2.1 BV functions and sets of finite perimeter

In the following we give some preliminary notions regarding functions of bounded variation and sets of finite perimeter. For a detailed treatment we refer to [Reference Ambrosio, Fusco and Pallara4].

Given $u \in L^1(\Omega ; {\mathbb {R}}^d)$ we let $\Omega _u$ be the set of Lebesgue points of $u$, i.e. $x\in \Omega _u$ if there exists $\widetilde u(x)\in {\mathbb {R}}^d$ such that

\[ \lim_{\varepsilon\to 0^+} \frac{1}{{\varepsilon}^N}\int_{B(x,\varepsilon)}|u(y)-\widetilde u(x)|\,{\rm d}y=0, \]

$\widetilde u(x)$ is called the approximate limit of $u$ at $x$. The Lebesgue discontinuity set $S_u$ of $u$ is defined as $S_u := \Omega {\setminus} \Omega _u$. It is known that ${\mathcal {L}}^{N}(S_u) = 0$ and the function $x \in \Omega \mapsto \widetilde u(x)$, which coincides with $u$ ${\mathcal {L}} ^N$-a.e. in $\Omega _u$, is called the Lebesgue representative of $u$.

The jump set of the function $u$, denoted by $J_u$, is the set of points $x\in \Omega {\setminus} \Omega _u$ for which there exist $a,\,b\in {\mathbb {R}}^d$ and a unit vector $\nu \in S^{N-1}$, normal to $J_u$ at $x$, such that $a\neq b$ and

\begin{align*} & \lim_{\varepsilon \to 0^+} \frac{1}{\varepsilon^N} \int_{\{ y \in B(x,\varepsilon) : (y-x)\cdot\nu > 0 \}} | u(y) - a|\,{\rm d}y = 0,\\ & \lim_{\varepsilon \to 0^+} \frac{1}{\varepsilon^N} \int_{\{ y \in B(x,\varepsilon) : (y-x)\cdot\nu < 0 \}} | u(y) - b|\,{\rm d}y = 0. \end{align*}

The triple $(a,b,\nu )$ is uniquely determined by the conditions above, up to a permutation of $(a,b)$ and a change of sign of $\nu$, and is denoted by $(u^+ (x),u^- (x),\nu _u (x)).$ The jump of $u$ at $x$ is defined by $[u](x) : = u^+(x) - u^-(x).$

We recall that a function $u\in L^{1}(\Omega ;{\mathbb {R}}^{d})$ is said to be of bounded variation, and we write $u\in BV(\Omega ;{\mathbb {R}}^{d})$, if all its first-order distributional derivatives $D_{j}u_{i}$ belong to $\mathcal {M}(\Omega )$ for $1\leq i\leq d$ and $1\leq j\leq N$.

The matrix-valued measure whose entries are $D_{j}u_{i}$ is denoted by $Du$ and $|Du|$ stands for its total variation. The space $BV(\Omega ; {\mathbb {R}}^d)$ is a Banach space when endowed with the norm

\[ \|u\|_{BV(\Omega ; {\mathbb{R}}^d)} = \|u\|_{L^1(\Omega ; {\mathbb{R}}^d)} + |Du|(\Omega ) \]

and we observe that if $u\in BV(\Omega ;\mathbb {R}^{d})$ then $u\mapsto |Du|(\Omega )$ is lower semicontinuous in $BV(\Omega ;\mathbb {R}^{d})$ with respect to the $L_{\mathrm {loc}}^{1}(\Omega ;\mathbb {R}^{d})$ topology.

By the Lebesgue decomposition theorem, $Du$ can be split into the sum of two mutually singular measures $D^{a}u$ and $D^{s}u$, the absolutely continuous part and the singular part, respectively, of $Du$ with respect to the Lebesgue measure $\mathcal {L}^N$. By $\nabla u$ we denote the Radon–Nikodým derivative of $D^{a}u$ with respect to $\mathcal {L}^N$, so that we can write

\[ Du= \nabla u \mathcal{L}^N \lfloor \Omega + D^{s}u. \]

If $u \in BV(\Omega )$ it is well known that $S_u$ is countably $(N-1)$-rectifiable, see [Reference Ambrosio, Fusco and Pallara4], and the following decomposition holds

\[ Du= \nabla u \mathcal{L}^N \lfloor \Omega + [u] \otimes \nu_u {\mathcal{H}}^{N-1}\lfloor S_u + D^cu, \]

where $D^cu$ is the Cantor part of the measure $Du$.

If $\Omega$ is an open and bounded set with Lipschitz boundary then the outer unit normal to $\partial \Omega$ (denoted by $\nu$) exists ${\mathcal {H}}^{N-1}$-a.e. and the trace for functions in $BV(\Omega ;{\mathbb {R}}^d)$ is defined.

Theorem 2.1 (Approximate differentiability)

If $u \in BV(\Omega ; {\mathbb {R}}^d),$ then for $\mathcal {L}^N$-a.e. $x \in \Omega$

(2.1)\begin{equation} \lim_{\varepsilon \rightarrow 0^+} \frac{1}{\varepsilon^{N+1}}\int_{Q(x, \varepsilon)} |u(y) - u(x) - \nabla u(x).(y-x)|\,{\rm d}y = 0. \end{equation}

Definition 2.2 Let $E$ be an $\mathcal {L}^{N}$-measurable subset of $\mathbb {R}^{N}$. For any open set $\Omega \subset \mathbb {R}^{N}$ the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega )$, is given by

(2.2)\begin{equation} P(E;\Omega):=\sup\left\{\int_{E} \mathrm{div}\,\varphi(x)\,{\rm d}x:\varphi\in C^{1}_{c}(\Omega;\mathbb{R}^{N}), \|\varphi\|_{L^{\infty}}\leq1\right\}. \end{equation}

We say that $E$ is a set of finite perimeter in $\Omega$ if $P(E;\Omega ) <+ \infty.$

Recalling that if $\mathcal {L}^{N}(E \cap \Omega )$ is finite, then $\chi _{E} \in L^{1}(\Omega )$, by [Reference Ambrosio, Fusco and Pallara4, proposition 3.6], it follows that $E$ has finite perimeter in $\Omega$ if and only if $\chi _{E} \in BV(\Omega )$ and $P(E;\Omega )$ coincides with $|D\chi _{E}|(\Omega )$, the total variation in $\Omega$ of the distributional derivative of $\chi _{E}$. Moreover, a generalized Gauss–Green formula holds:

\[ {\int_{E}\mathrm{div}\,\varphi(x)\,{\rm d}x=\int_{\Omega}\left\langle\nu_{E}(x),\varphi(x)\right\rangle\,{\rm d}|D\chi_{E}|,\quad\forall\,\varphi\in C_{c}^{1}(\Omega;\mathbb{R}^{N})}, \]

where $D\chi _{E}=\nu _{E}|D\chi _{E}|$ is the polar decomposition of $D\chi _{E}$.

The following approximation result can be found in [Reference Baldo9].

2.2 BD and LD functions

We now recall some facts about functions of bounded deformation. More details can be found in [Reference Ambrosio, Coscia and Dal Maso3, Reference Barroso, Fonseca and Toader12, Reference Bellettini, Coscia and Dal Maso16, Reference Temam51, Reference Temam and Strang52].

A function $u\in L^{1}(\Omega ;{\mathbb {R}}^{N})$ is said to be of bounded deformation, and we write $u\in BD(\Omega )$, if the symmetric part of its distributional derivative $Du$, $Eu: = {(Du + Du^T)}/{2},$ is a matrix-valued bounded Radon measure. The space $BD(\Omega )$ is a Banach space when endowed with the norm

\[ \|u\|_{BD(\Omega)} = \|u\|_{L^1(\Omega; {\mathbb{R}}^N)} + |Eu|(\Omega). \]

We denote by $LD(\Omega )$ the subspace of $BD(\Omega )$ comprised of functions $u$ such that $Eu \in L^1(\Omega ;{\mathbb {R}}^{N\times N}_s)$, a counterexample due to [Reference Ornstein44] shows that $W^{1,1}(\Omega ;\mathbb {R}^N) \subsetneq LD(\Omega )$.

The intermediate topology in the space $BD(\Omega )$ is the one determined by the distance

\[ d(u,v) := \|u-v\|_{L^1(\Omega;{\mathbb{R}}^N)} + \left| |Eu|(\Omega) - |Ev |(\Omega)\right|,\;u, v \in BD(\Omega). \]

Hence, a sequence $\{u_n\} \subset BD(\Omega )$ converges to a function $u \in BD(\Omega )$ with respect to this topology, written $u_n \stackrel {i}{\to }u$, if and only if, $u_n \to u$ in $L^1(\Omega ;{\mathbb {R}}^N)$, $Eu_n \stackrel {*}{\rightharpoonup } Eu$ in the sense of measures and $|Eu_n|(\Omega ) \to |Eu|(\Omega )$.

Recall that if $u_n \to u$ in $L^1(\Omega ;{\mathbb {R}}^N)$ and there exists $C > 0$ such that $|Eu_n|(\Omega ) \leq C, \forall n \in {\mathbb {N}}$, then $u \in BD(\Omega )$ and

(2.3)\begin{equation} |Eu|(\Omega) \leq \liminf_{n\to +\infty}|Eu_n|(\Omega). \end{equation}

By the Lebesgue decomposition theorem, $Eu$ can be split into the sum of two mutually singular measures $E^{a}u$ and $E^{s}u$, the absolutely continuous part and the singular part, respectively, of $Eu$ with respect to the Lebesgue measure $\mathcal {L}^N$. The Radon–Nikodým derivative of $E^{a}u$ with respect to $\mathcal {L}^N$, is denoted by ${\mathcal {E}} u$ so we have

\[ Eu= {\mathcal{E}} u\,\mathcal{L}^N \lfloor \Omega + E^{s}u. \]

With these notations we may write

\[ LD(\Omega):=\{u \in BD(\Omega): E^s u=0\} \]

and (cf. [Reference Temam51]) $LD(\Omega )$ is a Banach space when endowed with the norm

\[ \|u\|_{LD(\Omega)}: = \|u\|_{L^1(\Omega;\mathbb{R}^N)} +\|{\mathcal{E}} u\|_{L^1(\Omega;\mathbb{R}^N)}. \]

If $\Omega$ is a bounded, open subset of ${\mathbb {R}}^N$ with Lipschitz boundary $\Gamma$, then there exists a linear, surjective and continuous, both with respect to the norm and to the intermediate topologies, trace operator

\[ \text{tr }: BD(\Omega) \to L^1(\Omega;{\mathbb{R}}^N) \]

such that tr $u = u$ if $u \in BD(\Omega ) \cap C(\overline {\Omega };{\mathbb {R}}^N)$. Furthermore, the following Gauss–Green formula holds

(2.4)\begin{equation} \int_{\Omega}(u \odot D \varphi)(x)\,{\rm d}x + \int_{\Omega}\varphi(x)\,{\rm d}Eu(x) =\int_{\Gamma}\varphi (x)(\text{tr}\,u \odot \nu)(x)\,{\rm d}\mathcal{H} ^{N-1}(x), \end{equation}

for every $\varphi \in C^1(\overline {\Omega })$ (cf. [Reference Ambrosio, Coscia and Dal Maso3, Reference Temam51]).

