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%  Ultime correzioni Gianni 19/2/96.
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\def\autori{L.\ AMBROSIO, A.\ COSCIA, G.\ DAL MASO}
\def\titolo{Fine properties of functions with bounded deformation}


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\autori \hfil}
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\null
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{\twelvepoint
\baselineskip=1.7\baselineskip
\centerline{\bf FINE PROPERTIES OF FUNCTIONS}
\centerline{\bf WITH BOUNDED DEFORMATION}
}
\vskip2truecm
\centerline{Luigi AMBROSIO ($^1$)}
\medskip
\centerline{Alessandra COSCIA ($^2$)}
\medskip
\centerline{Gianni DAL MASO ($^3$)}
\medskip
\vfil

{\eightpoint
\baselineskip=1.2\baselineskip

\centerline{\bf Abstract}
\bigskip
\noindent
The paper is concerned with the fine properties of functions $u$ in
$BD$, the space of functions with bounded deformation.
We analyse the set of Lebesgue points and the set where
these functions have one sided approximate limits. Moreover,
following the analogy with $BV$, we decompose the symmetric
distributional derivative $E u$ into an absolutely continuous part
$E^au=\E u\,{\cal L}^n$, a jump part $\Jp u$, and a Cantor part $\Cp u$. 
The main result of the paper is a structure theorem for $BD$ functions, 
showing that these parts of the derivative can be recovered from the 
corresponding ones of the one dimensional sections. Moreover, we prove 
that $BD$ functions are approximately differentiable in almost
every point of their domain.
\par
%
\vfil
\item{$(^1)$} Dipartimento di Matematica, Universit\`a di Pavia,
\item{} Via Abbiategrasso 205, 27100 Pavia, Italy.
\item{} e-mail: ambrosio@dragon.ian.pv.cnr.it
\smallskip
\item{$(^2)$} Dipartimento di Matematica, Universit\`a di Parma,
\item{} via D'Azeglio 85, 43100 Parma, Italy.
\item{} e-mail: coscia@prmat.math.unipr.it
\smallskip
\item{$(^3)$} SISSA, Via Beirut 4, 34013 Trieste, Italy.
\item{} e-mail: dalmaso@neumann.sissa.it
\par
}
\vskip 1truecm
\centerline {Ref. S.I.S.S.A.
8/96/M (January 96)}
\vskip 1truecm
\eject
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%

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                % per la numerazione automatica

\parag{\chp{secint}}{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%

The space $BD(\Om)$ of functions with {\it bounded deformation\/} in 
an open
set $\Om\subset\Rn$ is defined as the set of functions $u\in 
L^1(\Om;\Rn)$
such that the symmetric distributional derivative
%
$$
E_{ij} u\,:=\, {1\over 2}\bigl(D_iu^j+D_j u^i\bigr)
$$
%
is a (matrix valued) measure with finite total variation in $\Om$.
This space, introduced in [\rf{MaStCh}] and [\rf{Su}],
has been widely considered
in the literature (see [\rf{Su79}], [\rf{AnGi}], [\rf{TeSt80}], [\rf{Te85}]) 
in connection with the mathematical theory of plasticity.

The goal of this paper is to study the fine properties of functions
in $BD(\Om)$.
%trying to see to what extent the properties of the
%space $BV(\Om;\Rn)$ can be extended to $BD(\Om)$.
%Indeed, the so-called Korn inequality (see ****), quite surprisingly,
%shows that in some cases where $E u$ is absolutely continuous
%there is the possibility to control the distributional derivative
%$D u$ by the $L^p$ norm of the density of $E u$.
%On the other hand, a counterexample by
%Ornstein in [\rf{Orn62}] shows that
%the Korn inequality is not true with the $L^1$ norm, and this
%counterexample can also be used to show (see [\rf{BeCoDM95}],
%Proposition~5.2) that the inclusion of
%$BV(\Om;\Rn)$ in $BD(\Om)$ is strict.
We will prove that many properties of $BV(\Om;\Rn)$, the space
of functions with {\it bounded variation\/}, can
be suitably extended to $BD(\Om)$. The crucial point is to choose
a ``correct'' definition of jump set, consistent, but not necessarily
equal, with the one usually adopted in the $BV$ theory.

