}^n$ and every $\nu \in {{\mathbb {S}}}^{n-1}$ it holds

  $$\begin{aligned} c_3 c_p \le g'(x, \nu ) \le c_4 c_p, \quad c_3 c_p \le g''(x,\nu ) \le c_4 c_p, \end{aligned}$$

  (4.18)

  where $c_p:=2(p-1)^{\frac{1-p}{p}}$.

Proof

We prove the statement only for $g'$, the proof for $g''$ being analogous.

We divide the proof into three steps.

Step 1: $g'$ is Borel measurable. Let $\rho >0$ and let $g'_\rho$ be the function defined in (4.1). Arguing as in the proof of Lemma 4.4 we deduce that the function $\rho \rightarrow g'_{\rho }(x,\nu )-c_4C_\textrm{v}\rho ^{n-1}$ is nonincreasing on $(0,+\infty )$. From this it follows that

$$\begin{aligned} \lim _{\rho '\rightarrow \rho ^-}g'_{\rho '}(x,\nu )\ge g'_{\rho }(x,\nu )\ge \lim _{\rho '\rightarrow \rho ^+} g'_{\rho '}(x,\nu ), \end{aligned}$$

for every $x\in {\mathbb {R}}^n$, $\nu \in {{\mathbb {S}}}^{n-1}$, and every $\rho >0$. Thus, if D is a countable dense subset of $(0,+\infty )$ we have

$$\begin{aligned} \limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}g'_{\rho }(x,\nu )= \limsup _{\begin{array}{c} \rho \rightarrow 0\\ \rho \in D \end{array}}\frac{1}{\rho ^{n-1}}g'_{\rho }(x,\nu ) \end{aligned}$$

and hence by Lemma 4.4 we get

$$\begin{aligned} g'(x,\nu )&=\limsup _{\begin{array}{c} \rho \rightarrow 0\\ \rho \in D \end{array}}\frac{1}{\rho ^{n-1}} g'_{\rho }(x,\nu ). \end{aligned}$$

Therefore the Borel measurability of $g'$ follows by Lemma 4.3 which guarantees, in particular, that the function $(x,\nu ) \mapsto g'_\rho (x,\nu )$ is Borel measurable for every $\rho >0$.

Step 2: $g'$ is symmetric in $\nu$. Property (4.17) immediately follows from the definition of $g'$ and from the fact that $u^\nu _x=-u^{-\nu }_x+e_1$ a.e. and $Q^\nu _\rho (x)=Q^{-\nu }_\rho (x)$ (see (f)), which implies in particular that $v\in {\mathscr {A}}_{\varepsilon _k,\rho }(x,\nu )$ with corresponding $u\in W^{1,p}(Q_\rho ^{\nu }(x);{\mathbb {R}}^m)$ if and only if $v\in {\mathscr {A}}_{\varepsilon _k,\rho }(x,-\nu )$ with corresponding $w{:}{=}-u+e_1\in W^{1,p}(Q_\rho ^{-\nu }(x);{\mathbb {R}}^m)$.

Step 3: $g'$ is bounded. To prove that $g'$ satisfies the bounds in (4.18) we start by observing that thanks to (g2) and (g3) we have

$$\begin{aligned} c_3 \int _{Q^\nu _\rho (x)} \bigg (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\bigg )\, \textrm{d}x\le & {} {\mathscr {F}}^s_{k}(v,Q^\nu _\rho (x)) \\\le & {} c_4 \int _{Q^\nu _\rho (x)} \bigg (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\bigg )\, \textrm{d}x, \end{aligned}$$

for every $v\in W^{1,p}(Q^\nu _\rho (x))$ with $0\le v \le 1$ a.e. in $Q^\nu _\rho (x)$. Therefore to establish (4.18) it is enough to show that

$$\begin{aligned} \lim _{k \rightarrow +\infty } {\textbf{m}}_{k,\rho }(x,\nu )=c_p\rho ^{n-1}, \end{aligned}$$

(4.19)

for every $x\in {\mathbb {R}}^n$ and $\rho >0$, where

$$\begin{aligned} {\textbf{m}}_{k,\rho }(x,\nu ) {:}{=}\min \bigg \{\int _{Q^\nu _\rho (x)} \bigg (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\bigg )\, \textrm{d}y:v\in {\mathscr {A}}_{\varepsilon _k,\rho }(x,\nu )\bigg \}. \end{aligned}$$

Let $x\in {\mathbb {R}}^n$, $\nu \in {\mathbb {S}}^{n-1}$, $\rho >0$, and let $k\in {\mathbb {N}}$ be such that $2\varepsilon _k<\rho$. By the homogeneity and rotation invariance of the Ambrosio-Tortorelli functional we have

$$\begin{aligned} {\textbf{m}}_{k,\rho }(x,\nu ) = {\textbf{m}}_{\varepsilon _k,\rho }(0,e_n), \end{aligned}$$

Let $\eta >0$ be arbitrary; reasoning as in the construction of a recovery sequence for the Ambrosio-Tortorelli functional we find a sequence $(u_k,v_k)\subset W^{1,p}(Q_\rho (0);{\mathbb {R}}^m)\times W^{1,p}(Q_\rho (0))$ satisfying $v_k\nabla u_k=0$ a.e. in $Q_\rho (0)$, $(u_k,v_k)=(u_0^{e_n},1)$ in $\{|y_n|>\varepsilon _k T_\eta \}$ for some $T_\eta >0$ and

$$\begin{aligned} \limsup _{k \rightarrow +\infty }\int _{Q_\rho (0)} \bigg (\frac{(1-v_k)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v_k|^p\bigg )\, \textrm{d}x= (c_p+\eta ) \rho ^{n-1}. \end{aligned}$$

(4.20)

Then, using a similar argument as in the proof of Proposition 2.6, we can modify $v_k$ to obtain a function $\widetilde{v}_k\in W^{1,p}(Q_\rho (0))$ satisfying $\widetilde{v}_k=1$ in $Q_{\rho }(0)\setminus {{\overline{Q}}}_{\rho -\varepsilon _k}(0)\cap \{|y_n|>\varepsilon _k\}$ and

$$\begin{aligned}{} & {} \int _{Q_\rho (0)} \bigg (\frac{(1-{\widetilde{v}}_k)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla {\widetilde{v}}_k|^p\bigg )\, \textrm{d}x\nonumber \\{} & {} \quad \le \int _{Q_\rho (0)} \bigg (\frac{(1-v_k)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v_k|^p\bigg )\, \textrm{d}x+C\varepsilon _k\rho ^{n-2}. \end{aligned}$$

(4.21)

In particular, since $\widetilde{v}_k\in {\mathscr {A}}_{\varepsilon _k,\rho }(0,e_n)$, gathering (4.20)–(4.21), by the arbitrariness of $\eta >0$ we conclude that

$$\begin{aligned} \limsup _{k\rightarrow +\infty }{\textbf{m}}_{k,\rho }(0,e_n) \le c_p \rho ^{n-1}. \end{aligned}$$

(4.22)

We now turn to the proof of the lower bound. By the Fubini Theorem we have

$$\begin{aligned} \begin{aligned}&\int _{Q_\rho (0)} \bigg (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\bigg )\, \textrm{d}x\\&\quad = \int _{Q'_{\rho }}\int _{-\frac{\rho }{2}}^{\frac{\rho }{2}} \bigg (\frac{(1-v(x',x_n))^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v (x',x_n)|^p\bigg )\, \textrm{d}x_n \, \textrm{d}x'\\&\quad \ge \int _{Q'_{\rho }}\int _{-\frac{\rho }{2}}^{\frac{\rho }{2}} \bigg (\frac{(1-v(x',x_n))^p}{\varepsilon _k}+\varepsilon _k^{p-1}\Big |\frac{\partial v (x',x_n)}{\partial x_n}\Big |^p\bigg )\, \textrm{d}x_n \, \textrm{d}x'. \end{aligned} \end{aligned}$$

(4.23)

Then, if $v\in {\mathscr {A}}_{\varepsilon _k, \rho }(0,e_n)$ the corresponding u coincides with $u_0^{e_n}$ in a neighbourhood of

$$\begin{aligned} \partial ^\pm Q_\rho (0){:}{=}\Big \{(x',x_n)\in {\mathbb {R}}^{n-1}\times {\mathbb {R}}:x'\in \overline{Q'_\rho }, \; x_n=\pm \frac{\rho }{2}\Big \}. \end{aligned}$$

Since it must hold that $v\nabla u=0$ a.e. in $Q_\rho (0)$, then almost every straight line intersecting $\partial ^\pm Q_\rho (0)$ and parallel to $e_n$ also intersects the level set $\{v=0\}$. Indeed, for ${\mathcal {L}}^{n-1}$-a.e. $x'\in Q'_{\rho }$ the pair $(u_{x'}(t),v_{x'}(t)){:}{=}\big (u(x',t)\cdot e_1,v(x',t)\big )$ belongs to $W^{1,p}(-\frac{\rho }{2},\frac{\rho }{2})\times W^{1,p}(-\frac{\rho }{2},\frac{\rho }{2})$ and satisfies

$$\begin{aligned} v_{x'}(t)u_{x'}'(t)=v(x',t)\frac{\partial u(x',t)}{\partial x_n}\cdot e_1=0\;\text { for }{\mathcal {L}}^1\text {-a.e.}\; t\in \Big (-\frac{\rho }{2},\frac{\rho }{2}\Big ), \end{aligned}$$

(4.24)

as well as $v_{x'}(\pm \frac{\rho }{2})=1$, $u_{x'}(-\frac{\rho }{2})=0$, and $u_{x'}(\frac{\rho }{2})=1$. Since $u_{x'}\in W^{1,p}(-\frac{\rho }{2},\frac{\rho }{2})$, the boundary conditions satisfied by $u_{x'}$ imply the existence of a subset of $(-\frac{\rho }{2},\frac{\rho }{2})$ with positive ${\mathcal {L}}^1$-measure on which $u_{x'}'\ne 0$, hence $v_{x'}=0$ in view of (4.24). In particular, for ${\mathcal {L}}^{n-1}$-a.e. $x'\in Q_\rho '$ there exists $s\in (-\frac{\rho }{2},\frac{\rho }{2})$ such that $v(x',s)=0$.

Therefore, the Young Inequality

$$\begin{aligned} \frac{(1-v(x',x_n))^p}{\varepsilon _k}+\varepsilon _k^{p-1}\bigg |\frac{\partial v (x',x_n)}{\partial x_n}\bigg |^p \ge \Big (\frac{p}{p-1}\Big )^{\frac{p-1}{p}}p^{\frac{1}{p}}(1-v(x',x_n))^{p-1}\bigg |\frac{\partial v (x',x_n)}{\partial x_n}\bigg |, \end{aligned}$$

applied for ${\mathcal {L}}^{n-1}$-a.e. $x'\in Q'_\rho$ together with the integration on $(-\frac{\rho }{2}, \frac{\rho }{2})$ give

$$\begin{aligned}&\int _{-\frac{\rho }{2}}^{\frac{\rho }{2}}\bigg (\frac{(1-v(x',x_n))^p}{\varepsilon _k}+\varepsilon _k^{p-1}\bigg |\frac{\partial v (x',x_n)}{\partial x_n}\bigg |^p\bigg )\, \textrm{d}x_n\nonumber \\&\quad = \int _{-\frac{\rho }{2}}^{s}\bigg (\frac{(1-v(x',x_n))^p}{\varepsilon _k}+\varepsilon _k^{p-1}\bigg |\frac{\partial v (x',x_n)}{\partial x_n}\bigg |^p\bigg )\, \textrm{d}x_n \nonumber \\&\qquad + \int _{s}^{\frac{\rho }{2}}\bigg (\frac{(1-v(x',x_n))^p}{\varepsilon _k}+\varepsilon _k^{p-1}\bigg |\frac{\partial v (x',x_n)}{\partial x_n}\bigg |^p\bigg )\, \textrm{d}x_n\nonumber \\&\quad \ge 2\Big (\frac{p}{p-1}\Big )^{\frac{p-1}{p}}p^{\frac{1}{p}}\int _0^1 (1-t)^{p-1}\, \textrm{d}t=c_p\,, \end{aligned}$$

(4.25)

for ${\mathcal {L}}^{n-1}$-a.e. $x'\in Q'_\rho$. Thus, gathering (4.23) and (4.25) we get

$$\begin{aligned} \int _{Q_\rho (0)} \bigg (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\bigg )\, \textrm{d}x\ge c_p \rho ^{n-1}, \end{aligned}$$

for every $v\in {\mathscr {A}}_{\varepsilon _k,\rho }(0,e_n)$. Passing to the infimum on v and to the liminf as $k\rightarrow +\infty$ we get

$$\begin{aligned} \liminf _{k\rightarrow +\infty }{\textbf{m}}_{k,\rho }(0,e_n) \ge c_p \rho ^{n-1}, \end{aligned}$$

(4.26)

for every $\rho >0$. Eventually, by combining (4.22) and (4.26) we get (4.19), and hence (4.18). $\square$

5 $\Gamma$-convergence and integral representation

In this section we show that, up to subsequences, the functionals ${\mathscr {F}}_k$ $\Gamma$-converge in $L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ to an integral functional of free-discontinuity type. This result is achieved by following a standard procedure which combines the localisation method of $\Gamma$-convergence (see e.g., [32, Chapters 14-18] or [23, Chapters 10, 11]) together with an integral-representation result in SBV [17, Theorem 1]. Though rather technical, this procedure is by now classical. For this reason here we only detail the adaptations of the theory to our specific setting, while we refer the reader to the literature for the more standard aspects.

We start by showing that the functionals ${\mathscr {F}}_k$ satisfy the so-called fundamental estimate, uniformly in k.

Proposition 5.1

(Fundamental estimate) Let ${\mathscr {F}}_k$ be as in (2.4). Then, for every $\eta >0$ and for every $A,\,A',\,B\in {\mathcal {A}}$ with $A \subset \subset A'$ there exists a constant $M_\eta >0$ (also depending on $A,A',B$) satisfying the following property: For every $k\in {\mathbb {N}}$ and for every $(u,v)\in W^{1,p}(A';{\mathbb {R}}^m)\times W^{1,p}(A')$, $({\widetilde{u}},{\widetilde{v}})\in W^{1,p}(B;{\mathbb {R}}^m)\times W^{1,p}(B)$, $0\le v,{\widetilde{v}}\le 1$, there exists a pair $({\hat{u}},{\hat{v}})\in W^{1,p}(A\cup B;{\mathbb {R}}^m)\times W^{1,p}(A\cup B)$ with $0\le {\hat{v}}\le 1$ such that $(\hat{u},\hat{v})=(u,v)$ a.e. in A, $(\hat{u},\hat{v})=(\widetilde{u},\widetilde{v})$ a.e. in $B\setminus {\overline{A}}'$ and

$$\begin{aligned} {\mathscr {F}}_k({\hat{u}}, {\hat{v}}, A\cup B)&\le (1+\eta )\big ({\mathscr {F}}_k(u,v,A')+ {\mathscr {F}}_k({\widetilde{u}}, {\widetilde{v}},B)\big )\nonumber \\&\quad +M_\eta \big (\Vert u-\widetilde{u}\Vert _{L^p(S;{\mathbb {R}}^m)}^p+\varepsilon _k^{p-1}\big )\,, \end{aligned}$$

(5.1)

where $S{:}{=}(A'\setminus A)\cap B$.

Proof

Fix $k\in {\mathbb {N}}$, $\eta >0$ and $A,\,A',\,B\in {\mathcal {A}}$ with $A\subset \subset A'$. Let $N \in {\mathbb {N}}$ and $A_1,\ldots ,A_{N+1}\in {\mathcal {A}}$ with

$$\begin{aligned} A\subset \subset A_1\subset \subset \dots \subset \subset A_{N+1} \subset \subset A'. \end{aligned}$$

For each $i=2,\ldots ,N+1$ let $\varphi _i$ be a smooth cut-off function between $A_{i-1}$ and $A_i$ and let

$$\begin{aligned} M:=\max _{2\le i\le N+1} \Vert \nabla \varphi _i\Vert _\infty . \end{aligned}$$

Let (u, v) and $({\widetilde{u}},{\widetilde{v}})$ be as in the statement and consider the function $w\in L^0({\mathbb {R}}^n)$ defined by

$$\begin{aligned} w:=\min \{v,\widetilde{v}\}, \end{aligned}$$

clearly $0\le w\le 1$. For $i=3,\ldots ,N$ we define $({\hat{u}}^i, {\hat{v}}^i) \in W^{1,p}(A\cup B;{\mathbb {R}}^m)\times W^{1,p}(A\cup B)$ as follows

$$\begin{aligned} {\hat{u}}^i:= \varphi _{i} u + (1-\varphi _{i}) {\widetilde{u}} \quad \text {and}\quad {\hat{v}}^i:= {\left\{ \begin{array}{ll} \varphi _{i-1} v + (1-\varphi _{i-1}) w &{} \text {in}\ A_{i-1}, \\ w &{} \text {in}\ A_i \setminus A_{i-1}, \\ \varphi _{i+1}\,w + (1-\varphi _{i+1}) {\widetilde{v}} &{} \text {in}\ {\mathbb {R}}^n\setminus A_{i}. \end{array}\right. } \end{aligned}$$

Then, setting $S_i{:}{=}A_i\setminus {\overline{A}}_{i-1}$ and taking into account the definition of $(\hat{u}^i,\hat{v}^i)$ we have

$$\begin{aligned} \begin{aligned} {\mathscr {F}}_k ({\hat{u}}^i,{\hat{v}}^i, A\cup B)&\le {\mathscr {F}}_k (u,v, A_{i-2}) + {\mathscr {F}}_k (u,{\hat{v}}^i, S_{i-1}\cap B) +{\mathscr {F}}_k ({\hat{u}}^i,w, S_i\cap B)\\&\quad + {\mathscr {F}}_k (\widetilde{u},{\hat{v}}^i, S_{i+1}\cap B)+ {\mathscr {F}}_k (\widetilde{u},\widetilde{v}, B\setminus {{\overline{A}}}_{i+1}). \end{aligned} \end{aligned}$$

(5.2)

We now come to estimate the three terms in (5.2) involving the sets $S_{i-1}, S_i$, and $S_{i+1}$. We start observing that thanks to (f3) and (f2), exploiting the definition of w and the fact that $\psi$ is increasing, we have

$$\begin{aligned} {\mathscr {F}}_k^b(u,{\hat{v}}^i, S_{i-1}\cap B)\le c_2\int _{S_{i-1}\cap B}\,\psi (v)|\nabla u|^p\, \textrm{d}x\le \frac{c_2}{c_1}{\mathscr {F}}_k^b(u,v,S_{i-1}\cap B). \end{aligned}$$

(5.3)

Analogously, there holds

$$\begin{aligned} {\mathscr {F}}_k^b(\widetilde{u},{\hat{v}}^i, S_{i+1}\cap B)\le \frac{c_2}{c_1}{\mathscr {F}}_k^b(\widetilde{u},\widetilde{v},S_{i+1}\cap B). \end{aligned}$$

(5.4)

We complete the estimate of the bulk part of the energy by noticing that on $S_i\cap B$ we have

$$\begin{aligned} |\nabla \hat{u}^i|^p&\le 3^{p-1}\big (|\nabla \varphi _i|^p|u-\widetilde{u}|^p+|\nabla u|^p+|\nabla \widetilde{u}|^p\big )\\&\le 3^{p-1}\big (M^p|u-\widetilde{u}|^p+|\nabla u|^p+|\nabla \widetilde{u}|^p\big )\,. \end{aligned}$$

Integrating over $S_i\cap B$, using (f3) and (f2), the definition of w, and the monotonicity of $\psi$, we infer

$$\begin{aligned} \begin{aligned}&{\mathscr {F}}_k^b(\hat{u}^i,w,S_i\cap B)\le c_2\int _{S_i\cap B}\,\psi (w)|\nabla \hat{u}^i|^p\, \textrm{d}x\\&\quad \le c_2 3^{p-1}\int _{S_i\cap B}\,\big (\psi (v)|\nabla u|^p+\psi (\widetilde{v})|\nabla \widetilde{u}|^p\big )\, \textrm{d}x+3^{p-1}M^pc_2\int _{S_i\cap B}\,|u-\widetilde{u}|^p\, \textrm{d}x\\&\quad \le \frac{3^{p-1}c_2}{c_1}\big ({\mathscr {F}}_k^b(u,v,S_i\cap B)+{\mathscr {F}}_k^b(\widetilde{u},\widetilde{v},S_i\cap B)\big )+3^{p-1}M^pc_2\int _{S_i\cap B}\,|u-\widetilde{u}|^p\, \textrm{d}x. \end{aligned} \end{aligned}$$

(5.5)

It remains to estimate the surface term in ${\mathscr {F}}_k$. Thanks to (g3) it holds

$$\begin{aligned} {\mathscr {F}}_k^s(w,S_i\cap B)\le c_4\int _{S_i\cap B}\hspace{-.5em}\bigg (\frac{(1-w)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla w|^p\bigg )\, \textrm{d}x. \end{aligned}$$

