\section{Introduction}
A classical problem in Calculus of Variations is to find minimal assumptions assuring the lower semicontinuity with respect to a suitable convergence of  integral functionals of the type
\begin{equation}
\label{functional}
F(u)=\int_\Omega f(x,u,\nabla u)\,dx,
\end{equation}
where $\Omega$ is an open subset of $\R^N$ and $u$ belongs to a given space of weakly differentiable functions. 
With this problem in mind,
it is well known that, if the integrand $f(x, s,\xi)$ admits a linear growth with respect to the gradient variable $\xi$, the natural  functional framework 
is the space $BV$ of functions of bounded variation. 


A fundamental result in this direction has been proved
by Dal Maso in \cite{DM1980}.
More precisely,
assuming that the integrand $f(x, s,\xi)$ is coercive, continuous and convex in the last variable, he introduced a proper lower semicontinuous extension of $F$ to $BV$ and proved that it coincides with the integral representation of the relaxed functional of $F$.
If we drop the coercivity assumption on $f$, the task of studying the lower semicontinuity and of finding the relaxation of $F$ is highly non-trivial and requires some additional regularity assumption on $f$ in the $x$ variable (see for instance \cite{FL,dcl,DFV,DFV1,ADCF1,ADCF2,BouDM}). 

Aim of this paper is to investigate 
the possibility of new progress
in this area, by confining our study to the model cases
\begin{gather*}
F_\varphi(u)=\int_\Omega \varphi\ \bsmall(x,u)\cdot\nabla u\,dx\,,
\quad 
G(u)=\int_\Omega |\bsmall(x,u)\cdot\nabla u|\,dx\,,
\qquad
\varphi\in C^1_0(\Omega)\,,
\end{gather*}
in the perspective to extend this study to more general cases. 
We remark that
all the results presented in this paper are new also 
in the case of a vector field $\bsmall$ 
independent of $u$. 

The lower semicontinuity of these functionals with respect to the $L^1$ convergence has been established in \cite{dcl} in the Sobolev space $W^{1,1}$ by requiring a very weak regularity assumption, i.e.\ that the divergence of the vector field $\bsmall(x,s)$ with respect to $x$ is an $L^1$ function. 
Our aim is to extend this result to the space $BV$, by considering a relaxed functional defined by an abstract relaxation procedure.
More precisely, for every 
fixed function $\varphi\in C_c(\Omega)$ and every
open set $A\subseteq\Omega$, let us consider the functional $F_\varphi(\cdot,A):\BVLloc\to[0,+\infty]$ defined by
\[
F_\varphi(u,A):=
\begin{cases}
\displaystyle\int_A \varphi\  \bsmall(x,u)\cdot\nabla u\,dx
& \text{if $u\in W^{1,1}_{{\rm loc}}(\Omega)$}\cap L_{{\rm loc}}^\infty(\Omega),
\\
+\infty
,
& \text{if}\ u\in (BV_{{\rm loc}}(\Omega)\
\setminus W_{{\rm loc}}^{1,1}(\Omega))\cap L_{{\rm loc}}^\infty(\Omega)\,,
\end{cases}
\]
and the associated relaxed functional
\begin{equation}
\label{relax1}
\rel{F_\varphi}(u,A)\!:=\inf\left\{\liminf_{n\to +\infty}F_\varphi(u_n,A)\!:\!
u_n\in W^{1,1}(\Om)\,,u_n\to 
\ {\rm weak}^*{ \rm \ in\  } BV(\Om)
\right\}\!.
\end{equation}
As it is customary,
this relaxed functional can be characterized as the greatest lower semicontinuous extension of  $F$ to $BV$, less than or equal to $F$. 
Besides this abstract definition, for the applications it is of paramount 
importance to
have an integral representation of $\rel{F_\varphi}$.
To this end,
the main difficulty is to find a precise representative for the singular part of the relaxed functional
(i.e., the representative where the measure $Du$ is singular).

The main contribution of this paper
is to find an integral representation for $\rel{F}_\varphi(u,A)$. 
More precisely, we prove that, 
for every 
$u\in  \BVLloc$ 
and for every open set $A\subseteq\Omega$,
it holds that
\begin{equation}
\label{t:generalenuovo1}
\rel{F_\varphi}(u,A)=\int_A\varphi\ (\bsmall(x,u),Du) \,,
\end{equation}
where $(\bsmall(x,u),Du)$ denotes the pairing measure defined in the recent paper \cite{CD4}. 
This measure extends the concept of pairing measure introduced by Anzellotti in the celebrated paper \cite{Anz} by establishing a pairing theory between 
weakly differentiable vector fields $\bsmall(x)$ and \(BV\) functions.
While the original definition of this measure starts from a  distributional viewpoint, our contribution shows that it can be regarded also in a variational sense as a relaxed functional.
This variational interpretation 
seems to be useful in order to study the $1$--Laplace operator,
both in the case of the associated Euler--Lagrange equations (see \cite{Mazon2016})
and in the study of the related Dirichlet problem with measure data.


Lower semicontinuity results and representation formulas for the relaxed functional in $BV(\Omega)$ have been obtained by many authors. 
In the already cited paper \cite{DM}, 
Dal Maso showed that, in order to prove lower semicontinuity,
in his general setting the coercivity assumption cannot be dropped.
In the spirit of the alternatives of Serrin
(see \cite{Serrin61}), in order to drop this assumption, 
Fonseca and Leoni in \cite{FL} assumed a uniform lower semicontinuity condition in $x$. Moreover, in \cite{ADCF1,ADCF2}, 
the authors required weak differentiability in $x$ and $BV$  in $x$ dependence, respectively. In these cases the precise representatives for the singular parts are the approximately continuous representative and the lower semicontinuous capacitary representative, respectively.


\smallskip


Before describing in more details our results,
a few words on the pairing measure are in order.

The pairing theory was initially used to extend the validity of the
Gauss--Green formula to divergence-measure vector fields and to 
non-smooth domains (see \cite{Anz,Cas,ChenFrid,ChTo,ChToZi,ComiPayne,CD3,LeoSar}). 
Moreover, it can be considered 
as a useful abstract tool in several contexts, 
ranging from applications in the theory of hyperbolic systems of conservation and balance laws
(see \cite{ChFr1,ChenFrid,ChTo2,ChTo,ChToZi,CD2,Pan1} and the references therein)
to the theory of capillarity and 
in the study of the Prescribed Mean Curvature problem (see e.g.\ \cite{LeoSar,LeoSar2})
and in the context of continuum mechanics 
(see e.g.\ \cite{DGMM,Silh,Schu}).
Another field of application is related to
the Dirichlet problem for equations involving the $1$--Laplacian operator (see \cite{ABCM,AVCM,Cas,DeGiOlPe,DeGiSe,HI,SchSch,SchSch2}).



In a recent paper \cite{CD4},
the authors introduced a nonlinear version of the pairing suitable for
applications to semicontinuity problems. 
This is the pairing appearing in the representation formula~\eqref{t:generalenuovo1}. The pairing $(\bsmall(x,u),Du)$ generalizes the Anzellotti pairing and inherits its properties. 
In particular, in that paper we characterized
the normal traces of the vector field
$\bsmall(x,u(x))$
and we performed an analysis of the singular part of the pairing measure. 
Moreover, we established a generalized Gauss-Green formula.

\smallskip
Let us now describe the contents of the present paper.
We underline that all our new results have been obtained assuming
that the divergence of the vector field with respect to $x$ is an $L^1$ function (see Section~\ref{ss:assumptions} for the detailed list of assumptions on $\bsmall$).
We believe that this can be considered as a first step in order to study the general case with a divergence--measure vector field.


In Section~\ref{ss:coarea},
we prove a coarea formula for the measure $(\bsmall(x,u),Du)$
and its variation
(see Theorems~\ref{p:coarea} and~\ref{p:coareavar}).


Then, in Section~\ref{repres},
we show that the pairing $(\bsmall(x,u),Du)$ admits a representation of the form
\begin{equation}
\label{represent1}
\begin{split}
(\bsmall(\cdot,u), Du) = {} &
\bsmall(x,u)\cdot\nabla u\,dx
+ \Cyl{\bsmall_{\ut}}{\nu_u}{\cdot}\, |D^c u|\\ & 
+ \left(\mean{u^-}^{u^+} \Cyl{\bsmall_t}{\nu_u}{\cdot}\, dt\right)
\, |D^j u|,\qquad u\in \BVLloc\,,
\end{split}
\end{equation}
where $\Cyl{\bsmall_t}{\nu_u}{\cdot}$
plays the role of a precise representative, defined by means of some cylindrical averages (see \eqref{defq} below). 
The above formula extends the representation formula for the pairing obtained by Anzellotti in the unpublished paper \cite{Anz2}
in the case of a vector field $\bsmall(x)$ independent of $u$.

In the same spirit, we prove a similar representation formula
\begin{equation}
\label{f:repr2}
\begin{split}
& (\bsmall(\cdot,u), Du)  =  
\bsmall(x,u)\cdot\nabla u\,dx
+
\Trace[]{\bsmall(\cdot,\widetilde u)}{\partial^* \{u>\widetilde u(x)\} }(x)|D^cu|\\
& +
\left(\mean{u^-(x)}^{u^+(x)}
\Trace[]{\bsmall(\cdot,t)}{\partial^* \{u>t\} }(x)
\, dt\right) |D^ju|,\quad u\in \BVLloc\,,
\end{split}
\end{equation}
based on the use of the weak normal traces as precise representatives
(see Section~\ref{distrtraces} for their definition).
This formula generalizes the representation obtained in the recent paper \cite{CCDM}
for vector fields $\bsmall(x)$ independent of $u$.

\smallskip
Sections~\ref{ss:lower} and~\ref{s:relax}
are devoted to the study of semicontinuity and relaxation.
The main result is the representation formula \eqref{t:generalenuovo1} for the relaxed functional.
Clearly, this representation formula, coupled with
\eqref{represent1} or \eqref{f:repr2},
gives a full integral representation of the relaxed functional.



In order to achieve \eqref{t:generalenuovo1}, we need to prove 
those which in relaxation theory are called
the liminf and the limsup inequalities. 
The first one is a consequence of the lower semicontinuity result (see Proposition \ref{p:cont} below), while the second one is obtained by using the blow-up method.



\section{Preliminaries}


Given $x_0\in\RN$ and $\rho>0$, $B_\rho(x_0)$ denotes the ball in $\RN$
centered in $x_0$ with radius $\rho$, while $\sfera^{N-1}$ is the
unit sphere of $\R^N$.

\noindent

In the following \(\Omega\) will always denote a nonempty open subset of \(\R^N\).
We denote by $\M(\Omega)$ the space of signed Radon measures
on $\Omega$.


\noindent
As usual,  ${\mathcal L}^{N}$ stands for the Lebesgue measure on $\RN$
and ${\H}^{k}$ for the $k$-dimensional Hausdorff
measure  on $\RN$.
The Lebesgue measure of the unit ball in $\RN$ is denoted by $\pallaunit$, hence
$\L^N(B_\rho(x_0))=\pallaunit\rho^N$.

For every $x\in \R^N$, $\Ixr(y) := (y-x) / r$ denotes the homothety with 
scaling 
factor $r$ mapping $x$ in $0$, and the pushforward $\Ixr[x, r_i]_\#\mu$ of 
a Radon measure $\mu$ in $\R^N$ is the 
measure  acting on a test function $\phi$ as 
\begin{equation*}
\int_{\R^N}\phi \,d(\Ixr[x, r]_\#\mu)=\int_{\R^N}\phi\circ \Ixr\, d\mu.
\end{equation*}

Let \(u\in L^1_{{\rm loc}}(\Omega, \R^m)\).
We say that \(u\) has an approximate limit at $x_{0}\in\Omega$ 
if there exists \(z\in\R^m\) such that
\begin{equation*}
	\lim_{r\rightarrow0^{+}}\frac{1}{\LLN\left(  B_r(x_0)\right)}\int_{B_r\left(  x_{0}\right)
	}\left|  u(x)  -z  \right|  \,dx=0.
\end{equation*}
The set $C_u\subset\Omega$ of points where this property holds is called the
\textsl{approximate continuity set} of $u$, whereas
the set \(S_u := \Omega\setminus C_u\)
is called the \textsl{approximate discontinuity set} of $u$.
For any $x\in C_u$ the approximate limit $z$ is uniquely 
determined and is denoted by $z:=\widetilde{u}(x)$.

