Proof. From (3.10) we clearly deduce that $e(u_{\rho })$ is bounded in $L^{2}(\Omega _1; \mathbb {M}^{n}_{s})$ and admits, up to a subsequence, a weak limit $f \in L^{2}(\Omega _1; \mathbb {M}^{n}_{s})$. Since $u_{\rho } \to u$ in measure in $\Omega _1$, from (3.10) and [Reference Dal Maso17, theorem 11.3] we deduce that $u \in GSBD^{2}(\Omega _1)$ with $e(u) = f$ and that $e(u_{\rho }) \rightharpoonup e(u)$ weakly in $L^{2}(\Omega _{1}; \mathbb {M}^{n}_{s})$.

By definition of $\mathcal {E}_{\rho }$ and by (3.2) we have that

\[ c_{1} \| e_{\alpha, n} (u_{\rho}) \|_{2}^{2} = c_{1} \rho^{2} \| e^{\rho}_{ \alpha, n} (u_{\rho}) \|_{2}^{2} \leq \rho^{2} \mathcal{E}_{\rho}(u_{\rho}) \]

and similarly $c_{1} \| e_{n,n} (u_{\rho }) \|_{2}^{2} \leq \rho ^{4} \mathcal {E}_{\rho } (u_{\rho })$. Hence, (3.10) implies that $e_{i,n} (u_{\rho }) \to 0$ in $L^{2}(\Omega _{1} ; \mathbb {M}^{n}_{s})$, from which we deduce that $e_{i,n} (u) = 0$ for $i = 1, \ldots, n$.

Finally, by [Reference Kholmatov and Piovano31, proposition 4.6], for every $\tilde {\rho }>0$ we have that

\begin{align*} \frac{1}{\tilde{\rho}} \int_{J_{u}} | ( \nu_{u})_{n} | \mathrm{d} \mathcal{H}^{n-1} & \leq \int_{J_{u}} \phi_{\tilde{\rho}}( \nu_{u}) \, \mathrm{d} \mathcal{H}^{n-1} \leq \liminf_{\rho \to 0} \int_{J_{u_{\rho}}} \phi_{\tilde\rho}(\nu_{u_{\rho}}) \,\mathrm{d} \mathcal{H}^{n-1}\\ & \leq \liminf_{\rho \to 0} \int_{J_{u_{\rho}}} \phi_{\rho}(\nu_{u_{\rho}}) \, \mathrm{d} \mathcal{H}^{n-1} \leq \liminf_{\rho \to 0} \mathcal{E}_{\rho} (u_{\rho}). \end{align*}

Letting $\tilde {\rho } \to 0$ in the previous inequality and using again (3.10) we infer that $(\nu _{u})_{n} = 0$ $\mathcal {H}^{n-1}$-a.e. on $J_{u}$.

Proof. Let $D$ be as in definition 4.2. For every $j \in \mathbb {N}$ let us denote by $J^{e}_j$ the directional half-neighbourhood of $\Gamma _j$ and define the discrete jump energy

\[ E^{y,h}(\Gamma_j) := h^{n} \sum_{e \in D} \sum_{z \in h\mathbb{Z}^{n}} \frac{\mathbb{1}_{J_j^{he}}(z + hy)}{h|e|}. \]

Notice that, by definition of bad hyper-cubes, we have

(4.3)\begin{equation} E^{y,h}(\Gamma_j) \geq C \# \mathcal{B}_{h,j,y} h^{n-1}, \end{equation}

for a positive constant $C$ independent of $h$ and $j$. Moreover for every $h$ we can give the following estimate

\begin{align*} \int_{[0,1)^{n}} \sum_{j=1}^{\infty} E^{y,h}(\Gamma_j) \, \mathrm{d} y & = \sum_{j=1}^{\infty} \sum_{e \in D} \sum_{z \in h\mathbb{Z}^{n}} h^{n}\int_{[0,1)^{n}} \frac{\mathbb{1}_{J^{he}_j}(z + hy)}{h|e|} \, \mathrm{d} y\\ & = \sum_{j=1}^{\infty} \sum_{e \in D} \int_{\mathbb{R}^{n}} \frac{\mathbb{1}_{J^{he}_j}(y)}{h|e|} \, \mathrm{d} y\\ & =\sum_{j=1}^{\infty} \sum_{e\in D} \int_{\Pi^{e}} \Bigg(\int_{\mathbb{R}} \frac{\mathbb{1}_{J_j^{he}}(\overline{y} + se)}{h|e|} \, \mathrm{d} s \Bigg) \mathrm{d} \overline{y}\\ & \leq \sum_{j=1}^{\infty} \sum_{e\in D}\int_{\Pi^{e}} \mathcal{H}^{0}((\Gamma_j)_{\overline{y}}^{e}) \, \mathrm{d} \overline{y} \leq c \sum_{j=1}^{\infty} \mathcal{H}^{n-1}(\Gamma_j) =c L, \end{align*}

where $c = \max _{| \nu | =1} (\sum _{e \in D}|\nu \cdot e|/|e|)$. Therefore, if we set $g(y) := \liminf _{h \to 0^{+}} \sum _{j=1}^{\infty } E^{y,h}(\Gamma _j)$ and define

\[ H := \{ y \in [0,1)^{n} \ | \ g(y) \leq cL/\delta \}, \]

by Fatou lemma and Chebyshev inequality we get that

\[ \mathcal{L}^{n}([0,1)^{n} \setminus H) \leq \delta. \]

Moreover, if $y \in H$, we have, up to passing to a subsequence depending on $y$, that

(4.4)\begin{equation} g(y) = \lim_{h\to 0^{+}} \sum_{j=1}^{\infty} E^{y, h}(\Gamma_{j}) \leq \frac{cL}{\delta}. \end{equation}

Again by Fatou lemma we have, along the same subsequence, that

(4.5)\begin{equation} \sum_{j=1}^{\infty} \liminf_{h \to 0^{+}} E^{y,h}(\Gamma_j) \leq g(y) \leq \frac{cL}{\delta}. \end{equation}

Therefore, for every $\epsilon _1>0$ there exists $j_1 \in \mathbb {N}$ such that

\[ \liminf_{h \to 0^{+}} E^{y,h}(\Gamma_{j_1}) \leq \epsilon_1. \]

In particular, we can find a subsequence $h^{1}_k \searrow 0$ such that

\[ \lim_{k \to \infty} E^{y,h^{1}_k}(\Gamma_{j_1}) = \liminf_{h \to 0^{+}} E^{y,h}(\Gamma_{j_1}) \leq \epsilon_1. \]

Since the bounds (4.4)–(4.5) are still valid along the subsequence $(h^{1}_k)_k$, given $\epsilon _2 >0$ we can find a sufficiently large $j_2 \in \mathbb {N}$ for which

\[ \liminf_{k \to \infty} E^{y,h^{1}_k}(\Gamma_{j_2}) \leq \epsilon_2. \]

As before, we can find a subsequence $(h^{2}_k)_k \subset (h^{1}_k)_k$ such that $h^{2}_k \searrow 0$ and

\[ \lim_{k \to \infty} \, E^{y,h^{2}_k}(\Gamma_{j_2}) = \liminf_{k \to \infty} \, E^{y,h^{1}_k}(\Gamma_{j_2}) \leq \epsilon_2. \]

By induction, given a sequence $\epsilon _m \searrow 0$, we can construct a sequence $j_m \nearrow \infty$ and, for every $m \in \mathbb {N}$, the subsequences $(h_k^{m})_k \subset (h_k^{m-1})_k$ satisfying

\[ \lim_{k \to \infty} \, E^{y,h^{m}_k}(\Gamma_{j_m}) = \liminf_{k \to \infty} \, E^{y,h^{m-1}_k}(\Gamma_{j_m}) \leq \epsilon_m. \]

Setting $h_k := h_k^{k}$ for every $k$, we infer that $h_k \searrow 0$ and

\[ \lim_{k \to \infty} \, E^{y,h_k}(\Gamma_{j_m}) \leq \epsilon_m \quad \text{ for every }m. \]

Finally, by (4.3) and by definition (4.1) of $A_{h_{k}, j_{m}}$ we estimate, for a suitable constants $c_{1}(n), c_{2}(n) >0$ depending only on the dimension $n$,

\begin{align*} \limsup_{k \to \infty} \, \mathcal{H}^{n-1}(\partial A_{h_k,j_m}) & \leq c_1(n) \, \limsup_{k \to \infty}\, \# \mathcal{B}_{h_k,j_m,y}\, h_k^{n-1}\\ & \leq c_{2}(n) \limsup_{k \to \infty} E^{y,h_k}(\Gamma_{j_m}) \leq c_{2}(n) \epsilon_m, \end{align*}

so that (4.2) holds. By setting $\epsilon _m := 1/c_2(n)m$ we infer (4.2).

Proof. First we prove that there exists a sequence $(v_{h_k})_{h >0} \subset GSBD^{2}(V) \cap W^{1,\infty }(V \setminus \overline {J_{v_{h_k}}}; \mathbb {R}^{n})$ satisfying $(i)$, $(ii)$, $(iv)$ and $(v)$ as $k \to \infty$, plus the fact that $J_{v_{h_k}} \subseteq \mathcal {Q}_{h_k}^{0} + h_ky_k$ for some $(y_k) \subset [0,1)^{n}$. In order to prove this, we proceed similarly to [Reference Iurlano30, theorem 5]: consider for a.e. $y \in [0,1)^{n}$ the $(n-1)$-dimensional $h$-grid $\mathcal {Q}_h^{0} +hy$ and consider the discretized function of $v$

\[ v_{h_k}^{y}(z) := v(z +hy), \quad z \in h\mathbb{Z}^{n} \cap (U -hy), \]

and define the continuous interpolation of $v_{h_k}^{y}$

\[ w_h^{y}(x) := \sum_{z \in h\mathbb{Z}^{n} \cap U -hy} v_{h_k}^{y}(z) \Delta \left(\frac{x - (z +hy)}{h} \right) \quad \text{for }x \in V, \]

where

\[ \Delta(x) := \prod_{i=1}^{n}(1-|x_i|)^{+}. \]

Let us fix $V\Subset V' \Subset V''\Subset V' \Subset U$ and let us define the piecewise constant strain in the direction $e \in D$ as

\[ E^{y,h}_e(x) := \sum_{z \in h \mathbb{Z}^{n} \cap V'} \frac{[(v_{h_k}^{y}(z +he) - v_{h_k}^{y}(z))\cdot e]}{h} c^{y}_{e,h}(z) \mathbb{1}_{z +hy + [0,h)^{n}}(x), \quad x \in V', \]

where $c^{y}_{e,h}(z) := 1-\mathbb{1}_{J^{eh}}(z +hy)$. Notice that, since $V \Subset V' \Subset V' \Subset U$, then $E^{y,h}_e$ is well defined in $V''$ for every sufficiently small $h >0$. We claim that

(4.6)\begin{equation} \lim_{h \to 0^{+}}\int_{[0,1)^{n}}\left(\int_{V''} |E_e^{y,h}(x) - f_e(x)|^{2} \, \mathrm{d} x \right)\mathrm{d} y =0 \quad \text{for every }e \in D, \end{equation}

where $f_e := e(v)e \cdot e$. In order to simplify the next computation let us set

\[ Q_h^{y}(z) := [z +hy+[0,h)^{n}] \cap V, \quad U_h^{y} := U -hy. \]

