Brittle fracture in linearly elastic plates

Abstract

In this work we derive by $\Gamma$-convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $\rho$ and study the limit as $\rho$ tends to $0$. The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.

1. Introduction

This paper is devoted to the rigorous derivation of a brittle fracture model for elastic plates by means of dimension reduction techniques. The target $(n-1)$-dimensional plate is represented by an open bounded subset $\omega$ of $\mathbb {R}^{n-1}$ with Lipschitz boundary $\partial \omega$. As it is typical in dimension reduction problems, the plate is first endowed with a fictitious thickness $\rho >0$, so that, in an $n$-dimensional setting, the initial reference configuration is given by the set $\Omega _{\rho }:= \omega \times (-\frac {\rho }{2}, \frac {\rho }{2})$. The starting point of our analysis is the by now classical variational model of brittle fracture in linearly elastic bodies [9]

(1.1)\begin{equation} \mathcal{F}_{\rho}(u) : = \frac{1}{2} \int_{\Omega_{\rho}} \mathbb{C} e(u) {\cdot} e(u) \, \mathrm{d} x + \mathcal{H}^{n-1}(J_{u}), \end{equation}

where the displacement $u\colon \Omega _{\rho } \to \mathbb {R}^{n}$ belongs to the space $GSBD^{2}(\Omega _{\rho })$ of generalized special functions of bounded deformation [17], $e(u)$ is the approximate symmetric gradient of $u$, $J_{u}$ stands for the jump set of $u$, $\mathcal {H}^{n-1}$ indicates the $(n-1)$-dimensional Hausdorff measure in $\mathbb {R}^{n}$, and $\mathbb {C}$ is the linear elasticity tensor. We further refer to § 2 and 3 for the notation and the precise assumptions.

The aim of our work is to study the limit, in terms of $\Gamma$-convergence, of the functional (1.1) as the thickness parameter $\rho$ tends to $0$. The literature related to dimension reduction problems in Continuum Mechanics is very rich. In a purely elastic regime, we mention [6, 16] for the derivation of reduced models of linearly elastic plates, and [1, 23–26, 33, 34] for a number of nonlinear models for plates and shells obtained as limit of $3$-dimensional nonlinear elasticity. Further applications to the theory of elastic plates and shells can be found in [28, 29, 38, 39], where the interplay between dimension reduction and homogenization is studied. In an elastoplastic setting, in [18–20, 35] the authors obtained models for thin elastoplastic plates, starting from either linearized or finite plasticity, and also proved the convergence of the corresponding quasistatic evolutions, in the spirit of evolutionary $\Gamma$-convergence [36, 37].

In the context of fracture mechanics, the study of the $\Gamma$-limit of free discontinuity functionals of the form (1.1) has been considered, for instance, in [2, 7, 8, 10, 22, 32]. In particular, [7, 10] are concerned with the nonlinearly elastic case, in which the stored elastic energy density obeys a $p$-growth condition of the form $W(F) \geq C(|F|^{p} - 1)$ which is incompatible with linear elasticity. The papers [2, 32] consider the antiplanar case, where the energy is in the form (1.1) but the displacement $u$ is supposed to be orthogonal to the middle surface $\omega$, so that the dimension reduction problem becomes scalar and is described in terms of $GSBV$-functions (see, e.g., [5, section 4.5]). In [22] the authors considered the convergence of quasistatic evolutions in the vectorial case, under the assumption that the crack path is known a priori, is transversal to the middle surface $\omega$, and cuts the whole of $\Omega _{\rho }$. In the static setting, the geometrical restriction on the fracture set was then removed in [8], where the $\Gamma$-limit of $\mathcal {F}_{\rho }$ in (1.1) has been studied under the restriction $u \in SBD(\Omega _{\rho })$, the space of special functions of bounded deformation [4]. In order to ensure that sequences equi-bounded in energy are sequentially relatively compact, the authors had however to assume an a priori bound on the $L^{\infty }$-norm of the displacement $u$, which is in general not guaranteed by the boundedness of functional $\mathcal {F}_{\rho }$.

The aim and main novelty of our work is to study the limit of $\mathcal {F}_{\rho }$ in a $GSBD$-setting, removing the unphysical a priori bound on the norm of the displacement. As in [8], we prove in theorem 3.4 that the $\Gamma$-limit writes

\[ \frac{1}{2} \int_{\Omega_1} \mathbb{C}_{0} e(u) {\, \cdot\,} e(u) \, \mathrm{d} x+ \mathcal{H}^{n-1} (J_{u}) \]

for $u \in GSBD^{2}(\Omega _1)$ such that $e_{i, n}(u) = 0$ for $i = 1, \ldots, n$ and $(\nu _{u})_{n} = 0$ on $J_u$. Here, $\nu _{u}$ is the approximate unit normal to $J_{u}$ and $\mathbb {C}_{0}$ is the reduced elasticity tensor of the Kirchhoff-Love theory of elastic plates [16] (we refer to (3.13) for the precise formulation). The most technical part of our result, which in particular influences the construction of a recovery sequence in the proof of theorem 3.4, is the characterization of the admissible displacement $u$ in the limit model. Indeed, in theorem 3.2 we show that $u$ has a Kirchhoff-Love type of structure: the out-of-plane component $u_{n}$ does not depend on the vertical variable $x_{n}$, while the in-plane components $u_{1},\ldots, u_{n-1}$ satisfy

(1.2)\begin{equation} u_{\alpha} (x', x_{n}) = \overline{u}_{\alpha} (x') - x_{n} \partial_{\alpha} u_{n}(x') \end{equation}

for $x= (x', x_{n}) \in \Omega _{1}$ and $\alpha = 1, \ldots, n-1$, where $\overline {u}_{\alpha } (x') := \int _{-1/2}^{1/2} u_{\alpha } (x', x_{n}) \, \mathrm {d} x_{n}$. In contrast to [8], due to the lack of integrability of $u$ we cannot conclude $u_n \in GSBV(\omega )$ while we can ensure that at a.e. $x' \in \omega$ $u_n$ is approximate differentiable. Moreover, the $L^{\infty }$-assumption used in [8] makes it possible to work in the $BD$-context, so that (1.2) is proven by convolution techniques combined with the study of the distributional symmetric gradient $\mathrm {E} u$ of $u$ (see [8, proposition 5.2]). In our setting, instead, such an approach is not feasible as $\mathrm {E} u$ is not a bounded Radon measure for $u \in GSBD^{2}(\Omega _{1})$.