(5.6)

We now want to bound the right-hand side of (5.6) in terms of ${\mathscr {F}}_k^s(v,S_i\cap B)+{\mathscr {F}}_k^s(\widetilde{v},S_i\cap B)$. To this end we first observe that by definition of w we have

$$\begin{aligned} |\nabla w|^p\le |\nabla v|^p+|\nabla \widetilde{v}|^p\quad \text {and}\quad (1-w)^p\le (1-v)^p+(1-\widetilde{v})^p. \end{aligned}$$

(5.7)

Thus, thanks to (g2), (5.6) becomes

$$\begin{aligned} {\mathscr {F}}_k^s(w,S_i\cap B)\le \frac{c_4}{c_3}\big ({\mathscr {F}}_k^s(v,S_i\cap B)+{\mathscr {F}}_k^s(\widetilde{v},S_i\cap B)\big ). \end{aligned}$$

(5.8)

Moreover, it holds

$$\begin{aligned} {\mathscr {F}}_k^s(\hat{v}^i,(S_{i-1}\cup S_{i+1})\cap B)\le c_4\int _{(S_{i-1}\cup S_{i+1})\cap B}\hspace{-.5em}\bigg (\frac{(1-\hat{v}^i)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla \hat{v}^i|^p\bigg )\, \textrm{d}x. \end{aligned}$$

(5.9)

By the definition of $\hat{v}^i$ and by the convexity of $z\mapsto (1-z)^p$ for $z\in [0,1]$, on $S_{i-1}\cap B$ we have

$$\begin{aligned} (1-\hat{v}^i)^p&=(\varphi _{i-1}(1-v)+(1-\varphi _{i-1})(1-w))^p\\&\le (1-v)^p+(1-w)^p\le 2\big ((1-v)^p+(1-\widetilde{v})^p\big )\,, \end{aligned}$$

where in the last step we used again (5.7). Similarly, there holds

$$\begin{aligned} |\nabla \hat{v}^i|^p&\le 3^{p-1}\big (|\nabla \varphi _i|^p|v-w|^p+|\nabla v|^p+|\nabla w|^p\big )\\&\le 3^{p-1}\big (M^p|v-\widetilde{v}|^p+2(|\nabla v|^p+|\nabla \widetilde{v}|^p)\big )\,. \end{aligned}$$

Since analogous arguments hold on $S_{i+1}\cap B$, from (5.9) and (g2) we deduce

$$\begin{aligned} \begin{aligned} {\mathscr {F}}_k^s(\hat{v}^i,(S_{i-1}\cup S_{i+1})\cap B)&\le 3^{p-1} \frac{2c_4}{c_3}\big ({\mathscr {F}}_k^s(v,(S_{i-1}\cup S_{i+1})\cap B)\\&\quad +{\mathscr {F}}_k^s(\widetilde{v},(S_{i-1}\cup S_{i+1})\cap B)\big )\\&\quad +3^{p-1}M^p c_4\int _{(S_{i-1}\cup S_{i+1})\cap B}\hspace{-3em}\varepsilon _k^{p-1}|v-\widetilde{v}|^p\, \textrm{d}x. \end{aligned} \end{aligned}$$

(5.10)

Now set ${\widehat{M}}{:}{=}\big (\frac{(1+3^{p-1}) c_2}{c_1}+\frac{(1+3^{p-1}2)c_4}{c_3}\big )$; then, summing up in (5.2) over all i, gathering (5.3)–(5.5), (5.8), and (5.10), by averaging we find an index $i^*\in \{3,\ldots ,N\}$ such that

$$\begin{aligned} {\mathscr {F}}_k(\hat{u}^{i^*},\hat{v}^{i^*},A\cup B)&\le \frac{1}{N-2} \sum _{i=3}^{N}{\mathscr {F}}_k ({\hat{u}}^i, {\hat{v}}^i, A\cup B)\\&\le \Big (1+\frac{{\widehat{M}}}{N-2}\Big )\big ({\mathscr {F}}_k(u,v,A')+{\mathscr {F}}_k(\widetilde{u},\widetilde{v},B)\big )\\&\quad +\frac{3^{p-1}M^pc_2}{N-2}\int _{S}|u-\widetilde{u}|^p\, \textrm{d}x+\frac{3^{p-1}M^p c_4}{N-2}\int _{S}\varepsilon _k^{p-1}|v-\widetilde{v}|^p\, \textrm{d}x\,. \end{aligned}$$

Thus, upon choosing N large enough so that $\frac{{\widehat{M}}}{N-2}<\eta$, since $0\le v,\widetilde{v}\le 1$ we obtain (5.1) by setting $(\hat{v},\hat{v}){:}{=}(\hat{u}^{i^*},\hat{v}^{i^*})$ and $M_\eta {:}{=}\frac{3^{p-1}M^p}{N-2}\big (c_2+2c_4{\mathcal {L}}^n(S)\big )$. $\square$

On account of the fundamental estimate, Proposition 5.1, we are now in a position to prove the following $\Gamma$-convergence result.

Theorem 5.2

Let ${\mathscr {F}}_k$ be as in (2.4). Then there exist a subsequence $({\mathscr {F}}_{k_j})$ of $({\mathscr {F}}_k)$ and a functional ${\mathscr {F}}:L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)\times {\mathcal {A}}\longrightarrow [0,+\infty ]$ such that for every $A\in {\mathcal {A}}$ the functionals ${\mathscr {F}}_{k_j}(\cdot \,,\cdot \,,A)$ $\Gamma$-converge in $L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ to ${\mathscr {F}}(\cdot ,\cdot ,A)$. Moreover, ${\mathscr {F}}$ is given by

$$\begin{aligned} {\mathscr {F}}(u,v,A){:}{=}{\left\{ \begin{array}{ll} \displaystyle \int _A {\hat{f}}(x,\nabla u)\, \textrm{d}x+\int _{S_u\cap A}{\hat{g}}(x,[u],\nu _u)\, \textrm{d}{\mathcal {H}}^{n-1}\, &{}\text {if}\, u \in GSBV^p(A;{\mathbb {R}}^m),\, v= 1\, \text {a.e. in } A,\\ +\infty &{}\text {otherwise}, \end{array}\right. } \end{aligned}$$

with $\hat{f}:{\mathbb {R}}^n\times {\mathbb {R}}^{m\times n}\rightarrow [0,+\infty )$, $\hat{g}:{\mathbb {R}}^n\times {\mathbb {R}}_0^m\times {\mathbb {S}}^{n-1}\rightarrow [0,+\infty )$ given by

$$\begin{aligned} {\hat{f}}(x,\xi )&:=\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n}}{\textbf{m}}(u_\xi , Q_\rho (x))\,, \end{aligned}$$

(5.11)

$$\begin{aligned} {\hat{g}}(x,\zeta ,\nu )&:=\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}{\textbf{m}}(u_{x,\zeta }^\nu ,Q_\rho ^\nu (x))\,, \end{aligned}$$

(5.12)

for every $x\in {\mathbb {R}}^n$, $\xi \in {\mathbb {R}}^{m\times n}$, $\zeta \in {\mathbb {R}}_0^m$, and $\nu \in {{\mathbb {S}}}^{n-1}$, where for $A\in {\mathcal {A}}$ and $\bar{u}\in SBV^p(A;{\mathbb {R}}^m)$

$$\begin{aligned} {\textbf{m}}(\bar{u},A){:}{=}\inf \{{\mathscr {F}}(u,1,A):u\in SBV^p(A;{\mathbb {R}}^m)\, ,\ u=\bar{u}\ \text {near}\ \partial A\}\,. \end{aligned}$$

Proof

Let ${\mathscr {F}}',{\mathscr {F}}'' :L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)\times {\mathcal {A}}\longrightarrow [0,+\infty ]$ be the functionals defined as

$$\begin{aligned} {\mathscr {F}}'(\,\cdot ,\,\cdot ,A) {:}{=}\Gamma \hbox {-}\liminf _{k \rightarrow +\infty }{\mathscr {F}}_{k}(\,\cdot ,\,\cdot ,A)\quad \text {and}\quad {\mathscr {F}}''(\,\cdot ,\,\cdot ,A) {:}{=}\Gamma \hbox {-}\limsup _{k \rightarrow +\infty }{\mathscr {F}}_{k}(\,\cdot ,\,\cdot ,A). \end{aligned}$$

In view of Remark 2.2 we can invoke [36, Theorem 3.1] to deduce the existence of a constant $C>0$ such that

$$\begin{aligned} \begin{aligned} \frac{1}{C}\Big (\int _A|\nabla u|^p\, \textrm{d}x+{\mathcal {H}}^{n-1}(S_u\cap A)\Big )&\le {\mathscr {F}}'(u,1,A)\\&\le {\mathscr {F}}''(u,1,A)\\&\le C\Big (\int _A|\nabla u|^p\, \textrm{d}x+{\mathcal {H}}^{n-1}(S_u\cap A)\Big ), \end{aligned} \end{aligned}$$

(5.13)

for every $A\in {\mathcal {A}}$ and every $u\in GSBV^p(A;{\mathbb {R}}^m)$; moreover

$$\begin{aligned} {\mathscr {F}}'(u,v,A)={\mathscr {F}}''(u,v,A)= +\infty \quad \text {if either } \; u\notin GSBV^p(A;{\mathbb {R}}^m) \; \text { or } \; v\ne 1. \end{aligned}$$

(5.14)

By the general properties of $\Gamma$-convergence we know that for every $A\in {\mathcal {A}}$ fixed the functionals ${\mathscr {F}}'(\cdot ,\cdot ,A)$ and ${\mathscr {F}}''(\cdot ,\cdot ,A)$ are $L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ lower semicontinuous [32, Proposition 6.8] and local [32, Proposition 16.15]. Further, the set functions ${\mathscr {F}}'(u,v,\cdot )$ and ${\mathscr {F}}''(u,v,\cdot )$ are increasing [32, Proposition 6.7] and ${\mathscr {F}}'(u,v,\cdot )$ is superadditive [32, Proposition 16.12].

Invoking [32, Theorem 16.9] we can deduce the existence of a subsequence $(k_j)$, with $k_j \rightarrow +\infty$ as $j \rightarrow +\infty$, such that the corresponding ${\mathscr {F}}'$ and ${\mathscr {F}}''$ also satisfy

$$\begin{aligned} \sup _{A'\subset \subset A,\, A'\in {\mathcal {A}}}{\mathscr {F}}'(u,v,A')\quad =\sup _{A'\subset \subset A,\, A'\in {\mathcal {A}}}{\mathscr {F}}''(u,v,A')=:{\mathscr {F}}(u,v,A), \end{aligned}$$

(5.15)

for every $(u,v)\in L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ and for every $A\in {\mathcal {A}}$. We notice that the set function ${\mathscr {F}}(u,v,\cdot )$ given by (5.15) is inner regular by definition. Moreover ${\mathscr {F}}$ satisfies the following properties: the functional ${\mathscr {F}}(\cdot ,\cdot ,A)$ is $L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ lower semicontinuous [32, Remark 15.10] and local [32, Remark 15.25], while the set function ${\mathscr {F}}(u,v,\cdot )$ is increasing and superadditive [32, Remark 15.10].

Thanks to the fundamental estimate Proposition 5.1 we can appeal to [32, Proposition 18.4] to deduce that ${\mathscr {F}}(u,v,\cdot )$ is also a subadditive set function. Here the only difference with a standard situation is that the reminder in (5.1) is infinitesimal with respect to the $L^p({\mathbb {R}}^n;{\mathbb {R}}^m)$ convergence in u while we are considering the $\Gamma$-convergence of ${\mathscr {F}}_{k_j}$ in $L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$. However, this issue can be easily overcome by resorting to a truncation argument together with a sequential characterisation of ${\mathscr {F}}$ (see e.g., [32, Proposition 16.4 and Remark 16.5]), which holds true on $SBV^p(A;{\mathbb {R}}^m)\cap L^\infty (A;{\mathbb {R}}^m)$. Hence, we can now invoke the measure-property criterion of De Giorgi and Letta (see e.g., [32, Theorem 14.23]) to deduce that for every $(u,v)\in L^0({\mathbb {R}}^n;{\mathbb {R}}^m) \times L^0({\mathbb {R}}^n)$ the set function ${\mathscr {F}}(u,v,\cdot )$ is the restriction to ${\mathcal {A}}$ of a Borel measure.

Furthermore, (5.13) together with [32, Proposition 18.6] and Proposition 5.1 imply that

$$\begin{aligned} {\mathscr {F}}(u,1,A)={\mathscr {F}}'(u,1,A)={\mathscr {F}}''(u,1,A) \quad \text {if}\quad u\in GSBV^p(A;{\mathbb {R}}^m), \end{aligned}$$

while, gathering (5.13) and (5.14) we may also deduce that

$$\begin{aligned} {\mathscr {F}}(u,v,A)={\mathscr {F}}'(u,v,A)={\mathscr {F}}''(u,v,A)=+\infty \quad \text {if either } \; u\notin GSBV^p(A;{\mathbb {R}}^m) \; \text { or } \; v\ne 1. \end{aligned}$$

As a consequence, ${\mathscr {F}}(\cdot ,\cdot , A)$ coincides with the $\Gamma$-limit of ${\mathscr {F}}_{k_j}(\cdot ,\cdot ,A)$ on $L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$, for every $A\in {\mathcal {A}}$.

By [17, Theorem 1] and a standard perturbation and truncation argument (see e.g., [27, Theorem 4.3]), for every $A\in {\mathcal {A}}$ and $u\in GSBV^p(A;{\mathbb {R}}^m)$ we can represent the $\Gamma$-limit ${\mathscr {F}}$ in an integral form as

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

for some Borel functions ${\hat{f}}$ and ${\hat{g}}$. Eventually, thanks to (5.13), it can be easily shown that ${\hat{f}}$ and ${\hat{g}}$ are given by the same derivation formulas provided by [17, Theorem 1], that is, they coincide with (5.11) and (5.12), respectively. $\square$

6 Identification of the volume integrand

In this section we identify the volume integrand ${\hat{f}}$. Namely, we prove that ${\hat{f}}$ coincides with both $f'$ and $f''$, given by (2.12) and (2.13), respectively. This shows, in particular, that the limit volume integrand ${\hat{f}}$ depends only on $f_k$, and hence only on ${\mathscr {F}}^b_k$.

Proposition 6.1

Let $(f_k) \subset {\mathcal {F}}$ and $(g_k)\subset {\mathcal {G}}$. Let $(k_j)$ and ${\hat{f}}$ be as in Theorem 5.2. Then it holds

$$\begin{aligned} {\hat{f}}(x,\xi )=f'(x,\xi ) = f''(x,\xi ), \end{aligned}$$

for a.e. $x\in {\mathbb {R}}^n$ and for every $\xi \in {\mathbb {R}}^{m\times n}$, where $f'$ and $f''$ are, respectively, as in (2.12) and (2.13) with k replaced by $k_j$.

Proof

For notational simplicity, in what follows we still let k denote the index of the sequence provided by Theorem 5.2.

By definition $f'\le f''$, hence to prove the claim it suffices to show that

$$\begin{aligned} f''(x,\xi )\le {\hat{f}}(x,\xi ) \le f' (x,\xi ), \end{aligned}$$

(6.1)

for a.e. $x\in {\mathbb {R}}^n$ and for every $\xi \in {\mathbb {R}}^{m\times n}$. We divide the proof of (6.1) into two steps.

Step 1: In this step we show that ${\hat{f}}(x,\xi ) \ge f''(x,\xi )$ for a.e. $x\in {\mathbb {R}}^n$ and for every $\xi \in {\mathbb {R}}^{m\times n}$.

By Theorem 5.2 we have that

$$\begin{aligned} \int _A{\hat{f}}(x,\xi )\, \textrm{d}x= {\mathscr {F}}(u_\xi ,1,A), \end{aligned}$$

(6.2)

for every $A\in {\mathcal {A}}$ and for every $\xi \in {\mathbb {R}}^n$.

Now let $x \in {\mathbb {R}}^n$ be arbitrary, let $A\in {\mathcal {A}}$ be such that $x\in A$, and let $\rho >0$ be so small that $Q_\rho (x)\subset A$. By $\Gamma$-convergence we can find $(u_k,v_k) \subset L^0({\mathbb {R}}^n;{\mathbb {R}}^m) \times L^0({\mathbb {R}}^n)$ which is a recovery sequence for ${\mathscr {F}}(u_\xi ,1,A)$ ; i.e., $(u_k, v_k) \subset W^{1,p}(A;{\mathbb {R}}^m) \times W^{1,p}(A)$, $0\le v\le 1$, $(u_k,v_k)$ converges to $(u_\xi ,1)$ in measure on bounded sets and

$$\begin{aligned} \lim _{k \rightarrow +\infty }{\mathscr {F}}_{k}(u_k,v_k,A)={\mathscr {F}}(u_\xi ,1,A). \end{aligned}$$

(6.3)

Moreover, by (g2) we also have $v_k \rightarrow 1$ in $L^p(A)$.

We notice that $(u_k,v_k)$ also satisfies

$$\begin{aligned} \lim _{k \rightarrow +\infty }{\mathscr {F}}_{k}(u_k,v_k,Q_{\rho }(x))={\mathscr {F}}(u_\xi ,1,Q_{\rho }(x))\,. \end{aligned}$$

(6.4)

Indeed, thanks to (6.2) we have

$$\begin{aligned} {\mathscr {F}}(u_\xi ,1,A)= {\mathscr {F}}(u_\xi ,1,Q_{\rho }(x))+{\mathscr {F}}(u_\xi ,1,A \setminus {{\overline{Q}}}_{\rho }(x))\,. \end{aligned}$$

Therefore, again by $\Gamma$-convergence we get

$$\begin{aligned} \liminf _{k\rightarrow +\infty } {\mathscr {F}}_{k}(u_k,v_k,Q_{\rho }(x))\ge {\mathscr {F}}(u_\xi ,1,Q_{\rho }(x)) \end{aligned}$$

and

$$\begin{aligned} \liminf _{k\rightarrow +\infty } {\mathscr {F}}_{k}(u_k,v_k,A \setminus {{\overline{Q}}}_{\rho }(x)) \ge {\mathscr {F}}(u_\xi ,1,A \setminus {{\overline{Q}}}_{\rho }(x))\,. \end{aligned}$$

Hence (6.4) follows by (6.3).

We now estimate separately the surface and bulk term in ${\mathscr {F}}_{k}$. We notice that by Young’s Inequality we have

$$\begin{aligned} \frac{p-1}{p}\frac{(1-v_k)^p}{\varepsilon _k}+\frac{1}{p}\varepsilon _k^{p-1}|\nabla v_k|^p \ge (1-v_k)^{p-1}|\nabla v_k|. \end{aligned}$$

Hence, thanks to (g2), by using the co-area formula we get that

$$\begin{aligned} {\mathscr {F}}_{k}^s(v_k,Q_\rho (x))&\ge c_3 \int _{Q_\rho (x)}(1-v_k)^{p-1}|\nabla v_k|\, \textrm{d}y= c_3 \int _0^{1} (1-t)^{p-1}\, {\mathcal {H}}^{n-1}(\partial ^*E^t_{k,\rho })\, \textrm{d}t\,, \end{aligned}$$

where $E_{k,\rho }^t{:}{=}\{y\in Q_\rho (x):v_k(y)<t\}$. Now let $\eta \in (0,1)$ be fixed and $\eta '\in (\eta ,1)$; then by the mean-value Theorem we deduce the existence of ${\bar{t}}={\bar{t}}(k,\rho ,\eta ,\eta ')$, ${\bar{t}} \in (\eta ,\eta ')$ such that

$$\begin{aligned} {\mathscr {F}}_{k}^s(v_k,Q_\rho (x)) \ge c_3\int _\eta ^{\eta '} (1-t)^{p-1} \, \textrm{d}t\, {\mathcal {H}}^{n-1}(\partial ^*E^{{\bar{t}}}_{k,\rho })\,. \end{aligned}$$

Set $w_k:=u_k \chi _{{\mathbb {R}}^n \setminus E^{{\bar{t}}}_{k,\rho }}$; since $u_k\in W^{1,p}(A;{\mathbb {R}}^m)$ and $E^{{\bar{t}}}_{k,\rho }$ is a set of finite perimeter, we have that $w_k \in GSBV^p(A;{\mathbb {R}}^m)$ and ${\mathcal {H}}^{n-1}(S_{w_k})\le {\mathcal {H}}^{n-1}(\partial ^*E^{{\bar{t}}}_{k,\rho })$. Moreover, since $v_k \rightarrow 1$ in $L^p(A)$ we have that ${\mathcal {L}}^n(E^{{\bar{t}}}_{k,\rho }) \rightarrow 0$ as $k \rightarrow +\infty$ so that $w_k \rightarrow u_\xi$ in measure on bounded sets. Eventually, since ${\mathscr {F}}_{k}$ is increasing as set function, using the fact that $\psi$ is increasing we obtain

$$\begin{aligned} {\mathscr {F}}_{k}(u_k,v_k,Q_{\rho }(x))&= {\mathscr {F}}_{k}^b(u_k,v_k,Q_\rho (x))+{\mathscr {F}}_{k}^s(v_k,Q_\rho (x))\\&\ge \psi (\eta )\int _{Q_\rho (x)\setminus E^{{\bar{t}}}_{k,\rho }}\hspace{-1em} f_{k}(y,\nabla u_k)\, \textrm{d}y+ c_3\int _\eta ^{\eta '} (1-t)^{p-1}\, \textrm{d}t\, {\mathcal {H}}^{n-1}(\partial ^*E^{{\bar{t}}}_{k,\rho })\,\\&\ge \psi (\eta )\int _{Q_\rho (x)}\hspace{-1em} f_{k}(y,\nabla w_k)\, \textrm{d}y+ c_3\int _\eta ^{\eta '} (1-t)^{p-1}\, \textrm{d}t\, {\mathcal {H}}^{n-1}(S_{w_k})\, , \end{aligned}$$

where the last inequality follows from the definition of $w_k$.