To illustrate our results, we briefly recall
standard notations and definitions from that theory. Denoting, for any
$u\in L^1(\Om;\R^m)$,
the complement of the Lebesgue set of $u$ by $S_u$,
it is well known that for any $u\in BV(\Om;{\bf R}^m)$ the set $S_u$ is
countably ${(\Hn,{n-1})}$ rectifiable and that for $\Hn$ almost every
$x\in S_u$ the function $u$ has one sided approximate limits
$u^+(x)\neq u^-(x)$ with respect to a suitable direction $\nu_u(x)$ 
normal
to $S_u$ at $x$ (see (\rf{twosided})). Moreover, the distributional
derivative $Du$ can be decomposed, with respect to the Lebesgue 
measure ${\cal L}^n$, into singular part $D^s u$ and absolutely 
continuous part $D^a u=\nabla u\,{\cal L}^n$, with density $\nabla u$ 
equal to the approximate differential of~$u$.
%by the Calderon-Zygmund theorem.
The singular part $D^s u$ can be written as
the sum of $D^s u\LL S_u$ and of $D^s u\LL (\Om\setminus S_u)$;
the first part, called {\it jump part\/} and denoted by $\DJp u$,
can be represented
by the integration of $(u^+-u^-){\otimes}\nu_u$ on $S_u$, while
the second part, called {\it Cantor part\/} and denoted by $\DCp u$,
vanishes on any Borel set $B\subset\Om$ which is $\sigma$-finite
with respect to $\Hn$. The space $SBV(\Om;\R^m)$
of {\it special functions\/} with bounded variation (i.e., functions
whose derivative has no Cantor part),
introduced in [\rf{DGAm88}], has been widely
studied in the last years in connection with the Mumford-Shah
image segmentation problem and, more generally, in connection with
free discontinuity problems (see [\rf{MuSh85}], [\rf{DG88}], 
[\rf{Am89a}], [\rf{Am89}], [\rf{DGCaLe}], [\rf{MuSh89}], [\rf{Am90}], 
[\rf{NiMu}], [\rf{MuNiSh93}], [\rf{MoSo}]).

In addition, it is well known that the space $BV(\Om;\R^m)$
can be characterized by
means of the one dimensional sections: a function $u$ belongs to 
$BV(\Om;\R^m)$
if and only if, for every direction
$\xi\in\Sn:= \{\zeta\in\Rn:{|\zeta|=1}\}$, we have
$u_{y,\xi}\in BV(\Om^\xi_y;\R^m)$ for $\Hn$ almost every
$y\in\pi_\xi$ and
%
$$
\int_{\pi_\xi}|Du_{y,\xi}|(\Om^\xi_y)\,d\Hn(y)<+\infty\,,
$$
%
where $\pi_\xi$ is the hyperplane orthogonal to $\xi$ passing
through the origin, $\Om^\xi_y=\{ t\in \R : y+t\xi \in \Om\}$,
and $u_{y,\xi}(t):=u(y+t\xi)$ for every $t\in \Om^\xi_y$.
The {\it structure theorem\/} for
$BV$ functions states that the scalar products
$(D^a u,\xi)$, $(\DJp u,\xi)$, and
$(\DCp u,\xi)$ can be recovered from the corresponding parts of the 
derivative of $u_{y,\xi}$, i.e.,
%
$$
(D^{\sigma} u,\xi)\,=\int_{\pi_\xi} D^\sigma u_{y,\xi}\,d\Hn(y)
\qquad \quad \hbox{for } \sigma=a,j,c
\leqno (\frm{fubi1})
$$
%
(see Section~\rf{secnot} for the main notations).
The structure theorem has been used by the first author in [\rf{Am89a}]
to prove the compactness properties of $SBV$, and later it has been 
used to get lower semicontinuity theorems in $BV$ in the isotropic case
(see [\rf{BBB}]) and $\Gamma$-convergence results (see [\rf{ATa}],
[\rf{ATb}], [\rf{Cha95}]).

For $BD$ functions the {\it discontinuity\/} set $S_u$ is not known to be
$\sigma$-finite with respect to $\Hn$. However, we discovered that
in the $BD$ theory the r\^ole of $S_u$ is played
by the {\it jump set\/} $J_u$ of $u$, defined as the set of discontinuity
points $x$
where $u$ has one sided approximate limits $u^+(x)\neq u^-(x)$
with respect to a suitable direction $\nu_u(x)$ normal to $J_u$
(see Section~\rf{secnot}). Let us remark that for $BV$ functions
the set $S_u \setminus J_u$ is $\Hn$ negligible.

