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\title[Decomposition of BV functions]
{Decomposition results for functions\\
with bounded variation 
}
\author[Gianni Dal Maso]{Gianni Dal Maso}
\address[Gianni Dal Maso]{SISSA, Via Beirut 2-4, 34014 Trieste,
Italy}
\email[Gianni Dal Maso]{dalmaso@sissa.it}

\author[Rodica Toader]{Rodica Toader}
\address[Rodica Toader]{Universit\`a di Udine, D.I.C.A., Via delle Scienze 208, 33100
 Udine, Italy}
\email[Rodica Toader]{toader@uniud.it}


\begin{document}

\maketitle

\hfill{\it Dedicated to the memory of Guido Stampacchia}
\bigskip

\begin{abstract}
Some decomposition results for functions with bounded variation are obtained by using 
Gagliardo's Theorem on the surjectivity of the trace operator from 
$W^{1,1}(\Om)$ into $L^1(\partial\Om)$. More precisely, we prove that every $BV$ function 
can 
be written as the sum of a $BV$ function without jumps and a $BV$ function without Cantor 
part. 
Alternatively, it can be written as the sum of a $BV$ function without jumps and a purely 
ingular $BV$ function (i.e., a function whose gradient is singular with respect to the Lebesgue 
measure). It can also be decomposed  as  the sum of a  purely singular $BV$ function  and a 
$BV$ function without Cantor part. We also prove similar results for the space $BD$ of 
functions 
with bounded deformation. In particular, we show that every $BD$ function can be written as 
the sum of a $BD$ function without jumps and a $BV$ function without Cantor part. Therefore, 
every $BD$ function without Cantor part is the sum of a function whose symmetrized gradient 
belongs to $L^1$ and a $BV$ function without Cantor part. 
\end{abstract}

{\small

\bigskip
\keywords{\noindent {\bf Keywords:} BV,  SBV, BD, LD, SBD.}


\bigskip
\subjclass{\noindent {\bf 2000 Mathematics Subject Classification:} 46E35.
}
}
\bigskip
\bigskip


\begin{section}{INTRODUCTION}\label{spaces}
\par

Throughout the paper $\Ln$  and $\hn$ denote the Lebesgue measure in $\Rn$ 
and the $n-1$ dimensional Hausdorff measure, respectively. Unless otherwise 
specified, the 
expression {\em almost everywhere\/} (abbreviated as {\em a.e.\/}) always refers 
to~$\Ln$.
If $1< r\le\infty$ and $E$ is a set, we use the notation $\|\cdot\|_r$ or $\|\cdot\|_{r,E}$ 
for the $L^r$ norm on $E$ with respect to $\Ln$ or $\hn$ (or to some other measure 
which is clear from the context), while $\|\cdot\|_1$ denotes the norm in $L^1$, as 
well as in the space $\M_b$ of bounded Radon measures. 
\par

For every measure $\mu$, the symbols $\mu^a$, $\mu^s$ will denote the absolutely
continuous and singular part of $\mu$ with respect to $\Ln$.  The former will always
be identified with its density.
\par

Throughout the paper $\Om$ is a bounded open set in $\Rn$ and
$BV(\Om)$ is the space of functions with bounded variation on $\Om$, i.e., 
the space of functions $u\in L^1(\Om)$ whose distributional partial 
derivatives $D_iu$, $i=1,\dots,n$, are bounded Radon measures on $\Om$. 
The following decomposition holds:
\begin{equation}\label{decompgrad}
Du=D^au+D^ju+D^cu\,,
\end{equation} 
where $D^au$ is the absolutely continuous part of $Du$ with respect to 
$\Ln$, $D^ju$ is  the jump part of $Du$ (defined as the restriction of $Du$ to 
the jump set $J_u$), and $D^cu$ is the Cantor part of 
$Du$. We refer to \cite{A-F-P} for the precise definitions of these measures 
and of $J_u$, as well as for other fine properties of the functions in $BV(\Om)$
 or in its subspace $SBV(\Om)$ of special functions of bounded variation.
\par

 We consider the following subspaces of $BV(\Om)$:
\begin{eqnarray*}
&& BV^j(\Om):=\{u\in BV(\Om): Du=D^ju\}, \\
&& BV^c(\Om):=\{u\in BV(\Om): Du=D^cu\},\\
&& BV^a(\Om):=\{u\in BV(\Om): Du=D^au\}=W^{1,1}(\Om). 
\end{eqnarray*}
\par