The following lemma is proved in [Reference Barroso, Fonseca and Toader12].

Lemma 2.4 Let $u \in BD(\Omega )$ and let $\rho \in C_0^{\infty }({\mathbb {R}}^N)$ be a non-negative function such that ${\rm supp}(\rho ) \subset \subset B(0,1)$, $\rho (-x) = \rho (x)$ for every $x \in {\mathbb {R}}^N$ and $\int _{{\mathbb {R}}^N}\rho (x)\,{\rm d}x = 1$. For any $n \in {\mathbb {N}}$ set $\rho _n(x) : = n^N\rho (nx)$ and

\[ u_n(x) := (u *\rho_n)(x) = \int_{\Omega}u(y)\rho_n(x-y)\,{\rm d}y,\quad\text{for}\ x \in \left\{y \in \Omega : {\rm dist}(y,\partial \Omega) > \frac{1}{n}\right\}. \]

Then $u_n \in C^{\infty }\left (\left \{y \in \Omega : {\rm dist}(y,\partial \Omega ) > \frac {1}{n}\right \};{\mathbb {R}}^N\right )$ and

1. (i) for any non-negative Borel function $h : \Omega \to {\mathbb {R}}$

   \[ \int_{B(x_0,\varepsilon)}h(x) |{\mathcal{E}} u_n(x)|\,{\rm d}x \leq\int_{B(x_0,\varepsilon+ {1}/{n})}(h*\rho_n)(x)\,{\rm d}|E u|(x), \]
   whenever ${\varepsilon } + \frac {1}{n} < {\rm dist}(x_0,\partial \Omega );$

2. (ii) for any positively homogeneous of degree one, convex function $\theta : {\mathbb {R}}^{N\times N}_{\rm sym} \to [0,+\infty [$ and any ${\varepsilon } \in \, ]0,{\rm dist}(x_0,\partial \Omega )[$ such that $|E u|(\partial B(x_0,{\varepsilon })) = 0$,

   \[ \lim_{n\rightarrow +\infty}\int_{B(x_0,\varepsilon)} \theta({\mathcal{E}} u_n(x))\,{\rm d}x =\int_{B(x_0,\varepsilon)}\theta\left(\frac{{\rm d}Eu}{{\rm d}|Eu|}\right)\,{\rm d}|Eu|, \]

3. (iii) $\lim _{n \to + \infty }u_n(x) = \widetilde u(x)$ and $\lim _{n \to + \infty }(|u_n -u| * \rho _n)(x) = 0$ for every $x \in \Omega {\setminus} S_u$, whenever $u \in L^{\infty }(\Omega ;{\mathbb {R}}^N)$.

The following result, proved in [Reference Temam51], see also [Reference Barroso, Fonseca and Toader12, theorem 2.6], shows that it is possible to approximate any $BD(\Omega )$ function $u$ by a sequence of smooth functions which preserve the trace of $u$.

It is also shown in [Reference Temam51] that if $\Omega$ is an open, bounded subset of ${\mathbb {R}}^N$, with Lipschitz boundary, then $BD(\Omega )$ is compactly embedded in $L^q(\Omega ;{\mathbb {R}}^N)$, for every $1 \leq q < {N}/{(N-1)}.$ In particular, the following result holds.

If $u \in BD(\Omega )$ then $J_u$ is countably $(N-1)$-rectifiable, see [Reference Ambrosio, Coscia and Dal Maso3], and the following decomposition holds

\[ Eu= {\mathcal{E}} u \mathcal{L}^N \lfloor \Omega + [u] \odot \nu_u {\mathcal{H}}^{N-1}\lfloor J_u + E^cu, \]

where $[u] = u^+ - u^-$, $u^{\pm }$ are the traces of $u$ on the sides of $J_u$ determined by the unit normal $\nu _u$ to $J_u$ and $E^cu$ is the Cantor part of the measure $Eu$ which vanishes on Borel sets $B$ with $\mathcal {H}^{N-1}(B) < + \infty.$

We end this subsection by pointing out that the equivalent of (2.1), with ${\mathcal {E}} u(x)$ replacing $\nabla u(x)$, is false (see [Reference Ambrosio, Coscia and Dal Maso3]). However the following result holds (cf. [Reference Ambrosio, Coscia and Dal Maso3, theorem 4.3] and [Reference Ebobisse28, theorem 2.5]).

Theorem 2.7 (Approximate symmetric differentiability)

If $u \in BD(\Omega ),$ then, for $\mathcal {L}^N$-a.e. $x \in \Omega$, there exists an $N\times N$ matrix $\nabla u(x)$ such that

(2.5)\begin{equation} \lim_{\varepsilon \rightarrow 0^+} \frac{1}{\varepsilon^{N+1}}\int_{B_\varepsilon(x)} |u(y) - u(x) - \nabla u(x)(y-x)|\,{\rm d}y = 0, \end{equation}

(2.6)\begin{equation} \lim_{\varepsilon \rightarrow 0^+} \frac{1}{\varepsilon^{N}}\int_{B_\varepsilon(x)} \frac{|\langle u(y) - u(x) - {\mathcal{E}} u(x)(y-x), y-x\rangle|}{|y-x|^2}\,{\rm d}y = 0, \end{equation}

for $\mathcal {L}^N$-a.e. $x \in \Omega$. Furthermore

\[ {\mathcal{L}}^N(\{x\in \Omega:|\nabla u(x)|>t \})\leq\frac{C(N,\Omega)}{t}\|u\|_{BD(\Omega)},\quad\forall\ t>0, \]

with $C(N,\Omega )>0$ depending only on $N$ and $\Omega$.

From (2.5) and (2.6) it follows that ${\mathcal {E}} u={(\nabla u+\nabla u^T)}/{2}$.

We denote by $\mathcal {R}$ the kernel of the linear operator $E$ consisting of the class of rigid motions in $\mathbb {R}^N$, i.e. affine maps of the form $Mx + b$ where $M$ is a skew-symmetric $N \times N$ matrix and $b\in \mathbb {R}^N$. $\mathcal {R}$ is therefore closed and finite-dimensional so it is possible to define the orthogonal projection $P : BD(\Omega )\to \mathcal {R}$. This operator belongs to the class considered in the following Poincaré–Friedrichs type inequality for $BD$ functions (see [Reference Ambrosio, Coscia and Dal Maso3, Reference Kohn38, Reference Temam51]).

Theorem 2.8 Let $\Omega$ be a bounded, connected, open subset of $\mathbb {R}^N$, with Lipschitz boundary, and let $R : BD(\Omega )\to \mathcal {R}$ be a continuous linear map which leaves the elements of $\mathcal {R}$ fixed. Then there exists a constant $C(\Omega, R)$ such that

\[ \int_\Omega |u(x) - R(u)(x)|\,{\rm d}x \leq C(\Omega,R)\,|E u|(\Omega),\ \text{ for every}\ u\in BD(\Omega). \]

2.3 Notions of quasiconvexity

Definition 2.9 ([Reference Barroso, Fonseca and Toader12], definition 3.1)

A Borel measurable function $f:\mathbb {R}^{N\times N}_s\to \mathbb {R}$ is said to be symmetric quasiconvex if

(2.7)\begin{equation} f(\xi)\leq\int_Q f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x, \end{equation}

for every $\xi \in \mathbb {R}^{N\times N}_s$ and for every $\varphi \in C^{\infty }_{\rm per}(Q;\mathbb {R}^N)$.

Given $f:\mathbb {R}^{N\times N}_s\to \mathbb {R}$, the symmetric quasiconvex envelope of $f$, $SQf$, is defined by

(2.8)\begin{equation} SQf(\xi): = \inf \left\{\int_Q f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x : \varphi \in C^{\infty}_{\rm per}(Q;\mathbb{R}^N)\right\}. \end{equation}

It is possible to show that $SQf$ is the greatest symmetric quasiconvex function that is less than or equal to $f$. Moreover, definition (2.8) is independent of the domain, i.e.

(2.9)\begin{equation} SQf(\xi): = \inf \left\{\frac{1}{{\mathcal{L}} ^N(D)}\int_{D} f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x : \varphi \in C^{\infty}_{0}(D;\mathbb{R}^N)\right\} \end{equation}

whenever $D \subset {\mathbb {R}}^N$ is an open, bounded set with ${\mathcal {L}} ^N(\partial D) = 0$.

In [Reference Ebobisse27], a Borel measurable function $f:\mathbb {R}^{N\times N}_s\to \mathbb {R}$ is said to be symmetric quasiconvex if and only if

(2.10)\begin{equation} f(\xi)\leq \frac{1}{\mathcal{L}^N(D)}\int_D f(\xi +{\mathcal{E}}\varphi(x))\,{\rm d}x\text{ for all }\varphi \in W^{1,\infty}_0(D;\mathbb{R}^N), \end{equation}

and it is stated that $f$ is symmetric quasiconvex if and only if $f\circ \pi$ is quasiconvex in the sense of Morrey, where $\pi$ is the projection of $\mathbb {R}^{N \times N}$ onto $\mathbb {R}^{N\times N}_{s}$.

Let us show that these two notions coincide. Observe first that, for any $\varphi \in C^\infty _0(D;\mathbb {R}^N)$,

(2.11)\begin{align} SQf(\xi)\leq \frac{1}{\mathcal{L}^N(D)}\int_D SQf(\xi+ {\mathcal{E}} \varphi(x))\,{\rm d}x=\frac{1}{\mathcal{L}^N(D)}\int_D (SQf \circ \pi)(\xi+ \nabla\varphi(x))\,{\rm d}x. \end{align}

If $f$ is upper semicontinuous and satisfies a growth condition from above as in (1.5), then $SQf$ in (2.9) is symmetric quasiconvex also in the sense of [Reference Ebobisse27]. Indeed, $SQf$ satisfies the same growth condition (1.5) and a density argument as in [Reference Ball and Murat10] shows that $SQf \circ \pi$ is $W^{1,1}$-quasiconvex, hence $W^{1,\infty }$-quasiconvex, i.e. $\varphi$ can be chosen in $W^{1,\infty }_0(D;\mathbb {R}^N)$. Thus,

(2.12)\begin{equation} SQf(\xi)\leq \frac{1}{\mathcal{L}^N(D)}\int_D SQf(\xi+ {\mathcal{E}} \varphi(x))\,{\rm d}x\leq \frac{1}{\mathcal{L}^N(D)}\int_D f(\xi+ {\mathcal{E}} \varphi(x))\,{\rm d}x, \end{equation}

for every $\varphi \in W^{1,\infty }_0(D;\mathbb {R}^N)$. Therefore, denoting by $SQf_E$ the symmetric quasiconvexification

(2.13)\begin{equation} SQf_E(\xi) : = \inf \left\{\frac{1}{{\mathcal{L}} ^N(D)}\int_{D} f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x : \varphi \in W^{1,\infty}_{0}(D;\mathbb{R}^N)\right\}, \end{equation}

and by $SQf$ the symmetric quasiconvexification defined through (2.9), trivially $SQf_E\leq SQf$ and by (2.12) we have equality.