We say that \(x\in\Omega\) is an {\sl approximate jump point} of \(u\) if
there exist \(a,b\in\R^m\), \(a\neq b\), and a unit vector \(\nu\in\R^N\) such that 
\begin{equation}
\label{f:disc}
\begin{gathered}
\lim_{r \to 0^+} \frac{1}{\LLN(B_r^i(x))}
\int_{B_r^i(x)} |u(y) - a|\, dy = 0,
\\
\lim_{r \to 0^+} \frac{1}{\LLN(B_r^e(x))}
\int_{B_r^e(x)} |u(y) - b|\, dy = 0,
\end{gathered}
\end{equation}
where \(B_r^i(x) := \{y\in B_r(x):\ (y-x)\cdot \nu > 0\}\), and 
\(B_r^e(x) := \{y\in B_r(x):\ (y-x)\cdot \nu < 0\}\).
The triplet \((a,b,\nu)\), uniquely determined by \eqref{f:disc} 
up to a permutation
of \((a,b)\) and a change of sign of \(\nu\),
is denoted by \((u^+(x_0), u^-(x_0), \nu_u(x_0))\).
The set of approximate jump points of \(u\) will be denoted by \(J_u\).




The space $BV(\Om)$ is defined as the space of all functions 
$u\colon\Omega\to \R$
belonging to $L^1(\Om)$ whose distributional gradient $Du$ is an
$\RN$-valued Radon measure
with total variation $\vert Du\vert$ bounded in $\Om$.
We indicate by $D^a u $ and $D^s u$ the {\it absolutely continuous} and
the {\it singular part} of the measure $Du$ with respect to
the Lebesgue measure. We recall that $D^a u$ and $D^s u$ are mutually
singular, moreover we can write
$$
Du = D^a u + D^s u \qquad {\rm and} \qquad D^a u = \nabla u ~\L^N,
$$
where $\nabla u$ is the {\it Radon-Nikod\'ym derivative} of $D^au$ with respect to
the Lebesgue measure.
In addition,
$$
D^su =  D^c u + (u^+-u^-)\nu_u~\H^{N-1}\ristretto{J_u}\,,
$$
where $J_u$ is a {\it countably $\H^{N-1}$-rectifiable} Borel set (see
\cite[Definition 2.57]{AFP}) contained in $S_u$, such that $\H^{N-1}(S_u\setminus J_u)=0$.
The remaining part $D^c u$ is called the {\it Cantor part} of $Du$.


A set $E\subset\Omega$ is of finite perimeter if 
its characteristic function $\chi_E$ belongs to $BV(\Omega)$.
If \(\Omega\subset\R^N\) is the largest open set such that \(E\)
is locally of finite perimeter in \(\Omega\),
we call \textsl{reduced boundary} \(\partial^* E\) of \(E\) the set of all 
points
\(x\in \Omega\) in the support of \(|D\chi_E|\) such that the limit
\[
\nuint_E(x) := \lim_{\rho\to 0^+} 
\frac{D\chi_E(B_\rho(x))}{|D\chi_E|(B_\rho(x))}
\]
exists in \(\R^N\) and satisfies \(|\nuint_E(x)| = 1\).
The function \(\nuint_E\colon\partial^* E\to S^{N-1}\) is called
the \textsl{measure theoretic unit interior normal} to \(E\).

A fundamental result of De Giorgi (see \cite[Theorem~3.59]{AFP}) states that
\(\partial^* E\) is countably \((N-1)\)-rectifiable,
\(|D\chi_E| = \hh\res \partial^* E\),
and $\nuint_E$ coincides (up to the sign) with the normal
$\nu_{\partial^* E}$ defined in Section~\ref{distrtraces}.
Moreover, the measure theoretic interior normal can be choosen as normal vector
to $\partial^* E$, in the sense of Section \ref{distrtraces}.

If $u\in BV(\Omega)$, then the level set
$E_t := \{u > t\}$ is of finite perimeter for a.e.\ $t\in\R$,
and we can choose the sign of the normal vectors so that
$\nuint_{E_t}(x) = \nu_{\Sigma_{t}}(x) = \nu_u(x)$
for $\hh$-a.e.\ $x\in\Sigma_t$,
where $\Sigma_t := \partial^*\{u>t\}$.

Moreover, we can choose an orientation on $J_u$ such that
$u^+(x) > u^-(x)$ for every $x\in J_u$.
We also set $u^-(x)=u^+(x) := \ut(x)$ for every $x\in C_u$,
and $u^*(x) := [u^+(x) + u^-(x)]/2$  for every $x\in C_u \cup J_u$.

The measure $Du$ can be disintegrated on the level sets of $u$
using the following coarea formula
(see \cite[Theorem 4.5.9]{Fed}).

\begin{theorem}[Coarea formula]\label{coarea}
If $u\in BV(\Omega)$, then for $\LLU$--a.e.\ $t\in\R$ the set $\{u>t\}$ has finite perimeter in $\Omega$ and  
the following coarea formula holds
\begin{equation*}
\int_\Omega
g\,d|Du|=\int_{-\infty}^{+\infty}\!\!dt\!\int_{\partial^* \{u>t\}
\cap\Omega}\!g\,d\hh
=
\int_{-\infty}^{+\infty}\!\!dt\!
\int_{\{u^-\leq t\leq u^+ \}}\!g\,d\hh\,,
\end{equation*}
for every Borel function $g:\Omega\to[0,+\infty]$.
Moreover, for $\LLU$--a.e. $t\in\R$,
\begin{itemize}
\item[(a)]
 $\partial^*{\{u>t\}} \subset \{u^-\leq t\leq u^+ \}$,
\item[(b)] $\hh\Bigl(\{u^-\leq t\leq u^+ \}\setminus
\bigl(\partial^*\{u>t\}\bigr)\Bigr)=0
$,
\end{itemize}
and, in particular,
\begin{itemize}
\item[(a$^\prime$)]
$
\partial^* \{u > t\} \cap (\Omega\setminus S_u)
\subseteq \{x\in \Omega\setminus S_u\colon \ut(x) = t\}
$,
\item[(b$^\prime$)]
$\hh\Bigl(\{x\in \Omega\setminus S_u\colon \ut(x)=t \}\setminus
\bigl((\Omega\setminus S_u)\cap\partial^*\{u>t\}\bigr)\Bigr)=0$.
\end{itemize}
\end{theorem}




\subsection{Divergence--measure fields }
\label{ss:div}

We will denote by \(\DM(\Omega)\) the space of all
vector fields 
\(\A\in L^\infty(\Omega, \R^N)\)
whose divergence in the sense of distributions is a bounded Radon measure in \(\Omega\).
Similarly, \(\DMloc[\Omega]\) will denote the space of
all vector fields \(\A\in L^\infty_{{\rm loc}}(\Omega, \R^N)\)
whose divergence in the sense of distribution is a Radon measure in \(\Omega\). 
We set \(\DM = \DM(\R^N)\).
Moreover, we denote by $\LDM$ (resp.\ $\LDMloc$)
the subset of $\DM(\Omega)$ (resp.\ $\DMloc$)
of vector fields whose divergence is in $L^1(\Omega)$
(resp.\ $L^1_{\text{loc}}(\Omega)$).

We recall that, if \(\A\in\DMloc[\Omega]\), then \(|\Div\A| \ll \hh\)
(see \cite[Proposition 3.1]{ChenFrid}).
As a consequence, the set
\begin{equation*}
	\jump{\A} 
	:= \left\{
	x\in\Omega:\
	\limsup_{r \to 0+}
	\frac{|\Div \A| (B_r(x))}{r^{N-1}} > 0
	\right\},
\end{equation*} 
is a Borel set, \(\sigma\)-finite with respect to \(\hh\),
and the measure \(\Div \A\) can be decomposed as
\[
\Div\A = \Div^a\A + \Div^c\A + \Div^j\A,
\]
where \(\Div^a\A\) is absolutely continuous with respect to \(\LLN\),
\(\Div^c\A (B) = 0\) for every set \(B\) with \(\hh(B) < +\infty\),
and
\[
\Div^j\A = h\, \hh\res\jump{\A}
\]
for some Borel function \(h\)
(see \cite[Proposition~2.3]{ADM}).


\subsection{Anzellotti's pairing}
As in Anzellotti \cite{Anz} (see also \cite{ChenFrid}),
for every \(\A\in\DMloc\) and \(u\in\BVLloc\) we define the linear functional
\((\A, Du) \colon C^\infty_0(\Omega) \to \R\) by
\begin{equation*}
	\pscal{(\A, Du)}{\varphi} :=
	-\int_\Omega u^*\varphi\, d \Div \A - \int_\Omega u \, \A\cdot \nabla\varphi\, dx. 
\end{equation*}
The distribution \((\A, Du)\) is a Radon measure in \(\Omega\),
absolutely continuous with respect to \(|Du|\)
(see \cite[Theorem 1.5]{Anz} and \cite[Theorem 3.2]{ChenFrid}),
hence the equation
\begin{equation*}
	\Div(u\A) = u^* \Div\A + (\A, Du)
\end{equation*}
holds in the sense of measures in \(\Omega\).
Furthermore, Chen and Frid in \cite{ChenFrid} proved that the absolutely continuous part
of this measure with respect to the Lebesgue measure is given by
\(
(\A, Du)^a = \A \cdot \nabla u\, \LLN
\).




In \cite{Anz2} it is proved that,
for every \(\A\in \LDMloc\),
\[
(\A, Du) = \Cyl{\A}{\nu_u}{\cdot}\, |Du|\,,
\qquad \text{$|Du|$--a.e.\ in $\Omega$,}
\]
where
\begin{equation}\label{defq}
\Cyl{\A}{\nu_u}{x}
:= \lim_{\rho\downarrow 0}
\lim_{r\downarrow 0}
\frac{1}{\LLN(C_{r,\rho}(x, \nu_u(x)))}
\int_{C_{r,\rho}(x, \nu_u(x))} \A(y) \cdot \nu_u(x)\, dy
\end{equation}
and, for every $\zeta\in\sfera^{N-1}$,
\begin{equation*}
C_{r,\rho}(x, \zeta) :=
\left\{
y\in\R^N:\ 
|(y-x)\cdot\zeta| < r,\
|(y-x) - [(y-x)\cdot\zeta]\zeta| < \rho
\right\}.
\end{equation*}
(The existence of the limit in \eqref{defq} for $|Du|$--a.e.\ $x\in\Omega$ is part of the statement.)

As a consequence, it holds that
\[
\lim_{r\downarrow 0} \frac{(\A, Du) (B_r(x))}{|Du|(B_r(x))}
= \Cyl{\A}{\nu_u}{x}
\qquad
\text{for $|Du|$--a.e.\ $x\in\Omega$}.
\]


\subsection{Weak normal traces on oriented countably $\Haus{N-1}$-rectifiable sets} \label{distrtraces}
We recall the notion of the traces of the normal component of a vector field \(\A\in 
\DMloc\) on an oriented countably \(\Haus{N-1}\)--rectifiable set
\(\Sigma\subset\Omega\),
introduced in 
\cite[Propositions~3.2, 3.4 and Definition~3.3]{AmbCriMan}.
In that paper the authors proved that there exist the normal traces $\Trp{\A}{\Sigma}, \Trm{\A}{\Sigma}$ belonging to
\(L^{\infty}(\Sigma, \Haus{N-1}\res\Sigma)\) 
and satisfying
\begin{equation}\label{f:trA}
\Div \A \res\Sigma =
\left[\Trp{\A}{\Sigma} - \Trm{\A}{\Sigma}\right]
\, {\mathcal H}^{N-1} \res\Sigma.
\end{equation}

In what follows we use the notation
\[
\Trace[*]{\A}{\Sigma}:= \frac{\Trp{\A}{\Sigma}+\Trm{\A}{\Sigma}}{2}\,.
\]
If $\A\in\LDMloc$, then $
\Trp{\A}{\Sigma}=\Trm{\A}{\Sigma}
$
and
$
\Div \A \res\Sigma =0.
$


\subsection{Representation formulas for the pairing measure}
In the following theorem, the pairing is characterized in terms of normal 
traces of the field $\A$ on level sets of $u$.


