Non-local approximation of free-discontinuity problems in linear elasticity and application to stochastic homogenisation

Abstract

We analyse the $\Gamma$-convergence of general non-local convolution type functionals with varying densities depending on the space variable and on the symmetrized gradient. The limit is a local free-discontinuity functional, where the bulk term can be completely characterized in terms of an asymptotic cell formula. From that, we can deduce an homogenisation result in the stochastic setting.

1. Introduction

This paper is focussed on the approximation of brittle fracture energies for linearly elastic materials, by means of non-local functionals defined on Sobolev spaces. The asymptotic behaviour of these functionals will simultaneously show the emergence both of effective energies for the elastic deformation (which may be, e.g., the output of homogenization), and of Griffith-type surface energies accounting for crack formation. In turn, this result can be further generalized to the setting of stochastic homogenization with fracture.

Precisely our results will extend the range of application of the recent papers [17, 22] while also providing some relevant technical improvement. We briefly comment on these previous contributions, in order to introduce our results. There, an approach originally devised by Braides and Dal Maso [13] for the approximation of the Mumford-Shah functional has been generalized to the linearly elastic setting. Namely, it was shown that, for a given bounded increasing function $f\colon \mathbb {R}^+\to \mathbb {R}^+$ the energies

(1.1)\begin{equation} \frac1{\varepsilon_k}\int_U f\left(\varepsilon_kW(e(u))*\rho_k(x) \right)\,{\rm d}x\,, \end{equation}

$\Gamma$-converge, in the $L^1(U)$-topology, to the functional

(1.2)\begin{equation} \alpha\int_{U} W(e(u))\,\mathrm{d}x + \beta \int_{J_u}\phi_\rho(\nu_u)\,{\rm d}\mathcal{H}^{n-1}, \end{equation}

with $\alpha =f^\prime (0)$ and $\beta =\lim _{t\to +\infty }f(t)$. Above, $\rho _k$ are rescaled convolution kernels with unit mass and compact convex symmetrical support $S$, $\phi _\rho$ is (twice) the support function of $S$ [see (2.4) for its precise definition], $W(e(u))$ is a convex elastic energy with superlinear $p$ growth depending on the symmetrized gradient $e(u)$ of a vector-valued displacement $u$, whose jump set is denoted by $J_u$. Notice that the effective domains of the approximations and of the limit are different. Actually (1.1) is finite on the Sobolev space $W^{1,p}(U;\mathbb {R}^n)$, while the energy space of (1.2) is the one of generalized functions with bounded deformation $GSBD^p(U)$, introduced in [11].

We stress that the above results allowed one for a general (convex) bulk energy. The proof strategy cannot rely, at least when estimating the bulk part, on any slicing procedure. The latter is instead successful in the particular case $W(\xi )=|\xi |^p$, considered for instance in [20]. We also remark that the results of [17, 22] were obtained under an additional structural assumption on the kernels $\rho _k$, which have to be radial with respect to the norm induced by $S$. In the particular case considered in [20], this restriction was instead not needed.

A natural extension of the aforementioned models allows one to include an explicit dependence on $k$ and on the space variable of the energy density. This amounts to consider functionals of the form

(1.3)\begin{equation} \frac1{\varepsilon_k}\int_U f\left(\varepsilon_kW_k({\cdot},e(u))*\rho_k(x) \right)\,{\rm d}x\,, \end{equation}

whose limit behaviour is the object of the present paper. Functionals of the form (1.3) can be used to approximate (1.2) with some gain in the ease of minimization, for a proper choice of $W_k$. Actually, this more general setting is also suitable for further applications, if one thinks about the mechanical counterpart of the model. Indeed energy densities of type $W_k(y,M)$, where $y$ is the position in the reference configuration, are customary when dealing with heterogeneous material with some microstructures. The prototypical example is the case of homogenisation, that is, when $W_k(y,M)=W\left (\frac {y}{\delta _k},M\right )$ with $\delta _k\searrow 0$. Taking this point of view amounts to regard (1.3) as a nonlocal linearly elastic model, with a truncated potential $\frac 1{\varepsilon _k}f(\varepsilon _k \cdot )$ accounting for the cost of breaking the elastic bonds on regions of size $\varepsilon _k$. In such a case, one is interested in deriving an effective asymptotic model for (1.3).

The main result of our paper is contained in theorem 2.1. There we show that the functionals in (1.3) $\Gamma$-converge to a limit energy of the form

(1.4)\begin{equation} \alpha\int_{U} W(x,e(u))\,\mathrm{d}x + \beta \int_{J_u}\phi_\rho(\nu_u)\,{\rm d}\mathcal{H}^{n-1}\,. \end{equation}

Above, the limit bulk density $W$ can be characterised in terms of cell formula [see (2.7)–(2.8)]. Remarkably, that coincides exactly with the asymptotic formula that one would obtain by considering the limit behaviour of the local energies $\int _UW_k(x,e(u))\,{\rm d}x$ in the Sobolev space $W^{1,p}(U)$. Hence, a decoupling effect between bulk and surface contribution occurs, since the volume energy only depends on $f$ through its derivative at the origin. A similar effect has been observed in [9] where the analogue of (1.3) for energies depending on the full deformation gradient was taken into account. On the one hand, the possibility of using smooth truncations (a tool which is not available in $GSBD$) allowed the author there to replace $f$ by a sequence $f_k$ and to derive more general surface energies in the limit. On the other hand, the precise characterisation of the volume energy density was obtained at the expense of an additional technical condition on the $W_k$'s (the so-called stable $\gamma$-convergence). It actually turns out, as an output of our proof strategy, that this extra assumption can be dropped (see Appendix). Thus, our results also permit some improvement in the previous literature about non-local approximation of free-discontinuity problems.

We now come to the description of our proof technique. The most difficult point is the lower bound for the bulk contribution. This is done in proposition 5.1, by means of a localisation and blow-up procedure which contains some elements of novelty in the non-local setting. More precisely, we consider the blow-up of sequences with equi-bounded energies at a Lebesgue point for the limit energy. A crucial task is to gain a uniform control on the $L^p$ norm of the symmetrized gradients of the blow-up functions up to sets with vanishing perimeter. This allows us to apply [18, lemma 5.1] (which relies on the Korn-type inequality of [6]): we can substitute, with almost no change in the energy, the above mentioned sequence with a more regular one bounded in $W^{1,p}$. Exploiting the properties of $f$, we are then reconducted to analyse the limit behaviour on small squares of a local energy in $W^{1,p}$, which can be estimated from below via a cell formula.

An optimal estimate from below for the surface term can be obtained by means of a slicing procedure (proposition 5.3). As for the $\Gamma$-limsup inequality it can be achieved by a direct construction for a class of competitors with regular jump set, which are dense in energy. Here, we use the classical approximation result of[7, 10].

We underline that even in the case of (1.1) (i.e., with $W$ not depending on $k$) we have some technical improvement in comparison with the result of [17, 22]. First of all, we do not need anymore to assume the kernels $\rho _k$ to be radially symmetric. Secondly, our $\Gamma$-convergence argument is carried out with respect to the convergence in measure instead of the $L^1$ convergence. This is (almost) the natural one for sequences with equibounded energy (see theorem 2.1-$(ii)$). It can be indeed shown that such sequences are compact in the measure convergence up to an exceptional set $U^\infty$, where their modulus diverges. However, this set can be easily made empty by adding a penalisation term in the energy (see the statement of theorem 3.5 and remark 2.2).

Eventually, we complement our analysis with a stochastic homogenisation result theorem 7.4. Namely, we consider functionals of type (1.3) with stationary random integrands

(1.5)\begin{equation} W_k(\omega,y,M)=W\left(\omega,\frac{y}{\delta_k},M\right)\,, \end{equation}

where $\omega$ belongs to the sample space $\Omega$ of a probability space $(\Omega,\mathcal {T},P)$ and $\delta _k\searrow 0$. Following the approach proposed by [12] (which relies on the Subadditive Ergodic Theorem in [19]) we show that, almost surely, such functionals $\Gamma$-converge to a free-discontinuity functional of the form (1.4) where the bulk energy density is independent of the space variable. A similar result was obtained in [2] in the context of elliptic approximation of free-discontinuity functionals.

Plan of the paper. The paper is structured as follows. After fixing the notation, in § 2, we introduce the problem, discuss the assumptions and state our main results. § 3 is devoted to recalling preliminary results which are useful for the analysis. The proof of theorem 2.1 is carried out through the § 4–6, dealing with compactness, lower, and upper bound, respectively. In § 7, we prove a stochastic homogenisation result theorem 7.4. Eventually in the Appendix, we briefly comment on the result of [9, theorem 3.2], highlighting that the assumptions made there can actually be weakened. A complete statement is given for the readers’ convenience in theorem A.1.

2. Setting of the problem and main results

2.1 Notation

We start by collecting the notation adopted throughout the paper.

1. (a) $n\geq 2$ is a fixed integer and $p>1$ is a fixed real number;

2. (b) $\mathbb {M}^{n\times n}$ denotes the space of $n\times n$ real matrices; $\mathbb {M}^{n\times n}_{\rm sym}$ and $\mathbb {M}^{n\times n}_{\rm skew}$ denote the spaces of symmetric and skew-symmetric matrices respectively;

3. (c) for a subset $A\subset \mathbb {R}^n$ $\partial ^*A$ denotes the essential boundary of $A$;

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

5. (e) for every $A\subset \mathbb {R}^n$ let $\chi _A$ denote the characteristic function of the set $A$;

6. (f) $U$ denotes an open bounded subset of $\mathbb {R}^n$ with Lipschitz boundary;

7. (g) we denote by $\mathcal {A}(U)$ and $\mathcal {A}$ the collection of all open and bounded subsets of $U$ and $\mathbb {R}^n$ respectively;

8. (h) If $A,B \in \mathcal {A}(U)$ (or $\mathcal {A}$) by $A \subset \subset B$ we mean that $A$ is relatively compact in $B$;

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

10. (j) for every $\xi \in {\mathbb {S}}^{n-1}$ let $R_\xi$ denote an orthogonal $(n\times n)$-matrix such that $R_\xi e_n=\xi$;

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

12. (l) 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 $\mathcal {B}$;

13. (m) we denote by $L^0(U;\mathbb {R}^n)$ the space of measurable functions;

14. (n) for $a,b\in \mathbb {R}^n$ the symbol $a \otimes b$ denotes the tensor product between $a$ and $b$, while $a\odot b:=\frac 12 (a\otimes b+b\otimes a)$.

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

2.2 (G)SBV and (G)SBD functions

We will work with the functional spaces $(G)SBV^p(U;\mathbb {R}^n)$ and $(G)SBD^p(U)$ for which we will recall the main properties and refer the reader to [1, 11] for a complete exposition of the subject. We say that $u\in L^1(U;\mathbb {R}^n)$ belongs to the space of special functions with bounded variation, i.e., $u\in SBV(U;\mathbb {R}^n)$, if its distributional gradient is a finite $\mathbb {M}^{n\times n}$-valued Radon measure without Cantor part, that is,

\[ Du=\nabla u\mathcal{L}^n+[u]\otimes \nu_u\mathcal{H}^{n-1}{\unicode{x221F}} J_u\,, \]

where $\nabla u$ is the approximate gradient, $J_u$ is the approximate jump set, $[u]=u^+-u^-$ the jump opening and $\nu _u$ the unit normal to $J_u$. A function $u\in L^0(U;\mathbb {R}^n)$ belongs to the space of generalised special functions with bounded variation, i.e., $u\in GSBV(U;\mathbb {R}^n)$, if for any $\varphi \in C^1(\mathbb {R}^n;\mathbb {R}^n)$ with support of $\nabla \varphi$ compact it holds $\varphi \circ u\in SBV_{\rm loc}(U;\mathbb {R}^n)$.

We say that $u\in L^1(U;\mathbb {R}^n)$ belongs to the space of special functions with bounded deformation, and we write $u\in SBD(U)$, if its symmetrized distributional gradient is a finite $\mathbb {M}^{n\times n}_{\rm sym}$-valued Radon measure without Cantor part, that is,

\[ Eu=\frac{Du+(Du)^T}{2}= e(u)\mathcal{L}^n+ [u]\odot \nu_u\mathcal{H}^{n-1}{\unicode{x221F}}J_u\,, \]

where $e(u)$ is the approximate symmetric gradient with respect to the Lebesgue measure. On the contrary, the space of generalised special functions with bounded deformation, $GSBD(U)$, cannot be defined analogously to the space $GSBV(U;\mathbb {R}^n)$ as if $u\in SBD(U)$ and $\varphi$ is as above, then in general $\varphi \circ u\notin SBD(U)$. To overcome this issue, Dal Maso in [11] proposed a definition of this space by relying on a slicing argument which we describe in the following.

For $\xi \in \mathbb {R}^n\setminus \{0\}$ we let $\Pi ^\xi := \{y\in \mathbb {R}^n\colon \langle \xi,y\rangle =0\}$; for any $y\in \Pi ^\xi$ and $A\in \mathcal {B}(U)$, we set

\[ A_{\xi,y}:=\{t\in \mathbb{R}\colon y+t\xi\in A\}\,. \]

Given $u\colon U\to \mathbb {R}^n$ we define $u^{\xi,y}\colon U_{\xi,y}\to \mathbb {R}$ by

\[ u^{\xi,y}(t):=\langle u(y+t\xi),\xi \rangle\,. \]

If $u^{\xi,y}\in SBV(U_{\xi,y};\mathbb {R})$, we set

\[ J_{u^{\xi,y}}^1:=\{t \in J_{u^{\xi,y}}\colon |[u^{\xi,y}](t)|\ge1\}\,. \]

We then say that $u\in L^0(U;\mathbb {R}^n)$ belongs to the space of generalised special functions with bounded deformation, and we write $u\in GSBD(U)$, if there exists a bounded Radon measure $\lambda$ on $U$ such that $u^{\xi,y}\in SBV_{\rm loc}(U_{\xi,y})$ for all $\nu \in \mathbb {S}^{n-1}$ and all $y\in \Pi ^\xi$ and

\[ \int_{\Pi^\xi}\left(|Du^{\xi,y}|(A_{\xi,y}\setminus J_{u^{\xi,y}}^1)+\mathcal{H}^0(A_{\xi,y}\cap J_{u^{\xi,y}}^1) \right) \,{\rm d}\mathcal{H}^{n-1}(t) \le \lambda(A)\,, \]

for all $A\in \mathcal {B}(U)$. Eventually, we set

\begin{align*} & GSBV^p(U)\nonumber\\ & \quad :=\{u\in (G)SBV(U;\mathbb{R}^n)\colon \nabla u\in L^p(U;\mathbb{M}^{n\times n})\ \text{and}\ \mathcal{H}^{n-1}(J_u)<{+}\infty \}\,; \end{align*}

and

\[ GSBD^p(U):=\{u\in (G)SBD(U)\colon e(u)\in L^p(U;\mathbb{M}^{n\times n}_{\rm sym})\ \text{and}\ \mathcal{H}^{n-1}(J_u)<{+}\infty \}\,, \]

where $\nabla u$ and $e(u)$ are well defined also in $GSBV(U;\mathbb {R}^n)$ and $GSBD(U)$ respectively.