In view of 2.2–(f4) the right-hand side belongs to the class of functionals considered in [27]. Then, thanks to [27, Theorem 3.5 and Theorem 5.2 (b)] we have

$$\begin{aligned} \lim _{k \rightarrow +\infty }{\mathscr {F}}_{k}(u_{k},v_{k},Q_{\rho }(x)) \ge \psi (\eta )\int _{Q_\rho (x)}f''(y,\xi )\, \textrm{d}y\,, \end{aligned}$$

where $f''$ is given by (2.13).

Hence appealing to (6.4) gives

$$\begin{aligned} {\mathscr {F}}(u_\xi ,1,Q_{\rho }(x))\ge \psi (\eta )\int _{Q_\rho (x)}f''(y,\xi )\, \textrm{d}y. \end{aligned}$$

Then, by dividing both terms in the above inequality by $\rho ^n$ and using (6.2), we obtain

$$\begin{aligned} \frac{1}{\rho ^n}\int _{Q_\rho (x)} {\hat{f}}(y,\xi )\, \textrm{d}y\ge \psi (\eta )\, \frac{1}{\rho ^n}\int _{Q_\rho (x)} f''(y,\xi )\, \textrm{d}y\,. \end{aligned}$$

Thus invoking the Lebesgue differentiation Theorem together with the continuity in $\xi$ of ${\hat{f}}$ and $f''$ (see [17] and Proposition 4.1) we deduce that

$$\begin{aligned} {\hat{f}}(x,\xi )\ge \psi (\eta )\, f''(x,\xi ), \end{aligned}$$

for a.e. $x\in {\mathbb {R}}^{n}$, for every $\xi \in {\mathbb {R}}^{m\times n}$, and for every $\eta \in (0,1)$. Eventually, since $\psi$ is continuous and $\psi (1)=1$, the claim follows by letting $\eta ,\eta ' \rightarrow 1$.

Step 2: In this step we show that ${\hat{f}}(x,\xi ) \le f'(x,\xi )$ for every $x\in {\mathbb {R}}^n$ and for every $\xi \in {\mathbb {R}}^{m\times n}$. The proof is similar to that of [27, Theorem 5.2]. However, we repeat it here for the readers’ convenience.

Let $x\in {\mathbb {R}}^n$, $\xi \in {\mathbb {R}}^{m\times n}$, $\rho >0$, and $\eta >0$ be fixed. By (2.8) for every $k\in {\mathbb {N}}$ fixed we can find $u_k\in W^{1,p}(Q_\rho (x);{\mathbb {R}}^m)$ with $u_k=u_\xi$ near $\partial Q_\rho (x)$ such that

$$\begin{aligned} {\mathscr {F}}_{k}^b(u_k,1, Q_\rho (x))\le {\textbf{m}}_{k}^b(u_\xi ,Q_\rho (x))+\eta \rho ^n\,. \end{aligned}$$

(6.5)

Combining (6.5) with (f2) and (f3) yields

$$\begin{aligned} c_1\Vert \nabla u_k\Vert _{L^{p}(Q_\rho (x);{\mathbb {R}}^{m\times n})}^p\le {\mathscr {F}}_{k}^b(u_k,1,Q_\rho (x))\le \rho ^n(c_2|\xi |^p+\eta ), \end{aligned}$$

(6.6)

where the second inequality follows by taking $u_\xi$ as a test in the definition of ${\textbf{m}}_{k}^b(u_\xi ,Q_\rho (x))$. Let now $(k_j)$ be a diverging sequence such that

$$\begin{aligned} \lim _{j\rightarrow +\infty }{\mathscr {F}}_{{k_j}}^b(u_{k_j},1,Q_\rho (x))=\liminf _{k\rightarrow +\infty }{\mathscr {F}}_{k}^b(u_k,1,Q_\rho (x))\, . \end{aligned}$$

Since $u_{k_j}-u_\xi \in W_0^{1,p}(Q_\rho (x);{\mathbb {R}}^m)$, the uniform bound (6.6) together with the Poincaré Inequality provide us with a further subsequence (not relabelled) and a function $u\in W^{1,p}(Q_\rho (x);{\mathbb {R}}^m)$ such that $u_{k_j}\rightharpoonup u$ weakly in $W^{1,p}(Q_\rho (x);{\mathbb {R}}^m)$. Then, by the Rellich Theorem $u_{k_j}\rightarrow u$ in $L^p(Q_\rho (x);{\mathbb {R}}^m)$. We now extend u and $u_k$ to functions $w,w_k\in W^{1,p}_{\textrm{loc}}({\mathbb {R}}^n;{\mathbb {R}}^m)$ by setting

$$\begin{aligned} w{:}{=}{\left\{ \begin{array}{ll} u &{}\text {in}\ Q_\rho (x)\,,\\ u_\xi &{}\text {in}\ {\mathbb {R}}^n\setminus Q_\rho (x)\,, \end{array}\right. } \quad w_k{:}{=}{\left\{ \begin{array}{ll} u_k &{}\text {in}\ Q_\rho (x)\,,\\ u_\xi &{}\text {in}\ {\mathbb {R}}^n\setminus Q_\rho (x)\,, \end{array}\right. } \end{aligned}$$

respectively; clearly, $w=u_\xi$ in a neighbourhood of $\partial Q_{(1+\eta )\rho }(x)$ and $w_{k_j}\rightarrow w$ in $L^p_{\textrm{loc}}({\mathbb {R}}^n;{\mathbb {R}}^m)$. Hence by $\Gamma$-convergence, by (2.3) and (6.5) we get

$$\begin{aligned} \begin{aligned} {\textbf{m}}(u_\xi ,Q_{(1+\eta )\rho }(x))&\le {\mathscr {F}}(w,1,Q_{(1+\eta )\rho }(x))\le \lim _{j\rightarrow +\infty }{\mathscr {F}}_{{k_j}}(w_{k_j},1,Q_{(1+\eta )\rho }(x))\\&\le \lim _{j\rightarrow +\infty }{\mathscr {F}}_{{k_j}}^b(u_{k_j},Q_\rho (x))+c_2|\xi |^p\big ((1+\eta )^n-1\big )\rho ^n\\&\le \liminf _{k\rightarrow +\infty }{\textbf{m}}_{k}^b(u_\xi ,Q_\rho (x))+\eta \rho ^n+c_2|\xi |^p\big ((1+\eta )^n-1\big )\rho ^n. \end{aligned} \end{aligned}$$

Eventually, dividing by $\rho ^n$, passing to the limsup as $\rho \rightarrow 0$, and recalling the definition of ${\hat{f}}$ and $f'$ we get

$$\begin{aligned} (1+\eta )^n\hat{f}(x,\xi )\le f'(x,\xi )+\eta +c_2|\xi |^p\big ((1+\eta )^n-1\big )\, , \end{aligned}$$

and hence the claim follows by the arbitrariness of $\eta >0$. $\square$

7 Identification of the surface integrand

In this section we identify the surface integrand ${\hat{g}}$. Namely, we show that ${\hat{g}}$ coincides with both $g'$ and $g''$, given by (2.14) and (2.15), respectively. This shows, in particular, that the limit surface integrand ${\hat{g}}$ is obtained by minimising only the surface term ${\mathscr {F}}_k^s$. We notice, however, that in this case the presence the bulk term ${\mathscr {F}}_k^b$ affects the class of test functions over which the minimisation is performed (cf. (2.9)–(2.10)).

We start by proving some preliminary lemmas. The first lemma concerns the approximation of a minimisation problem involving the $\Gamma$-limit ${\mathscr {F}}$.

Lemma 7.1

(Approximation of minimum values) Let $(f_k) \subset {\mathcal {F}}$ and $(g_k)\subset {\mathcal {G}}$. Let $\rho >0$; for $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}_0^m$, $\nu \in {{\mathbb {S}}}^{n-1}$, and $k\in {\mathbb {N}}$ such that $2\varepsilon _k<\rho$ set

$$\begin{aligned} {\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)):= & {} \inf \{{\mathscr {F}}_k(u,v,Q_\rho ^\nu (x)):(u,v)\in W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)\times W^{1,p}(Q_\rho ^\nu (x)),\; \\ 0\le & {} v\le 1, (u,v)=({\bar{u}}_{x,\zeta ,\varepsilon _k}^{\nu },{\bar{v}}_{x,\varepsilon _k}^{\nu })\ \text {near}\ \partial Q_\rho ^\nu (x)\}\,. \end{aligned}$$

Let $(k_j)$ be as in Theorem 5.2 and $\hat{g}$ be as in (5.12). Then for every $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}_0^m$, and $\nu \in {{\mathbb {S}}}^{n-1}$ it holds

$$\begin{aligned} \hat{g}(x,\zeta ,\nu )&=\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\liminf _{j\rightarrow +\infty }{\textbf{m}}_{k_j}({\bar{u}}_{x,\zeta ,\varepsilon _{k_j}}^\nu ,Q_\rho ^\nu (x))\nonumber \\&=\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\limsup _{j\rightarrow +\infty }{\textbf{m}}_{k_j}({\bar{u}}_{x,\zeta ,\varepsilon _{k_j}}^\nu ,Q_\rho ^\nu (x)). \end{aligned}$$

(7.1)

Proof

For notational simplicity, in what follows we still denote with k the index of the (sub)sequence provided by Theorem 5.2.

We divide the proof into two steps.

Step 1: In this step we show that

$$\begin{aligned} \hat{g}(x,\zeta ,\nu )\le \limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\liminf _{k\rightarrow +\infty }{\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)), \end{aligned}$$

(7.2)

for every $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}_0^m$, and $\nu \in {{\mathbb {S}}}^{n-1}$.

Let $\rho >0$ and $\eta >0$ be fixed; by definition of ${\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$ there exists $(u_k,v_k) \subset W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)\times W^{1,p}(Q_\rho ^\nu (x))$ such that $(u_k,v_k)=({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu \,,{\bar{v}}_{x,\varepsilon _k}^\nu )$ in a neighbourhood of $\partial Q_\rho ^\nu (x)$, and

$$\begin{aligned} {\mathscr {F}}_{k}(u_k,v_k,Q_\rho ^\nu (x)) \le {\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)) +\eta \rho ^{n-1}. \end{aligned}$$

(7.3)

Since the pair $({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,{\bar{v}}_{x,\varepsilon _k}^\nu )$ is admissible for ${\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$, then (2.11) and (2.19) readily give

$$\begin{aligned} {\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\le {\mathscr {F}}_k(\bar{u}_{x,\zeta ,\varepsilon _k}^\nu ,\bar{v}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))={\mathscr {F}}_k^s(\bar{v}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\le c_4C_\textrm{v}\rho ^{n-1}. \end{aligned}$$

(7.4)

By a truncation argument (see e.g., [23, Lemma 3.5] or [27, Lemma 4.1]) it is not restrictive to assume that $\sup _k \Vert u_k\Vert _{L^\infty (Q^\nu _{\rho }(x);{\mathbb {R}}^m)}<+\infty$. We now extend $u_k$ to a $W^{1,p}_{\textrm{loc}}({\mathbb {R}}^n;{\mathbb {R}}^m)$-function by setting

$$\begin{aligned} w_k{:}{=}{\left\{ \begin{array}{ll} u_k &{}\text {in } Q_\rho ^\nu (x)\,,\\ {\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu &{}\text {in } {\mathbb {R}}^n\setminus Q_\rho ^\nu (x)\,, \end{array}\right. } \end{aligned}$$

then, clearly, $\sup _k \Vert w_k\Vert _{L^\infty ({\mathbb {R}}^n;{\mathbb {R}}^m)}<+\infty$. Now let $({k_j})$ be such that

$$\begin{aligned} \lim _{j\rightarrow +\infty } {\mathscr {F}}_{{k_j}}(u_{k_j},v_{k_j},Q_\rho ^\nu (x)) = \liminf _{k \rightarrow +\infty }{\mathscr {F}}_{k}(u_k,v_k,Q_\rho ^\nu (x)). \end{aligned}$$

In view of (7.4), (2.5), and the uniform $L^\infty ({\mathbb {R}}^n;{\mathbb {R}}^m)$-bound on $w_{k_j}$ we can invoke [36, Lemma 4.1] to deduce the existence of a subsequence (not relabelled) such that

$$\begin{aligned} (w_{k_j}, v_{k_j}) \rightarrow (u, 1)\quad \text {in}\quad L^p_{\textrm{loc}}({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^p_{\textrm{loc}}({\mathbb {R}}^n), \end{aligned}$$

for some $u\in L^p_{\textrm{loc}}({\mathbb {R}}^n;{\mathbb {R}}^m)$ also belonging to $SBV^p(Q^\nu _{(1+\eta )\rho }(x);{\mathbb {R}}^m)$. Moreover, we also have $u=u_{x,\zeta }^\nu$ in a neighbourhood of $\partial Q^\nu _{(1+\eta )\rho }(x)$, so that

$$\begin{aligned} {\textbf{m}}(u^\nu _{x,\zeta },Q^\nu _{(1+\eta )\rho }(x)) \le {\mathscr {F}}(u,1,Q^\nu _{(1+\eta )\rho }(x)). \end{aligned}$$

(7.5)

Eventually, by $\Gamma$-convergence together with (7.3) we obtain

$$\begin{aligned} {\mathscr {F}}(u,1,Q^\nu _{(1+\eta )\rho }(x))&\le \liminf _{j\rightarrow +\infty } {\mathscr {F}}_{{k_j}}(w_{k_j},v_{k_j},Q_{(1+\eta )\rho }^\nu (x))\\&\le \lim _{j \rightarrow +\infty } {\mathscr {F}}_{k_j}(u_{k_j},v_{k_j},Q_{\rho }^\nu (x))+c_4C_\textrm{v}\big ((1+\eta )^{n-1}-1)\rho ^{n-1}\\&\le \liminf _{k\rightarrow +\infty }{\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))+\eta \rho ^{n-1}+c_4C_\textrm{v}\big ((1+\eta )^{n-1}-1\big )\rho ^{n-1}\,. \end{aligned}$$

Thus, using (7.5), dividing the above inequality by $\rho ^{n-1}$ and passing to the limsup as $\rho \rightarrow 0$ we obtain

$$\begin{aligned} (1+\eta )^{n-1}\hat{g}(x,\zeta ,\nu )\le \limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\liminf _{k\rightarrow +\infty }{\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))+\eta +c_4C_\textrm{v}\big ((1+\eta )^{n-1}-1\big ), \end{aligned}$$

hence (7.2) follows by the arbitrariness of $\eta >0$.

Step 2: In this step we show that

$$\begin{aligned} \limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\limsup _{k\rightarrow +\infty }{\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)) \le \hat{g}(x,\zeta ,\nu ), \end{aligned}$$

(7.6)

for every $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}_0^m$, $\nu \in {{\mathbb {S}}}^{n-1}$, and $\rho >0$. To this end, we fix $\eta >0$ and we choose $u\in SBV^p(Q_\rho ^\nu (x);{\mathbb {R}}^m)$ with $u=u_{x,\zeta }^\nu$ near $\partial Q_\rho ^\nu (x)$ and

$$\begin{aligned} {\mathscr {F}}(u,1,Q_\rho ^\nu (x))\le {\textbf{m}}(u_{x,\zeta }^\nu ,Q_\rho ^\nu (x))+\eta . \end{aligned}$$

(7.7)

We extend u to the whole ${\mathbb {R}}^n$ by setting $u=u_{x,\zeta }^\nu$ in ${\mathbb {R}}^n\setminus Q_\rho ^\nu (x)$. Then, by $\Gamma$-convergence there exists a sequence $(u_k,v_k)$ converging to (u, 1) in measure on bounded sets such that

$$\begin{aligned} \lim _{k \rightarrow +\infty }{\mathscr {F}}_{k}(u_k,v_k,Q_\rho ^\nu (x))= {\mathscr {F}}(u,1,Q_\rho ^\nu (x)). \end{aligned}$$

(7.8)

We notice, moreover, that thanks to a truncation argument (both on u and $u_k$) and to the bound (g2), it is not restrictive to assume that $(u_k,v_k)$ converges to (u, 1) in $L^p_{\textrm{loc}}({\mathbb {R}}^n;{\mathbb {R}}^m) \times L^p_{\textrm{loc}}({\mathbb {R}}^n)$. We now modify the sequence $(u_k,v_k)$ in such a way that it satisfies the boundary conditions required in the definition of ${\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$. This will be done by resorting to the fundamental estimate Proposition 5.1. Namely, we choose $0<\rho ''<\rho '<\rho$ such that $u=u_{x,\zeta }^\nu$ on $Q_\rho ^\nu (x)\setminus {\overline{Q}}_{\rho ''}^\nu (x)$ and we apply Proposition 5.1 with $A=Q_{\rho ''}^\nu (x)$, $A'=Q_{\rho '}^\nu (x)$, $B=Q_\rho ^\nu (x)\setminus {\overline{Q}}_{\rho ''}^\nu (x)$. In this way, we obtain a sequence $(\hat{u}_k,\hat{v}_k)\subset W^{1,p}_\textrm{loc}({\mathbb {R}}^n;{\mathbb {R}}^m)\times W^{1,p}_\textrm{loc}({\mathbb {R}}^n)$ converging to (u, v) in $L^p(Q_\rho ^\nu (x);{\mathbb {R}}^m)\times L^p(Q_\rho ^\nu (x))$ such that $(\hat{u}_k,\hat{v}_k)=(u_k,v_k)$ in $Q_{\rho ''}^\nu (x)$, $(\hat{u}_k,\hat{v}_k)=({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,{\bar{v}}_{x,\varepsilon _k}^\nu )$ in $Q_\rho ^\nu (x)\setminus {\overline{Q}}_{\rho '}^\nu (x)$, and

$$\begin{aligned} \begin{aligned} \limsup _{k\rightarrow +\infty }{\mathscr {F}}_{k}(\hat{u}_k,\hat{v}_k,Q_\rho ^\nu (x))&\le (1+\eta )\limsup _{k\rightarrow +\infty } \Big ({\mathscr {F}}_{k}(u_k,v_k,Q_{\rho '}^\nu (x))\\&\quad +{\mathscr {F}}_{k}^s({\bar{v}}_{x,\varepsilon _k}^\nu , Q_{\rho }^\nu (x)\setminus {\overline{Q}}_{\rho ''}^\nu (x))\Big )\\&\le (1+\eta ){\mathscr {F}}(u,1,Q_\rho ^\nu (x)) +c_4C_\textrm{v}{\mathcal {L}}^{n-1}\big (Q_{\rho }'\setminus \overline{Q'}_{\!\!\rho ''}\big ), \end{aligned} \end{aligned}$$

(7.9)

where the second inequality follows from (7.8) and (2.19) together with (2.3), respectively.

Eventually, since $({\hat{u}}_k,{\hat{v}}_k)$ is admissible in the definition of ${\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$, gathering (7.7) and (7.9) we deduce that

$$\begin{aligned} \limsup _{k\rightarrow +\infty }{\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))&\le (1+\eta )\big ({\textbf{m}}(u_{x,\zeta }^\nu ,Q_\rho ^\nu (x))+\eta \big )+C (\rho ^{n-1}-(\rho '')^{n-1})\,. \end{aligned}$$

Then letting $\rho ''\rightarrow \rho$, by the arbitrariness of $\eta >0$ we get

$$\begin{aligned} \limsup _{k\rightarrow +\infty }{\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\le {\textbf{m}}(u_{x,\zeta }^\nu ,Q_\rho ^\nu (x)), \end{aligned}$$

(7.10)

therefore (7.6) follows by dividing both sides of (7.10) by $\rho ^{n-1}$ and passing to the limsup as $\rho \rightarrow 0$. $\square$

Remark 7.2

Thanks to the $(W^{1,p}(A;{\mathbb {R}}^m)\times W^{1,p}(A))$-continuity of ${\mathscr {F}}_k(\,\cdot ,\,\cdot ,A)$ (cf. Remark 2.3), a standard convolution argument shows that (7.1) holds also true if the minimisation in ${\textbf{m}}_k$ is carried over $C^1(Q_\rho ^\nu (x);{\mathbb {R}}^m)\times C^1(Q_\rho ^\nu (x))$.