Now we write

(4.7)\begin{align} & \int_{[0,1)^{n}}\Bigg(\int_{V''} |E_e^{y,h}(x) - f_e(x)|^{2} \, \mathrm{d} x \Bigg) \mathrm{d} y\nonumber \\ & \quad\leq \int_{[0,1)^{n}} \Bigg( \!\sum_{z \in h \mathbb{Z}^{n} \cap V'} \int_{Q^{y}_h(z)} \! \Bigg| \frac{(v (z + hy + he) - v(z +hy)) \cdot e}{h}c_{e,h}^{y}(z) - f_e(x) \Bigg|^{2} \, \mathrm{d} x \Bigg) \mathrm{d} y \nonumber\\ & \quad= \int_{V} \Bigg( \!\sum_{z \in h \mathbb{Z}^{n} \cap V'} \int_{[0,1)^{n}} \! \Bigg| \frac{(v(z + hy + he) - v( z + hy)) \cdot e}{h}c_{e,h}^{y}(z) - f_e(x) \Bigg|^{2} \mathbb{1}_{Q^{y}_h(z)}(x) \, \mathrm{d} y \Bigg) \mathrm{d} x \nonumber\\ & \quad\leq \int_{V} \Bigg( \!\sum_{z \in h \mathbb{Z}^{n} \cap V' } \unicode{x2A0D}_{z +[0,h)^{n}} \! \Bigg| \frac{(v(\zeta + he) - v(\zeta)) \cdot e}{h}\mathbb{1}_{U \setminus J^{eh}}(\zeta) - f_e(x) \Bigg|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x - \zeta}{h}\Bigg) \, \mathrm{d} \zeta \Bigg)\mathrm{d} x, \end{align}

where in the last inequality we have performed the change of variable $\zeta = z + hy$ and we have used the trivial inclusion $[0,1)^{n} \cap (\frac {V - z}{h} - y) \subset [0,1)^{n}$. We can continue the estimate (4.7) by noticing that the cubes $z + [0,h)^{n}$ are pairwise disjoints, so that

(4.8)\begin{align} & \int_{[0,1)^{n}}\Bigg(\int_{V} |E_e^{y,h}(x) - f_e(x)|^{2} \, \mathrm{d} x \Bigg)\mathrm{d} y\nonumber \\ & \quad \leq \int_{V} \Bigg( \frac{1}{h^{n}} \int_{U} \Bigg| \frac{(v( \zeta + he) - v ( \zeta )) \cdot e}{h}\mathbb{1}_{U \setminus J^{eh}}(\zeta) - f_e(x) \Bigg|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-\zeta}{h}\Bigg) \, \mathrm{d} \zeta \Bigg)\mathrm{d} x \nonumber\\ & \quad \leq \int_{V} \Bigg( \frac{1}{h^{n}} \int_{U \setminus J^{eh}} \Bigg| \frac{(v(\zeta + he) - v(\zeta)) \cdot e}{h} - f_e(x) \Bigg|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-\zeta}{h}\Bigg) \, \mathrm{d} \zeta \Bigg)\mathrm{d} x \nonumber\\ & \qquad+ \int_V \Bigg( \frac{1}{h^{n}} \int_{J^{eh}} |f_e(x)|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-\zeta}{h}\Bigg)\mathrm{d} \zeta \Bigg)\mathrm{d} x. \end{align}

We treat the last two integrals in (4.8) separately. For the first we have that

\begin{align*} & \int_{V} \Bigg( \frac{1}{h^{n}} \int_{U \setminus J^{eh}} \Bigg| \frac{(v(\zeta + he) - v(\zeta)) \cdot e}{h} - f_e(x) \Bigg|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x - \zeta }{h} \Bigg) \, \mathrm{d} \zeta \Bigg)\mathrm{d} x\\ & \quad=\int_{V} \Bigg( \frac{1}{h^{n}} \int_{\Pi^{e} } \Bigg( \int_{(U \setminus J^{eh})_y^{e}} \Bigg| \frac{(v(y+te+he) - v(y+te)) \cdot e}{h} - f_e(x) \Bigg|^{2}\\ & \qquad \times \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-y -te}{h} \Bigg) \, \mathrm{d} t\Bigg)\mathrm{d} y\Bigg)\mathrm{d} x\\ & \quad\leq \int_{V} \Bigg( \frac{1}{h^{n}} \int_{\Pi^{e} } \Bigg( \int_{(U \setminus J^{eh})_y^{e}} \Bigg(\unicode{x2A0D}_0^{h} |D_s v(y+te+se) \cdot e - f_e(x)|^{2} \, \mathrm{d} s\Bigg)\\ & \qquad \times\mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-y -te}{h} \Bigg) \, \mathrm{d} t\Bigg)\mathrm{d} y\Bigg)\mathrm{d} x\\ & \quad= \int_{V} \Bigg( \frac{1}{h^{n}} \int_{\Pi^{e} } \Bigg( \int_{(U \setminus J^{eh})_y^{e}} \Bigg(\unicode{x2A0D}_0^{h} |f_e(y+te+se) - f_e(x)|^{2} \, \mathrm{d} s\Bigg) \\ & \qquad \times \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-y -te}{h} \Bigg) \mathrm{d} t\Bigg)\mathrm{d} y\Bigg)\mathrm{d} x, \end{align*}

where in the last equality we have used the fact that $t \notin (U \setminus J^{eh})_y^{e}$ implies $\{t+s :\, \ s \in [0,h)\} \cap (J_u)_y^{e} = \emptyset$. We can continue the previous estimate with

\begin{align*} & \int_{V} \Bigg( \frac{1}{h^{n}} \int_{U \setminus J^{eh}} \Bigg| \frac{(v(\zeta + he) - v (\zeta )) \cdot e}{h} - f_e(x) \Bigg|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-\zeta }{h} \Bigg) \, \mathrm{d} \zeta \Bigg) \mathrm{d} x\\ & \quad\leq \int_{V''}\Bigg(\unicode{x2A0D}_0^{h} \Bigg( \frac{1}{h^{n}} \int_{U} |f_e(\zeta + se) - f_e(x)|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-\zeta }{h}\Bigg)\mathrm{d} \zeta \Bigg)\mathrm{d} s\Bigg)\mathrm{d} x \nonumber\\ & \quad= \unicode{x2A0D}_0^{h}\Bigg( \int_{V''}\Bigg( \unicode{x2A0D}_{x - [0,h)^{n}} |f_e(\zeta + se) - f_e(x)|^{2} \, \mathrm{d} \zeta \Bigg)\mathrm{d} x\Bigg)\mathrm{d} s \nonumber\\ & \quad= \unicode{x2A0D}_0^{h}\Bigg( \int_{V''}\Bigg( \unicode{x2A0D}_{[0,h)^{n}} |f_e(x -\zeta + se) - f_e(x)|^{2} \, \mathrm{d} \zeta \Bigg)\mathrm{d} x \Bigg)\mathrm{d} s \nonumber\\ & \quad= \int_0^{1}\Bigg(\int_{[0,1)^{n}}\Bigg( \int_{V} |f_e(x + h( se - \zeta )) - f_e(x)|^{2} \, \mathrm{d} x\Bigg)\mathrm{d} \zeta \Bigg)\mathrm{d} s. \nonumber \end{align*}

The continuity property of the translations in $L^{2}(U)$ plus the Dominated Convergence Theorem allow us to deduce that

(4.9)\begin{align} & \lim_{h \to 0^{+}} \int_{V''} \Bigg( \frac{1}{h^{n}} \int_{U \setminus J^{eh}} \Bigg| \frac{(v(\zeta +he) - v(\zeta )) \cdot e}{h} - f_e(x) \Bigg|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-\zeta}{h} \Bigg) \, \mathrm{d} \zeta \Bigg) \mathrm{d} x\nonumber \\ & \quad \leq \lim_{h \to 0^{+}} \int_0^{1}\Bigg(\int_{[0,1)^{n}}\Bigg( \int_{V''} |f_e(x+h(se - \zeta)) - f_e(x)|^{2} \, \mathrm{d} x\Bigg)\mathrm{d} \zeta \Bigg)\mathrm{d} s=0. \end{align}

The second term on the right-hand side of (4.8) can be estimated as follows

\begin{align*} & \int_{V''} \Bigg( \frac{1}{h^{n}} \int_{J^{eh}} |f_e(x)|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-\zeta }{h} \Bigg)\mathrm{d} \zeta \Bigg)\mathrm{d} x\\ & \quad \leq \int_{J^{eh}}\Bigg( \unicode{x2A0D}_{[0,h)^{n}} |f_e(\zeta + x)|^{2} \, \mathrm{d} x \Bigg)\mathrm{d} \zeta \\ & \quad = \int_{[0,1)^{n}}\Bigg(\int_{J^{eh} +hx} |f_e(\zeta )|^{2} \, \mathrm{d} \zeta \Bigg)\mathrm{d} x. \end{align*}

Being $\mathcal {L}^{n}(J^{eh})$ infinitesimal as $h \to 0^{+}$ (see the proof of proposition 4.3), we easily deduce that

(4.10)\begin{align} & \lim_{h \to 0^{+}} \int_V \Bigg( \frac{1}{h^{n}} \int_{J^{eh}} |f_e(x)|^{2} \mathbb{1}_{[0,1)^{n}}\Bigg(\frac{x-\zeta }{h} \Bigg)\mathrm{d} \zeta \Bigg)\mathrm{d} x\nonumber \\ & \quad\leq \lim_{h \to 0^{+}} \int_{[0,1)^{n}}\Bigg(\int_{J^{eh} +hx} |f_e(\zeta )|^{2} \, \mathrm{d} \zeta \Bigg)\mathrm{d} x =0. \end{align}

As a consequence of (4.8)–(4.10) we obtain the claim (4.6). Moreover, by looking at the proof of [Reference Iurlano30, theorem 5, formulas (1’)–(3’b)], thanks to the fact that $V \Subset U$ and $\mathcal {H}^{n-1}(\partial V \cap J_v)=0$, we deduce that

(4.11)\begin{align} & \lim_{h \to 0^{+}}\int_{[0,1)^{n}} \Bigg(\int_V|w^{y}_h(x) - v(x)| \wedge 1 \, \mathrm{d} x \Bigg)\mathrm{d} y=0, \end{align}

(4.12)\begin{align} & \lim_{h \to 0^{+}}\int_{[0,1)^{n}} \Bigg(\int_{\partial V} | Tr( w^{y}_h) (x) - Tr( v) (x) | \wedge 1\, \mathrm{d}\mathcal{H}^{n-1}(x) \Bigg) \mathrm{d} y =0, \end{align}

(4.13)\begin{align} & \lim_{h \to 0^{+}}\int_{[0,1)^{n}} E^{y,h}_2( (\partial V)_{2nh}) \, \mathrm{d} y =0, \end{align}

where $(\partial V)_{2nh} := \{ x \in \mathbb {R}^{n} : \, d (x, \partial V) <2nh\}$ and $E^{y,h}_2 ( (\partial V)_{2nh})$ is defined as in [Reference Iurlano30, formula (32)] as

\[ E^{y, h}_{2} ((\partial V)_{2nh}) := h^{n} \sum_{ e \in D} \sum_{\substack{z \in (\partial V)_{2nh} - hy \\ z \in (\partial V)_{2nh} - hy - he}} \frac{\mathbb{1}_{J^{he} ( z + hy)}}{h |e|}. \]

We recall that since $J_v$ is countably $(\mathcal {H}^{n-1},n-1)$-rectifiable and has finite measure, arguing similarly to [Reference Federer21, lemma 3.2.18] we find a sequence $K_j$ of compact subsets of $\mathbb {R}^{n-1}$ with associated Lipschitz maps $\psi _j \colon K_j \to \mathbb {R}^{n}$ such that $\psi _{j_1}(K_{j_1}) \cap \psi _{j_2}(K_{j_2}) = \emptyset$ for $j_1 \neq j_2$ and

(4.14)\begin{equation} \mathcal{H}^{n-1}\Bigg(J_v \setminus \bigcup_{j=1}^{\infty} \psi_j (K_j)\Bigg) = 0 \text{ and } \mathcal{H}^{n-1}(\psi_j (K_j) \setminus J_v) = 0 \text{ for }j \in \mathbb{N}. \end{equation}

In addition, being $\mathcal {H}^{n-1}(\partial V \cap J_v)=0$ we may also suppose that

(4.15)\begin{equation} \psi_{j}(K_j) \Subset V \text{ or } \psi_{j}(K_j) \Subset U \setminus \overline{V}\text{ for }j \in \mathbb{N}. \end{equation}

For every $m \in \mathbb {N}\setminus \{0\}$, let $j_m$ be such that

(4.16)\begin{equation} \sum_{j > j_m} \mathcal{H}^{n-1}(\psi_{j}(K_j)) \leq \frac{1}{m^{2}}. \end{equation}