Actually, under linear growth conditions and upper semicontinuity of $f$, we may also conclude that

\[ SQf_E(\xi) : = \inf \left\{\frac{1}{{\mathcal{L}}^N(D)}\int_{D} f(\xi+{\mathcal{E}} \varphi(x))\,{\rm d}x : \varphi \in W^{1,1}_{0}(D;\mathbb{R}^N)\right\}. \]

4.1 The bulk term

Proposition 4.1 Let $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and let $W_0$ and $W_1$ be continuous functions satisfying (3.4). Assume that $f$ given by (3.5) is symmetric quasiconvex. Then, for ${\mathcal {L}} ^N$ a.e. $x_0 \in \Omega$,

\[ \mu^a(x_0) = \dfrac{{\rm d}\mathcal{F}(\chi, u; \cdot)}{{\rm d}{\mathcal{L}}^N}(x_0) \geq f(\chi(x_0), \mathcal{E} u(x_0)). \]

Proof. Let $x_0 \in \Omega$ be a point satisfying

(4.1)\begin{equation} \mu^a(x_0) = \frac{{\rm d}\mu}{{\rm d}\mathcal{L}^N}(x_0)= \lim_{{\varepsilon} \to 0^+}\frac{\mu(Q(x_0,{\varepsilon}))}{{\varepsilon}^ N}\quad \text{exists and is finite} \end{equation}

and

(4.2)\begin{equation} \frac{{\rm d}|E^su|}{{\rm d}\mathcal{L}^N}(x_0)= 0, \quad\frac{{\rm d}|D\chi|}{{\rm d}\mathcal{L}^N}(x_0)=0. \end{equation}

Furthermore, we choose $x_0$ to be a point of approximate continuity for $u$, for ${\mathcal {E}} u$ and for $\chi$, namely we assume that

(4.3)\begin{align} & \lim_{{\varepsilon}\rightarrow 0^+}\frac{1}{{\varepsilon}^{N}}\int_{Q\left(x_0,{\varepsilon}\right)}\left\vert u(x)-u(x_0)\right\vert\,{\rm d}x = 0, \end{align}

(4.4)\begin{align} & \lim_{{\varepsilon} \to 0^+} \frac{1}{{\varepsilon}^N} \int_{Q(x_0, {\varepsilon})}|\mathcal{E} u(x) - \mathcal{E}u(x_0)|\,{\rm d}x = 0 \end{align}

and

(4.5)\begin{equation} \lim_{{\varepsilon}\rightarrow 0^+}\frac{1}{{\varepsilon}^{N}}\int_{Q\left(x_0,{\varepsilon}\right)}\left\vert \chi(x)-\chi(x_0)\right\vert\,{\rm d}x = 0. \end{equation}

We observe that the above properties hold for ${\mathcal {L}}^N$ a.e. $x_0 \in \Omega$ (applying, for instance, [Reference Ambrosio, Coscia and Dal Maso3, equation (2.5)] to $u$, $\mathcal {E} u$ and $\chi$).

Assuming that the sequence ${\varepsilon} _k \to 0^+$ is chosen in such a way that $\mu (\partial Q(x_0,{\varepsilon} _k)) = 0$, we have

\begin{align*} \mu^a(x_0) & = \lim_{\varepsilon_k \to 0^+}\frac{\mu (Q(x_0,\varepsilon_k))}{\varepsilon_k^N}\\ & = \lim_{\varepsilon_k,n}\left[\frac{1}{\varepsilon_k^N}\int_{Q(x_0,\varepsilon_k)} f(\chi_n(x),\mathcal{E} u_n(x))\,{\rm d}x + |D \chi_n|(Q(x_0,\varepsilon_k))\right]\\ & \geq \lim_{\varepsilon_k,n}\int_Q f(\chi_n(x_0+\varepsilon_k y), \mathcal{E} u_n(x_0+\varepsilon_k y))\,{\rm d}y, \end{align*}

where $\chi _n \in BV(Q(x_0,{\varepsilon} _k);\{0,1\})$, $\chi _n \to \chi$ in $L^1(Q(x_0,{\varepsilon} _k);\{0,1\})$ and $u_n \in W^{1,1}(Q(x_0,{\varepsilon} _k);{\mathbb {R}}^N)$, $u_n \to u$ in $L^1(Q(x_0,{\varepsilon} _k);{\mathbb {R}}^N)$.

Defining

\[ \chi_{n,\varepsilon_k}(y):=\chi_n(x_0+\varepsilon_k y)- \chi(x_0), \]

it follows by (4.5) that

(4.6)\begin{align} \lim_{\varepsilon_k,n}\|\chi_{n,\varepsilon_k}\|_{L^1(Q)}& = \lim_{\varepsilon_k,n}\int_Q|\chi_n(x_0+\varepsilon_k y)-\chi(x_0)|\,{\rm d}y\nonumber\\ & =\lim_{\varepsilon_k,n}\frac{1}{\varepsilon_k^{N}}\int_{Q(x_0,\varepsilon_k)}|\chi_n(x)-\chi(x_0)|\,{\rm d}x\nonumber\\ & =\lim_{\varepsilon_k \to 0^+}\frac{1}{\varepsilon_k^{N}}\int_{Q(x_0,\varepsilon_k)}|\chi(x)-\chi(x_0)|\,{\rm d}x =0. \end{align}

Analogously, letting

\[ u_{n,\varepsilon_k}(y):=\frac{u_n(x_0+\varepsilon_k y)- u(x_0)}{\varepsilon_k}, \]

then $\mathcal {E} u_{n,{\varepsilon} _k}(y)= \mathcal {E} u_n(x_0+{\varepsilon} _k y)$ and, since $u_{n,{\varepsilon} _k}\in W^{1,1}(\Omega ;{\mathbb {R}}^N)$, $E u_{n,{\varepsilon} _k}= \mathcal {E} u_{n,{\varepsilon} _k} {\mathcal {L}}^N$.

Moreover, arguing as in the proof of [Reference Barroso, Fonseca and Toader12, proposition 4.1], exploiting the coercivity of $f$ in the second variable and theorems 2.8 and 2.6, we conclude that there exists a function $v \in BD(\Omega )$, such that

\[ \lim_{\varepsilon_k,n}\|u_{n,\varepsilon_k}- P(u_{n,\varepsilon_k})-v\|_{L^1(Q;\mathbb{R}^N)} = 0, \]

where $P$ is the projection of $BD(\Omega )$ onto the kernel of the operator $E$. Furthermore, given that the point $x_0$ was chosen to satisfy (4.2) and (4.4), it was shown in [Reference Barroso, Fonseca and Toader12, proposition 4.1, (4.8)] that

(4.7)\begin{equation} Ev = \mathcal{E} u(x_0)\mathcal{L}^N. \end{equation}

Therefore, a diagonalization argument allows us to extract subsequences $u_k := u_{n_k,{\varepsilon} _k} - P(u_{n_k,{\varepsilon} _k})$ and $\chi _k := \chi _{n_k,{\varepsilon} _k}$, such that

(4.8)\begin{equation} \begin{aligned} & \lim_{k \to +\infty}\|\chi_{k}\|_{L^1(Q)} = 0,\\ & \lim_{k \to +\infty} \|u_{k}- v\|_{L^1(Q;\mathbb{R}^N)} = 0 \end{aligned} \end{equation}

and

(4.9)\begin{equation} \mu^a(x_0) = \frac{{\rm d}\mu}{{\rm d}\mathcal{L}^N}(x_0)\geq \lim_{k \to + \infty}\int_Q f(\chi(x_0)+ \chi_{k}(y),\mathcal{E} u_{k}(y))\,{\rm d}y. \end{equation}

Our next step is to fix $\chi (x_0)$ in the first argument of $f$ in the previous integral. To this end we make use of Chacon's biting lemma (see [Reference Ambrosio, Fusco and Pallara4, lemma 5.32]). Indeed, by the coercivity hypothesis (3.4) and (4.9), the sequence $\{{\mathcal {E}} u_k\}$ is bounded in $L^1(Q;{\mathbb {R}}^{N\times N}_s)$ so the biting lemma guarantees the existence of a (not relabelled) subsequence of $\{u_k\}$ and of a decreasing sequence of Borel sets $D_r$, such that $\lim _{r \to + \infty }{\mathcal {L}}^N(D_r) = 0$ and the sequence $\{\mathcal {E}u_k\}$ is equiintegrable in $Q{\setminus} D_r$, for any $r \in \mathbb {N}$.

Since $f \geq 0$, by (3.7) and (4.9), we have

(4.10)\begin{align} \mu^a(x_0) & \geq \lim_{k \to + \infty}\int_{Q{\setminus} D_r}f(\chi(x_0) + \chi_k(y), \mathcal{E} u_{k}(y))\,{\rm d}y\nonumber\\ & \geq \lim_{k \to + \infty}\left\{\int_{Q{\setminus} D_r} f(\chi(x_0), \mathcal{E} u_{k}(y))\,{\rm d}y - \int_{Q{\setminus} D_r} C |\chi_{k}(y)|\cdot (1 + |\mathcal{E} u_k(y)|)\,{\rm d}y \right\}\nonumber\\ & \geq \lim_{k \to + \infty} \int_{Q {\setminus} D_r}f(\chi(x_0), \mathcal{E} u_{k}(y))\,{\rm d}y - \limsup_{k \to + \infty} \int_{Q{\setminus} D_r} C |\chi_{k}(y)| \cdot |\mathcal{E} u_k(y)|\,{\rm d}y, \end{align}

where we used (4.6).