The following lemma shows that if v is “small” in some region, then it can be replaced by a function which is equal to zero in that region, without essentially increasing ${\mathscr {F}}_k^s$.

Lemma 7.3

Let $(g_k)\subset {\mathcal {G}}$, $A\in {\mathcal {A}}$, $v\in W^{1,p}(A)$, and $\eta \in (0,1)$ be fixed. Let $v^\eta \in W^{1,p}(A)$ be defined as

$$\begin{aligned} v^\eta {:}{=}\min \{((1+\sqrt{\eta })v-\eta )^+\, ,\ v\}. \end{aligned}$$

(7.11)

Then for a.e. $x\in A$ we have

$$\begin{aligned} v^\eta (x)=0\;\text { iff }\; v(x)\le \frac{\eta }{1+\sqrt{\eta }}\qquad \text {and}\qquad v^\eta (x)=v(x)\;\text { iff }\; v(x)\ge \sqrt{\eta }. \end{aligned}$$

(7.12)

Moreover, $v^\eta$ satisfies

$$\begin{aligned} {\mathscr {F}}_k^s(v^\eta ,A)\le (1+c_\eta ){\mathscr {F}}_k^s(v,A)\,, \end{aligned}$$

(7.13)

where $c_\eta >0$ is independent of v and A and such that $c_\eta \rightarrow 0$ as $\eta \rightarrow 0$.

Proof

A direct computation shows that $((1+\sqrt{\eta })v-\eta )^+\le v$ if and only if $v\le \sqrt{\eta }$ and that $((1+\sqrt{\eta })v-\eta )^+= 0$ if and only if $v\le \frac{\eta }{1+\sqrt{\eta }}$ ; i.e., (7.12) is satisfied. Thus it remains to show that (7.13) holds true. To this end we introduce the sets

$$\begin{aligned} A^\eta {:}{=}\Big \{x\in A:v(x)\le \frac{\eta }{1+\sqrt{\eta }}\Big \}\quad \text {and}\quad B^\eta {:}{=}\Big \{x\in A:\frac{\eta }{1+\sqrt{\eta }}< v(x) < \sqrt{\eta }\Big \}\, , \end{aligned}$$

so that $v^\eta =v$ in $A{\setminus }(A^\eta \cup B^\eta )$; we have

$$\begin{aligned} {\mathscr {F}}_k^s(v^\eta ,A)= & {} \frac{1}{\varepsilon _k}\int _{A^\eta }g_k(x,0,0)\, \textrm{d}x+\frac{1}{\varepsilon _k}\int _{B^\eta }g_k(x,v^\eta ,\varepsilon _k\nabla v^\eta )\, \textrm{d}x\nonumber \\{} & {} +\frac{1}{\varepsilon _k}\int _{A\setminus (A^\eta \cup B^\eta )}\hspace{-2em} g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x. \end{aligned}$$

(7.14)

We start by estimating the first term on the right-hand side of (7.14). Since $v\le \frac{\eta }{1+\sqrt{\eta }}<\eta <1$ in $A^\eta$, using (g4) and (g6) we get

$$\begin{aligned} \begin{aligned} g_k(x,0,0)&\le g_k(x,v,0)+L_2(1+v^{p-1})v\\&\le g_k(x,v,\varepsilon _k\nabla v)+2L_2\eta \le g_k(x,v,\varepsilon _k\nabla v)+\frac{2L_2\eta }{(1-\eta )^p}(1-v)^p, \end{aligned} \end{aligned}$$

in $A^\eta$, which together with (g2) yields

$$\begin{aligned} \frac{1}{\varepsilon _k}\int _{A^\eta }g_k(x,0,0)\, \textrm{d}x\le \Big (1+\frac{2 L_2\eta }{c_3(1-\eta )^p}\Big )\frac{1}{\varepsilon _k}\int _{A^\eta }g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x\, . \end{aligned}$$

(7.15)

We now come to estimate the second term on the right-hand side of (7.14). By definition, we have $v^\eta =(1+\sqrt{\eta })v-\eta<v<\sqrt{\eta }$ in $B^\eta$, thus from (g4) we deduce that

$$\begin{aligned} \begin{aligned} g_k(x,v^\eta ,\varepsilon _k\nabla v^\eta )&\le g_k(x,v,\varepsilon _k\nabla v) + L_2\big (1+2\eta ^{\frac{p-1}{2}}\big )(\eta -\sqrt{\eta }v)\\&\quad + L_2\big (1+\varepsilon _k^{p-1}|\nabla v|^{p-1}+\varepsilon _k^{p-1}|\nabla v^\eta |^{p-1}\big )\varepsilon _k|\nabla v-\nabla v^\eta |, \end{aligned} \end{aligned}$$

(7.16)

in $B^\eta$. Furthermore, since $v<\sqrt{\eta }<1$ in $B^\eta$, it also holds

$$\begin{aligned} \big (1+2\eta ^{\frac{p-1}{2}}\big )(\eta -\sqrt{\eta }v)\le 3\eta \le \frac{3\eta }{(1-\sqrt{\eta })^p}(1-v)^p\,, \end{aligned}$$

(7.17)

and similarly

$$\begin{aligned} |\nabla v-\nabla v^\eta |= & {} \sqrt{\eta }|\nabla v|\le \frac{\sqrt{\eta }}{(1-\sqrt{\eta })^{p-1}}(1-v)^{p-1}|\nabla v|\nonumber \\\le & {} \frac{\sqrt{\eta }}{(1-\sqrt{\eta })^{p-1}}\Big (\frac{(1-v)^p}{\varepsilon _k}+\varepsilon _k^{p-1}|\nabla v|^p\Big ), \end{aligned}$$

(7.18)

where the last estimate follows by Young’s inequality. Eventually, we also have

$$\begin{aligned} \varepsilon _k^{p}\big (|\nabla v|^{p-1}+|\nabla v^\eta |^{p-1}\big )|\nabla v-\nabla v^\eta |= & {} \varepsilon _k^p\big (1+(1+\sqrt{\eta })^{p-1}\big )\sqrt{\eta }|\nabla v|^p\nonumber \\\le & {} \varepsilon _k^p\big (1+2^{p-1}\big )\sqrt{\eta }|\nabla v|^p. \end{aligned}$$

(7.19)

Gathering (7.16)–(7.19), dividing by $\varepsilon _k$ and using (g2) give

$$\begin{aligned} \begin{aligned} \frac{1}{\varepsilon _k}\int _{B^\eta }g_k(x,v^\eta ,\varepsilon _k\nabla v^\eta )\, \textrm{d}x&\le \frac{1}{\varepsilon _k}\int _{B^\eta }g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x\\&\quad +c_\eta \frac{1}{\varepsilon _k}\int _{B^\eta }g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x\,, \end{aligned} \end{aligned}$$

(7.20)

where $c_\eta {:}{=}\frac{L_2}{c_3}\max \{\frac{3\eta }{(1-\sqrt{\eta })^p}+(1+2^{p-1})\sqrt{\eta }\,,\frac{\sqrt{\eta }}{(1-\sqrt{\eta })^{p-1}}\}$. Then, since $(1-\eta )^p>(1-\sqrt{\eta })^p$ for $\eta \in (0,1)$, gathering (7.14), (7.15), and (7.20) we obtain

$$\begin{aligned} {\mathscr {F}}_k^s(v^\eta ,A)\le (1+c_\eta ){\mathscr {F}}_k^s(v,A)\,. \end{aligned}$$

Since $c_\eta \rightarrow 0$ as $\eta \rightarrow 0$ this concludes the proof. $\square$

With the help of Lemma 7.1 and Lemma 7.3 we are now in a position to identify the surface integrand ${\hat{g}}$.

Proposition 7.4

Let $(f_k) \subset {\mathcal {F}}$ and $(g_k) \subset {\mathcal {G}}$; let $(k_j)$ and ${\hat{g}}$ be as in Theorem 5.2. Then, it holds

$$\begin{aligned} {\hat{g}}(x,\zeta ,\nu )=g'(x,\nu ) = g''(x,\nu )\,, \end{aligned}$$

for every $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}^m_0$, and $\nu \in {{\mathbb {S}}}^{n-1}$, where $g'$ and $g''$ are, respectively, as in (2.14) and (2.15), with k replaced by $k_j$.

Proof

For notational simplicity, in what follows we still denote with k the index of the sequence provided by Theorem 5.2.

By definition we have $g'\le g''$; hence to prove the claim it suffices to show that

$$\begin{aligned} g''(x,\nu )\le \hat{g}(x,\zeta ,\nu )\le g'(x,\nu ), \end{aligned}$$

(7.21)

for every $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}^m_0$, and $\nu \in {{\mathbb {S}}}^{n-1}$. We divide the proof of (7.21) into two steps.

Step 1: In this step we show that $\hat{g}(x,\zeta ,\nu )\ge g''(x,\nu )$, for every $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}^m_0$, and $\nu \in {{\mathbb {S}}}^{n-1}$.

In view of Lemma 7.1 we have

$$\begin{aligned} \hat{g}(x,\zeta ,\nu )=\limsup _{\rho \rightarrow 0}\frac{1}{\rho ^{n-1}}\limsup _{k\rightarrow +\infty }{\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)). \end{aligned}$$

Thanks to Remark 7.2 the minimisation in the definition of ${\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$ can be carried over $C^1$-pairs $(u_k,v_k)$. Now let $\rho >0$ and $\eta \in (0,1)$ be fixed and for every k such that $2\varepsilon _k<\rho$ let $(u_k,v_k) \subset C^1(Q_\rho ^\nu (x);{\mathbb {R}}^m)\times C^1(Q_\rho ^\nu (x))$ satisfy

$$\begin{aligned} (u_k,v_k)=({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu \,,{\bar{v}}_{x,\varepsilon _k}^\nu )\;\text { in }\; U_k, \end{aligned}$$

(7.22)

where $U_k$ is a neighbourhood of $\partial Q_\rho ^\nu (x)$ and

$$\begin{aligned} {\mathscr {F}}_{k}(u_k,v_k,Q_\rho ^\nu (x))\le {\textbf{m}}_k({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x)) + \eta \rho ^{n-1}, \end{aligned}$$

(7.23)

then, (7.4) readily gives

$$\begin{aligned} {\mathscr {F}}_{k}(u_k,v_k,Q_\rho ^\nu (x))\le C \rho ^{n-1}. \end{aligned}$$

(7.24)

We now modify $v_k$ in order to obtain a new function $\widetilde{v}_k$ for which there exists a corresponding $\widetilde{u}_k$ such that the pair $(\widetilde{u}_k,\widetilde{v}_k)$ satisfies both

$$\begin{aligned} ({\widetilde{u}}_k,{\widetilde{v}}_k)=( u_{x}^\nu \,,1) \; \text { in }\; {\widetilde{U}}_k\cap \{|(y-x)\cdot \nu |>\varepsilon _k\} , \end{aligned}$$

(7.25)

where ${\widetilde{U}}_k$ is a a neighbourhood of $\partial Q_\rho ^\nu (x)$, and the constraint

$$\begin{aligned} \widetilde{v}_k\,\nabla \widetilde{u}_k=0\;\text { a.e. in }\; Q_\rho ^\nu (x). \end{aligned}$$

(7.26)

In this way we have ${\mathscr {F}}_{k}({\widetilde{u}}_k,{\widetilde{v}}_k, Q^\nu _{\rho }(x))={\mathscr {F}}^s_{k}({\widetilde{v}}_k, Q^\nu _{\rho }(x))$ with $\widetilde{v}_k \in {\mathscr {A}}_{\varepsilon _k,\rho }(x,\nu )$.

The modification as above shall be performed without essentially increasing the energy ${\mathscr {F}}_{k}$.

To this end, set $u_k:=(u_k^1,\ldots ,u_k^m)$ and $\zeta :=(\zeta ^1,\ldots ,\zeta ^m)$. Since $\zeta \in {\mathbb {R}}^m_0$ we can find $i\in \{1,\ldots ,m\}$ so that $\zeta ^i\ne 0$; without loss of generality we assume that $\zeta ^i>0$. We now consider the open set

$$\begin{aligned} S^\rho _k:=\left\{ y\in Q^\nu _\rho (x):0<u_k^i(y)<\zeta ^i\right\} . \end{aligned}$$

Moreover, let $\sigma \in (0,1)$ be fixed and consider the following partition of $S^\rho _k$:

$$\begin{aligned} S^\rho _k=\bigcup _{\ell =0}^{h_k^\sigma -1}S_{k,\ell }^\rho \end{aligned}$$

with

$$\begin{aligned} S_{k,\ell }^\rho&:=\big \{y\in Q_\rho ^\nu (x):\ell \tfrac{\zeta ^i}{h_k^\sigma }<u_k^i(y)\le (\ell +1)\tfrac{\zeta ^i}{h_k^\sigma }\big \}\,\quad \ell =0,\ldots ,h_k^\sigma -2\,,\\ S_{k,h_{k}^\sigma -1}^\rho&:=\big \{y\in Q_\rho ^\nu (x):(h_k^\sigma -1)\tfrac{\zeta ^i}{h_k^\sigma }<u_k^i(y)<\zeta ^i\big \}\, \end{aligned}$$

and $h_k^\sigma \in {\mathbb {N}}$ to be chosen later.

Then, there exists ${\bar{\ell }}={\bar{\ell }}(k,\rho ,\sigma )\in \{0,\ldots ,h_k^\sigma -1\}$ such that

$$\begin{aligned} \int _{{\widetilde{S}}^\rho _k}\left( \sigma \psi (v_k)|\nabla u_k^i|^p+(1-\sigma )\right) \,dy\le \frac{1}{h_k^\sigma }\int _{S^\rho _k}\left( \sigma \psi (v_k)|\nabla u_k^i|^p+(1-\sigma )\right) \, \textrm{d}y, \end{aligned}$$

(7.27)

where ${\widetilde{S}}^\rho _k:=S_{k,{\bar{\ell }}}^\rho$.

Therefore, gathering (f2), (7.24), and (7.27) yields

$$\begin{aligned} \int _{{\widetilde{S}}^\rho _k}\sigma \psi (v_k)|\nabla u_k^i|^p\,dy + (1-\sigma ){\mathcal {L}}^n ({\widetilde{S}}^\rho _k) \le \frac{\rho ^{n-1}}{h_k^\sigma } C (\sigma +(1-\sigma )\rho ). \end{aligned}$$

(7.28)

In view of (7.22) we have that

$$\begin{aligned} {\widetilde{S}}^\rho _k\cap U_k\subset \{|(y-x)\cdot \nu |\le \varepsilon _k\}\cap U_k. \end{aligned}$$

(7.29)

In this way any modification to $v_k$ performed in the set ${\widetilde{S}}^\rho _k$ will not affect the boundary conditions. In order to modify $v_k$ in $\widetilde{S}_k^\rho$, we introduce an auxiliary function $\hat{v}_k$ which interpolates in a suitable way between the values 0 and 1 in $\widetilde{S}_k^\rho$. To this end, we set $\gamma _k{:}{=}({\bar{\ell }}+\frac{1}{2})\frac{\zeta ^i}{h_k^\sigma }$ and $\tau _k{:}{=}\frac{\zeta ^i}{4h_k^\sigma }$ and we define $\hat{v}_k\in W^{1,p}(Q_\rho ^\nu (x))$ as follows:

$$\begin{aligned} {\hat{v}}_k:={\left\{ \begin{array}{ll} \min \biggl \{\dfrac{u^i_k-\gamma _k-\tau _k}{\tau _k}, 1\biggr \}&{} \text {in}\ \{ u^i_k \ge \gamma _k+\tau _k\},\\ 0&{}\text {in}\ \{\gamma _k-\tau _k< u^i_k < \gamma _k+\tau _k\},\\ \min \biggl \{\dfrac{\gamma _k-\tau _k-u^i_k}{\tau _k},1\biggr \}&{} \text {in}\ \{ u^i_k \le \gamma _k-\tau _k\}, \end{array}\right. } \end{aligned}$$

Eventually, we let $v_k^\eta \in W^{1,p}(Q_\rho ^\nu (x))$ be the function defined in (7.11) in Lemma 7.3, with $v_k$ in place of v and we define ${\widetilde{v}}_k\in W^{1,p}(Q_\rho ^\nu (x))$ as

$$\begin{aligned} {\widetilde{v}}_k:=\min \{v_k^\eta , {\hat{v}}_k\}. \end{aligned}$$

We notice that by definition

$$\begin{aligned} {\hat{v}}_k\equiv 1 \quad \text {in } Q_\rho ^\nu (x)\setminus {\widetilde{S}}^\rho _k, \end{aligned}$$

so that in particular ${\widetilde{v}}_k\equiv v_k^\eta$ in $Q_\rho ^\nu (x)\setminus {\widetilde{S}}^\rho _k$; moreover,

$$\begin{aligned} {\widetilde{v}}_k\equiv 0 \quad \text {in }\{\gamma _k-\tau _k< u^i_k <\gamma _k+\tau _k\}. \end{aligned}$$

By the regularity of $u_k^i$ we can find $\xi _k>0$ (possibly small) such that

$$\begin{aligned} \{y \in Q^\nu _{\rho }(x) :{{\,\textrm{dist}\,}}(y,\{u_k^i> \gamma _k\})<\xi _k \}\subset \{y \in Q^\nu _{\rho }(x) :u^i_k(y)>\gamma _k-\tau _k\}. \end{aligned}$$

Then, we define ${\widetilde{u}}_k\in W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)$ as

$$\begin{aligned} \widetilde{u}_k(y){:}{=}{\left\{ \begin{array}{ll} \biggl (1-\dfrac{{{\,\textrm{dist}\,}}(y,\{u^i_k> \gamma _k\})}{\xi _k}\biggr )e_1 &{}\text {if}\ {{\,\textrm{dist}\,}}(y,\{u^i_k> \gamma _k\})<\xi _k\,,\\ 0 &{}\text {otherwise}\,. \end{array}\right. } \end{aligned}$$

Thus the pair $({\widetilde{u}}_k,{\widetilde{v}}_k)$ belongs to $W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)\times W^{1,p}(Q_\rho ^\nu (x))$ and by construction satisfies (7.26). Moreover, (7.29) together with (7.22) ensures that (7.25) is satisfied. Eventually, we have

$$\begin{aligned} \frac{1}{\rho ^{n-1}}{\textbf{m}}_{k,\mathrm N}^s(u_x^\nu ,Q_\rho ^\nu (x))\le \frac{1}{\rho ^{n-1}}{\mathscr {F}}_{k}^s(\widetilde{v}_k,Q_\rho ^\nu (x))\,. \end{aligned}$$

(7.30)

To conclude the proof it only remains to show that, up to a small error, ${\mathscr {F}}_{k}^s(\widetilde{v}_k,Q_\rho ^\nu (x))$ is a lower bound for ${\mathscr {F}}_{k}(u_k,v_k,Q_\rho ^\nu (x))$. To this end we consider the following partition of $Q^\nu _\rho (x)$:

$$\begin{aligned} S^1_k:=\left\{ y\in Q^\nu _\rho (x):v^\eta _k(y)\le {\hat{v}}_k(y) \right\} ,\quad S^2_k:=\left\{ y\in Q^\nu _\rho (x):v_k^\eta (y)> {\hat{v}}_k(y) \right\} ; \end{aligned}$$

then, appealing to (7.13) in Lemma 7.3, we deduce

$$\begin{aligned} \begin{aligned} {\mathscr {F}}^s_{k}(\widetilde{v}_k,Q^\nu _\rho (x))&=\frac{1}{\varepsilon _k}\int _{S^1_k}g_k(y,v_k^\eta ,\varepsilon _k\nabla v_k^\eta )\, \textrm{d}y+\frac{1}{\varepsilon _k}\int _{S_k^2}g_k(y,{\hat{v}}_k,\varepsilon _k\nabla \hat{v}_k)\, \textrm{d}y\\&\le (1+c_\eta ){\mathscr {F}}_{k}^s(v_k,Q_\rho ^\nu (x))+\frac{1}{\varepsilon _k}\int _{S_k^2}g_k(y,{\hat{v}}_k,\varepsilon _k\nabla \hat{v}_k)\, \textrm{d}y, \end{aligned} \end{aligned}$$

(7.31)

where $c_\eta \rightarrow 0$ as $\eta \rightarrow 0$. Hence, it remains to estimate the second term on the right-hand side of (7.31).