Let us set $\Gamma _{J_{0}} := J_{v}$ and $\Gamma _{j_{m}} := J_v \setminus \cup _{j \leq j_m }\psi _j(K_j)$ for $m \geq 1$. In view of (4.14)–(4.16) we can apply proposition 4.3 from which we deduce, in combination with (4.6) and (4.11)–(4.13), that there exists $y \in [0,1)^{n}$, a subsequence of $(j_m)_m$, which with abuse of notation we still denote by $(j_m)_m$, and a subsequence $(h_k)_k$ for which we have

(4.17)\begin{align} & \lim_{k \to \infty}\int_{V''} |E_e^{y,h_k}(x) - f_e(x)|^{2} \, \mathrm{d} x =0 \text{ for }e \in D, \end{align}

(4.18)\begin{align} & \lim_{k \to \infty}\int_V|w^{y}_{h_k}(x) - v(x)| \wedge 1 \, \mathrm{d} x =0, \end{align}

(4.19)\begin{align} & \lim_{k \to \infty}\int_{\partial V} |Tr( w^{y}_{h_k}) (x) - Tr( v) (x) | \wedge 1\, \mathrm{d} \mathcal{H}^{n-1}(x) =0, \end{align}

(4.20)\begin{align} & \lim_{k \to \infty} \, E^{y,h_k}_2( (\partial V)_{2nh_k}) =0, \end{align}

(4.21)\begin{align} & \limsup_{k \to \infty} \, \mathcal{H}^{n-1}( \partial A_{h_{k}, j_{m}} ) <\frac{1}{m} \quad \text{for every }m, \end{align}

where $A_{h_k,j_m}$ is the union of bad hyper-cubes of $\mathcal {Q}_{h_k}^{0}+h_ky$ relative to $\Gamma _{j_m}$. We further notice that, following the proof of proposition 4.3, we may assume that the first term of the subsequence $\Gamma _{j_{0}} = J_{v}$. Since $y$ is fixed, in what follows we omit the dependence on $y$.

Now we proceed with the construction of $(v_k)_{k=1}^{\infty }$. Arguing similarly to [Reference Iurlano30, theorem 5] we define the function $v_k$ equals to $0$ on each bad hyper-cube of $\mathcal {Q}^{0}_{h_k}$ relative to $J_v$ and $v_k := w_{h_k}$ otherwise in $V$. In this way (4.18)–(4.20) imply $(i)$ and $(iv)$ by arguing in a very same way as in [Reference Iurlano30, theorem 5], while $(v)$ comes by construction. Now we need an intermediate step which allows us to prove at the end $(ii)$. If we introduce the discretized bulk energy

(4.22)\begin{equation} E_{h_k}(v_{h_k}) := h_k^{n} \sum_{e \in D} \sum_{z \in h_{k} \mathbb{Z}^{n} \cap V'''} \alpha(e) \frac{((v_{h_k}(z+h_{k}e)-v_{h_k}(z))\cdot e)^{2}}{h_k^{2}}c_{e,h_k}(z), \end{equation}

where $\alpha (e) = n-1$ whenever $e=e_i$, $i=1,\dotsc,n$, and $1/4$ for $e=e_i\pm e_j$, $1 \leq i < j \leq n$, then [Reference Chambolle13] tells us that the contribution of a good hyper-cube $Q$ of $\mathcal {Q}^{0}_{h_{k}}$ having non-empty intersection with $V$ to the discretized bulk energy is exactly

\begin{align*} I_Q& = \frac{(n-1)h_k^{n}}{2^{n-1}} \sum_{i=1}^{n}\sum_{\substack{\eta\in \{0,1\}^{n} \\ \eta_i = 0}} \frac{((v_{h_k}(z+h_k\eta+h_ke_i)-v_{h_k}^{y}(z+h_k\eta))\cdot e_i)^{2}}{h_k^{2}} \\ & \quad+ \frac{h_k^{n}}{2^{n-2}} \sum_{1\leq i < j \leq n} \sum_{\substack{\eta\in \{0,1\}^{n} \\ \eta_i =\eta_j= 0}} \\ & \quad \times\Bigg(\frac{((v_{h_k}(z+h_k\eta+h_k(e_i+e_j))-v_{h_k}(z+h_k\eta))\cdot (e_i+e_j))^{2}}{4h_k^{2}}\\ & \quad+\frac{((v_{h_k}(z+h_k\eta+h_ke_j)-v_{h_k}(z +h_k\eta+ h_ke_i))\cdot (e_i-e_j))^{2}}{4h_k^{2}} \Bigg), \end{align*}

and that the following fundamental relation holds true

\[ I_Q \geq \int_Q \sum_{e \in D} \alpha(e) |e(v_{h_k})e\cdot e|^{2}\, \mathrm{d} x. \]

By definition of piecewise constant strain we can thus write

(4.23)\begin{align} \sum_{e \in D} \int_{V}\alpha(e) |e(v_{h_k})e\cdot e|^{2}\, \mathrm{d} x & \leq \sum_{\substack{Q \in \mathcal{Q}^{good}_{h_{k}} \nonumber \\ Q\cap V \neq \emptyset}}\int_Q \sum_{e \in D} \alpha(e) |e(v_{h_k})e\cdot e|^{2}\, \mathrm{d} x\\ & \leq \sum_{\substack{Q \in \mathcal{Q}^{good}_{h_{k}} \\ Q\cap V \neq \emptyset}} I_Q \leq E_{h_k}(v_{h_k}) \leq \sum_{e \in D}\int_{V''}\alpha(e)|E_e^{h_k}|^{2} \, \mathrm{d} x, \end{align}

where we have denoted by $\mathcal {Q}^{good}_{h_{k}}$ the set of good hyper-cubes relative of $\mathcal {Q}^{0}_{h_k}$ relative to $J_v$. Inequality (4.23) together with (4.17) immediately implies

(4.24)\begin{equation} \limsup_{k \to \infty} \sum_{e \in D} \int_{V}\alpha(e) |e(v_{h_k})e\cdot e|^{2}\, \mathrm{d} x \leq \sum_{e \in D} \int_{V''}\alpha(e) |e(v)e\cdot e|^{2}\, \mathrm{d} x. \end{equation}

To prove $(iii)$ we fix $\xi \in \mathbb {S}^{n-1}$. To simplify the notation we denote by $A_k$ and $A'_k$ the union of bad hyper-cubes of $\mathcal {Q}^{0}_{h_k}$ relative to $J_v$ and $J_v \cap V$, respectively. By construction, $J_{v_k}$ is contained in $\partial A_k \cap V$. We proceed as follows: first we estimate the measure of the projection of $A'_k$ onto $\xi ^{\bot }$, then we show that the measure of the projection of $(A_k \setminus A'_k) \cap V$ onto $\xi ^{\bot }$ is infinitesimal as $k \to \infty$, and finally we deduce $(iii)$.

In what follows we consider only those indices $j$ for which $\psi _{j} (K_j) \Subset V$ (see (4.15)). Let us denote by $\mathcal {B}_{h_k,j}$ the set of bad hyper-cubes relative to $\psi _j (K_j)$ and let $\mathcal {B}_{h_k,j}'$ be the set of hyper-cubes for which one of their edges is contained in the set $\{x \in V \ | \ \mathrm {dist} (x,\psi _j (K_j)) \leq h_k \}$. Then, $\mathcal {B}_{h_k,j} \subseteq \mathcal {B}'_{h_k, j}$. Now fix a direction $\xi \in \mathbb {S}^{n-1}$. If we set $B''_{k,j} := \pi _\xi (\cup _{Q \in \mathcal {B}'_{h_k,j}} \overline {Q})$, we have that

(4.25)\begin{equation} \mathcal{H}^{n-1}(B''_{k,j}\setminus \pi_\xi(\psi_{j}(K_j)) ) = O(1/k). \end{equation}

Indeed, equality (4.25) follows from the fact that $B''_{k,j} \subset \{y \in \Pi ^{\xi } \ | \ \text {dist}(y,\pi _\xi (\psi _j(K_j))) \leq (1 + \sqrt {n})h_k \}$ and clearly, since $\pi _\xi (\psi _j(K_j))$ is compact, it holds true

\[ \lim_{k \to \infty} \mathcal{H}^{n-1}\Big(\{y \in \Pi^{\xi} \ | \ \text{dist}(y,\pi_\xi(\psi_j(K_j))) \leq (1 + \sqrt{n})h_k \} \setminus \pi_\xi(\psi_j(K_j))\Big) = 0. \]

In view of (4.25) given $m$ we can find $k_m$ such that for every $j \leq j_m$ and for every $k \geq k_m$

(4.26)\begin{equation} \mathcal{H}^{n-1}(B''_{k,j} \setminus \pi_\xi(\psi_{j}(K_j)) ) \leq \frac{\epsilon}{j_m}. \end{equation}

Let us define $B_{k,1} := B''_{k,1}$ and, by induction, $B_{k,j} := B''_{k,j} \setminus \cup _{l=1}^{j-1} B_{k,l}$ for every $1 < j \leq j_m$ and for every $k \geq k_m$. Notice that (4.26) implies

(4.27)\begin{equation} \mathcal{H}^{n-1}(B_{k,j} \setminus \pi_\xi(\psi_{j}(K_j)) ) \leq \frac{\epsilon}{j_m} \quad \text{for }1 \leq j \leq j_{m}\text{ and }k \geq k_{m}.\end{equation}

Now for every $k \geq k_m$, by construction we have that if $Q \in \mathcal {B}'_{h_{k},j}$ for some $1 \leq j \leq j_m$, then $\pi _\xi (\overline {Q}) \subset \bigcup _{j=1}^{j_m} B_{k,j}$. Therefore, we can use (4.14) and (4.27) to estimate for every $k \geq k_m$

(4.28)\begin{align} & \mathcal{H}^{n-1} \Bigg( \Bigg( \bigcup_{j=1}^{j_{m}} \bigcup_{ Q \in \mathcal{B}_{h_k,j} } \pi_\xi ( \overline{Q} ) \Bigg) \setminus \pi_{\xi}( J_{v} \cap V ) \Bigg)\nonumber\\ & \quad \leq \mathcal{H}^{n-1} \Bigg ( \Bigg( \bigcup_{j=1}^{j_{m} } \bigcup_{ Q \in \mathcal{B}_{h_k,j} } \pi_\xi ( \overline{Q} ) \Bigg) \setminus \Bigg(\bigcup_{j=1}^{\infty} \pi_{\xi}(\psi_{j}(K_{j})) \Bigg) \Bigg ) \nonumber\\ & \quad \leq \mathcal{H}^{n-1} \Bigg ( \Bigg(\bigcup_{j=1}^{j_{m}} \bigcup_{Q \in \mathcal{B}'_{h_k,j}} \pi_\xi(\overline{Q}) \Bigg) \setminus \Bigg(\bigcup_{j=1}^{\infty} \pi_{\xi}(\psi_{j}(K_{j})) \Bigg) \Bigg ) \nonumber\\ & \quad \leq \mathcal{H}^{n-1} \Bigg ( \Bigg( \bigcup_{j=1}^{j_{m}} B_{k, j} \Bigg)\setminus \Bigg(\bigcup_{j=1}^{\infty} \pi_{\xi}(\psi_{j}(K_{j})) \Bigg) \Bigg ) \nonumber\\ & \quad \leq \sum_{j = 1}^{j_m} \mathcal{H}^{n-1} \Big(B_{k, j} \setminus \pi_{\xi}( \psi_{j}(K_{j})) \Big) \leq \epsilon. \end{align}

To estimate the $\mathcal {H}^{n-1}$-measure of the projection of the bad hyper-cubes relative to $J_v \cap V$ which do not belong to $\mathcal {B}_{h_k,j}$ for some $1 \leq j \leq j_m$, we can notice that such hyper-cubes are contained in the family of bad hyper-cubes relative to $\Gamma _{j_{m}} = (J_v\cap V) \setminus \cup _{j \leq j_m} \psi _j(K_j)$. If we denote by $A'_{h_{k}, j_{m}}$ the union of such bad hyper-cubes, we can use relation (4.21) to write

