Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence and stochastic homogenisation of singularly-perturbed elliptic functionals

We study the limit behaviour of singularly-perturbed elliptic functionals of the form Fk(u,v)=∫Av2fk(x,∇u)dx+1εk∫Agk(x,v,εk∇v)dx,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathscr {F}}_k(u,v)=\int _A v^2\,f_k(x,\nabla u)\, \textrm{d}x+\frac{1}{\varepsilon _k}\int _A g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x, \end{aligned}$$\end{document}where u is a vector-valued Sobolev function, v∈[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \in [0,1]$$\end{document} a phase-field variable, and εk>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _k>0$$\end{document} a singular-perturbation parameter; i.e., εk→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _k \rightarrow 0$$\end{document}, as k→+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\rightarrow +\infty $$\end{document}. Under mild assumptions on the integrands fk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_k$$\end{document} and gk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_k$$\end{document}, we show that if fk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_k$$\end{document} grows superlinearly in the gradient-variable, then the functionals Fk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}_k$$\end{document}Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-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 Fk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}_k$$\end{document}decouple in the limit. The abstract Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.

In this paper we study the -convergence, as k → +∞, of general elliptic functionals of the form where ε k 0 is a singular-perturbation parameter. The set A ⊂ R n is open bounded and with Lipschitz boundary, u : A → R m is a vectorial function, v : A → [0, 1] is a phase-field variable, and ψ : [0, 1] → [0, 1] is an increasing and continuous function satisfying ψ(0) = 0, ψ(1) = 1, and ψ(s) > 0 for s > 0. For every k ∈ N, the integrands f k : R n × R m×n → [0, +∞) and g k : R n ×[0, 1]×R n → [0, +∞) belong to suitable classes of functions denoted by F and G, respectively (see Sect. 2.2 for their definition). The requirement ( f k ) ⊂ F and (g k ) ⊂ G in particular ensures the existence of an exponent p > 1 such that for every k ∈ N and every x ∈ R n : for every ξ ∈ R m×n and for some 0 < c 1 ≤ c 2 < +∞, and for every v ∈ [0, 1] and w ∈ R n , and for some 0 < c 3 ≤ c 4 < +∞. As a consequence, the functionals F k are finite in W 1, p (A; R m ) × W 1, p (A) and are bounded both from below and from above by Ambrosio-Tortorelli functionals of the form Therefore if (u k , v k ) ⊂ W 1, p (A; R m )×W 1, p (A) is a pair satisfying sup k F k (u k , v k ) < +∞, the lower bound on F k immediately yields v k → 1 in L p (A), as k → +∞. On the other hand, |∇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 -convergence, of the free-discontinuity functional of Mumford-Shah type given by A |∇u| p dx + c p H n−1 (S u ), (1.5) where now the variable u belongs to the space of generalised special functions of bounded variation G S BV p (A; R m ). As in the case of the Modica-Mortola approximation of the perimeter-functional [48,49], the effect of the singular perturbation, ε p−1 k |∇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 ψ 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 ≥ 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 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 -convergence result for 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 -convergence [23,32] with a careful local analysis which eventually allows us to completely characterise the integrands of the -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 freediscontinuity 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 -convergence and an integral representation result for the -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 F k -converge to a free-discontinuity functional of the form where now u ∈ G S BV p (A; R m ) and ν u denotes the generalised normal to S u . We observe that the surface term in F is both inhomogeneous and anisotropic, however it does not depend on the jump-opening [u] = u + − u − ; in other words, 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 ∞ and g ∞ . In fact, in Theorem 3.1 we also provide formulas for f ∞ and g ∞ . Namely, we prove that . The surface energy density is given instead by where the cube Q ν ρ (x) is a suitable rotation of Q ρ (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 , and u ν x is the jump function given by In (1.8) the boundary datum (u ν x , 1) cannot be prescribed in the vicinity of {y ∈ R n : (y − x) · ν = 0} due to the discontinuity of u ν 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 ν ρ (x), up to replacing u ν 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 ψ, the constraint v∇u = 0 satisfied a.e. in Q ν ρ (x) is equivalent to which makes apparent why the bulk term in F k does not contribute to g ∞ . We notice, however, that due to the nature of the problem, the variable u must enter in the definition of g ∞ , 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 ∞ 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, 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 ν ρ (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 ∇u k = 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 -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 for some f ∈ F and g ∈ G. More specifically, in Theorem 3.5 we prove a homogenisation result for 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 -convergence result for functionals with oscillating integrands of type (1.9) when the heterogeneity scale does not necessarily coincide with the Ambrosio-Tortorelli parameter ε k , but is rather given by a different infinitesimal scale δ 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 ε k δ k and ε k δ 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 -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 -convergence proving, in particular, a fundamental estimate for the functionals F k (Proposition 5.1) and a compactness and integral representation result for the -limit of F k (Theorem 5.2). In Sect. 6 we characterise the volume integrand of the -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.