To this end, we start noticing that

$$\begin{aligned} S^2_k\subset \left\{ v_k>\frac{\eta }{1+\sqrt{\eta }}\right\} . \end{aligned}$$

(7.32)

Indeed, since $v_k^\eta >0$ a.e. in $S^2_k$, by definition of $v_k^\eta$ we readily get (7.32).

Therefore, by (g3), using that ${\hat{v}}_k\equiv 1$ in $Q_\rho ^\nu (x)\setminus {\widetilde{S}}^\rho _k$ we obtain

$$\begin{aligned} \begin{aligned} \frac{1}{\varepsilon _k}\int _{S_k^2}g_{k}(y,{\hat{v}}_k,\varepsilon _k\nabla \hat{v}_k)\, \textrm{d}y&\le c_4\int _{S_k^2}\left( \frac{(1- {\hat{v}}_k)^p}{\varepsilon _k} +\varepsilon _k^{p-1}|\nabla {\hat{v}}_k|^p\right) \, \textrm{d}y\\&=c_4\int _{S_k^2\cap {\widetilde{S}}^\rho _k}\left( \frac{(1-{\hat{v}}_k)^p}{\varepsilon _k} +\varepsilon _k^{p-1}|\nabla {\hat{v}}_k|^p\right) \, \textrm{d}y\\&\le c_4 \left( \frac{{\mathcal {L}}^n({\widetilde{S}}_k^\rho )}{\varepsilon _k}+\frac{\varepsilon _k^{p-1}}{\psi (\frac{\eta }{1+\sqrt{\eta }})} \left( \frac{4h_k^\sigma }{\zeta ^i}\right) ^{p}\int _{S_k^2\cap {\widetilde{S}}^\rho _k}\psi (v_k)|\nabla u_k|^p\, \textrm{d}y\right) , \end{aligned}\nonumber \\ \end{aligned}$$

(7.33)

where the last inequality follows by (7.32), the definition of ${\hat{v}}_k$, and the monotonicity of $\psi$.

From (7.28) we deduce both that

$$\begin{aligned} {\mathcal {L}}^n({\widetilde{S}}_k^\rho ) \le C\frac{\rho ^{n-1}}{ h_k^\sigma }\left( \frac{\sigma }{1-\sigma }+\rho \right) \end{aligned}$$

(7.34)

and

$$\begin{aligned} \int _{S_k^2\cap {\widetilde{S}}_k}\psi (v_k)|\nabla u_k|^p\, \textrm{d}x\le C \frac{\rho ^{n-1}}{h_k^\sigma }\Big (1+\frac{1-\sigma }{\sigma }\rho \Big ). \end{aligned}$$

(7.35)

Hence, gathering (7.33), (7.34), and (7.35) we obtain

$$\begin{aligned} \frac{1}{\varepsilon _k}\int _{S_k^2}g_{k}(y,{\hat{v}}_k,\varepsilon _k\nabla \hat{v}_k)\, \textrm{d}y\le & {} C\,c_4\,\rho ^{n-1}\bigg (\frac{1}{ \varepsilon _kh_k^\sigma }\Big (\frac{\sigma }{1-\sigma }+\rho \Big )\nonumber \\{} & {} +K_\eta (\varepsilon _kh_k^\sigma )^{p-1}\Big (1+\frac{1-\sigma }{\sigma }\rho \Big )\bigg ), \end{aligned}$$

(7.36)

with $K_\eta := 4^p (\psi (\frac{\eta }{1+\sqrt{\eta }})(\zeta ^i)^p)^{-1}$. Now set $h_k^\sigma :=\lfloor {\widetilde{h}}_k^\sigma \rfloor$ where

$$\begin{aligned} {\widetilde{h}}_k^\sigma :=\frac{1}{\varepsilon _k}\Big (\frac{\sigma }{1-\sigma }\Big )^{\frac{1}{p}}. \end{aligned}$$

(7.37)

Using that $h_k^\sigma \le {\widetilde{h}}_k^\sigma \le h_k^\sigma +1$ we infer

$$\begin{aligned} \frac{1}{\varepsilon _k}\int _{S_k^2}g_{k}(y,{\hat{v}}_k,\varepsilon _k\nabla \hat{v}_k)\, \textrm{d}y\le C\,c_4\rho ^{n-1} \biggl (\frac{{\widetilde{h}}_k^\sigma }{{\widetilde{h}}_k^\sigma -1}+K_\eta \biggr )\bigg (\Big (\frac{\sigma }{1-\sigma }\Big )^{\frac{p-1}{p}}+\rho \Big (\frac{\sigma }{1-\sigma }\Big )^{-\frac{1}{p}}\bigg ). \end{aligned}$$

Plugging (7.36) into (7.31) gives

$$\begin{aligned} \frac{1}{\rho ^{n-1}}{\mathscr {F}}^s_{k}(\widetilde{v}_k,Q^\nu _\rho (x))\le & {} (1+c_\eta )\frac{1}{\rho ^{n-1}}{\mathscr {F}}_{k}(u_k,v_k,Q^\nu _\rho (x))\nonumber \\{} & {} + {{\widehat{K}}}_\eta \bigg (\Big (\frac{\sigma }{1-\sigma }\Big )^{\frac{p-1}{p}}+\rho \Big (\frac{\sigma }{1-\sigma }\Big )^{-\frac{1}{p}}\bigg ), \end{aligned}$$

(7.38)

where ${{\widehat{K}}}_\eta :=C\,c_4(1+ K_\eta )$. We notice that $c_\eta \rightarrow 0$ as $\eta \rightarrow 0$, while ${{\widehat{K}}}_\eta \rightarrow +\infty$ as $\eta \rightarrow 0$. Finally, by combining (7.23), (7.30), and (7.38) we get

$$\begin{aligned} \frac{1}{\rho ^{n-1}}{\textbf{m}}_{k,\mathrm N}^s(u_x^\nu ,Q_\rho ^\nu (x))\le & {} (1+c_\eta )\bigg (\frac{1}{\rho ^{n-1}}{\textbf{m}}_{k}({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))+ \eta \bigg ) \nonumber \\{} & {} + {{\widehat{K}}}_\eta \bigg (\Big (\frac{\sigma }{1-\sigma }\Big )^{\frac{p-1}{p}}+\rho \Big (\frac{\sigma }{1-\sigma }\Big )^{-\frac{1}{p}}\bigg ). \end{aligned}$$

(7.39)

Eventually, the claim follows by passing to the limit in (7.39) in the following order: first as $k\rightarrow +\infty$, then as $\rho \rightarrow 0$, $\sigma \rightarrow 0$, and finally as $\eta \rightarrow 0$.

Step 2: In this step we show that

$$\begin{aligned} \hat{g}(x,\zeta ,\nu )\le g'(x,\nu ), \end{aligned}$$

for every $x\in {\mathbb {R}}^n$, $\zeta \in {\mathbb {R}}^m_0$ and $\nu \in {{\mathbb {S}}}^{n-1}$. Thanks to (5.12), in view of Lemma 7.1 and Proposition 2.6 it suffices to show that

$$\begin{aligned} {\textbf{m}}_{k}({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\le {\textbf{m}}_{k}^s({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\,, \end{aligned}$$

(7.40)

for every $\varepsilon >0$, where ${\textbf{m}}_{k}^s({\bar{u}}_{x,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$ is defined in (2.16). To prove (7.40) let $v \in {\mathscr {A}}({\bar{u}}^\nu _{x,\varepsilon _k},Q^\nu _\rho (x))$ with corresponding $u\in W^{1,p}(Q_\rho ^\nu (x);{\mathbb {R}}^m)$ and notice that the pair $(\widetilde{u},v)$ with $\widetilde{u}{:}{=}(u\cdot e_1)\zeta$ is an admissible competitor for ${\textbf{m}}_{k}({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))$. Therefore, we obtain

$$\begin{aligned} {\textbf{m}}_{k}({\bar{u}}_{x,\zeta ,\varepsilon _k}^\nu ,Q_\rho ^\nu (x))\le {\mathscr {F}}_{k}(\widetilde{u},v,Q_\rho ^\nu (x))={\mathscr {F}}_{k}^s(v,Q_\rho ^\nu (x)), \end{aligned}$$

from which we deduce (7.40) by passing to the infimum in $v \in {\mathscr {A}}({\bar{u}}^\nu _{x,\varepsilon _k},Q^\nu _\rho (x))$. $\square$

Remark 7.5

We observe that the second term in the right hand side of (7.39) is infinitesimal as $\rho , \sigma \rightarrow 0$ thanks to the fact that $p>1$. We notice, moreover, that in (7.39) the presence of the exponent p in the reminder is a consequence of the p-growth of the volume integrand $f_k$. Indeed if $g_k$ satisfied conditions (g2)–(g4) with p replaced by some exponent $q\in (1,p]$, then arguing as in the proof of Proposition 7.4, in place of (7.36) we would get

$$\begin{aligned}{} & {} \frac{1}{\varepsilon _k}\int _{S_k^2}g_{k}(y,{\hat{v}}_k,\varepsilon _k\nabla \hat{v}_k)\, \textrm{d}y\\{} & {} \quad \le C c_4\rho ^{n-1}\bigg (\frac{1}{ \varepsilon _kh_k^\sigma }\Big (\frac{\sigma }{1-\sigma }+\rho \Big ) +K_\eta (\varepsilon _kh_k^\sigma )^{q-1}\Big (1+\frac{1-\sigma }{\sigma }\rho \Big )^\frac{q}{p}\Big (\frac{\sigma }{1-\sigma }+\rho \Big )^\frac{p-q}{p}\bigg ), \end{aligned}$$

with $K_\eta := 4^q (\psi (\frac{\eta }{1+\sqrt{\eta }})(\zeta ^i)^q)^{-1}$. Then, an easy computation shows that choosing $h_k^\sigma =\lfloor {\widetilde{h}}_k^\sigma \rfloor$, with ${\widetilde{h}}_k^\sigma$ given by (7.37), exactly yields (7.39).

We are now in a position to prove the main result of this paper, namely, Theorem 3.1.

Proof of Theorem 3.1

The proof follows by combining Proposition 4.1, Proposition 4.5, Theorem 5.2, Proposition 6.1, and Proposition 7.4. $\square$

8 Stochastic homogenisation

In this section we study the $\Gamma$-convergence of the functionals ${\mathscr {F}}_k$ when $f_k$ and $g_k$ are random integrands of type

$$\begin{aligned} f_k(\omega ,x,\xi )=f\Big (\omega ,\frac{x}{\varepsilon _k},\xi \Big ), \quad g_k(\omega , x,v,w)=g\Big (\omega ,\frac{x}{\varepsilon _k},v,w\Big ), \end{aligned}$$

where $\omega$ belongs to the sample space $\Omega$ of a probability space $(\Omega ,{\mathcal {T}},P)$.

Before stating the main result of this section, we need to recall some useful definitions.

Definition 8.1

(Group of P-preserving transformations) Let $d\in {\mathbb {N}}$, $d\ge 1$. A group of P-preserving transformations on $(\Omega ,{\mathcal {T}},P)$ is a family $(\tau _z)_{z\in {\mathbb {Z}}^d}$ of mappings $\tau _z:\Omega \rightarrow \Omega$ satisfying the following properties:

• (1) (measurability) $\tau _z$ is ${\mathcal {T}}$-measurable for every $z\in {\mathbb {Z}}^d$;

• (2) (invariance) $P(\tau _z(E))=P(E)$, for every $E\in {\mathcal {T}}$ and every $z\in {\mathbb {Z}}^d$;

• (3) (group property) $\tau _0=\textrm{id}_\Omega$ and $\tau _{z+z'}=\tau _z\circ \tau _{z'}$ for every $z,z'\in {\mathbb {Z}}^d$.

If, in addition, every $(\tau _z)_{z\in {\mathbb {Z}}^d}$-invariant set (i.e., every $E\in {\mathcal {T}}$ with $\tau _z(E)=E$ for every $z\in {\mathbb {Z}}^d$) has probability 0 or 1, then $(\tau _z)_{z\in {\mathbb {Z}}^d}$ is called ergodic.

Let $a{:}{=}(a_1,\dots ,a_d),\,b{:}{=}(b_1,\dots ,b_d)\in {\mathbb {Z}}^d$ with $a_i<b_i$ for all $i\in \{1,\ldots ,d\}$; we define the d-dimensional interval

$$\begin{aligned}{}[a,b):=\{x\in {\mathbb {Z}}^d:a_i\le x_i<b_i\,\,\text {for}\,\,i=1,\ldots ,d\} \end{aligned}$$

and we set

$$\begin{aligned} {\mathcal {I}}_d:=\{[a,b):a,b\in {\mathbb {Z}}^d,\, a_i<b_i\,\,\text {for}\,\,i=1,\ldots ,d\}. \end{aligned}$$

Definition 8.2

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

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

• (2) (covariance) for every $\omega \in \Omega$, $A\in {\mathcal {I}}_d$, and $z\in {\mathbb {Z}}^d$ we have $\mu (\omega ,A+z)=\mu (\tau _z(\omega ),A)$;

• (3) (subadditivity) for every $A\in {\mathcal {I}}_d$ and for every finite family $(A_i)_{i\in I}\subset {\mathcal {I}}_d$ of pairwise disjoint sets such that $A=\cup _{i\in I}A_i$, we have

  $$\begin{aligned} \mu (\omega ,A)\le \sum _{i\in I}\mu (\omega ,A_i)\quad \text {for every}\,\,\omega \in \Omega ; \end{aligned}$$

• (4) (boundedness) there exists $c>0$ such that $0\le \mu (\omega ,A)\le c{\mathcal {L}}^d(A)$ for every $\omega \in \Omega$ and $A\in {\mathcal {I}}_d$.

Definition 8.3

(Stationarity) Let $(\tau _z)_{z\in {\mathbb {Z}}^n}$ be a group of P-preserving transformations on $(\Omega ,{\mathcal {T}},P)$. We say that $f:\Omega \times {\mathbb {R}}^n\times {\mathbb {R}}^{m\times n}\rightarrow [0,+\infty )$ is stationary with respect to $(\tau _z)_{z\in {\mathbb {Z}}^n}$ if

$$\begin{aligned} f(\omega ,x+z,\xi )=f(\tau _z(\omega ),x,\xi ) \end{aligned}$$

for every $\omega \in \Omega$, $x\in {\mathbb {R}}^n$, $z\in {\mathbb {Z}}^n$ and $\xi \in {\mathbb {R}}^{m\times n}$.

Analogously, we say that $g:\Omega \times {\mathbb {R}}^n\times {\mathbb {R}}\times {\mathbb {R}}^n\rightarrow [0,+\infty )$ is stationary with respect to $(\tau _z)_{z\in {\mathbb {Z}}^n}$ if

$$\begin{aligned} g(\omega ,x+z,v,w)=g(\tau _z(\omega ),x,v,w) \end{aligned}$$

for every $\omega \in \Omega$, $x\in {\mathbb {R}}^n$, $z\in {\mathbb {Z}}^n$, $v\in [0,1]$ and $w\in {\mathbb {R}}^n$.

In all that follows we consider random integrands $f:\Omega \times {\mathbb {R}}^n\times {\mathbb {R}}^{m\times n}\rightarrow [0,+\infty )$ satisfying

(F1):

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

(F2):

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

and random integrands $g:\Omega \times {\mathbb {R}}^n\times {\mathbb {R}}\times {\mathbb {R}}^n\rightarrow [0,+\infty )$ satisfying

(G1):

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

(G2):

$g(\omega ,\,\cdot \,,\,\cdot \,,\,\cdot \,)\in {\mathcal {G}}$ for every $\omega \in \Omega$.

Let f and g be random integrands satisfying (F1)–(F2) and (G1)–(G2), respectively. We consider the sequence of random elliptic functionals ${\mathscr {F}}_k(\omega ) :L^0({\mathbb {R}}^n;{\mathbb {R}}^m) \times L^0({\mathbb {R}}^n) \times {\mathcal {A}}\longrightarrow [0,+\infty ]$ given by

$$\begin{aligned} {\mathscr {F}}_k(\omega )(u,v, A){:}{=}\int _A \psi (v)\,f\left( \omega ,\frac{x}{\varepsilon _k},\nabla u\right) \, \textrm{d}x+\frac{1}{\varepsilon _k}\int _A g\left( \omega ,\frac{x}{\varepsilon _k},v,\varepsilon _k\nabla v\right) \, \textrm{d}x, \end{aligned}$$

(8.1)

if $(u,v)\in W^{1,p}(A;{\mathbb {R}}^m)\times W^{1,p}(A)$, $0\le v\le 1$ and extended to $+\infty$ otherwise. We also let ${\mathscr {F}}^b(\omega )$ be as in (3.4) and ${\mathscr {F}}^s(\omega )$ as in (3.5), with $f(\,\cdot \,,\,\cdot \,)$ and $g(\,\cdot \,,\,\,\cdot \,,\,\cdot \,)$ replaced by $f(\omega ,\,\cdot ,\,\cdot \,)$ and $g(\omega ,\,\cdot ,\,\cdot ,\,\cdot \,)$, respectively. Moreover, for $\omega \in \Omega$ and $A\in {\mathcal {A}}$ we set

$$\begin{aligned} {\textbf{m}}^b_\omega (u_\xi ,A){:}{=}\inf \{{\mathscr {F}}^b(\omega )(u,A):u\in W^{1,p}(A;{\mathbb {R}}^m)\,,\ u=u_\xi \ \text {near}\ \partial A\} \end{aligned}$$

(8.2)

and

$$\begin{aligned} {\textbf{m}}^s_{\omega }({\bar{u}}_{x}^\nu ,A){:}{=}\inf \{{\mathscr {F}}^s(\omega )(v,A):v\in {\mathscr {A}}(\bar{u}_x^\nu ,A)\}, \end{aligned}$$

(8.3)

where ${\mathscr {A}}(\bar{u}_x^\nu ,A)$ is as in (2.17) with $\bar{u}_x^\nu$ in place of $\bar{u}_{x,\varepsilon _k}^\nu$ ; i.e., with $\varepsilon _k=1$.

Eventually, we extend the definition of ${\textbf{m}}^b_{\omega }(u_\xi ,\cdot )$, ${\textbf{m}}^s_{\omega }(\bar{u}_x^\nu ,\cdot )$, and ${\mathscr {A}}(\bar{u}_0^\nu ,\cdot )$ to any $A\subset {\mathbb {R}}^n$ with $\mathrm int\, A\in {\mathcal {A}}$ by setting ${\textbf{m}}^b_{\omega }(u_\xi ,A){:}{=}{\textbf{m}}^b_{\omega }(u_\xi ,\mathrm int\, A)$, ${\textbf{m}}^s_{\omega }(\bar{u}_x^\nu ,A){:}{=}{\textbf{m}}^s_{\omega }(\bar{u}_x^\nu ,\mathrm int\, A)$, and ${\mathscr {A}}(\bar{u}_0^\nu ,A){:}{=}{\mathscr {A}}(\bar{u}_0^\nu ,\mathrm int\, A)$.

We are now ready to state the main result of this section.

Theorem 8.4

(Stochastic homogenisation) Let f and g be random integrands satisfying (F1)–(F2) and (G1)–(G2), respectively. Assume moreover that f and g are stationary with respect to a group $(\tau _z)_{z\in {\mathbb {Z}}^n}$ of P-preserving transformations on $(\Omega ,{\mathcal {T}},P)$. For every $\omega \in \Omega$ let ${\mathscr {F}}_k(\omega )$ be as in (8.1) and ${\textbf{m}}^b_\omega$, ${\textbf{m}}^s_\omega$ be as in (8.2) and (8.3), respectively. Then there exists $\Omega '\in {\mathcal {T}}$, with $P(\Omega ')=1$ such that for every $\omega \in \Omega '$, $x\in {\mathbb {R}}^n$, $\xi \in {\mathbb {R}}^{m\times n}$, $\nu \in {{\mathbb {S}}}^{n-1}$ the limits

$$\begin{aligned} \lim _{r\rightarrow +\infty } \frac{{\textbf{m}}^b_\omega (u_\xi ,Q_r(rx))}{r^n}= & {} \lim _{r\rightarrow +\infty } \frac{{\textbf{m}}^b_\omega (u_\xi ,Q_r(0))}{r^n}=:f_\textrm{hom}(\omega ,\xi )\,, \end{aligned}$$

(8.4)

$$\begin{aligned} \lim _{r\rightarrow +\infty } \frac{{\textbf{m}}^s_\omega ({\bar{u}}^\nu _{rx},Q^\nu _r(rx))}{r^{n-1}}= & {} \lim _{r\rightarrow +\infty } \frac{{\textbf{m}}^s_\omega ({\bar{u}}^\nu _{0},Q^\nu _r(0))}{r^{n-1}}=:g_\textrm{hom}(\omega ,\nu ) \end{aligned}$$

(8.5)

exist and are independent of x. The function $f_\textrm{hom}:\Omega \times {\mathbb {R}}^{m\times n} \rightarrow [0,+\infty )$ is $({\mathcal {T}}\otimes {\mathcal {B}}^{m\times n})$-measurable and $g_\textrm{hom}:\Omega \times {{\mathbb {S}}}^{n-1} \rightarrow [0,+\infty )$ is $({\mathcal {T}}\otimes {\mathcal {B}}({{\mathbb {S}}}^{n-1}))$-measurable.