To overcome this obstacle, we obtain (1.2) through an approximation result similar to [12, theorem 1] and [30, theorem 5], which therefore allows us to work with functions that are $W^{1, \infty }$ out of the closure of their jump set. The crucial point in such an approximation is that we need to

1. (1) guarantee that on large part of the domain $\Omega _{1}$ the $n$-th component of the approximating function $u_{k}$ is still independent of $x_{n}$;

2. (2) control the $\mathcal {H}^{n-1}$-measure of the projection $\pi _{n}(\overline {J_{u_{k}}})$ of the closure of the jump set of the approximating sequence $u_{k}$ on $\omega$ by means of $\mathcal {H}^{n-1}( \pi _{n} (J_{u}))$.

The two properties above, together with the fact that actually $\mathcal {H}^{n-1}( \pi _{n} (J_{u}))= 0$, allow us to apply the Fundamental Theorem of Calculus in the direction $x_{n}$ to the sequence $u_{k}$, obtain a first version of (1.2) for $u_{k}$, and then conclude by passing to the limit in $k$ and by further exploiting that the jump set $J_u$ is transversal to the middle surface $\omega$. This argument is made rigorous in propositions 4.4, 4.7 and 4.8.

In a similar way to [8], we show that the jump set $J_{u}$ takes the form

\[ J_{u} = (J_{\overline{u}} \cup J_{u_n} \cup J_{\nabla u_n} ) \times \Bigg( -\frac 12, \frac 12 \Bigg), \]

where $\overline {u} : = ( \overline {u}_{1}, \ldots, \overline {u}_{n-1})$, concluding the description of the admissible displacements.

Finally, we extend the $\Gamma$-convergence result of theorem 3.4 to the case of non-homogeneous Dirichlet boundary conditions in corollary 4.10 and further discuss the convergence of minima and minimizers in theorem 4.12 and corollary 4.13. With respect to [8], we notice that in the proof of convergence of minima and minimizers we can not rely on the (higher) integrability of the displacement. Hence, we apply the recent compactness result in $GSBD^{p}$, $p>1$, obtained in [15] (see also [3] for an alternative proof and for the case $p=1$).

2. Preliminaries and notation

We briefly recall here the notation used throughout the paper. For $n, k \in \mathbb {N}$, we denote by $\mathcal {L}^{n}$ the Lebesgue measure in $\mathbb {R}^{n}$ and by $\mathcal {H}^{k}$ the $k$-dimensional Hausdorff measure in $\mathbb {R}^{n}$. The symbol $\mathbb {M}^{n}$ stands for the space of square matrices of order $n$ with real coefficients, while $\mathbb {M}^{n}_{s}$ indicates the subspace of $\mathbb {M}^{n}$ of squared symmetric matrices of order $n$. For every $r>0$ and every $x \in \mathbb {R}^{n}$, we denote by $B_{r}(x)$ the open ball in $\mathbb {R}^{n}$ of radius $r$ and centre $x$. We will indicate with $\{e_{1}, \ldots, e_{n}\}$ the canonical basis of $\mathbb {R}^{n}$ and with $\mathbb{1}_{E}$ the characteristic function of a set $E \subseteq \mathbb {R}^{n}$. For every $\xi \in \mathbb {S}^{n-1}$, $\pi _{\xi }$ stands for the projection over the subspace $\xi ^{\perp }$ orthogonal to $\xi$. If $\xi = e_{i}$ for $i = 1, \ldots, n$, we use the symbol $\pi _{i}$.

For every $U\subseteq \mathbb {R}^{n}$ open, we denote by $\mathcal {M}_{b}(U)$ and $\mathcal {M}^{+}_{b}(U)$ the set of bounded Radon measures and of positive bounded Radon measures in $U$, respectively. Let $m \in \mathbb {N}$ with $m\geq 1$. For every $\mathcal {L}^{n}$-measurable function $v \colon U \to \mathbb {R}^{m}$ and every $x \in U$ such that

\[ \limsup_{r \searrow 0} \frac{\mathcal{L}^{n}( U \cap B_{r} (x)) }{r^{n}} >0, \]

we say that $a \in \mathbb {R}^{m}$ is the approximate limit of $v$ at $x$ if

\[ \lim_{r \searrow 0} \frac{\mathcal{L}^{n}( U \cap B_{r} (x) \cap \{ | v - a| > \epsilon\} ) }{r^{n}} = 0 \quad \text{for every }\epsilon > 0. \]

In this case, we write

\[ {\mathop {\text{ap- lim}}\limits_{y\to x}}\,v(y) = a. \]

We say that $x \in U$ is an approximate jump point of $v$, and we write $x \in J_v$, if there exist $a, b \in \mathbb {R}^{m}$ with $a \neq b$ and $\nu \in \mathbb {S}^{n-1}$ such that

\[ \mathop {\text{ap- lim}}\limits_{\scriptstyle y\to x \atop \scriptstyle (y-x)\cdot \nu > 0} v(x) = a\quad {\rm and}\quad \mathop {\text{ap- lim}}\limits_{\scriptstyle y\to x \atop \scriptstyle (y-x)\cdot \nu < 0} v(x) = b.\]

In particular, for every $x \in J_v$ the triple $(a, b, \nu )$ is uniquely determined up to a change of sign of $\nu$ and a permutation of $a$ and $b$. We indicate such triple by $(v^{+}(x), v^{-}(x), \nu _{v}(x))$. The jump of $v$ at $x \in J_v$ is defined as $[v](x) := v^{+}(x) - v^{-}(x)$. We denote by $(\nu _{v})_{i}$ the components of $\nu _{v}$, for $i = 1, \ldots, n$.