We claim that for each $j \in \mathbb {N}$, there exist $k=k(j)$ and $r_j \in \mathbb {N}$, such that

(4.11)\begin{equation} \int_{Q{\setminus} D_{r_j}} f(\chi(x_0), {\mathcal{E}} u_{k(j)}(y))\,{\rm d}y \geq\int_Q f(\chi(x_0), {\mathcal{E}} u_{k(j)}(y))\,{\rm d}y - \frac{C}{j}. \end{equation}

In light of (3.4), in order to guarantee that (4.11) holds, it suffices to show that for each $j \in {\mathbb {N}}$, there exist $k=k(j)$ and $r_j \in {\mathbb {N}}$, such that

(4.12)\begin{equation} \int_{D_{r_j}} 1 + |\mathcal{E} u_{k(j)}(y)|\,{\rm d}y \leq \frac{1}{j}. \end{equation}

Suppose not. Then, there exists $j_0 \in {\mathbb {N}}$ such that, for all $r, k \in {\mathbb {N}}$,

(4.13)\begin{equation} \int_{D_r}1 + |\mathcal{E} u_{k}(y)|\,{\rm d}y > \frac{1}{j_0} \end{equation}

which contradicts the equiintegrability of the constant sequence $\{1 + |{\mathcal {E}} u_k|\}$, for $k$ fixed, and the fact that $\lim _{r \to + \infty }{\mathcal {L}}^N(D_r) = 0$.

For this choice of $k(j)$ and $r_j$, we now estimate the last term in (4.10). Since $|\chi _{k(j)}| \to 0$, as $j \to + \infty$, in $L^1(Q)$, this sequence also converges to zero in measure. Thus, denoting by

\[ A_{k(j)} : = \left\{x \in Q {\setminus} D_{r_j} : |\chi_{k(j)}(x)| = 1 \right\}, \]

it follows that for every $\delta > 0$, there exists $j_0 \in {\mathbb {N}}$ such that ${\mathcal {L}}^N(A_{k(j)}) < \delta$, for all $j > j_0$.

On the other hand, because the sequence $\{\mathcal {E}u_{k(j)}\}$ is equiintegrable in $Q{\setminus} D_{r_j}$, we know that for every ${\varepsilon } > 0$, there exists $\delta = \delta ({\varepsilon }) > 0$ such that for any measurable set $A \subset Q{\setminus} D_{r_j}$ with ${\mathcal {L}}^N(A) < \delta ({\varepsilon })$ we have $\int _{A}|\mathcal {E}u_{k(j)}(y)|\,{\rm d}y < {\varepsilon }$. Choosing $j$ large enough so that ${\mathcal {L}}^N(A_{k(j)}) < \delta ({\varepsilon })$ we obtain $\int _{A_{k(j)}}|\mathcal {E}u_{k(j)}(y)|\,{\rm d}y < {\varepsilon }$ and hence

(4.14)\begin{equation} \int_{Q{\setminus} D_{r_j}}|\chi_{k(j)}(y) \cdot |\mathcal{E}u_{k(j)}(y)|\,{\rm d}y < {\varepsilon}, \end{equation}

for every sufficiently large $j$.

Therefore, up to the extraction of a further subsequence, and denoting in what follows $\chi _j := \chi _{k(j)}$, $v_j : = u_{k(j)}$ and $D_j:=D_{r_j}$, (4.10), (4.11) and (4.14) yield

\begin{align*} \mu^a(x_0) = \frac{{\rm d}\mu}{{\rm d}\mathcal{L}^N}(x_0) & \geq \liminf_{j \to + \infty} \left(\int_Q f(\chi(x_0),\mathcal{E} v_j(y))\,{\rm d}y - \frac{C}{j}\right)\\ & \quad -\limsup_{j \to + \infty}\int_{Q{\setminus} D_{j}} C |\chi_j(y)| \cdot |\mathcal{E} v_j(y)|\,{\rm d}y \\ & \geq \liminf_{j \to + \infty} \int_Q f(\chi(x_0),\mathcal{E} v_j(y))\,{\rm d}y - {\varepsilon}. \end{align*}

Since $v_j \to v$ in $L^1(Q;{\mathbb {R}}^N)$, proposition 3.6 allows us to assume, without loss of generality, that $v_j = v$ on $\partial Q$. Hence, using the symmetric quasiconvexity of $f$ in the second variable, which also holds for test functions in $LD_{\rm per}(Q)$ (cf. remark 2.10), and (4.7), we obtain

\begin{align*} \mu ^a(x_0) & \geq\liminf_{j \to + \infty} \int_Q f(\chi(x_0),\mathcal{E} v_j(y))\,{\rm d}y - {\varepsilon} \\ & \geq \liminf_{j \to + \infty}\int_Q f(\chi(x_0),{\mathcal{E}} u(x_0) + \mathcal{E} (v_j - v)(y))\,{\rm d}y - {\varepsilon} \\ & \geq f(\chi(x_0),\mathcal{E} u(x_0)) - {\varepsilon}, \end{align*}

so to conclude it suffices to let ${\varepsilon } \to 0^+$.

Proposition 4.2 Let $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and let $W_0$ and $W_1$ be continuous functions satisfying (3.4). Let $f$ be given by (3.5) and assume that $f$ is symmetric quasiconvex. Then, for ${\mathcal {L}}^N$ a.e. $x_0 \in \Omega$,

\[ \mu^a(x_0) = \dfrac{{\rm d}\mathcal{F}(\chi, u; \cdot)}{{\rm d}{\mathcal{L}}^N}(x_0) \leq f(\chi(x_0), \mathcal{E} u(x_0)). \]

Proof. Choose a point $x_0 \in \Omega$ such that (4.3), (4.4) and (4.5) hold,

(4.15)\begin{align} & \lim_{\varepsilon \to 0^+}\frac{1}{{\varepsilon} ^N}|E^s u|(Q(x_0, {\varepsilon})) = 0, \end{align}

(4.16)\begin{align} & \lim_{\varepsilon \to 0^+}\frac{1}{{\varepsilon} ^N}|D \chi|(Q(x_0, {\varepsilon})) = 0, \end{align}

and, furthermore, such that

(4.17)\begin{equation} \mu^a(x_0) = \lim_{\varepsilon \to 0^+}\frac{\mathcal{F}(\chi, u;Q(x_0,{\varepsilon}))}{{\varepsilon} ^N}\quad \text{exists and is finite}, \end{equation}

where the sequence of ${\varepsilon } \to 0^+$ is chosen so that $|E u|(\partial Q(x_0,{\varepsilon })) = 0$. Notice that ${\mathcal {L}}^N$ almost every point $x_0 \in \Omega$ satisfies the above properties.

For the purposes of this proof we assume that $\chi (x_0) = 1$, the case $\chi (x_0) = 0$ is treated in a similar fashion. Thus, it follows from (4.5) that

(4.18)\begin{equation} \lim_{\varepsilon \to 0^+}\frac{1}{{\varepsilon} ^N}{\mathcal{L}}^N\left(Q(x_0, {\varepsilon}) \cap \{\chi = 0\}\right) = 0. \end{equation}

Using the symmetric quasiconvexity of $f$, fix $\delta > 0$ and let $\phi \in C^{\infty }_{\rm per}(Q;{\mathbb {R}}^N)$ be such that

(4.19)\begin{equation} \int_Qf(\chi(x_0),{\mathcal{E}} u(x_0) + {\mathcal{E}} \phi(x))\,{\rm d}x \leq f(\chi(x_0),{\mathcal{E}} u(x_0)) + \delta. \end{equation}

We extend $\phi$ to ${\mathbb {R}}^N$ by periodicity, define $\phi _n(x) := \frac {1}{n}\phi (nx)$ and consider the sequence of functions in $W^{1,1}(Q(x_0,{\varepsilon });{\mathbb {R}}^N)$ given by

\[ u_{n,{\varepsilon}}(x) : = (\rho_n * u)(x) + {\varepsilon} \phi_n\left(\frac{x - x_0}{{\varepsilon}}\right). \]

The periodicity of $\phi$ ensures that, as $n \to + \infty$, $u_{n,{\varepsilon }} \to u$ in $L^1(Q(x_0,{\varepsilon });{\mathbb {R}}^N)$ and so, letting $\chi _n = \chi$, $\forall n \in {\mathbb {N}}$, the sequences $\{u_{n,{\varepsilon }}\}_n$ and $\{\chi _n\}_n$ are admissible for $\mathcal {F}(\chi,u;Q(x_0,{\varepsilon }))$. Hence, by (4.16), we have

\begin{align*} \mu^a(x_0) & = \lim_{\varepsilon \to 0^+}\frac{\mathcal{F}(\chi, u;Q(x_0,{\varepsilon}))}{{\varepsilon} ^N}\\ & \leq \liminf_{\varepsilon \to 0^+}\liminf_{n \to +\infty}\frac{1}{{\varepsilon}^N}\left(\int_{Q(x_0, {\varepsilon})}f(\chi(x),{\mathcal{E}} u_{n,{\varepsilon}}(x))\,{\rm d}x + |D\chi|(Q(x_0,{\varepsilon}))\right)\\ & =\liminf_{\varepsilon \to 0^+}\liminf_{n \to +\infty}\frac{1}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon})}f\left(\chi(x),{\mathcal{E}} u_{n,{\varepsilon}}(x)\right)\,{\rm d}x \\ & \leq \limsup_{\varepsilon \to 0^+}\limsup_{n \to +\infty}\frac{1}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon})}f\left(\chi(x_0),{\mathcal{E}} u(x_0) + {\mathcal{E}} \phi_n\left(\frac{x-x_0}{{\varepsilon}}\right)\right)\,{\rm d}x \\ & \quad+ \limsup_{\varepsilon \to 0^+}\limsup_{n \to +\infty}\frac{1}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon})} f\left(\chi(x),{\mathcal{E}} u_{n,{\varepsilon}}(x)\right) - f\left(\chi(x_0),{\mathcal{E}} u_{n,{\varepsilon}}(x))\right)\,{\rm d}x \\ & \quad+ \limsup_{\varepsilon \to 0^+}\limsup_{n \to +\infty}\frac{1}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon})} f\left(\chi(x_0),{\mathcal{E}} u_{n,{\varepsilon}}(x)\right)\\ & \quad - f\left(\chi(x_0),{\mathcal{E}} u(x_0) + {\mathcal{E}} \phi_n\left(\frac{x-x_0}{{\varepsilon}}\right)\right)\,{\rm d}x \\ & =: I_1 + I_2 + I_3. \end{align*}

By changing variables, using the periodicity of $\phi$ and (4.19), it follows that

\begin{align*} I_1 & = \limsup_{n \to +\infty}\int_Q f(\chi(x_0),{\mathcal{E}} u(x_0) + {\mathcal{E}} \phi_n(y))\,{\rm d}y\\ & = \limsup_{n \to +\infty}\int_Q f(\chi(x_0),{\mathcal{E}} u(x_0) + {\mathcal{E}} \phi(ny))\,{\rm d}y\\ & = \limsup_{n \to +\infty}\int_Q f(\chi(x_0),{\mathcal{E}} u(x_0) + {\mathcal{E}} \phi(x))\,{\rm d}x\leq f(\chi(x_0),{\mathcal{E}} u(x_0)) + \delta. \end{align*}