If $n=1$ a classical result states that any function $u\in BV(\Om)$ can be written as 
\begin{equation}\label{123}
u=v+w+z\,,
\end{equation}
with $v\in BV^a(\Om)$, $w\in BV^j(\Om)$, and $z\in BV^c(\Om)$ (see, e.g., 
\cite[Corollary 3.33]{A-F-P}). The same result cannot hold true for $n>1$.
 Indeed, given a function $f\in L^1(\Om;\Rn)$ with ${\rm curl} f\neq0$, 
Alberti \cite{A} proved  that  there exists  $u\in BV(\Om)$ such that $D^au=f$. 
It is clear that this function $u$ cannot be decomposed as in \eqref{123}, since 
in this case we would have $v\in W^{1,1}(\Om)$ and $Dv=f$, hence 
${\rm curl} (Dv)\neq0$. For another example see \cite[Example 4.1]{A-F-P}.
\par


We therefore introduce the following subspaces of $BV(\Om)$:
\begin{eqnarray}
&& BV^{aj}(\Om):=\{u\in BV(\Om): Du=D^au+D^ju\}=SBV(\Om),\nonumber \\
&& BV^{ac}(\Om):=\{u\in BV(\Om): Du=D^au+D^cu\},\label{111}\\
&& BV^{jc}(\Om):=\{u\in BV(\Om): Du=D^ju+D^cu\}. \nonumber
\end{eqnarray}
The argument mentioned after \eqref{123} shows that in dimension $n>1$ 
we cannot decompose $BV(\Om)$ as $BV^a(\Om)+BV^{jc}(\Om)$. Moreover, 
we cannot decompose 
$BV(\Om)$ as $BV^j(\Om)+BV^{ac}(\Om)$, see \cite[Example 4.1]{A-F-P}, 
nor as $BV^c(\Om)+BV^{aj}(\Om)$, as the following example shows. 
 Let $\Om={]0,1[}^{n-1}{\times}{]-1,1[}$, let 
$\psi\colon[0,1]\to[0,1]$ be the Cantor-Vitali function (see, e.g., 
\cite[Example 1.67]{A-F-P}), and let $u(x)=\psi(x_1)$ for $x_n>0$, and 
$u(x)=0$ for $x_n\le0$. If $u=v+w$ with $v\in BV^c(\Om)$ and $w\in BV^{aj}(\Om)$, 
we would have $Dv=D^cu$, hence $D_iv=0$ for $i>1$ and $Dv=0$ for 
$x_n<0$. These properties  imply  that $v$ is constant, hence $D^cu=Dv=0$, 
which contradicts the fact that $|D^cu|(\Om)=1$.
\par

The main result of the paper is that any function $u\in BV(\Om)$ can be written 
as the sum of two functions which belong to two of the three subspaces
 introduced in \eqref{111}, namely
$$
u=v+w\,,
$$
with $v\in BV^{aj}(\Om)$ and $w\in BV^{ac}(\Om)$ (Theorem \ref{1}), or 
$v\in BV^{aj}(\Om)$ and $w\in BV^{jc}(\Om)$ (Theorem  \ref{2}), or
$v\in BV^{jc}(\Om)$ and $w\in BV^{ac}(\Om)$ (Theorem \ref{3}).
\par

To this end, we first prove in Section~\ref{ext} some extension results analogue 
to Gagliardo's theorem, showing that, if $\Om$ has a $C^1$-boundary $\partial\Om$,
 then any function in $L^1(\partial\Om)$ can be obtained  as the trace of a function in
  any of the three subspaces $BV^a(\Om)$, $BV^c(\Om)$, or ~$BV^j(\Om)$.
\par