The space $BV(U;\mathbb {R}^{n})$ of functions of bounded variation is the set of $u\in L^{1}(U;\mathbb {R}^{n})$ whose distributional gradient $Du$ is a bounded Radon measure on $U$ with values in $\mathbb {M}^{n}$. Given $u\in BV(U;\mathbb {R}^{n})$, we can write $Du=D^{a}u+D^{s}u$, where $D^{a}u$ is absolutely continuous and $D^{s}u$ is singular w.r.t. $\mathcal {L}^{n}$. The set $J_{u}$ is countably $(\mathcal {H}^{n-1}, n-1)$-rectifiable and has approximate unit normal vector $\nu _{u}$, while the density $\nabla {u} \in L^{1}(U;\mathbb {M}^{n})$ of $D^{a}u$ w.r.t. $\mathcal {L}^{n}$ coincides a.e. in $U$ with the approximate gradient of $u$, that is, for a.e. $x \in U$ it holds

\[ \mathop {\text{ap- lim}}\limits_{y\to x} \displaystyle{{u(y)-u(x)-\nabla u(x)\cdot (y-x)} \over {|x-y|}} = 0.\]

The space $SBV(U;\mathbb {R}^{n})$ of special functions of bounded variation is defined as the set of all $u\in BV(U;\mathbb {R}^{n})$ such that $|D^{s}u|(U\setminus J_{u})=0$. Moreover, we denote by $SBV_{loc}(U;\mathbb {R}^{n})$ the space of functions belonging to $SBV(V;\mathbb {R}^{n})$ for every $V\Subset U$. For $p\in [1,+\infty )$, $SBV^{p}(U;\mathbb {R}^{n})$ stands for the set of functions $u\in SBV(U;\mathbb {R}^{n})$ with approximate gradient $\nabla {u}\in L^{p}(U;\mathbb {M}^{n})$ and $\mathcal {H}^{n-1}(J_{u})<+\infty$.

We say that $u \in GSBV( U; \mathbb {R}^{n})$ if $\varphi (u)\in SBV_{loc}(U; \mathbb {R}^{n})$ for every $\varphi \in C^{1}(\mathbb {R}^{n};\mathbb {R}^{n})$ whose gradient has compact support. Also for $u\in GSBV(U;\mathbb {R}^{n})$ the approximate gradient $\nabla {u}$ exists $\mathcal {L}^{n}$-a.e. in $U$ and the jump set $J_{u}$ is countably $(\mathcal {H}^{n-1}, n-1)$-rectifiable, with approximate unit normal vector $\nu _{u}$. For $p\in [1,+\infty )$, we define $GSBV^{p}(U;\mathbb {R}^{n})$ as the set of functions $u\in GSBV(U;\mathbb {R}^{n})$ such that $\nabla {u}\in L^{p}(U;\mathbb {M}^{n})$ and $\mathcal {H}^{n-1}(J_{u})<+\infty$. We refer to [5, sections 3.6, 3.9 and 4.5] for more details on the above spaces.

In a similar fashion, the space $BD(U)$ of functions of bounded deformation is defined as the set of functions $u \in L^{1}(U;\mathbb {R}^{n})$ whose distributional symmetric gradient $Eu$ is a bounded Radon measure on $U$ with values in $\mathbb {M}^{n}_{s}$. In particular, we can split $Eu$ as $Eu = E^{a}u + E^{s}u$, where $E^{a}u$ is absolutely continuous and $E^{s}u$ is singular w.r.t. $\mathcal {L}^{n}$. Furthermore, the density $e(u) \in L^{1}(U; \mathbb {M}^{n}_{s})$ of $E^{a}u$ is the approximate symmetric gradient of $u$, meaning that for a.e. $x \in U$ it holds

(2.1)\[ \mathop {\text{ap- lim}}\limits_{y\to x} \displaystyle{{(u(y)-u(x)-e(u)(x)(y-x))\cdot (y-x)} \over {|x-y|^2}} = 0.\]

The space $SBD(U)$ of special functions of bounded deformation is the set of $u \in BD(U)$ such that $| E^{s}u| (U \setminus J_u)=0$. For $p \in (1, +\infty )$, we further denote by $SBD^{p}(U)$ the space of functions $u \in SBD (U)$ such that $\mathcal {H}^{n-1}(J_u)<+\infty$ and $e(u) \in L^{p}(U; \mathbb {M}^{n}_{s})$.

We now give the definition of $GSBD(U)$, the space of generalized special functions of bounded deformation [17]. For $u \colon U \to \mathbb {R}^{n}$ measurable, $\xi \in \mathbb {S}^{n-1}$, $y \in \mathbb {R}^{n}$ and $V \subseteq \mathbb {R}^{n}$, we set

\begin{align*} & \Pi^{\xi} := \{ z \in \mathbb{R}^{n}: z \cdot \xi = 0\}, \quad V^{\xi}_{y} := \{t \in \mathbb{R}: y + t\xi \in V\},\\ & \hat{u}^{\xi}_{y} := u(y + t\xi) \cdot \xi \quad \text{for every }t \in V^{\xi}_{y}, \quad J^{1}_{\hat{u}^{\xi}_{y}} := \{ t \in V^{\xi}_{y} : \,| [\hat{u}^{\xi}_{y}]| >1\}. \end{align*}

Then, we say that $u \in GSBD(U)$ if there exists $\lambda \in \mathcal {M}^{+}_{b}(U)$ such that for every $\xi \in \mathbb {S}^{n-1}$ one of the two equivalent conditions is satisfied [17, theorem 3.5]:

• • for every $\theta \in C^{1}(\mathbb {R}; [-\tfrac {1}{2}; \tfrac {1}{2}])$ such that $0 \leq \theta ' \leq 1$, the partial derivative $D_{\xi } (\theta (u \cdot \xi ))$ belongs to $\mathcal {M}_{b}(U)$ and $| D_{\xi } (\theta (u \cdot \xi )) | (B) \leq \lambda (B)$ for every Borel subset $B$ of $U$;

• • for $\mathcal {H}^{n-1}$-a.e. $y \in \Pi _{\xi }$ the function $\hat {u}^{\xi }_{y}$ belongs to $SBV_{loc}(U^{\xi }_{y})$ and

  \[ \int_{\Pi^{\xi}} \Big| (D \hat{u}^{\xi}_{y}) \Big| \Big( B^{\xi}_{y} \setminus J^{1}_{\hat{u}^{\xi}_{y}} \Big) + \mathcal{H}^{0} \Big( B^{\xi}_{y} \cap J^{1}_{\hat{u}^{\xi}_{y}} \Big) \mathrm{d} \mathcal{H}^{n-1}(y) \leq \lambda(B) \]
  for every Borel subset $B$ of $U$.

For $u \in GSBD(U)$, the approximate symmetric gradient $e(u)$ in (2.1) exists a.e. in $U$ and belongs to $L^{1}(U;\mathbb {M}_{s}^{n})$. Its components are denoted by $e_{i,j}(u)$ for $i, j \in \{1, \ldots, n\}$. The jump set $J_u$ is countably $(\mathcal {H}^{n-1}, n-1)$-rectifiable with approximate unit normal vector $\nu _{u}$.

Finally, if $U$ has a Lipschitz boundary $\partial U$ and $v \in GSBD(U)$, there exists a function $Tr(v) \colon \partial U \to \mathbb {R}^{n}$ such that for $\mathcal {H}^{n-1}$-a.e. $x \in \partial U$

\[ Tr(v)(x) = \mathop {\text{ap- lim}}\limits_{\scriptstyle y\to x y\in U} v(y). \]

We refer to $Tr(v)$ as the trace of $v$ on $\partial U$. Finally, for $p \in (1, +\infty )$ we say that $u \in GSBD^{p}(U)$ if $e(u) \in L^{p}(U; \mathbb {M}^{n}_{s})$ and $\mathcal {H}^{n-1}(J_u) <+\infty$. We further refer to [17] for an exhaustive discussion on the fine properties of functions in $GSBD(U)$.

3. Setting of the problem and main results

In this section we present the setting of the problem and the main results of the paper. We start by discussing the energy functional that we consider in the non-rescaled reference configuration. Let $\omega$ be an open bounded subset of $\mathbb {R}^{n-1}$ with Lipschitz boundary $\partial \omega$. As we aim at deducing a model of brittle fracture on thin films moving from the variational theory of brittle fractures in linearly elastic materials [9], we endow $\omega$ with a fictitious thickness $\rho >0$ and define $\Omega _{\rho } := \omega \times (- \frac {\rho }{2}, \frac {\rho }{2})$. Therefore, the starting point of our analysis is the functional

(3.1)\begin{equation} \mathcal{F}_{\rho}(u) : = \frac{1}{2} \int_{\Omega_{\rho}} \mathbb{C} e(u) {\cdot} e(u) \, \mathrm{d} x + \mathcal{H}^{n-1}(J_{u}), \end{equation}

where the displacement $u \colon \Omega _{\rho } \to \mathbb {R}^{n}$ belongs to $GSBD^{2}(\Omega _{\rho })$ and $\mathbb {C}$ stands for the usual linear elasticity tensor. In a fracture mechanics setting [9, 27], the volume integral in (3.1) is the stored elastic energy, while the surface term denotes the energy dissipated by the production of a fracture set $J_u$. We assume in (3.1) that the elastic body $\Omega _{\rho }$ is homogeneous outside the crack $J_{u}$. Thus, the elasticity tensor $\mathbb {C}$ is supposed to be constant in space. As usual, we assume that $\mathbb {C}$ is positive definite, that is, there exist $0 < c_{1} \leq c_{2} < +\infty$ such that

(3.2)\begin{equation} c_{1}| \mathrm{E}|^{2} \leq \mathbb{C} \mathrm{E} {\cdot} \mathrm{E} \leq c_{2} | \mathrm{E}|^{2} \quad \text{for every }\mathrm{E} \in \mathbb{M}^{n}_{s}. \end{equation}

As it is customary in dimension reduction, we rescale the energy functional $\mathcal {F}_{\rho }$ to the fixed domain $\Omega _{1} = \omega \times ( -\frac 12, \frac 12 )$, the so-called rescaled configuration. Proceeding as in [8, section 3.2], for every $v \in GSBD^{2}(\Omega _{\rho })$ we define the rescaled function $u$ in the rescaled configuration $\Omega _{1}$ as

(3.3)\begin{align} u (x) := ( v_1 ( \psi_{\rho}(x) ), \ldots, \rho v_n (\psi_{\rho}(x) ) ), \quad \psi_{\rho}(x) := (x', \rho x_{n}),\quad \text{for }(x', x_{n}) \in \Omega_{1}. \end{align}

We notice that for $x \in \Omega _1$ and $\alpha, \beta = 1, \ldots, n-1$ it holds

(3.4)\begin{align} \displaystyle e_{\alpha, \beta} (v) (\psi_{\rho}(x)) & = e_{\alpha, \beta} (u) (x) = : e^{\rho}_{\alpha, \beta}(u) (x), \end{align}

(3.5)\begin{align} \displaystyle e_{\alpha, n} (v) ( \psi_{\rho}(x)) & = \frac{1}{\rho} e_{\alpha, n} (u) (x) = : e^{\rho}_{\alpha, n}(u) (x), \end{align}

(3.6)\begin{align} \displaystyle e_{n,n} (v) (\psi_{\rho}(x)) & = \frac{1}{\rho^{2}} e_{n,n}(u) (x) = : e^{\rho}_{n,n}(u) (x). \end{align}