Consequently, to complete the proof it remains to show that $I_2 = I_3 = 0$ and finally to let $\delta \to 0^+$. To conclude that $I_3 = 0$ we reason exactly as in [Reference Barroso, Fonseca and Toader12, proposition 4.2] since $\chi (x_0)$ is fixed in both terms of the integrand. As for $I_2$, since $\chi (x_0) = 1$, we have by (3.7),

\begin{align*} I_2 & = \limsup_{\varepsilon \to 0^+}\limsup_{n \to +\infty}\frac{1}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon}) \cap \{\chi = 0\}} f\left(0,{\mathcal{E}} u_{n,{\varepsilon}}(x)\right) - f\left(1,{\mathcal{E}} u_{n,{\varepsilon}}(x))\right)\,{\rm d}x\\ & \leq \limsup_{\varepsilon \to 0^+}\limsup_{n \to +\infty}\frac{C}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon}) \cap \{\chi = 0\}} 1 + \left|{\mathcal{E}} (u*\rho_n)(x) + {\mathcal{E}} \phi_n\left(\frac{x-x_0}{{\varepsilon}}\right)\right|\,{\rm d}x, \end{align*}

where, by periodicity and the Riemann–Lebesgue lemma,

\begin{align*} & \limsup_{\varepsilon \to 0^+}\limsup_{n \to +\infty}\frac{C}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon}) \cap \{\chi = 0\}} \left|{\mathcal{E}} \phi_n\left(\frac{x-x_0}{{\varepsilon}}\right)\right|\,{\rm d}x\\ & \quad = \limsup_{\varepsilon \to 0^+}\limsup_{n \to +\infty} C \int_{Q \cap \{y : \chi(x_0 + {\varepsilon} y) = 0\}}\left |{\mathcal{E}}\phi(ny)\right|\,{\rm d}y\\ & \quad = \limsup_{\varepsilon \to 0^+}C \int_{Q \cap \{y : \chi(x_0 + {\varepsilon} y) = 0\}}\left(\int_Q|{\mathcal{E}} \phi(x)|\,{\rm d}x\right)\,{\rm d}y\\ & \quad = \limsup_{\varepsilon \to 0^+}\frac{C}{{\varepsilon}^N}{\mathcal{L}}^N\left(Q(x_0,{\varepsilon}) \cap \{\chi = 0\}\right)\int_Q|{\mathcal{E}} \phi(x)|\,{\rm d}x = 0 \end{align*}

by (4.18). On the other hand, since $|E u|$ does not charge the boundary of $Q(x_0,{\varepsilon })$, using lemma 2.4, (4.15), (4.4) and (4.18), it follows that

\begin{align*} & \limsup_{\varepsilon \to 0^+}\limsup_{n \to +\infty}\frac{C}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon}) \cap \{\chi = 0\}} \left |{\mathcal{E}} (u*\rho_n)(x)\right|\,{\rm d}x \\ & \quad \leq \limsup_{\varepsilon \to 0^+}\limsup_{n \to +\infty}\frac{C}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon} + {1}/{n}) \cap \{\chi = 0\}}\,{\rm d}|E u|(x)\\ & \quad= \limsup_{\varepsilon \to 0^+}\frac{C}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon}) \cap \{\chi = 0\}}|{\mathcal{E}} u(x)|\,{\rm d}x\\ & \quad \leq \limsup_{\varepsilon \to 0^+}\frac{C}{{\varepsilon}^N}\int_{Q(x_0, {\varepsilon})}|{\mathcal{E}} u(x) - {\mathcal{E}} u(x_0)|\,{\rm d}x\\ & \qquad +\limsup_{\varepsilon \to 0^+}\frac{C |{\mathcal{E}} u(x_0)|}{{\varepsilon}^N} {\mathcal{L}}^N\left(Q(x_0, {\varepsilon}) \cap \{\chi = 0\}\right) = 0. \end{align*}

Therefore, a final application of (4.18) allows us to conclude that $I_2 = 0$.

4.2 The Cantor term

This section is devoted to the identification of the density of $\mathcal {F}$ in (1.8) with respect to $|E^c u|$. To this end, we start by observing that, by virtue of proposition 3.10, there is no loss of generality in assuming that $f$ is symmetric quasiconvex. If this symmetric quasiconvexity hypothesis on $f$ is omitted, the result of the next proposition holds provided we replace $f^\infty$ by $(SQf)^\infty$, whereas, due to the inequality $(SQf)^{\infty } \leq f^{\infty }$, (4.30) holds as stated.

Proposition 4.4 Let $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and let $W_0$ and $W_1$ be continuous functions satisfying (3.4). Assume that $f$ given by (3.5) is symmetric quasiconvex. Then, for $|E^cu|$ a.e. $x_0 \in \Omega$,

\[ \mu^c(x_0) = \frac{{\rm d}\mathcal{F}(\chi, u; \cdot)}{{\rm d}|E^cu|}(x_0) \geq f^{\infty}\left(\chi(x_0), \frac{{\rm d}E^c u}{{\rm d}|E^c u|}(x_0)\right). \]

Proof. Let $x_0 \in \Omega$ be a point satisfying (4.3), (4.5) and

(4.20)\begin{align} \mu^c(x_0) & = \dfrac{{\rm d}\mathcal{F}(\chi, u; \cdot)}{{\rm d}|E^cu|}(x_0)= \frac{{\rm d}\mu}{{\rm d}|E^cu|}(x_0)\notag\\ & =\lim_{{\varepsilon} \to 0^+}\frac{\mu(Q(x_0,{\varepsilon}))}{|E^cu|(Q(x_0,{\varepsilon}))}\quad \text{exists and is finite,} \end{align}

these properties hold for $|E^c u|$ a.e. $x_0 \in \Omega$. Indeed, by [Reference Ambrosio, Coscia and Dal Maso3, theorem 6.1], $|E u|(S_u{\setminus} J_u)=0$, thus $|E^c u|(S_u{\setminus} J_u)=0$. Hence, by [Reference Ambrosio, Coscia and Dal Maso3, propositions 3.5 and 4.4], we have

\[ |E^c u|(S_u)= |E^c u|(J_u)+ |E^c u|(S_u{\setminus} J_u)=0, \]

which justifies the validity of (4.3). As for (4.5), this is a well-known property of $BV$ functions (cf. [Reference Ambrosio, Fusco and Pallara4]).

We define

\[ f_0(\xi) = f(0, \xi) \text{ and } f_1(\xi) = f(1, \xi), \forall \xi \in {\mathbb{R}}^{N \times N}_s \]

and we consider the auxiliary functionals

(4.21)\begin{align} \mathcal{F}_i(u; A) & := \inf \left\{\liminf_{n \to + \infty}\int_A f_i ({\mathcal{E}} u_n(x))\,{\rm d}x : u_n \in W^{1, 1}(A; {\mathbb{R}}^N),\right.\notag\\ & \quad \left. u_n \to u \; {\rm in} \; L^1(A;{\mathbb{R}}^N)\vphantom{\int_A}\right\},\quad i=0,1. \end{align}

Referring to theorem 6.1, remark 6.4 and corollary 6.8 in [Reference Caroccia, Focardi and Van Goethem23], $\mathcal {F}_i(u;\cdot )$, $i=0,1$, are the restriction to $\mathcal {O}(\Omega )$ of Radon measures whose densities with respect to $|E^c u|$ are given by

(4.22)\begin{equation} \frac{{\rm d}\mathcal{F}_i(u;\cdot)}{{\rm d}|E^c u|}(x_0)= f_{i}^\infty \left(\frac{{\rm d} E^c u}{{\rm d}|E^c u|}(x_0)\right) = f^\infty \left(i,\frac{{\rm d}E^c u}{{\rm d}|E^c u|}(x_0)\right) \end{equation}

for $|E^c u|$ a.e. $x_0 \in \Omega$. Choose $x_0$ so that it also satisfies (4.22), $i=0,1$.

In what follows we assume, without loss of generality, that $\chi (x_0)=1$, the case $\chi (x_0)=0$ can be treated similarly. Bearing this choice in mind we work with the functional (4.21) and we will make use of (4.22), with $i=1$. Selecting the sequence ${\varepsilon} _k \to 0^+$ in such a way that $\mu (\partial Q(x_0,{\varepsilon} _k)) = 0$ and $Q(x_0,{\varepsilon} _k) \subset \Omega$, we have

\begin{align*} \mu^c(x_0) & = \lim_{k \to +\infty}\frac{\mu (Q(x_0,\varepsilon_k))}{|E^cu|(Q(x_0,\varepsilon_k))} \\ & = \lim_{k,n}\left[\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f(\chi_n(x),\mathcal{E} u_n(x))\,{\rm d}x + |D \chi_n|(Q(x_0,\varepsilon_k))\right] \end{align*}

where $\chi _n \in BV(Q(x_0,{\varepsilon} _k);\{0,1\})$, $\chi _n \to \chi$ in $L^1(Q(x_0,{\varepsilon} _k);\{0,1\})$, $u_n \in W^{1,1} (Q(x_0,{\varepsilon} _k);{\mathbb {R}}^N)$, $u_n \to u$ in $L^1(Q(x_0,{\varepsilon} _k);{\mathbb {R}}^N)$. Taking into account that we are searching for a lower bound for $\mu ^c(x_0)$, we neglect the perimeter term $|D \chi _n|(Q(x_0,{\varepsilon} _k))$ and obtain

(4.23)\begin{align} \mu^c(x_0) & \geq\liminf_{k,n}\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f(\chi_n(x),\mathcal{E} u_n(x))\,{\rm d}x\\ & \geq\liminf_{k,n}\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f_1(\mathcal{E} u_n(x))\,{\rm d}x \nonumber\\ & \quad+ \liminf_{k,n}\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f(\chi_n(x),\mathcal{E} u_n(x)) - f(1,\mathcal{E} u_n(x))\,{\rm d}x\nonumber \\ & \geq\liminf_k \frac{\mathcal{F}_1(u;Q(x_0,\varepsilon_k))}{|E^cu|(Q(x_0,\varepsilon_k))}+ \liminf_{k,n}I_{k,n}\nonumber\\ & \geq\frac{{\rm d}\mathcal{F}_{1}(u;\cdot)}{{\rm d}|E^c u|}(x_0) + \liminf_{k,n}I_{k,n}\nonumber\end{align}

(4.24)\begin{align} & \geq f^\infty \left(1,\frac{{\rm d}E^c u}{{\rm d}|E^c u|}(x_0)\right) + \liminf_{k,n}I_{k,n} \end{align}

where

\[ I_{k,n} = \frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f(\chi_n(x),\mathcal{E} u_n(x)) - f(1,\mathcal{E} u_n(x))\,{\rm d}x. \]