We conclude the paper by proving a similar decomposition result (Theorem~\ref{bd}) for the 
space $BD(\Om)$ of functions with bounded deformation.
\end{section}
\par

\begin{section}{EXTENSION RESULTS}\label{ext} 
\par


Let $Q_{n-1}:={]0,1[}^{n-1}{\times}\{0\}$, $Q:={]0,1[}^{n-1}{\times}\,]0,1[$, and let $x:=(\overline 
x,x_n)$, with $\overline x\in {]0,1[}^{n-1}$ denote the coordinates of a generic point in $Q$.
Let us recall Gagliardo's extension theorem (\cite{Ga}, see also, e.g., 
\cite[Proposition 2.15]{Giu}).
\par

\begin{theorem}\label{g1}
Let $\varphi\in L^1(Q_{n-1};\Rm)$ with compact support in $Q_{n-1}$.  Then for every $\e>0$ 
there exists $u\in BV^a(Q;\Rm)=W^{1,1}(Q;\Rm)$ with trace $\varphi$ on $Q_{n-1}$ such that
\begin{eqnarray}
&\displaystyle \vphantom{\int_{Q}}{\rm supp}(u)\cap\partial Q\subset Q_{n-1}\,,\\
& \displaystyle\int_{Q}|u|\,dx\le\e\int_{Q_{n-1}}|\varphi|\,d\hn\,,\\
& \displaystyle\int_{Q}|Du|\,dx\le(1+\e)\int_{Q_{n-1}}|\varphi|\,d\hn\,.
\end{eqnarray}
\end{theorem}
\par

We prove now the analogue of Gagliardo's result for purely Cantorian functions. 
\begin{theorem}\label{g2}
Let $\varphi\in L^1(Q_{n-1};\Rm)$ with compact support in $Q_{n-1}$. Then for every $\e>0$ 
there exists $u\in BV^c(Q;\Rm)$ 
with trace $\varphi$ on $Q_{n-1}$ such that
\begin{eqnarray}
&\displaystyle \vphantom{\int_{Q}}{\rm supp}(u)\cap\partial Q\subset Q_{n-1}\,,\label{u20}\\
&\displaystyle \int_Q |u|\,dx\le\e\int_{Q_{n-1}}|\varphi|\,d\hn\,,\label{u21}\\
&\displaystyle |Du|(Q)\le(1+\e)\int_{Q_{n-1}}|\varphi|\,d\hn\,.\label{u22}
\end{eqnarray}
\end{theorem}
\par

\begin{proof} We begin by observing that, 
 if $A$ and $B$ are open subsets of $Q_{n-1}$ and ${]0,1[}$,  $v\in BV^c(A;\Rm)$, $w\in BV^c
(B)$, and $u(\overline x,x_n)=v(\overline x) w(x_n)$, then $u\in BV^c(A{\times} B;\Rm)$ and
\begin{equation}\label{101}
D_iu= D_iv\otimes w\quad i=1,\dots,n-1\,,\qquad D_nu=v\otimes Dw\,,
\end{equation}
where $v$ and $w$ are interpreted as measures on $A$ and $B$, respectively.
\par

Following the proof of Gagliardo's extension theorem,
we approximate $\varphi$ with purely Cantorian functions $\varphi_k$. 
More precisely, given $\e\in{]0,1[}$ we may choose a compact set $K\subset Q_{n-1}$ and a 
sequence of 
functions
$\varphi_k\in BV^c(Q_{n-1};\Rm)$, such that ${\rm supp}(\varphi_k)\subset K$ and 
$\varphi_k$ converges to $\varphi$ strongly in $L^1(Q_{n-1};\Rm)$. Moreover, we may 
assume that $\varphi_0=0$, $\|\varphi_k\|_1\le 2\|\varphi\|_1$,
and
\begin{equation}\label{155}
\sum_{k=1}^\infty \|\varphi_k-\varphi_{k-1}\|_1<( 1+\tfrac \e n)\|\varphi\|_1\,.
\end{equation}
Given a decreasing sequence $t_k$ converging to $0$, 
we consider the stripes $t_k\le x_n\le t_{k-1}$ and we interpolate between $\varphi_k$ and $
\varphi_{k-1}$ using the Cantor-Vitali function $\psi\colon[0,1]\to[0,1]$ instead of the linear 
interpolation. For 
$t_k\le x_n\le t_{k-1}$ we define
$$
u(\overline x,x_n)=\varphi_{k}(\overline x)+
(\varphi_{k-1}(\overline x)-\varphi_{k}(\overline x))\,\psi_k(x_n)\,,
$$
where
$$
\psi_k(x_n):=\psi\Big(\frac{x_n-t_{k}}{t_{k-1}-t_{k}}\Big)\,,
$$
and we set $u(\overline x,x_n)=0$ for $x_n\ge t_0$.
\par