We further define

(3.7)\begin{equation} \phi_{\rho} (\nu) := \Bigg| \Bigg( \nu_{1}, \ldots, \frac{1}{\rho} \nu_{n} \Bigg) \Bigg| \quad \text{for every }\rho>0\text{ and every }\nu \in \mathbb{R}^{n}. \end{equation}

By a change of coordinate and using the notation (3.4)–(3.7), we rewrite (3.1) computed for $v \in GSBD^{2}(\Omega _{\rho })$ as

(3.8)\begin{equation} \mathcal{G}_{\rho} (u) := \frac{\rho}{2} \int_{\Omega_1} \mathbb{C} e^{\rho} (u) {\cdot} e^{\rho}(u) \mathrm{d} x + \rho \int_{J_{v}} \phi_{\rho}(\nu_{u}) \mathrm{d} \mathcal{H}^{n-1}. \end{equation}

Considering the functional (3.8) for $u \in GSBD^{2}(\Omega _1)$, we define

(3.9)\begin{equation} \mathcal{E}_{\rho}(u) := \left\{ \begin{array}{ll} \displaystyle \dfrac{1}{\rho} \mathcal{G}_{\rho}(u) & \text{for }u \in GSBD^{2}(\Omega_1),\\ \displaystyle \vphantom{\int}+\infty & \text{otherwise in }L^{0}(\Omega_1). \end{array}\right. \end{equation}

We now study the limit of $\mathcal {E}_{\rho }$ as the thickness parameter $\rho$ tends to $0$. Before giving the exact expression of the limit functional, however, we investigate the closedness of a converging sequence $u_{\rho } \in GSBD^{2}(\Omega _1)$ equi-bounded in energy.

Proposition 3.1 Let $u_{\rho } \in GSBD^{2}(\Omega _1)$ and $u \colon \Omega _1 \to \mathbb {R}^{n}$ measurable be such that

(3.10)\begin{equation} \sup_{\rho>0} \mathcal{E}_{\rho}(u_{\rho}) <{+}\infty \end{equation}

and $u_{\rho } \to u$ in measure as $\rho \to 0$. Then, $u \in GSBD^{2}(\Omega _1),$ $e(u_{\rho }) \rightharpoonup e(u)$ weakly in $L^{2}(\Omega _1; \mathbb {M}^{n}_{s})$, $e_{i,n}(u)=0$ and $e_{i, n}(u_{\rho }) \to 0$ in $L^{2}(\Omega _{1})$ for $i = 1, \ldots, n,$ and $(\nu _{u})_{n} = 0$ $\mathcal {H}^{n-1}$-a.e. on $J_u$.

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 [17, 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 [31, 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 $J_{u}$.

In view of proposition 3.1, we expect the limit functional to be defined on the space

(3.11)\begin{align} \mathcal{KL} (\Omega_{1}) & := \{ u \in GSBD^{2}(\Omega_{1}): e_{i, n} (u) = 0 \text{ in } \Omega_{1} \text{for }i = 1, \ldots, n,\\ & \text{ and }(\nu_{u})_{n} = 0 \text{ on }J_{u}\}. \nonumber \end{align}

We further denote by $\mathcal {KL}(U)$ the same space defined on a generic open subset $U$ of $\mathbb {R}^{n}$.

In the next theorem we complete the description of $\mathcal {KL}(\Omega _{1})$. The proof of the theorem is given in § 4 (see, in particular, propositions 4.3–4.7).

Theorem 3.2 Let $u \in \mathcal {KL}(\Omega _{1})$. Then, the following facts hold:

1. (i) $u_n$ does not depend on $x_n$ and it is approximately differentiable for $\mathcal {H}^{n-1}$-a.e. $x' \in \omega$. Moreover, denoting by $\nabla u_n$ its approximate gradient, we have $\nabla u_n \in GSBD^{2}(\omega );$

2. (ii) for $\mathcal {L}^{n}$-a.e. $(x',x_n) \in \Omega _{1}$ we have

   (3.12)\begin{equation} u_\alpha(x',x_n) = \overline{u}_{\alpha}(x') - x_n \partial_\alpha u_n(x'), \quad\alpha = 1, \dotsc, n-1, \end{equation}
   where $\overline {u}_\alpha (x'):= \int _{- 1/2}^{1/2} u_\alpha (x',x_n) \, \mathrm {d} x_n$ and $\overline {u} := (\overline {u}_1, \dotsc, \overline {u}_{n-1}) \in GSBD^{2}(\omega );$

3. (iii) $J_{u} = (J_{\overline {u}} \cup J_{u_n} \cup J_{\nabla u_n} ) \times ( -\frac 12,\frac 12 )$.

Remark 3.3 In view of theorem 3.2 we have that the space $\mathcal {KL}(\Omega _{1})$ in (3.11) is $n$-dimensional in nature, as the out of plane component $u_{n}$ depends on the planar coordinates $x'$ and the planar components $u_{\alpha }$, $\alpha = 1, \ldots, n-1$ depend linearly on $x_{n}$ through (3.12). However, the approximate symmetric gradient $e(u) \in \mathbb {M}^{n}_{s}$ can be identified with an element of $\mathbb {M}^{n-1}_{s}$, since the $n$-th column and the $n$-th row are zero. The structure highlighted in theorem 3.2 is typical of the so-called Kirchhoff-Love plate, which appears in many dimension reduction problems in elasticity.