It remains to estimate this term. Changing variables we get

(4.25)\begin{align} \left|I_{k,n}\right| & = \left|\frac{\varepsilon_k^N}{|E^cu|(Q(x_0,\varepsilon_k))}\int_Q f(\chi_n(x_0+\varepsilon_k y),\right.\notag\\ & \qquad \left.\mathcal{E} u_n(x_0+\varepsilon_k y)) - f(1, \mathcal{E} u_n(x_0+\varepsilon_k y))\,{\rm d}y\right| \nonumber\\ & = \left|\delta_k\int_Q f(\chi_{n,k}(y) + 1,\mathcal{E} u_{n,k}(y)) -f(1, \mathcal{E} u_{n,k}(y))\,{\rm d}y\right| \end{align}

where

\begin{align*} \displaystyle \delta_k & := \frac{\varepsilon_k^N}{|E^cu|(Q(x_0,\varepsilon_k))}, \quad\chi_{n,k}(y) := \chi_n(x_0+\varepsilon_k y) - 1, \\ u_{n,k}(y) & := \frac{u_n(x_0+\varepsilon_k y) - u(x_0)}{\varepsilon_k}. \end{align*}

By (4.5) it follows that $\lim _{k,n}\|\chi _{n,k}\|_{L^1(Q)} = 0$ [see (4.6)] and $\lim _k \delta _k = 0$. Thus, using also (3.7), we have from (4.25)

(4.26)\begin{align} \liminf_{k,n}\left|I_{k,n}\right| & \leq \limsup_{k,n} \delta_k\int_Q\left| f(\chi_{n,k}(y) + 1,\mathcal{E} u_{n,k}(y)) - f(1, \mathcal{E} u_{n,k}(y))\right|\,{\rm d}y \nonumber\\ & \leq \limsup_{k,n} C \delta_k\int_Q |\chi_{n,k}(y)|\left(1 + |\mathcal{E} u_{n,k}(y)|\right)\,{\rm d}y\nonumber\\ & = \limsup_{k,n} C \delta_k\int_Q |\chi_{n,k}(y)|\,|\mathcal{E} u_{n,k}(y)|\,{\rm d}y. \end{align}

From the growth condition from below on $f$, (4.23) and (4.20) we conclude that

\begin{align*} & \limsup_{k,n} C \delta_k\int_Q |\chi_{n,k}(y)| \, |\mathcal{E} u_{n,k}(y)| \,{\rm d}y\\ & \quad \leq \limsup_{k,n} \frac{C}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}|\mathcal{E} u_{n}(x)|\,{\rm d}x\\ & \quad\leq \limsup_{k,n} \frac{C}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f(\chi_n(x),\mathcal{E} u_{n}(x))\,{\rm d}x\\ & \quad \leq C \mu^c(x_0) <{+} \infty. \end{align*}

Using a diagonalization argument, let $\chi _k := \chi _{n(k),k}$, $u_k := u_{n(k),k}$ be such that $\chi _k \to 0$ in $L^1(Q)$ and

(4.27)\begin{align} \limsup_{k,n}C \delta_k\int_Q |\chi_{n,k}(y)| \, |\mathcal{E} u_{n,k}(y)|\,{\rm d}y = \lim_{k}C \delta_k\int_Q |\chi_{k}(y)|\,|\mathcal{E} u_{k}(y)|\,{\rm d}y <{+} \infty. \end{align}

Therefore, the sequence $\{\delta _k \chi _k \, \mathcal {E} u_{k}\}$ is bounded in $L^1(Q;{\mathbb {R}}^{N \times N}_s)$ so, by the biting lemma, there exists a subsequence (not relabelled) and there exist sets $D_r \subset Q$ such that $\lim _{r \to + \infty }{\mathcal {L}}^N(D_r) = 0$ and the sequence $\{\delta _k \chi _k \, \mathcal {E}u_k\}$ is equiintegrable in $Q{\setminus} D_r$, for any $r \in \mathbb {N}$. Following the reasoning in the proof of proposition 4.1 [see (4.12)], for any $j \in {\mathbb {N}}$ there exist $k(j), r(j) \in {\mathbb {N}}$ such that

(4.28)\begin{equation} \delta_k(j)\int_{D_{r(j)}} |\chi_{k(j)}(y)| \, |\mathcal{E} u_{k(j)}(y)|\,{\rm d}y \leq \frac{1}{j}. \end{equation}

The fact that $\chi _{k(j)} \to 0$, as $j \to + \infty$, in $L^1(Q)$ and the equiintegrability of $\{\delta _{k(j)} \chi _{k(j)} \, \mathcal {E}u_{k(j)}\}$ in $Q{\setminus} D_{r(j)}$ ensures that, for any ${\varepsilon } > 0$,

(4.29)\begin{equation} \delta_k(j)\int_{Q {\setminus} D_{r(j)}} |\chi_{k(j)}(y)|\,|\mathcal{E} u_{k(j)}(y)|\,{\rm d}y < {\varepsilon}, \end{equation}

provided $j$ is large enough [see the argument used to obtain (4.14)]. Hence, from (4.24), (4.26), (4.27), (4.28) and (4.29) we conclude that

\[ \mu^c(x_0) \geq f^\infty \left(1,\frac{{\rm d}E^c u}{{\rm d}|E^c u|}(x_0)\right), \]

which completes the proof.

Proposition 4.5 Let $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and let $W_0$ and $W_1$ be continuous functions satisfying (3.4). Assume that $f$ given by (3.5) is symmetric quasiconvex. Then, for $|E^cu|$ a.e. $x_0 \in \Omega$,

(4.30)\begin{equation} \mu^c(x_0) = \dfrac{{\rm d}\mathcal{F}(\chi, u; \cdot)}{{\rm d}|E^cu|}(x_0) \leq f^{\infty}\left(\chi(x_0), \frac{{\rm d}E^c u}{{\rm d}|E^c u|}(x_0)\right). \end{equation}

Proof. Let $x_0 \in \Omega$ be a point satisfying (4.20), (4.3) and (4.5) (which hold for $|E^cu|$ a.e. $x \in \Omega$, as observed in the proof of proposition 4.4) and, in addition,

(4.31)\begin{equation} \lim_{{\varepsilon} \to 0^+}\frac{|D\chi|(Q(x_0,{\varepsilon}))}{|E^cu|(Q(x_0,{\varepsilon}))} = 0. \end{equation}

Assuming, once again, that $\chi (x_0) = 1$, we also require that $x_0$ satisfies (4.22). Choosing the sequence ${\varepsilon} _k \to 0^+$ in such a way that $\mu (\partial Q(x_0,{\varepsilon} _k)) = 0$ and $Q(x_0,{\varepsilon} _k) \subset \Omega$, let $u_n \in W^{1,1}(Q(x_0,{\varepsilon} _k);{\mathbb {R}}^N)$ be such that $u_n \to u$ in $L^1(Q(x_0,{\varepsilon} _k);{\mathbb {R}}^N)$ and

(4.32)\begin{align} \dfrac{{\rm d}\mathcal{F}_1(u; \cdot)}{{\rm d}|E^cu|}(x_0) & =\lim_{k \to +\infty}\frac{\mathcal{F}_1(u;Q(x_0,\varepsilon_k))}{|E^cu|(Q(x_0,\varepsilon_k))} \notag\\ & = \lim_{k,n}\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f_1(\mathcal{E} u_n(x))\,{\rm d}x. \end{align}

Then, as the constant sequence $\chi _n = \chi$ is admissible for $\mathcal {F}(\chi,u;Q(x_0,{\varepsilon} _k))$, from (4.31), (4.32) and (4.22) with $i=1$, it follows that

\begin{align*} \mu^c(x_0) & = \lim_{k \to +\infty}\frac{\mathcal{F}(\chi,u;Q(x_0,\varepsilon_k))}{|E^cu|(Q(x_0,\varepsilon_k))} \\ & \leq \liminf_{k,n}\left[\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f(\chi(x),\mathcal{E} u_n(x))\,{\rm d}x+|D \chi|(Q(x_0,\varepsilon_k))\right]\\ & \leq \lim_{k,n}\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f(1,\mathcal{E} u_n(x))\,{\rm d}x\\ & \quad + \limsup_{k,n}\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f(\chi(x),\mathcal{E} u_n(x)) - f(1,\mathcal{E} u_n(x))\,{\rm d}x\\ & = f^{\infty}\left(\chi(x_0), \frac{{\rm d}E^c u}{{\rm d}|E^c u|}(x_0)\right) + \limsup_{k,n}\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\\ & \quad \times \int_{Q(x_0,\varepsilon_k)} \ f(\chi(x),\mathcal{E} u_n(x)) - f(1,\mathcal{E} u_n(x))\,{\rm d}x. \end{align*}

The same argument used in the proof of proposition 4.4, now applied to the sequences

\[ \chi_{k}(y) = \chi(x_0 + \varepsilon_ky) - 1, \quad u_{n,k}(y) := \frac{u_n(x_0+\varepsilon_k y) - u(x_0)}{\varepsilon_k}, \]

yields

\[ \limsup_{k,n}\frac{1}{|E^cu|(Q(x_0,\varepsilon_k))}\int_{Q(x_0,\varepsilon_k)}f(\chi(x),\mathcal{E} u_n(x)) - f(1,\mathcal{E} u_n(x))\,{\rm d}x = 0 \]

from which the conclusion follows.

4.3 The surface term

Given $x_0 \in J_\chi \cup J_u$ we denote by $\nu (x_0)$ the vector $\nu _u(x_0)$, if $x_0 \in J_u {\setminus} J_\chi$, whereas $\nu (x_0) := \nu _{\chi }(x_0)$ if $x_0 \in J_\chi {\setminus} J_u$, these vectors are well defined as Borel measurable functions for ${\mathcal {H}}^{N-1}$ a.e. $x_0 \in J_\chi \cup J_u$. Due to the rectifiability of both $J_\chi$ and $J_u$ (cf. [Reference Ambrosio, Fusco and Pallara4, theorems 3.77 and 3.78] and [Reference Ambrosio, Coscia and Dal Maso3, proposition 3.5 and remark 3.6]), for ${\mathcal {H}}^{N-1}$ a.e. $x_0 \in J_\chi \cap J_u$ we may select $\nu (x_0) := \nu _{\chi }(x_0) = \nu _u(x_0)$ where the orientation of $\nu _{\chi }(x_0)$ is chosen so that $\chi ^+(x_0) = 1$, $\chi ^-(x_0) = 0$ and then $u^+(x_0)$ and $u^-(x_0)$ are selected according to this orientation.

Thus, in the sequel for $\mathcal {H}^{N-1}$ a.e. $x_0 \in J_{\chi } \cup J_u$, the vector $\nu (x_0)$ is defined according to the above considerations.