By \eqref{101} $u\in BV^c(Q_{n-1}{\times}{]t_k,t_{k-1}[};\Rm)$ and 
$D_n u=(\varphi_{k-1}-\varphi_{k})\otimes D\psi_k$
so that 
\begin{eqnarray*}
|D_n u|({]0,1[}^{n-1}{\times}{]t_k,t_{k-1}[})=|D\psi_k|({]t_k,t_{k-1}[})
\|\varphi_k-\varphi_{k-1}\|_1\,,
\end{eqnarray*} 
and, for $i<n$, 
$D_i u=D_i\varphi_{k-1}\otimes\psi_k+D_i\varphi_k\otimes(1-\psi_k)$,
so that, as $0\le \psi_k\le1$, 
$$
|D_i u|({]0,1[}^{n-1}{\times}{]t_k,t_{k-1}[})\le(t_{k-1}-t_{k})
(|D_i\varphi_{k-1}|(Q_{n-1})+|D_i\varphi_k|(Q_{n-1}))\,.
$$
Since there is no jump at $x_n=t_k$, we have also 
$|Du|({]0,1[}^{n-1}{\times}\{t_k\})=0$. 
\par

As in the proof of Gagliardo's extension theorem, it suffices now to choose the decreasing 
sequence~$(t_k)$ so that 
\begin{eqnarray*}
&& t_{k-1}-t_{k}\le \frac{\e}{n2^k}\,\frac{\min\{1,\|\varphi\|_1\}}{1+\|D\varphi_k\|_1+
\|D\varphi_{k-1}\|_1}\,.
\end{eqnarray*}
Under these assumptions we get not only that $u\in BV^c(Q;\Rm)$, but also that  \eqref{u20}, 
(\ref{u21}), and~(\ref{u22}) hold.
\par


Since, by construction, $u(\cdot,x_n)\to \varphi$ in $L^1(Q_{n-1};\Rm)$, as $x_n\to0$, the trace 
of $u$ on $Q_{n-1}$ coincides with $\varphi$.
\end{proof}
\par

The analogue extension result to a purely jump function holds.
\par

\begin{theorem}\label{g3}
Let $\varphi\in L^1(Q_{n-1};\Rm)$ with compact support in $Q_{n-1}$. Then for every $\e>0$ 
there exists $u\in BV^j(Q;\Rm)$ 
with trace $\varphi$ on $Q_{n-1}$  such that
\begin{eqnarray}
&\displaystyle \vphantom{\int_{Q}}{\rm supp}(u)\cap\partial Q\subset Q_{n-1}\,,\label{u30}\\
&\displaystyle \int_Q |u|dx\le\e\int_{Q_{n-1}}|\varphi|d\hn\,,\label{u31}\\
&\displaystyle|Du|(Q)\le(1+\e)\int_{Q_{n-1}}|\varphi|d\hn\,.\label{u32}
\end{eqnarray}
\end{theorem}
\begin{proof}
We approximate now the boundary datum $\varphi$ with piecewise constant functions  
$\varphi_k\in BV^j(Q_{n-1};\Rm)$ such that their support is contained in the same compact 
subset of $Q_{n-1}$, $\varphi_k$ converges to $\varphi$ in $L^1(Q_{n-1};\Rm)$, $\varphi_0=0
$,
$\|\varphi_k\|_1\le2\|\varphi\|_1$, and \eqref{155} holds.
Let $t_k$ be as in the previous proof, and set $u(\overline x, x_n)=\varphi_k(\overline x)$ if 
$t_k<x_n\le t_{k-1}$, and $u(\overline x, x_n)=0$ if $x_n>t_0$.
Then $u\in BV^j(Q;\Rm)$ and \eqref{u30}, (\ref{u31}), and (\ref{u32}) hold.
\end{proof}
\par