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.

Notation
The present subsection is devoted to the notation we employ throughout.
If A, B ∈ A by A ⊂⊂ B we mean that A is relatively compact in B; (e) Q denotes the open unit cube in R n with sides parallel to the coordinate axis, centred at the origin; for x ∈ R n and r > 0 we set Q r (x) := r Q + x. Moreover, Q denotes the open unit cube in 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 ν ∈ S n−1 let R ν denote an orthogonal (n × n)-matrix such that R ν e n = ν; we also assume that R −ν Q = R ν Q for every ν ∈ S n−1 , R ν ∈ Q n×n if ν ∈ S n−1 ∩ Q n , and that the restrictions of the map ν → R ν to S n−1 ± are continuous. For an explicit example of a map ν → R ν satisfying all these properties we refer the reader, e.g., to [27, Example A.1]; (g) for x ∈ R n , r > 0, and ν ∈ S n−1 , we define Q ν r (x) := R ν Q r (0) + x. (h) for ξ ∈ R m×n we let u ξ be the linear function whose gradient is equal to ξ ; i.e., u ξ (x) := ξ x, for every x ∈ R n ; (i) for x ∈ R n , ζ ∈ R m 0 , and ν ∈ S n−1 we denote with u ν x,ζ the piecewise constant function taking values 0, ζ and jumping across the hyperplane ν (x) := {y ∈ R n : (y−x)·ν = 0}; i.e., when ζ = e 1 we simply write u ν x in place of u ν x,e 1 ; (j) let u ∈ C 1 (R), v ∈ C 1 (R), with 0 ≤ v ≤ 1, be one-dimensional functions satisfying the following two properties: (k) for x ∈ R n and ν ∈ S n−1 we set (l) for x ∈ R n , ν ∈ S n−1 , ζ ∈ R m 0 and ε > 0 we set When ζ = e 1 we simply writeū ν x,ε in place ofū ν x,e 1 ,ε . We notice that in particular, For every L n -measurable set A ⊂ R n we define L 0 (A; R m ) as the space of all R m -valued Lebesgue measurable functions. We endow L 0 (A; 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 ∈ A; in this paper we deal with the functional space S BV (A; R m ) (resp. G S BV (A; 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 ∈ S BV (A; R m ) then its distributional derivative can be represented as for every B ∈ B n . In (2.1) ∇u denotes the approximate gradient of u (which makes sense also for u ∈ G S BV ), S u the set of approximate discontinuity points of u, [u] := u + − u − where u ± are the one-sided approximate limit points of u at S u , and ν u is the measure theoretic normal to S u . Let p > 1; we also consider We recall that G S BV p (A; R m ) is a vector space; moreover, if u ∈ G S BV p (A; R m ) then we have that φ(u) ∈ S BV p (A; R m ) ∩ L ∞ (A; R m ), for every φ ∈ C 1 c (R m ; 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.
(g4) (continuity in v and w) for every x ∈ R n we have for every v 1 , v 2 ∈ R and every w 1 , w 2 ∈ R n ; (g5) (monotonicity in v) for every x ∈ R n and every w ∈ R n , g(x, · , w) is decreasing on (−∞, 1) and increasing on [1, +∞); (g6) (minimum in w) for every x ∈ R n and every v ∈ R it holds for every w ∈ R n . For k ∈ N let ( f k ) ⊂ F and (g k ) ⊂ G and let (ε k ) be a decreasing sequence of strictly positive real numbers converging to zero, as k → +∞.
We consider the sequence of elliptic functionals F k :

Remark 2.2
Assumptions (f2)-(f3) and (g2)-(g3) imply that for every A ∈ A and every that is, up to a multiplicative constant, the functionals F k are bounded from below and from above by the Ambrosio-Tortorelli functionals

Remark 2.3
For later use, we notice that the assumptions on ψ, f k and g k ensure that for every A ∈ A the functionals F k ( ·, ·, A) are continuous in the strong For every A ∈ A, u ∈ L 0 (R n ; R m ) and v ∈ L 0 (R n ) it is also convenient to write +∞] denote the bulk and the surface part of F k , respectively; i.e., (2.7) For ρ > 2ε k , x ∈ R n , ξ ∈ R m×n , and ν ∈ S n−1 we consider the following two minimisation problems We observe that in (2.8) by "u = u ξ near ∂ Q ρ (x)" we mean that the boundary datum is attained in a neighbourhood of ∂ Q ρ (x). Whereas in (2.10) the boundary datum is prescribed only in