2.3 Setting of the problem

Let $1< p<+\infty$; let $c_1,c_2$ be given positive constants such that $0< c_1\le c_2<+\infty$. Let $\mathcal {W}:= \mathcal {W}(p,c_1,c_2)$ be the collection of all functions $W\colon \mathbb {R}^n\times \mathbb {M}^{n\times n}\to \mathbb {R}$ satisfying the following conditions:

1. (W1) $W$ is a Carathéodory function on $\mathbb {R}^n\times \mathbb {M}^{n\times n}$;

2. (W2) ($W2$) $W(x,0)=0$ for every $x\in \mathbb {R}^n$;

3. (W3) for every $x\in \mathbb {R}^n$, $M\in \mathbb {M}^{n\times n}$ and $S\in \mathbb {M}^{n\times n}_{\rm skew}$

   \[ W(x,M+S)=W(x,M)\,; \]

4. (W4) for every $x\in \mathbb {R}^n$ and every $M\in \mathbb {M}^{n\times n}$

   \[ c_1|M+M^T|^p\le W(x,M)\le c_2(|M+M^T|^p+1)\,. \]

Let $f\colon [0,+\infty )\to [0,+\infty )$ be a concave¹ increasing function such that there exist $\alpha,\beta >0$ with

(2.1)\begin{equation} \lim_{t\to 0^+}\frac{f(t)}{t}=\alpha\,,\quad\lim_{t\to+\infty}f(t)=\beta\,. \end{equation}

Note that for such $f$ it holds

(2.2)\begin{equation} f(t)\le \hat\alpha t\quad \forall \hat{\alpha}>\alpha\,; \end{equation}

moreover by [22, lemma 2.10] there exist $(\alpha _i)_{i\in \mathbb {N}}$, $(\beta _i)_{i\in \mathbb {N}}$ sequences of positive numbers with $\sup _i\alpha _i=\alpha$, $\sup _i\beta _i=\beta$ such that

(2.3)\begin{equation} f(t)\ge f_i(t):=\alpha_it\wedge\beta_i\quad\forall i\in\mathbb{N}\,,\ t\in\mathbb{R}\,. \end{equation}

Let $\rho \in L^\infty (\mathbb {R}^n;[0,+\infty ))$ be a lower semi-continuous convolution kernel with $\int _{\mathbb {R}^n}\rho \,{\rm d}x=1$ and $S:= {\{\rho >0\}}$ bounded, convex, symmetrical and with $0\in S$. We denote by $|\cdot |_S$ the norm induced by $S$, namely,

\[ |x|_S:=\inf\{\lambda>0\colon x\in\lambda S\}\,. \]

Under the above assumptions, $|\cdot |_S$ is a norm and $S=\{|x|_S<1\}$. Then for any bounded set $K\subset \mathbb {R}^n$ and $x\in \mathbb {R}^n$, we let

\[ \,\mathrm{d}_S(x,K):=\inf_{y\in K}|x-y|_S\,. \]

For any Borel set $E$ and any $r>0$, we denote by $E_r$ and $E_{-r}$ respectively the sets

\[ E_r:=\{x\in\mathbb{R}^n\colon\,\mathrm{d}_S(x,E)< r\}\,,\quad E_{{-}r}:=\{x\in\mathbb{R}^n\colon\,\mathrm{d}_S(x,E^c)>r\}\,. \]

Finally, we let $\phi _\rho \colon \mathbb {R}^n\to [0,+\infty )$ be given by

(2.4)\begin{equation} \phi_\rho(\nu):=2\sup_{y\in S}|y\cdot \nu|\,. \end{equation}

For $\delta >0$, we set $\rho _\delta (x):= \frac 1{\delta ^n}\rho \left (\frac x{\delta }\right )$, $S_\delta (x):= x+\delta S$.

For $k\in \mathbb {N}$ let $(W_k)\subset \mathcal {W}$, let $(\varepsilon _k)$ be a decreasing sequence of strictly positive real numbers converging to zero, as $k\to +\infty$ and let $\rho _k:= \rho _{\varepsilon _k}$. We consider the family of functionals $F_k\colon L^0(U;\mathbb {R}^n)\to [0,+\infty ]$ defined as

(2.5)\begin{equation} F_k(u):=\begin{cases} \displaystyle\frac1{\varepsilon_k}\int_U f\left(\varepsilon_kW_k({\cdot},e(u))*\rho_k(x) \right)\,{\rm d}x & \text{ if } u\in W^{1,p}(U;\mathbb{R}^n)\,,\\ + \infty & \text{ otherwise }. \end{cases} \end{equation}

Here and henceforth, it remains understood that each $u\in W^{1,p}(U;\mathbb {R}^n)$ is extended to a fixed neighbourhood of $U$ to have a well-defined functional. The $\Gamma$-limit, as we will see, is independent of the considered extension. Let $x\in \mathbb {R}^n$, $M\in \mathbb {M}^{n\times n}$, $A\in \mathcal {A}$ and $u\in W^{1,p}(A;\mathbb {R}^n)$ be fixed. Set $u_M(y):= My$. We then define the minimisation problem

(2.6)\begin{equation} \textbf{m}_k(u_M,A):=\inf\left\{\int_AW_k(x,e(v))\,{\rm d}x \colon v\in W^{1,p}(A;\mathbb{R}^n),\ v=u_M\ \text{near}\ \partial A \right\}\,, \end{equation}

and the cell formulas

(2.7)\begin{gather} W'(x,M):= \limsup_{r\searrow0^+}\liminf_{k\to+\infty}\frac{\textbf m_k(u_M,Q_r(x))}{r^n}\,, \end{gather}

(2.8)\begin{gather} W''(x,M):= \limsup_{r\searrow0^+}\limsup_{k\to+\infty}\frac{\textbf m_k(u_M,Q_r(x))}{r^n}\,. \end{gather}

Notice that $W'$ and $W''$ depend on the given sequence of $k$ and are to be modified accordingly if one takes subsequences. This will be highlighted in the statement of our main result.

2.4 Main results

In this Section we state our main results. The first one is a $\Gamma$-convergence theorem for the energies $F_k$.

Theorem 2.1 $\Gamma$-convergence of $F_k$

Let $F_k$ be as in (2.5). Then the following hold:

1. (i) There exists a subsequence, not relabelled, such that for every $x\in U$ and every $M\in \mathbb {M}^{n\times n}$, and for $W'$, $W''$ as in (2.7) and (2.8) (calculated for the given subsequence), one has

   (2.9)\begin{equation} W'(x,M)=W''(x,M):=W(x, M)\,. \end{equation}
   and it holds $W(x,M)=W(x,{\rm sym}(M))$. Moreover, $F_k$ $\Gamma$-converges with respect to the convergence in measure to the functional $F\colon L^0(U;\mathbb {R}^n) \to [0,+\infty ]$ given by
   (2.10)\begin{equation} F(u):=\begin{cases} \displaystyle\ \alpha\int_UW(x,e(u))\,{\rm d}x+\beta \int_{J_u}\phi_\rho(\nu_u)\,{\rm d}\mathcal{H}^{n-1} & \text{if }u\in GSBD^p(U)\,,\\ + \infty & \text{otherwise}\,, \end{cases} \end{equation}
   with $\phi _\rho$ as in (2.4);

2. (ii) Let $(u_k)\subset L^0(U;\mathbb {R}^n)$ be such that $\sup _kF_k(u_k)<+\infty$. Set $U^\infty := ~\{x\in U\colon |u_k(x)|\to +\infty \}$. Then there exists $u\in GSBD^p(U)$ such that, up to subsequence, it holds $u_k\to u$ in measure on $U\setminus U^\infty$. If in addition

   \[ \sup_{k\in\mathbb{N}}\int_U\psi(|u_k|)\,{\rm d}x<{+}\infty\,, \]
   for some $\psi \colon [0,+\infty )\to [0,+\infty )$, continuous, increasing with $\lim _{s\to +\infty }\psi (s)=+\infty$, then $U^\infty =\emptyset$, so that $|u|$ is finite a.e., and $u_k\to u$ in measure on $U$.

Remark 2.2 The addition of a penalty term of the form $\int _U\psi (|u|)\,{\rm d}x$ to the energy enforces then compactness in measure, while causing no troubles in the $\Gamma$-convergence analysis. Indeed, such a term is clearly lower semicontinuous, hence the corresponding lower bound follows immediately. As for the upper bound, if one takes $\psi$ as in theorem 3.4, the argument of proposition 6.1 can be readily adapted also in presence of such an additional term. As this is not the core of the argument, we will neglect lower order terms in our statements and proofs, directly assuming that convergence in measure holds everywhere. The technical details left to prove the upper bound are summarized in remark 6.2 for the readers convenience.

The proof of theorem 2.1 is divided into three main steps contained respectively in § 4, 5 and 6. As a consequence of theorem 2.1 and the Urysohn property of $\Gamma$-convergence [14, proposition 8.3], we deduce the following corollary.

Corollary 2.3 Let $(W_k)\subset \mathcal {W}$ and let $F_k$ be the functionals as in (2.5). Let $W'$, $W''$ be as in (2.7) and (2.8), respectively. Assume that

\[ W'(x,M)=W''(x,M)=:W(x,M)\,, \quad \text{for a.e. } x\in\mathbb{R}^n\text{ and for every }M\in\mathbb{M}^{n\times n}\,, \]

for some Borel function $W\colon \mathbb {R}^n\times \mathbb {M}^{n\times n}\to [0,+\infty )$. Let $F$ defined as in (2.10) accordingly. Then the functionals $F_k$ $\Gamma$-converge with respect to the convergence in measure to $F$. Moreover

\[ W(x,M)=W(x,{\rm sym}(M) )=W'(x,M)=W''(x,M)\,, \]

for every $x\in U$ and every $M\in \mathbb {M}^{n\times n}$.

We now state a homogenisation theorem without assuming any spatial periodicity of the energy densities $W_k$. We start by introducing some notation. We fix $W\in \mathcal {W}$ and set

(2.11)\begin{equation} \mathbf{m}(u_M,A):=\inf\left\{\int_AW(x,e(v))\,{\rm d}x\colon v\in W^{1,p}(A;\mathbb{R}^n),\ v=u_M\ \text{near}\ \partial A \right\}\,, \end{equation}

for all $A\in \mathcal {A}$ and all $M\in \mathbb {M}^{n\times n}$. Let also $(W_k)\subset \mathcal {W}$ be given by

(2.12)\begin{equation} W_k(x,M):= W\left(\frac{x}{\delta_k},M\right)\,, \end{equation}

with $\delta _k\searrow 0$ when $k\to +\infty$.

Theorem 2.4 Deterministic homogenisation

Let $W\in \mathcal {W}$ and let $\mathbf {m}(u_M,Q_t(tx))$ be as in (2.11) with $A=Q_t(tx)$. Assume that for every $x\in \mathbb {R}^n$, $M\in \mathbb {M}^{n\times n}$ the following limit

(2.13)\begin{equation} \lim_{t\to+\infty}\frac{\mathbf{m}(u_M,Q_t(tx))}{t^n}=:W_{\rm hom}(M)\,, \end{equation}

exists and is independent of $x$. Then the functionals $F_k$ defined in (2.5) with $W_k$ as in (2.12) $\Gamma$-converge with respect to the convergence in measure to the functional $F_{\rm hom}\colon L^0(U;\mathbb {R}^n)\to [0,+\infty ]$ given by

(2.14)\begin{equation} F_{\rm hom}(u):=\begin{cases} \displaystyle\ \alpha\int_U W_{\rm hom}(e(u))\,{\rm d}x+\beta \int_{J_u}\phi_\rho(\nu_u)\,{\rm d}\mathcal{H}^{n-1} & \text{if }u\in GSBD^p(U)\,,\\ + \infty & \text{otherwise}\,, \end{cases} \end{equation}

with $\phi _\rho$ as in (2.4). Moreover $W_{\rm hom}(M)=W_{\rm hom}({\rm sym}(M))$ for all $M\in \mathbb {M}^{n\times n}$.