\begin{theorem}[see \cite{CCDM}, Thm.~3.9]
\label{t:repr}
Let $\A \in \DMloc[\R^N]$ and $u\in\BVLloc[\R^N]$.
Then, the following equality holds in the sense of measures
\begin{equation}
\label{f:repr}
(\A, Du) =
\mean{u^-(x)}^{u^+(x)}
\Trace[*]{\A}{\partial^* \{u>t\} }(x)
\, dt|Du|\,,
\end{equation}
where we use the convention $\mean{a}^a f(t)\, dt := f(a)$.
Moreover, 
\begin{itemize}
	\item[(i)]
	absolutely continuous part: 
	\((\A, Du)^a = \A \cdot \nabla u\, \LLN\);

	
	\item[(ii)]
	jump part:
	$
(\A,Du)^j = \Trace[*]{\A}{J_u}(x)|D^j u|;
$

	\item[(iii)]
	Cantor part:
$
(\A, Du)^c =
\Trace[*]{\A}{\partial^* \{u>\widetilde u(x)\} }(x)|D^cu|.
$
\end{itemize}

\end{theorem}


\begin{corollary}\label{c:paironch}
Let \(\A\in\DMloc\), and let $E\subseteq \Omega$ be a set of finite perimeter,
with $\overline{E}\subset\Omega$.
Then
\[
(\A,D\chi_E)=\Trace[*]{\A}{\partial^* E}(x)\hh\res\partial^* E.
\]
\end{corollary} 






\section{Assumptions on  the vector field \(\bsmall\)}
\label{ss:assumptions}

Let \(\bsmall\colon \Omega\times \R \to \R^N\) be a function
satisfying the following assumptions:
\begin{itemize}
\item[(i)]
\(\bsmall\) is a locally bounded Borel function;

\item[(ii)]
the function \(\bsmall(x, \cdot)\) is Lipschitz continuous in \(\R\),
uniformly with respect to $x$, i.e.\
there exists a constant $L>0$ such that
\[
|\bsmall(x,t) - \bsmall(x, s)| \leq L\, |t-s|,
\qquad
\forall t,s\in\R,\
\text{for \(\LLN\)--a.e.\ \(x\in\Omega\)}\,;
\]



\item[(iii)]
for every \(t\in\R\), \(\bsmall_t := \bsmall(\cdot, t)\in\LDMloc\);


\item[(iv)]
the least upper bound
\[
\sigma := \sup_{t\in \R} |\Div_x \bsmall_t|
\]
belongs to $L^1_{\rm {loc}}(\Omega)$.
\end{itemize}

\smallskip
We remark that, at the price of some additional
technicality, 
assumption (iv) could be replaced by the weaker assumption
\begin{itemize}
\item[(iv$^\prime$)]
for every $m>0$,
the least upper bound
\[
\sigma_m := \sup_{|t|\leq m} |\Div_x \bsmall_t|
\]
belongs to $L^1_{\rm {loc}}(\Omega)$.

\end{itemize}

The results of Section~\ref{ss:coarea} 
will be mainly proved replacing (ii) with the following weaker assumption:
\begin{itemize}
\item[(ii$^\prime$)]
for \(\LLN\)--a.e.\ \(x\in\Omega\),
the function \(\bsmall(x, \cdot)\) is continuous in \(\R\).
\end{itemize}



\bigskip

Let us extend $\bsmall$ to $0$ in $(\R^N\setminus\Omega)\times\R$,
so that the vector field
\begin{equation}\label{f:B}
\B(x,t) := \int_0^t \bsmall(x,s)\, ds,
\qquad x\in\R^N,\ t\in\R,
\end{equation}
is defined for all $(x,t)\in\R^N\times\R$.
Moreover \(\B(x,0) = 0\) for every \(x\in\R^N\)
and, from (ii$^\prime$), for every \(x\in\R^n\)
one has \(\bsmall(x,t) = \partial_t \B(x,t)\) for every \(t\in\R\).




\begin{theorem}[See \cite{CD4}]
\label{t:chainb4}
Let $\bsmall$
satisfy assumptions (i)-(ii$^\prime$)-(iii)-(iv),
let \(\B\) be defined by~\eqref{f:B},
and let $u\in  \BVLloc$. 
Then the distribution $(\bsmall(\cdot,u), Du)$,
defined by
\begin{equation}
\label{f:bxu}
\begin{split}
 \langle(\bsmall(\cdot,u), Du),\varphi\rangle  := {} &
-\int_\Omega
\varphi(x)\, (\Div_x \B) (x,u(x)) \, dx
\\ & -\int_\Omega \B(x,u(x))\cdot\nabla\varphi(x)\, dx,
\qquad \forall\varphi\in C_c^\infty(\Omega),
\end{split}
\end{equation}
is a Radon measure in $\Omega$, and satisfies
\begin{equation}\label{f:mubdd}
|(\bsmall(\cdot,u), Du)|(E)\leq
\|\bsmall\|_{L^\infty(K, \R^N)}
|Du|(E),
\qquad
\text{for every Borel set}\
E\Subset\Omega\,,
\end{equation}
where $K := \overline{E} \times \left[-\|u\|_{L^\infty(\overline{E})}, \|u\|_{L^\infty(\overline{E})}\right]$.

In other words,
the composite function $\vv\colon\Omega\rightarrow\R^N$,
defined by
$\vv(x):=\B(x,u(x))$,
belongs to $L^\infty_{\rm loc}(\Omega, \R^N)$,
and the following equality holds in the sense of measures:
\begin{equation}\label{lolo}
\begin{split}
\Div\,\vv =
(\Div_x\B)(x,u(x))\, \LLN
+(\bsmall(\cdot,u), Du).
\end{split}
\end{equation}
\end{theorem}

From \eqref{f:mubdd} it follows that
\((\bsmall(\cdot,u), Du) \ll |Du| \),
hence there exists a function
\(\Theta(\bsmall, u; \cdot) \in L^1(\Omega, |Du|)\)
such that
\begin{equation}\label{f:Theta}
(\bsmall(\cdot,u), Du) = \Theta(\bsmall, u; \cdot)\, |Du|\,,
\qquad \text{$|Du|$--a.e.\ in $\Omega$.}
\end{equation}


\begin{remark}
\label{r:repr}
By the definition \eqref{f:bxu} of the pairing and the definition \eqref{f:B} of $\B$, it follows that,
for every $\varphi\in C_c^\infty(\Omega)$,
\begin{equation}
\label{f:bxurepr}
\begin{split}
 \langle(\bsmall(\cdot,u), Du),\varphi\rangle  = {} &
-\int_\Omega \varphi(x) \int_0^{u(x)} \Div_x \bsmall_t(x)\, dt\, dx
\\ & - \int_\Omega \int_0^{u(x)} \bsmall_t(x)\cdot \nabla\varphi(x)\, dt\, dx.
\end{split}
\end{equation}
\end{remark}












\section{Coarea formula for the pairing measure}
\label{ss:coarea}

In this section we establish a coarea formula for the pairing measure
$(\bsmall(\cdot, u), Du)$,
and we draw some consequences
that will be used in order to prove its integral representation (see Theorem \ref{t:repr2} below).


\begin{theorem}[Coarea formula for the pairing measure]
\label{p:coarea}
Let $\bsmall$
satisfy assumptions (i)-(ii$^\prime$)-(iii)-(iv),
and let $u\in  \BVLloc$. 
Then
\begin{gather}
 \langle(\bsmall(\cdot,u), Du),\varphi\rangle
= \int_{\R}  \langle(\bsmall_t, D\chi_{\{u>t\}}),\varphi\rangle\,dt,
\qquad \forall\varphi\in C_c^\infty(\Omega)\,,
\label{f:coarea}
\\
(\bsmall(\cdot,u), Du) (B)
= \int_{\R}  (\bsmall_t, D\chi_{\{u>t\}}) (B)\,dt,
\qquad \forall\ \text{Borel set}\ B\subset\Omega\,.
\label{f:coareaB}
\end{gather}
\end{theorem}


\begin{proof}
Assume, for simplicity, that $u\geq 0$ and let $C> \|u\|_\infty$.
Using the representation \eqref{f:bxurepr}, we have that
\begin{equation*}
\begin{split}
 \langle(\bsmall(\cdot,u), Du),\varphi\rangle  = {} &
-\int_0^C \int_\Omega \chi_{\{u > t\}} \varphi \Div_x \bsmall_t \, dx\, dt
-\int_0^C \int_\Omega  \chi_{\{u > t\}} \bsmall_t\cdot\nabla\varphi\, dx\, dt
\\ = {} &
\int_0^C \langle(\bsmall_t, D\chi_{\{u>t\}}),\varphi\rangle\,dt,
\end{split}
\end{equation*}
where, in the last equality, we have used the fact that,
for $\LLU$-a.e.\ $t\in\R$,
\[
\Div (\chi_{\{u>t\}} \bsmall_t) = 
\chi_{\{u>t\}}^* \, \Div \bsmall_t + (\bsmall_t, D \chi_{\{u>t\}}).
\]
The general case follows with minor modifications.

Finally, since both sides of \eqref{f:coarea} are real measures in $\Omega$,
they coincide not only as distributions, but also as measures,
hence \eqref{f:coareaB} follows.
\end{proof}






The following approximation result is in the spirit of
\cite[Proposition~4.11]{CDM}, \cite[Proposition~4.15]{CD3},
\cite[Theorem~1.2]{ChenFrid}, \cite[Lemma~2.2]{Anz}.
\begin{theorem}[Approximation by $C^\infty$ fields]
\label{t:convolu}
Let $\bsmall$
satisfy assumptions (i)-(ii$^\prime$)-(iii)-(iv).
Then there exists a sequence of vector fields $\bsmall^k\colon\Omega\times\R\to\R^N$
satisfying the same assumptions, such that
$\bsmall^k_t \in C^\infty(\Omega, \R^N)$ for every $t\in\R$ and 
\[
(\bsmall^k(\cdot, u), Du) \stackrel{*}{\rightharpoonup}
(\bsmall(\cdot, u), Du),
\qquad \forall u\in\BVLloc,
\]
locally in the weak${}^*$ sense of measures in $\Omega$.
If, in addition, $\bsmall$ satisfies (ii),
then also the vector fields $\bsmall^k$ satisfy (ii).
\end{theorem}

\begin{proof}
Using the same construction described in the proof of
\cite[Proposition~4.11]{CDM},
we obtain locally uniformly bounded vector fields 
$\bsmall^k_t \in C^\infty(\Omega, \R^N)$ satisfying
(i)-(ii$^\prime$)-(iii)-(iv),  and, for every $t\in\R$,
$\bsmall^k_t \to \bsmall_t$ in $L^1_{loc}(\Omega)$.
If, in addition, $\bsmall$ satisfies (ii),
then it is verified that also the vector fields $\bsmall^k$ satisfy (ii).