Moreover, for every $\omega \in \Omega '$ and for every $A\in {\mathcal {A}}$ the functionals ${\mathscr {F}}_k(\omega )(\cdot ,\cdot ,A)$ $\Gamma$-converge in $L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)$ to the functional ${\mathscr {F}}_{\textrm{hom}}(\omega )(\cdot ,\cdot ,A)$ with ${\mathscr {F}}_{\textrm{hom}}(\omega ):L^0({\mathbb {R}}^n;{\mathbb {R}}^m)\times L^0({\mathbb {R}}^n)\times {\mathcal {A}}\longrightarrow [0,+\infty ]$ given by

$$\begin{aligned} {\mathscr {F}}_{\textrm{hom}}(\omega )(u,v,A){:}{=}{\left\{ \begin{array}{ll} \displaystyle \int _A f_{\textrm{hom}}(\omega ,\nabla u)\, \textrm{d}x+\int _{S_u \cap A} g_{\textrm{hom}}(\omega ,\nu _u)\, \textrm{d}{\mathcal {H}}^{n-1}&{}\text {if}\ u \in GSBV^p(A;{\mathbb {R}}^m)\,,\\ &{} v= 1\, \text {a.e. in } A\,,\\ +\infty &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

If, in addition, $(\tau _z)_{z\in {\mathbb {Z}}^n}$ is ergodic, then $f_{\textrm{hom}}$ and $g_{\textrm{hom}}$ are independent of $\omega$ and

$$\begin{aligned} f_{\textrm{hom}}(\xi )&= \lim _{r\rightarrow +\infty } \frac{1}{r^n}\int _\Omega {\textbf{m}}^b_\omega (u_\xi ,Q_r(0))\, \textrm{d}P(\omega )\,,\\ g_{\textrm{hom}}(\nu )&= \lim _{r\rightarrow +\infty } \frac{1}{r^{n-1}}\int _\Omega {\textbf{m}}^s_\omega ({\bar{u}}^\nu _{0},Q^\nu _r(0))\, \textrm{d}P(\omega )\,, \end{aligned}$$

thus, ${\mathscr {F}}_\textrm{hom}$ is deterministic.

The almost sure $\Gamma$-convergence result in Theorem 8.4 is an immediate consequence of Theorem 3.5 once we show the existence of a ${\mathcal {T}}$-measurable set $\Omega ' \subset \Omega$, with $P(\Omega ')=1$, such that for every $\omega \in \Omega '$ the limits in (8.4)–(8.5) exist and are independent of x. Therefore, the rest of this section is devoted to prove the existence of such a set.

8.1 Homogenisation formulas

In this subsection we prove that conditions (F1), (F2), (G1), and (G2) together with the stationarity of the random integrands f and g ensure that the assumptions of Theorem 3.5 are satisfied almost surely.

The proof of the following result is based on the pointwise Subbaditive Ergodic Theorem [1, Theorem 2.4] applied to the function $(\omega ,I) \mapsto {\textbf{m}}^b_\omega (u_\xi ,I)$, which defines a subadditive process on $\Omega \times {\mathcal {I}}_n$, as shown in [47, Proposition 3.2].

Proposition 8.5

(Homogenised volume integrand) Let f satisfy (F1)–(F2) and assume that it is stationary with respect to a group $(\tau _z)_{z\in {\mathbb {Z}}^n}$ of P-preserving transformations on $(\Omega ,{\mathcal {T}},P)$. For $\omega \in \Omega$ let ${\textbf{m}}^b_\omega$ be as in (8.2). Then there exists $\Omega '\in {\mathcal {T}}$, with $P(\Omega ')=1$ and a $({\mathcal {T}}\otimes {\mathcal {B}}^{m\times n})$-measurable function $f_{\textrm{hom}}:\Omega \times {\mathbb {R}}^{m\times n}\rightarrow [0,+\infty )$ such that

$$\begin{aligned} \lim _{r\rightarrow +\infty } \frac{{\textbf{m}}^b_\omega (u_\xi ,Q_r(rx))}{r^n}= \lim _{r\rightarrow +\infty } \frac{{\textbf{m}}^b_\omega (u_\xi ,Q_r(0))}{r^n}= f_{\textrm{hom}}(\omega ,\xi ). \end{aligned}$$

for every $\omega \in \Omega '$, $x\in {\mathbb {R}}^n$ and every $\xi \in {\mathbb {R}}^{m\times n}$. If, in addition, $(\tau _z)_{z\in {\mathbb {Z}}^n}$ is ergodic, then $f_{\textrm{hom}}$ is independent of $\omega$ and given by

$$\begin{aligned} f_{\textrm{hom}}(\xi )= \lim _{r\rightarrow +\infty } \frac{1}{r^n}\int _\Omega {\textbf{m}}^b_\omega (u_\xi ,Q_r(0))\, \textrm{d}P(\omega ). \end{aligned}$$

Proof

The proof follows by [47, Proposition 3.2] arguing as in [35, Theorem 1] (see also [47, Corollary 3.3]). $\square$

We now deal with the existence of the homogenised surface integrand $g_\textrm{hom}$. Unlike the case of $f_\textrm{hom}$, the existence and x-homogeneity of the limit in (8.5) cannot be deduced by a direct application of the Subadditive Ergodic Theorem [1, Theorem 2.4]. In fact, due to the x-dependent boundary datum appearing in ${\textbf{m}}^s_\omega ({\bar{u}}_{rx}^\nu ,Q_r^\nu (rx))$ (cf. (8.3)), the proof of the x-homogeneity of $g_\textrm{hom}$ is rather delicate and follows by ad hoc arguments, which are typical of surface functionals [3, 28].

The proof of the existence of $g_\textrm{hom}$ will be carried out in several step. In a first step we prove that when $x=0$ the minimisation problem (8.3) defines a subadditive process on $\Omega \times {\mathcal {I}}_{n-1}$. To do so we follow the same procedure as in [28, Section 5] (see also [3, 22]). Namely, given $\nu \in {{\mathbb {S}}}^{n-1}$ we let $R_\nu$ be an orthogonal matrix as in (f). Then $\{R_\nu e_i:i=1\,,\ldots \,,n-1\}$ is an orthonormal basis for $\Pi ^\nu$, further $R_\nu \in {\mathbb {Q}}^{n\times n}$, if $\nu \in {{\mathbb {S}}}^{n-1}\cap {\mathbb {Q}}^n$. Now let $M_\nu >2$ be an integer such that $M_\nu R_\nu \in {\mathbb {Z}}^{n\times n}$; therefore $M_\nu R_\nu (z',0)\in \Pi ^\nu \cap {\mathbb {Z}}^n$ for every $z'\in {\mathbb {Z}}^{n-1}$.

Let $I\in {\mathcal {I}}_{n-1}$ ; i.e., $I=[a,b)$ with $a,b\in {\mathbb {Z}}^{n-1}$. Starting from I we define the n-dimensional interval $I_\nu$ as

$$\begin{aligned} I_\nu {:}{=}M_\nu R_\nu \big (I\times [-c,c)\big )\quad \text {where}\quad c{:}{=}\frac{1}{2}\max _{i=1,\ldots ,n-1}(b_i-a_i). \end{aligned}$$

(8.6)

Correspondingly, for fixed $\nu \in {{\mathbb {S}}}^{n-1}\cap {\mathbb {Q}}^n$ we define the function $\mu _\nu :\Omega \times {\mathcal {I}}_{n-1} \mapsto {\mathbb {R}}$ as

$$\begin{aligned} \mu _\nu (\omega ,I){:}{=}\frac{1}{M_\nu ^{n-1}}{\textbf{m}}^s_\omega (\bar{u}_{0}^\nu , I_\nu ), \end{aligned}$$

(8.7)

where ${\textbf{m}}^s_\omega (\bar{u}_{0}^\nu , I_\nu )$ is as in (8.3) with $x=0$ and $A=I_\nu$.

The following result asserts that $\mu _\nu$ defines a subadditive process on $\Omega \times {\mathcal {I}}_{n-1}$.

Proposition 8.6

Let g satisfy (G1)–(G2) and assume that it is stationary with respect to a group $(\tau _z)_{z\in {\mathbb {Z}}^n}$ of P-preserving transformations on $(\Omega ,{\mathcal {T}},P)$. Let $\nu \in {{\mathbb {S}}}^{n-1}\cap {\mathbb {Q}}^n$ and let $\mu _\nu :\Omega \times {\mathcal {I}}_{n-1} \mapsto {\mathbb {R}}$ be as in (8.7). Then there exists a group of P-preserving transformations $(\tau _{z'}^\nu )_{z'\in {\mathbb {Z}}^{n-1}}$ on $(\Omega ,{\mathcal {T}},P)$ such that $\mu _\nu$ is a subadditive process on $(\Omega ,{\mathcal {T}},P)$ with respect to $(\tau _{z'}^\nu )_{z'\in {\mathbb {Z}}^{n-1}}$. Moreover, it holds

$$\begin{aligned} 0\le \mu _\nu (\omega , I)\le c_4 C_\textrm{v}{\mathcal {L}}^{n-1}(I), \end{aligned}$$

(8.8)

for P-a.e. $\omega \in \Omega$ and for every $I\in {\mathcal {I}}_{n-1}$.

Proof

Let $\nu \in {\mathbb {S}}^{n-1}\cap {\mathbb {Q}}^n$ be fixed; below we show that $\mu _\nu$ satisfies conditions (1)–(4) in Definition 8.2, for some group of P-preserving transformations $(\tau _{z'}^\nu )_{z'\in {\mathbb {Z}}^{n-1}}$.

We divide the proof into four steps, each of them corresponding to one of the four conditions in Definition 8.2.

Step 1: measurability. Let $I\in {\mathcal {I}}_{n-1}$ and let $I_\nu \subset {\mathbb {R}}^n$ be as in (8.6). Let $v\in W^{1,p}(I_\nu )$ be fixed. In view of (G1) the Fubini Theorem ensures that the map $\omega \mapsto {\mathscr {F}}^s(\omega )(v,I_\nu )$ is ${\mathcal {T}}$-measurable. On the other hand the space $W^{1,p}(I_\nu )$ is separable, hence the set of functions

$$\begin{aligned} {\mathscr {A}}(\bar{u}_0^\nu ,I_\nu )= & {} \{v\in W^{1,p}(I_\nu )\,,\; 0\le v \le 1, \ v= {\bar{v}}^\nu _0\ \text { near }\ \partial I_\nu \ \text {and}\ \exists \, u\in W^{1,p}(I_\nu ;{\mathbb {R}}^m),\\ u= & {} {\bar{u}}^\nu _0\ \text { near }\, \partial I_\nu ,\; \text {such that}\; v\,\nabla u=0\ \text {a.e. in}\ I_\nu \} \end{aligned}$$

defines a separable metric space, when endowed with the distance induced by the $W^{1,p}(I_\nu )$-norm. Then, by the continuity of ${\mathscr {F}}^s(\omega )(\cdot , I_\nu )$ with respect to the strong $W^{1,p}(I_\nu )$-topology, the infimum in the definition of ${\textbf{m}}^s_\omega ({\bar{u}}^\nu _0,I_\nu )$ can be equivalently expressed as an infimum on a countable subset of ${\mathscr {A}}(\bar{u}_0^\nu ,I_\nu )$, thus ensuring the ${\mathcal {T}}$-measurability of $\omega \mapsto {\textbf{m}}^s_\omega ({\bar{u}}^\nu _0,I_\nu )$ and consequently that of $\omega \mapsto \mu _\nu (\omega ,I)$, as desired.

Step 2: covariance. Let $z'\in {\mathbb {Z}}^{n-1}$ be fixed. Let $I\in {\mathcal {I}}_{n-1}$; by (8.6) we have

$$\begin{aligned} (I+z')_\nu =I_\nu +M_\nu R_\nu (z',0)=I_\nu +z'_\nu , \end{aligned}$$

where $z'_{\nu }:=M_\nu R_\nu (z',0) \in {\mathbb {Z}}^n\cap \Pi ^\nu$. Then, by (8.7) we get

$$\begin{aligned} \mu _\nu (\omega ,I+z')=\frac{1}{M_\nu ^{n-1}}{\textbf{m}}^s_\omega ({\bar{u}}_0^\nu ,I_\nu +z'_\nu ). \end{aligned}$$

(8.9)

Now let $v\in {\mathscr {A}}(\bar{u}_0^\nu ,I_\nu +z_\nu ')$ with its corresponding $u\in W^{1,p}(I'_\nu +z'_\nu ; {\mathbb {R}}^m)$. Setting ${\widetilde{v}}(x){:}{=}v(x+z'_\nu )$ for $x\in I_\nu$, a change of variables together with the stationarity of g yield

$$\begin{aligned} {\mathscr {F}}^s(\omega )(v,\textrm{int}\, (I_\nu +z'_\nu ))= & {} \int _{I_\nu +z'_\nu }g(\omega ,x,v,\nabla v)\, \textrm{d}x=\int _{I_\nu }g(\omega ,x+z'_\nu ,{\widetilde{v}},\nabla {\widetilde{v}})\, \textrm{d}x\\= & {} \int _{I_\nu }g(\tau _{z'_\nu }(\omega ),x,{\widetilde{v}},\nabla {\widetilde{v}})\, \textrm{d}x={\mathscr {F}}^s(\tau _{z'_\nu }(\omega ))({\widetilde{v}}, \textrm{int}\, I_\nu ). \end{aligned}$$

Set $(\tau ^\nu _{z'})_{z'\in {\mathbb {Z}}^{n-1}}:=(\tau _{z'_\nu })_{z'\in {\mathbb {Z}}^{n-1}}$; we notice that $(\tau ^\nu _{z'})_{z'\in {\mathbb {Z}}^{n-1}}$ is well defined since $z'_\nu \in {\mathbb {Z}}^n$ and it defines a group of P-preserving transformations on $(\Omega , P,{\mathcal {T}})$. Then, the equality above can be rewritten as

$$\begin{aligned} {\mathscr {F}}^s(\omega )(v, \textrm{int}\,(I_\nu +z'_\nu ))={\mathscr {F}}^s(\tau _{z'}^\nu (\omega ))({\widetilde{v}},\textrm{int}\, I_\nu ). \end{aligned}$$

(8.10)

Moreover, if we set $\widetilde{u}(x){:}{=}u(x+z'_\nu )$ for $x\in I_\nu$, then $\widetilde{v}\, \nabla \widetilde{u}=0$ a.e. in $I_\nu$, while since $z_\nu '\in \Pi ^\nu$, we also have

$$\begin{aligned} \big (\widetilde{u},\widetilde{v}\big )=\big (\bar{u}_0^\nu (\cdot +z'_\nu ),\bar{v}_0^\nu (\cdot +z'_\nu )\big )=\big (\bar{u}_0^\nu ,\bar{v}_0^\nu \big ) \end{aligned}$$

near $\partial I_\nu$ ; i.e., $\widetilde{v}\in {\mathscr {A}}(\bar{u}_0^\nu , I_\nu )$. Thus, gathering (8.9) and (8.10), by the arbitrariness of v we infer

$$\begin{aligned} \mu _\nu (\omega ,I+z')= \mu _\nu (\tau ^\nu _{z'}(\omega ),I), \end{aligned}$$

and hence the covariance of $\mu _\nu$ with respect to $(\tau ^\nu _{z'})_{z'\in {\mathbb {Z}}^{n-1}}$.

Step 3: subadditivity. Let $\omega \in \Omega$, $I\in {\mathcal {I}}_{n-1}$, and let $\{I_1,\dots ,I_N\}\subset {\mathcal {I}}_{n-1}$ be a finite family of pairwise disjoint sets such that $I=\bigcup _i I_i$. Let $\eta >0$ be fixed; for every $i=1,\dots ,N$ let $v_i\in W^{1,p}((I_i)_\nu )$ be admissible for ${\textbf{m}}^s_\omega ({\bar{u}}_0^\nu ,(I_i)_\nu )$ and such that

$$\begin{aligned} {\mathscr {F}}^s(\omega )(v_i,\textrm{int}\,(I_i)_\nu )\le {\textbf{m}}^s_\omega ({\bar{u}}_0^\nu ,(I_i)_\nu )+\eta . \end{aligned}$$

(8.11)

Therefore for every $i=1,\dots ,N$ there exists a corresponding $u_i\in W^{1,p}((I_i)_\nu ;{\mathbb {R}}^m)$ such that

$$\begin{aligned} v_i \nabla u_i =0\;\text { a.e. in }\;(I_i)_\nu \;\text { with }\; (u_i,v_i)=({\bar{u}}^\nu _0,{\bar{v}}^\nu _0)\quad \text {near}\,\,\partial (I_i)_\nu . \end{aligned}$$

We define

$$\begin{aligned} v:={\left\{ \begin{array}{ll} v_i&{}\text {in }\;(I_i)_\nu ,\ i=1,\ldots ,N,\\ {\bar{v}}_0^\nu &{}\text {in }\; I_\nu \setminus \bigcup _i (I_i)_\nu , \end{array}\right. } \quad u:= {\left\{ \begin{array}{ll} u_i&{}\text {in }\;(I_i)_\nu ,\ i=1,\ldots ,N,\\ {\bar{u}}_0^\nu &{}\text {in }\; I_\nu \setminus \bigcup _i(I_i)_\nu , \end{array}\right. } \end{aligned}$$

so that $(u,v)\in W^{1,p}(I_\nu )\times W^{1,p}(I_\nu ;{\mathbb {R}}^m)$ and

$$\begin{aligned} v \nabla u=0\;\text { a.e. in }\; I_\nu \;\text { with }\; (u,v)=({\bar{u}}^\nu _0,{\bar{v}}^\nu _0)\;\text { near }\;\partial I_\nu . \end{aligned}$$

Therefore $v\in {\mathscr {A}}(\bar{u}_0^\nu ,I_\nu )$. Furthermore, we have

$$\begin{aligned} {\mathscr {F}}^s(\omega )(v,\textrm{int}\,I_\nu )=\sum _{i=1}^N{\mathscr {F}}^s(\omega )(v_i,\textrm{int}\,(I_i)_\nu )+{\mathscr {F}}^s(\omega )\big ({\bar{v}}^\nu _0,\textrm{int}\,(I_\nu \setminus \bigcup _{i=1}^N(I_i)_\nu )\big ). \end{aligned}$$

(8.12)

Since $M_\nu >2$ and $c\ge \frac{1}{2}$ in (8.6), it follows that $\{y\in I_\nu :|y\cdot \nu |\le 1\}\subset \bigcup _{i}(I_i)_\nu$. Thus, ${\bar{v}}_0^\nu \equiv 1$ in $I_\nu {\setminus }\bigcup _i(I_i)_\nu$ which, thanks to (2.3), gives

$$\begin{aligned} {\mathscr {F}}^s(\omega )\big ({\bar{v}}^\nu _0,\textrm{int}\,(I_\nu \setminus \bigcup _{i=1}^N(I_i)_\nu )\big )=0. \end{aligned}$$

Eventually, gathering (8.11)–(8.12) we obtain

$$\begin{aligned} {\textbf{m}}^s_\omega ({\bar{u}}_0^\nu ,I_\nu )\le {\mathscr {F}}^s(\omega )(v,\textrm{int}\,I_\nu )=\sum _{i=1}^N{\mathscr {F}}^s(\omega )(v_i,\textrm{int}\,(I_i)_\nu )\le \sum _{i=1}^N{\textbf{m}}^s_\omega ({\bar{u}}_0^\nu ,(I_i)_\nu )+N\eta , \end{aligned}$$

thus the subadditivity of $\mu _\nu$ follows from (8.7) and the arbitrariness of $\eta >0$.

Step 4: boundedness. Let $\omega \in \Omega$ and $I\in {\mathcal {I}}_{n-1}$. Then (8.6) and (2.19) readily imply

$$\begin{aligned} 0\le \mu _\nu (\omega , I)=\frac{1}{M_\nu ^{n-1}}{\textbf{m}}^s_\omega ({\bar{u}}_0^\nu ,I_\nu )\le c_4 C_\textrm{v}{\mathcal {L}}^{n-1}(I). \end{aligned}$$

$\square$

Having at hand Proposition 8.6, with the help of Lemma A.1 and Lemma A.2 we now prove the following result, which establishes the almost sure existence of the limit defining $g_\textrm{hom}$ when $x=0$.