Proof. Let $W'$, $W''$ be respectively as in (2.7) and (2.8). By corollary 2.3, it is sufficient to show that

(2.15)\begin{equation} W_{\rm hom}(M)=W'(x,M)=W''(x,M)\,, \end{equation}

for all $x\in \mathbb {R}^n$ and $M\in \mathbb {M}^{n\times n}$. We fix $x\in \mathbb {R}^n$, $M\in \mathbb {M}^{n\times n}$, $r>0$ and $k\in \mathbb {N}$. For any $u\in W^{1,p}(Q_r(x);\mathbb {R}^n)$ with $u=u_M$ near $\partial Q_r(x)$, we let $u_k\in W^{1,p}\left (Q_{\frac r{\delta _k}}\left (\frac x{\delta _k}\right );\mathbb {R}^n\right )$ be given by $u_k(y):= \frac 1{\delta _k}u(\delta _ky)$. Then clearly $u_k=u_M$ near $\partial Q_{\frac r{\delta _k}}\left (\frac x{\delta _k}\right )$. Moreover by performing the change of variable $\hat y=\frac y{\delta _k}$, we find

\[ \int_{Q_r(x)}W\left(\frac y{\delta_k},e(u)\right)\,{\rm d}y=\delta_k^n \int_{Q_{\frac r{\delta_k}}(\frac x{\delta_k})} W(y,e(u_k))\,{\rm d}y\,. \]

Hence in particular

\[ \mathbf{m}_k(u_M,Q_r(x))=\delta_k^n\mathbf{m}\left(u_M,Q_{\frac r{\delta_k}}\left(\tfrac{x}{\delta_k}\right)\right)=\frac{r^n}{t_k^n} \mathbf{m}\left(u_M,Q_{t_k}\left(t_k\tfrac{x}{r}\right)\right)\,, \]

with $t_k:= \frac r{\delta _k}$. Eventually passing to the limit as $k\to +\infty$ by (2.13), we deduce

\[ \lim_{k\to+\infty} \frac{\mathbf{m}_k(u_M,Q_r(x))}{r^n}= \lim_{k\to+\infty} \frac{\mathbf{m}\left(u_M,Q_{t_k}\left(t_k\tfrac{x}{r}\right)\right)}{t_k^n}=W_{\rm hom}(M)\,. \]

3. Some preliminary results

In this section, we collect some useful results that will be employed throughout the paper. We start by recalling a $\Gamma$-convergence result for the bulk energies defined in (3.1) (theorem 3.1) and a $\Gamma$-convergence result for one-dimensional non-local energies (theorem 3.3). To follow we recall a density and a compactness result (cf., theorems 3.4 and 3.5). We conclude this section with a series of technical lemmas (cf. lemmas 3.6, 3.7, 3.8 and corollary 3.9).

We consider the family of functionals $E_k\colon L^0(\mathbb {R}^n;\mathbb {R}^n)\times \mathcal {A}\to [0,+\infty ]$ given by

(3.1)\begin{equation} E_k(u,A):=\begin{cases} \displaystyle\int_A W_k(x,e(u)) \,{\rm d}x & \text{ if } u\in W^{1,p}(A;\mathbb{R}^n)\,,\\ + \infty & \text{ otherwise }. \end{cases} \end{equation}

Theorem 3.1 $\Gamma$-convergence of $E_k$

Let $E_k$ be as in (3.1). Then there exists a subsequence, not relabelled, such that for every $A\in \mathcal {A}$ the functionals $E_k(\cdot,A)$ $\Gamma$-converge, with respect to the convergence in measure, to the functional $E(\cdot,A)$ with $E\colon L^0(\mathbb {R}^n;\mathbb {R}^n)\times \mathcal {A}\to [0,+\infty ]$ given by

(3.2)\begin{equation} E(u,A)=\begin{cases} \displaystyle\int_A W(x,e(u)) \,{\rm d}x & \text{ if } u\in W^{1,p}(A;\mathbb{R}^n)\,,\\ + \infty & \text{ otherwise }\,, \end{cases} \end{equation}

where for every $x\in \mathbb {R}^n$ and every $M\in \mathbb {M}^{n\times n}$

(3.3)\begin{equation} W(x,M)=W(x,{\rm sym}(M) )=W'(x,M)=W''(x,M)\,, \end{equation}

with $W'$, $W''$ as in (2.7) and (2.8) for the given subsequence. The same $\Gamma$-convergence holds with respect to the $L^p_{\rm loc}(\mathbb {R}^n;\mathbb {R}^n)$ convergence.

Observe that the above theorem yields in particular a subsequence for which (2.9) holds. From now on, it remains understood that such a subsequence has been fixed, without relabelling. The proof of theorem (3.1) is rather standard and follows by the localisation method (see e.g., [14, Sections 18,19]) and by suitably adapting the integral representation result in [3, theorem 2] to our setting with the help of Korn-Poincaré inequality. For this reason we omit the proof here and we refer the reader to [18, proposition 3.13] for more details. We only highlight that the result holds also for non regular open bounded subsets of $\mathbb {R}^n$. Since this may not be immediately clear from the statement given in [18, proposition 3.13], we discuss this point in the remark below.

Remark 3.2 Let $A$ be any open bounded subset of $\mathbb {R}^n$ and $u\in W^{1,p}(A ;\mathbb {R}^n)$. We show that there exists a sequence $(u_k)\subset W^{1,p}(A;\mathbb {R}^n)$ such that $u_k\to u$ in $L^p(A;\mathbb {R}^n)$ and $E_k(u_k,A)\to E(u,A)$. With the use of Korn-Poincaré inequality, it is clear that this can be done if $A$ is an extension domain. In the general case, consider smooth relatively compact subsets $A'\subset \subset A''\subset \subset A$, and fix $\eta >0$. We find a sequence $(v_k)\subset W^{1,p}(A'';\mathbb {R}^n)$ such that $E_k(v_k,A'')\to E(u,A'')$. By the liminf inequality, this also gives $E_k(v_k,A''\setminus A')\to E(u,A''\setminus A')$. With ($W4$) we have

\[ \limsup_{k\to +\infty}\int_{A''\setminus A'}|e(v_k)|^p\,{\rm d}x\le \frac{c_2}{c_1}\int_{A''\setminus A'}(1+|e(u)|^p)\,{\rm d}x \]

Then, considering a cut-off $\varphi$ between $A'$ and $A''$, we set $u_k:= \varphi v_k+(1-\varphi )u$. Clearly $u_k\to u$ in $L^p(A;\mathbb {R}^n)$. Furthermore, by ($W4$) one has

\begin{align*} \limsup_{k\to +\infty}E_k(u_k,A)& =\limsup_{k\to +\infty}E_k(v_k,A')+E_k(u_k,A\setminus A')\\ & \le \limsup_{k\to +\infty}E_k(v_k,A'')+ c_2\left[\int_{A\setminus A'}(1+|e(u)|^p)\,{\rm d}x\right. \\ & \left.+\int_{A''\setminus A'}(1+|\nabla \phi \odot(v_k-u)|^p+|e(v_k)|^p)\,{\rm d}x\right] \\ & \le E(u,A'')+c_2\left(1+\frac{c_2}{c_1}\right)\int_{A\setminus A'}(1+|e(u)|^p)\,{\rm d}x \le E(u,A)+\eta\,, \end{align*}

provided $\mathcal {L}^n(A\setminus A')$ is sufficiently small. The limsup inequality, which is the only relevant one, follows by a diagonal argument.

We recall now the following one-dimensional result for non-local energies given in [4, theorem 3.30].

Theorem 3.3 $\Gamma$-convergence in 1d

Let $I\subset \mathbb {R}$ be a bounded interval. Let $f\colon [0,+\infty )\mapsto [0,+\infty )$ be a lower semi-continuous function satisfying (2.1) for some $\alpha,\beta >0$. Consider the family of functionals $G_k\colon L^0(I)\to [0,+\infty ]$ defined by

\[ G_k(w):=\frac1{\varepsilon_k}\int_If\left(\frac12\int_{x-\varepsilon_k}^{x+\varepsilon_k}|\dot w(y)|^p\,{\rm d}y\right)\,{\rm d}x\,, \]

if $u\in W^{1,p}(I)$ and $+\infty$ otherwise. Then $G_k$ $\Gamma$-converge with respect to the convergence in measure, to the functional $G\colon L^0(I)\to [0,+\infty ]$ given by

\[ G(w):=\alpha\int_I|\dot w|^p\,{\rm d}x+ 2\beta\#(J_w)\,, \]

if $w\in SBV(I)$ and $+\infty$ otherwise.

We next recall an approximation result [7, theorem 1.1] and a compactness result in $GSBD^p$ in [8] (which generalises [11, theorem 11.3]). To this aim we denote by $\mathcal {W}^\infty _{\rm pw}(U;\mathbb {R}^{n})\subset GSBD^p(U)$ the space of ‘piecewise smooth’ $SBV$-functions, that is,

(3.4)\begin{align} \mathcal{W}^\infty_{\rm pw}(U;\mathbb{R}^{n})& := \left\{ u\!\in\! GSBD^p(U)\colon u\!\in\! SBV(U;\mathbb{R}^n)\cap W^{m,\infty}(U\setminus \overline{J_u};\mathbb{R}^n), \ \forall m\in \mathbb{N}, \right.\nonumber\\ & \mathcal{H}^{n-1}(\overline{J_u}\setminus J_u)=0,\ \overline{J_u}=\cup_{i=1}^kK_i\subset{\subset} U \nonumber\\ & \left.\text{with $K_i$ connected (n-1)-rectifiable set}, \forall\, 1\le i\le k \right\} \end{align}

Theorem 3.4 Density in $GSBD^p$

Let $\phi$ be a norm on $\mathbb {R}^n$. Let $u\in GSBD^p(U)$. Then there exists a sequence $(u_j)\subset \mathcal {W}^\infty _{\rm pw}(U;\mathbb {R}^n)$ such that

1. (i) $u_j\to u\ \text {in measure on }U$ ;

2. (ii) $e(u_j)\to e(u)\ \text {in } L^p(U;\mathbb {M}^{n\times n}_{\rm sym})$ ;

3. (iii) $\lim _{j\to \infty } \int _{J_{u_j}}\phi (\nu _{u_j})\,{\rm d}\mathcal {H}^{n-1}= \int _{J_{u}}\phi (\nu _{u})\,{\rm d}\mathcal {H}^{n-1}$ .

Moreover, if

\[ \int_U\psi(|u|)\,{\rm d}x<{+}\infty\,, \]

for some $\psi \colon [0,+\infty )\to [0,+\infty )$, continuous, increasing with

\[ \psi(0)=0\,,\ \psi(s+t)\le C(\psi(s)+\psi(t))\,,\ \psi(s)\le C(1+s^p)\,, \lim_{s\to+\infty}\psi(s)={+}\infty\,; \]

then

\[ \lim_{j\to+\infty}\int_U\psi(|u_j-u|)\,{\rm d}x=0\,. \]

We notice that the approximating class considered above fulfils the additional requirement of having a jump set compactly contained in $U$. This is possible, as shown in [15, theorem C].

Theorem 3.5 Compactness in $GSBD^p$

Let $(u_j)\subset GSBD^p(U)$ be a sequence satisfying

\[ \sup_{j\in\mathbb{N}}\left(\|e(u_j)\|_{L^p(U)}+\mathcal{H}^{n-1}(J_{u_j}) \right)<{+}\infty\,. \]

Then there exist a subsequence, still denoted by $(u_j)$, and $u\in GSBD^p(U)$ with the following properties:

1. (i) the set $U^\infty := \{x\in U\colon |u_j|\to +\infty \}$ has finite perimeter;

2. (ii) $u_j\to u$ in measure on $U\setminus U^\infty$ and $u=0$ on $U^\infty$  ;

3. (iii) $e(u_j)\rightharpoonup e(u)$ in $L^p(U\setminus U^\infty ;\mathbb {M}^{n\times n}_{\rm sym})$ ;

4. (iv) $\displaystyle \liminf _{j\to +\infty }\mathcal {H}^{n-1}(J_{u_j})\!\ge\! \mathcal {H}^{n-1}(J_u\cap (U\setminus U^\infty ))+\mathcal {H}^{n-1}(U\cap \partial ^*U^\infty )\!\ge\! \mathcal {H}^{n-1} (J_u\cup (U\cap \partial ^*U^\infty ))$ .

In the statement above, the last semicontinuity property is stated in a stronger form than in the original paper, but is also proved there (see [8, Formula (3.25)]). In the rest of this section we give some technical Lemmas.

Lemma 3.6 Let $g_j\colon \mathbb {R}^n\to [0,+\infty )$ be a sequence of equi-integrable functions. Let $E_j\subset \mathbb {R}^n$ be such that $\mathcal {L}^n(E_j)\to 0$ and let $\delta _j\searrow 0$ as $j\to +\infty$. Then $(g_j\chi _{E_j})*\rho _{\delta _j}\to 0$ strongly in $L^1(\mathbb {R}^n)$.

Proof. By properties of convolution it holds that

\[ \|g_j\chi_{E_j}*\rho_{\delta_j}\|_{L^1(\mathbb{R}^n)}\le \|g_j\chi_{E_j}\|_{L^1(\mathbb{R}^n)}. \]

By equi-integrability we have that for every $\varepsilon >0$ there is $J\in \mathbb {N}$ such that for every $j\ge J$

\[ \|g_j\chi_{E_j}\|_{L^1(\mathbb{R}^n)}=\int_{E_j}g_j\,{\rm d}x\le \varepsilon, \]

from which the thesis follows.