(4.29)\begin{align} \limsup_{k \to \infty} \mathcal{H}^{n-1}(\pi_\xi(\partial A'_{h_{k}, j_{m}})) & \leq \limsup_{k \to \infty} \mathcal{H}^{n-1}(\pi_\xi(\partial A_{h_{k}, j_{m}})) \\ & \leq \limsup_{k \to \infty} \mathcal{H}^{n-1}(\partial A_{h_{k}, j_{m}}) =0, \nonumber \end{align}

where in the first inequality we have used the following general fact

\[ A' \subset A \Rightarrow \pi_\xi(\partial A') \subset \pi_\xi(\partial A), \]

for every couple of sets $A',A \subset \mathbb {R}^{n}$ with $A' \subset A$ and $A$ bounded. Now we define

\begin{align*} \mathcal{A}^{1}_k & := \{Q \in \mathcal{Q}_{h_k}^{0} : Q \text{ is a bad hyper-cubes for } J_v, \, Q \cap V \neq \emptyset,\\ & \qquad \overline{Q} \cap (V \setminus (\partial V)_{2nh_k}) \neq \emptyset \} \\ \mathcal{A}^{2}_k & := \{Q \in \mathcal{Q}_{h_k}^{0} : Q \text{ is a bad hyper-cubes for } J_v, \, Q \cap V \neq \emptyset, \\ & \qquad \overline{Q} \cap (V \setminus (\partial V)_{2nh_k}) = \emptyset\}. \end{align*}

Notice that if $Q \in \mathcal {A}^{1}_k$ then $Q \subset V$ and $\overline {Q} \cap (\partial V)_{h_k} = \emptyset$; but this implies that actually $Q$ is a bad hyper-cube relative to $J_v \cap V$. Namely, the following implication holds true

(4.30)\begin{equation} Q \in \mathcal{A}_k^{1} \Rightarrow Q \subset A'_k. \end{equation}

On the other hand, if $Q \in \mathcal {A}^{2}_k$ then $Q$ is a bad hyper-cube relative to $J_v$ such that $Q \subset (\partial V)_{2nh_k}$ which means that each of its edges is contained in $(\partial V)_{2nh_k}$. A similar argument to the proof of (3’) [Reference Iurlano30, theorem 3.5] shows that there exists a dimensional constant $c>0$ for which

\[ (\# \mathcal{A}^{2}_k) h_{k}^{n-1} \leq c E^{h_k}_2( (\partial V)_{2nh_k}). \]

In particular we can infer

\[ \mathcal{H}^{n-1} \Bigg(\partial \Bigg(\bigcup_{Q \in \mathcal{A}^{2}_k} Q \Bigg) \Bigg) \leq (\# \mathcal{A}^{2}_k) h_{k}^{n-1} \leq c E^{h_k}_2( (\partial V)_{2nh_k}). \]

Condition (4.20) ensures that

(4.31)\begin{equation} \lim_{k \to \infty} \, \mathcal{H}^{n-1} \Bigg(\partial \Bigg(\bigcup_{Q \in \mathcal{A}^{2}_k} Q \Bigg) \Bigg) =0. \end{equation}

Every bad hyper-cube relative to $J_v$ which has non-empty intersection with $V$ is contained in $\mathcal {A}^{1}_k \cup \mathcal {A}^{2}_k$. Therefore, if we set

\[ A^{1}_k := \bigcup_{Q \in \mathcal{A}_k^{1}} Q \text{ and } A^{2}_k:= \bigcup_{Q \in \mathcal{A}_k^{2}} Q \]

we can give the following estimate

(4.32)\begin{align} & \mathcal{H}^{n-1} (\pi_\xi(\partial A_k \cap V) \setminus \pi_\xi(J_v \cap V))\\ & \quad\leq \mathcal{H}^{n-1}(\pi_\xi(\partial A^{1}_k) \setminus \pi_\xi(J_v \cap V)) + \mathcal{H}^{n-1}(\pi_\xi(\partial A^{2}_k )) \nonumber\\ & \quad\leq \mathcal{H}^{n-1}(\pi_\xi(\partial A'_k)\setminus \pi_\xi(J_v \cap V)) + \mathcal{H}^{n-1}(\pi_\xi(\partial A^{2}_k)), \nonumber \end{align}

where for the last inequality we have used (4.30) to deduce that $\pi _\xi (\partial A^{1}_k) \subset \pi _\xi (\partial A'_k)$. We estimate separately the limsup of the last two terms of (4.32). Concerning the first term we can use implication (4.30) to write

\begin{align*} \mathcal{H}^{n-1}(\pi_\xi(\partial A'_k)\setminus \pi_\xi(J_v \cap V)) & \leq \mathcal{H}^{n-1} (\pi_\xi ( \partial A_{h_{k}, j_{m}} ) ) \\ & \quad+ \mathcal{H}^{n-1} \Bigg( \Bigg( \bigcup_{j=1}^{j_{m} } \bigcup_{ Q \in \mathcal{B}_{h_k,j} } \pi_\xi ( \overline{Q} ) \Bigg) \setminus \pi_{\xi}( J_{v} \cap V ) \Bigg), \end{align*}

for every $m$, where we have used that

\[ \pi_\xi(\partial (A'_k \setminus A_{h_k,j_m})) \subset \bigcup_{j=1}^{j_{m} } \bigcup_{ Q \in \mathcal{B}_{h_k,j} } \pi_\xi ( \overline{Q} ). \]

Hence, we can make use of (4.28) and (4.29) to write

(4.33)\begin{equation} \limsup_{k \to \infty}\mathcal{H}^{n-1} (\pi_\xi (\partial A'_k) \setminus \pi_{\xi}(J_{v} \cap V) )\leq O(1/m) + \epsilon. \end{equation}

The second term on the right-hand side of (4.32) can be estimated by using (4.31), i.e.

(4.34)\begin{equation} \limsup_{k \to \infty} \mathcal{H}^{n-1}(\pi_\xi(\partial A^{2}_k)) \leq \limsup_{k \to \infty} \mathcal{H}^{n-1} \Bigg(\partial \Bigg(\bigcup_{Q \in \mathcal{A}^{2}_k} Q\Bigg) \Bigg) =0. \end{equation}

Thanks to (4.33)–(4.34) and the arbitrariness of $m \in \mathbb {N}$ and $\epsilon >0$ we obtain from (4.32)

\[ \lim_{k \to \infty}\mathcal{H}^{n-1}(\pi_\xi(\partial A_k \cap V) \setminus \pi_\xi(J_v \cap V)) =0. \]

Finally, $(iii)$ is proved since $\overline {J_{v_k}} \subset \partial A_k \cap V$.

Now we are in position to prove $(ii)$. Thanks to the arbitrariness of the sets $V' \Subset V' \Subset V'$ we can consider sequences $V'''_{l+1} \Subset V''_{l+1} \Subset V'_{l+1} \Subset V'''_l \Subset V''_l \Subset V'_l$ and $V \Subset V'''_l$, $l=1,2,\dotsc$ with $\cap _l V'''_l = \overline {V}$. As above we associate to every $l$ an approximating sequence $v^{l}_{h_k}$ satisfying $(i)$, $(iii)$–$(v)$, and (4.24). By eventually using a diagonal argument we can find a sequence which with abuse of notation we still denote by $v_{h_k}$ still preserving items $(i)$, $(iii)$–$(v)$ but with the following refinement of (4.24)

(4.35)\begin{equation} \limsup_{k \to \infty} \sum_{e \in D} \int_{V}\alpha(e) |e(v_{h_k})e\cdot e|^{2}\, \mathrm{d} x \leq \sum_{e \in D} \int_{V}\alpha(e) |e(v)e\cdot e|^{2}\, \mathrm{d} x.\end{equation}

This immediately gives $(ii)$ and the conclusion of the proof.

Proof. Combining the fact that $e_{n,n}(u) = 0$ with $(\nu _{u})_n =0$ we easily deduce that $D_n u_n =0$, so that $u_n$ does not depend on $x_n$.

To show formula (4.36) we consider a Lipschitz-regular open set $\omega ' \Subset \omega$ such that $\mathcal {H}^{n-1}( ( \partial \omega ' \times (-\frac 12, \frac 12)) \cap J_u ) = 0$. For $0 < \delta < \frac 12$, we apply proposition 4.4 to the function $u$ on the open sets $\omega \times (-\frac 12, \frac 12)$ and $\omega ' \times (- \delta, \delta )$, taking care to have chosen $\delta >0$ such that $\mathcal {H}^{n-1}(\partial (\omega ' \times (- \delta, \delta )) \cap J_u)=0$ (a.e. choice of $\delta$ does the job). We denote by $(u_h)_h \subset GSBD^{2} (\omega ' \times (- \delta, \delta )) \cap W^{1,\infty }(\omega ' \times (- \delta, \delta ) \setminus \overline {J_{u_h}}; \mathbb {R}^{n})$ the approximating sequence given by proposition 4.4.

First of all notice that since $(\nu _u)_n=0$, by property $(iii)$ of proposition 4.4 we know that $\mathcal {H}^{n-1}(\pi _n(\overline {J_{u_h}})) \to 0$ as $h \to \infty$. By passing eventually through a subsequence we may suppose $\sum _h \mathcal {H}^{n-1}(\pi _n(\overline {J_{u_h}})) < \infty$. Hence, if we define

\[ A_h := \bigcup_{k \geq h} \pi_n (\overline{J_{u_k}}) \quad\text{ and }\quad A := \bigcap_{h=1}^{\infty} A_h, \]

then $\mathcal {H}^{n-1}(A) =0$. Moreover, from $(i)$ and $(iv)$ of proposition 4.4 we deduce that there exists a set $I \subset (-\frac 12,\frac 12)$ with $\mathcal {H}^{1}(I)=0$ such that for $\alpha = 1, \dotsc, n-1$ the following holds true:

1. (1) $\displaystyle \lim _{h \to \infty } \int _{ \omega '} |u_h(x',x_n) - u(x',x_n)| \wedge 1 \, \mathrm {d} \mathcal {H}^{n-1}(x') =0$, $x_n \in (-\frac 12, \frac 12) \setminus I$;

2. (2) $\displaystyle \vphantom {\int } \lim _{h \to \infty } \int _{ \omega '} |Tr ( (u_h)_\alpha ) (x', - \delta ) - Tr (u_\alpha ) (x',- \delta )| \wedge 1 \, \mathrm {d} \mathcal {H}^{n-1}(x') =0$.