Moreover, for every $t\in\R$ and $v\in\BVLloc$,
\[
\lim_{k\to +\infty}\int_{\Omega} v \, \varphi \, \Div\bsmall^k_t\, dx
= \int_{\Omega} v^* \,\varphi \, d \Div\bsmall_t,
\qquad \forall \varphi\in C_c(\Omega)
\]
(see \cite{CDM}, formula (4.8)).
We underline that, since by assumption $\Div\bsmall_t\in L^1_{loc}(\Omega)$,
then the above relation can be written as
\begin{equation}
\label{f:ivp0}
\lim_{k\to +\infty}\int_{\Omega} v \, \varphi \, \Div\bsmall^k_t\, dx
= \int_{\Omega} v \,\varphi \, \Div\bsmall_t\, dx,
\qquad \forall \varphi\in C_c(\Omega)\,.
\end{equation}
Let us fix $u\in \BVLloc$ and $\varphi\in C_c(\Omega)$.
To simplify the notation, we assume without loss of generality
that $u\geq 0$.
By the representation formula \eqref{f:bxurepr} and Fubini's Theorem, we have that
\begin{equation*}
\begin{split}
 \langle(\bsmall^k(\cdot,u), Du),\varphi\rangle  = {} &
-\int_\Omega \varphi(x) \int_0^{u(x)} \Div_x \bsmall^k_t(x)\, dt\, dx
\\ & - \int_\Omega \int_0^{u(x)} \bsmall^k_t(x)\cdot \nabla\varphi(x)\, dt\, dx
\\ = {} &
- \int_0^{\infty} \int_\Omega \chi_{\{u > t\}}(x)\, \varphi(x)\,\Div_x \bsmall^k_t(x)\,dx\, dt
\\ & 
- \int_0^{\infty} \int_\Omega \chi_{\{u > t\}}(x)\, \bsmall^k_t(x)\cdot
\nabla\varphi(x)\, dx \, dt
\\ =: {} & -I^k_1 - I^k_2.
\end{split}
\end{equation*}
For every $t\in\R$, by \eqref{f:ivp0} with $v = \chi_{\{u > t\}}$ we 
deduce that, as $k\to\infty$,
\[
\zeta^k(t) := 
\int_\Omega \chi_{\{u > t\}}(x)\, \varphi(x)\,\Div_x \bsmall^k_t(x)\,dx
\to
\int_\Omega \chi_{\{u > t\}}(x)\, \varphi(x)\,\Div_x \bsmall_t(x)\,dx
\,.
\]
Let $K\Subset \Omega$ denote the support of $\varphi$
and let $a := \|u\|_{L^\infty(K)}$.
Since
\[
|\zeta^k(t)| \leq \chi_{[0,a]}(t) \, \|\varphi\|_\infty \int_K \sigma\,,
\]
by the Dominated Convergence Theorem we deduce that
\begin{equation}
\label{f:Ik1}
\lim_{k\to \infty} I^k_1 =
\int_0^{\infty} \int_\Omega \chi_{\{u > t\}}(x)\, \varphi(x)\,\Div_x \bsmall^k_t(x)\,dx\, dt
= 
\int_\Omega \varphi(x) \int_0^{u(x)} \Div_x \bsmall^k_t(x)\, dt\, dx\,.
\end{equation}

Let us compute the limit of $I^k_2$.
Since, for every $t\in\R$, $\bsmall^k_t\to \bsmall_t$ in $L^1_{loc}(\Omega)$,
it holds that
\[
\psi^k(t) := \int_\Omega \chi_{\{u > t\}}(x)\, \bsmall^k_t(x)\cdot
\nabla\varphi(x)\, dx \to
\int_\Omega \chi_{\{u > t\}}(x)\, \bsmall_t(x)\cdot
\nabla\varphi(x)\, dx\,.
\]
Moreover, 
there exists a constant $M> 0$ such that
$\|\bsmall^k\|_{L^\infty(K\times [-a, a])}\leq M$ for every $k\in\N$,
so that
\[
\psi^k(t)| \leq M\, \|\nabla\varphi\|_{1}\,,
\]
and hence, by the Dominated Convergence Theorem,
\begin{equation}
\label{f:Ik2}
\lim_{k\to +\infty} I^k_2
=
 \int_0^{\infty} \int_\Omega \chi_{\{u > t\}}(x)\, \bsmall_s(x)\cdot
\nabla\varphi(x)\, dx \, dt
=
\int_\Omega \int_0^{u(x)} \bsmall^k_t(x)\cdot \nabla\varphi(x)\, dt\, dx
\,.
\end{equation}
The conclusion now follows from \eqref{f:Ik1} and \eqref{f:Ik2}.
\end{proof}



\begin{proposition}\label{l:theta}
Let $\bsmall$
satisfy assumptions (i)-(ii$^\prime$)-(iii)-(iv),
and let $u\in \BVLloc[\Omega]$.
Then
\begin{equation*}
\text{for $\mathcal{L}^1$-a.e.}\ t\in\R:
\quad
\Theta(\bsmall, u; x) =
\Theta(\bsmall_t, \chiut; x)
\quad
\text{for \(|D\chiut|\)-a.e.}\ x\in\Omega\,.
\end{equation*}
\end{proposition}

\begin{proof}
The proof is essentially the same of Proposition~5.2 in
\cite{CDM}, and it is based on the use
of the coarea formula (Theorem~\ref{p:coarea}) and the approximation result by smooth fields (Theorem~\ref{t:convolu}).
\end{proof}


\begin{theorem}[Coarea formula for the variation]
\label{p:coareavar}
Let $\bsmall$
satisfy assumptions (i)-(ii$^\prime$)-(iii)-(iv),
and let $u\in \BVLloc[\Omega]$.
Then
\begin{equation*}
 \langle\left|(\bsmall(\cdot,u), Du)\right|\,,\,\varphi\rangle
= \int_{\R}  \langle\left|(\bsmall_t, D\chi_{\{u>t\}})\right|\,,\,\varphi\rangle\,dt,
\qquad \forall\varphi\in C_c^\infty(\Omega).
\end{equation*}
\end{theorem}

\begin{proof}
To simplify the notation let $\mu := (\bsmall(\cdot,u), Du)$ and
$\mu_t := (\bsmall_t, D\chi_{\{u>t\}})$, $t\in\R$.
By \eqref{f:Theta}, we have that
$\mu = \Theta(\bsmall, u) \, |Du|$, so that 
\[
|\mu| = |\Theta(\bsmall, u)| \, |Du|\,,
\qquad 
|\mu_t| = |\Theta(\bsmall_t, \chiut)|\,  |D\chiut|
\]
(see \cite[Proposition~1.23]{AFP}).
Let $B\subset\Omega$ be a Borel set.
By the coarea formula in BV (see \cite[Theorem~3.40]{AFP}) and
Proposition~\ref{l:theta}
it holds that
\[
\begin{split}
|\mu|(B) & = \int_B |\Theta(\bsmall, u)| \, d|Du|
= \int_{\R} dt \int_B |\Theta(\bsmall, u)|\, d |D\chiut|
\\ &
= \int_{\R} dt \int_B |\Theta(\bsmall_t, \chiut)|\, d |D\chiut|
= \int_{\R} |\mu_t|(B)\, dt\,,
\end{split}
\]
concluding the proof.
\end{proof}


\begin{lemma}
\label{l:lip}
Let $\bsmall$ satisfy (i)--(iv), and let
$u\in  \BVLloc$.
Then, for every $\tau\in\R$
and every $\varphi\in C_c^\infty(\Omega)$, it holds that
\begin{equation*}
\begin{split}
& \left|\langle(\bsmall(\cdot,u), Du),\varphi\rangle
- \langle(\bsmall_\tau, Du),\varphi\rangle\right|
\\ & \leq
L \, \|\varphi\|_\infty \left[
\int_{\spt\varphi} |\ut-\tau|\, d |D^d u|
+ \int_{J_u\cap\spt\varphi} \left(\int_{u^-}^{u^+} |t-\tau|\, dt\right) d \Haus{N-1}
\right],
\end{split}
\end{equation*}
where $\spt\varphi \Subset \Omega$ denotes the support of $\varphi$.
\end{lemma}

\begin{proof}
Using the coarea formula \eqref{f:coarea} and (ii)
we obtain that
\begin{equation*}
\begin{split}
I_\tau & :=  \left|\langle(\bsmall(\cdot,u), Du),\varphi\rangle
- \langle(\bsmall_\tau, Du),\varphi\rangle\right|
= \left| \int_\R  \langle(\bsmall_t - \bsmall_\tau, D\chi_{\{u>t\}}),\varphi\rangle\,dt \right|
\\ & \leq
\|\varphi\|_\infty
\int_\R \int_{\spt\varphi} \|\bsmall_t - \bsmall_\tau\|_\infty\, d |D\chi_{\{u>t\}}|\, dt
\\ & \leq
L\, \|\varphi\|_\infty
\int_\R \int_{\spt\varphi} |t - \tau| \, d |D\chi_{\{u>t\}}|\, dt\,.
\end{split}
\end{equation*}
We now consider $\spt\varphi$ as the disjoint union of
$\spt\varphi\setminus J_u$ and  $J_u\cap {\spt\varphi}$,
and we use the coarea formula in $BV$, obtaining
\begin{equation*}
\begin{split}
I_\tau & \leq
L\, \|\varphi\|_\infty
\left[\int_\R \int_{{\spt\varphi}\setminus J_u} |t - \tau| \, d |D\chi_{\{u>t\}}|\, dt
+ \int_\R \int_{J_u\cap {\spt\varphi}} |t - \tau| \, d |D\chi_{\{u>t\}}|\, dt\right]
\\ & =
L \, \|\varphi\|_\infty \left[\int_{{\spt\varphi}} |\ut-\tau|\, d |D^d u|
+ \int_{J_u\cap {\spt\varphi}} \left(\int_{u^-}^{u^+} |t-\tau|\, dt\right) d \Haus{N-1}
\right].
\qedhere
\end{split}
\end{equation*}
\end{proof}



\section{Integral representation of the pairing}
\label{repres}

In this section we are interested in finding an integral representation of the pairing measure and of its total variation. 
We prove that the pairing measure can be represented by an integral functional defined on the space $BV(\Omega)$,
provided that in the support of the singular part of the measure
we choose a suitable precise representative of the vector field $\bsmall$. 

We recall the general form of an integral functional defined in $BV(\Omega)$.
Given the integrand $f(x,t,\xi)=\bsmall(x,t)\cdot\xi$,
for every open set $A\subset\Omega$, let us define the functional 
$\F(\cdot,A)\colon BV(\Om)\to]-\infty,+\infty]$ by setting
\begin{equation*}
\begin{split}
\F(u,A) = {} & \int_A\bsmall(x,u)\cdot \nabla u \, dx 
\\ & +
\int_A\overline f\Bigl(x,\preciso{u},\frac{D^c u}{|D^c u|}\Bigr)\, d|D^c u|
+ \int_{J_u\cap A}d\H^{N-1}\int_{u^-}^{u^+}\overline f(x,t,\nu_u)\,dt,
\end{split}
\end{equation*}
where $\overline f(\cdot,s,\xi)$ is a proper precise representative of $f(\cdot,t,\xi)=\bsmall_t\cdot\xi$.
\medskip


We show that, in our case, this representative is 
the limit of cylindrical averages introduced in \cite{Anz2} for vector fields $\bsmall(x)$ whose divergence belongs to $L^1$. 




\begin{theorem}[Integral representation of the pairing measure]
\label{t:repr2}
Let $\bsmall$
satisfy assumptions (i)--(iv),
and let $u\in  \BVLloc$.
Then it holds that
\begin{equation*}
\begin{split}
(\bsmall(\cdot,u), Du) = {} &
\Cyl{\bsmall_{\ut}}{\nu_u}{\cdot}\, |D^d u|
+ \left(\mean{u^-}^{u^+} \Cyl{\bsmall_t}{\nu_u}{\cdot}\, dt\right)
\, |D^j u|.
\end{split}
\end{equation*}
In other words,
the density \(\Theta\) defined at \eqref{f:Theta} is given by
\begin{equation}
\label{represent2}
\Theta(\bsmall, u; x) = 
\begin{cases}
\Cyl{\bsmall_{\ut(x)}}{\nu_u}{x}\,,
& \text{$|D^d u|$--a.e.\ $x\in\Omega$},
\\
\displaystyle\mean{u^-(x)}^{u^+(x)} \Cyl{\bsmall_t}{\nu_u}{x}\, dt
\,,
& \text{$\H^{N-1}$--a.e.\  $x\in J_u$}.
\end{cases}
\end{equation}
Moreover,
\(
\Cyl{\bsmall_{\ut(x)}}{\nu_u}{x} |\nabla u(x)| = 
{\bsmall(x, u(x))}\cdot {\nabla u(x)}
\)
for $\LLN$-a.e.\ $x\in\Omega$.
\end{theorem}

\begin{proof}
By assumption (iii), for every $t\in\R$ we have that
\begin{gather*}
\frac{d (\bsmall_t, Du)}{d |D u|}
= \Cyl{\bsmall_t}{\nu_u}{x},
\qquad
\text{$|D^d u|$--a.e.\ in $\Omega$},
\\
\Trp{\B_t}{J_u} = \Trm{\B_t}{J_u} =
\Cyl{\B_t}{\nu_u}{\cdot},
\qquad
\text{$\H^{N-1}$--a.e.\ in $J_u$}
\end{gather*}
(see \cite[Theorems 2.6 and 3.6]{Anz2}).