Lemma 3.7 Let $A'$ be an open bounded subset of $\mathbb {R}^n$. Let $g_j\colon A'\to [0,+\infty )$ be a sequence of equi-integrable functions. Let $\delta _j\searrow 0$ as $j\to +\infty$. Then for every $A\subset \subset A'$ there holds

\[ \liminf_{j\to+\infty}\int_{A}g_j*\rho_{\delta_j}\,{\rm d}x\ge \liminf_{j\to+\infty}\int_{A}g_j\,{\rm d}x\,. \]

Proof. We consider the sequence of positive measures $\nu _j:= g_j*\rho _{\delta _j}\mathcal {L}^n{\unicode{x221F}} A$. Since $A'$ is bounded $g_j$ turn out to be equi-bounded in $L^1(A')$, hence we get

\[ \nu_j(A)= \int_Ag_j*\rho_{\delta_j}\,{\rm d}x \le \int_{A'}g_j\,{\rm d}x \le C\,. \]

Therefore there exist a positive measure $\nu \in \mathcal {M}_b(A)$, a function $g\in L^1(A')$, and a not-relabelled subsequence such that $\nu _j\stackrel {*}{\rightharpoonup }\nu$ weakly $*$ in $\mathcal {M}_b(A)$ and $g_j\to g$ weakly in $L^1(A')$. It remains to show that $\nu =g\mathcal {L}^n{\unicode{x221F}} A$, indeed this would imply

\[ \liminf_{j\to+\infty}\int_Ag_j*\rho_{\delta_j}\,{\rm d}x= \liminf_{j\to+\infty}\nu_j(A)\ge \nu(A)=\int_Ag\,{\rm d}x=\liminf_{j\to+\infty}\int_Ag_j\,{\rm d}x\,, \]

and we could conclude. Let $\varphi \in C^\infty _c(A)$ and let $A\subset \subset A''\subset \subset A'$ be fixed. By Fubini's theorem we have

\begin{align*} \int_A\varphi \,\mathrm{d}\nu =\lim_{j\to+\infty}\int_A\varphi \,\mathrm{d}\nu_j& =\lim_{j\to+\infty}\int_A\varphi (g_j*\rho_{\delta_j})\,{\rm d}x\\ & =\lim_{j\to+\infty}\int_{A''}(\varphi*\hat\rho_{\delta_j} )g_j\,{\rm d}x=\int_{A''}\varphi g\,{\rm d}x =\int_{A}\varphi g\,{\rm d}x, \end{align*}

where $\hat \rho _{\delta _j}(x):= \rho _{\delta _j}(-x)$ and the last equality follows since $g_j\rightharpoonup g$ weakly in $L^1(A')$ and $\varphi *\hat \rho _{\delta _j}(x)\to \varphi$ strongly in $L^\infty (A')$. Thus, we deduce $\nu =g\mathcal {L}^n {\unicode{x221F}} A$ and the proof is concluded.

Lemma 3.8 Let $A\subset \mathbb {R}^{n-1}$. Let $(u_k)\subset L^1(A)$ be a sequence converging to $u$ in $L^1(A)$. Let $A'\subset \subset A$ and let $w_k\colon A'\times Q'\to \mathbb {R}$ be given by $w_k(x,y):= u_k(x+\varepsilon _k y)$. Then $w_k$ converges to $u$ in $L^1(A'\times Q')$.

Proof. By Frechet-Kolmogoroff's Theorem, for every $\eta >0$ there is $h\in \mathbb {N}$ such that for all $k\ge h$ and $y\in Q'$ there holds

\[ \int_{A'}|u_k(x+\varepsilon_k y)-u(x)|\,{\rm d}x\le \eta\,. \]

This together with Fubini's theorem yield

\[ \int_{A'\times Q'}|w_k(x,y)-u(x)|\,{\rm d}x\,{\rm d}y\le \int_{Q'}\int_{A'}|u_k(x+\varepsilon_k y)-u(x)|\,{\rm d}x\,{\rm d}y\le \eta \,, \]

for all $k\ge h$. Eventually by letting $\eta \to 0$, we conclude.

Corollary 3.9 Let $A\subset \mathbb {R}^{n-1}$. Let $(u_k)\subset L^0(A)$ be a sequence converging to $u$ in measure. Let $A'\subset \subset A$ and let $w_k\colon A'\times Q'\to \mathbb {R}$ be given by $w_k(x,y):= u_k (x+\varepsilon _k y)$. Then $w_k$ converges to $u$ in measure.

Proof. Since ${\rm arctan}(u_k)$ converges to ${\rm arctan}(u)$ in $L^1(A)$ by lemma 3.8 we have that ${\rm arctan}(w_k)$ converges to ${\rm arctan}(u)$ in $L^1(A'\times Q')$. Hence $w_k$ converges to $u$ in measure.

4. Compactness

In this section, we prove point $(ii)$ of theorem 2.1.

Proposition 4.1 Compactness

Let $F_k$ be as in (2.5). Let $(u_k)\subset L^0(U;\mathbb {R}^n)$ be such that $\sup _kF_k(u_k)<+\infty$. Then there exist $\bar {u}_k \in GSBV^p(U; \mathbb {R}^n)$ and $u\in GSBD^p(U)$ such that $\bar {u}_k-u_k \to 0$ in measure on $U$ and, up to a subsequence, it holds

\begin{align*} & \bar{u}_k\to u\quad \text{in measure on }U\setminus U^\infty\,, \end{align*}

\begin{align*} & e(\bar{u}_k)\rightharpoonup e(u)\quad \text{in } L^p_{\rm loc}(U\setminus U^\infty;\mathbb{M}^{n\times n}_{\rm sym})\,, \end{align*}

\begin{align*} & \liminf_{k\to+\infty}\mathcal{H}^{n-1}(J_{\bar{u}_k})\ge \mathcal{H}^{n-1}(J_u\cup(\partial^*U^\infty\cap U))\,, \end{align*}

where $U^\infty := \{x\in U\colon |u_k(x)|\to +\infty \}$. If in addition

\[ \sup_{k\in\mathbb{N}}\int_U\psi(|u_k|)\,{\rm d}x<{+}\infty\,, \]

for some $\psi \colon [0,+\infty )\to [0,+\infty )$, continuous, increasing with $\lim _{s\to +\infty }\psi (s)=+\infty$, then $U^\infty =\emptyset$, and all implications hold on $U$.

Proof. The proof is inspired by that of [22, proposition 4.1]. Let $(u_k)$ be as in the statement and let $U'\subset \subset U$ be fixed. We will prove the following claim: there exist ($\bar u_k)\subset GSBV^p(U;\mathbb {R}^n)$ and $c_0>0$ (independent of $k$) such that

(4.1)\begin{gather} \bar u_k-u_k\to0\ \text{in measure on }U\,, \end{gather}

(4.2)\begin{gather} \liminf_{k \to +\infty}F_k(u_k)\ge c_0\limsup_{k \to +\infty}\left(\int_{U'} |e(\bar u_k)|^p\,{\rm d}x+ \mathcal{H}^{n-1}(J_{\bar u_k})\right)\,. \end{gather}

Now, if (4.2) holds, we fix a sequence $U_i\nearrow U$ and apply theorem 3.5 to each $U_i$. With a diagonal argument we deduce the existence of $u\in GSBD(U)$ with $u=0$ on $U^\infty$ such that, up to a subsequence,

\[ \bar u_k\to u\ \text{in measure on } U\setminus U^\infty\,, \]

\[ e(\bar u_k)\rightharpoonup e(u)\ \text{in}\ L^p_{\rm loc}(U\setminus {U}^\infty;\mathbb{M}^{n\times n}_{\rm sym})\,. \]

We remark that since the constant $c_0$ is independent of $k$, we indeed have $u\in GSBD(U)$ by the very same argument of [8, Formula (3.33)], with the minor difference that the Radon measure $\lambda$ in the definition of $GSBD$ is first defined as local weak$^*$-limit, but turns eventually out to be uniformly bounded by (4.2). We also get $e(u)\in L^p(U)$, since $c_0$ is independent of $k$. Concerning the remaining inequality, for fixed $i$ we set $U_i^\infty : =U^\infty \cap U_i$, then by theorem 3.5 $(iv)$ we have

(4.3)\begin{align} & \liminf_{k\to+\infty}\mathcal{H}^{n-1}(J_{\bar u_k})\ge \liminf_{k\to+\infty}\mathcal{H}^{n-1}(J_{\bar u_k}\cap U_i)\ge \mathcal{H}^{n-1}(J_u\cap (U_i \setminus U^\infty))\nonumber\\ & \quad + \mathcal{H}^{n-1}(U_i\cap\partial^*{U_i}^\infty)\,. \end{align}

Observing that $U_j^\infty \nearrow U^\infty$ by lower semicontinuity of the relative perimeter we have

\[ \liminf_{j\to+\infty} \mathcal{H}^{n-1}(U_i\cap\partial^*{U_j}^\infty)\ge \mathcal{H}^{n-1}(U_i\cap\partial^*{U}^\infty) \quad \forall i\,. \]

Letting $i \to +\infty$, by monotonicity we get

(4.4)\begin{equation} \lim_{i\to+\infty} \mathcal{H}^{n-1}(J_u\cap (U_i \setminus U^\infty))= \mathcal{H}^{n-1}(J_u\cap (U\setminus U^\infty))\,, \end{equation}

and being $\mathcal {H}^{n-1}(U_i\cap \partial ^*U_i^\infty )=\mathcal {H}^{n-1}(U_i\cap \partial ^*U_j^\infty )$ for all $j\ge i$ it holds

(4.5)\begin{equation} \liminf_{i\to+\infty} \mathcal{H}^{n-1}(U_i\cap\partial^*{U_i}^\infty)\!\ge\! \liminf_{i\to+\infty}\liminf_{j\to+\infty}\mathcal{H}^{n-1}(U_i\cap\partial^*{U_j}^\infty)\!\ge\! \mathcal{H}^{n-1}(U\cap\partial^*{U}^\infty)\,. \end{equation}

Eventually, combining (4.3) with (4.4) and (4.5)

\begin{align*} & \liminf_{k\to+\infty}\mathcal{H}^{n-1}(J_{\bar u_k})\ge \mathcal{H}^{n-1}(J_u\cap (U \setminus U^\infty))+ \mathcal{H}^{n-1}(U\cap\partial^*{U}^\infty))\\ & \quad \ge \mathcal{H}^{n-1}(J_u\cup(\partial^*U^\infty\cap U))\,. \end{align*}

This also gives $u\in GSBD^p(U)$. As the remaining part of the statement follows directly from Chebycheff inequality and Fatou's lemma, we are only left to prove the claim.

For fixed $i\in \mathbb {N}$ let $f_i(t)=\alpha _i t\wedge \beta _i$ be as in (2.3). Choose $\eta \in (0,1)$ such that $Q_\eta (0)\subset \subset S$ and let

(4.6)\begin{equation} m_\eta:=\min_{x\in\overline Q_\eta(0)}\rho(x)>0\quad \text{and}\quad f_i^\eta(t):= f_i(m_\eta\eta^nt)=\alpha_im_\eta\eta^nt\wedge \beta_i\,. \end{equation}

Then we have

(4.7)\begin{align} F_k(u_k)& \ge \frac1{\varepsilon_k}\int_U f_i\left(\varepsilon_kW_k({\cdot},e(u_k))*\rho_k(x) \right)\,{\rm d}x\nonumber\\ & \ge \frac1{\varepsilon_k} \int_{ U}f_i^\eta\left(\varepsilon_k{\unicode{x2A0D}}_{Q_{\eta\varepsilon_k}(x)} W_{k}(y,e(u_{k}))\,{\rm d}y\right)\,{\rm d}x \,. \end{align}

We set

\[ A_k^1:=\left\{x\in U\colon \varepsilon_k{\unicode{x2A0D}}_{Q_{\eta\varepsilon_k(x)}} W_{k}(y,e(u_{k}))\,{\rm d}y\ge \frac{\beta_i}{\alpha_im_\eta\eta^{2n}} \right\}\,, \]

\[ A_k^2:=\left\{x\in U\colon \operatorname{dist}(x,A_k^1)\le (1-\eta)\varepsilon_{k} \right\}\,. \]

Note that

(4.8)\begin{equation} A_k^1\subset A_k^2\subset\left\{x\in U\colon \varepsilon_k{\unicode{x2A0D}}_{Q_{\varepsilon_{k}}(x)} W_{k}(y,e(u_{k}))\,{\rm d}y\ge \frac{\beta_i}{\alpha_im_\eta\eta^{n}} \right\}\,. \end{equation}

Indeed if $x\in A_k^2$ there is $z\in A_k^1$ with $Q_{\eta \varepsilon _{k}}(z)\subset Q_{\varepsilon _{k}}(x)$ and therefore

\[ \varepsilon_k{\unicode{x2A0D}}_{Q_{\varepsilon_k}(x)} W_{k}(y,e(u_{k}))\,{\rm d}y\ge \eta^n \varepsilon_{k}{\unicode{x2A0D}}_{Q_{\eta\varepsilon_{k}}(z)} W_{k}(y,e(u_{k}))\,{\rm d}y\ge \frac{\beta_i}{\alpha_im_\eta\eta^{n}}\,. \]

By combining together (4.7) and (4.8) we find

(4.9)\begin{equation} F_k(u_k)\ge\frac{\beta_i}{\varepsilon_k}\mathcal{L}^n(A_k^2) \,. \end{equation}

By the coarea formula (see e.g., [16, theorem 3.14]) and the mean value theorem there exists $t_k\in (0,(1-\eta )\varepsilon _{k})$ such that the set $A_k^3:= \{\operatorname {dist}(\cdot,A_k^1)\le t_k \}\subset A_k^2$ satisfies

(4.10)\begin{equation} \mathcal{L}^n(A_k^2)\ge (1-\eta)\varepsilon_{k}\mathcal{H}^{n-1}(\partial A_k^3 )\,. \end{equation}

Let now

\[ \bar u_k(x):=\begin{cases} 0 & \text{ if } x\in A_k^3\,,\\ u_{k} & \text{ otherwise in }U\,. \end{cases} \]

By construction $\bar u_k\in GSBV^p(U;\mathbb {R}^n)$. By (4.9) and the fact that $A_k^3\subset A_k^2$ we have $\mathcal {L}^n(A_k^3)\to 0$ as $k\to +\infty$ from which (4.1) follows. On the other hand as $J_{\bar u_k}\subseteq \partial A_k^3$ (4.9) and (4.10) yield

(4.11)\begin{equation} F_k(u_k)\ge (1-\eta)\beta_i\mathcal{H}^{n-1}(J_{\bar u_k})\,. \end{equation}

We next show that there exists $K(n)\ge 1$ such that for every $x\in U$

(4.12)\begin{equation} c_1\varepsilon_k{\unicode{x2A0D}}_{Q_{\eta\varepsilon_{k}}(x)} |e(\bar u_{k}(y))|^p \,{\rm d}y\le K \frac{\beta_i}{\alpha_im_\eta\eta^n}\,. \end{equation}

By ($W4$) we have

(4.13)\begin{equation} W_{k}(x, e(u_{k}(x)))\ge c_1|e(u_{k}(x))|^p\ge c_1|e(\bar u_k(x))|^p\quad \text{for a.e. }x\in U\,. \end{equation}