We claim that for every $t_1,t_2 \in (-\frac 12, \frac 12 ) \setminus I$ we have

(4.37)\begin{equation} \frac{u_{\alpha}(x',t_1) - Tr (u_\alpha) (x', - \delta)}{(t_1 + \delta )} =\frac{u_{\alpha}(x',t_2) - Tr (u_\alpha) (x', - \delta)}{(t_2 + \delta )} \quad \mathcal{H}^{n-1}\text{-a.e. in }\omega'. \end{equation}

To show (4.37) fix $\epsilon >0$. We use conditions (1) and (2) together with Egoroff's Theorem to deduce that, up to subsequences, there exists a measurable set $E \subset \omega '$ with $\mathcal {H}^{n-1}(\omega ' \setminus E) \leq \epsilon$ such that

(4.38)\begin{align} & \lim_{h \to \infty} \|u_h({\cdot},t_1) -u({\cdot},t_1)\|_{L^{\infty}(\omega'\setminus E)} =0, \end{align}

(4.39)\begin{align} & \lim_{h \to \infty} \|u_h({\cdot},t_2) -u({\cdot},t_2)\|_{L^{\infty}(\omega'\setminus E)} =0, \end{align}

(4.40)\begin{align} & \lim_{h \to \infty} \|Tr((u_h)_\alpha) ({\cdot}, - \delta ) -Tr(u_\alpha) ({\cdot}, - \delta )\|_{L^{\infty}(\omega'\setminus E)} =0. \end{align}

Now let $x' \in \omega ' \setminus (A \cup E)$. Then, there exists $h$ for which $x' \notin A_h$, that is, $x' \in \bigcap _{k \geq h} [ \omega ' \setminus \pi _n(\overline {J_{u_k}})]$. Therefore, being $\pi _n(\overline {J_{u_k}})$ closed sets, for every $k \geq h$ there exists $r >0$ (depending on $k$) for which

(4.41)\begin{equation} B^{n-1}_{r}(x') \times ( - \delta, \delta) \cap \overline{J_{u_k}} = \emptyset, \end{equation}

where $B^{n-1}_{r}(x') \subseteq \omega '$ denotes here the $(n-1)$-dimensional ball of radius $r$ and centre $x'$. In particular, being $u_n$ independent of $x_n$, by (4.41) and by $(v)$ of proposition 4.4 we have that the approximating functions $u_k$ is such that $(u_k)_n$ does not depend on $x_n$ in the set $B^{n-1}_r(x') \times ( - \delta, \delta )$. Moreover, since $u_k$ is Lipschitz continuous on $B_{r}^{n-1}(x') \times ( - \delta, \delta )$, we can apply the Fundamental Theorem of Calculus on the segment $\{x'\} \times ( - \delta,t_1)$ $(x_n < \delta )$ to deduce that, for $\alpha = 1, \ldots, n-1$,

\[ (u_k)_\alpha(x',t_1) - Tr \Big ( (u_k)_\alpha \Big) (x', - \delta) = 2 \int_{-\delta}^{t_1} e_{\alpha,n}(u_k)(x',t) \, \mathrm{d} t - (t_1 + \delta) D_{\alpha}(u_k)_n(x'). \]

Hence, by using (4.38), (4.40), the convergence $(ii)$ of proposition 4.4, and the fact that $e_{\alpha,n}(u) =0$, we can take the integral on an arbitrary measurable set $B \subset \omega ' \setminus (A \cup E)$ on both sides of the previous inequality and let $k \to \infty$ to deduce that

(4.42)\begin{align} \int_{B} \frac{u_\alpha(x',t_1) - Tr ( u_\alpha) (x', - \delta)}{t_1 + \delta} \, \mathrm{d} \mathcal{H}^{n-1}(x') = \lim_{k \to \infty} \int_{B} D_\alpha (u_k)_n(x') \, \mathrm{d} \mathcal{H}^{n-1}(x'). \end{align}

Notice that the uniform convergence (4.38)–(4.40) together with the fact that $u_k \in W^{1,\infty }([\omega ' \times ( - \delta, \delta )] \setminus \overline {J_{u_h}};\mathbb {R}^{n})$ guarantee that the integrand in the left-hand side of (4.42) belongs to $L^{1}(\omega ' \setminus (A \cup E))$. Thanks to (4.39), the same argument shows that for every measurable set $B \subset \omega ' \setminus (A \cup E)$ it holds true

(4.43)\begin{align} \int_{B} \frac{u_\alpha(x',t_2) - Tr ( u_\alpha) (x', - \delta)}{t_2+\delta} \, \mathrm{d} \mathcal{H}^{n-1}(x') = \lim_{k \to \infty} \int_{B} D_\alpha (u_k)_n(x') \, \mathrm{d} \mathcal{H}^{n-1}(x'). \end{align}

Finally, putting together (4.42) with (4.43) we deduce that

\begin{align*} & \frac{u_\alpha(x',t_1) - Tr ( u_\alpha) (x', - \delta)}{t_1 + \delta}\\ & \quad = \frac{u_\alpha(x',t_2) - Tr ( u_\alpha) (x', - \delta)}{t_2 + \delta}, \quad \mathcal{H}^{n-1}\text{-a.e. in } \omega' \setminus (A \cup E). \end{align*}

Letting $\epsilon \searrow 0$ in the construction of $E$, we deduce (4.37) since $\mathcal {H}^{n-1}(A) = 0$.

Now fix $t \in (-\frac 12, \frac 12) \setminus I$ and define the measurable set

\begin{align*} H & := \Bigg\{x \in \omega' \times (-\delta, \delta) \ | \ \frac{u_{\alpha}(x',x_n) - Tr (u_\alpha) (x', - \delta)}{(x_n + \delta)}\\ & \quad \times =\frac{u_{\alpha}(x',t) - Tr (u_\alpha) (x', - \delta)}{(t + \delta)} \Bigg\}. \end{align*}

We claim that $H$ has full measure in $\omega ' \times ( - \delta, \delta )$. Indeed by using Fubini's Theorem we can write

\[ \mathcal{L}^{n}(H) = \int_{- \delta}^{\delta} \mathcal{H}^{n-1}(\{x' \in \omega \ | \ (x',x_n) \in H \}) \, \mathrm{d} x_n, \]

which immediately implies our claim thanks to (4.37). By applying again Fubini's Theorem we infer that

\[ \mathcal{H}^{1}(\{x_n \in (-\delta, \delta) \ | \ (x',x_n) \in H \}) = 2\delta \quad \mathcal{H}^{n-1}\text{-a.e. }x' \in \omega'. \]

Thus, defining

\[ \psi_{\alpha}^{\delta}(x') := \frac{ Tr (u_\alpha ) (x', - \delta) - u_\alpha(x',t) }{(t + \delta)} \quad \text{for }\mathcal{H}^{n-1}\text{-a.e. }x' \in \omega', \]

we obtain exactly that for $\mathcal {L}^{n}\text {-a.e. }x=(x',x_n) \in \omega ' \times (- \delta, \delta )$

(4.44)\begin{equation} u_\alpha(x',x_n) = Tr(u_\alpha)(x', - \delta) -(x_n + \delta) \psi_\alpha^{\delta}(x'),\end{equation}

for every $\alpha = 1, \dotsc, n-1$. Moreover, since $Tr (u_\alpha ) (x', - \delta ) \to Tr ( u_\alpha ) (x', -\frac 12)$ as $\delta \to \frac {1}{2}^{+}$, defining

\[ \psi_{\alpha}(x') := \frac{ Tr ( u_\alpha) (x', - \frac 12 ) - u_\alpha(x',t) }{t + \frac 12} \quad \text{for }\mathcal{H}^{n-1}\text{-a.e. }x' \in \omega' \]

and passing to the limit as $\delta \to \frac 12^{+}$ in (4.44) (this can be done since a.e. $\delta >0$ is admissible) we obtain (4.36) for $\mathcal {L}^{n}$-a.e. $(x',x_n) \in \omega ' \times (-\frac 12,\frac 12)$. Finally, (4.36) is achieved by letting $\omega ' \nearrow \omega$.

Proof of proposition 4.8. By [Reference Dal Maso17, theorem 4.19] we know that for $\mathcal {L}^{1}$-a.e. $x_n \in (-\frac 12,\frac 12)$ it holds true

\[ (u_1({\cdot},x_n), \dotsc, u_{n-1}({\cdot},x_n)) \in GSBD^{2} (\omega). \]

In order to simplify the notation, set $w(x',x_n) := (u_1(x',x_n), \dotsc, u_{n-1}(x',x_n))$. Thus, by (4.36) there exist $y_n \neq z_n$ such that

\[ \frac{w(x',y_n) - w(x',z_n)}{(z_n - y_n)}= \psi(x') \in GSBD^{2} (\omega), \]

which in turn, by using again formula (4.36), also implies $v \in GSBD^{2} (\omega )$. This gives the second part of the proposition. In particular, we notice that $w(\cdot, x_{n}) \in GSBD^{2}(\omega )$ for every $x_{n} \in (-\frac 12, \frac 12)$.

In order to prove $J_u = \Gamma ' \times (-\frac 12,\frac 12)$ for some $\Gamma ' \subseteq \omega$, it is enough to prove that for $\mathcal {H}^{n-1}$-a.e. $x= (x', x_{n}) \in J_u$ we have

(4.46)\begin{equation} \mathcal{H}^{1} \Bigg( \Bigg(\{x'\} \times \Bigg(- \frac 12, \frac 12 \Bigg)\Bigg) \cap J_u \Bigg) =1. \end{equation}

Suppose $x= (x',x_n) \in J_u$. Since, by proposition 4.7, $u_{n}$ does not depend on $x_{n}$, (4.46) is satisfied whenever $x' \in J_{u_{n}}$. Thus, without loss of generality we may assume $x' \notin J_{u_{n}}$. Then, there are two possibilities:

1. (1) there exists $y_n \in (-\frac 12, \frac 12) \setminus \{x_{n}\}$ such that $(x',y_n) \in J_u$;

2. (2) $(x',t) \notin J_u$ for every $t \neq x_n$.

In case (1), we further distinguish two subcases: either $\nu _u((x',y_n)) = \pm \nu _u((x',x_n))$ or $\nu _u((x',y_n)) \neq \pm \nu _u((x',x_n))$. In the first case, by using formula (4.36) together with the fact that $u_n$ does not depend on $x_n$ we have

(4.47)\begin{equation} \frac{u(x',t) - u(x',s)}{s-t} = (\psi(x'),0) \quad \text{for } (x',t,s) \in \omega \times \Bigg(-\frac 12,\frac 12 \Bigg) \times \Bigg(-\frac 12,\frac 12 \Bigg). \end{equation}

This implies that $x'$ is a point of approximate continuity for $\psi$ or a jump point for $\psi$ with $\nu _{\psi }(x') = \pm \nu _u(x',x_n)$. In particular, the last relation follows from (4.47) written for $(t, s) = (y_{n}, x_{n})$, from the equality $\nu _u((x',y_n)) = \pm \nu _u((x',x_n))$, and from the fact that $\psi$ does not depend on $x_{n}$ and $(\nu _{u})_{n} = 0$.

Suppose now that $x'$ is a jump point of $\psi$ (in the case of a point of approximate continuity one can argue in the very same way). Then, there exist $a \neq b \in \mathbb {R}^{n}$ and $a' \neq b' \in \mathbb {R}^{n-1}$ such that

\[ u(x + ry) \to a\mathbb{1}_{\{\nu_u(x) \cdot z >0\}}(y) + b\mathbb{1}_{\{-\nu_u(x) \cdot z >0\}}(y), \]

locally in $\mathcal {L}^{n}$-measure as $r \to 0^{+}$, and

\[ \psi(x' + ry') \to a'\mathbb{1}_{\{\nu_u(x) \cdot z' >0\}}(y') + b'\mathbb{1}_{\{-\nu_u(x) \cdot z' >0\}}(y'), \]

locally in $\mathcal {H}^{n-1}$-measure as $r \to 0^{+}$. These two convergences imply that if we set $x_0 := (x',t)$ with $t \neq x_n$, by using

\[ \frac{u(x',t) - u(x',x_n)}{x_n - t} = (\psi(x'),0), \]

we deduce

(4.48)\begin{align} u(x_0 + ry) & \to \ [a + (x_n - t) ( a', 0) ]\mathbb{1}_{\{\nu_u(x) \cdot z >0\}}(y)\\ & \quad+ [b + (x_n - t) ( b', 0) ]\mathbb{1}_{\{-\nu_u(x) \cdot z >0\}}(y), \nonumber \end{align}

locally in $\mathcal {H}^{n-1}$-measure as $r \to 0^{+}$. Since

\[ a +(x_n -t) (a', 0) \neq b + (x_n-t) ( b', 0) \quad \text{for a.e. }t \in (-\frac 12, \frac 12 ) \]

the convergence in (4.48) implies that $(x',t) \in J_u$ for a.e. $t \in (-\frac 12, \frac 12)$. Hence, (4.46) is satisfied if $(1)$ holds and $\nu _u(x',y_n) = \pm \nu _u(x',x_n)$.