Remark 2.4 Clearly, the class of competitors
. In view of (2.2) and of the properties satisfied by ψ, we also observe that in (2.10) the constraint v ∇u = 0 a.e. in Q ν ρ (x) can be equivalently replaced by Hence, in particular, for every x ∈ R n , ζ ∈ R m 0 , ν ∈ S n−1 , ρ > 2ε k , and k ∈ N we have Finally, for every x ∈ R n and every ξ ∈ R m×n we define while for every x ∈ R n and every ν ∈ S n−1 we set

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 ν ρ (x). More precisely, we consider the minimum values defined as follows: For every x ∈ R n , ν ∈ S n−1 , and A ∈ A we set

Remark 2.5 Let
In particular from (g3), Remark 2.4, and (2.18) we infer 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 ∂ Q ν ρ (x) while keeping the nonconvex constraint v ∇u = 0 a.e. in Q ν ρ (x).
Proof We only prove the equality for g , the proof of the equality for g being analogous. Let x ∈ R n and ν ∈ S n−1 be fixed and set and in gray the sets R k (dark gray) and {d k < 1} (light gray) In view of Remark 2.4 we readily have for every x ∈ R n and every ν ∈ S n−1 . To prove the opposite inequality, let ρ > 0 and α ∈ (0, 1) be fixed and letk ∈ N be such that ε k < αρ 2 , for every k ≥k.
where R ν is as in (f). By construction we have that Fig. 1). Now let ϕ k ∈ C 1 c (Q ρ ) be a cut-off function between Q ρ−β k and Q ρ ; i.e., 0 ≤ ϕ k ≤ 1, and ϕ k ≡ 1 on Q ρ−β k . Eventually, for y = (y , y n ) ∈ Q ν (1+α)ρ (x) we define the pair ( u k , v k ) by setting To this end, we start

Statements of the main results
In this section we state the main results of this paper, namely, a -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).

0-convergence
The following result asserts that, up to subsequences, the functionals F k -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.

be the functionals as in (2.4). Then there exists a subsequence, not relabelled, such that for every
with f , f as in (2.12) and (2.13), respectively; ii. for every x ∈ R n and every ν ∈ S n−1 with g , g as in (2.14) and (2.15), respectively.

Remark 3.2
The choice of considering functionals F k which are finite when the variable v satisfies the bounds 0 ≤ v ≤ 1 is a choice of convenience and it is not restrictive. Indeed, in view of (g5) and (g6) and of the properties of ψ, the functionals F k decrease under the transformation v → min{max{v, 0}, 1}. Hence a -convergence result for functionals F k defined on functions v with values in 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 → +∞ as j → +∞, such that for every A ∈ A the corresponding functionals F k j ( · , · , A) -converge to a free-discontinuity functional which is finite in G S BV p (A; R m ) × {1} and is of the form for some Borel functionsf andĝ (see Theorem 5.2).
In the third step we identifyf by showing that it is equal both to f and f (see Proposition 6.1). Eventually, in the final step we identifyĝ by proving that it coincides with both g and g (see Proposition 7.4). The representation result forĝ 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 -convergence [32,Proposition 8.3].
and let 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 , ξ), for a.e. x ∈ R n and for every ξ ∈ R m×n and g (x, ν) = g (x, ν) =: g ∞ (x, ν), for every x ∈ R n and every ν ∈ S n−1 , otherwise.

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 F k . Other minimisation problems can be treated similarly. .

3)
then v k → 1 in L p (A) and there exists a subsequence of (u k ) which converges in L q (A; R m ) to a solution of (3.2).

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.
We are now ready to state the homogenisation result; the latter corresponds to the choice We stress again that we will not assume any spatial periodicity of f and g.
Assume that for every x ∈ R n , ξ ∈ R m×n , ν ∈ S n−1 the two following limits

10)
exist and are independent of x. Then, for every A ∈ A the functionals F k (·, ·, A) defined in (2.4) with f k and g k as in (3.8 otherwise. 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 for every x ∈ R n , ξ ∈ R m×n , and ν ∈ S n−1 .
We start by proving the first two equalities in (3.11). To this end, fix and thus (3.9) applied with x replaced by x/ρ yields Eventually, by (2.12) and (2. We now prove the second two equalities in (3.11). To this end, for fixed x ∈ R n , ν ∈ S n−1 , . Further, by a change of variables we immediately obtain that Hence, again setting r k := ρ ε k , we infer Hence invoking (3.10) applied with x replaced by x/ρ we get Eventually, Proposition 2.6 gives g (x, ν) = g (x, ν) = g hom (ν), for every x ∈ R n , ν ∈ S n−1 and thus the claim.