Now if $x\in U\setminus A_k^3$, then $x\notin A_k^1$ and

\[ c_1\varepsilon_k{\unicode{x2A0D}}_{Q_{\eta\varepsilon_{k}}(x)} |e(\bar u_{k}(y))|^p \,{\rm d}y\le \varepsilon_{k}{\unicode{x2A0D}}_{Q_{\eta\varepsilon_{k}}(x)}W_{k}(y, e(u_{k}(y)))\,{\rm d}y\le \frac{\beta_i}{\alpha_im_\eta\eta^n}\,. \]

Assume instead that $x\in A_k^3$. Observe that $\bar u_k=0$ in $Q_{\eta \varepsilon _{k}}(x)\cap A_k^3$, so that

\[ \int_{Q_{\eta\varepsilon_{k}}(x)} |e(\bar u_{k}(y))|^p \,{\rm d}y=\int_{Q_{\eta\varepsilon_{k}}(x)\setminus A_k^3} |e(\bar u_{k}(y))|^p \,{\rm d}y\,. \]

Furthermore, we can cover $Q_{\eta \varepsilon _{k}}(x)\cap (Q\setminus A_k^3)$ with a finite number $K(n)\ge 1$ of balls of radius $\eta \varepsilon _{k}$ and centres $x_1,\ldots,x_{ K}\in U\setminus A_k^3$ (see e.g. [22, remark 2.8]). Hence, we find

\begin{align*} c_1\varepsilon_k{\unicode{x2A0D}}_{Q_{\eta\varepsilon_{k}}(x)} |e(\bar u_{k}(y))|^p \,{\rm d}y& \le c_1\varepsilon_k\sum_{i=1}^{K}{\unicode{x2A0D}}_{Q_{\eta\varepsilon_{k}}(x_i)} |e(\bar u_{k}(y))|^p \,{\rm d}y\\ & \le \varepsilon_k\sum_{i=1}^{K}{\unicode{x2A0D}}_{Q_{\eta\varepsilon_{k}}(x_i)}W_{k}(y, e(u_{k}(y)))\,{\rm d}y\le K \frac{\beta_i}{\alpha_im_\eta\eta^n}\,, \end{align*}

and (4.12) follows. Finally by (4.7), the monotonicity of $f_i^\eta$, (4.13) we infer

(4.14)\begin{align} F_k(u_k) & \ge \frac1{\varepsilon_k} \int_{ U}f_i^\eta\left(\varepsilon_k{\unicode{x2A0D}}_{Q_{\eta\varepsilon_k}(x)}c_1 |e(\bar u_{k}(y))|^p\,{\rm d}y\right)\,{\rm d}x\nonumber\\ & \ge c_1 \frac{\alpha_im_\eta\eta^n}{K} \int_{ U} {\unicode{x2A0D}}_{Q_{\eta\varepsilon_{k}}(x)} |e(\bar u_{k}(y))|^p \,{\rm d}y\,{\rm d}x\,, \end{align}

where the last inequality follows from (4.12) and the fact that $f_i^\eta (t)\ge \frac {\alpha _im_\eta \eta ^n}{K} t$ when $t\le K \frac {\beta _i}{m_\eta \eta ^n \alpha _i}$. Moreover by using in order the change of variable $y=x-\eta \varepsilon _{k} z$, Fubini's theorem, and the change of variable $\hat x=x-\eta \varepsilon _kz$ (for $k$ large enough), we find

(4.15)\begin{align} \int_{ U}{\unicode{x2A0D}}_{Q_{\eta\varepsilon_{k}}(x)} |e(\bar u_{k}(y))|^p \,{\rm d}y\,{\rm d}x& =\int_{ Q}\int_{ U}|e(\bar u_{k}(x-\eta\varepsilon_{k} z))|^p \,{\rm d}x\,{\rm d}z\nonumber\\ & \ge \int_{U'} |e(\bar u_k(x))|^p\,{\rm d}x\,. \end{align}

Eventually gathering together (4.10), (4.14), and (4.15), we deduce (4.2) with

\[ c_0:= \frac12\left(\frac{c_1 \alpha_im_\eta\eta^n}{K}\wedge(1-\eta)\beta_i\right)\,. \]

5. Lower bound

In this section we prove the lower bound. To this purpose it is convenient to localise the functionals $F_k$, namely we set

(5.1)\begin{equation} F_k(u,A):= \frac1{\varepsilon_k}\int_A f\left(\varepsilon_kW_k({\cdot},e(u))*\rho_k(x) \right)\,{\rm d}x \,,\quad \text{for } u\in W^{1,p}(U)\,,\ A\subset U\,. \end{equation}

Proposition 5.1 Lower bound: bulk contribution

Let $(u_k)\subset L^0(U;\mathbb {R}^n)$ be a sequence that converges in measure to $u\in L^0(U;\mathbb {R}^n)$. Assume moreover that $F_k(u_k)\le C$ and that (2.9) holds. Then, for $W$ as (2.9)

\[ \liminf_{k\to+\infty}F_k(u_{k},A)\ge \alpha\int_A W(x, e(u))\,{\rm d}x\quad\forall A\in\mathcal{A}(U)\,, \]

Proof. Let $(u_k)$ and $u$ be as in the statement. By proposition 4.1 $u\in GSBD^p(U)$. For every $k\in \mathbb {N}$ let $\mu _k$ be the Radon measure on $(U, \mathcal {B}(U))$ given by

(5.2)\begin{equation} \mu_k(A):= F_k(u_k,A)\,,\quad \forall A \in \mathcal{B}(U)\,. \end{equation}

As $\mu _k(A)\le C$, by [1, theorem 1.59] we deduce the existence of a subsequence, not relabelled, and of a Radon measure $\mu$ on $(A, \mathcal {B}(A))$ such that

(5.3)\begin{equation} \mu_k\stackrel{ *}{\rightharpoonup}\mu\quad \text{and}\quad \liminf_{k\to+\infty}\mu_k(A)\ge \mu(A)\,. \end{equation}

By Radon-Nikodym's Theorem (in the version of [1, theorem 1.28]) there exist two measures $\mu ^a, \mu ^s$ with $\mu ^a\ll \mathcal {L}^n$ and $\mu ^s\perp \mathcal {L}^n$, and a function $h\in L^1(A)$ such that $\mu =\mu ^a+\mu ^s$ and $\mu ^a=h \mathcal {L}^n$. This together with (5.3) imply that

\[ \liminf_{k\to+\infty}F_k(u_k,A)\ge\int_A h(x)\,{\rm d}x\,. \]

Hence to conclude we need to show that

(5.4)\begin{equation} h(x)\ge \alpha W(x,e(u(x)))\quad \text{for a.e.}\ x\in U\,. \end{equation}

with $W$ as in (3.3). For $i\in \mathbb {N}$ fixed let $f_i(t)=\alpha _it\wedge \beta _i$ be as in (2.3). Then it is enough to show that

(5.5)\begin{equation} h(x)\ge \alpha_i W(x,e(u(x)))\quad \text{for a.e.}\ x\in U\,, \end{equation}

We divide the proof of (5.5) into four steps. Step 1: In this step we show that for a.e. $x_0\in U$ there exists a sequence $(k_j,r_j)\to (+\infty,0)$ as $j\to +\infty$ such that setting $\delta _j:= \frac {\varepsilon _{k_j}}{r_j}$,

(5.6)\begin{equation} u_{k}^{r}(v):=\frac{u_k(x_0+r v)-u_k(x_0)}{r}\quad \text{and}\quad W_k^r(x,M):= W(x_0+r x,M)\,, \end{equation}

there hold

(5.7)\begin{equation} h(x_0)\ge \lim_{j\to+\infty}\frac{1}{r_j} \frac{1}{\delta_{j}}\int_{ Q}f_i\left(r_j{\delta_{j}}W_{k_j}^{r_j}({\cdot},e(u_{k_j}^{r_j}))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x\,, \end{equation}

and

(5.8)\begin{equation} u_{k_j}^{r_j}\to \nabla u(x_0)({\cdot})\quad \text{in measure on }Q\,, \end{equation}

together with

(5.9)\begin{equation} \lim_{j\to +\infty} \frac{\mathbf{m}_{k_j}(u_{\nabla u(x_0)},Q_{r_j}(x_0))}{r_j^n}=W(x_0,e(u(x_0))) \end{equation}

By Besicovitch differentiation theorem and [6, corollary 5.2] we have that for a.e. $x_0\in U$ the following hold:

(5.10)\begin{gather} h(x_0)=\lim_{r\searrow0^+}\frac{\mu(\overline Q_r(x_0))}{|Q_r(x_0)|}\,, \end{gather}

(5.11)\begin{gather} \lim_{r\searrow0^+}\frac{1}{r^n}\mathcal{L}^n\left(\left\{y\in Q_r(x_0)\colon\frac{|u(y)-u(x_0)-\nabla u(x_0)(y-x_0)|}{|y-x_0|}>\delta\right\}\right)=0\quad\forall\delta>0\,. \end{gather}

We fix $x_0\in \Omega$ for which (5.10) and (5.11) hold. By [1, proposition 1.62] we have

\[ \mu(\overline Q_r(x_0))\ge\limsup_{k\to+\infty}\mu_k(\overline Q_r(x_0)) \,, \]

for every $r>0$, which together with (5.10) yield

(5.12)\begin{equation} h(x_0)\ge\limsup_{r\searrow0^+}\limsup_{k\to+\infty} \frac{\mu_k(\overline Q_r(x_0))}{|Q_r(x_0)|}\,. \end{equation}

Moreover from (5.2) and the change of variable $x=x_0+r x'$ we get

(5.13)\begin{align} \mu_k(\overline Q_r(x_0))& =\frac{1}{\varepsilon_k}\int_{\overline Q_r(x_0)}f\left(\varepsilon_k W_k({\cdot},e(u_k))*\rho_k(x)\right)\,{\rm d}x\nonumber\\ & =\frac{r^n}{\varepsilon_k}\int_{\overline Q}f\left(\varepsilon_k W_k({\cdot},e(u_k))*\rho_k(x_0+rx)\right)\,{\rm d}x\,. \end{align}

From (5.6) and the change of variable $y=ry'$ we may deduce that

(5.14)\begin{equation} W_k({\cdot},e(u_k))*\rho_k(x_0+rx)=W_{k}^r({\cdot},e(u_{k}^r))*\rho_{\frac{\varepsilon_k}r}(x)\,. \end{equation}

Gathering together (5.12), (5.13), (5.14) and using (2.3) we obtain

(5.15)\begin{equation} h(x_0)\ge\limsup_{r\searrow0^+}\limsup_{k\to+\infty}\frac{1}{r} \frac{r}{\varepsilon_k}\int_{\overline Q}f_i\left(r{\frac{\varepsilon_k}r}W_{k}^r({\cdot},e(u_{k}^r))*\rho_{\frac{\varepsilon_k}r}(x)\right)\,{\rm d}x\,. \end{equation}

Now, from (5.11) and the fact that $u_k$ converges to $u$ in measure we can deduce that

\[ \lim_{r\to0}\lim_{k\to+\infty}\mathcal{L}^n\left(\left\{v\in Q\colon|u_{k}^r(v)-\nabla u(x_0)(v)|>\delta\right\}\right)=0\quad\forall\delta>0\,. \]

If we fix a diagonal subsequence $(k_j,r_j)\to (+\infty,0)$ as $j\to +\infty$ for which (5.8) and (5.9) hold, from (5.15) we also get (5.7) for $\delta _j:= \tfrac {\varepsilon _{k_j}}{r_j}$ (up to taking a further subsequence to have a limit in place of a limsup). With this, the proof of step 1 is concluded.

Step 2: In this step we show that for any $0<\zeta <1$ and a.e. $x_0\in U$ there exist $(\bar u_j)\subset GSBV^p(Q;\mathbb {R}^n)$ and $c_0>0$ independent of $j$ such that

(5.16)\begin{gather} \lim_{j\to+\infty}\mathcal{L}^n\{\bar u_j\ne u_{k_j}^{r_j}\}=0; \end{gather}

(5.17)\begin{gather} \bar u_j\to\nabla u(x_0)({\cdot})\quad \text{in measure on } Q\,; \end{gather}

(5.18)\begin{gather} \mathcal{H}^{n-1}(J_{\bar u_j}\cap Q)\to 0\,; \end{gather}

(5.19)\begin{gather} \int_{Q_{1- \zeta }(0)}|e(\bar u_j)|^p\,{\rm d}x\le c_0 \,. \end{gather}

By step 1 we have that $u_{k_j}^{r_j}$ converges in measure to $\nabla u(x_0)(\cdot )$ in $Q$ as $j\to +\infty$ and for $j$ large enough it satisfies

(5.20)\begin{equation} \frac{1}{r_j} \frac{1}{\delta_{j}}\int_{ Q}f_i\left(r_j{\delta_{j}}W_{k_j}^{r_j}({\cdot},e(u_{k_j}^{r_j}))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x\le C\,. \end{equation}

Next we fix $\eta \in (0,1)$ such that $Q_\eta (0)\subset \subset S$ and let $m_\eta$ and $f_i^\eta$ be as in (4.6). Then we get

(5.21)\begin{equation} \int_{ Q}f_i\left({r_j\delta_{j}}W_{k_j}^{r_j}({\cdot},e(u_{k_j}^{r_j}))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x\ge \int_{ Q}f_i^\eta\left({r_j\delta_{j}}{\unicode{x2A0D}}_{Q_{\eta\delta_{j}}(x)} W_{k_j}^{r_j}(y,e(u_{k_j}^{r_j}))\,{\rm d}y\right)\,{\rm d}x\,. \end{equation}

We define the sets

\[ A_j^1:=\left\{x\in Q\colon r_j\delta_{j}{\unicode{x2A0D}}_{Q_{\eta\delta_{j}}(x)} W_{k_j}^{r_j}(y,e(u_{k_j}^{r_j}))\,{\rm d}y\ge \frac{\beta_i}{\alpha_im_\eta\eta^{2n}} \right\}\,, \]

\[ A_j^2:=\left\{x\in Q\colon \operatorname{dist}(x,A_j^1)\le (1-\eta)\delta_{j} \right\}\,. \]