 By a partition of unity argument we can extend the previous results from $Q$ to a bounded 
open set $\Om$ with $C^1$-boundary, preserving the constant $1$ in the estimates. 
\par

\begin{theorem}\label{c1} Assume that $\Om$ has a $C^1$-boundary and 
let $\varphi\in L^1(\partial\Om;\Rm)$. For every $\e>0$ there exists a function $u\in BV^a(\Om;
\Rm)=W^{1,1}(\Om;\Rm)$ with trace $\varphi$ on $\partial\Om$ and such that
\begin{eqnarray}
&& \int_\Om |u|dx\le\e\int_{\partial\Om}|\varphi|d\hn\,,\label{u111}\\
&& |Du|(\Om)\le(1+\e)\int_{\partial\Om}|\varphi|d\hn\,.\label{u112}
\end{eqnarray}
The same result holds for $BV^c(\Om;\Rm)$ and $BV^j(\Om;\Rm)$.
\end{theorem}
\begin{proof}
The result for $BV^a(\Om;\Rm)=W^{1,1}(\Om;\Rm)$ is already known, see, e.g., 
\cite[Theorem~2.16, Remark 2.17]{Giu}. We give an outline of the proof only in the case $BV^j
(\Om;\Rm)$.
\par

We begin by observing that the construction of the extension theorems above can be repeated 
in any cube and that the estimates do not depend on the size of the cube. 
For every $x_0\in\partial\Om$, let 
$R_\delta(x_0)$ be an open parallelepiped centred in $x_0$, 
whose base is an $n{-}1$-dimensional cube of size 
$\delta$ parallel to the tangent hyperplane to $\partial\Om$ at $x_0$, and having height~$2
\delta$. As $\partial\Om$ is of class $C^1$, if $\delta$ is small enough there exists a 
diffeomorphism, close to an isometry in a $C^1$-sense, which maps $R_\delta(x_0)\cap\Om$ 
onto ${]0,\delta[}^n$ and $R_\delta(x_0)\cap\partial\Om$ onto ${]0,\delta[}^{n-1}{\times}\{0\}$. 
By using this change of variables we obtain from Theorem~\ref{g3} that 
for every $\e>0$ there exists $\delta>0$ such that, if $\varphi\in L^1(\partial\Om;\Rm)$ has 
compact support in 
$\partial\Om\cap R_\delta(x_0)$, then 
there exists a function $u\in BV^j(\Om;\Rm)$, with trace $\varphi$ on $\partial\Om$ and ${\rm 
supp}(u)\subset R_\delta(x_0)\cap\overline\Om$, satisfying  \eqref{u111} and 
\eqref{u112}.
\par

Given $\e>0$, by compactness we can cover $\partial\Om$ with a finite number of such 
parallelepipeds. Let $\alpha_i$ be a partition of unity associated to this covering. By the 
previous step of the proof for every $i$ there exists a function  $u_i\in BV^j(\Om;\Rm)$, with 
trace $\alpha_i\,\varphi$ on $\partial\Om$, satisfying  \eqref{u111} and 
\eqref{u112} with $\varphi$ replaced by $\alpha_i\,\varphi$.
 Then the function $u=\sum_iu_i$ belongs to $BV^j(\Om;\Rm)$, has trace $\varphi$ on $\partial
\Om$, and satisfies
\eqref{u111} and \eqref{u112}.
\end{proof}
\par

\end{section}
\par

\begin{section}{DECOMPOSITION RESULTS}\label{decomp}
\par

For every $u\in BV(\Om;\Rm)$ the set $J_u$ is countably $(\hn,n-1)$-rectifiable. Therefore we 
can define a Borel function $\nu_u:J_u\to \Rn$ such that $\nu_u(x)$ is  the approximate unit 
normal to $J_u$ at $x$. The approximate limits $u^+(x)$ and $u^-(x)$ associated with $\nu_u
(x)$ are defined according to \cite[Definition 3.67]{A-F-P}.
\par

We apply now Gagliardo's extension Theorem to prove the following decomposition result, see 
also \cite[Theorem 4.6]{A-F-P}.
\par


\par

\begin{theorem}\label{1}
Let $u\in BV_{loc}(\Om;\Rm)$ with $|Du|(\Om)<+\infty$. For every $\e>0$ there exists 
$v\in BV^{aj}(\Om;\Rm)=SBV(\Om;\Rm)$ such that
$J_v= J_u$ and $v^+-v^-=u^+-u^-$ $\hn$-a.e. on $J_u$, and 
\begin{equation}\label{v}
\begin{array}{c}
\displaystyle\int_\Om|v|\,dx\le\e\int_{J_u}|u^+-u^-|\,d\hn\,,\\
\displaystyle\int_\Om|D^av|\,dx\le(1+\e)\int_{J_u}|u^+-u^-|\,d\hn\,.
\end{array}
\end{equation}
In particular,  $u-v\in BV_{loc}^{ac}(\Om;\Rm)$.
\end{theorem}
\begin{proof}
As $J_u$ is countably $(\hn,n-1)$-rectifiable, we may write
$$
J_u=N_0\cup\bigcup_{i=1}^\infty N_i\,,
$$
with $\hn(N_0)=0$ and with each $N_i$ contained in an $(n-1)$-dimensional $C^1$-manifold 
$M_i$ for $i>0$.  We may assume without loss of generality that the sets $N_i$ are pairwise 
disjoint and that each manifold $M_i$ is the boundary of an open set $U_i$ which lies on one 
side of $M_i$. We may choose 
$\nu_u$ so that $\nu_u(x)$ points towards $U_i$ for each $x\in N_i$. 
\par