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.
Before proving the corresponding result for g , g we need to prove the two following technical lemmas.
and (ū ν x,ε k ,v ν x,ε k ) are as in (l). Moreover, let g ρ , g ρ : R n × S n−1 → [0, +∞] be the functions defined as Then the restrictions of g ρ , g ρ to the sets R n × S n−1 + and R n × S n−1 − are upper semicontinuous.
Proof We only show that the restriction of the function g ρ to the set R n × S n−1 + is upper semicontinuous, the other proofs being analogous.

Lemma 4.4
Let g and g be as in (2.14) and (2.15), respectively, and let g ρ , g ρ be as in (4.1). Then for every x ∈ R n and every ν ∈ S n−1 it holds Proof We prove the statement only for g , the proof for g being analogous. We notice that 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 ρ > 0, x ∈ R n , and ν ∈ S n−1 and for every k ∈ N such that ε k ∈ (0, ρ and let u k ∈ W 1, p (Q ν ρ (x); R m ) be the corresponding u-variable. Let α > 0 be arbitrary; thanks to the boundary conditions satisfied by (u k , v k ) we can extend the pair
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 ) ⊂ 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 ∈ R n and every ν ∈ S n−1 it holds g (x, ν) = g (x, −ν), g (x, ν) = g (x, −ν); (4.17) (2) (boundedness) for every x ∈ R n and every ν ∈ S n−1 it holds where c p := 2( p − 1) 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 ρ > 0 and let g ρ be the function defined in (4.1). Arguing as in the proof of Lemma 4.4 we deduce that the function ρ → g ρ (x, ν)−c 4 C v ρ n−1 is nonincreasing on (0, +∞). From this it follows that for every x ∈ R n , ν ∈ S n−1 , and every ρ > 0. Thus, if D is a countable dense subset of (0, +∞) we have lim sup Therefore the Borel measurability of g follows by Lemma 4.3 which guarantees, in particular, that the function (x, ν) → g ρ (x, ν) is Borel measurable for every ρ > 0.
Step 2: g is symmetric in ν. Property (4.17) immediately follows from the definition of g and from the fact that u ν x = −u −ν x + e 1 a.e. and Q ν 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 for every v ∈ W 1, p (Q ν ρ (x)) with 0 ≤ v ≤ 1 a.e. in Q ν ρ (x). Therefore to establish (4.18) it is enough to show that lim k→+∞ m k,ρ (x, ν) = c p ρ n−1 , for every x ∈ R n and ρ > 0, where Let x ∈ R n , ν ∈ S n−1 , ρ > 0, and let k ∈ N be such that 2ε k < ρ. By the homogeneity and rotation invariance of the Ambrosio-Tortorelli functional we have Let η > 0 be arbitrary; reasoning as in the construction of a recovery sequence for the Ambrosio-Tortorelli functional we find a sequence ( Then, using a similar argument as in the proof of Proposition 2.6, we can modify v k to obtain a function We now turn to the proof of the lower bound. By the Fubini Theorem we have Then, if v ∈ A ε k ,ρ (0, e n ) the corresponding u coincides with u e n 0 in a neighbourhood of Since it must hold that v∇u = 0 a.e. in Q ρ (0), then almost every straight line intersecting ∂ ± Q ρ (0) and parallel to e n also intersects the level set {v = 0}. Indeed, for L n−1 -a.e. x ∈ Q ρ the pair (u as well as v x (± ρ 2 ) = 1, u x (− ρ 2 ) = 0, and u x ( ρ 2 ) = 1. Since u x ∈ W 1, p (− ρ 2 , ρ 2 ), the boundary conditions satisfied by u x imply the existence of a subset of (− ρ 2 , ρ 2 ) with positive L 1 -measure on which u x = 0, hence v x = 0 in view of (4.24). In particular, for L n−1 -a.e. x ∈ Q ρ there exists s ∈ (− ρ 2 , ρ 2 ) such that v(x , s) = 0. Therefore, the Young Inequality for L n−1 -a.e. x ∈ Q ρ . Thus, gathering (4.23) and (4.25) we get