In view of theorem 3.2 and of remark 3.3, it is convenient to introduce the following reduced linear elasticity tensor:

(3.13)\begin{equation} \mathbb{C}_{0} \mathrm{E} {\, \cdot\,} \mathrm{E} := \min_{\xi \in \mathbb{R}^{n}} \mathbb{C} \mathrm{E}_{\xi} {\, \cdot\,} \mathrm{E}_{\xi} \quad \text{for every }\mathrm{E} \in \mathbb{M}^{n-1}_{s}, \end{equation}

where for every $\xi \in \mathbb {R}^{n}$ we have set

(3.14)\begin{equation} \mathrm{E}_{\xi} := \left( \begin{array}{ccc|c} e_{1,1} & \cdots & e_{1, n-1} & \xi_{1} \\ \vdots & \ddots & \vdots & \vdots \\ e_{n-1, 1} & \cdots & e_{n-1, n-1} & \xi_{n-1} \\ \hline \xi_{1} & \cdots & \xi_{n-1} & \xi_{n} \end{array} \right) \end{equation}

With this notation at hand, the $\Gamma$-limit of $\mathcal {E}_{\rho }$ writes

\[ \mathcal{E}_{0} (u) := \left\{ \begin{array}{ll} \displaystyle \dfrac{1}{2} \int_{\Omega} \mathbb{C}_{0} e(u){\, \cdot\,} e(u) \, \mathrm{d} x + \mathcal{H}^{n-1} ( J_{u}) & \text{if }u \in \mathcal{KL} (\Omega_{1}),\\ \displaystyle \vphantom{\int} + \infty & \text{otherwise in } L^{0}(\Omega_{1}), \end{array}\right. \]

and we have the following convergence result.

Theorem 3.4 The sequence $\mathcal {E}_{\rho }$ $\Gamma$-converges to $\mathcal {E}_{0}$ w.r.t. the topology induced by the convergence in measure.

The proof of theorem 3.4 is given in § 4.

4. Proofs of theorems 3.2 and 3.4

We start by proving theorem 3.2. Its proof is articulated in the next four propositions. The first two give an approximation result in the spirit of [12, section 4, theorem 1] and [30, theorem 5]. The last two propositions, instead, provide intermediate results for the proof of items $(i)$–$(iii)$ of theorem 3.2.

We now recall the definition (cf. [30, formulas (39)–(41)]) of good/bad hyper-cubes of an $(n-1)$-dimensional grid of $\mathbb {R}^{n}$ in relation with a rectifiable set with finite $(n-1)$-dimensional Hausdorff measure.

Definition 4.1 Let $h \in \mathbb {R}^{+}$. The $(n-1)$-dimensional $h$-grid $\mathcal {Q}_h^{0}$ centred at zero and parallel to the coordinate axis is defined as

\[ \mathcal{Q}_{h}^{0} := \bigcup_{i=1}^{n} \bigcup_{z \in h \mathbb{Z}} \{ x \in \mathbb{R}^{n} : \,x_i = z \}. \]

A generic $(n-1)$-dimensional $h$-grid $\mathcal {Q}_h$ parallel to the coordinate axis is obtained simply by translating of a generic vector $y \in [0,1)^{n}$, i.e., $\mathcal {Q}_h = \mathcal {Q}_{h}^{0} +hy$.

We say that $Q$ is a hyper-cube of $\mathcal {Q}_h = \mathcal {Q}_h^{0} +hy$ if there exists $z \in h\mathbb {Z}^{n}$ such that

\[ Q = z + hy + (0,h)^{n}. \]

Definition 4.2 Let $\Gamma \subset \mathbb {R}^{n}$ be a countably $(\mathcal {H}^{n-1},n-1)$-rectifiable set with $\mathcal {H}^{n-1}(\Gamma ) <\infty$. For every $y \in \mathbb {R}^{n}$ we introduce the directional half-neighbourhood $J^{y}$ of $\Gamma$

\[ J^{y} := \bigcup_{x \in \Gamma} [x, x-y], \]

where $[a, b]$ denotes the segment joining $a, b \in \mathbb {R}^{n}$. Set $D := \{e_i, \ e_i \pm e_j, \ i,j =1, \dotsc,n, \ i\neq j \}$. Given an $(n-1)$-dimensional $h$-grid $\mathcal {Q}_h$, we say that a hyper-cube $Q^{y}_{h} = z +hy +(0,h)^{n}$ of $\mathcal {Q}_h$ is a bad hyper-cube relative to $\Gamma$ if there exist $e \in D$ and $\eta \in \{0,1 \}^{n}$ such that

\[ \begin{cases} z + hy+ h\eta \in J^{he}, \text{ with } \eta_i=0, & \text{ if } e=e_i, \\ z + hy + h\eta \in J^{he}, \text{ with } \eta_i=\eta_j=0, & \text{ if } e=e_i + e_j,\\ z + hy + h\eta + he_j \in J^{he}, \text{ with } \eta_i=\eta_j=0, & \text{ if } e=e_i - e_j. \end{cases} \]

Otherwise, we say that a hyper-cube of $\mathcal {Q}_h$ is a good hyper-cube relative to $\Gamma$.

The following proposition provides an estimate of the $\mathcal {H}^{n-1}$-measure of the boundaries of the bad hyper-cubes, which will be useful in view of the approximating result of proposition 4.4.

Proposition 4.3 Let $\Gamma \subset \mathbb {R}^{n}$ be a countably $(\mathcal {H}^{n-1},n-1)$-rectifiable set with $\mathcal {H}^{n-1}(\Gamma ) <\infty,$ and let $\Gamma _{j}$ be a sequence of measurable sets such that

\[ \Gamma_j \subset \Gamma \text{ for every }j \in \mathbb{N} \text{ and } \sum_{j=1}^{\infty} \mathcal{H}^{n-1}(\Gamma_j) = L < \infty. \]

Moreover, for every $j \in \mathbb {N},$ every $h>0,$ and every $y \in [0,1)^{n}$, let $\mathcal {B}_{h,j,y}$ be the family of bad hyper-cubes of $\mathcal {Q}_{h}^{0} + hy$ relative to $\Gamma _j$ and define

(4.1)\begin{equation} A_{h,j} := \bigcup_{Q \in \mathcal{B}_{h,j,y}} Q. \end{equation}

Then, for every $\delta >0$ there exists a subset $H \subset (0,1)^{n}$ with $\mathcal {L}^{n}((0,1)^{n} \setminus H) \leq \delta$ for which for every $y \in H$ there exist a sequence $h_k \searrow 0$ and a sequence $j_m \nearrow \infty$ such that

(4.2)\begin{equation} \limsup_{k \to \infty} \mathcal{H}^{n-1}(\partial A_{h_k,j_m}) <\frac{1}{m} \quad \text{for every }m, \end{equation}

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).