The set of jump points $J_u$ turns out to be useful in the
decomposition of $E u$.
If $E^a u =\E u\,{\cal L}^n$ and $E^s u$ are the absolutely continuous and
the singular part of $E u$ with respect to the Lebesgue measure, we set
%
$$
E u\,=\, E^a u + E^s u \,=\,\E u \,{\cal L}^n+\Jp u+\Cp u\,,
$$
%
where $\Jp u$ and $\Cp u$ are the restrictions of $E^s u$ to
$J_u$ and to the complement of $J_u$ respectively.
First of all we prove that the density $\E u$
of $E^a u$ with respect to ${\cal L}^n$ can still be interpreted as an 
approximate symmetric differential (Theorem~\rf{symmdi}).
Moreover we prove that $J_u$ is countably $(\Hn,{n-1})$ rectifiable and 
that the {\it jump part\/} $\Jp u$ can be represented on every Borel 
subset of
$\Om$ by the formula
%
$$
\Jp u(B) \,=\int_{B\cap J_u}
\bigl[{(u^+ - u^-) {\otimes} \nu_u
+\nu_u {\otimes} (u^+ - u^-)\over 2}\bigr]\,d\Hn\,,
$$
%
where $\nu_u$ is the unit normal to $J_u$ in an approximate sense.
Let us remark that
$J_u$ coincides, up to an $\Hn$ negligible set, with the set
$\Theta_u$ introduced by Kohn in~[\rf{Ko}] and defined as
the set of points in $\Om$ where
the ${n-1}$ dimensional upper density of $\v Eu\v$ is positive, i.e.,
%
$$
\Theta_u\,:=\, \Bigl\{x\in\Om:
\limsup_{\ro\to 0^+} {|Eu|(B_\ro(x))\over \ro^{n-1}}>0\Bigr\}\,.
\leqno (\frm{Thetau})
$$
%
Finally we prove in Theorem~\rf{cantor0} that the {\it Cantor part\/} 
$\Cp u$ vanishes on any
Borel set $B\subset\Om$ which is $\sigma$-finite with respect to $\Hn$.

One of the main results of our paper is the extension of the
structure theorem to $BD(\Om)$. Let $u\in L^1(\Om;\Rn)$, let 
$\xi\in\Sn$, and let $u^\xi_y(t)= (u(y+t\xi), \xi)$,
where $(\cdot ,\cdot)$ denotes the scalar product. 
Using a result of~[\rf{Mi64}] it follows that $u\in BD(\Om)$
if and only if, for every $\xi\in\Sn$, the function $u^\xi_y$ belongs to
$BV(\Om_y^\xi)$ for $\Hn$ almost every $y\in\pi_\xi$ and
%
$$
\int_{\pi_\xi}|Du^\xi_y|(\Om_y^\xi)\,d\Hn(y)<+\infty
$$
%
(see~[\rf{BeCo93}] and~[\rf{BeCoDM95}]). We prove that
%
$$
(E^{\sigma} u \xi,\xi)\,=\int_{\pi_\xi} D^\sigma u_y^\xi \,d\Hn(y)
\qquad \quad \hbox{for } \sigma=a,j,c\,.
\leqno (\frm{fubi2})
$$
In particular, the equality for $\sigma=j$ implies
that, generically, the section in the $\xi$ direction
of the jump set of $u$ coincides with
the jump set of $u^\xi_y$, i.e.,
%
$$
\bigl\{t:y+t\xi\in J_u\bigr\}\,=\,J_{u^\xi_y}\,=\,S_{u^\xi_y}
\leqno(\frm{inge})
$$
%
for $\Hn$ almost every $\xi\in\Sn$ and for $\Hn$ almost every 
$y\in\pi_\xi$. More generally, using the 
structure
theorem and the Lebesgue representative of $u$,
we can relate
$u^+(x)$ and $u^-(x)$ to left and right limits
of the one dimensional sections $u^\xi_y$, and 
$(\E u \xi,\xi)$ to the classical derivative of $u^\xi_y$
(Theorem~\rf{structure}).