Given that $\mathcal {H}^{N-1}(S_\chi {\setminus} J_\chi ) = 0$ and that all points in $\Omega {\setminus} S_\chi$ are Lebesgue points of $\chi$, in what follows we take $\chi ^+(x_0) = \chi ^-(x_0) = \widetilde \chi (x_0)$ for $\mathcal {H}^{N-1}$-a.e. $x_0 \in J_u {\setminus} J_\chi$, where $\tilde v$ denotes the precise representative of a field $v$ in $BV$, cf. § 2.1. On the other hand, for a $BD$ function $u$ it is not known whether $\mathcal {H}^{N-1}(S_u {\setminus} J_u) = 0$. However, given that all points in $\Omega {\setminus} S_u$ are Lebesgue points of $u$ and that, by [Reference Ambrosio, Coscia and Dal Maso3, remark 6.3] and the $\mathcal {H}^{N-1}$ rectifiability of $J_\chi$, $\mathcal {H}^{N-1}(S_u {\setminus} J_u)\cap J_\chi )=0$, we may consider $u^+(x_0) = u^-(x_0) = \widetilde u(x_0)$ for $\mathcal {H}^{N-1}$ a.e. $x_0 \in J_\chi {\setminus} J_u$, where $\tilde v$ denotes the Lebesgue representative of a field $v$ in $BD$ (cf. [Reference Ambrosio, Coscia and Dal Maso3, p. 206]), see also [Reference Babadjian6].

In order to describe $\mu ^j$ we will follow the ideas of the global method for relaxation introduced in [Reference Bouchitté, Fonseca and Mascarenhas17] (see also [Reference Barroso, Fonseca and Toader12, Reference Caroccia, Focardi and Van Goethem23]), the sequential characterization of $K(a,b,c,d, \nu )$, obtained in proposition 3.11, will also be used.

Given $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and $V \in \mathcal {O}_{\infty }(\Omega )$ we define

(4.33)\begin{align} m(\chi,u;V) & := \inf\left\{\mathcal{F} (\theta,v;V): \theta \in BV(\Omega;\{0,1\}), v \in BD(\Omega),\right.\notag\\ & \quad \left.\theta = \chi \text{ on } \partial V, v = u \text{ on } \partial V\right\}. \end{align}

Our goal is to show the following result.

Proposition 4.6 Let $f$ be given by (3.5), where $W_0$ and $W_1$ are continuous functions satisfying (3.4). Given $u \in SBD(\Omega )$ and $\chi \in BV(\Omega ;\{0,1\})$, we have

\begin{align*} & \mathcal{F}(\chi, u; V \cap (J_\chi \cup J_u))\\ & \quad =\int_{V \cap (J_\chi \cup J_u)}g(x,\chi^+(x),\chi^-(x),u^+(x),u^-(x),\nu(x))\,{\rm d}{\mathcal{H}}^{N-1}(x), \end{align*}

where

(4.34)\begin{equation} g(x_0,a,b,c,d,\nu) := \limsup_{\varepsilon \to 0^+}\frac{m(\chi_{a,b,\nu}({\cdot}{-} x_0),u_{c,d,\nu}({\cdot}{-} x_0);Q_\nu(x_0,{\varepsilon}))} {{\varepsilon}^{N-1}} \end{equation}

and $\chi _{a,b,\nu }$, $u_{c,d,\nu }$ were defined in (3.24).

The proof of the above proposition relies on a series of auxiliary results, based on lemmas 3.1, 3.3 and 3.5 in [Reference Bouchitté, Fonseca and Mascarenhas17] and which were adapted to the $BD$ case in [Reference Barroso, Fonseca and Toader12, lemmas 3.10–3.12]. The properties of $\mathcal {F}(\chi,u;A)$ established in proposition 3.3, and the fact that $\mathcal {F}(\chi,u;\cdot )$ is a Radon measure, ensure that we can apply the reasoning given in their respective proofs.

Lemma 4.7 Let $f$ be given by (3.5), where $W_0$ and $W_1$ are continuous functions satisfying (3.4). Then there exists a positive constant $C$ such that

\begin{align*} & |m(\chi_1,u_1;V) - m(\chi_2,u_2;V)|\\ & \quad \leq C\left[\int_{\partial V}|{\rm tr\ } \, \chi_1(x) - {\rm tr}\,\chi_2(x)| + |{\rm tr}\,u_1(x) - {\rm tr}\,u_2(x)|\,{\rm d}{\mathcal{H}}^{N-1}(x) \right], \end{align*}

for every $\chi _1,\chi _2 \in BV(\Omega ;\{0,1\})$, $u_1,u_2 \in BD(\Omega )$ and any $V \in \mathcal {O}_{\infty }(\Omega ).$

Proof. The proof follows that of lemma 3.10 in [Reference Barroso, Fonseca and Toader12]. Given $\delta > 0$ let $V_\delta := \{x \in V : {\rm dist}(x,\partial V) > \delta \}$ and select $\theta \in BV(\Omega ;\{0,1\})$ and $v \in BD(\Omega )$ such that $\theta = \chi _2$ and $v = u_2$ on $\partial V$. Now define $\theta _\delta \in BV(\Omega ;\{0,1\})$ and $v_\delta \in BD(\Omega )$ by

\[ \theta_\delta : = \begin{cases} \theta, & {\rm in}\; V_\delta\\ \chi_1, & {\rm in}\; \Omega {\setminus} V_\delta \end{cases}\quad {\rm and}\quad v_\delta : = \begin{cases} v, & {\rm in}\; V_\delta\\ u_1, & {\rm in}\; \Omega {\setminus} V_\delta. \end{cases} \]

The definition of $m(\cdot,\cdot ;\cdot )$ and the additivity and locality of $\mathcal {F}(\cdot,\cdot ;\cdot )$, as well as the inequality from above in proposition 3.3 $(i)$, lead to the conclusion.

Fixing $\chi \in BV(\Omega ;\{0,1\})$, $u \in BD(\Omega )$ and $\nu \in S^{N-1}$, we define $\lambda := {\mathcal {L}}^N + |E^s u| + |D\chi |$ and, following [Reference Bouchitté, Fonseca and Mascarenhas17], we let

\[ \mathcal{O}^*:= \left\{Q_\nu(x,{\varepsilon}) : x \in \Omega,\, {\varepsilon} > 0\right\} \]

and, for $\delta > 0$ and $V \in \mathcal {O}(\Omega )$, set

\begin{align*} m^\delta(\chi,u;V) & := \inf\left\{\sum_{i = 1}^{+\infty}m(\chi,u;Q_i) : Q_i \in \mathcal{O}^*, Q_i \cap Q_j = \emptyset \ {\rm if} \ i \neq j,\right.\\ & \quad \left. Q_i \subset V, {\rm diam} \, Q_i < \delta, \lambda\left(V {\setminus} \displaystyle\bigcup_{i=1}^{+\infty}Q_i\right) = 0\right\}. \end{align*}

Clearly, $\delta \mapsto m^\delta (\chi,u;V)$ is a decreasing function, so we define

\[ m^*(\chi,u;V) := \sup \left\{m^\delta(\chi,u;V) : \delta > 0\right\}= \lim_{\delta \to 0^+}m^\delta(\chi,u;V). \]

Lemma 4.8 Let $f$ be given by (3.5), where $W_0$ and $W_1$ are continuous functions satisfying (3.4). Given $\chi \in BV(\Omega ;\{0,1\})$, $u \in BD(\Omega )$, we have

\[ \mathcal{F}(\chi,u;V) = m^*(\chi,u;V), \; \text{ for every } V \in \mathcal{O}(\Omega). \]

Proof. The inequality

\[ m^*(\chi,u;V) \leq \mathcal{F}(\chi,u;V) \]

is an immediate consequence of the fact that $m(\chi,u;Q_i) \leq \mathcal {F}(\chi,u;Q_i)$ and that $\mathcal {F}(\chi,u;\cdot )$ is a Radon measure.

The proof of the reverse inequality relies on the lower semicontinuity of $\mathcal {F}(\cdot,\cdot ;V)$ obtained in proposition 3.3 $(iv)$ and on the definitions of $m^\delta (\chi,u;V)$, $m(\chi,u;V)$ and $m^*(\chi,u;V)$. Indeed, fixing $\delta > 0$, we consider $(Q_i^\delta )$ an admissible family for $m^\delta (\chi,u;V)$ such that, letting $N^\delta := V {\setminus} \displaystyle \cup _{i=1}^{+\infty }Q_i^\delta$,

\[ \sum_{i =1}^{+\infty}m(\chi,u;Q_i^\delta) < m^\delta(\chi,u;V) + \delta\text{ and }\lambda(N^\delta) = 0, \]

and we now let $\theta _i^\delta \in BV(\Omega ;\{0,1\})$ and $v_i^\delta \in BD(\Omega )$ be such that $\theta _i^\delta = \chi$ on $\partial Q_i^\delta$, $v_i^\delta = u$ on $\partial Q_i^\delta$ and

\[ \mathcal{F}(\theta_i^\delta,v_i^\delta;Q_i^\delta) \leq m(\chi,u;Q_i^\delta) + \delta {\mathcal{L}}^N(Q_i^\delta). \]

Setting $N_0^\delta := \Omega {\setminus} \cup _{i=1}^{+\infty }Q_i^\delta$, we define

\[ \theta^\delta := \sum_{i =1}^{+\infty}\theta_i^\delta \,\chi_{Q_i^\delta}+ \chi \,\chi_{N_0^\delta}\quad \text{and}\quad v^\delta := \sum_{i =1}^{+\infty}v_i^\delta \,\chi_{Q_i^\delta}+ u \,\chi_{N_0^\delta}. \]

Following the computations in the proof of [Reference Barroso, Fonseca and Toader12, lemma 3.11], we may show that $\theta ^\delta \in BV(\Omega ;\{0,1\})$, $v^\delta \in BD(\Omega )$, $\theta ^\delta \to \chi$ in $L^1(V;\{0,1\})$ and $v^\delta \to u$ in $L^1(V;{\mathbb {R}}^N)$, as $\delta \to 0^+$, and also

\[ \mathcal{F}(\theta^\delta,v^\delta;N^\delta) \leq C \lambda(N^\delta) = 0. \]

Using the additivity of $\mathcal {F}(\theta ^\delta,v^\delta ;\cdot )$ we have

\begin{align*} \mathcal{F}(\theta^\delta,v^\delta;V) & = \sum_{i =1}^{+\infty}\mathcal{F}(\theta_i^\delta,v_i^\delta;Q_i^\delta) + \mathcal{F}(\theta^\delta,v^\delta;N^\delta)\\ & \leq \sum_{i =1}^{+\infty}m(\chi,u;Q_i^\delta) + \delta {\mathcal{L}}^N(V)\leq m^\delta(\chi,u;V) + \delta + \delta {\mathcal{L}}^N(V), \end{align*}

so that the lower semicontinuity of $\mathcal {F}(\cdot,\cdot ;V)$ yields

\begin{align*} \mathcal{F}(\chi,u;V) & \leq \liminf_{\delta \to 0^+}\mathcal{F}(\theta^\delta,v^\delta;V) \\ & \leq \liminf_{\delta \to 0^+}\left(m^\delta(\chi,u;V) + \delta + \delta {\mathcal{L}}^N(V)\right) = m^*(\chi,u;V) \end{align*}

and this completes the proof.

