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.

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.

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.

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.

Proof. The proof is inspired by that of [Reference Scilla and Solombrino22, 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 [Reference Chambolle and Crismale8, 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., [Reference Evans and Gariepy16, 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. [Reference Scilla and Solombrino22, 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

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

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 [Reference Ambrosio, Fusco and Pallara1, 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 [Reference Ambrosio, Fusco and Pallara1, 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 [Reference Cagnetti, Chambolle and Scardia6, 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 [Reference Ambrosio, Fusco and Pallara1, 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 [Reference Friedrich, Perugini and Solombrino18, 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.

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 [Reference Braides5, 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 [Reference Negri20, 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).

Proof. The result can be obtained exactly as in [Reference Scilla and Solombrino22, proposition 5.4].

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 [Reference Villa and Lussardi23, 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.

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 [Reference Ruf and Ruf21, 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 [Reference Ruf and Ruf21, lemma C.2.] imply that $\omega \mapsto \mu _M(u_M,A)$ is $\mathcal {T}$-measurable.