Following
the analogy with the theory of $SBV$ functions,
we can define the space of {\it special functions\/} with bounded
deformation $SBD(\Om)$ by
%
$$
SBD(\Om)\,:=\,\bigl\{u\in BD(\Om): \Cp u=0\bigr\}\,.
$$
%
As a consequence of the structure theorem, it turns out that
a function $u\in BD(\Om)$ belongs to $SBD(\Om)$ if and only
if for any $\xi\in\Sn$ the function $u^\xi_y$ belongs to
$SBV(\Om^\xi_y)$ for $\Hn$ almost every $y\in\pi_\xi$
(Proposition~\rf{equiva}).
This condition has previously
been used in [\rf{BeCo93}] and [\rf{BeCoDM95}] to define $SBD(\Om)$.
Moreover, using (\rf{inge}) we obtain in Theorem~\rf{teoluBD}
that the integralgeometric measure $\lambda_u$,
introduced in [\rf{BeCoDM95}] to study compactness 
and semicontinuity properties in $SBD(\Om)$,
coincides with $\Hn\LL J_u$.

%In Section~\rf{isotro} we show one more application of the
%structure theorem estabilishing in
%Theorem~\rf{1dsci} a lower semicontinuity theorem
%for isotropic functionals on $BD(\Om)$ with different energy
%densities on the different parts of $E u$.

In general we have that $J_u\subset S_u$, but the inclusion can be 
strict for a function $u\in BD(\Omega)$.
In Section~\rf{sumenfu} we study the set $S_u\setminus J_u$,
using a ``Poincar\'e type'' inequality for $BD$ functions 
(Theorem~\rf{kohn1}) that we found
in the unpublished part of Kohn's Ph.D. thesis~[\rf{Ko}].
The main result, proved in Theorem~\rf{suju}, is
that the set $S_u\setminus J_u$ of
discontinuity points of~$u$ which are not jump points is
$|Ev|$ negligible for every $v\in BD(\Om)$, i.e., $|Ev|$ almost every $x$ 
in $\Omega$ is either a Lebesgue point or a jump point for $u$.

Finally, in the last section of the paper we prove that any function
$u\in BD(\Om)$
is approximately differentiable in $\Le^n$ almost every point of $\Om$.
A similar result has been proved independently by Haj\l asz 
in~[\rf{Ha}].
If either $\Om=\Rn$ or $\Om$ is bounded with Lipschitz boundary,
the approximate differential $\nabla u$ satisfies the weak
$L^1$ estimate
%
$$
{\cal L}^n\bigl(\{x\in\Om: |\nabla u(x)|>t\}\bigr) \leq
{c(n,\Omega)\over t} \bigl[\,|Eu|(\Om)+ \int_\Om |u(x)|\,dx\,\bigr]
$$
%
for every $t >0$.
In case $E^s u=0$ and $\E u\in L^p(\Om)$ for some $p>1$,
our proof of the approximate differentiability can also be used,
with minor modifications, to
obtain Korn's inequality (Remark~\rf{korn}).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\parag{\chp{secnot}}{Preliminaries and notation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
Let $\Om$ be an open subset of $\Rn$, $n \geq 1$. The
$\sigma$-algebra of all Borel subsets of $\Om$ is denoted by
$\B(\Om)$. The symbols $(\cdot,\cdot)$ and $\vert \cdot \vert$
stand for the euclidean scalar product and
the euclidean norm.
The Lebesgue measure in $\Rn$ is denoted by $dx$ or by ${\cal L}^n$,
and the~$k$ dimensional Hausdorff measure by~$\H^k$
(see~[\rf{Fed68}], 2.10.2).
We recall that for any two vectors $a$ and $b$ in $\Rn$, the symmetric
product $a {\odot} b$ is the symmetric $n{\times} n$ matrix defined by
$a {\odot} b \,=\,(a {\otimes} b + b{\otimes} a) /2$, where $\otimes$ 
denotes the tensor product.

If $\mu$ is a Borel measure on $\Om$ with values
in $\R^m$, ${m\ge1}$,
its total variation is denoted by $\vert \mu\vert$.
If $F$ is a Borel subset of $\Om$, the measure $\mu \LL F $ is
defined by $(\mu \LL F)(B) = \mu(B \cap F)$ for any $B
\in \B(\Om)$.
For every $\mu$ integrable function $f$, the measure $f \mu$
is defined by $(f \mu)(B)= \int_B f \,d\mu$ for every $B \in \BOm$.
% We shall use also the notation $\int_B f \mu$.
The absolutely continuous and singular parts of~$\mu$ with respect to
the Lebesgue measure ${\cal L}^n$ are denoted by $\mu^a$ and
$\mu^s$ respectively.

