\documentclass[ reqno]{amsart} \usepackage{amsfonts} \textwidth14cm \textheight23cm \flushbottom \theoremstyle{plain} \begingroup \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \endgroup \theoremstyle{definition} \begingroup \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \endgroup \theoremstyle{remark} \begingroup \endgroup \mathsurround=1pt \mathchardef\emptyset="001F \numberwithin{equation}{section} \newcommand{\e}{\varepsilon} \newcommand{\Om}{\Omega} \newcommand{\R}{{\mathbb R}} \newcommand{\Rn}{{\R}^n} \newcommand{\Rm}{{\R}^m} \newcommand{\wto}{\rightharpoonup} \newcommand{\setmeno}{\!\setminus\!} \newcommand{\into}{{\int_{\Omega}}} \newcommand{\hn}{{\mathcal H}^{n-1}} \newcommand{\Ln}{{\mathcal L}^n} \newcommand{\M}{{\mathcal M}} \newcommand{\LM}[2]{\hbox{\vrule width.4pt \vbox to#1pt{\vfill \hrule width#2pt height.4pt}}} \newcommand{\LLL}{{\mathchoice {\>\LM{7}{5}\>}{\>\LM{7}{5}\>}{\,\LM{5}{3.5}\,}{\,\LM{3.35} {2.5}\,}}} % L symbol \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 $i0$ 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 $t_kt_0$. 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 \begin{thebibliography}{99} \par \bibitem{A}Alberti G.: A Lusin type property of gradients. {\it J. Funct. Anal.\/} {\bf 100} (1991), 110-118. \par \bibitem{ACDM}Ambrosio L., Coscia A., Dal Maso G.: Fine properties of functions with bounded deformation. {\it Arch. Rational Mech. Anal.\/} {\bf 139} (1997), 201-238. \par \bibitem{A-F-P}Ambrosio L., Fusco, N., Pallara D.: Functions of bounded variation and free-discontinuity problems. Oxford Mathematical Monographs, 2000. \par \bibitem{Ga}Gagliardo E.: Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili. {\it Rend. Sem. Mat. Univ. Padova\/} {\bf 27} (1957) 284-305. \par \bibitem{Giu}Giusti E.: Minimal Surfaces and Functions of Bounded Variations. Monographs in Mathematics, Birkh\"auser, Boston, 1984. \par \bibitem{Tem}Temam R.: Probl\`emes math\'ematiques en plasticit\'e. Gauthier-Villars, Paris, 1983. \par \end{thebibliography} \end{document}