In order to show that the set of $x'$ satisfying $(1)$ and $\nu _u( x',y_n) \neq \pm \nu _u(x',x_n)$ is $\mathcal {H}^{n-2}$-negligible, we recall that $\psi$, $\frac {w(\cdot, x_{n})}{y_{n} - x_{n}}$, $\frac {w(\cdot, y_{n})}{y_{n} - x_{n}} \in GSBD^{2}(\omega )$, and notice that, since $x' \in J_{\psi }\setminus J_{u_{n}}$ and $(\nu _{u})_{n}=0$, it holds $x' \in J_{w(\cdot, x_{n})} \cap J_{w(\cdot, y_{n})}$ and $\nu _{u} (x', x_{n}) = (\nu _{w (\cdot, x_{n})}(x'), 0)$. Hence, applying for instance [Reference Ambrosio, Fusco and Pallara5, proposition 2.85],

\begin{align*} 0 & = \mathcal{H}^{n-2} ( \{ x' \in J_{w({\cdot}, x_{n})} \cap J_{w({\cdot}, y_{n})}: \, \nu_{w({\cdot}, x_{n})} (x') \neq{\pm} \nu_{w({\cdot}, y_{n})} (x')\} )\\ & = \mathcal{H}^{n-2}(\{x' \in \omega \setminus J_{u_{n}} \ | \ \text{(1) holds and } \pm\nu_{u}(x',x_n)\neq \nu_{u}(x',y_n)\}). \end{align*}

Therefore, $\mathcal {H}^{n-1}$-a.e. $x$ satisfying case (1) also fulfills (4.46).

Finally, suppose (2) holds. Such points are a subset of $J_u$, denoted here by $A$, satisfying $\mathcal {H}^{0}((A)_{x'}^{e_n}) =1$ for every $x' \in \pi _{n} (A)$. Since $(\nu _u)_{n} = 0$, by the Area Formula we have

\[ \mathcal{H}^{n-1} (\pi_{n}(A)) = \int_{\Pi^{e_{n}}} \mathcal{H}^{0}( (A)_{x'}^{e_n}) \, \mathrm{d} \mathcal{H}^{n-1} (x') = \int_{A} | \nu_{u} \cdot e_{n}| \, \mathrm{d} \mathcal{H}^{n-1} =0, \]

and we conclude (4.46).

Proof of theorem 3.2. First we prove that $u_n$ is approximately differentiable $\mathcal {L}^{n}$-a.e. in $\Omega _{1}$. In view of [Reference Federer21, theorem 3.1.4] it is enough to prove that the approximate partial derivatives $\partial _i u_n$ exist $\mathcal {L}^{n}$-a.e. in $\Omega _{1}$ for every $i =1,\dotsc,n$. Since we already know that $u_n$ does not depend on $x_n$, we need only to prove $\partial _\alpha u_n$ exist $\mathcal {L}^{n}$-a.e. in $\Omega _{1}$ for every $\alpha = 1, \dotsc, n-1$.

Given $\alpha$, we notice that since $u \in GSBD^{2} (\Omega _{1})$, setting $\xi := (e_n + e_\alpha )/\sqrt {2}$ we have that $\partial _\xi (u\cdot \xi )$ and $\partial _\alpha u_\alpha$ exist $\mathcal {L}^{n}$-a.e. in $\Omega _{1}$ and by formula (4.36) also $\partial _n u_\alpha$ exists $\mathcal {L}^{n}$-a.e. in $\Omega _{1}$.

We now claim that

(4.49)\begin{equation} \partial_\alpha u_n = 2\partial_\xi (u\cdot \xi) - \partial_\alpha u_\alpha -\partial_n u_\alpha \quad \mathcal{L}^{n}\text{-a.e. in }\Omega_{1}. \end{equation}

Indeed, up to a set of $\mathcal {L}^{n}$-measure zero we have that for every $x \in \Omega _{1}$ the following holds true:

(4.50)\begin{align} & {\mathop {\text{ap- lim}}\limits_{h \to 0}} \frac{u(x + h\xi)\cdot \xi - u(x) \cdot \xi }{h} = \partial_\xi (u \cdot \xi)(x), \end{align}

(4.51)\begin{align} & {\mathop {\text{ap- lim}}\limits_{h \to 0}} \frac{ u_\alpha(x + he_n) - u_\alpha(x)}{h} = \partial_n u_\alpha(x), \end{align}

(4.52)\begin{align} & {\mathop {\text{ap- lim}}\limits_{h \to 0}} \frac{ u_\alpha(x + he_\alpha) - u_\alpha(x) }{h} = \partial_\alpha u_\alpha(x), \end{align}

(4.53)\begin{align} & {\mathop {\text{ap- lim}}\limits_{h \to 0 }} \psi (x' + h e_{\alpha}) = \psi(x'). \end{align}

By a simple algebraic computation we can write

(4.54)\begin{align} & u_n (x + he_\alpha) - u_n(x)\nonumber\\ & \quad = u_n(x + he_\alpha) - u_n(x + he_\alpha + h e_n) + u_n(x + he_\alpha + h e_n) - u_n(x) \nonumber\\ & \quad = u_n(x + he_\alpha) -u_n(x + he_\alpha + h e_n) + \sqrt{2}u(x + h\sqrt{2} \xi) \cdot \xi - \sqrt{2}u(x) \cdot \xi \nonumber\\ & \qquad - ( u_\alpha(x + h\sqrt{2}\xi) - u_\alpha(x) ). \end{align}

By proposition 4.7, $u_{n}$ does not depend on $x_{n}$. Thus,

(4.55)\begin{equation} u_n(x + he_\alpha) -u_n(x + he_\alpha + h e_n) =0. \end{equation}

By (4.50) we have that for $\mathcal {L}^{n}$-a.e. $x \in \Omega _{1}$

(4.56)\begin{equation} {\mathop {\text{ap- lim}}\limits_{h \to 0}} \frac{\sqrt{2} u ( x + h \sqrt{2} \xi ) \cdot \xi - \sqrt{2} u(x) \cdot \xi }{h} = 2 \partial_\xi (u \cdot \xi)(x). \end{equation}

We re-write the last term on the right-hand side of (4.54) as

\begin{align*} u_\alpha(x + h\sqrt{2}\xi) - u_\alpha(x) & = u_\alpha(x + h (e_n + e_\alpha) ) - u_\alpha(x + h e_\alpha)\\ & \quad + u_\alpha(x + h e_\alpha) - u_\alpha(x). \end{align*}

Using formula (4.36) we have that

\[ u_\alpha(x + h (e_n + e_\alpha) ) - u_\alpha(x + h e_\alpha) ={-} h \psi_\alpha(x'+ h e_\alpha), \]

which implies, together with (4.53), that for $\mathcal {L}^{n}$-a.e. $x \in \Omega _{1}$

(4.57)\begin{equation} {\mathop {\text{ap- lim}}\limits_{h\to 0}} \, \frac{u_\alpha(x + h (e_\alpha + e_n) ) - u_\alpha(x + h e_\alpha)}{h} ={-} \psi_{\alpha} (x') = \partial_{n} u_{\alpha} (x), \end{equation}

where $\psi _{\alpha }$, $\alpha = 1, \ldots, n-1$ are the functions determined in (4.36). Therefore, combining (4.51), (4.52) and (4.57) we deduce that for $\mathcal {L}^{n}$-a.e. $x \in \Omega _{1}$

(4.58)\begin{equation} {\mathop {\text{ap- lim}}\limits_{h \to 0}} \, \frac{ u_\alpha ( x + h\sqrt{2} \xi ) - u_\alpha(x) }{h} = \partial_n u_\alpha(x) + \partial_\alpha u_\alpha(x). \end{equation}

Inserting (4.55)–(4.58) in (4.54) we obtain (4.49).

Since $\alpha \in \{1, \dotsc, n-1\}$ was arbitrary, we deduce that $u_n$ is approximately differentiable $\mathcal {L}^{n}$-a.e. in $\Omega _{1}$. Furthermore, since $u_n$ does not depend on $x_n$, $u_n$ is approximately differentiable $\mathcal {H}^{n-1}$-a.e. on $\omega$. If we denote (with abuse of notation) $\nabla u_n = (\partial _1 u_n, \dotsc, \partial _{n-1} u_n)$, then $\nabla u_n$ is the approximate gradient of $u_{n}$.

In order to prove that $\nabla u_n \in GSBD^{2} (\omega )$, we claim that

(4.59)\begin{equation} \nabla u_n(x') = \Big(\psi_1(x'), \dotsc, \psi_{n-1}(x') \Big) \quad \text{for }\mathcal{H}^{n-1}\text{ -a.e. }x'\in \omega. \end{equation}

Once we show (4.59), the fact that $\nabla u_n \in GSBD^{2} (\omega )$ will follow from proposition 4.8. The equality (4.59) is a consequence of the hypothesis $e_{i,n}(u) = 0$ and of (4.36). The latter yields that $\partial _n u_\alpha = -\psi _\alpha$ $\mathcal {L}^{n}$-a.e.. Hence, being $e_{\alpha, n}(u)= 0$, we infer exactly $\partial _{\alpha } u_n = \psi _\alpha$ $\mathcal {L}^{n}$-a.e., which is (4.59).

In order to prove (3.12) notice that formula (4.36) becomes now

(4.60)\begin{equation} u_{\alpha}(x',x_n) = Tr ( u_\alpha) \Bigg(x', - \frac 12 \Bigg) - \Bigg( x_n + \frac 12 \Bigg) \partial_{\alpha} u_{n} (x') \quad \text{for }\mathcal{L}^{n}\text{-a.e. }(x',x_n) \in \Omega_{1}. \end{equation}

Recalling that $\Omega _{1} = \omega \times ( -\frac 12, \frac 12)$, by integrating both sides of (4.60) with respect to $x_n \in (-\frac 12,\frac 12)$ we obtain

\[ \overline{u}_{\alpha}(x') = Tr ( u_\alpha) \Bigg(x', - \frac 12 \Bigg) - \frac{1}{2}\partial_{\alpha} u_{n} (x' ) \quad \text{for }\mathcal{L}^{n}\text{-a.e. }(x',x_n) \in \Omega_{1}. \]

Combining the last two equalities we deduce exactly (3.12). The fact that $\overline {u} \in GSBD^{2} (\omega )$ simply follows now by (3.12).

We are finally left to prove that $J_u = (J_{\overline {u}} \cup J_{u_n} \cup J_{\nabla u_n}) \times (-\frac 12,\frac 12)$, for which we follow the lines of [Reference Babadjian and Henao8, proposition 5.2, step 4]. By proposition 4.8 we already know that $J_u = \Gamma ' \times (-\frac 12,\frac 12)$ for some $\Gamma ' \subset \omega$. Thus, we only need to show that $\Gamma ' = J_{\overline {u}} \cup J_{u_n} \cup J_{\nabla u_n}$ up to a set of $\mathcal {H}^{n-2}$-measure zero. First, we prove $\Gamma ' \subset J_{\overline {u}} \cup J_{u_n} \cup J_{\nabla u_n}$. By formula (3.12) and by proposition 4.7 we have that

\[ (\nabla u_{n} (x'), 0) = \frac{u(x', t) - u(x', s) }{s-t} \quad \text{for }(x', t, s) \in \omega \times \left(-\frac 12, \frac 12 \right) \times \left(-\frac 12, \frac 12\right). \]

Hence, for $\mathcal {H}^{n-2}$-a.e. $x' \in \Gamma '$, either $x' \in J_{\nabla u_n}$ or $x'$ is an approximate continuity point for $\nabla u_n$. In the first case, we clearly have $x' \in J_{\overline {u}} \cup J_{u_n} \cup J_{\nabla u_n}$.

Let us suppose, instead, that $x'$ is an approximate continuity point of $\nabla u_{n}$. By rewriting formula (3.12) in the vectorial form as

\[ u = (\overline{u}_1, \dotsc, \overline{u}_{n-1}, u_n) - x_n (\partial_1 u_n, \dotsc, \partial_{n-1} u_n, 0), \]

then, it is easy to see that, being $x'$ a point of approximate continuity for $\nabla u_n$, the fact that $(x', x_{n}) \in J_u$ for $x_{n} \in (-\frac 12, \frac 12)$ forces $x' \in J_{\overline {u}} \cup J_{u_n}$. This gives the first inclusion $\Gamma ' \subset J_{\overline {u}} \cup J_{u_n} \cup J_{\nabla u_n}$.