The $k$ {\it dimensional density\/} of a non-negative measure $\mu$ 
at a point $x\in\Omega$ is defined as
$$
\limro {\mu(B_\ro(x))\over \omega_k\ro^k} \,,
$$
where $B_\ro(x)$ is the open ball in $\Rn$ with centre~$x$ and 
radius~$\ro$, and 
$\omega_k$ is the $k$ dimensional measure of the unit ball in $\R^k$.

If $\mu\,=\, \{\mu_{ij}\}$ is a Radon measure on $\Om$ with
values in the space of symmetric $n{\times} n$ matrices and
$\xi =(\xi^1,\dots,\xi^n) \in \Rn$,
then the real valued Radon measure $(\mu \xi,\xi)$ is defined by
%
$$
(\mu \xi,\xi)\,:=\, \sum_{i,j} \xi^i \xi^j \mu_{ij}\,,
\leqno(\frm{muxixi})
$$
%
so that $(\mu \xi,\xi)(B)=(\mu (B)\xi,\xi)$ for every $B\in \BOm$.
It is easy to check that for any basis $\xi_1,\ldots,\xi_n$ of
$\Rn$ there exists a constant $c$, depending on the basis, such
that
%
$$
|\mu|\leq c\sum_{i,j=1}^n \bigl|
(\mu (\xi_i+\xi_j),\xi_i+\xi_j)\bigr|\,.
\leqno (\frm{matrici})
$$
%
For any $y,\xi \in \Rn$, $\xi\neq 0$, and any
$B \subset \Rn$ we define
%
$$
\eqalign
{
\pix &:=\{y \in \Rn : (y,\xi) = 0\}\,,
\cr
B_y^{\xi} &:= \{t \in \R :
y + t \xi \in B\}\,,
\cr
B^\xi &:=
\{y \in \pix : B_y^{\xi} \neq \emptyset\}\,.
}
\leqno(\frm{sezioni})
$$
%
In other words $\pix$ is the hyperplane orthogonal to $\xi$ passing
through the origin and $B^\xi=p_\xi(B)$, where $p_\xi$ denotes the
orthogonal projection onto $\pix$. Moreover $B_y^{\xi}$ is the one 
dimensional section of~$B$ on the
straight line passing through~$y$ in the direction of~$\xi$.

Let $\xi\in\Rn$ with $\xi\neq 0$.
For every $y\in \Om^\xi$ let $\nu_y\colon\B(\Om_y^\xi)\to\R$ be a 
bounded measure. Suppose
that for every $\varphi\in \C_c(\R)$ the map
%
$$
y\mapsto\int_{\Om_y^\xi}\varphi(t)\,d\nu_y(t)
$$
%
is Borel measurable on $\Om^\xi$ and that
%
$$\int_{\Om^\xi}|\nu_y|(\Om_y^\xi)\,d\Hn(y) <+\infty\,.
$$
%
Then for every $B\in \BOm$ the map $y\mapsto \nu_y(B_y^\xi)$
is Borel measurable and $\Hn$ integrable on $\Om^\xi$,
and the set function $\lambda \colon  \BOm \to \R$ defined by
%
$$
\lambda (B)\,:=\,\int_{\Om^\xi}
\nu_y(B_y^\xi)\,d\Hn(y) \qquad \forall B\in \BOm
$$
%
is a measure, which will be denoted by
%
$$
\int_{\Om^\xi} \nu_y \,d\Hn(y)\,.
$$
%
It is easy to see that its total variation $\v \lambda \v$ is given by
%
$$
\v \lambda \v = \int_{\Om^\xi} \v \nu_y \v \,d\Hn(y)\,.
\leqno(\frm{vartot})
$$
%
If each measure $\nu_y$ is absolutely continuous (resp. singular) with
respect to $\Le^1$, then $\int_{\Om^\xi} \nu_y \,d\Hn(y)$ is absolutely
continuous (resp. singular) with respect to $\Le^n$.

For simplicity we have considered only the case of measures obtained, 
by integration, from their one dimensional sections. 
In the proof of Proposition~\rf{projBD} and Theorem~\rf{structure} we 
shall use similar constructions by slicing $\Omega$ by parallel
$k$~dimensional planes.