The representation of the jump part (i.e.\ of $\Theta$ on $J_u$)
follows directly from \cite[Theorem~5.6]{CD4}
and the simple computation
\[
\Cyl{\B_{u^+(x)}}{\nu_u}{x} - \Cyl{\B_{u^-(x)}}{\nu_u}{x}
= \int_{u^-(x)}^{u^+(x)} \Cyl{\bsmall_t}{\nu_u}{x}\, dt\,.
\]
It remains to prove that
\[
\Theta(\bsmall, Du; \cdot) = 
\Cyl{\bsmall_{\ut}}{\nu_u}{\cdot}
\qquad \text{$|D^d u|$--a.e.\ in $\Omega$}\,.
\]
First, we remark that there exists a Borel set
$N\subset\Omega$, with $|D^d u|(N) = 0$,
such that the limit of cylindrical averages $\Cyl{\bsmall_t}{\nu_u}{\cdot}$
exists for every $x\in\Omega\setminus N$ and every $t\in\R$
(see e.g.\ the proof of \cite[Lemma~4.2]{CD4}).
As a consequence, the map 
$x\mapsto \Cyl{\bsmall_{\ut(x)}}{\nu_u}{x}$
belongs to $L^\infty_{\rm loc}(\Omega, |D^d u|)$. 


To simplify the notation, let us denote by
$\mu := (\bsmall(\cdot, u), Du)$ the pairing measure.
We have to prove that
\[
\frac{d\mu^d}{d|Du|}(x) =
\Cyl{\bsmall_{\ut(x)}}{\nu_u}{x}\,,
\quad
\text{for $|D^d u|$--a.e.\ $x\in \Omega$.}
\]

Let us choose $x\in \Omega$ such that
\begin{itemize}
	\item[(a)]
	\(x\) belongs to the support of \(D^d u\), that is
	\(|D^d u|(B_r(x)) > 0\) for every \(r>0\);
	\item[(b)] 
	there exists the limit
	\(\displaystyle
\lim_{r\downarrow 0}
\frac{\mu^d(B_r(x))}{|D u|(B_r(x))} = \frac{d\mu^d}{d|Du|}(x)\,;
	\)
	\item[(c)]
	\(\displaystyle
\lim_{r\downarrow 0}\frac{|D^j u|(B_r(x))}{|D u|(B_r(x))}=0
	\);

	\item[(d)]
	\(\displaystyle
\lim_{r\downarrow 0}\frac{(\bsmall_{\ut(x)}, Du)(B_r(x))}{|D u|(B_r(x))}
=	\Cyl{\bsmall_{\ut(x)}}{\nu_u}{x}
	\);
\item[(e)]
	\(\displaystyle
\lim_{r\downarrow 0}
\frac{1}{|Du|(B_r(x))}
\int_{B_r(x)} |\ut(y)-\ut(x)|\, d |D^d u|(y) = 0
\).
\end{itemize}
We remark that these conditions are satisfied
for $|D^d u|$-a.e.\ $x\in\Omega$.
In particular, (e) holds since $|D^d u|$-a.e.\ $x\in\Omega$
is a Lebesgue point of $\ut$ with respect to $|Du|$.

Since
\[
\begin{split}
& \left|
\frac{(\bsmall(\cdot, u), Du)(B_r(x))}{|Du|(B_r(x))}
-  \Cyl{\bsmall_{\ut(x)}}{\nu_u}{x}
\right|
\\ & \leq
\left|
\frac{(\bsmall(\cdot, u), Du)(B_r(x))}{|Du|(B_r(x))}
-  \frac{(\bsmall_{\ut(x)}, Du)(B_r(x))}{|Du|(B_r(x))}
\right|
+
\left|
\frac{(\bsmall_{\ut(x)}, Du)(B_r(x))}{|Du|(B_r(x))}
-  \Cyl{\bsmall_{\ut(x)}}{\nu_u}{x}
\right|,
\end{split}
\]
by (d) it is enough to prove that
\begin{equation}\label{f:diffuse}
I_r := \left|\frac{(\bsmall(\cdot, u), Du)(B_r(x))}{|Du|(B_r(x))}
-  \frac{(\bsmall_{\ut(x)}, Du)(B_r(x))}{|Du|(B_r(x))}
\right| \longrightarrow 0\,,
\qquad\text{as}\ r\searrow 0,
\end{equation}
i.e.
\[
\frac{d(\bsmall(\cdot, u), Du)}{d|Du|}(x)
=  \frac{d(\bsmall_{\ut(x)}, Du)}{d|Du|}(x)\,.
\]


By Lemma~\ref{l:lip},
choosing $\tau = \ut(x)$ and
taking a sequence $\phi_j\in C^\infty_c(B_r(x))$,
$\phi_j(y)\to 1$ in $B_r(x)$, with \(0\leq \phi_j \leq 1\), we get
\[
\begin{split}
I_r \leq {} &
\frac{L}{|Du|(B_r(x))}
\bigg[
\int_{B_r(x)} |\ut(y)-\ut(x)|\, d |D^d u|(y)
\\ & 
+ \int_{B_r(x) \cap J_u} \left(
\int_{u^-(y)}^{u^+(y)} |t - \ut(x)|\, dt
\right)
\, d\H^{N-1}(y)
\bigg]
\\ \leq {} &
\frac{L}{|Du|(B_r(x))}
\int_{B_r(x)} |\ut(y)-\ut(x)|\, d |D^d u|(y)
+
2L \|u\|_\infty \frac{|D^j u|(B_r(x))}{|Du|(B_r(x))}\,.
\end{split}
\]
Finally, by (c) and (e)
we conclude that \eqref{f:diffuse} holds.
\end{proof}

\begin{corollary}
[Integral representation of the pairing functional]
Let $\bsmall$
satisfy assumptions (i)--(iv),
and let $u\in  \BVLloc$.
Then it holds that
\begin{equation}
\begin{split}
\label{modulo}
\int_\Omega\phi(\bsmall(x,u),Du)= 
\int_\Omega\phi\mean{u^-}^{u^+} \Cyl{\bsmall_t}{\nu_u}{x}\, dt
|Du|,
\qquad \phi\in C_c(\Omega)\,,
\end{split}
\end{equation}
where we use the compact notation
\begin{equation}
\begin{split}
\label{modulo1}
\mean{u^-}^{u^+} \Cyl{\bsmall_t}{\nu_u}{x}\, dt
|Du|:= 
{} &
\bsmall(x,u)\cdot\nabla u\,dx
+ \Cyl{\bsmall_{\ut}}{\nu_u}{x}\, |D^c u|
\\ & 
+ \mean{u^-}^{u^+} \Cyl{\bsmall_t}{\nu_u}{x}\, dt
|D^j u|.
\end{split}
\end{equation}

\end{corollary}


\begin{remark}
As a direct consequence of the above corollary, it holds that
\begin{equation*}
\frac{(\bsmall(\cdot,u),Du)}{|D^c u|}(x)=\Cyl{\bsmall_{\ut}}{\nu_u}{x}
\end{equation*}
for $|D^c u|$-a.e. $\in\Omega$.
\end{remark}

\begin{theorem}\label{t:repr3}
Let $\bsmall$
satisfy assumptions (i)--(iv), and $u\in\BVLloc$.
Then,
for $|Du|$-a.e.\ $x\in \Omega$,
\begin{equation*}
\Theta(\bsmall, u; x) =
\mean{u^-(x)}^{u^+(x)}
\Trace[]{\bsmall_t}{\partial^* \{u>t\} }(x)
\, dt\,,
\end{equation*}
where we use the convention $\mean{a}^a f(t)\, dt := f(a)$.
In particular, 
\begin{equation}
\label{f:reprC}
\Theta(\bsmall, u; x) =
\Trace[]{\bsmall(\cdot,u)}{\partial^* \{u>\widetilde u(x)\} }(x),
\qquad
\text{for $|D^du|$-a.e.\ $x\in \Omega$}.
\end{equation}
\end{theorem}

\begin{proof}
It suffices to prove that, for every Borel set $B\subset \Omega$,
it holds that
\begin{equation}\label{f:repr177}
\int_B\bsmall(\cdot,u), Du) =\int_B
\mean{u^-(x)}^{u^+(x)}
\Trace[]{\bsmall_t}{\partial^* \{u>t\} }(x)
\, dt\,|Du|.
\end{equation}
For every $t\in\R$ such that $\partial^* \{u > t\}$ is locally of finite perimeter
(and hence for a.e.\ $t\in\R$),
by Corollary \ref{c:paironch} 
we deduce that
\begin{equation}\label{f:repr100}
\Theta(\bsmall_t, D\chi_{\{u > t\}}, x)
= \Trace[]{\bsmall_t}{\partial^* \{u > t\} }(x)
\quad
\text{for $\hh$-a.e.}\ x\in \partial^* \{u > t\}.
\end{equation}
Then, by using the coarea formula \eqref{f:coarea} for the pairing measure, formula \eqref{f:repr100}, Fubini's theorem and Theorem~\ref{coarea}  we get
\[
\begin{split}
\int_B & \bsmall(\cdot,u), Du) =\int_{\R} \int_B (\bsmall_t, D\chi_{\{u>t\}})\, dt
=\int_{\R} \int_{B\cap \partial^*\{u>t\}} \Trace[]{\bsmall_t}{\partial^*\{u>t\}}
	d\hh\, dt
\\
& =\int_{B\setminus J_u}
\Trace[]{\bsmall_t}{\partial^* \{u>\ut(x)\} }
\,d\H^{N-1}+
\int_{B\cap J_u}
\int_{u^-(x)}^{u^+(x)}
\Trace[]{\bsmall_t}{\partial^* \{u>t\} }
\, dt\,d\H^{N-1}
\\
& =
\int_B
\mean{u^-(x)}^{u^+(x)}
\Trace[]{\bsmall_t}{\partial^* \{u>t\} }
\, dt\,|Du|\,,
\end{split}
\]
so that \eqref{f:repr177} is proved.
\end{proof}



\section{Lower semicontinuity of the pairing}
\label{ss:lower}

In this section, by using the nonautonomous chain rule formula \eqref{lolo} for the divergence, we study the lower semicontinuity with respect to the 
$L^1$
convergence 
of the functionals
$F, G^+\colon \BVL\to\R$ defined by
\[
F(u):= \int_\Omega |(\bsmall(x, u),Du)|\,,
\quad
G(u):= \int_\Omega (\bsmall(x, u),Du)\,,
\quad
G^+(u):= \int_\Omega (\bsmall(x, u),Du)^+\,.
\]






We start by proving the following continuity result (see \cite{dcl} for the
analogous result in $W^{1,1}$).

\begin{proposition}\label{p:cont}
Let $\bsmall$
satisfy assumptions (i)--(iv),
let $\varphi\in C^1_c(\Omega)$ be a fixed test function,
and let $G_\varphi\colon \BVL\to\R$ be the functional defined by
\[
G_\varphi(u) :=
\pscal{(\bsmall(x, u), Du)}{\varphi}= \int_\Omega \varphi\, d (\bsmall(x, u),Du)\,,
\qquad u\in\BVL.
\]
Then,  
for every sequence $(u_j) \subset  \BVL$ converging to
$u\in \BVL$ in the $L^1$ convergence,
and satisfying 
\begin{equation*}
L := \sup_j \|u_j\|_{L^\infty(\Omega)} < +\infty,
\end{equation*}
it holds that
\begin{equation}\label{continu}
\lim_{j\to +\infty} G_\varphi(u_j) = G_\varphi(u).
\end{equation}
Moreover, by assuming, instead of (iv), the stronger condition $\sigma\in L^N_{\rm {loc}}(\Omega)$ (respectively $\sigma\in L^\infty_{\rm {loc}}(\Omega)$), the continuity \eqref{continu} holds 
if $(u_j)$ converges to $u$ weakly$^*$ in $BV(\Omega)$
(respectively strongly in $L^1_{\rm {loc}}(\Omega)$).
\end{proposition}

\begin{proof}
Using \eqref{f:bxurepr} we have that
\[
G_\varphi(u_j) - G_\varphi(u) =
-\int_\Omega \varphi(x) \int_{u(x)}^{u_j(x)} \Div_x \bsmall_t(x)\, dt\, dx
- \int_\Omega \int_{u(x)}^{u_j(x)} \bsmall_t(x)\cdot \nabla\varphi(x)\, dt\, dx.
\]
Since $\bsmall$ is a locally bounded vector field, the second integral converges to $0$
by the Lebesgue Dominated Convergence theorem.
The first integral can be written as
\begin{equation}\label{f:fi}
\iint_{K\times[-L, L]} \sign(u(x) - u_j(x))
\chi_{D_j}(x,s)\, \Div_x\bsmall_s(x)\, dx\, ds,
\end{equation}
where $K\subset\Omega$ is the support of $\varphi$ and 
$D_j \subset\Omega\times [-L, L]$ is the set of pairs $(x,s)$ such that
$s$ belongs to the segment of endpoints $u(x)$ and $u_j(x)$.
Since
\[
\left|
\chi_{D_j}(x,s) \varphi(x)\, \Div_x \bsmall_s(x)
\right|
\leq
\|\varphi\|_{L^\infty(\Omega)}
\, |\Div_x \bsmall_s(x)|
\leq
\|\varphi\|_{L^\infty(\Omega)}
\, \sigma(x)
\in L^1(K\times[-L, L]),
\]
the integral \eqref{f:fi} converges to $0$ by the Lebesgue Dominated Convergence theorem.