0-convergence and integral representation
In this section we show that, up to subsequences, the functionals F k -converge in L 0 (R n ; R m ) × L 0 (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 -convergence (see e.g., [32, or [23, Chapters 10, 11]) together with an integral-representation result in S BV [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 F k satisfy the so-called fundamental estimate, uniformly in k. Proposition 5.1 (Fundamental estimate) Let F k be as in (2.4). Then, for every η > 0 and for every A, A , B ∈ A with A ⊂⊂ A there exists a constant M η > 0 (also depending on A, A , B) satisfying the following property: For every k ∈ N and for every (u, v . in A, (û,v) = ( u, v) a.e. in B \ A and For each i = 2, . . . , N + 1 let ϕ i be a smooth cut-off function between A i−1 and A i and let Let (u, v) and ( u, v) be as in the statement and consider the function w ∈ L 0 (R n ) defined by w := min{v, v}, as followŝ Then, setting S i :=A i \ A i−1 and taking into account the definition of (û i ,v i ) we have 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 ψ is increasing, we have We complete the estimate of the bulk part of the energy by noticing that on S i ∩ B we have Integrating over S i ∩ B, using (f3) and (f2), the definition of w, and the monotonicity of ψ, we infer It remains to estimate the surface term in F k . Thanks to (g3) it holds We now want to bound the right-hand side of (5.6) in terms of F s . To this end we first observe that by definition of w we have Thus, thanks to (g2), (5.6) becomes By the definition ofv i and by the convexity of z → (1 − z) p for z ∈ [0, 1], on S i−1 ∩ B we have where in the last step we used again (5.7). Similarly, there holds Since analogous arguments hold on S i+1 ∩ B, from (5.9) and (g2) we deduce On account of the fundamental estimate, Proposition 5.1, we are now in a position to prove the following -convergence result.
Theorem 5.2 Let F k be as in (2.4). Then there exist a subsequence (F k j ) of (F k ) and a functional F : ·, A). Moreover, F is given by for every x ∈ R n , ξ ∈ R m×n , ζ ∈ R m 0 , and ν ∈ S n−1 , where for A ∈ A and u ∈ S BV p (A; R m ) In view of Remark 2.2 we can invoke [36, Theorem 3.1] to deduce the existence of a constant C > 0 such that By the general properties of -convergence we know that for every A ∈ A fixed the functionals F (·, ·, A) and F (·, ·, A) are L 0 (R n ; R m ) × L 0 (R n ) lower semicontinuous [ Invoking [32,Theorem 16.9] we can deduce the existence of a subsequence (k j ), with k j → +∞ as j → +∞, such that the corresponding F and F also satisfy u, v, A), (5.15) for every (u, v) ∈ L 0 (R n ; R m )× L 0 (R n ) and for every A ∈ A. We notice that the set function F (u, v, ·) given by (5.15) is inner regular by definition. Moreover F satisfies the following properties: the functional  F (u, v, ·) 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 (R n ; R m ) convergence in u while we are considering the -convergence of F k j in L 0 (R n ; R m ) × L 0 (R n ). However, this issue can be easily overcome by resorting to a truncation argument together with a sequential characterisation of F (see e.g., [32,Proposition 16.4 and Remark 16.5]), which holds true on S BV p (A; R m ) ∩ L ∞ (A; 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) ∈ L 0 (R n ; R m ) × L 0 (R n ) the set function F (u, v, ·) is the restriction to A of a Borel measure.
Furthermore, (5.13) together with [32, Proposition 18.6] and Proposition 5.1 imply that while, gathering (5.13) and (5.14) we may also deduce that As a consequence, F (·, ·, A) coincides with the -limit of F k j (·, ·, A) on L 0 (R n ; R m ) × L 0 (R n ), for every A ∈ A. By [17, Theorem 1] and a standard perturbation and truncation argument (see e.g., [27,Theorem 4.3]), for every A ∈ A and u ∈ G S BV p (A; R m ) we can represent the -limit F in an integral form as for some Borel functionsf andĝ. Eventually, thanks to (5.13), it can be easily shown thatf andĝ are given by the same derivation formulas provided by [17,Theorem 1], that is, they coincide with (5.11) and (5.12), respectively.

Identification of the volume integrand
In this section we identify the volume integrandf . Namely, we prove thatf coincides with both f and f , given by (2.12) and (2.13), respectively. This shows, in particular, that the limit volume integrandf depends only on f k , and hence only on F b k .