To prove $J_{\overline {u}} \cup J_{u_n} \cup J_{\nabla u_n} \subset \Gamma '$ we argue as follows: if $x' \in J_{u_n}$, then, by definition of $J_{u_{n}}$, we have

\[ \mathcal{H}^{1} \Bigg( \Bigg(\{x'\} \times \Bigg(-\frac 12,\frac 12 \Bigg) \Bigg) \cap J_u \Bigg) = 1. \]

Hence, we can reduce ourselves to prove the inclusion in the case $x' \in J_{\overline {u}} \cup J_{\nabla u_n}$. Since $J_{u} = \Gamma ' \times (-\frac 12, \frac 12)$, we can choose $\tilde {x}_n \in (-\frac 12, \frac 12)$ such that $v(\cdot ) := u(\cdot,\tilde {x}_n) \in GSBD^{2} (\omega )$ and $J_v = \Gamma '$ up to a set of $\mathcal {H}^{n-2}$-measure zero in $\omega$. Moreover, formula (3.12) implies that

\[ J_{\nabla u_n} \subset J_{\overline{u}} \cup J_{v} \quad \text{and} \quad J_{\overline{u}} \subset J_{\nabla u_n} \cup J_{v}. \]

Thus, we deduce that, up to an $\mathcal {H}^{n-2}$-negligible set in $\omega$,

(4.61)\begin{equation} J_{\nabla u_n} \setminus J_{\overline{u}} \subset J_v = \Gamma' \quad \text{and} \quad J_{\overline{u}} \setminus J_{\nabla u_n} \subset J_v = \Gamma'. \end{equation}

It remains to prove that

(4.62)\begin{equation} J_{\nabla u_n} \cap J_{\overline{u}} \subset \Gamma'. \end{equation}

If $x' \in J_{\nabla u_n} \cap J_{\overline {u}}$ and $J_{\nabla u_n}, J_{\overline {u}}$ have the same tangent plane at $x'$, i.e., $\nu := \nu _{\overline {u}} (x') = \pm \nu _{\nabla {u_{n}}} (x')$, for $\alpha = 1, \ldots, n-1$ there exist $\xi ^{\pm }, \eta ^{\pm } \in \mathbb {R}^{n-1}$ with $\xi ^{+} \neq \xi ^{-}$ and $\eta ^{+} \neq \eta ^{-}$ such that, by (3.12),

\begin{align*} & (u_{1}, \ldots, u_{n-1}) ((x', x_{n}) + ry) \to (\xi^{+} - x_{n} \eta^{+})\mathbb{1}_{\{\nu \cdot z >0\}}(y)\\ & \quad + (\xi^{-} - x_{n} \eta^{-})\mathbb{1}_{\{-\nu \cdot z >0\}}(y) \end{align*}

locally in $\mathcal {L}^{n}$-measure as $r \to 0$. Since $\xi ^{+} - x_{n} \eta ^{+} \neq \xi ^{-} - x_{n} \eta ^{-}$ for a.e. $x_{n} \in (-\frac 12, \frac 12)$, we deduce that $\mathcal {H}^{1} ((\{x'\} \times (-\frac 12,\frac 12)) \cap J_u)=1$ and $x' \in \Gamma '$.

Finally, applying [Reference Ambrosio, Fusco and Pallara5, proposition 2.85] to the functions $\overline {u}, x_{n} \nabla u_{n} \in GSBD^{2}(\omega )$ for $x_{n} \in (-\frac 12, \frac 12)$, we deduce that

\[ \mathcal{H}^{n-2} \Big( \{x' \in J_{\nabla u_n} \cap J_{\overline{u}}: \nu_{\overline{u}} (x') \neq{\pm} \nu_{\nabla u_{n}} (x') \} \Big) = 0. \]

This gives (4.62) and the conclusion of the theorem.

Proof of theorem 3.4. We follow here the steps of [Reference Babadjian and Henao8, theorem 5.1]. Since the convergence in measure is metrizable, we can show the $\Gamma$-convergence in terms of converging sequences. As for the $\Gamma$-liminf, for every infinitesimal sequence $\rho _{k}$, every $u\colon \Omega _{1} \to \mathbb {R}^{n}$, and every $u_{k} \in GSBD^{2}(\Omega _{1})$ such that $u_{k} \to u$ in measure and

\[ \liminf_{k\to\infty}\, \mathcal{E}_{\rho_{k}}(u_{k}) <{+}\infty, \]

we have, in view of proposition 3.1, that $u \in \mathcal {KL} (\Omega _{1})$. Furthermore, since $(\nu _{u})_{n}=0$ $\mathcal {H}^{n-1}$-a.e. in $J_{u}$, by [Reference Kholmatov and Piovano31, proposition 4.6] for every $\tilde {\rho }>0$ we have that

(4.63)\begin{align} \mathcal{H}^{n-1}(J_u) & = \int_{J_{u}} \phi_{\tilde{\rho}} (\nu_{u} ) \, \mathrm{d} \mathcal{H}^{n-1} \leq \liminf_{k \to \infty} \int_{J_{u_{k}}} \phi_{\tilde{\rho}} (\nu_{u_{k}} ) \, \mathrm{d} \mathcal{H}^{n-1}\\ & \leq \liminf_{k\to\infty} \int_{J_{u_{k}}} \phi_{\rho_{k}} ( \nu_{u_{k}}) \, \mathrm{d} \mathcal{H}^{n-1}. \nonumber \end{align}

For every $v \in GSBD^{2}(\Omega _{1})$ let us set $\overline {e}(v): = (e_{\alpha \beta }(v))_{\alpha, \beta =1}^{n-1}$. Then, by definition (3.8)–(3.9) of $\mathcal {E}_{\rho }$ we have

(4.64)\begin{equation} \begin{aligned} \int_{\Omega_1} \mathbb{C}_{0} e(u) {\, \cdot\,} e(u) \, \mathrm{d} x & \leq \liminf_{k \to \infty} \int_{\Omega_1} \mathbb{C}_{0} \overline{e}(u_{k}) {\, \cdot \,} \overline{e}(u_{k}) \, \mathrm{d} x\\ & \leq \liminf_{k\to\infty} \int_{\Omega_1} \mathbb{C} e^{\rho_{k}}(u_{k}) {\, \cdot\,} e^{\rho_{k}}(u_{k}) \, \mathrm{d} x. \end{aligned} \end{equation}

Hence, combining (4.63) and (4.64) we infer that

\[ \mathcal{E}_{0}(u) \leq \liminf_{k\to \infty}\, \mathcal{E}_{\rho_{k}} (u_{k}), \]

which in turn implies that $\mathcal {E}_{0} \leq \Gamma$-$\liminf _{\rho \to 0} \mathcal {E}_{\rho }$.

We conclude with the $\Gamma$-limsup inequality. Let $u \in GSBD^{2}( \Omega _1 )$. If $u \notin \mathcal {KL} (\Omega _1)$, then $\mathcal {E}_{0}(u) = +\infty$ and there is nothing to show. If $u \in \mathcal {KL} (\Omega _1)$, let us fix $\lambda = (\lambda _{1}, \ldots, \lambda _{n}) \in L^{2}(\Omega _{1}; \mathbb {R}^{n})$ such that

(4.65)\begin{equation} \mathbb{C}_{0} e(u) {\,\cdot\,} e(u) = \mathbb{C} (e(u))_{\lambda} {\, \cdot\,} (e(u))_{\lambda} \quad \text{ a.e. in }\Omega_{1}, \end{equation}

where we recall the notation introduced in (3.13)–(3.14). Let $h_{\rho, 1}, \ldots, h_{\rho, n} \in C^{\infty }_{c}(\Omega _{1})$ be such that

(4.66)\begin{align} & h_{\rho, \alpha} \to 2\lambda_{\alpha} \quad \text{ in }L^{2}(\Omega_{1}), \quad\text{for }\alpha \in \{1, \ldots, n-1\}, \end{align}

(4.67)\begin{align} & h_{\rho, n} \to \lambda_{n} \quad \text{ in }L^{2}(\Omega_{1}), \end{align}

(4.68)\begin{align} & \rho h_{\rho, i},\quad \rho \nabla h_{\rho, i} \to 0 \quad \text{in }L^{2}(\Omega_{1}) \text{ for }i \in \{1, \ldots, n\}. \end{align}

In particular, (4.66)–(4.68) imply that the sequences

\begin{align*} & H_{\rho, \alpha} (x', x_{n} ) := \rho \int_{0}^{x_{n}} h_{\rho, \alpha} (x', t) \, \mathrm{d} t \quad \in L^{2}(\Omega_{1}) \text{ for }\alpha \in \{1, \ldots, n-1\},\\ & H_{\rho, n} (x', x_{n} ) := \rho \int_{0}^{x_{n}} h_{\rho, n} (x', t) \, \mathrm{d} t \quad \in L^{2}(\Omega_{1}) \end{align*}

satisfy $H_{\rho, i}, \, \partial _{j} H_{\rho, i} \to 0$ in $L^{2}(\Omega _{1})$ for every $i, j \in \{1, \ldots, n\}$.

For every $x = ( x', x_{n}) \in \Omega _1$ we define

(4.69)\begin{equation} u_{\rho} (x) := u (x) + \left( H_{\rho, 1}, \ldots, H_{\rho, n}\right) (x). \end{equation}

Then, $u_{\rho } \in GSBD^{2}(\Omega _1)$, $J_{u_{\rho }} = J_{u}$ for every $\rho >0$, and $(\nu _{u_{\rho }})_{n} = 0$ on $J_{u_{\rho }}$. Moreover, we have that $u_{\rho } \to u$ in measure on $\Omega _1$.

We now write the components of $e^{\rho }(u_{\rho })$. Since $u \in \mathcal {KL}(\Omega _1)$, for every $\alpha, \beta = 1, \ldots, n-1$ we have

\begin{align*} \displaystyle e_{\alpha, \beta}^{\rho} (u_{\rho}) & = e_{\alpha, \beta} (u) +\frac 12 (\partial_{\alpha} H_{\rho, \beta} + \partial_{\beta} H_{\rho, \alpha}), \\ \displaystyle e^{\rho}_{\alpha, n} (u_{\rho}) & = \frac{1}{2} h_{\rho, \alpha}(x', x_{n}) + \frac{1}{2} \partial_{\alpha} H_{\rho, n} (x', x_{n}), \\ \displaystyle e^{\rho}_{n,n} (u_{\rho}) & = h_{\rho, n} (x', x_{n}). \end{align*}

Thus, from (4.65)–(4.69) we deduce that

\begin{align*} \lim_{\rho \to 0}\, \mathcal{E}_{\rho}(u_{\rho}) & = \lim_{\rho \to 0} \, \frac{1}{2} \int_{\Omega_1} \mathbb{C} e^{\rho} (u_{\rho}) {\, \cdot\,} e^{\rho}(u_{\rho}) \, \mathrm{d} x + \mathcal{H}^{n-1}(J_{u})\\ & = \frac{1}{2} \int_{\Omega_{1}} \mathbb{C} (e(u))_{\lambda}{\, \cdot\,} (e(u))_{\lambda} \, \mathrm{d} x + \mathcal{H}^{n-1}(J_{u}) = \mathcal{E}_{0}(u), \end{align*}

and the proof is thus complete.