Let us prove the second part of the theorem.
If the sequence $(u_j) \subset  \BVL$ weak$^*$-converges to
$u\in \BVL$, by the Poincaré inequality
(see \cite[Remark~3.50]{AFP})
 we have that $(u_j)$ weakly converges to
$u$ in $L^{\frac N{N-1}}(\Omega)$.
Since
$$
\left|\int_\Omega \varphi(x) \int_{u(x)}^{u_j(x)} \Div_x \bsmall_t(x)\, dt\, dx\right|\leq
\|\varphi\|_\infty
\int_K |u_j(x)-u(x)|  \sigma(x)\, dx,
$$
then, if $\sigma\in L^N_{\rm {loc}}(\Omega)$,
 the integral on the right-hand side converges to $0$. 
The same conclusion holds if $(u_j)$ converges to
$u$ strongly in $L^1_{\rm {loc}}(\Omega)$ and $\sigma\in L^\infty_{\rm {loc}}(\Omega)$.
\end{proof}

\begin{remark}
In \cite{DM} Dal Maso proved the lower semicontinuity
of integral functionals with coercive integrands and he showed, by exploiting Aronszajn's example, that
this result is sharp, in the sense that, in general, 
the coercivity assumption cannot be dropped. 
Indeed, Dal Maso constructed a continuous function
$\omega\colon\Omega\to \R$, where $\Omega=(0,1)\times (0,1)$ and
$x=(x_1,x_2)$, and a sequence of functions $\{u_n\}$
 converging to $u(x)=x_2$ in $L^\infty(\Omega)$, such that
\[
\int_\Omega |(\sin \omega (x),\cos \omega(x))\cdot\nabla u(x))|\,dx
>\liminf_{n\to\infty}
\int_\Omega |(\sin \omega(x),\cos \omega(x))\cdot\nabla u_n(x))|\,dx.
\]
Let us remark that the integrand $|\bsmall(x)\cdot\xi|$ of Dal Maso's example does not satisfy our condition $\Div\bsmall\in L^1$.
\end{remark}

\begin{theorem}\label{p:lsc}
[Lower semicontinuity]
Let $\bsmall$
satisfy assumptions (i)--(iv). 
Then the functionals $F, G^+$ are lower semicontinuous on $\BVL$ 
with respect to the
$L^1$ convergence. 
\end{theorem}


\begin{proof}
Let us define the auxiliary functionals
$H, H^+\colon\BVL\to\R$ by
\[
H(u):= -\int_\Omega (\bsmall(x, u),Du)\,,
\qquad
H^+(u):= \int_\Omega (-(\bsmall(x, u),Du))^+\,.
\]
Since $F(u)=[G^+(u)+H^+(u)]/2$,
it suffices to prove that $G^+(u)$ and $H^+(u)$ are lower semicontinuous on $\BVL$ with respect to the
$L^1$ convergence. 
We shall prove the claim only for $G^+$, being the proof for $H^+$ similar.




Let us prove that $\liminf_n G^+(u_n) \geq G^+(u)$
for every sequence $(u_n) \subset \BVL$ 
converging to $u\in  \BVL$ in $L^1(\Omega)$. 


For every $k\in\mathbb{N}$
let us consider the Lipschitz function $\sigma_{k}\colon\R\to\R$ defined by
\begin{equation*}
\sigma_{k}(t):=
\begin{cases}
1\quad & |t|\leq k-1,\\
-|t|+k\quad & k-1<|t|\leq k,\\
0\quad & |t|>k.
\end{cases}
\label{2.5}
\end{equation*}
The vector field
\[
\bsmall^{k}\left(  x,t\right)  :=\sigma_{k}\left(  t\right)  \,\bsmall(x,t)
\]
satisfies 
\begin{equation*}
\bsmall^k(x,t) =0 {\hbox{ for }}|t|\geq k,\qquad 
\bsmall^k(x,t) =\bsmall(x,t)  {\hbox{ for }}|t|\leq k-1.
\end{equation*}

We claim that, for every $v\in \BVL$, 
\begin{gather}
\Theta(\bsmall^k, v; x) = \sigma_k(\widetilde{v}(x))\, \Theta(\bsmall, v; x)\,,
\qquad 
\text{for $|D^d v|$-a.e.\ $x\in\Omega$},
\label{f:thetak1}
\\
\Theta(\bsmall^k, v; x) =
\mean{v^-(x)}^{v^+(x)}\sigma_k(t)
\Cyl{\bsmall_t}{\nu_v}{x}\, dt\,,
\qquad
\text{ for $\hh$-a.e. $x\in\Omega\cap J_v$}.
\label{f:thetak2}
\end{gather}
Specifically, both relations are a consequence of the representation formula \eqref{represent2} and
of the equality
\[
\begin{split}
\Cyl{\bsmall^k_t}{\nu_v}{x}
& =
\lim_{\rho\downarrow 0}
\lim_{r\downarrow 0}
\frac{1}{\LLN(C_{r,\rho}(x, \nu_v(x)))}
\int_{C_{r,\rho}(x, \nu_v(x))} \sigma_k(t)\, \bsmall_t(y) \cdot \nu_v(x)\, dy
\\ & 
=
\sigma_k(t)\, \Cyl{\bsmall_t}{\nu_v}{x}\,.
\end{split}
\]

We now proceed as in \cite{DCbumi}.
Given $\varphi\in C^1_c(\Omega)$,
let us consider the following functionals, defined in $\BVL$:
\[
\begin{split}
F^1_\varphi(v) := {} &
\int_{\Omega}
\varphi(x)
\Cyl{\bsmall_{\widetilde{v}(x)}}{\nu_{v}}{x}|D^d v|
+ \int_{\Omega\cap J_{v}}
\varphi(x)
\mean{v^-(x)}^{v^+(x)} 
\Cyl{\bsmall_t}{\nu_{v}}{x}\, dt\, 
d\hh\,,
\\
F^2_\varphi(v) := {} &
\int_{\Omega}
\varphi(x)
\Big[\Cyl{\bsmall_{\widetilde{v}(x)}}{\nu_{v}}{x}\Big]^+|D^d v|
\\ &
+\int_{\Omega\cap J_{v}}
\varphi(x)
\mean{v^-(x)}^{v^+(x)} \Big[ 
\Cyl{\bsmall_t}{\nu_{v}}{x}\, dt\Big]^+\, 
d\hh\,,
\\
F^3_\varphi(v) := {} &
\int_{\Omega}
\varphi(x)
\Big[\Cyl{\bsmall_{\widetilde{v}(x)}}{\nu_{v}}{x}\Big]^+|D^d v|
\\ &
+\int_{\Omega\cap J_{v}}
\varphi(x)
\Big[\mean{v^-(x)}^{v^+(x)} 
\Cyl{\bsmall_t}{\nu_{v}}{x}\, dt\Big]^+\, 
d\hh\,.
\end{split}
\]

Let $\Phi$ denote the  set of all functions $\varphi\in C^1_c(\Omega)$ such that $0\leq\varphi\leq 1$.
Then it holds that
\begin{equation}
\label{fff266}
\begin{split}
G^+(u_n)
& = \sup_{\varphi\in\Phi}\int_{\Omega}
\varphi(x)
(\bsmall(x, u_n), Du_n)
=
\sup_{\varphi\in\Phi}\int_{\Omega}
\varphi(x)
(\bsmall(x, u_n), Du_n)^+
.
\end{split}
\end{equation}
Using the representation formulas \eqref{f:thetak1} and \eqref{f:thetak2},
from \eqref{fff266}
we deduce that
\begin{equation}\label{f:13}
G^+(u_n)
=
\sup_{\varphi\in\Phi}\int_{\Omega} F^1_\varphi(u_n)
=
\sup_{\varphi\in\Phi}\int_{\Omega} F^3_\varphi(u_n)\,.
\end{equation}
Since $F^1_\varphi \leq F^2_\varphi \leq F^3_\varphi$, 
from \eqref{f:13} it follows that
\[
G^+(u_n)
= 
\sup_{\varphi\in\Phi}\int_{\Omega} F^2_\varphi(u_n)\,.
\]
As a consequence, recalling that $0\leq \sigma_k\leq 1$, for every $\varphi\in\Phi$ we deduce that
\[
\begin{split}
G^+(u_n)
\geq  {} &
F^2_\varphi(u_n)
\\ \geq {} &
\int_{\Omega}
\varphi(x) \sigma_k(\widetilde{u}_n(x))\,
\Big[\Cyl{\bsmall_{\widetilde{u}_n(x)}}{\nu_{u_n}}{x}\Big]^+|D^d u_n|
\\ &
+\int_{\Omega\cap J_{u_n}}
\varphi(x)
\mean{u_n^-(x)}^{u_n^+(x)} 
\sigma_k(t)\,
\Big[ 
\Cyl{\bsmall_t}{\nu_{u_n}}{x}\, dt\Big]^+\, 
d\hh
\\ \geq {} &
\int_{\Omega}
\varphi(x) \sigma_k(\widetilde{u}_n(x))\,
\Cyl{\bsmall_{\widetilde{u}_n(x)}}{\nu_{u_n}}{x}\,|D^d u_n|
\\ &
+\int_{\Omega\cap J_{u_n}}
\varphi(x)
\mean{u_n^-(x)}^{u_n^+(x)} 
\sigma_k(t)\,
\Cyl{\bsmall_t}{\nu_{u_n}}{x}\, dt\, 
d\hh
\\ = {} &
\int_{\Omega}
\varphi(x) 
\Cyl{\bsmall^k_{\widetilde{u}_n(x)}}{\nu_{u_n}}{x}\,|D^d u_n|
\\ &
+\int_{\Omega\cap J_{u_n}}
\varphi(x)
\mean{u_n^-(x)}^{u_n^+(x)} 
\Cyl{\bsmall^k_t}{\nu_{u_n}}{x}\, dt\, 
d\hh
\\ = {} &
\int_{\Omega} \varphi(x)\, (\bsmall^k(x, u_n), Du_n)
\,.
\end{split}
\]

Let us choose $k > \|u\|_\infty + 1$,
let $T_k z := \min\{\max\{z, -k\} , k\}$, $z\in\R$,
and let us define the functional
\[
G^k_\varphi(v) := \int_{\Omega} \varphi(x)\, (\bsmall^k(x, v), Dv),
\qquad v\in \BVL,
\]
so that the previous inequality reads
\begin{equation}
\label{f:ineq}
G^+(u_n) \geq G^k_\varphi(u_n)\,.
\end{equation}
From the representation formula \eqref{f:bxurepr} it holds that
\[
G^k_\varphi(v) = G^k_\varphi(T_k v),
\qquad \forall v\in \BVL,
\]
and, by our choice of $k$,
$G^k_\varphi(u) = G_\varphi(u)$.