\bigskip
\noindent
{\it Lebesgue points, discontinuity points, and jump points.\/}
Given a function $u\in L^1(\Om;{\bf R}^m)$,
the set of Lebesgue points of $u$ in $\Om$ is denoted by $\Om_u$.
In other words, $x\in \Om_u$ if and only if there exists
$\tilde u(x) \in {\bf R}^m$ such that
%
$$
\limro {1\over \ro^n} \int_{\Brx} |u(y) - \tu (x)|\,dy \,=\,0\,.
\leqno(\frm{puntoLeb})
$$
%
The {\it Lebesgue discontinuity\/} set $S_u$ of $u$ is defined as the set 
of points which are not Lebesgue points, i.e.,
$S_u:=\Om\setminus \Om_u$.
By Lebesgue's differentiation theorem the set $S_u$ is negligible and
the function $\tilde{u} \colon \Om_u \to \R^m$
coincides with $u$ $\Le^n$ almost everywhere in $\Om_u 
=\Om\setminus S_u$.
The function $\tilde u$ is the so-called {\it Lebesgue representative\/} 
of~$u$.
We say that $u$ has {\it one sided Lebesgue limits\/} $u^+(x)$ and
$u^-(x)$ at $x\in \Om$, with respect to a suitable direction
$\nu_u(x)\in{\bf S}^{n-1} := \{ \xi\in\Rn : {\v \xi \v =1}\}$, if
%
$$
\lim_{\ro\to 0^+} {1\over \ro^n} \int_{B_\ro^{\pm}(x,\nu_u(x))}
|u(y)-u^\pm(x)|\,dy\,=\,0\,,
\leqno(\frm{twosided})
$$
%
where $B_\ro^{\pm}(x,\nu_u(x)):=\{y\in B_\ro(x):(y-x,\pm\nu_u(x)) > 
0\}$.
In this case the rescaled
functions $u_\ro(y):=u(x+\ro y)$ converge in $L^1(B_1(0);\R^m)$,
as $\ro \to 0^+$, to the function
%
$$
u_0(y)\,:=\,\cases{u^+(x)\,, & if $(y,\nu_u(x))>0\,$,\cr\cr
u^-(x)\,, & if $(y,\nu_u(x))<0\,$.}
\leqno (\frm{uoo})
$$
%
This shows that,
if $u^+(x)\neq u^-(x)$, then the triplet $(u^+(x),u^-(x),\nu_u(x))$
is uniquely determined up to a change of sign of $\nu_u(x)$ and a 
permutation of $(u^+(x),u^-(x))$. If $u^+(x)=u^-(x)$, then $x\in \Om_u$ 
and the one sided Lebesgue limits coincide with the Lebesgue 
representative. The
{\it jump set\/} $J_u$ of $u$ is defined as the set of all points
$x\in S_u$ such that
$u$ has one sided Lebesgue limits at $x$. Equivalently, a point
$x\in\Om$ belongs to $J_u$ if and only if $u$ has two different one
sided Lebesgue limits at~$x$. It is easy to prove that $J_u$ and $S_u$ 
are Borel sets and that $\tilde u$, $u^+$, and $u^-$ are Borel functions.