Then arguing as in the proof of proposition 4.1 we find that

(5.22)\begin{equation} A_j^1\subset A_j^2\subset\left\{x\in Q\colon r_j\delta_{j}{\unicode{x2A0D}}_{Q_{\delta_{j}}(x)} W_{k_j}^{r_j}(y,e(u_{k_j}^{r_j}))\,{\rm d}y\ge \frac{\beta_i}{\alpha_im_\eta\eta^{n}} \right\}\,. \end{equation}

(5.22) together with (5.20) and (5.21) imply that (for $j$ large enough)

(5.23)\begin{equation} C\ge\frac{\beta_i}{r_j\delta_{j}}\mathcal{L}^n(A_j^2)=\frac{\beta_i}{\varepsilon_j}\mathcal{L}^n(A_j^2) \,. \end{equation}

By the coarea formula and the mean value theorem we can find $t_j\in (0,(1-\eta )\delta _{j})$ such that setting $A_j^3:= \{\operatorname {dist}(\cdot,A_j^1)\le t_j \}\subset A_j^2$

(5.24)\begin{equation} \mathcal{L}^n(A_j^2)\ge (1-\eta)\delta_{j}\mathcal{H}^{n-1}(\partial A_j^3 )\,. \end{equation}

We finally define

\[ \bar u_j(x):=\begin{cases} 0 & \text{ if } x\in A_j^3,\\ u_{k_j}^{r_j} & \text{ otherwise in }Q. \end{cases} \]

Recall that, by definition, $\frac {\varepsilon _j}{\delta _j}\to 0$. With this, as a consequence of (5.23) and (5.24) we have that both $\mathcal {L}^n( A_j^3)$ and $\mathcal {H}^{n-1}(\partial A_j^3)=\mathcal {H}^{n-1}(J_{\bar u_j})$ converge to $0$ as $j\to +\infty$. Hence $\bar u_j\subset GSBV^p(Q;\mathbb {R}^n)$ and $\bar u_j-u_{k_j}^{r_j}\to 0$ in measure on $Q$ which combined with (5.8) yield $\bar u_j\to \nabla u(x_0)(\cdot )$ in measure on $Q$. It remains to show (5.19). To this aim notice that arguing exactly as in the proof of proposition 4.1 one can find $K(n)\ge 1$ such that for every $x\in \overline Q$

(5.25)\begin{equation} c_1{r_j\delta_{j}}{\unicode{x2A0D}}_{Q_{\eta\delta_{j}}(x)} |e(\bar u_{j}(y))|^p \,{\rm d}y\le K \frac{\beta_i}{\alpha_im_\eta\eta^n}\,. \end{equation}

Next from (5.21) and the monotonicity of $f_i^\eta$ we infer

(5.26)\begin{align} & \frac{1}{r_j\delta_{j}} \int_{ Q}f_i\left({r_j\delta_{j}}W_{k_j}^{r_j}({\cdot},e(u_{k_j}^{r_j}))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x \nonumber\\ & \quad \ge \frac{1}{r_j\delta_{j}}\int_{ Q}f_i^\eta\left(c_1{r_j\delta_{j}}{\unicode{x2A0D}}_{Q_{\eta\delta_{j}}(x)} |e(\bar u_{j}(y))|^p \,{\rm d}y\right)\,{\rm d}x\nonumber\\ & \quad\ge c_1 \frac{\alpha_im_\eta\eta^n}{K} \int_{ Q} {\unicode{x2A0D}}_{Q_{\eta\delta_{j}}(x)} |e(\bar u_{j}(y))|^p \,{\rm d}y\,{\rm d}x \, \end{align}

where the last inequality follows from (5.25) and fact that $f_i^\eta (t)\ge \frac {\alpha _im_\eta \eta ^n}{K} t$ when $t\le K \frac {\beta _i}{m_\eta \eta ^n \alpha _i}$. Finally, for a fixed $0<\zeta <1$, arguing exactly as for (4.15) we get

(5.27)\begin{align} \int_{ Q}{\unicode{x2A0D}}_{Q_{\eta\delta_{j}}(x)} |e(\bar u_{j}(y))|^p \,{\rm d}y\,{\rm d}x\ge \int_{Q_{1-\zeta}(0)} |e(\bar u_j(x))|^p\,{\rm d}x\, \end{align}

when $j$ is sufficiently large. Eventually gathering together (5.20), (5.26), and (5.27), we deduce (5.19) with $c_0:= \frac {CK}{c_1\alpha _im_\eta \eta ^n}$.

Step 3: In this step show that for a.e. $x_0\in U$ there exists a sequence $(w_j)\subset W^{1,p}(Q;\mathbb {R}^n)$ such that:

(5.28)\begin{gather} (|\nabla w_j|^p)\quad \text{ is equi-integrable;} \end{gather}

(5.29)\begin{gather} \lim_{j\to+\infty}\mathcal{L}^n(\left\{w_j\ne\bar u_j \right\}=0\,; \end{gather}

(5.30)\begin{gather} \lim_{j\to+\infty} \|w_j-\nabla u(x_0)({\cdot}) \|_{ L^p(Q)}=0\,; \end{gather}

(5.31)\begin{gather} h(x_0)\ge \liminf_{j\to+\infty}\frac{1}{r_j\delta_{j}}\int_{ Q_{1-\zeta}(0)}f_i\left({r_j\delta_{j}}W_{k_j}^{r_j}({\cdot},e(w_j))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x\quad \forall i\in\mathbb{N}\,. \end{gather}

From step 2 we can apply [18, lemma 5.1] to the sequence $\bar u_j$ and get the existence of $(w_j)\subset W^{1,p}(Q;\mathbb {R}^n)$ that satisfies (5.28)–(5.30). Moreover recalling ($W4$) and the equi-integrability of $(|\nabla w_j|^p)$ we have that $W_{k_j}^{r_j}(x,e(w_j))$ is equi-integrable as well, while from the inclusion

\[ E_j:= \left\{ e(w_j)\ne e( u_{k_j}^{r_j} )\right\}\subset \{w_j\ne u_{k_j}^{r_j}\}\subset\left\{ w_j\ne \bar u_j \right\}\cup\{\bar u_j\ne u^{r_j}_{k_j}\} \,, \]

it follows that $\mathcal {L}^n( \left \{ e(w_j)\ne e( u_{k_j}^{r_j} )\right \})\to 0$. Thus, we can apply lemma 3.6 with $g_j=W_{k_j}^{r_j}(x,e(w_j))$, and $E_j=\left \{ e(w_j)\ne e( u_{k_j}^{r_j} )\right \}$, and deduce that

(5.32)\begin{equation} \int_{Q_{1-\zeta}(0)}(W_{k_j}^{r_j}({\cdot}, e(w_j))\chi_{E_j})*\rho_{\delta_{j}}(x)\,{\rm d}x\to0\,. \end{equation}

We also remark that by the definition of $E_j$ we have

\begin{align*} W_{k_j}^{r_j}({\cdot},e(w_j))*\rho_{\delta_{j}}& =(W_{k_j}^{r_j}({\cdot},e(w_j))\chi_{E_j}^c)*\rho_{\delta_{j}}+(W_{k_j}^{r_j}({\cdot}, e(w_j))\chi_{E_j})*\rho_{\delta_{j}}\\ & \le W_{k_j}^{r_j}({\cdot},e( u_{k_j}^{r_j}))*\rho_{\delta_{j}}+(W_{k_j}^{r_j}({\cdot}, e(w_j))\chi_{E_j})*\rho_{\delta_{j}} \end{align*}

By monotonicity of $f_i$ and since $f_i(t)\le \alpha _it$ we obtain the following estimate

\begin{align*} \frac{1}{r_j\delta_{j}}& \int_{ Q_{1-\zeta}(0)}f_i\left({r_j}\delta_jW_{k_j}^{r_j}({\cdot},e(w_j))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x\\ & \le\frac{1}{r_j\delta_{j}}\int_{ Q_{1-\zeta}(0)}f_i\left({r_j\delta_{j}}W_{k_j}^{r_j}({\cdot},e( u_{k_j}^{r_j}))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x\\ & +\alpha_i\int_{Q_{1-\zeta}(0)}(W_{k_j}^{r_j}({\cdot}, e(w_j))\chi_{E_j})*\rho_{\delta_{j}}(x)\,{\rm d}x \,. \end{align*}

Passing to the limit as $j\to +\infty$ in the above inequality and using (5.7) and (5.32) we infer (5.31).

Step 4: In this step we show that for a.e. $x_0\in U$

(5.33)\begin{align} & \liminf_{j\to+\infty}\frac{1}{r_j\delta_{j}}\int_{ Q_{1-\zeta}(0)}f_i\left({r_j\delta_{j}}W_{k_j}^{r_j}({\cdot},e(w_j))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x\nonumber\\& \quad\ge \alpha_iW(x_0,e(u(x_0)))\quad\forall i\in\mathbb{N}\,. \end{align}

We define the following partition

\[ B_j^1:= \left\{x\in Q_{1-\zeta}(0)\colon {r_j\delta_{j}}W_{k_j}^{r_j}({\cdot},e(w_j))*\rho_{\delta_{j}}(x)\ge \frac{\beta_i}{\alpha_i}\right\}\,,\quad B_j^2:= Q_{1-\zeta}(0)\setminus B_j^1\,. \]

Since $f_i(t)=\alpha _it$ when $t\le \frac {\beta _i}{\alpha _i}$, and $(W_{k_j}^{r_j}(\cdot,e(w_j))\chi _{B_j^2})*\rho _{\delta _{j}}\le \frac {\beta _i}{\alpha _i}$ by definition of $B^2_j$ and standard properties of the convolution, we have

(5.34)\begin{align} & \frac{1}{r_j\delta_{j}}\int_{ Q_{1-\zeta}(0)}f_i\left({r_j\delta_{j}}W_{k_j}^{r_j}({\cdot},e(w_j))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x\nonumber\\ & \quad\ge {\alpha_i}\int_{Q_{1-\zeta}(0)}(W_{k_j}^{r_j}({\cdot},e(w_j))\chi_{B_j^2})*\rho_{\delta_{j}}(x)\,{\rm d}x\,. \end{align}

As for $j$ large enough there holds $\mathcal {L}^n( B_j^1)\le Cr_j\delta _{j}\to 0$, lemma 3.6 implies

(5.35)\begin{equation} \int_{Q_{1-\zeta}(0)}(W_{k_j}^{r_j}({\cdot},e(w_j))\chi_{B_j^1})*\rho_{\delta_{j}}(x)\,{\rm d}x\to0\,. \end{equation}

Now, taking the liminf as $j\to +\infty$ in (5.34), and adding the vanishing term in (5.35) to the right-hand side, we get

(5.36)\begin{align} & \liminf_{j\to+\infty} \frac{1}{r_j\delta_{j}}\int_{ Q_{1-\zeta}(0)}f_i\left({r_j\delta_{j}}W_{k_j}^{r_j}({\cdot},e(w_j))*\rho_{\delta_{j}}(x)\right)\,{\rm d}x\nonumber\\ & \quad \ge \liminf_{j\to+\infty} {\alpha_i}\int_{Q_{1-\zeta}(0)}W_{k_j}^{r_j}({\cdot},e(w_j))*\rho_{\delta_{j}}(x)\,{\rm d}x\,. \end{align}

From this, applying lemma 3.7 with $g_j=W_{k_j}^{r_j}(x,e(w_j))$ we have

(5.37)\begin{align} & \liminf_{j\to+\infty} {\alpha_i}\int_{Q_{1-\zeta}(0)}W_{k_j}^{r_j}({\cdot},e(w_j))*\rho_{\delta_{j}}(x)\,{\rm d}x\nonumber\\ & \quad \ge{\alpha_i} \liminf_{j\to+\infty}\int_{Q_{1-\zeta}(0)}W_{k_j}(x_0+r_jx,e(w_j))\,{\rm d}x\,. \end{align}

Next we modify $w_j$ so that it coincides with $\nabla u(x_0)(\cdot )$ on $\partial Q_{1-\zeta }(0)$ without essentially increasing the energy. This can be achieved by relying on the following Fundamental Estimate than can be proved with standard arguments: for given $\gamma >0$, there exist $C(\gamma )$ and a sequence $(\overline w_j)\subset W^{1,p}(Q_{1-\zeta }(0);\mathbb {R}^d)$ with $\overline w_i=\nabla u(x_0)(\cdot )$ in a neighbourhood of $\partial Q_{1-\zeta }(0)$ such that

(5.38)\begin{align} & \int_{Q_{1-\zeta}(0)}W_{k_j}(x_0+r_jx,e(\overline w_j))\,{\rm d}x \le (1+\gamma)\int_{Q_{1-\zeta}(0)}W_{k_j}(x_0+r_jx,e(w_j))\,{\rm d}x\nonumber\\ & \qquad+ (1+\gamma) \int_{Q_{1-\zeta}(0)\setminus Q_{1-\zeta-\gamma}(0)}W_{k_j}(x_0+r_jx,e(u(x_0)))\,{\rm d}x \nonumber\\ & \qquad + C(\gamma)\|w_j-\nabla u(x_0)({\cdot})\|^p_{L^p(Q_{1-\zeta}(0))}+\gamma. \end{align}

By (5.30) we know that $w_j$ converges to $\nabla u(x_0)(\cdot )$ in $L^p(Q)$, moreover from ($W4$) there holds

\begin{align*} \int_{Q_{1-\zeta}(0)}W_{k_j}(x_0+r_jx,e(u(x_0)))\,{\rm d}x & \le c_2(|e(u(x_0))|^p+1) \mathcal{L}^n( {Q_{1-\zeta}(0)\setminus Q_{1-\zeta-\gamma}(0) }) \\ & \le c_2(|e(u(x_0))|^p+1)n\gamma\,. \end{align*}