Let us fix $\e>0$ and define 
$\varphi_i=u^+-u^-$ on $N_i$ and $\varphi_i=0$ on $\partial U_i\setminus N_i$.
By Gagliardo's extension theorem for sets with $C^1$-boundary (Theorem \ref{c1}) there exists 
$v_i\in W^{1,1}(U_i;\Rm)$ such that 
\begin{eqnarray*}
&\displaystyle\int_{U_i}|v_i|\,dx\le\e\int_{\partial U_i}|\varphi_i|d\hn\,,\\
&\displaystyle\int_{U_i}|Dv_i|\,dx\le(1+\e)\int_{\partial U_i}|\varphi_i|d\hn\,.
\end{eqnarray*}
We set $v_i=0$ on $\Om\setminus U_i$. In this way $v_i \in SBV(\Om;\Rm)$, $J_{v_i}= N_i$, 
$v_i^+=\varphi_i$, $v_i^-=0$, so that $v_i^+-v_i^-=u^+-u^-$ $\hn$-a.e. on $N_i$. 
Moreover,
\begin{equation}\label{v1}
\begin{array}{c}
\displaystyle\int_\Om|v_i|\,dx\le\e\int_{N_i}|u^+-u^-|d\hn\,,\\
\displaystyle\int_\Om|D^a v_i|\,dx\le(1+\e) \int_{N_i}|u^+-u^-|d\hn\,.
\end{array}
\end{equation}
Since $|Du|(\Om)<+\infty$, we have
$$
\sum_{i=1}^\infty\int_{N_i}|u^+-u^-|d\hn=\int_{J_u}|u^+-u^-|d\hn<+\infty\,,
$$
hence the sequence
$w_k=v_1+\dots+v_k$ converges strongly in $L^1(\Om;\Rm)$ to some function~$v$,
while $D^a w_k$ converges strongly in $L^1(\Om;\R^{m{\times}n})$ to some function $\psi$.
Moreover, $w_k\in SBV(\Om;\Rm)$ and, since the sets $N_i$ are pairwise disjoint, we have 
$J_{w_k}=N_1\cup\dots\cup N_k$, and $w_k^+-w_k^-=u^+-u^-$ on $N_1\cup\dots\cup N_k$, 
hence 
$D^j w_k=(u^+-u^-)\otimes\nu_u\hn\LLL{\bigcup_{i=1}^k N_i}$. This shows that $D^j w_k$ 
converges to $(u^+-u^-)\otimes\nu_u\hn\LLL{J_u}$ strongly in the space of bounded Radon 
measures on $\Om$ and thus 
$Dw_k$ converges to the measure $\mu:=\psi+(u^+-u^-)\otimes\nu_u\hn\LLL{J_u}$.
We conclude that $Dv=\mu$, 
hence $v\in SBV(\Om;\Rm)$, $D^av=\psi$, $D^jv=(u^+-u^-)\hn\LLL{J_u}$, which implies $J_v= 
J_u$ and $v^+-v^-=u^+-u^-$ $\hn$-a.e. on $J_u$, and thus $D^j(u-v)=0$. This gives $u-v\in 
BV_{loc}^{ac}(\Om)$. Inequalities \eqref{v} follow from \eqref{v1}.
\end{proof}
\par