\bigskip
\noindent
{\it Pointwise variation and distributional derivative.\/}
Let $A$ be an open subset of $\R$, let $E$ be a Borel subset of $A$
such that $\Le^1(A\setminus E)=0$, and let $u\colon  E\to \R^m$
be an integrable function. As $u$ is defined $\Le^1$ almost everywhere
in $A$, it can be identified with a distribution on $A$, whose derivative
is denoted by $Du$.
For every open interval $I\subset A$,
the {\it pointwise variation \/} $Vu(I)$ of $u$ on $I$ is defined by
%
$$
Vu(I)\,:=\,\sup\biggl\{\sum_{i=1}^{k-1}|u(t_{i+1})-u(t_i)|:
t_1<t_2<\cdots<t_k,\,\, t_i\in E\cap I\biggr\}\,.
\leqno(\frm{defVu})
$$
%
Let us remark that $Vu$, unlike $Du$, is sensitive to changes of
the values (or of the domain) of $u$ on
sets of Lebesgue measure zero.
If $Vu(I)<+\infty$, then $Du$ can be represented by a bounded measure
on $I$, whose total variation satisfies
%
$$
|Du|(I)\leq Vu(I)\,.
\leqno(\frm{DuleVu})
$$
%
Moreover, if $Vu(I)<+\infty$ for every open interval $I\subset A$,
then it is possible to extend $Vu$ to a non-negative
Radon measure defined on $\B (A)$, which will still be
denoted by $Vu$. Indeed, given
an open set $U\subset A$, it is enough to define $Vu(U)$ as the
sum of $Vu(I)$ over all connected components $I$ of $U$. It turns out
that the set function $U\mapsto Vu(U)$ is increasing, subadditive,
additive on disjoint open sets, and inner regular. These properties imply
(see, e.g., [\rf{DM}], Theorem~14.23) that
%
$$
Vu(B)\,:=\,\inf\Bigl\{Vu(U): U\,\,\hbox{\rm open },\,\,
B\subset U \subset A\Bigr\}\qquad\qquad \forall B\in{\cal B}(A)
$$
%
is a Radon measure on $A$ which coincides with the function defined
by (\rf{defVu}) on all open intervals $I\subset A$.
It is easy to prove that, if $Vu(I)<+\infty$ for every open
interval $I\subset A$, then there exists an extension $v$ of $u$,
defined on $A$, such that $Vv=Vu$ as measures on~$A$. Therefore,
using the classical theory of functions of bounded variation
on bounded intervals, that applies to~$v$ on each interval $I\subset A$, 
we obtain that~$u$ has left and right limits
at every $t\in A$ and that these limits coincide except on a countable
subset of $A$. Moreover the classical derivative
%
$$
u'(t)= \lim_{{\scriptstyle s\to t}\atop {\scriptstyle s\in E}}
{{u(s)-u(t)} \over {s-t}}
\leqno(\frm{rappincr})
$$
%
exists for $\Le^1$ almost every $t\in E$ and coincides with the
density of the absolutely continuous part of the measure $Du$
with respect to $\Le^1$.

It is well known that, if $E=A$ and $u\in \C^1(A)$, then
%
$$
Vu= \v Du\v = \v u'\v \Le^1
\leqno(\frm{asscont})
$$
%
as measures on $A$.

\bigskip
\noindent
{\it The spaces $BV(\Om;\R^m)$ and $SBV(\Om;\R^m)$.\/}
The space $BV(\Om;\R^m)$, $m \geq 1$, of {\it functions of bounded
variation\/} in $\Om$ with values in $\R^m$ is defined as the set of
all functions $u \in L^1(\Om;\R^m)$ whose distributional gradient
$Du$ is a Radon measure (with values in the space of
${m{\times}n}$ matrices)  with bounded total variation in $\Om$.
We set $BV(\Om) = BV(\Om;\R)$.
It is well known that for every $u\in BV(\Om;\Rn)$ the discontinuity 
set
$S_u$ is a countably $(\Hn,{n-1})$
rectifiable Borel set, i.e., $\Hn$ almost all of $S_u$ can be covered by
a sequence of $\C^1$ hypersurfaces $M_i$ (see, e.g.,
[\rf{Fed68}], 3.2.14). Hence, we can define
$\Hn$ almost everywhere in $S_u$
an approximate unit normal
$\nu_u$ (see [\rf{Si83}], Section~11), characterized by the fact that 
for every~$i$ 
%
$$
\nu_u(x)=\pm\nu_{_{M_i}}(x)
\qquad\hbox{\rm for $\Hn$ a.e. }x\in
S_u\cap M_i\,,
$$
%
where $\nu_{_{M_i}}(x)$ is a normal to $M_i$ at the point 
$x$.
Moreover, for $\Hn$ almost every $x\in S_u$, the function $u$
has one sided Lebesgue limits $u^+(x)\neq u^-(x)$
with respect to the approximate unit normal
$\nu_u(x)$, which is uniquely determined up to the sign. In particular, 
${\cal H}^{n-1}(S_u\setminus J_u)=0$.