Finally, a straightforward adaptation of [Reference Bouchitté, Fonseca and Mascarenhas17, lemma 3.5] leads to the following result.

Lemma 4.9 Let $f$ be given by (3.5), where $W_0$ and $W_1$ are continuous functions satisfying (3.4). Given $\chi \in BV(\Omega ;\{0,1\})$, $u \in BD(\Omega )$, we have

\[ \lim_{{\varepsilon} \to 0^+}\frac{\mathcal{F}(\chi,u;Q_\nu(x_0,{\varepsilon}))}{\lambda(Q_\nu(x_0,{\varepsilon}))} = \lim_{{\varepsilon} \to 0^+}\frac{m(\chi,u;Q_\nu(x_0,{\varepsilon}))}{\lambda(Q_\nu(x_0,{\varepsilon}))}, \]

for $\lambda$ a.e. $x_0 \in \Omega$ and for every $\nu \in S^{N-1}$.

We now proceed with the proof of proposition 4.6.

Proof of proposition 4.6. In the sequel, for simplicity of notation, we will write $\nu = \nu (x_0)$.

Let $x_0 \in \Omega \cap (J_\chi \cup J_u)$ be a point satisfying

(4.35)\begin{align} & \lim_{\varepsilon \rightarrow 0^+}\frac{1}{{\varepsilon}^N}\int_{Q_{\nu}(x_0,{\varepsilon})}|\chi(x) - \widetilde{\chi}(x_0)|\,{\rm d}x = 0, \ {\rm if}\ x_0 \in \Omega {\setminus} J_{\chi}, \end{align}

(4.36)\begin{align} & \lim_{\varepsilon \rightarrow 0^+}\frac{1}{{\varepsilon}^N}\int_{Q^+_{\nu}(x_0,{\varepsilon})}\ |\chi(x) - {\chi}^+(x_0)|\,{\rm d}x\notag\\ & \quad = \lim_{\varepsilon \rightarrow 0^+}\frac{1}{{\varepsilon}^N}\int_{Q^-_{\nu}(x_0,{\varepsilon})}\ |\chi(x) - {\chi}^-(x_0)|\,{\rm d}x =0, \quad {\rm if}\ x_0 \in \Omega \cap J_{\chi}, \end{align}

(4.37)\begin{align} & \lim_{\varepsilon \rightarrow 0^+}\frac{1}{{\varepsilon}^N}\int_{Q_{\nu}(x_0,{\varepsilon})}|u(x) - \widetilde{u}(x_0)|\,{\rm d}x = 0, \quad {\rm if}\ x_0 \in \Omega {\setminus} J_{u}, \end{align}

(4.38)\begin{align} & \lim_{\varepsilon \rightarrow 0^+}\frac{1}{{\varepsilon}^N}\int_{Q^+_{\nu}(x_0,{\varepsilon})} |u(x) - {u}^+(x_0)|\,{\rm d}x\notag\\ & \quad = \lim_{\varepsilon \rightarrow 0^+}\frac{1}{{\varepsilon}^N}\int_{Q^-_{\nu}(x_0,{\varepsilon})} |u(x) - {u}^-(x_0)|\,{\rm d}x =0, \quad{\rm if}\ x_0 \in \Omega \cap J_{u}, \end{align}

where

\[ Q^{{\pm}}_{\nu}(x_0,{\varepsilon}) =\left\{x \in Q_{\nu}(x_0,{\varepsilon}) : (x - x_0) \cdot ({\pm} \nu) > 0\right\}, \]

and

(4.39)\begin{align} \mu^j(x_0) & = \lim_{\varepsilon \rightarrow 0^+}\frac{\mathcal{F}(\chi,u;Q_{\nu}(x_0,{\varepsilon}))}{\mathcal{H}^{N-1} \lfloor (J_\chi \cup J_u)(Q_{\nu}(x_0,{\varepsilon}))}\notag\\ & = \lim_{\varepsilon \rightarrow 0^+}\frac{1}{{\varepsilon}^{N-1}}\int_{Q_{\nu}(x_0,{\varepsilon})}\,{\rm d}\mu(x)\ \text{exists and is finite}. \end{align}

In view of the considerations made at the beginning of this subsection, these properties hold for $\mathcal {H}^{N-1}$ a.e. $x_0 \in \Omega \cap (J_\chi \cup J_u)$. Furthermore, we require that $x_0$ also satisfies

(4.40)\begin{equation} \lim_{{\varepsilon} \to 0^+}\frac{1}{{\varepsilon}^{N-1}}|Eu|(Q_{\nu}(x_0,{\varepsilon})) =|([u]\odot \nu)(x_0)| = |Eu_0|(Q_\nu) \end{equation}

and

(4.41)\begin{equation} \lim_{{\varepsilon} \to 0^+}\frac{1}{{\varepsilon}^{N-1}}|D\chi|(Q_{\nu}(x_0,{\varepsilon})) = 1 = |D\chi_0|(Q_\nu), \end{equation}

where we are denoting by $\chi _0$ and $u_0$ the functions given by (3.24) with $\nu = \nu (x_0)$ and $a=\chi ^+(x_0)$, $b=\chi ^-(x_0)$, $c=u^+(x_0)$ and $d=u^-(x_0)$. Letting $\sigma := {\mathcal {H}}^{N-1}\lfloor (J_\chi \cup J_u)$, by lemma 4.9 it follows that, for $\sigma$ a.e. $x_0 \in \Omega$,

(4.42)\begin{equation} \frac{{\rm d}\mathcal{F}(\chi,u;\cdot)}{{\rm d}\sigma}(x_0) =\lim_{\varepsilon \to 0^+}\frac{\mathcal{F}(\chi,u;Q_\nu(x_0,{\varepsilon}))}{\sigma(Q_\nu(x_0,{\varepsilon}))} = \lim_{\varepsilon \to 0^+}\frac{m(\chi,u;Q_\nu(x_0,{\varepsilon}))}{\sigma(Q_\nu(x_0,{\varepsilon}))}. \end{equation}

Let $\chi _{{\varepsilon }}: Q_\nu \to \{0,1\}$ and $u_\varepsilon : Q_\nu \to {\mathbb {R}}^N$ be defined by $\chi _{{\varepsilon }}(y) := \chi (x_0 + {\varepsilon } y)$, $u_\varepsilon (y) :=u(x_0 + {\varepsilon } y)$. Properties (4.35) or (4.36), and (4.37) or (4.38), respectively, guarantee that $\chi _{{\varepsilon }} \to \chi _0$ in $L^1(Q_\nu ;\{0,1\})$ and $u_\varepsilon \to u_0$ in $L^1(Q_\nu ;{\mathbb {R}}^N)$. On the other hand, by (4.40) and (4.41) we have

\[ \lim_{{\varepsilon} \to 0^+}|Eu_\varepsilon|(Q_\nu) =\lim_{{\varepsilon} \to 0^+}\frac{1}{{\varepsilon}^{N-1}}|Eu|(Q_\nu(x_0,{\varepsilon})) = |([u]\odot \nu)(x_0)| = |Eu_0|(Q_\nu) \]

and

\[ \lim_{{\varepsilon} \to 0^+}|D\chi_\varepsilon|(Q_\nu) =\lim_{{\varepsilon} \to 0^+}\frac{1}{{\varepsilon}^{N-1}}|D\chi|(Q_\nu(x_0,{\varepsilon}))= |D\chi_0|(Q_\nu). \]

Due to the continuity of the trace operator with respect to the intermediate topology we conclude that

(4.43)\begin{align} & \lim_{{\varepsilon} \to 0^+}\frac{1}{{\varepsilon}^{N-1}}\int_{\partial Q_{\nu}(x_0,{\varepsilon})}|{\rm tr}\,\chi(x) - {\rm tr}\,\chi_0(x-x_0)|\notag\\ & \qquad + |{\rm tr}\,u(x) - {\rm tr}\,u_0(x-x_0)|\,{\rm d}{\mathcal{H}}^{N-1}(x) \nonumber\\ & \quad = \lim_{{\varepsilon} \to 0^+}\int_{\partial Q_{\nu}}|{\rm tr}\,\chi_\varepsilon(y) - {\rm tr}\,\chi_0(y)| +|{\rm tr}\,u_\varepsilon(y) - {\rm tr}\,u_0(y)|\,{\rm d}{\mathcal{H}}^{N-1}(y) = 0. \end{align}

Hence, from (4.42), (4.39), lemma 4.7 and (4.43), we obtain

\begin{align*} & \frac{{\rm d}\mathcal{F}(\chi,u;\cdot)}{{\rm d}\sigma}(x_0) =\lim_{\varepsilon \to 0^+}\frac{m(\chi,u;Q_\nu(x_0,{\varepsilon}))}{\sigma(Q_\nu(x_0,{\varepsilon}))}\\ & \quad= \lim_{\varepsilon \to 0^+}\frac{\begin{matrix}m(\chi,u;Q_\nu(x_0,{\varepsilon}))-m(\chi_0({\cdot}{-} x_0),u_0({\cdot}{-} x_0);Q_\nu(x_0,{\varepsilon}))\\ + m(\chi_0({\cdot}{-} x_0),u_0({\cdot}{-} x_0);Q_\nu(x_0,{\varepsilon}))\end{matrix}}{{\varepsilon}^{N-1}}\\ & \quad= \lim_{\varepsilon \to 0^+}\frac{m(\chi_0({\cdot}{-} x_0),u_0({\cdot}{-} x_0);Q_\nu(x_0,{\varepsilon}))}{{\varepsilon}^{N-1}} \end{align*}

and, therefore,

\begin{align*} & \mathcal{F}(\chi, u; V \cap (J_\chi \cup J_u))\\ & \quad =\int_{V \cap (J_\chi \cup J_u)}\frac{{\rm d}\mathcal{F}(\chi,u;\cdot)}{{\rm d}\sigma}(x)\,{\rm d}\sigma(x)\\ & \quad =\int_{V \cap (J_\chi \cup J_u)}g(x,\chi^+(x),\chi^-(x),u^+(x),u^-(x),\nu(x))\,{\rm d}{\mathcal{H}}^{N-1}(x). \end{align*}

In the final two propositions we will show that, under assumption (3.9), the surface energy density $g(x_0,a,b,c,d,\nu )$ may be more explicitly characterized. For this purpose we need an additional lemma which states that more regular functions can be considered in the definition of the Dirichlet functional $m(\chi,u;V)$ in (4.33). In what follows, for $u \in BD(\Omega )$, $\chi \in BV(\Omega ;\{0,1\})$ and $V \in \mathcal {O}_{\infty }(\Omega )$ we define