Proposition 6.1 Let ( f k ) ⊂ F and (g k ) ⊂ G. Let (k j ) andf be as in Theorem 5.2. Then it holdsf
for a.e. x ∈ R n and for every ξ ∈ R m×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 ≤ f , hence to prove the claim it suffices to show that for a.e. x ∈ R n and for every ξ ∈ R m×n . We divide the proof of (6.1) into two steps.
Step 1: In this step we show thatf (x, ξ) ≥ f (x, ξ) for a.e. x ∈ R n and for every ξ ∈ R m×n . By Theorem 5.2 we have that for every A ∈ A and for every ξ ∈ R n . Now let x ∈ R n be arbitrary, let A ∈ A be such that x ∈ A, and let ρ > 0 be so small that converges to (u ξ , 1) in measure on bounded sets and Moreover, by (g2) we also have v k → 1 in L p (A). We notice that (u k , v k ) also satisfies Indeed, thanks to (6.2) we have Therefore, again by -convergence we get Hence (6.4) follows by (6.3). We now estimate separately the surface and bulk term in F k . We notice that by Young's Inequality we have Hence, thanks to (g2), by using the co-area formula we get that where E t k,ρ :={y ∈ Q ρ (x) : v k (y) < t}. Now let η ∈ (0, 1) be fixed and η ∈ (η, 1); then by the mean-value Theorem we deduce the existence oft =t(k, ρ, η, η ),t ∈ (η, η ) such that Set w k := u k χ R n \Et k,ρ ; since u k ∈ W 1, p (A; R m ) and Et k,ρ is a set of finite perimeter, we have we have that L n (Et k,ρ ) → 0 as k → +∞ so that w k → u ξ in measure on bounded sets. Eventually, since F k is increasing as set function, using the fact that ψ is increasing we obtain 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 where f is given by (2.13).
Hence appealing to (6.4) gives Then, by dividing both terms in the above inequality by ρ n and using (6.2), we obtain Thus invoking the Lebesgue differentiation Theorem together with the continuity in ξ off and f (see [17] and Proposition 4.1) we deduce that for a.e. x ∈ R n , for every ξ ∈ R m×n , and for every η ∈ (0, 1). Eventually, since ψ is continuous and ψ(1) = 1, the claim follows by letting η, η → 1.
Step 2: In this step we show thatf (x, ξ) ≤ f (x, ξ) for every x ∈ R n and for every ξ ∈ R m×n . The proof is similar to that of [27,Theorem 5.2]. However, we repeat it here for the readers' convenience. Let x ∈ R n , ξ ∈ R m×n , ρ > 0, and η > 0 be fixed. By (2.8) for every k ∈ N fixed we can find Combining (6.5) with (f2) and (f3) yields where the second inequality follows by taking u ξ as a test in the definition of m b k (u ξ , Q ρ (x)). Let now (k j ) be a diverging sequence such that Since u k j − u ξ ∈ W 1, p 0 (Q ρ (x); R m ), the uniform bound (6.6) together with the Poincaré Inequality provide us with a further subsequence (not relabelled) and a function u ∈ W 1, p (Q ρ (x); R m ) such that u k j u weakly in W 1, p (Q ρ (x); R m ). Then, by the Rellich Theorem u k j → u in L p (Q ρ (x); R m ). We now extend u and u k to functions w, w k ∈ W 1, p loc (R n ; R m ) by setting respectively; clearly, w = u ξ in a neighbourhood of ∂ Q (1+η)ρ (x) and w k j → w in L p loc (R n ; R m ). Hence by -convergence, by (2.3) and (6.5) we get Eventually, dividing by ρ n , passing to the limsup as ρ → 0, and recalling the definition of f and f we get and hence the claim follows by the arbitrariness of η > 0.

Identification of the surface integrand
In this section we identify the surface integrandĝ. Namely, we show thatĝ coincides with both g and g , given by (2.14) and (2.15), respectively. This shows, in particular, that the limit surface integrandĝ is obtained by minimising only the surface term F s k . We notice, however, that in this case the presence the bulk term F b k 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 -limit F .

Lemma 7.1 (Approximation of minimum values) Let
Let (k j ) be as in Theorem 5.2 andĝ be as in (5.12). Then for every x ∈ R n , ζ ∈ R m 0 , and ν ∈ S n−1 it holdsĝ 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 for every x ∈ R n , ζ ∈ R m 0 , and ν ∈ S n−1 . Let ρ > 0 and η > 0 be fixed; by definition of m k (ū ν Since the pair (ū ν x,ζ,ε k ,v ν x,ε k ) is admissible for m k (ū ν x,ζ,ε k , Q ν ρ (x)), then (2.11) and (2.19) readily give 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 u k L ∞ (Q ν ρ (x);R m ) < +∞. We now extend u k to a W 1, p loc (R n ; R m )-function by setting In view of (7.4), (2.5), and the uniform L ∞ (R n ; 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 for some u ∈ L p loc (R n ; R m ) also belonging to S BV p (Q ν (1+η)ρ (x); R m ). Moreover, we also have u = u ν x,ζ in a neighbourhood of ∂ Q ν (1+η)ρ (x), so that ). (7.5) Eventually, by -convergence together with (7.3) we obtain Thus, using (7.5), dividing the above inequality by ρ n−1 and passing to the limsup as ρ → 0 we obtain 2) follows by the arbitrariness of η > 0.
Step 2: In this step we show that lim sup for every x ∈ R n , ζ ∈ R m 0 , ν ∈ S n−1 , and ρ > 0. To this end, we fix η > 0 and we choose We extend u to the whole R n by setting u = u ν x,ζ in R n \ Q ν ρ (x). Then, by -convergence there exists a sequence (u k , v k ) converging to (u, 1) in measure on bounded sets such that 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 loc (R n ; R m ) × L p loc (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 m k (ū ν x,ζ,ε k , Q ν ρ (x)). This will be done by resorting to the fundamental estimate Proposition 5.1. Namely, we choose 0 < ρ < ρ < ρ such that u = u ν x,ζ on Q ν ρ (x) \ Q ν ρ (x) and we apply Proposition 5.