We now provide an approximation result for a function $v \in GSBD^{2}(\Omega _{1})$ in terms of more regular functions $v_{k}$ whose jump $J_{v_{k}}$ is contained in an $(n-1)$-dimensional $h$-grid $\mathcal {Q}_h$ and that are $W^{1, \infty }$ out of $\overline {J_{v_{k}}}$, as in [30, theorem 5]. The main difference is that here, in order to later prove theorem 3.2, we have to carefully estimate the measure $\mathcal {H}^{n-1}(\pi _{n}(\overline {J_{v_{k}}}))$, where $\pi _n$ denotes the orthogonal projection of $\mathbb {R}^{n}$ onto $e_n^{\bot }$. We further point out that our approximation is local in space, i.e., for $\Omega ' \Subset \Omega _{1}$, and that we do not need to approximate $v$ in energy, as done in [14]. For these reasons, a construction similar to those in [30, theorem 5] can be performed in our setting without the additional assumptions $v \in L^{2}(\Omega _1;\mathbb {R}^{n})$ and $e(v) \in L^{2}(\Omega _1;\mathbb {M}^{n}_s)$, which were instead crucial in [12, 30] to guarantee the convergence in energy and to construct a recovery sequence.

Proposition 4.4 Let $U \subset \mathbb {R}^{n}$ be open, $v \in GSBD^{2}(U)$ with $\mathcal {H}^{n-1}(J_v) < \infty$, and $V \Subset U$ a Lipschitz regular domain with $\mathcal {H}^{n-1}(\partial V \cap J_v)=0$. Then, there exists $(v_k)_{k=1}^{\infty } \subset GSBD^{2}(V) \cap W^{1,\infty }(V \setminus \overline {J_{v_k}}; \mathbb {R}^{n})$ such that

1. (i) $v_k \to v$ in measure in $V$ as $k \to \infty ;$

2. (ii) $\|e(v_k) -e(v)\|_{L^{2}(V; \mathbb {M}^{n}_{s})} \to 0$ as $k \to \infty ;$

3. (iii) for every $\xi \in \mathbb {S}^{n-1}$

   \[ \lim_{k \to \infty} \mathcal{H}^{n-1}(\pi_\xi (\overline{J_{v_k}}) \setminus \pi_\xi (J_v \cap V) ) = 0; \]

4. (iv) $Tr(v_k) \to Tr(v)$ in $\mathcal {H}^{n-1}$-measure on $\partial V$ as $k \to \infty ;$

5. (v) If $v \cdot e_{j}$ is independent of $x_{i},$ then for $\mathcal {H}^{n-1}$-a.e. $x' \notin \pi _{e_{i}} (\overline {J_{v_{k}}})$ the function $t \mapsto v_{k}(x' + t e_{i})\cdot e_{j}$ is constant.

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 [30, 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 [30, 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 [30, 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 [21, 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 [30, 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 [30, 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 [13] 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’) [30, 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.

Remark 4.5 We remark that the results of [30, theorem 6] and [12, theorem 2] can not be used to prove proposition 4.4, as they do not comply with item $(v)$, i.e., with the independence of the approximating sequence of the variable $x_{i}$. We further mention that items $(i)$–$(iv)$ of proposition 4.4 could also be deduced from the recent results of [11, theorem 5.1]. As for property $(v)$, instead, it could be proven that $t \mapsto v_{k}(x' + t e_{i})\cdot e_{j}$ is constant for $\mathcal {H}^{n-1}$-a.e. $x' \notin \pi _{e_{i}} (\Lambda )$, for a set $\Lambda \subseteq \Omega _{1}$ which in general is not (essentially) closed.

Remark 4.6 Here we limit ourselves to observe that, in point $(iii)$ of the previous theorem, also $\mathcal {H}^{n-1}(\pi _\xi (J_v \cap V) \setminus \pi _\xi (\overline {J_{v_k}}))$ goes to zero as $k \to \infty$ but possibly only for a.e. $\xi \in \mathbb {S}^{n-1}$.

In the next proposition we show $(i)$ of theorem 3.2 and do a first step towards the proof of formula (3.12).

Proposition 4.7 Let $u \in \mathcal {KL}(\Omega _{1})$. Then, $u_n$ does not depend on $x_n$. Moreover, for every $\alpha = 1, \dotsc, n-1$ there exists an $\mathcal {H}^{n-1}$-measurable function $\psi _{\alpha } \colon \omega \to \mathbb {R}$ such that

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

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 $\omega ' \nearrow \omega$.

Proposition 4.8 Let $u \in \mathcal {KL}(\Omega _{1} )$. Then, there exists $\Gamma ' \subset \omega$ such that

(4.45)\begin{equation} J_u = \Gamma' \times \Bigg( -\frac 12, \frac 12 \Bigg). \end{equation}

Moreover, if $\psi _{\alpha }$ are as in proposition 4.7, then the functions

\begin{align*} & v(x') := \Bigg(Tr (u_1) \Bigg(x',-\frac 12 \Bigg), \dotsc, Tr ( u_{n-1} ) \Bigg(x',-\frac 12\Bigg) \Bigg), \\ & \psi(x') := (\psi_1(x'), \dotsc, \psi_{n-1}(x')) \end{align*}

belong to $GSBD^{2} (\omega )$.

Remark 4.9 Notice that being the jump of $u$ of the form $J_u = \Gamma ' \times (-\frac 12, \frac 12)$ and being $u_n$ independent of $x_n$, then also $J_{u_n}$ is of the form $\Gamma ' \times (-\frac 12,\frac 12)$ for some $\Gamma ' \subset \Gamma '$.