Proof. We consider $\widetilde {\omega } \subseteq \mathbb {R}^{n-1}$ smooth, bounded, and such that $\omega \Subset \widetilde {\omega }$, and define $\widetilde {\Omega } := \widetilde {\omega } \times ( -\frac 12, \frac 12 )$. For every $u \in GSBD^{2}(\Omega _1)$, we consider the extension

(4.72)\begin{equation} \widetilde{u} := \left\{ \begin{array}{@{}ll} u & \text{in }\Omega_1,\\ g & \text{in }\widetilde{\Omega} \setminus\Omega_1. \end{array}\right. \end{equation}

Then, we can rewrite $\mathcal {E}^{g}_{\rho } (u)$ as

\[ \displaystyle \mathcal{E}^{g}_{\rho}(u) := \frac 12 \int_{\Omega_{1}} \mathbb{C} e^{\rho} (\tilde{u}) {\, \cdot\,} e^{\rho} (\tilde{u}) \, \mathrm{d} x + \int_{J_{\widetilde{u}}\cap \widetilde{\Omega} } \phi_{\rho}(\nu_{\widetilde{u}}) \, \mathrm{d} \mathcal{H}^{n-1}. \]

With this notation at hand, we can show the $\Gamma$-liminf inequality by following step by step the proof of theorem 3.4. Given $u_{\rho } \in GSBD^{2}(\Omega _1)$ such that $u_{\rho }$ converges in measure to $u \in GSBD^{2}(\Omega _1)$, we consider their extensions $\widetilde {u}_{\rho }, \widetilde {u} \in GSBD^{2}(\widetilde {\Omega })$. If

\[ \sup_{\rho>0} \mathcal{E}^{g}_{\rho}(u_{\rho}) <{+}\infty, \]

we deduce that $e (u_{\rho }) \rightharpoonup e(u)$ weakly in $L^{2}( \Omega _1 ; \mathbb {M}^{n}_{s})$ and $u \in \mathcal {KL} (\Omega _1)$, so that also $\widetilde {u} \in \mathcal {KL} (\widetilde {\Omega })$. Furthermore, the bulk energy satisfies

\[ \int_{\Omega_1} \mathbb{C}_{0} e (u) {\, \cdot\,} e (u) \, \mathrm{d} x \leq \liminf_{\rho \to 0} \int_{\Omega_1} \mathbb{C} e^{\rho}(\tilde{u}_{\rho}) {\, \cdot\,} e^{\rho} (\tilde{u}_{\rho}) \, \mathrm{d} x. \]

As in (4.63) we have that

\[ \mathcal{H}^{n-1} (J_{\widetilde{u}} \cap \widetilde{\Omega} ) \leq \liminf_{\rho \to 0} \int_{J_{\widetilde{u}_{\rho}} \cap \widetilde{\Omega}} \phi_{\rho} (\nu_{\widetilde{u}_{\rho}}) \, \mathrm{d} \mathcal{H}^{n-1}. \]

Noticing that $\mathcal {H}^{n-1} (J_{\tilde {u}} \cap (\widetilde {\Omega } \setminus \Omega _{1} ) ) = 0$ and

\[ J_{\tilde{u}} \cap \partial \omega \times \Bigg( -\frac 12, \frac 12 \Bigg) = \Big \{ Tr (u) \neq Tr (g) \Big \} \cap \Bigg(\partial\omega \times \Bigg( -\frac 12, \frac 12 \Bigg) \Bigg), \]

we deduce that $\mathcal {E}^{g}_{0}(u) \leq \liminf _{\rho \to 0} \mathcal {E}^{g}_{\rho }(u_{\rho })$.

A recovery sequence can be constructed as in (4.69), where we modify a function $u \in \mathcal {KL} (\Omega _{1})$ within $\Omega _{1}$ by considering $h_{\rho, i} \in C_{c}^{\infty } (\Omega _{1})$ as in (4.66)–(4.68), so that $u$ remains unchanged on $\partial \omega \times (-\frac 12, \frac 12 )$.

Proof. Let $\widetilde {\omega }$ and $\widetilde {\Omega }$ be as in the proof of corollary 4.10. Along the proof, we denote by $\partial ^{*}E$ and $\widetilde {\partial }^{*} E$ the reduced boundary of a set $E \subseteq \widetilde {\Omega }$ in $\Omega$ and $\widetilde {\Omega }$, respectively.

The existence of the set $A$ and of a limit function $u \in GSBD^{2}(\Omega _1)$ such that (4.75)–(4.76) holds follows from theorem 4.11 applied to the sequence $\widetilde {u}_{\rho } \in GSBD^{2}(\widetilde {\Omega })$ defined as in (4.72). Precisely, there exists $A \subseteq \widetilde {\Omega }$ and $\widetilde {u} \in GSBD^{2}(\widetilde {\Omega })$ such that (4.75)–(4.76) hold for $\widetilde {u}_{\rho }$ and $\widetilde {u}$ in $\widetilde {\Omega }$. Since $\widetilde {u}_{\rho } = \widetilde {u} = g$ in $\widetilde {\Omega } \setminus \Omega _1$ and $g \in H^{1}(\mathbb {R}^{n}; \mathbb {R}^{n})$, we clearly have that $u:= \tilde {u} \mathbb{1}_{\Omega _{1}} \in GSBD^{2}(\Omega _1)$ and $A \subseteq \overline {\Omega }_1$.

Let us denote by $\nu _{\tilde {u} \cup \widetilde {\partial }^{*} A}$ the approximate unit normal to $J_{\tilde {u}} \cup \widetilde {\partial }^{*} A$. By [Reference Kholmatov and Piovano31, proposition 4.6], $\widetilde {u}$ and $A$ are such that

(4.78)\begin{equation} \int_{J_{\widetilde u} \cup \widetilde{\partial}^{*} A} \phi( x, \nu_{\widetilde u \cup \widetilde{\partial}^{*}A}) \, \mathrm{d} \mathcal{H}^{n-1} \leq \liminf_{\rho \to 0} \int_{J_{\widetilde{u}_{\rho}}} \phi(x, \nu_{\widetilde{u}_{\rho}}) \, \mathrm{d} \mathcal{H}^{n-1} \end{equation}

for every $\phi \in C(\widetilde {\Omega } \times \mathbb {R}^{n})$ such that $\phi (x, \cdot )$ is a norm on $\mathbb {R}^{n}$ for every $x \in \widetilde {\Omega }$ and

\[ c_{1} | \nu| \leq \phi (x, \nu) \leq c_{2} | \nu| \quad \text{for every }x \in \widetilde{\Omega}\text{ and every }\nu \in \mathbb{R}^{n}, \]

for some $0 < c_{1} \leq c_{2} < + \infty$.

Recalling (3.7), we deduce from (4.78) that for every $\tilde {\rho }>0$

(4.79)\begin{align} \int_{J_{\widetilde u} \cup \widetilde{\partial}^{*} A} & \phi_{\tilde{\rho}} ( \nu_{\widetilde u \cup \widetilde{\partial}^{*} A} )\, \mathrm{d} \mathcal{H}^{n-1} \leq \liminf_{\rho \to 0} \int_{J_{\widetilde{u}_{\rho}}} \phi_{\tilde{\rho}} ( \nu_{\widetilde{u}_{\rho}}) \, \mathrm{d} \mathcal{H}^{n-1}\\ & \leq \liminf_{\rho \to 0} \int_{J_{\widetilde{u}_{\rho}}} \phi_{\rho} (\nu_{\widetilde{u}_{\rho}}) \, \mathrm{d} \mathcal{H}^{n-1} \nonumber\\ & = \liminf_{\rho \to 0} \int_{J_{u_{\rho}}} \phi_{\rho} ( \nu_{u_{\rho}}) \, \mathrm{d} \mathcal{H}^{n-1} \nonumber\\ & \qquad + \mathcal{H}^{n-1} \Bigg( \Big \{ Tr (u_{\rho}) \neq Tr (g) \Big \} \cap \Bigg( \partial\omega \times \Bigg( - \frac 12, \frac 12 \Bigg) \Bigg ) \Bigg) <{+}\infty. \nonumber \end{align}

Passing to the limsup in (4.79) as $\tilde {\rho } \to 0$ we deduce that $(\nu _{\widetilde {\partial }^{*} A})_{n} = (\nu _{u})_{n} = 0$ $\mathcal {H}^{n-1}$-a.e. in $J_{u} \cup \widetilde {\partial }^{*} A$. It follows that there exists $A' \subseteq \omega$ such that (4.74) holds.

As a consequence of (4.73), we infer that $e_{i,n}(u) = 0$ in $\Omega _{1}$ for every $i=1, \ldots, n$. Hence, $u \in \mathcal {KL} (\Omega _{1})$. Taking into account that $(\nu _{u})_{n} = ( \nu _{\widetilde {\partial }^{*} A})_{n} = 0$ and that

\[ J_{\tilde{u}} \cap \partial \omega \times \left( -\frac 12, \frac 12 \right) = \left\{ Tr (u) \neq Tr (g) \right\} \cap \Big( \partial\omega \times \left( -\frac 12, \frac 12 \right) \Big), \]

we infer (4.77) by rewriting (4.79), and the proof is thus concluded.

Proof. Let $u_{\rho }$ be as in the statement of the corollary. Then, it is easy to check that (4.73) is satisfied. Hence, theorem 4.12 implies that there exist $A$ and $u \in \mathcal {KL} (\Omega _1)$ such that (4.74)–(4.77) hold. The minimality of $u$ follows from theorem 3.4 by a $\Gamma$-convergence argument. Indeed, by (4.76)–(4.77) we have that

(4.81)\begin{align} & \int_{\Omega_1} \mathbb{C}_{0} e(u) {\, \cdot\,} e(u) \, \mathrm{d} x \leq \liminf_{\rho \to 0} \int_{\Omega_1} \mathbb{C} e^{\rho}(u_{\rho}) {\, \cdot\,} e^{\rho} (u_{\rho}) \, \mathrm{d} x, \end{align}

(4.82)\begin{align} & \vphantom{\int} \mathcal{H}^{n-1}(J_{u} \cup \partial^{*} A ) + \mathcal{H}^{n-1} \Bigg( \Big \{ Tr (u) \neq Tr (g) \Big \} \cap \Bigg( \partial\omega \times \Bigg(- \frac 12, \frac 12 \Bigg) \Bigg) \Bigg) \nonumber\\ & \quad \leq \liminf_{\rho \to 0} \, \int_{J_{u_{\rho}}} \phi_{\rho}( \nu_{u_{\rho}}) \, \mathrm{d} \mathcal{H}^{n-1}\notag\\ & \qquad + \mathcal{H}^{n-1} \Bigg( \Big \{ Tr (u_{\rho}) \neq Tr (g) \Big \} \cap \Bigg( \partial\omega \times \Bigg( - \frac 12, \frac 12 \Bigg) \Bigg) \Bigg). \end{align}

Thanks to corollary 4.10, for every $v \in \mathcal {KL} (\Omega _{1})$ there exists a sequence $v_{\rho } \in GSBD^{2}(\Omega _{1})$ converging to $v$ in measure such that

(4.83)\begin{equation} \mathcal{E}^{g}_{0} (v) = \lim_{\rho \to 0} \mathcal{E}^{g}_{\rho}(v_{\rho}). \end{equation}

Combining (4.81), (4.82), and (4.83) we deduce that

\begin{align*} \mathcal{E}^{g}_{0}(u) & \leq \int_{\Omega_1} \mathbb{C}_{0} e(u) {\, \cdot\,} e(u) \, \mathrm{d} x + \mathcal{H}^{n-1}(J_{u} \cup \partial^{*} A )\\ & \qquad + \mathcal{H}^{n-1} \Bigg( \Big\{ Tr (u) \neq Tr (g) \Big\} \cap \Bigg( \partial\omega \times \Bigg( - \frac 12, \frac 12 \Bigg) \Bigg) \Bigg)\\ & \leq \liminf_{\rho \to 0} \mathcal{E}^{g}_{\rho} (u_{\rho}) \leq \liminf_{\rho \to 0} \, \mathcal{E}^{g}_{\rho}(v_{\rho}) = \mathcal{E}^{g}_{0}(v), \end{align*}

which yields the minimality of $u$. Since we can construct a recovery sequence $w_{\rho } \in GSBD^{2}(\Omega _1)$ for $u$ such that $\mathcal {E}^{g}_{\rho }(w_{\rho }) \to \mathcal {E}^{g}_{0}(u)$ and $u_{\rho }$ is a minimizer of $\mathcal {E}^{g}_{\rho }$ for every $\rho$, we deduce that, along a suitable not relabelled subsequence, the inequalities (4.81)–(4.82) are actually equalities. This implies that $\partial ^{*} A \subseteq J_{u}$, (4.80), and that

\[ \int_{\Omega_1} \mathbb{C}_{0} e(u) {\, \cdot\,} e(u) \, \mathrm{d} x = \lim_{\rho \to 0} \, \int_{\Omega_{1}} \mathbb{C}_{0} e(u_{\rho}) {\, \cdot\,} e(u_{\rho}) \, \mathrm{d} x. \]

From the last equality and from proposition 3.1 we infer that $e(u_{\rho }) \to e(u)$ in $L^{2}(\Omega _{1}; \mathbb {M}^{n}_{s})$.