Proposition 8.7

(Homogenised surface integrand for $x=0$) Let g satisfy (G1)–(G2) and assume that it is stationary with respect to a group $(\tau _z)_{z\in {\mathbb {Z}}^n}$ of P-preserving transformations on $(\Omega ,{\mathcal {T}},P)$. For $\omega \in \Omega$ let ${\textbf{m}}^s_\omega$ be as in (8.3). Then there exist ${\widetilde{\Omega }}\in {\mathcal {T}}$ with $P({\widetilde{\Omega }})=1$ and a $({\mathcal {T}}\otimes {\mathcal {B}}({{\mathbb {S}}}^{n-1}))$-measurable function $g_{\textrm{hom}}:\Omega \times {{\mathbb {S}}}^{n-1}\rightarrow [0,+\infty )$ such that

$$\begin{aligned} \lim _{r\rightarrow +\infty }\frac{{\textbf{m}}^s_\omega (\bar{u}_{0}^\nu ,Q_{r}^\nu (0))}{r^{n-1}}=g_\textrm{hom}(\omega ,\nu ) \end{aligned}$$

(8.13)

for every $\omega \in {\widetilde{\Omega }}$ and every $\nu \in {{\mathbb {S}}}^{n-1}$. Moreover, ${{\widetilde{\Omega }}}$ and $g_\textrm{hom}$ are $(\tau _z)_{z\in {\mathbb {Z}}^n}$-translation invariant; i.e., $\tau _{z}({\widetilde{\Omega }})={\widetilde{\Omega }}$ for every $z\in {\mathbb {Z}}^n$ and

$$\begin{aligned} g_\textrm{hom}(\tau _z(\omega ),\nu )=g_\textrm{hom}(\omega ,\nu ), \end{aligned}$$

(8.14)

for every $z\in {\mathbb {Z}}^n$, for every $\omega \in {\widetilde{\Omega }}$, and every $\nu \in {{\mathbb {S}}}^{n-1}$. Therefore, if $(\tau _z)_{z\in {\mathbb {Z}}^n}$ is ergodic then $g_{\textrm{hom}}$ is independent of $\omega$ and given by

$$\begin{aligned} g_{\textrm{hom}}(\nu )= \lim _{r\rightarrow +\infty } \frac{1}{r^{n-1}}\int _\Omega {\textbf{m}}^s_\omega ({\bar{u}}^\nu _0,Q^\nu _r(0))\, \textrm{d}P(\omega ). \end{aligned}$$

(8.15)

Proof

We divide the proof into three steps.

Step 1: existence of the limit for $\nu \in {{\mathbb {S}}}^{n-1}\cap {\mathbb {Q}}^n$. Let $\nu \in {{\mathbb {S}}}^{n-1}\cap {\mathbb {Q}}^n$ be fixed. Thanks to Proposition 8.6 we can apply the Subadditive Ergodic Theorem [1, Theorem 2.4] to the subadditive process $\mu _\nu$ defined on $(\Omega ,{\mathcal {T}},P)$ by (8.7). Then choosing $I=[-1,1)^{n-1}$ we get the existence of a set $\Omega _\nu \in {\mathcal {T}}$, with $P(\Omega _\nu )=1$, and of a ${\mathcal {T}}$-measurable function $g_\nu :\Omega \rightarrow [0,+\infty )$ such that

$$\begin{aligned} \lim _{j\rightarrow +\infty }\frac{{\mu _\nu }(\omega , j I)}{(2j)^{n-1}} = g_\nu (\omega ), \end{aligned}$$

for every $\omega \in \Omega _\nu$. By (8.7), since $I_\nu =2M_\nu Q^\nu (0)$ this yields

$$\begin{aligned} g_\nu (\omega )=\lim _{j\rightarrow +\infty }\frac{{\textbf{m}}^s_\omega ({\bar{u}}_0^\nu , j2M_\nu Q^\nu (0))}{(j2M_\nu )^{n-1}}, \end{aligned}$$

(8.16)

for every $\omega \in \Omega _\nu$. Let $(r_j)$ be a sequence of strictly positive real numbers with $r_j \rightarrow +\infty$, as $j \rightarrow +\infty$, and consider the two sequences of integers defined as follows:

$$\begin{aligned} r_j^-{:}{=}2M_\nu \Big (\Big \lfloor \frac{r_j}{2M_\nu }\Big \rfloor -1\Big )\quad \text {and}\quad r_j^+{:}{=}2M_\nu \Big (\Big \lfloor \frac{r_j}{2M_\nu }\Big \rfloor +2\Big ). \end{aligned}$$

Let $j \in N$ be such that $r_j>4(1+M_\nu )$, and thus $r_j^->4$. We clearly have

$$\begin{aligned} Q^\nu _{r_j^-+2}(0)\subset \subset Q_{r_j}^\nu (0)\subset \subset Q_{r_j+2}^\nu (0)\subset \subset Q_{r_j^+}^\nu (0), \end{aligned}$$

hence we can apply Lemma A.1 with $x={\widetilde{x}}=0$ first choosing $r=r_j^-$ and ${\widetilde{r}}=r_j$ to get

$$\begin{aligned} \frac{{\textbf{m}}^s_\omega (\bar{u}_0^\nu ,Q_{r_j}^\nu (0))}{r_j^{n-1}}\le \frac{{\textbf{m}}^s_\omega (\bar{u}_{0}^\nu ,Q_{r_j^-}^\nu (0))}{(r_j^-)^{n-1}} + \frac{L(r_j-r_j^-+1)}{r_j} , \end{aligned}$$

(8.17)

and then $r=r_j$ and ${\widetilde{r}}= r_j^+$ to obtain

$$\begin{aligned} \frac{{\textbf{m}}^s_\omega (\bar{u}_{0}^\nu ,Q_{r_j}^\nu (0))}{r_j^{n-1}}\ge \frac{{\textbf{m}}^s_\omega (\bar{u}_0^\nu ,Q_{r_j^+}^\nu (0))}{(r_j^+)^{n-1}}- \frac{L(r_j^+-r_j+1)}{r_j}. \end{aligned}$$

(8.18)

Clearly, $r_j^+-r_j\le 4M_\nu$ and $r_j-r_j^-\le 4M_\nu$, thus, if $\omega \in \Omega _\nu$, thanks to (8.16), passing to the limsup in as $j \rightarrow +\infty$ in (8.17) gives

$$\begin{aligned} \limsup _{j \rightarrow +\infty } \frac{{\textbf{m}}^s_\omega (\bar{u}_{0}^\nu ,Q_{r_j}^\nu (0))}{r_j^{n-1}} \le g_\nu (\omega ), \end{aligned}$$

(8.19)

while passing to the liminf in as $j \rightarrow +\infty$ in (8.18) yields

$$\begin{aligned} \liminf _{j \rightarrow +\infty } \frac{{\textbf{m}}^s_\omega (\bar{u}_{0}^\nu ,Q_{r_j}^\nu (0))}{r_j^{n-1}} \ge g_\nu (\omega ). \end{aligned}$$

(8.20)

Hence gathering (8.19) and (8.20) gives for every $\omega \in \Omega _\nu$ the existence of the limit along $(r_j)$ together with the equality

$$\begin{aligned} \lim _{j \rightarrow +\infty } \frac{{\textbf{m}}^s_\omega (\bar{u}_{0}^\nu ,Q_{r_j}^\nu (0))}{r_j^{n-1}}=g_\nu (\omega ). \end{aligned}$$

Therefore setting

$$\begin{aligned} {{\widetilde{\Omega }}}:=\,\bigcap _{\nu \in {\mathbb {S}}^{n-1}\cap {\mathbb {Q}}^n}\,\Omega _\nu , \end{aligned}$$

clearly gives ${{\widetilde{\Omega }}} \in {\mathcal {T}}$ as well as $P({{\widetilde{\Omega }}})=1$; moreover we have

$$\begin{aligned} g_\nu (\omega )=\lim _{r\rightarrow +\infty }\frac{{\textbf{m}}^s_\omega ({\bar{u}}_0^\nu ,Q_{r}^\nu (0))}{r^{n-1}} \end{aligned}$$

for every $\omega \in {{\widetilde{\Omega }}}$ and every $\nu \in {\mathbb {S}}^{n-1}\cap {\mathbb {Q}}^n$.

Step 2: existence of the limit for $\nu \in {{\mathbb {S}}}^{n-1}\setminus {\mathbb {Q}}^n$. Consider the two functions $\underline{g}, {\overline{g}} :{{\widetilde{\Omega }}} \times {{\mathbb {S}}}^{n-1} \rightarrow [0,+\infty ]$ defined as

$$\begin{aligned} \underline{g}(\omega ,\nu ){:}{=}\liminf _{r\rightarrow +\infty }\frac{{\textbf{m}}^s_\omega (\bar{u}_0^\nu ,Q_r^\nu (0))}{r^{n-1}},\quad {\overline{g}}(\omega ,\nu ){:}{=}\limsup _{r\rightarrow +\infty }\frac{{\textbf{m}}^s_\omega (\bar{u}_0^\nu ,Q_r^\nu (0))}{r^{n-1}}. \end{aligned}$$

By the previous step we have that $\underline{g}(\omega ,\nu )={\overline{g}}(\omega ,\nu )=g_\nu (\omega )$, for every $\omega \in {{\widetilde{\Omega }}}$ and for every $\nu \in {{\mathbb {S}}}^{n-1}\cap {\mathbb {Q}}^n$. Therefore, if we show that the restrictions of the functions $\nu \mapsto {\underline{g}}(\omega ,\nu )$ and $\nu \mapsto {{\overline{g}}}(\omega ,\nu )$ to the sets ${\widehat{{\mathbb {S}}}}_\pm ^{n-1}$ are continuous, by the density of ${\widehat{{{\mathbb {S}}}}}_\pm ^{n-1}\cap {\mathbb {Q}}^n$ in ${\widehat{{{\mathbb {S}}}}}_\pm ^{n-1}$ we can readily deduce that ${\underline{g}}(\omega ,\nu )={{\overline{g}}}(\omega ,\nu )=g_\nu (\omega )$ for every $\omega \in {\widetilde{\Omega }}$ and every $\nu \in {{\mathbb {S}}}^{n-1}$, and thus the claim.

Then it remains to show that ${\underline{g}}(\omega ,\cdot )$ and ${{\overline{g}}}(\omega ,\cdot )$ are continuous in ${{\widehat{{\mathbb {S}}}}}^{n-1}_\pm$. We only prove that ${{\overline{g}}}(\omega ,\cdot )$ is continuous in ${{\widehat{{\mathbb {S}}}}}^{n-1}_+$, the other proofs being analogous. To this end, let $\nu \in {\widehat{{\mathbb {S}}}}_+^{n-1}$, $(\nu _j)\subset {\widehat{{\mathbb {S}}}}_+^{n-1}$ be such that $\nu _j\rightarrow \nu$, as $j \rightarrow +\infty$. Then, for every $\alpha \in (0,\frac{1}{2})$ there exists $j_\alpha \in {\mathbb {N}}$ such that the assumptions of Lemma A.2 are satisfied for every $j\ge j_\alpha$. Thus, choosing $x=0$ and ${{\widetilde{\nu }}} =\nu _j$ in (A.7) we obtain

$$\begin{aligned} {\textbf{m}}^s_\omega ({\bar{u}}^{\nu _j}_0, Q^{\nu _j}_{(1+\alpha )r}(0)) -c_\alpha r^{n-1} \le {\textbf{m}}^s_\omega ({\bar{u}}^{\nu }_0, Q^{\nu }_{r}(0)) \le {\textbf{m}}^s_\omega ({\bar{u}}^{\nu _j}_0, Q^{\nu _j}_{(1-\alpha )r}(0)) +c_\alpha r^{n-1}, \end{aligned}$$

where $c_\alpha \rightarrow 0$, as $\alpha \rightarrow 0$. Therefore, by definition of ${{\overline{g}}}$ passing to the limsup as $r \rightarrow +\infty$ we get the two following inequalities

$$\begin{aligned} (1+\alpha )^{n-1}\,{\overline{g}}(\omega ,\nu _j)&\le {{\overline{g}}}(\omega ,\nu )+c_\alpha \,, \end{aligned}$$

(8.21)

$$\begin{aligned} (1-\alpha )^{n-1}\,{\overline{g}}(\omega ,\nu _j)&\ge {{\overline{g}}}(\omega ,\nu )-c_\alpha \,. \end{aligned}$$

(8.22)

Passing to the limsup as $j\rightarrow +\infty$ in (8.21) and to the liminf as $j\rightarrow +\infty$ in (8.22) and eventually letting $\alpha \rightarrow 0$ we infer

$$\begin{aligned} \limsup _{j\rightarrow +\infty }{\overline{g}}(\omega ,\nu _j)\le {{\overline{g}}}(\omega ,\nu )\le \liminf _{j\rightarrow +\infty }{\overline{g}}(\omega ,\nu _j), \end{aligned}$$

and hence the claim.

For every $\omega \in \Omega$ and $\nu \in {{\mathbb {S}}}^{n-1}$ set

$$\begin{aligned} g_{\textrm{hom}}(\omega ,\nu ):={\left\{ \begin{array}{ll} {{\overline{g}}}(\omega ,\nu ) &{} \text {if }\; \omega \in {{\widetilde{\Omega }}}, \\ c_2 c_p &{} \text {if }\; \omega \in \Omega \setminus {{\widetilde{\Omega }}}, \end{array}\right. } \end{aligned}$$

(8.23)

then, for every $\omega \in {{\widetilde{\Omega }}}$ and every $\nu \in {{\mathbb {S}}}^{n-1}$ we have

$$\begin{aligned} g_\textrm{hom}(\omega , \nu )=\lim _{r\rightarrow +\infty }\frac{{\textbf{m}}^s_\omega ({\bar{u}}_0^\nu ,Q_{r}^\nu (0))}{r^{n-1}}. \end{aligned}$$

(8.24)

Moreover

$$\begin{aligned} \omega \mapsto {{\overline{g}}} (\omega ,\nu ) \; \text { is }{\mathcal {T}}\text {-measurable in }{{\widetilde{\Omega }}}\text {, for every }\nu \in {{\mathbb {S}}}^{n-1} \end{aligned}$$

and

$$\begin{aligned} \nu \mapsto {{\overline{g}}} (\omega ,\nu ) \; \text { is continuous in }{{\widehat{{{\mathbb {S}}}}}}^{n-1}_\pm \text {, for every }\omega \in {{\widetilde{\Omega }}}. \end{aligned}$$

Therefore the restriction of ${{\overline{g}}}$ to ${{\widetilde{\Omega }}} \times {{\widehat{{{\mathbb {S}}}}}}^{n-1}_\pm$ is measurable with respect to the $\sigma$-algebra induced in ${{\widetilde{\Omega }}} \times {{\widehat{{{\mathbb {S}}}}}}^{n-1}_\pm$ by ${\mathcal {T}} \otimes {\mathcal {B}}({{\mathbb {S}}}^{n-1})$ thus, finally, $g_\textrm{hom}$ is $({\mathcal {T}} \otimes {\mathcal {B}}({{\mathbb {S}}}^{n-1}))$-measurable on $\Omega \times {{\mathbb {S}}}^{n-1}$.

Step 3: $(\tau _z)_{z\in {\mathbb {Z}}^n}$-translation invariance. Let $z\in {\mathbb {Z}}^n$, $\omega \in {{\widetilde{\Omega }}}$, and $\nu \in {\mathbb {S}}^{n-1}$ be fixed. Let $r>4$ and $v\in {\mathscr {A}}({\bar{u}}^\nu _0,Q^\nu _r(0))$ with

$$\begin{aligned} {\mathscr {F}}^s(\omega )(v,Q_r^\nu (0))\le {\textbf{m}}^s_\omega ({\bar{u}}^\nu _0,Q^\nu _r(0))+1. \end{aligned}$$

(8.25)

Set ${\widetilde{v}}(y):=v(y+z)$; by the stationarity of g we have

$$\begin{aligned} {\mathscr {F}}^s(\omega )(v,Q_r^\nu (0))={\mathscr {F}}^s(\tau _z(\omega ))({\widetilde{v}},Q_r^\nu (-z)), \end{aligned}$$

further, since ${\widetilde{v}}\in {\mathscr {A}}({\bar{u}}^\nu _{-z},Q^\nu _r(-z))$, by (8.25) we immediately get

$$\begin{aligned} {\textbf{m}}^s_{\tau _z(\omega )}({\bar{u}}^\nu _{-z},Q^\nu _r(-z))\le {\textbf{m}}^s_\omega ({\bar{u}}^\nu _0,Q^\nu _r(0))+1. \end{aligned}$$

(8.26)

Now let $r, {\widetilde{r}}$ be such that ${\widetilde{r}}> r$ and

$$\begin{aligned} Q^\nu _{r+2}(-z)\subset \subset Q^\nu _{{\widetilde{r}}}(0)\; \text { and }\; {{\,\textrm{dist}\,}}(0,\Pi ^\nu (-z))\le \frac{r}{4}, \end{aligned}$$

therefore applying Lemma A.1 with $x=-z$ and ${\widetilde{x}}=0$ we may deduce the existence of a constant $L>0$ such that

$$\begin{aligned} {\textbf{m}}^s_{\tau _z(\omega )}(\bar{u}_{0}^\nu ,Q_{\widetilde{r}}^\nu (0)) \le {\textbf{m}}^s_{\tau _z(\omega )}(\bar{u}_{-z}^\nu ,Q_{r}^\nu (-z))+L\big (|z|+|r-\widetilde{r}|+1\big )(\widetilde{r})^{n-2}. \end{aligned}$$

(8.27)

Hence, gathering (8.26) and (8.27) we obtain

$$\begin{aligned} \frac{{\textbf{m}}^s_{\tau _z(\omega )}(\bar{u}_{0}^\nu ,Q_{\widetilde{r}}^\nu (0))}{{\widetilde{r}}^{n-1}}\le \frac{{\textbf{m}}^s_{\omega }({\bar{u}}^\nu _0,Q^\nu _r(0))+1}{r^{n-1}}+\frac{L\big (|z|+|r-\widetilde{r}|+1\big )}{{\widetilde{r}}}. \end{aligned}$$

(8.28)

An analogous argument, now replacing $\omega$ with $\tau _z(\omega )$ and z with $-z$, yields

$$\begin{aligned} \frac{{\textbf{m}}^s_\omega (\bar{u}_{0}^\nu ,Q_{\widetilde{r}}^\nu (0))}{{\widetilde{r}}^{n-1}}\le \frac{{\textbf{m}}^s_{\tau _z(\omega )}({\bar{u}}^\nu _0,Q^\nu _r(0))+1}{r^{n-1}}+\frac{L\big (|z|+|r-\widetilde{r}|+1\big )}{{\widetilde{r}}}. \end{aligned}$$

(8.29)

Taking in (8.28) the limsup as ${\widetilde{r}}\rightarrow +\infty$ and the limit as $r\rightarrow +\infty$ gives

$$\begin{aligned} \limsup _{{\widetilde{r}}\rightarrow +\infty } \frac{{\textbf{m}}^s_{\tau _z(\omega )}(\bar{u}_{0}^\nu ,Q_{\widetilde{r}}^\nu (0))}{{\widetilde{r}}^{n-1}}\le g_\textrm{hom}(\omega ,\nu ), \end{aligned}$$

(8.30)

while taking in (8.29) the limit as ${\widetilde{r}}\rightarrow +\infty$ and the liminf as $r\rightarrow +\infty$ entails

$$\begin{aligned} g_\textrm{hom}(\omega ,\nu )\le \liminf _{r\rightarrow +\infty }\frac{{\textbf{m}}^s_{\tau _z(\omega )}(\bar{u}_{0}^\nu ,Q_{r}^\nu (0))}{r^{n-1}}. \end{aligned}$$

(8.31)

Eventually, by combining (8.30) and (8.31) we both deduce that $\tau _z(\omega )\in {{\widetilde{\Omega }}}$ and that

$$\begin{aligned} g_\textrm{hom}(\tau _z(\omega ),\nu )=g_\textrm{hom}(\omega ,\nu ), \end{aligned}$$

for every $z\in {\mathbb {Z}}$, $\omega \in {\widetilde{\Omega }}$, and $\nu \in {{\mathbb {S}}}^{n-1}$. Then, we observe that thanks to the group properties of $(\tau _z)_{z\in {\mathbb {Z}}^n}$ we also have that $\omega \in \tau _z({{\widetilde{\Omega }}})$, for every $z\in {\mathbb {Z}}^n$. Indeed we have $\omega = \tau _{z}(\tau _{-z}(\omega ))$ and $\tau _{-z}(\omega ) \in {{\widetilde{\Omega }}}$.

If $(\tau _z)_{z\in {\mathbb {Z}}^n}$ is ergodic, then the independence of $\omega$ of the function of $g_\textrm{hom}$ is a direct consequence of (8.14) (cf. [28, Corollary 6.3]). Furthermore, (8.15) can be obtained by integrating (8.24) on $\Omega$ and using the Dominated Convergence Theorem, thanks to (8.8) (see also (8.7)). $\square$

The following result is of crucial importance in our analysis as it extends Proposition 8.7 to the case of an arbitrary $x\in {\mathbb {R}}^n$. More precisely, Proposition 8.8 below establishes the existence of the limit in (8.13) when $x=0$ is replaced by any $x\in {\mathbb {R}}^n$; moreover it shows that this limit is x-independent, and hence it coincides with (8.13).