We are now ready to prove proposition 4.8.

Proof of proposition 4.8. By [17, 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 [5, 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).

We are now in a position to conclude the proof of theorem 3.2.

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 [21, 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 [8, 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 [5, 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.

We are now in a position to prove the $\Gamma$-convergence result of theorem 3.4.

Proof of theorem 3.4. We follow here the steps of [8, 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 [31, 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.

In the following corollary we show that we can naturally handle the presence of boundary conditions satisfying the properties of (3.11). Although the result follows directly from theorem 3.4, it justifies the study of convergence of minima and minimizers, considered in theorem 4.12 and corollary 4.13 below.

Corollary 4.10 Let $g \in \mathcal {KL} (\mathbb {R}^{n} ) \cap H^{1}(\mathbb {R}^{n}; \mathbb {R}^{n})$, and let us define, for $u \in GSBD^{2} ( \Omega _1 ),$

(4.70)\begin{align} & \mathcal{E}_{\rho}^{g} (u) := \mathcal{E}_{\rho} (u) + \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), \end{align}

(4.71)\begin{align} & \mathcal{E}_{0}^{g} (u) := \mathcal{E}_{0} (u) + \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). \end{align}

Then, $\mathcal {E}_{\rho }^{g}$ $\Gamma$-converges to $\mathcal {E}^{g}_{0}$ w.r.t. the topology induced by the convergence in measure in $\Omega _1$.

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 $\partial \omega \times (-\frac 12, \frac 12 )$.

We now discuss the convergence of minimizers of the functionals $\mathcal {E}^{g}_{\rho }$. To do this, we recall here the $GSBD$-compactness result obtained in [15, theorem 1.1] (see also [3]).

Theorem 4.11 Let $U \subseteq \mathbb {R}^{n}$ be an open bounded subset of $\mathbb {R}^{n},$ let $\phi \colon \mathbb {R}^{+} \to \mathbb {R}^{+}$ be an increasing function such that

\[ \lim_{t \to +\infty} \frac{\phi(t)}{t} ={+}\infty, \]

and let $u_\rho \in GSBD^{2}(U)$ be such that

\[ \sup_{\rho>0} \int_{U} \phi(|e(u_\rho) |) \, \mathrm{d} x + \mathcal{H}^{n-1}(J_{u_\rho}) < \infty. \]

Then, there exists a subsequence, still denoted by $u_\rho,$ such that the set

\[ A := \{x \in U : \ |u_\rho(x)| \to +\infty \text{ as } \rho \to 0^{+} \} \]

has finite perimeter, $u_\rho \to u$ a.e. in $U \setminus A$ and $e(u_{\rho }) \rightharpoonup e(u)$ weakly in $L^{1} (U\setminus A; \mathbb {M}^{n}_{s})$ for some function $u \in GSBD^{2}(U)$ with $u=0$ in $A$. Furthermore,

\[ \mathcal{H}^{n-1}(J_{u} \cup \partial^{*}A) \leq \liminf_{\rho \to 0} \, \mathcal{H}^{n-1}(J_{u_{\rho}}). \]

From theorem 4.11 we deduce the convergence of minima and minimizers.

Theorem 4.12 Let $g \in \mathcal {KL} ( \mathbb {R}^{n} ) \cap H^{1}(\mathbb {R}^{n}; \mathbb {R}^{n} ),$ and let $\mathcal {E}^{g}_{\rho }$ be the sequence of functionals defined in (4.70). Assume that $u_{\rho } \in GSBD^{2}( \Omega _1)$ satisfies

(4.73)\begin{equation} \liminf_{\rho \to 0} \mathcal{E}^{g}_{\rho} (u_{\rho}) <{+}\infty. \end{equation}

Then, there exists a subsequence, still denoted by $u_{\rho },$ such that the set

\[ A := \{ x \in \Omega_1 : \, | u_{\rho} (x) | \to +\infty \text{ as }\rho \to 0\} \]

is a set of finite perimeter. Moreover, there exist $A' \subseteq \omega$ and $u \in \mathcal {KL} (\Omega _1)$ with $u=0$ in $A$ such that

(4.74)\begin{align} & A = A' \times \Bigg( - \frac 12, \frac 12 \Bigg), \end{align}

(4.75)\begin{align} & u_{\rho} \to u \quad \text{a.e. in }\Omega_1 \setminus A, \end{align}

(4.76)\begin{align} & e (u_{\rho}) \rightharpoonup e(u) \quad \text{weakly in }L^{2}(\Omega_1 \setminus A; \mathbb{M}^{n}_{s}), \end{align}

(4.77)\begin{align} & \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}

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 [31, 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.

Corollary 4.13 Under the assumptions of theorem 4.12, let $u_{\rho } \in GSBD^{2}(\Omega _1)$ be a sequence of minimizers of $\mathcal {E}_{\rho }^{g}$. Then, there exist a subsequence, still denoted by $u_{\rho },$ such that the set $A := \{ x \in \Omega _1: \, | u_{\rho } (x) | \to +\infty \}$ is of finite perimeter, and a minimizer $u \in \mathcal {KL} (\Omega _1)$ of $\mathcal {E}^{g}_{0}$ with $u=0$ on $A$ such that (4.75)–(4.76) hold. Moreover, $\partial ^{*} A \subseteq J_{u},$ $e(u_{\rho }) \to e(u)$ in $L^{2}(\Omega _{1}; \mathbb {M}^{n}_{s}),$ and

(4.80)\begin{align} & \mathcal{H}^{n-1} (J_{u} ) + \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= \lim_{\rho \to 0} \, \int_{J_{u_{\rho}}} \phi_{\rho}( \nu_{u_{\rho}}) \, \mathrm{d} \mathcal{H}^{n-1}\notag\\ & \quad + \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}

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 $L^{2}(\Omega _{1}; \mathbb {M}^{n}_{s})$.