For every $\varphi\in\Phi$,
from \eqref{f:ineq} and
Proposition~\ref{p:cont}
applied to the functional $G^k_\varphi$ and the
uniformly bounded sequence $(T_k u_n)_n$, 
we deduce that
\[
\begin{split}
\liminf_{n\to +\infty} 
G^+(u_n)
& 
\geq
\liminf_{n\to +\infty} 
G^k_\varphi(u_n)
=
\liminf_{n\to +\infty} 
G^k_\varphi(T_k u_n)
\geq
G^k_\varphi(u) = G_\varphi(u).
\end{split}
\]
Taking the supremum for $\varphi\in\Phi$ we finally conclude that
$\liminf\limits_{n\to +\infty} G^+(u_n) \geq G^+(u)$.
\end{proof}





\section{Pairing as relaxed functional}
\label{s:relax}

For every function $\varphi\in C^1_c(\Omega)$ and for every 
open set $A\subset\Omega$
let us consider  the functional 
$F^\varphi(\cdot,A):\BVLloc\to]-\infty,+\infty]$ 
defined by
\[
F^\varphi(u,A):=
\begin{cases}
\displaystyle\int_A \varphi\,\bsmall(x,u)\cdot\nabla u\,dx
& \text{$u\in W^{1,1}_{{\rm loc}}(\Omega)$}\cap L^\infty(\Omega),
\\
+\infty
,
& u\in (BV_{{\rm loc}}(\Omega)\
\setminus W_{{\rm loc}}^{1,1}(\Omega))\cap L_{{\rm loc}}^\infty(\Omega).
\end{cases}
\]
Moreover, for every $u\in\BVLloc$
we define the relaxations
with respect to the weak$^*$ convergence in $BV$ and the $L^1$ convergence in $BV$
as
\[
\rel{F^\varphi}(u,A) :=
\inf_{\{u_n\}} 
\left\{
\liminf_{n\to +\infty}
F^\varphi(u_n, A)\colon
u_n\in W^{1,1}_{\text{loc}}(A),\
u_n\weakstarto u\ \text{in}\ BV_{{\rm loc}}(A)
\right\}
\,,
\]
\[
\rel{F_1^\varphi}(u,A) :=
\inf_{\{u_n\}} 
\left\{
\liminf_{n\to +\infty}
F^\varphi(u_n, A)\colon
u_n\in W^{1,1}_{\text{loc}}(A),\
u_n\to u\ \text{strongly in }L^1(A)
\right\}
\,.
\]





\begin{theorem}
\label{t:limsupG}
[Integral representation of the relaxed functionals of $F^\varphi$]
Let $\bsmall$
satisfy assumptions (i)--(iv).
Then for every
$u\in  \BVLloc$ 
and for every 
open set $A\subset\Omega$,
if we assume $\sigma\in L^N_{\rm {loc}}(\Omega)$, then it holds that
\begin{equation*}
\rel{F^\varphi}(u,A)=\int_A\varphi\, d (\bsmall(x,u),Du)
\end{equation*}
and, if we assume $\sigma\in L^\infty_{\rm {loc}}(\Omega)$, then it holds that
\begin{equation*}
\rel{F_1^\varphi}(u,A)=\int_A\varphi\, d (\bsmall(x,u),Du).
\end{equation*}

\end{theorem}

\begin{proof}
Thanks to the continuity results proved
in Proposition~\ref{p:cont} 
and the argument in \cite[Theorem 1.3, part (i)]{FL1},
it is enough to prove the following two inequalities:
\begin{itemize}
\item[(J)] for $\H^{N-1}$-a.e. $x_0\in J_u$, it holds that
\[
\frac{d\rel{F^\varphi}(u,\cdot)}{d\H^{N-1}\ristretto{J_u}} (x_0)
\leq \varphi(x_0)\int_{u^-(x_0)}^{u^+(x_0)}\!
\Cyl{\bsmall(\cdot,t)}{\nu_u(x_0)}{x_0}\,dt\,;
\]
\item[(C)] for $\vert D^c u\vert$-a.e. $x_0\in \Omega$, it holds that
\[
\frac{d\rel{F^\varphi}(u,\cdot)}{d\vert D^c u\vert} (x_0)
\leq  \varphi(x_0)\Cyl{\bsmall(\cdot,\preciso{u}(x_0))}{\nu_u(x_0)}{x_0}\,.
\] 
\end{itemize}
Since both results are of local nature,
it is not restrictive to assume that $\Omega = \R^N$
and that
$u\in BV(\R^N)\cap L^\infty(\R^N)$.
Moreover, to simplify the notation we denote
\(\mu:=(\bsmall(\cdot, u),Du)\).

\bigskip
\noindent
\textbf{Proof of (J).}
By the definition of relaxed functional we have that
\[
\begin{split}
\frac{d\rel{F^\varphi}(u,\cdot)}{d\H^{N-1}\ristretto{J_u}} (x_0)
& = \lim_{r\searrow 0} \frac{\rel{F^\varphi}(u, B_r(x_0))}{\omega_{N-1} r^{N-1}}
\\ & \leq 
\lim_{r\searrow 0} 
\liminf_{\varepsilon\searrow 0}
\frac{1}{\omega_{N-1} r^{N-1}}
\int_{B_r(x_0)} \varphi(y)\, \bsmall(y, \rho_\varepsilon\ast u (y))\cdot
\nabla(\rho_\varepsilon\ast u)(y)\, dy\,.
\end{split}
\]
As $\varepsilon\to 0^+$,
the integral above converges to
$\int_{B_r(x_0)} \varphi \, d\mu$
(see the proof of Theorem~4.3 in \cite{CD4}, where this convergence
is stated in formula (38)).
Hence,
\[
\begin{split}
\frac{d\rel{F^\varphi}(u,\cdot)}{d\H^{N-1}\ristretto{J_u}} (x_0)
& \leq
\lim_{r\searrow 0} 
\frac{1}{\omega_{N-1} r^{N-1}}
\int_{B_r(x_0)} \varphi\, d\mu
\\ & =
\varphi(x_0) \,
\lim_{r\searrow 0}
\frac{|Du|(B_r(x_0))}{\omega_{N-1} r^{N-1}}
\cdot
 \frac{\mu(B_r(x_0))}{|Du|(B_r(x_0))}
\\ & =
\varphi(x_0) \, [u^+(x_0) - u^-(x_0)] \, \Theta(\bsmall, u, x_0)\,,
\end{split}
\]
so that (J) follows from 
Theorem~\ref{t:repr2}.

\bigskip
\noindent
\textbf{Proof of (C).}
Reasoning as in the proof of (J) above, we have that
\[
\begin{split}
\frac{d\rel{F^\varphi}(u,\cdot)}{d |D^c u|} (x_0)
& = \lim_{r\searrow 0} \frac{\rel{F^\varphi}(u, B_r(x_0))}{\omega_{N-1} r^{N-1}}
\\ & \leq 
\varphi(x_0) \,
\lim_{r\searrow 0}
\frac{\mu(B_r(x_0))}{|D u| (B_r(x_0))}
= 
\varphi(x_0)\, \Theta(\bsmall, u, x_0)\,,
\end{split}
\]
and again the conclusion follows from 
Theorem~\ref{t:repr2}.
\end{proof}




\def\cprime{$'$}
\begin{bibdiv}
\begin{biblist}

\bib{ADCF1}{article}{
      author={Amar, {M.}},
      author={De~Cicco, {V.}},
      author={Fusco, {N.}},
       title={A relaxation result in {BV} for integral functionals with
  discontinuous integrands},
        date={2007},
        ISSN={1292-8119},
     journal={ESAIM Control Optim. Calc. Var.},
      volume={13},
      number={2},
       pages={396\ndash 412},
         url={https://doi.org/10.1051/cocv:2007015},
      review={\MR{2306643}},
}

\bib{ADCF2}{article}{
      author={Amar, {M.}},
      author={De~Cicco, {V.}},
      author={Fusco, {N.}},
       title={Lower semicontinuity and relaxation results in {BV} for integral
  functionals with {BV} integrands},
        date={2008},
        ISSN={1292-8119},
     journal={ESAIM Control Optim. Calc. Var.},
      volume={14},
      number={3},
       pages={456\ndash 477},
         url={https://doi.org/10.1051/cocv:2007061},
      review={\MR{2434061}},
}

\bib{AmbCriMan}{article}{
      author={Ambrosio, {L.}},
      author={Crippa, {G.}},
      author={Maniglia, {S.}},
       title={Traces and fine properties of a {$BD$} class of vector fields and
  applications},
        date={2005},
        ISSN={0240-2963},
     journal={Ann. Fac. Sci. Toulouse Math. (6)},
      volume={14},
      number={4},
       pages={527\ndash 561},
         url={http://afst.cedram.org/item?id=AFST_2005_6_14_4_527_0},
      review={\MR{2188582}},
}

\bib{ADM}{incollection}{
      author={Ambrosio, {L.}},
      author={De~Lellis, {C.}},
      author={Mal\'y, {J.}},
       title={On the chain rule for the divergence of {BV}-like vector fields:
  applications, partial results, open problems},
        date={2007},
   booktitle={Perspectives in nonlinear partial differential equations},
      series={Contemp. Math.},
      volume={446},
   publisher={Amer. Math. Soc., Providence, RI},
       pages={31\ndash 67},
         url={http://dx.doi.org/10.1090/conm/446/08625},
      review={\MR{2373724}},
}

\bib{AFP}{book}{
      author={Ambrosio, {L.}},
      author={Fusco, {N.}},
      author={Pallara, {D.}},
       title={Functions of bounded variation and free discontinuity problems},
      series={Oxford Mathematical Monographs},
   publisher={The Clarendon Press Oxford University Press},
     address={New York},
        date={2000},
        ISBN={0-19-850245-1},
      review={\MR{MR1857292 (2003a:49002)}},
}

\bib{ABCM}{article}{
      author={Andreu, F.},
      author={Ballester, C.},
      author={Caselles, V.},
      author={Maz\'{o}n, J.M.},
       title={The {D}irichlet problem for the total variation flow},
        date={2001},
        ISSN={0022-1236},
     journal={J. Funct. Anal.},
      volume={180},
      number={2},
       pages={347\ndash 403},
         url={https://doi.org/10.1006/jfan.2000.3698},
      review={\MR{1814993}},
}

\bib{AVCM}{book}{
      author={Andreu-Vaillo, {F.}},
      author={Caselles, {V.}},
      author={Maz\'on, {J.M.}},
       title={Parabolic quasilinear equations minimizing linear growth
  functionals},
      series={Progress in Mathematics},
   publisher={Birkh\"auser Verlag, Basel},
        date={2004},
      volume={223},
        ISBN={3-7643-6619-2},
         url={http://dx.doi.org/10.1007/978-3-0348-7928-6},
      review={\MR{2033382}},
}

\bib{Anz}{article}{
      author={Anzellotti, {G.}},
       title={Pairings between measures and bounded functions and compensated
  compactness},
        date={1983},
        ISSN={0003-4622},
     journal={Ann. Mat. Pura Appl. (4)},
      volume={135},
       pages={293\ndash 318 (1984)},
         url={http://dx.doi.org/10.1007/BF01781073},
      review={\MR{750538}},
}

\bib{Anz2}{misc}{
      author={Anzellotti, {G.}},
       title={Traces of bounded vector--fields and the divergence theorem},
        date={1983},
        note={Unpublished preprint},
}

\bib{BouDM}{article}{
      author={Bouchitt\'e, G.},
      author={Dal~Maso, G.},
       title={Integral representation and relaxation of convex local
  functionals on {${\rm BV}(\Omega)$}},
        date={1993},
        ISSN={0391-173X},
     journal={Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)},
      volume={20},
      number={4},
       pages={483\ndash 533},
         url={http://www.numdam.org/item?id=ASNSP_1993_4_20_4_483_0},
      review={\MR{1267597}},
}

\bib{Cas}{article}{
      author={Caselles, V.},
       title={On the entropy conditions for some flux limited diffusion
  equations},
        date={2011},
        ISSN={0022-0396},
     journal={J. Differential Equations},
      volume={250},
      number={8},
       pages={3311\ndash 3348},
         url={http://dx.doi.org/10.1016/j.jde.2011.01.027},
      review={\MR{2772392}},
}

\bib{ChenFrid}{article}{
      author={Chen, {G.-Q.}},
      author={Frid, {H.}},
       title={Divergence-measure fields and hyperbolic conservation laws},
        date={1999},
        ISSN={0003-9527},
     journal={Arch. Ration. Mech. Anal.},
      volume={147},
      number={2},
       pages={89\ndash 118},
         url={http://dx.doi.org/10.1007/s002050050146},
      review={\MR{1702637}},
}

\bib{ChFr1}{article}{
      author={Chen, {G.-Q.}},
      author={Frid, {H.}},
       title={Extended divergence-measure fields and the {E}uler equations for
  gas dynamics},
        date={2003},
        ISSN={0010-3616},
     journal={Comm. Math. Phys.},
      volume={236},
      number={2},
       pages={251\ndash 280},
         url={http://dx.doi.org/10.1007/s00220-003-0823-7},
      review={\MR{1981992}},
}

\bib{ChTo2}{article}{
      author={Chen, {G.-Q.}},
      author={Torres, {M.}},
       title={Divergence-measure fields, sets of finite perimeter, and
  conservation laws},
        date={2005},
        ISSN={0003-9527},
     journal={Arch. Ration. Mech. Anal.},
      volume={175},
      number={2},
       pages={245\ndash 267},
         url={http://dx.doi.org/10.1007/s00205-004-0346-1},
      review={\MR{2118477}},
}