Remark 7.2 Thanks to the
, a standard convolution argument shows that (7.1) holds also true if the minimisation in m k is carried over ).
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 F s k .
where c η > 0 is independent of v and A and such that c η → 0 as η → 0.
Proof A direct computation shows that i.e., (7.12) is satisfied. Thus it remains to show that (7.13) holds true. To this end we introduce the sets We start by estimating the first term on the right-hand side of (7.14). Since v ≤ η 1+ √ η < η < 1 in A η , using (g4) and (g6) we get in A η , which together with (g2) yields Step 1: In this step we show thatĝ(x, ζ, ν) ≥ g (x, ν), for every x ∈ R n , ζ ∈ R m 0 , and ν ∈ S n−1 .
In view of Lemma 7.1 we havê Thanks to Remark 7.2 the minimisation in the definition of m k (ū ν x,ζ,ε k , Q ν ρ (x)) can be carried over C 1 -pairs (u k , v k ). Now let ρ > 0 and η ∈ (0, 1) be fixed and for every k such that where U k is a neighbourhood of ∂ Q ν ρ (x) and then, (7.4) readily gives We now modify v k in order to obtain a new function v k for which there exists a corresponding u k such that the pair ( u k , v k ) satisfies both where U k is a a neighbourhood of ∂ Q ν ρ (x), and the constraint v k ∇ u k = 0 a.e. in Q ν ρ (x). (7.26) In this way we have ρ (x, ν). The modification as above shall be performed without essentially increasing the energy F k .
To this end, set u k := (u 1 k , . . . , u m k ) and ζ := (ζ 1 , . . . , ζ m ). Since ζ ∈ R m 0 we can find i ∈ {1, . . . , m} so that ζ i = 0; without loss of generality we assume that ζ i > 0. We now consider the open set Moreover, let σ ∈ (0, 1) be fixed and consider the following partition of S ρ k : and h σ k ∈ N to be chosen later. (7.27) where S ρ k := S ρ k,¯ . Therefore, gathering (f2), (7.24), and (7.27) yields In view of (7.22) we have that In this way any modification to v k performed in the set S ρ k will not affect the boundary conditions. In order to modify v k in S ρ k , we introduce an auxiliary functionv k which interpolates in a suitable way between the values 0 and 1 in S ρ k . To this end, we set γ k :=(¯ + 1 2 ) ζ i h σ k and τ k := ζ i 4h σ k and we definev k ∈ W 1, p (Q ν ρ (x)) as follows: We notice that by definitionv By the regularity of u i k we can find ξ k > 0 (possibly small) such that Then, we define u k ∈ W 1, p (Q ν ρ (x); R m ) as Thus the pair ( u k , v k ) belongs to W 1, p (Q ν ρ (x); R m )× W 1, p (Q ν ρ (x)) and by construction satisfies (7.26). Moreover, (7.29) together with (7.22) ensures that (7.25) is satisfied. Eventually, we have ) . (7.30) To conclude the proof it only remains to show that, up to a small error, F s k ( v k , Q ν ρ (x)) is a lower bound for F k (u k , v k , Q ν ρ (x)). To this end we consider the following partition of Q ν ρ (x): : v η k (y) >v k (y) ; then, appealing to (7.13) in Lemma 7.3, we deduce where c η → 0 as η → 0. Hence, it remains to estimate the second term on the right-hand side of (7.31).
To this end, we start noticing that Indeed, since v η k > 0 a.e. in S 2 k , by definition of v η k we readily get (7.32). Therefore, by (g3), using thatv where the last inequality follows by (7.32), the definition ofv k , and the monotonicity of ψ. From (7.28) we deduce both that Hence, gathering (7.33), (7.34), and (7.35) we obtain Moreover, for every ω ∈ and for every A ∈ A the functionals F k (ω)(·, ·, A) -converge If, in addition, (τ z ) z∈Z n is ergodic, then f hom and g hom are independent of ω and The almost sure -convergence result in Theorem 8.4 is an immediate consequence of Theorem 3.5 once we show the existence of a T -measurable set ⊂ , with P( ) = 1, such that for every ω ∈ 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.