\begin{theorem}\label{2}
Let $u\in BV_{loc}(\Om;\Rm)$ with $|Du|(\Om)<+\infty$. For every $\e>0$ there exists $v\in 
BV^{jc}(\Om;\Rm)$ such that
$J_v=J_u$ and $v^+-v^-=u^+-u^-$ $\hn$-a.e. on $J_u$, and 
\begin{equation}\label{v2}
\begin{array}{c}
\displaystyle\int_\Om|v|\,dx\le\e\int_{J_u}|u^+-u^-|\,d\hn\,,\\
\displaystyle|D^cv|(\Om)\le(1+\e)\int_{J_u}|u^+-u^-|\,d\hn\,.
\end{array}
\end{equation}
In particular,
$u-v\in BV_{loc}^{ac}(\Om;\Rm)$.
\end{theorem}
\begin{proof}
In the previous proof it is enough to replace the extension to $W^{1,1}(U_i;\Rm)$ by the  
extension to $BV^c(U_i;\Rm)$, also provided by Theorem~\ref{c1}.
\end{proof} 
\par

The last decomposition result of this type is just Alberti's Theorem~\cite{A} for $BV$ functions.
\begin{theorem}\label{3}
For every $u\in BV_{loc}(\Om;\Rm)$ with $|Du|(\Om)<+\infty$ there exists $v\in BV^{aj}(\Om;
\Rm)$ such that
$D^a v=D^a u$ a.e. in $\Om$, and hence $u-v\in BV_{loc}^{jc}(\Om;\Rm)$.
\end{theorem}
\par

\end{section}
\par

\begin{section}{Other decomposition results}
\par

Let us consider now the space $BD(\Om)$ of functions of bounded deformation (see 
\cite{Tem}), i.e., the  functions $u\in L^1(\Om;\Rn)$ such that the symmetric part of the gradient  
$Eu:=\frac 1 2 (Du+(Du)^T)$ is a matrix-valued bounded Radon measure.  As for functions in 
$BV(\Om)$ we have the decomposition $Eu=E^au+E^cu+E^ju$. We refer to \cite{ACDM} for the 
precise definition of  $E^au$, $E^cu$, $E^ju$, as well as for other fine properties of the 
functions in $BD(\Om)$ and in its subspace $SBD(\Om)$. As for $BV(\Om)$ we introduce the 
following subspaces of $BD(\Om)$:
\begin{eqnarray}
&& BD^{aj}(\Om):=\{u\in BD(\Om): Eu=E^au+E^ju\}=SBD(\Om),\nonumber \\
&& BD^{ac}(\Om):=\{u\in BD(\Om): Eu=E^au+E^cu\},\label{bd1}\\
&& BD^{jc}(\Om):=\{u\in BD(\Om): Eu=E^ju+E^cu\}. \nonumber
\end{eqnarray}
The following result can be obtained by adapting the proof of Theorem~\ref{1}.
\begin{theorem}\label{bd}
 Let $u\in BD_{loc}(\Om)$ with $|Eu|(\Om)<+\infty$. For every $\e>0$ there exists $v\in BV^{aj}
(\Om;\Rn)=SBV(\Om;\Rn)$ such that
$J_v=J_u$, $v^+-v^-=u^+-u^-$ $\hn$-a.e. on $J_u$, and 
\eqref{v} holds.
In particular,
$u-v\in BD_{loc}^{ac}(\Om)$.
\end{theorem}
\begin{proof}
Since $J_u$ is countably $(\hn,n-1)$-rectifiable and $u^+-u^-$ is $\hn$-integrable on $J_u$ 
(see \cite{ACDM}), as in the proof of Theorem~\ref{1}, we find a function $v\in BV^{aj}(\Om;\Rn)
$ such that
$J_v= J_u$, $v^+-v^-=u^+-u^-$ $\hn$-a.e. on $J_u$, and \eqref{v} holds. Then $E^j(u-v)=0$, 
so that  
$u-v\in BD_{loc}^{ac}(\Om)$.
\end{proof} 
\par

We recall that $LD(\Om)=\{u\in BD(\Om):Eu=E^au\}$.
The following corollary can be easily deduced from Theorem \ref{bd}.
\par

\begin{corollary}
Given any $u\in SBD(\Om)$ there exist  $v\in LD(\Om)$ and $w\in SBV(\Om;\Rn)$ such that
$u=v+w$.
\end{corollary}
\end{section}
\bigskip
\noindent {\bf Acknowledgments.} {This work is part of the Project ``Variational problems with 
multiple scales" 2006, 
supported by the Italian Ministry of University and Research.}
\par

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\par


\end{thebibliography}
\end{document}