\bib{ChTo}{article}{
      author={Chen, {G.-Q.}},
      author={Torres, {M.}},
       title={On the structure of solutions of nonlinear hyperbolic systems of
  conservation laws},
        date={2011},
        ISSN={1534-0392},
     journal={Commun. Pure Appl. Anal.},
      volume={10},
      number={4},
       pages={1011\ndash 1036},
         url={http://dx.doi.org/10.3934/cpaa.2011.10.1011},
      review={\MR{2787432 (2012c:35263)}},
}

\bib{ChToZi}{article}{
      author={Chen, {G.-Q.}},
      author={Torres, {M.}},
      author={Ziemer, {W.P.}},
       title={Gauss-{G}reen theorem for weakly differentiable vector fields,
  sets of finite perimeter, and balance laws},
        date={2009},
        ISSN={0010-3640},
     journal={Comm. Pure Appl. Math.},
      volume={62},
      number={2},
       pages={242\ndash 304},
         url={http://dx.doi.org/10.1002/cpa.20262},
      review={\MR{2468610}},
}

\bib{CCDM}{misc}{
      author={Comi, {G.E.}},
      author={Crasta, {G.}},
      author={De~Cicco, {V.}},
      author={Malusa, {A.}},
       title={Representation formulas of pairings between divergence--measure
  vector fields and {BV} functions},
        date={2022},
        note={In preparation},
}

\bib{ComiPayne}{article}{
      author={Comi, {G.E.}},
      author={Payne, {K.R.}},
       title={On locally essentially bounded divergence measure fields and sets
  of locally finite perimeter},
        date={2017},
     journal={Adv. Calc. Var.},
        note={DOI 10.1515/acv-2017-0001},
}

\bib{CD2}{article}{
      author={Crasta, {G.}},
      author={De~Cicco, {V.}},
       title={On the chain rule formulas for divergences and applications to
  conservation laws},
        date={2017},
        ISSN={0362-546X},
     journal={Nonlinear Anal.},
      volume={153},
       pages={275\ndash 293},
         url={https://doi.org/10.1016/j.na.2016.10.005},
      review={\MR{3614672}},
}

\bib{CD3}{article}{
      author={Crasta, {G.}},
      author={De~Cicco, {V.}},
       title={Anzellotti's pairing theory and the {G}auss--{G}reen theorem},
        date={2019},
        ISSN={0001-8708},
     journal={Adv. Math.},
      volume={343},
       pages={935\ndash 970},
         url={https://doi.org/10.1016/j.aim.2018.12.007},
      review={\MR{3892346}},
}

\bib{CD4}{article}{
      author={Crasta, {G.}},
      author={De~Cicco, {V.}},
       title={An extension of the pairing theory between divergence-measure
  fields and {BV} functions},
        date={2019},
        ISSN={0022-1236},
     journal={J. Funct. Anal.},
      volume={276},
      number={8},
       pages={2605\ndash 2635},
         url={https://doi.org/10.1016/j.jfa.2018.06.007},
      review={\MR{3926127}},
}

\bib{CDM}{article}{
      author={Crasta, {G.}},
      author={De~Cicco, {V.}},
      author={Malusa, {A.}},
       title={Pairings between bounded divergence-measure vector fields and bv
  functions},
        date={2020},
     journal={Adv. Calc. Var.},
        note={arXiv: 1902.06052},
}

\bib{DM1980}{article}{
      author={Dal~Maso, {G.}},
       title={Integral representation on {${\rm BV}(\Omega )$} of {$\Gamma
  $}-limits of variational integrals},
        date={1979/80},
        ISSN={0025-2611},
     journal={Manuscripta Math.},
      volume={30},
      number={4},
       pages={387\ndash 416},
         url={https://doi.org/10.1007/BF01301259},
      review={\MR{567216}},
}

\bib{DM}{book}{
      author={Dal~Maso, {G.}},
       title={An introduction to {$\Gamma$}--convergence},
   publisher={Birkh\"auser},
     address={Boston},
        date={1993},
}

\bib{DCbumi}{article}{
      author={De~Cicco, {V.}},
       title={Lower semicontinuity for certain integral functionals on {${\rm
  BV}(\Omega)$}},
        date={1991},
     journal={Boll. Un. Mat. Ital. B (7)},
      volume={5},
      number={2},
       pages={291\ndash 313},
      review={\MR{1111124}},
}

\bib{DFV1}{article}{
      author={De~Cicco, {V.}},
      author={Fusco, {N.}},
      author={Verde, {A.}},
       title={On {$L^1$}-lower semicontinuity in {BV}},
        date={2005},
        ISSN={0944-6532},
     journal={J. Convex Anal.},
      volume={12},
      number={1},
       pages={173\ndash 185},
      review={\MR{2135805}},
}

\bib{DFV}{article}{
      author={De~Cicco, {V.}},
      author={Fusco, {N.}},
      author={Verde, {A.}},
       title={A chain rule formula in {$BV$} and application to lower
  semicontinuity},
        date={2007},
        ISSN={0944-2669},
     journal={Calc. Var. Partial Differential Equations},
      volume={28},
      number={4},
       pages={427\ndash 447},
         url={https://doi.org/10.1007/s00526-006-0048-7},
      review={\MR{2293980}},
}

\bib{DeGiOlPe}{article}{
      author={De~Cicco, {V.}},
      author={Giachetti, {D.}},
      author={Oliva, {F.}},
      author={Petitta, {F.}},
       title={The {D}irichlet problem for singular elliptic equations with
  general nonlinearities},
        date={2019},
        ISSN={0944-2669},
     journal={Calc. Var. Partial Differential Equations},
      volume={58},
      number={4},
       pages={Paper No. 129, 40},
         url={https://doi.org/10.1007/s00526-019-1582-4},
      review={\MR{3978950}},
}

\bib{DeGiSe}{article}{
      author={De~Cicco, {V.}},
      author={Giachetti, {D.}},
      author={Segura De~Le\'on, {S.}},
       title={Elliptic problems involving the 1-{L}aplacian and a singular
  lower order term},
        date={2019},
        ISSN={0024-6107},
     journal={J. Lond. Math. Soc. (2)},
      volume={99},
      number={2},
       pages={349\ndash 376},
         url={https://doi.org/10.1112/jlms.12172},
      review={\MR{3939259}},
}

\bib{dcl}{article}{
      author={De~Cicco, V.},
      author={Leoni, G.},
       title={A chain rule in {$L^1({\rm div};\Omega)$} and its applications to
  lower semicontinuity},
        date={2004},
        ISSN={0944-2669},
     journal={Calc. Var. Partial Differential Equations},
      volume={19},
      number={1},
       pages={23\ndash 51},
         url={http://dx.doi.org/10.1007/s00526-003-0192-2},
      review={\MR{MR2027846 (2005c:49030)}},
}

\bib{DGMM}{article}{
      author={Degiovanni, {M.}},
      author={Marzocchi, {A.}},
      author={Musesti, {A.}},
       title={Cauchy fluxes associated with tensor fields having divergence
  measure},
        date={1999},
        ISSN={0003-9527},
     journal={Arch. Ration. Mech. Anal.},
      volume={147},
      number={3},
       pages={197\ndash 223},
         url={http://dx.doi.org/10.1007/s002050050149},
      review={\MR{1709215}},
}

\bib{Fed}{book}{
      author={Federer, H.},
       title={Geometric measure theory},
      series={Die Grundlehren der mathematischen Wissenschaften, Band 153},
   publisher={Springer-Verlag New York Inc., New York},
        date={1969},
      review={\MR{MR0257325 (41 \#1976)}},
}

\bib{FL}{article}{
      author={Fonseca, {I.}},
      author={Leoni, {G.}},
       title={Some remarks on lower semicontinuity},
        date={2000},
        ISSN={0022-2518},
     journal={Indiana Univ. Math. J.},
      volume={49},
      number={2},
       pages={617\ndash 635},
         url={https://doi.org/10.1512/iumj.2000.49.1791},
      review={\MR{1793684}},
}

\bib{FL1}{article}{
      author={Fonseca, {I.}},
      author={Leoni, {G.}},
       title={On lower semicontinuity and relaxation},
        date={2001},
        ISSN={0308-2105},
     journal={Proc. Roy. Soc. Edinburgh Sect. A},
      volume={131},
      number={3},
       pages={519\ndash 565},
         url={https://doi.org/10.1017/S0308210500000998},
      review={\MR{1838501}},
}

\bib{HI}{article}{
      author={Huisken, {G.}},
      author={Ilmanen, {T.}},
       title={The inverse mean curvature flow and the {R}iemannian {P}enrose
  inequality},
        date={2001},
        ISSN={0022-040X},
     journal={J. Differential Geom.},
      volume={59},
      number={3},
       pages={353\ndash 437},
         url={http://projecteuclid.org/euclid.jdg/1090349447},
      review={\MR{1916951}},
}

\bib{LeoSar2}{misc}{
      author={Leonardi, {G.P.}},
      author={Saracco, {G.}},
       title={Rigidity and trace properties of divergence-measure vector
  fields},
        date={2017},
        note={Preprint},
}

\bib{LeoSar}{article}{
      author={Leonardi, {G.P.}},
      author={Saracco, {G.}},
       title={The prescribed mean curvature equation in weakly regular
  domains},
        date={2018},
        ISSN={1021-9722},
     journal={NoDEA Nonlinear Differential Equations Appl.},
      volume={25},
      number={2},
       pages={Art. 9, 29},
         url={https://doi.org/10.1007/s00030-018-0500-3},
      review={\MR{3767675}},
}

\bib{Mazon2016}{article}{
      author={Maz\'{o}n, {J.M.}},
       title={The {E}uler-{L}agrange equation for the anisotropic least
  gradient problem},
        date={2016},
        ISSN={1468-1218},
     journal={Nonlinear Anal. Real World Appl.},
      volume={31},
       pages={452\ndash 472},
         url={https://doi.org/10.1016/j.nonrwa.2016.02.009},
      review={\MR{3490852}},
}

\bib{Pan1}{article}{
      author={Panov, E.~Yu.},
       title={Existence and strong pre-compactness properties for entropy
  solutions of a first-order quasilinear equation with discontinuous flux},
        date={2010},
        ISSN={0003-9527},
     journal={Arch. Ration. Mech. Anal.},
      volume={195},
      number={2},
       pages={643\ndash 673},
         url={http://dx.doi.org/10.1007/s00205-009-0217-x},
      review={\MR{2592291 (2011h:35039)}},
}

\bib{SchSch}{article}{
      author={Scheven, {C.}},
      author={Schmidt, {T.}},
       title={B{V} supersolutions to equations of 1-{L}aplace and minimal
  surface type},
        date={2016},
        ISSN={0022-0396},
     journal={J. Differential Equations},
      volume={261},
      number={3},
       pages={1904\ndash 1932},
         url={http://dx.doi.org/10.1016/j.jde.2016.04.015},
      review={\MR{3501836}},
}

\bib{SchSch2}{misc}{
      author={Scheven, {C.}},
      author={Schmidt, {T.}},
       title={An {A}nzellotti type pairing for divergence-measure fields and a
  notion of weakly super-1-harmonic functions},
        date={2017},
        note={Preprint},
}

\bib{Schu}{article}{
      author={Schuricht, {F.}},
       title={A new mathematical foundation for contact interactions in
  continuum physics},
        date={2007},
        ISSN={0003-9527},
     journal={Arch. Ration. Mech. Anal.},
      volume={184},
      number={3},
       pages={495\ndash 551},
         url={http://dx.doi.org/10.1007/s00205-006-0032-6},
      review={\MR{2299760}},
}

\bib{Serrin61}{article}{
      author={Serrin, {J.}},
       title={On the definition and properties of certain variational
  integrals},
        date={1961},
        ISSN={0002-9947},
     journal={Trans. Amer. Math. Soc.},
      volume={101},
       pages={139\ndash 167},
         url={https://doi.org/10.2307/1993416},
      review={\MR{138018}},
}

\bib{Silh}{article}{
      author={\v{S}ilhav\'{y}, {M.}},
       title={Divergence measure fields and {C}auchy's stress theorem},
        date={2005},
        ISSN={0041-8994},
     journal={Rend. Sem. Mat. Univ. Padova},
      volume={113},
       pages={15\ndash 45},
      review={\MR{2168979}},
}

\end{biblist}
\end{bibdiv}
\end{document}