This fact and (5.38) imply that

(5.39)\begin{align} \liminf_{j\to+\infty}& \int_{Q_{1-\zeta}(0)}W_{k_j}(x_0+r_jx,e(w_j))\,{\rm d}x\nonumber\\ & \quad\ge \frac{1}{1+\gamma} \liminf_{j\to+\infty} \int_{Q_{1-\zeta}(0)}W_{k_j}(x_0+r_jx,e(\overline w_j))\,{\rm d}x \nonumber\\ & \quad -c_2(|e(u(x_0))|^p+1)n\gamma-\frac{\gamma}{1+\gamma}\,. \end{align}

We now set $\widetilde w_j(x):= r_j\overline w_j((x-x_0)/r_j)$, which is admissible for $\mathbf {m}_{k_j}(u_{\nabla u(x_0)}, Q_{(1-\zeta )r_j}(0))$ in (2.6). Hence, by a change of variable in (5.39) we obtain

(5.40)\begin{align} \liminf_{j\to+\infty}{\alpha_i} \int_{Q_{1-\zeta}(0)}W_{k_j}(x_0+r_jx,e(\overline w_j))\,{\rm d}x& \ge \liminf_{j\to+\infty}\frac{\alpha_i}{r_j^n}\int_{Q_{(1-\zeta)r_j}(x_0)}W_{k_j}(x,e(\widetilde w_j))\,{\rm d}x\nonumber\\ & \ge\liminf_{j\to+\infty}\alpha_i \frac{\mathbf{m}_{k_j}(u_{\nabla u(x_0)},Q_{(1-\zeta)r_j}(x_0))}{r_j^n}\nonumber\\ & =(1-\zeta)\alpha_iW(x_0,e(u(x_0)))\,. \end{align}

Gathering together (5.39) and (5.40), with (5.9) we deduce

\begin{align*} & \liminf_{j\to+\infty}{\alpha_i}\int_{Q_{1-\zeta}(0)}W_{k_j}(x_0+r_jx,e(w_j))\,{\rm d}x\\ & \quad \ge \frac{(1-\zeta)^n}{1+\gamma}\alpha_iW(x_0,e(u(x_0))) -C\left(\gamma+ \frac{\gamma}{1+\gamma}\right)\,. \end{align*}

With this, (5.36), and (5.37), we eventually deduce (5.33) by arbitrariness of $\zeta$ and $\gamma$.

Conclusion: from step 3 and step 4 we deduce the validity of (5.5) and the proof is concluded.

Remark 5.2 We observe en passant that proposition 5.1 indeed holds also for a sequence of functionals

\[ F_k(u,A):= \frac1{\varepsilon_k}\int_A f_k\left(\varepsilon_kW_k({\cdot},e(u))*\rho_k(x) \right)\,{\rm d}x \,,\quad \text{for } u\in W^{1,p}(U)\,,\ A\subset U\,, \]

provided the functions $f_k$ satisfy an estimate of the form

\[ f_k(t)\ge \alpha_k t \wedge \beta \]

for all $t\in [0,+\infty )$, where $\beta$ is a uniform constant and $\alpha =\lim _{k\to +\infty }\alpha _k$.

Proposition 5.3 Lower bound: surface contribution

Let $(u_k)\subset L^0(U;\mathbb {R}^n)$ be a sequence that converges to in measure to $u\in L^0(U;\mathbb {R}^n)$. Assume moreover that $F_k(u_k)\le C$. Then there holds

\[ \liminf_{k\to+\infty}F_k(u_k,A)\ge \beta \int_{J_u\cap A}\phi_\rho(\nu_u)\,{\rm d}\mathcal{H}^{n-1}\quad \forall A\in \mathcal{A}(U)\,. \]

Proof. Let $(u_k)$ and $u$ be as in the statement, so that by proposition 4.1 $u\in GSBD^p(U)$. Let $A\in \mathcal {A}(U)$ be fixed. We claim that it suffices to show that for any $\xi \in \mathbb {S}^{n-1}$ fixed there holds

(5.41)\begin{equation} \liminf_{k\to+\infty}F_k^\xi(u_k,A) \ge \beta \int_{J_u^\xi\cap A}\mu_\xi|\langle\nu_u,\xi\rangle|\,{\rm d}\mathcal{H}^{n-1}\,, \end{equation}

with

\[ F_k^\xi(u_k,A):=\frac{1}{\varepsilon_k}\int_A f\left(c_1 \varepsilon_k |\langle e(u_k)\xi,\xi\rangle|^p *\rho_k(x)\right)\,{\rm d}x\,, \]

\[ J_u^\xi\!:=\!\{x\!\in\! J_u\colon \langle u^+(x)-u^-(x),\xi\rangle\ne0\}\quad\text{and}\quad \mu_\xi\!:=\!\mathcal{H}^1(\{x\!\in\! S\colon x\!=\!t\xi\ \text{ for }\ t\!\in\!\mathbb{R}\})\,. \]

Indeed, assume for the moment (5.41) holds true. Then ($W4$) gives

\[ W_k(x,e(u_k))\ge c_1|e(u_k)|^p\ge c_1 |\langle(e(u_k))\xi,\xi\rangle|^p \,. \]

Since $f$ is nondecreasing, the above implies

\begin{align*} & \liminf_{k\to+\infty} F_k(u_k,A)\ge\liminf_{k\to+\infty} F_k^\xi(u_k,A)\\ & \quad \ge \beta \int_{J_u^\xi\cap A}\mu_\xi|\langle\nu_u,\xi\rangle|\,{\rm d}\mathcal{H}^{n-1}=\beta\int_{J_u\cap A}\varphi_\xi \,{\rm d}\mathcal{H}^{n-1}\,, \end{align*}

with $\varphi _\xi \colon J_u\to [0,+\infty ]$ given by

\[ \varphi_\xi(x):= \begin{cases} \mu_{\xi}|\langle\nu_u(x),\xi\rangle| & \text{if }x\in J_u^{\xi}\,,\\ 0 & \text{otherwise}\,. \end{cases} \]

Now let $(\xi _h)\subset \mathbb {S}^{n-1}$ be a dense subset, in this way by [5, proposition 1.16] it holds

\[ \liminf_{k\to+\infty} F_k(u_k,A)\ge \beta\int_{J_u\cap A}\sup_h\varphi_{\xi_h}\,{\rm d}\mathcal{H}^{n-1}\,. \]

On the other hand by [20, lemma 4.5], we have

(5.42)\begin{equation} \phi_\rho(\nu)=\sup_{\xi\in\mathbb{S}^{n-1}}\mu_\xi |\langle\nu,\xi\rangle| \,, \end{equation}

which in turn implies $\phi _\rho (\nu _u(x))=\sup _h\varphi _{\xi _h}(x)$ and the thesis follows. It remains to show (5.41) for which we will argue by slicing. As the set $S$ is convex for $\delta \in (0,1)$ fixed we can find $r=r(\delta,S)$ such that the cylinder

\[ C_\xi^{r,\delta}:= R_\xi(Q'_r(0)\times \left(-{\mu_\xi\delta}/2,{\mu_\xi\delta}/2\right))\subset{\subset} S\,, \]

where $R_\xi \in SO(n)$ is such that $R_\xi e_n=\xi$ [see (j)]. Let now $m_\xi := \min _{x\in \overline {C}_\xi ^{r,\delta }}\rho (x)$ and

\[ C_{\xi,k}^{r,\delta}(x):= \varepsilon_kC_\xi^{r,\delta}+x\,. \]

Next for any $x\in A$ we denote by $\hat x_\xi$ the projection of $x$ onto $\Pi ^\xi := \{y\in \mathbb {R}^n\colon \langle y,\xi \rangle =0\}$ and

\[ I_\xi:=\{\tau\in\mathbb{R}\colon \hat x_\xi+\tau\xi\in A \}\,. \]

Then we have

(5.43)\begin{align} F_k^\xi(u_k,A) & \ge\frac{1}{\varepsilon_k} \int_{A}f\left(\frac{c_1m_\xi}{\varepsilon_k^{n-1}}\int_{C_{\xi,k}^{r,\delta}(x)} |\langle e(u_k(z))\xi,\xi\rangle|^p \,{\rm d}z\right)\,{\rm d}x\nonumber\\ & =\frac{1}{\varepsilon_k}\int_{\Pi^\xi}\int_{I_\xi} f\left(\frac{c_1m_\xi}{\varepsilon_k^{n-1}}\int_{C_{\xi,k}^{r,\delta}(\hat x_\xi+\tau\xi)} |\langle e(u_k(z))\xi,\xi\rangle|^p \,{\rm d}z\right) {\rm d}\tau\,{\rm d}\mathcal{H}^{n-1}(\hat x_\xi), \end{align}

where the last equality follows by Fubini's Theorem. Noticing that $f$ is concave and using the change of variable $z=\hat x_\xi +s\xi +\varepsilon _krz'$ with

\[ z'\in Q'_{\xi}:=R_\xi\left(Q'\times\{0\}\right)\quad\text{and}\quad s\in \left(\tau-\frac{\mu_\xi\delta\varepsilon_k}2,\tau+\frac{\mu_\xi\delta\varepsilon_k}2\right)\,, \]

together with Jensen's inequality yield

(5.44)\begin{align} & f\left(\frac{c_1m_\xi}{\varepsilon_k^{n-1}}\int_{C_{\xi,k}^{r,\delta}(\hat x_\xi+\tau\xi)} |\langle e(u_k(z))\xi,\xi\rangle|^p \,{\rm d}z\right)\nonumber\\ & \quad \ge{\unicode{x2A0D}}_{Q'_{\xi}}\widetilde f\left( \int_{\tau-\frac{\mu_\xi\delta\varepsilon_k}{2}}^{\tau+\frac{\mu_\xi\delta\varepsilon_k}{2}} |\langle e(u_k(\hat x_\xi+\varepsilon_krz'+s\xi))\xi,\xi\rangle|^p \,{\rm d}s\right)\,{\rm d}z'\nonumber\\ & \quad ={\unicode{x2A0D}}_{Q'_{\xi}}\widetilde f\left( \int_{\tau-\frac{\mu_\xi\delta\varepsilon_k}{2}}^{\tau+\frac{\mu_\xi\delta\varepsilon_k}{2}} \Big|\frac{\partial}{\partial s}w_{\xi,k}(\hat x_\xi, z',s)\Big|^p \,{\rm d}s\right)\,{\rm d}z' \end{align}

with $\widetilde f(t):=f\left (\frac {c_1m_\xi }{r^{n-1}}t\right )$ and $w_{\xi,k}(\hat x_\xi, z',s):= \langle u_k(\hat x_\xi +\varepsilon _krz'+s\xi ),\xi \rangle$. Observe now that applying corollary 3.9 and Fubini's Theorem to the functions $w_{\xi,k}(\hat x_\xi, z',s)$ we have that, for a.e. $(\hat x_\xi, z')\in \Pi _\xi \times Q'_{\xi }$ the functions $s\mapsto w_{\xi,k}(\hat x_\xi, z',s)$ converge to the section $u^{\hat x_\xi }(s):=\langle u(\hat x_\xi +s\xi ), \xi \rangle$ in measure on $I_\xi$. Further, gathering together (5.43) and (5.44), and exchanging the order of integration it holds

(5.45)\begin{equation} F_k^\xi(u_k,A)\ge\int_{\Pi^\xi}{\unicode{x2A0D}}_{Q'_\xi}\left(\frac{1}{\varepsilon_k}\int_{ I_\xi }\widetilde f\left(\int_{\tau-\frac{\mu_\xi\delta\varepsilon_k}{2}}^{\tau+\frac{\mu_\xi\delta\varepsilon_k}{2}} |\dot w^{\hat x_\xi, z'}_{\xi,k}(s)|^p \,{\rm d}s\right)\,{\rm d}\tau\right) \,{\rm d}z' \,{\rm d}\mathcal{H}^{n-1}(\hat x_\xi)\,, \end{equation}

where the shortcut $w^{\hat x_\xi, z'}_{\xi,k}(s)$ denotes the function $s\mapsto w_{\xi,k}(\hat x_\xi, z',s)$ for fixed $(\hat x_\xi, z')$. By theorem 3.3 we get

(5.46)\begin{equation} \liminf_{k\to+\infty}\frac{1}{\varepsilon_k}\int_{ I_\xi }\widetilde f\left(\int_{\tau-\frac{\mu_\xi\delta\varepsilon_k}{2}}^{\tau+\frac{\mu_\xi\delta\varepsilon_k}{2}} |\dot w^{\hat x_\xi, z'}_{\xi,k}(s)|^p \,{\rm d}s\right) {\rm d}\tau\ge \beta\delta \mu_\xi\#(J_{u^{\hat x_\xi}}\cap I_\xi )\,. \end{equation}

Combining (5.45) with (5.46), we finally obtain

\[ \liminf_{k\to+\infty} F_k^\xi(u_k,A)\!\ge\!\delta\beta \int_{\Pi^\xi} \mu_\xi \#(J_{u^{\hat x_\xi} }\cap I_\xi) \,{\rm d}\mathcal{H}^{n-1}(\hat x_\xi) \!=\! \delta \beta \int_{J_u^\xi\cap A}\mu_\xi|\langle\nu_u,\xi\rangle|\,{\rm d}\mathcal{H}^{n-1}\,. \]

Eventually by the arbitrariness of $\delta$, we deduce (5.41).

With the help of propositions 5.1 and 5.3 we can now prove the following lower bound.

Proposition 5.4 Lower-bound

Let $F_k$ and $F$ be as in (2.5) and (2.10) respectively. Let $(u_k)\subset L^0(U;\mathbb {R}^n)$ and $u$ be such that $u_k$ converges to $u$ in measure. Then there exists a subsequence, not relabelled, such that

\[ \liminf_{k\to+\infty}F_k(u_k)\ge F(u)\,. \]

Proof. The result can be obtained exactly as in [22, proposition 5.4].

6. Upper bound

In this section we prove the upper bound.