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 For ω ∈ let m b ω be as in (8.2). Then there exists ∈ T , with P( ) = 1 and a (T ⊗B m×n )measurable function f hom : × R m×n → [0, +∞) such that for every ω ∈ , x ∈ R n and every ξ ∈ R m×n . If, in addition, (τ z ) z∈Z n is ergodic, then f hom is independent of ω and given by We now deal with the existence of the homogenised surface integrand g hom . Unlike the case of f 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 m s ω (ū ν r x , Q ν r (r x)) (cf. (8.3)), the proof of the x-homogeneity of g 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 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 × I n−1 . To do so we follow the same procedure as in [28,Section 5] (see also [3,22]). Namely, given ν ∈ S n−1 we let R ν be an orthogonal matrix as in (f). Then {R ν e i : i = 1 , . . . , n − 1} is an orthonormal basis for ν , further R ν ∈ Q n×n , if ν ∈ S n−1 ∩ Q n . Now let M ν > 2 be an integer such that M ν R ν ∈ Z n×n ; therefore M ν R ν (z , 0) ∈ ν ∩ Z n for every z ∈ Z n−1 .
Let I ∈ I n−1 ; i.e., I = [a, b) with a, b ∈ Z n−1 . Starting from I we define the ndimensional interval I ν as Correspondingly, for fixed ν ∈ S n−1 ∩ Q n we define the function μ ν : × I n−1 → R as where m s ω (ū ν 0 , I ν ) is as in (8.3) with x = 0 and A = I ν . The following result asserts that μ ν defines a subadditive process on × I n−1 .
Proof Let ν ∈ S n−1 ∩ Q n be fixed; below we show that μ ν satisfies conditions (1)-(4) in Definition 8.2, for some group of P-preserving transformations (τ ν z ) z ∈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 ∈ I n−1 and let I ν ⊂ R n be as in (8.6). Let v ∈ W 1, p (I ν ) be fixed. In view of (G1) the Fubini Theorem ensures that the map ω → F s (ω)(v, I ν ) is T -measurable. On the other hand the space W 1, p (I ν ) is separable, hence the set of functions A (ū ν 0 , I ν ) = {v ∈ W 1, p (I ν ) , 0 ≤ v ≤ 1, v =v ν 0 near ∂ I ν and ∃ u ∈ W 1, p (I ν ; R m ), u =ū ν 0 near ∂ I ν , such that v ∇u = 0 a.e. in I ν } defines a separable metric space, when endowed with the distance induced by the W 1, p (I ν )norm. Then, by the continuity of F s (ω)(·, I ν ) with respect to the strong W 1, p (I ν )-topology, the infimum in the definition of m s ω (ū ν 0 , I ν ) can be equivalently expressed as an infimum on a countable subset of A (ū ν 0 , I ν ), thus ensuring the T -measurability of ω → m s ω (ū ν 0 , I ν ) and consequently that of ω → μ ν (ω, I ), as desired.
Step 4: boundedness. Let ω ∈ and I ∈ I n−1 . Then (8.6) and ( 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 hom when x = 0. By the previous step we have that g(ω, ν) = g(ω, ν) = g ν (ω), for every ω ∈ and for every ν ∈ S n−1 ∩ Q n . Therefore, if we show that the restrictions of the functions ν → g(ω, ν) and ν → g(ω, ν) to the sets S n−1 ± are continuous, by the density of S n−1 ± ∩ Q n in S n−1 ± we can readily deduce that g(ω, ν) = g(ω, ν) = g ν (ω) for every ω ∈ and every ν ∈ S n−1 , and thus the claim.
Therefore the restriction of g to × S n−1 ± is measurable with respect to the σ -algebra induced in × S n−1 ± by T ⊗ B(S n−1 ) thus, finally, g hom is (T ⊗ B(S n−1 ))-measurable on × S n−1 .
For ω ∈ let m s ω be as in (8.3). Then there exists ∈ T with P( ) = 1 such that lim r →+∞ m s ω (ū ν r x , Q ν r (r x)) r n−1 = g hom (ω, ν) (8.32) for every ω ∈ , every x ∈ R n , and every ν ∈ S n−1 , where g hom is given by (8.13). In particular, the limit in (8.32) is independent of x. Moreover, if (τ z ) z∈Z n is ergodic, then g hom is independent of ω 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.