The proof of the following proposition can be obtained arguing exactly as in [28, Theorem 6.1] (see also [3, Theorem 5.5]), now appealing to Proposition 8.7, Lemma A.1, and Lemma A.2. For this reason we omit its proof here.

Proposition 8.8

(Homogenised surface integrand) Let g satisfy (G1)–(G2) and assume that it is stationary with respect to a group $(\tau _z)_{z\in {\mathbb {Z}}^n}$ of P-preserving transformations on $(\Omega ,{\mathcal {T}},P)$. For $\omega \in \Omega$ let ${\textbf{m}}^s_\omega$ be as in (8.3). Then there exists $\Omega '\in {\mathcal {T}}$ with $P(\Omega ')=1$ such that

$$\begin{aligned} \lim _{r\rightarrow +\infty }\frac{{\textbf{m}}^s_\omega (\bar{u}_{rx}^\nu ,Q_{r}^\nu (rx))}{r^{n-1}}=g_\textrm{hom}(\omega ,\nu ) \end{aligned}$$

(8.32)

for every $\omega \in \Omega '$, every $x\in {\mathbb {R}}^n$, and every $\nu \in {{\mathbb {S}}}^{n-1}$, where $g_\textrm{hom}$ is given by (8.13). In particular, the limit in (8.32) is independent of x. Moreover, if $(\tau _z)_{z\in {\mathbb {Z}}^n}$ is ergodic, then $g_\textrm{hom}$ is independent of $\omega$ and given by (8.15).

We conclude this section with the proof of Theorem 8.4.

Proof of Theorem 8.4

The proof follows by Theorem 3.5 now invoking Proposition 8.5 and Proposition 8.8. $\square$

Appendix

In this last section we state and prove two technical lemmas which are used in Sect. 8.1.

For $A \in {\mathcal {A}}$, $x\in {\mathbb {R}}^n$, and $\nu \in {{\mathbb {S}}}^{n-1}$, in what follows ${\textbf{m}}^s({\bar{u}}^\nu _x, A)$ denotes the infimum value given by (3.7).

Lemma A.1

Let $g \in {\mathcal {G}}$; let $\nu \in {{\mathbb {S}}}^{n-1}$, $x,\widetilde{x}\in {\mathbb {R}}^n$, and $\widetilde{r}> r>4$ be such that

$$\begin{aligned} \mathrm{(i)}\ Q_{r+2}^\nu (x)\subset \subset Q_{\widetilde{r}}^\nu ({\widetilde{x}}),\qquad \mathrm{(ii)}\ {{\,\textrm{dist}\,}}({\widetilde{x}},\Pi ^\nu (x))\le \frac{r}{4}. \end{aligned}$$

Then there exists a constant $L>0$ (independent of $\nu ,x,\widetilde{x},r,\widetilde{r}$) such that

$$\begin{aligned} {\textbf{m}}^s(\bar{u}_{\widetilde{x}}^\nu ,Q_{\widetilde{r}}^\nu ({\widetilde{x}}))\le {\textbf{m}}^s(\bar{u}_{x}^\nu ,Q_{r}^\nu (x))+L\big (|x-{\widetilde{x}}|+|r-\widetilde{r}|+1\big )\widetilde{r}^{n-2}. \end{aligned}$$

(A.1)

Proof

To prove (A.1) we are going to perform a construction which is similar to that used in Proposition 2.6.

Let $\nu \in S^{n-1}$, $\eta >0$ be fixed and let $v\in {\mathscr {A}}(\bar{u}^\nu _x,Q^\nu _r(x))$ with

$$\begin{aligned} {\mathscr {F}}^s(v,Q^\nu _{r}(x))\le {\textbf{m}}^s(\bar{u}^\nu _{x},Q^\nu _{r}(x))+\eta . \end{aligned}$$

(A.2)

Let $u\in W^{1,p}(Q^\nu _{r}(x);{\mathbb {R}}^m)$ correspond to the v as above; i.e., $v \nabla u=0$ a.e. in $Q^\nu _{r}(x)$ and $(u,v)=(\bar{u}^\nu _{x},\bar{v}^\nu _{x})$ a.e. in U, where U is a neighbourhood of $\partial Q^\nu _{r}(x)$. Let moreover $\beta \in (0,1)$ be such that $Q^\nu _{r}(x)\setminus {\overline{Q}}^\nu _{r-\beta }(x)\subset U$. Let ${\widetilde{x}} \in {\mathbb {R}}^n$ and ${\widetilde{r}}>r>4$ satisfy (i)–(ii). We set

$$\begin{aligned} R{:}{=}R_\nu \biggl ( \big (Q'_{r}\setminus \overline{Q'}_{r-\beta }\big )\times \Big (-1-\frac{|(x- \widetilde{x})\cdot \nu |}{2},1+\frac{|(x-\widetilde{x})\cdot \nu |}{2}\Big )\biggr )+x+\frac{({\widetilde{x}}-x)\cdot \nu }{2} \nu , \end{aligned}$$

where $R_\nu$ be as in (f) (see Fig. 3a). Then let $\varphi \in C_c^\infty (Q_{r}')$ be a cut-off function between $Q'_{r-\beta }$ and $Q'_{r}$ ; i.e., $0\le \varphi \le 1$, and $\varphi \equiv 1$ on $Q_{r-\beta }'$. Eventually, for $y=(y',y_n)\in Q_{{\widetilde{r}}}^\nu ({\widetilde{x}})$ we define the pair $({\widetilde{u}},{\widetilde{v}})$ by setting

$$\begin{aligned} {\widetilde{u}}(y):= & {} \varphi ((R_\nu ^T(y- x))')u(y)+(1-\varphi ((R_\nu ^T(y- x))')){\bar{u}}_{{\widetilde{x}}}^\nu (y), \\ {\widetilde{v}}(y):= & {} {\left\{ \begin{array}{ll} \min \{v(y),d(y)\}&{}\text {in}\ Q^\nu _{r}(x),\\ \min \{{\bar{v}}^\nu _{{\widetilde{x}}}(y),d(y)\}&{}\text {in}\ Q^\nu _{\widetilde{r}}({\widetilde{x}})\setminus {\overline{Q}}^\nu _{r}(x), \end{array}\right. } \end{aligned}$$

where $d(y){:}{=}{{\,\textrm{dist}\,}}(y,R)$. Clearly, ${\widetilde{u}}\in W^{1,p}(Q^\nu _{\widetilde{r}}({\widetilde{x}});{\mathbb {R}}^m)$, moreover, by construction we have

$$\begin{aligned} |(y-x)\cdot \nu |>1\;\text { and }\;|(y-{\widetilde{x}})\cdot \nu |>1\;\text { for every }\; y\in (Q_r^\nu (x)\setminus {\overline{Q}}_{r-\beta }^\nu (x))\setminus {\overline{R}}. \end{aligned}$$

Hence, the boundary conditions satisfied by v imply that

$$\begin{aligned} v=\bar{v}^\nu _{x}=\bar{v}^\nu _{{\widetilde{x}}}=1 \; \text { in }\; (Q^\nu _r(x)\setminus {\overline{Q}}^\nu _{r-\beta }(x))\setminus {\overline{R}}, \end{aligned}$$

thus $\widetilde{v}\in W^{1,p}(Q^\nu _{\widetilde{r}}({\widetilde{x}}))$.

Thanks to (ii) and to the fact that $r>4$ we have $R\subset Q^\nu _{r}(x)$ so that $\{d<1\}\subset Q^\nu _{r+2}(x)$. Therefore, in view of (i) we get that $\widetilde{v}=\bar{v}^\nu _{\widetilde{x}}$ in a neighbourhood of $\partial Q^\nu _{\widetilde{r}}(\widetilde{x})$, further $\widetilde{u}=\bar{u}^\nu _{{\widetilde{x}}}$ in $Q^\nu _{\widetilde{r}}({\widetilde{x}})\setminus {\overline{Q}}^\nu _r(x)$. Then, to show that $\widetilde{v}$ is admissible for ${\textbf{m}}^s(\bar{u}^\nu _{{\widetilde{x}}},Q^\nu _{\widetilde{r}}({\widetilde{x}}))$ it only remains to check that $\widetilde{v}\,\nabla \widetilde{u}=0$ a.e. in $Q^\nu _{\widetilde{r}}({\widetilde{x}})$. Clearly, $\widetilde{v}\,\nabla \widetilde{u}=0$ a.e. in R. On the other hand, in $Q^\nu _r(x)\setminus {\overline{R}}$ we have $\widetilde{v} |\nabla \widetilde{u}| \le v|\nabla u|=0$, while in $\big (Q^\nu _{\widetilde{r}}({\widetilde{x}})\setminus Q^\nu _r(x)\big )\setminus {\overline{R}}$ we get $\widetilde{v}|\nabla \widetilde{u}|\le \bar{v}^\nu _{\widetilde{x}}|\nabla \bar{u}^\nu _{\widetilde{x}}|=0$, and hence the claim.

Eventually, arguing as in Proposition 2.6 we obtain

$$\begin{aligned} \begin{aligned} {\mathscr {F}}^s\big (\widetilde{v},Q^\nu _{\widetilde{r}}({\widetilde{x}})\big )&\le {\mathscr {F}}^s(v,Q^\nu _r(x))+{\mathscr {F}}^s\big (\bar{v}^\nu _{{\widetilde{x}}},Q^\nu _{\widetilde{r}}({\widetilde{x}})\setminus {\overline{Q}}^\nu _r(x)\big )+2c_4{\mathcal {L}}^n(\{d<1\}). \end{aligned} \end{aligned}$$

(A.3)

Thanks to (2.3) and (2.19) we have

$$\begin{aligned} {\mathscr {F}}^s\big (\bar{v}^\nu _{{\widetilde{x}}},Q^\nu _{\widetilde{r}}({\widetilde{x}})\setminus {\overline{Q}}^\nu _r(x)\big )\le c_4C_{\textrm{v}}{\mathcal {L}}^{n-1}\big (Q'_{\widetilde{r}}\setminus {\overline{Q}}'_r\big )\le C|r-\widetilde{r}|\widetilde{r}^{n-2}, \end{aligned}$$

(A.4)

moreover

$$\begin{aligned} {\mathcal {L}}^n(\{d<1\})\le C(|x-{\widetilde{x}}|+1)(\beta +1) r^{n-2}. \end{aligned}$$

(A.5)

We observe that both in (A.4) and (A.5) the positive constant $C>0$ does not depend on any of the parameters and on $\nu$. Therefore, gathering (A.2) and (A.3) we finally obtain

$$\begin{aligned} {\textbf{m}}^s(\bar{u}^\nu _{{\widetilde{x}}},Q^\nu _{\widetilde{r}}({\widetilde{x}}))\le {\mathscr {F}}^s\big (\widetilde{v},Q^\nu _{\widetilde{r}}({\widetilde{x}})\big )\le {\textbf{m}}^s(\bar{u}^\nu _{x},Q^\nu _r(x))+L\big (|x-{\widetilde{x}}|+|r-\widetilde{r}|+1\big )\widetilde{r}^{n-2}+\eta , \end{aligned}$$

for some $L>0$ independent of $x,{\widetilde{x}}, r, {\widetilde{r}}, \nu$, thus (A.1) follows by the arbitrariness of $\eta >0$.

$\square$

Fig. 3

The sets used in the construction of $(\widetilde{u},\widetilde{v})$ in Lemma A.1 and Lemma A.2

Lemma A.2

Let $g \in {\mathcal {G}}$; let $\alpha \in (0,\frac{1}{2})$ and $\nu ,{\widetilde{\nu }}\in {{\mathbb {S}}}^{n-1}$ be such that

$$\begin{aligned} \max _{1\le i\le n-1}|R_\nu e_i-R_{{{\widetilde{\nu }}}} e_i| + |\nu -{{\widetilde{\nu }}}| <\frac{\alpha }{\sqrt{n}}, \end{aligned}$$

(A.6)

where $R_\nu$ and $R_{{{\widetilde{\nu }}}}$ are orthogonal $(n\times n)$-matrices as in (f). Then there exists a constant $c_\alpha >0$ (independent of $\nu ,{\widetilde{\nu }}$), with $c_\alpha \rightarrow 0$ as $\alpha \rightarrow 0$, such that for every $x\in {\mathbb {R}}^n$ and every $r>2$ we have

$$\begin{aligned} \begin{aligned} {\textbf{m}}^s\big (\bar{u}_{rx}^{{\widetilde{\nu }}},Q_{(1+\alpha )r}^{{\widetilde{\nu }}}(rx)\big )-c_\alpha r^{n-1}&\le {\textbf{m}}^s\big (\bar{u}_{rx}^\nu , Q_{r}^\nu (rx)\big )\\&\le {\textbf{m}}^s\big (\bar{u}_{rx}^{{\widetilde{\nu }}},Q_{(1-\alpha )r}^{{\widetilde{\nu }}}(rx)\big )+c_\alpha r^{n-1}. \end{aligned} \end{aligned}$$

(A.7)

Proof

We only prove that

$$\begin{aligned} {\textbf{m}}^s\big (\bar{u}_{rx}^{{\widetilde{\nu }}},Q_{(1+\alpha )r}^{{\widetilde{\nu }}}(rx)\big )-c_\alpha r^{n-1} \le {\textbf{m}}^s\big (\bar{u}_{rx}^\nu , Q_{r}^\nu (rx)\big ), \end{aligned}$$

(A.8)

for some $c_\alpha >0$, with $c_\alpha \rightarrow 0$, as $\alpha \rightarrow 0$; the proof of the other inequality is analogous.

Let $x\in {\mathbb {R}}^n$, $r>2$, and set $r_\alpha ^\pm {:}{=}(1\pm \alpha )r$. We notice that condition (A.6) readily implies that

$$\begin{aligned} Q_{ r_\alpha ^-}^{{\widetilde{\nu }}}(rx) \subset \subset Q_r^\nu (rx)\subset \subset Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx). \end{aligned}$$

(A.9)

Moreover, we let r be so large that $Q_{r+2}^\nu (rx)\subset \subset Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx)$. This allows us to use a similar argument as in the proofs of Proposition 2.6 and Lemma A.1, which we repeat here for the readers’ convenience.

Let $\eta >0$ be fixed and let $v\in W^{1,p}(Q^\nu _r(rx))$ be a test function for ${\textbf{m}}^s({\bar{u}}_{rx}^\nu ,Q^\nu _r(rx))$ satisfying

$$\begin{aligned} {\mathscr {F}}^s(v,Q^\nu _r(rx))\le {\textbf{m}}^s({\bar{u}}_{rx}^\nu ,Q^\nu _r(rx))+\eta . \end{aligned}$$

(A.10)

Therefore we know that there exist $u\in W^{1,p}(Q_r^\nu (rx);{\mathbb {R}}^m)$ and a neighbourhood U of $\partial Q_r^\nu (rx)$ such that

$$\begin{aligned} (u,v)=({\bar{u}}_{rx}^\nu ,{\bar{v}}_{rx}^\nu )\quad \text {in}\quad U\;\text { and }\; v \nabla u=0\;\text { a.e. in }\; Q_r^\nu (rx). \end{aligned}$$

(A.11)

Now let $\beta \in (0,1)$ be such that $Q_r^\nu (rx)\setminus {\overline{Q}}_{r-\beta }^\nu (rx)\subset U$ and set

$$\begin{aligned} R:=R_\nu \Big (Q'_{r} \setminus \overline{Q'}_{r-\beta }\times (-1-\alpha r,1+\alpha r)\Big )+rx, \end{aligned}$$

where $R_\nu$ is as in (f) (see Fig. 3b). Let $\varphi \in C^\infty _c(Q'_r)$ be a cut-off function between $Q_{r-\beta }'$ and $Q_{r}'$; we define

$$\begin{aligned} {\widetilde{u}}(y)&:=\varphi \big (\big (R_\nu ^T(y-rx)\big )'\big )u(y)+\Big (1-\varphi \big (\big (R_\nu ^T(y-rx)\big )'\big )\Big ){\bar{u}}_{ rx}^{{\widetilde{\nu }}}(y)\,,\\ {\widetilde{v}}(y)&:={\left\{ \begin{array}{ll} \min \{v(y),d(y)\}&{}\text {in }\; Q_r^\nu (rx)\,,\\ \min \{{\bar{v}}^{{\widetilde{\nu }}}_{ rx}(y),d(y)\}&{}\text {in }\;Q^\nu _{ r_\alpha ^+}(rx)\setminus {\overline{Q}}^\nu _r(rx)\,, \end{array}\right. } \end{aligned}$$

where $d(y):={{\,\textrm{dist}\,}}(y,R)$. We now observe that in view of (A.6) and (A.11), by definition of R we get

$$\begin{aligned} v={\bar{v}}_{rx}^\nu =\bar{v}_{ rx}^{{\widetilde{\nu }}}=1\; \text { in }\; (Q_{r}^\nu (rx)\setminus {\overline{Q}}_{r-\beta }(rx))\setminus {\overline{R}}, \end{aligned}$$

so that ${\widetilde{v}}\in W^{1,p}\big (Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx)\big )$. Since $Q_{r+2}^\nu (rx)\subset \subset Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx)$, arguing as in Lemma A.1 it is easy to check that the pair $({\widetilde{u}},{\widetilde{v}}) \in W^{1,p}\big (Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx);{\mathbb {R}}^m\big )\times W^{1,p}\big (Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx)\big )$ also satisfies

$$\begin{aligned} ({\widetilde{u}},{\widetilde{v}})=({\bar{u}}_{rx}^{{\widetilde{\nu }}},{\bar{v}}_{rx}^{{\widetilde{\nu }}})\;\text { near }\; \partial Q^{{\widetilde{\nu }}}_{ r_\alpha ^+}(rx)\quad \text {and}\quad {\widetilde{v}}\, \nabla {\widetilde{u}} =0\; \text { a.e. in }\; Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx). \end{aligned}$$

Therefore, ${\widetilde{v}}$ is admissible for ${\textbf{m}}^s({\bar{u}}_{ rx}^{{\widetilde{\nu }}},Q^{{\widetilde{\nu }}}_{ r_\alpha ^+}(rx))$. Then, using the same arguments as in Proposition 2.6 we deduce

$$\begin{aligned} \begin{aligned} {\mathscr {F}}^s({\widetilde{v}},Q^{{\widetilde{\nu }}}_{ r_\alpha ^+}(rx))&\le {\mathscr {F}}^s(v,Q_r^\nu (rx))+{\mathscr {F}}^s(\bar{v}_{ rx}^{{\widetilde{\nu }}},Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx)\setminus {\overline{Q}}_{r}^\nu (rx))+2c_4{\mathcal {L}}^n(\{d<1\})\\&\le {\mathscr {F}}^s(v,Q_r^\nu (rx))+{\mathscr {F}}^s(\bar{v}_{ rx}^{{\widetilde{\nu }}},Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx)\setminus {\overline{Q}}_{ r_\alpha ^-}^{{\widetilde{\nu }}}(rx))+2c_4{\mathcal {L}}^n(\{d<1\}), \end{aligned} \end{aligned}$$

(A.12)

where to obtain the second inequality we used the first inclusion in (A.9). Furthermore by (2.3) and (2.19) we have

$$\begin{aligned} {\mathscr {F}}^s(\bar{v}_{ rx}^{{\widetilde{\nu }}},Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx)\setminus {\overline{Q}}_{ r_\alpha ^-}^{{\widetilde{\nu }}}(rx))&\le c_4C_{\textrm{v}}{\mathcal {L}}^{n-1}\big (Q_{ r_\alpha ^+}'(rx)\setminus Q_{ r_\alpha ^-}'(rx)\big )\nonumber \\&= c_4C_{\textrm{v}} ((1+\alpha )^{n-1}-(1-\alpha )^{n-1})r^{n-1} \end{aligned}$$

(A.13)

whereas

$$\begin{aligned} {\mathcal {L}}^n(\{d<1\})\le C (\beta +1) r^{n-2}\,\alpha r\le C \alpha r^{n-1}, \end{aligned}$$

(A.14)

for some $C>0$ independent of $x, \nu$, and r. Thus, gathering (A.10), (A.12), (A.13), and (A.14), thanks to (A.11) we infer

$$\begin{aligned} {\textbf{m}}^s(\bar{u}_{ rx}^{{\widetilde{\nu }}},Q_{ r_\alpha ^+}^{{\widetilde{\nu }}}(rx)) \le {\textbf{m}}^s(\bar{u}_{rx}^\nu ,Q_r^\nu (rx))+c_\alpha r^{n-1} +\eta \,, \end{aligned}$$

where $c_\alpha {:}{=}c_4C_{\textrm{v}}\big ((1+\alpha )^{n-1}-(1-\alpha )^{n-1}\big )+C\alpha$. Eventually, (A.8) follows by the arbitrariness of $\eta >0$. $\square$