Proposition 6.1 Let $F_k$ and $F$ be as in (2.5) and (2.10) respectively. Then for each $u\in L^0(U;\mathbb {R}^n)$ there is $(u_k)\subset L^0(U;\mathbb {R}^n)$ that converges in measure to $u$ and such that

\[ \limsup_{k\to\infty}F_k(u_k)\le F(u)\,. \]

Proof. Without loss of generality we assume $F(u)< C$ so that $u\in GSBD^p(U)$. Moreover, since $W$ has $p$-growth from above, by theorem 3.4 we can assume that $u\in \mathcal {W}^\infty _{\rm pw}(U;\mathbb {R}^n)$ and that $J_u$ is an essentially closed connected $(n-1)$-rectifiable set compactly contained in $U$, since the above subspace is dense in energy. We fix $U'\in \mathcal {A}$ with $U\subset \subset U'$ and consider an extension of $u$ on $U'$, not relabelled, such that $u\in \mathcal {W}^\infty _{\rm pw}(U';\mathbb {R}^n)$. Then by theorem 3.1 and remark 3.2 we can find $(v_k)\subset W^{1,p}(U'\setminus \overline {J_u};\mathbb {R}^n)$ such that $v_k$ converges strongly to $u$ in $L^p(U'\setminus J_u;\mathbb {R}^n)$ and

(6.1)\begin{equation} \lim_{k\to\infty}E_k(v_k,U'\setminus \overline{J_u})= E(u,U'\setminus \overline{J_u})=\int_{U'}W(x, e(u))\,{\rm d}x\,. \end{equation}

where the last equality clearly holds as $J_u$ is a null set. For every $h>0$ we set

\[ (J_u)_h:=\{x\in U\colon {\rm d}_S(x,J_u)< h\}\,, \]

so that for $h$ small enough $(J_u)_h\subset \subset U$. Fix now $0<\delta _k<<\varepsilon _k$ and take $\varphi _k\in C^\infty _c(U')$ a cutoff between $(J_u)_{\delta _k}$ and $(J_u)_{2\delta _k}$. Next define $(u_k)\subset W^{1,p}(U';\mathbb {R}^n)$ as

\[ u_k:= v_k(1-\varphi_k)\to u\quad\text{strongly in }L^p(U'\setminus J_u;\mathbb{R}^n)\,, \]

and in particular $u_k\to u$ in measure on $U'$. Then using that $u_k=v_k$ in $U'\setminus (J_u)_{2\delta _k}$ we have

(6.2)\begin{equation} F_k(u_k)\le F_k(v_k, U\setminus \overline{J_u})+ \beta\frac{\mathcal{L}^n((J_u)_{2\delta_k+\varepsilon_k})}{\varepsilon_k}\,. \end{equation}

Now invoking [23, theorem 3.7] we have

(6.3)\begin{equation} \lim_{k\to\infty}\frac{\mathcal{L}^n((J_u)_{2\delta_k+\varepsilon_k})}{\varepsilon_k}=\int_{J_u}\phi_\rho(\nu_u)\,{\rm d}\mathcal{H}^{n-1}\,. \end{equation}

Fix $\hat \alpha >\alpha$. With (2.2), the change of variable $y=x-\varepsilon _kz$, Fubini's theorem, and the change of variable $\hat x=x-\varepsilon _k z$ we have

\begin{align*} F_k(v_k, U\setminus \overline{J_u}) & \le \hat\alpha \int_{U\setminus J_u}\int_{\mathbb{R}^n}W_k(y,e(v_k))\rho_k(x-y)\,{\rm d}y\,{\rm d}x\\ & = \hat{\alpha}\int_{U}\int_{\mathbb{R}^n}W_k\left(x-\varepsilon_kz, e(v_k(x-\varepsilon_k\cdot))\right)\rho(z)\,{\rm d}z\,{\rm d}x\\ & = \hat{\alpha}\int_{\mathbb{R}^n}\rho(z)\int_{U} W_k\left(x-\varepsilon_kz, e(v_k(x-\varepsilon_k\cdot))\right)\,{\rm d}x\,{\rm d}z\\ & \le\hat\alpha \int_{U'}W_k(x,e(v_k))\,{\rm d}x=E_k(v_k,U'\setminus \overline{J_u}). \end{align*}

Hence passing to the limit in $k$ in the above inequality and using (6.1) we get

(6.4)\begin{equation} \limsup_{k\to\infty}F_k(v_k, {U\setminus \overline{J_u}})\le \hat\alpha E(u,U'\setminus \overline{J_u})=\hat\alpha\int_{U'}W(x, e(u))\,{\rm d}x\,, \end{equation}

for all $\hat \alpha >\alpha$. Finally gathering together (6.2)–(6.4), we obtain

\[ \limsup_{k\to\infty}F_k(u_k)\le \hat{\alpha}\int_{U'}W(x, e(u))\,{\rm d}x+\beta \int_{J_u}\phi_\rho(\nu_u)\,{\rm d}\mathcal{H}^{n-1}\,. \]

Eventually by the arbitrariness of $U'$ and $\hat \alpha$, we conclude.

Remark 6.2 If a lower order term $\int _U\psi (|u|)\,{\rm d}x$, is added to the energy, the density argument above can still be applied if $\psi$ complies with the assumptions of theorem 3.4. Also observe that within the same assumptions, $\int _U\psi (|u_k|)\,{\rm d}x$ is equiintegrable whenever $(u_k)$ is converging in $L^p$. For $u_k$ and $u$ as in the proof above, this entails the convergence $\int _U\psi (|u_k|)\,{\rm d}x\to \int _U\psi (|u|)\,{\rm d}x$.

We are now in a position to prove theorem 2.1.

Proof Proof of theorem 2.1

Theorem 3.1 provides a subsequence for which (2.9) holds. Point $(i)$ follows by combining propositions 5.4 and 6.1, while $(ii)$ is a consequence of proposition 4.1.

7. Stochastic homogenisation

In this section we are concerned with the $\Gamma$-convergence analysis of the functionals $F_k$ when $W_k$ are random integrands of type

\[ W_k(\omega,y,M)=W\left(\omega,\frac{y}{\delta_k},M\right)\,, \]

with $\omega$ belonging to the sample space $\Omega$ of a complete probability space $(\Omega,\mathcal {T},P)$ and $\delta _k\searrow 0$. In order to do that we first give some definitions.

Definition 7.1 Group of $P$-preserving transformations

A group of $P$-preserving transformations on $(\Omega,\mathcal {T},P)$ is a family $(\tau _z)_{z\in \mathbb {Z}^n}$ of mappings $\tau _z\colon \Omega \to \Omega$ satisfying the following:

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

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

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

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

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

\[ [a,b):=\{x\in\mathbb{Z}^n\colon a_i\le x_i< b_i\ \text{for}\ i=1,\ldots,n\} \]

and we set

\[ \mathcal{I}_n:=\{[a,b)\colon a,b\in\mathbb{Z}^n\,,\, a_i< b_i\ \text{for}\ i=1,\ldots,n\}\,. \]

Definition 7.2 Subadditive process

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

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

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

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

   \[ \mu(\omega,A)\le\sum_{i\in I}\mu(\omega,A_i)\quad\text{for every}\,\,\omega\in\Omega\,; \]

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

Definition 7.3 Stationarity

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

\[ W(\omega,x+z,M)=W(\tau_z(\omega),x,M) \]

for every $\omega \in \Omega$, $x\in \mathbb {R}^n$, $z\in \mathbb {Z}^n$ and $M\in \mathbb {M}^{n\times n}$. Moreover we say that a stationary random integrand $W$ is ergodic if $(\tau _z)_{z\in \mathbb {Z}^n}$ is ergodic.

For our purposes we consider random integrands $W\colon \Omega \times \mathbb {R}^n\times \mathbb {M}^{n\times n}\to [0,+\infty )$ satisfying the following assumptions:

1. (W1) $W$ is $(\mathcal {T}\otimes \mathcal {B}^n\otimes \mathcal {B}^{n\times n})$-measurable;

2. (W2) $W(\omega,\cdot,\cdot )\in \mathcal {W}$ for every $\omega \in \Omega$;

3. (W3) the map $M\mapsto W(\omega,x,M)$ is lower semicontinuous for every $\omega \in \Omega$ and every $x\in \mathbb {R}^n$.

Let $W$ be a random integrand satisfying ($w1$)–($w3$) and $\delta _k\searrow 0$. We consider the family of functionals $F_k(\omega )\colon L^0(U;\mathbb {R}^n)\to [0,+\infty ]$ defined as

(7.1)\begin{equation} F_k(\omega)(u):=\frac1{\varepsilon_k}\int_Uf\left(\varepsilon_kW\left(\omega,\frac{\cdot}{\delta_k},e(u)\right)*\rho_k(x)\right)\,{\rm d}x\,, \end{equation}

if $u\in W^{1,p}(U;\mathbb {R}^n)$, and extended to $+\infty$ otherwise. Let also for $\omega \in \Omega$ and $A\in \mathcal {A}$

(7.2)\begin{equation} \mathbf{m}_\omega(u_M,A):=\inf\left\{\int_AW(\omega,x,e(v))\,{\rm d}x\colon v\in W^{1,p}(A;\mathbb{R}^n),\ v=u_M\ \text{near}\ \partial A \right\}\,. \end{equation}

We now state the main theorem of this section.

Theorem 7.4 Stochastic homogenisation

Let $W$ be a random integrand satisfying ($w1$)–($w3$). Assume moreover $W$ is 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 $F_k(\omega )$ be as in (7.1) and $\mathbf {m}_\omega$ be as in (7.2). Then there exists $\Omega '\in \mathcal {T}$, with $P(\Omega ')=1$ such that for every $\omega \in \Omega '$, $x\in \mathbb {R}^n$, $M\in \mathbb {M}^{n\times n}$ the limit

(7.3)\begin{equation} \lim_{t\to+\infty}\frac{\mathbf{m}_\omega(u_M,Q_t(tx))}{t^n}= \lim_{t\to+\infty}\frac{\mathbf{m}_\omega(u_M,Q_t(0))}{t^n}=:W_{\rm hom}(\omega,M) \end{equation}

exists and is independent of $x$. The function $W_{\rm hom}\colon \Omega \times \mathbb {M}^{n\times n}\to [0,+\infty )$ is $({\mathcal {T}\otimes \mathcal {B}^{n\times n}})$-measurable. Moreover, for every $\omega \in \Omega '$ the functionals $F_k(\omega )$ $\Gamma$-converge in measure to the functional $F_{\rm hom}(\omega )\colon L^0(U;\mathbb {R}^n)\to [0,+\infty ]$ given by

\[ F_{\rm hom}(\omega)(u):=\begin{cases} \alpha\int_U W_{\rm hom}(\omega,e(u))\,{\rm d}x+\beta \int_{J_u}\phi_\rho(\nu_u)\,{\rm d}\mathcal{H}^{n-1} & \text{if }u\in GSBD^p(U)\,,\\ + \infty & \text{otherwise}\,. \end{cases} \]

If, in addition, $W$ is ergodic, then $W_{\rm hom}$ is independent of $\omega$ and

(7.4)\begin{equation} W_{\rm hom}(M)=\lim_{t\to+\infty}\frac{1}{t^n}\int_{\Omega}\mathbf{m}_\omega(u_M,Q_t(0))\,{\rm d}P(\omega)\,, \end{equation}

and thus $F_{\rm hom}$ is deterministic.

The proof of theorem 7.4 is quite standard and can be achieved as in [12] (see also [21]). For this reason here we only detail the main adaptations.

Proposition 7.5 Let $W$ be a stationary random integrand satisfying ($w1$)–($w3$) and let $\mathbf {m}_\omega$ be as in (7.2). Then for every $M\in \mathbb {M}^{n\times n}$ the function $\mu _M\colon \Omega \times \mathcal {I}_n\to \mathbb {R}$ given by $\mu _M(\omega,A):= \mathbf {m}_\omega (u_M,A)$ defines a subadditive process on $(\Omega,\mathcal {T},P)$. Moreover

(7.5)\begin{equation} 0\le \mu_M(\omega,A)\le c_2(|M+M^T|^p+1)\mathcal{L}^n(A)\,, \end{equation}

for $P$-a.e. $\omega \in \Omega$ and for every $A\in \mathcal {I}_n$.

Proof. Let $M\in \mathbb {M}^{n\times n}$ be fixed. Then we need to show that $\mu _M$ satisfies properties (a)–(d). The proof of properties (b)–(d) and of (7.5) are standard and therefore we omit it here. It then remains to prove (a). Let $A\in \mathcal {I}_n$ be fixed. For $N\in \mathbb {N}$ let

\[ W^N(\omega,x,M):= \inf_{\xi\in\mathbb{M}^{n\times n}}\{W(\omega,x,\xi)+N|\xi-M|\} \]

be the Moreau-Yosida regularisation of $M\mapsto W(\omega,x,M)$ which is $N$-Lipschitz. Let also

\[ F^N(\omega)\colon W^{1,p}(A)\to[0,+\infty)\,, \]

be defined as

\[ F^N(\omega)(u):=\int_AW^N(\omega,x,e(u))\,{\rm d}x\,. \]

Arguing as in the proof of [21, lemma C.1.] it can be shown that $(\omega,u)\mapsto F^N(\omega )(u)$ is $\mathcal {T}\otimes \mathcal {B}(W^{1,p}(A))$-measurable. By ($w3$) $W^N\nearrow W$ pointwise, and in particular $F^N(\omega )(u)$ converges to $\int _AW(\omega,x,e(u))\,{\rm d}x$ pointwise. As a consequence $(\omega,u)\mapsto \int _AW(\omega,x,e(u))\,{\rm d}x$ is also $\mathcal {T}\otimes \mathcal {B}(W^{1,p}(A))$-measurable. Now we note that $F(\omega )(u_M)<+\infty$. This together with ($w3$) and [21, lemma C.2.] imply that $\omega \mapsto \mu _M(u_M,A)$ is $\mathcal {T}$-measurable.

The proof of theorem 7.4 follows by proposition 7.5 and the Subadditive Ergodic Theorem [19, theorem 2.4] arguing as in [12].