If $u \in BV(\Om;\R^m)$, then the Lebesgue decomposition of the
measure $Du$ can be written as
%
$$
Du \,=\, D^au + D^s u \,=\, \nabla u\,dx+ D^s u\,,
$$
%
where $\nabla u$ denotes the density of the absolutely continuous part
of $Du$; the notation is motivated by the fact that $\grad u$ can be
interpreted as an approximate differential for $u$, i.e.,
%
$$
\lim_{\ro\to 0^+} {1\over \ro^n} \int_{B_\ro(x)}
{|u(y)-u(x)-(\nabla u(x),y-x)|\over |y-x|}\,dy\,=\,0
\leqno(\frm{apdi})
$$
%
for $\Le^n$ almost every $x\in\Om$.
The singular part $D^su$ can be further decomposed into
two mutually singular measures as
%
$$
D^su \,=\, \DJp u + \DCp u \,=\, (u^+-u^-) {\otimes} \nu_u
\,{{\cal H}^{n-1}} \LL  S_u+ \DCp u\,,
\leqno(\frm{decBV})
$$
%
where $\DJp u=D^su\LL S_u=Du\LL S_u$  is the ${n-1}$ dimensional part 
of $Du$ and $\DCp u=D^su\LL \Omega_u$ is the Cantor part of $Du$,
vanishing on all Borel sets $B$ with $\Hn(B)<+\infty$.

We say that $u$ is a {\it special function of bounded
variation\/} in $\Om$ with values in $\R^m$,
and we write $u\in SBV(\Om;\R^m)$, if
$u\in BV(\Omega;\R^m)$ and $\DCp u= 0$ in~$\Om$; again, we write 
$SBV(\Om)$ for $SBV(\Om;\R)$. For a general exposition of the theory of
functions of bounded variation we refer to
[\rf{Giu84}], [\rf{Zie89}], [\rf{EvGa93}];
for an introduction to the properties of the space
$SBV(\Om;\R^m)$ we refer to
[\rf{DGAm88}], [\rf{Am89a}], [\rf{Am89}],~[\rf{Am90}].

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
\parag{\chp{secbd}}{The space $\bf BD(\Om)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
Let $u=(u^1,\dots,u^n) \in L^1(\Om;\Rn)$, and let $E u$ be the
symmetric part of the distributional gradient
of $u$,  i.e.,
%
$$
E u \,:=\, \{E_{ij}u\}\,, \qquad
E_{ij}u \,:=\, {1\over 2}\bigl(D_i u^j + D_j u^i\bigr)\,.
$$
%
The space $BD(\Om)$ of vector fields of {\it bounded deformation\/}
in $\Om$ is defined as the set of all functions $u \in L^1(\Om;\Rn)$
such that $E_{ij}u$ is a Radon measure with
bounded total variation in $\Om$ for any $i,j=1,\dots,n$.
The space $BD(\Om)$ is a Banach space with the norm
$\Vert u\Vert_{BD(\Om)} = \Vert u\Vert_{L^1(\Om)} + \vert E
u\vert(\Om)$.

For a general exposition of the theory of functions of bounded
deformation we refer to [\rf{Su}], [\rf{Su79}], [\rf{Ko}], [\rf{MaStCh}], 
[\rf{AnGi}], [\rf{TeSt80}],~[\rf{Te85}].

Now we summarize some results
on functions of bounded deformation which will be useful in the sequel.

(i) For every $\xi=(\xi^1, \ldots, \xi^n)\in\Rn$ let $D_\xi$ be the 
distributional derivative in the direction~$\xi$ defined by
$D_\xi v =\sum_j \xi^j D_j v$. For every function $u\colon\Om \rga \Rn$
let us define the function $u^\xi\colon \Om \rga \R$ by
$u^\xi(x)=(u(x),\xi)$. If $u\in BD(\Om)$, then $D_\xi u^\xi$
is a bounded Radon measure on $\Om$ for every $\xi\in\Rn$ and
$$
D_\xi u^\xi=(Eu\xi,\xi)\,.
\leqno(\frm{Dxiuxi})
$$
Conversely, let $\xi_1, \ldots, \xi_n$ be a basis of $\Rn$ and
let $u\in L^1(\Om;\Rn)$; then $u\in BD(\Om)$ if $D_\xi u^\xi$
is a bounded Radon measure on $\Om$ for every $\xi$ of the form 
$\xi_i+\xi_j$, $i,\,j=1,\ldots,n$
(see [\rf{Te85}], Chapter~II, Section~2.2).

(ii) Let $u \in BD(\Om)$ and let $M$ be a
$\C^1$ hypersurface contained in $\Om$; it is well known (see
[\rf{Te85}], Chapter~II) that the one sided Lebesgue limits
$u^\pm(x)$ on both sides of $M$ (i.e., with respect to~$\nu_M^{}(x)$) 
exist 
for $\Hn$ almost every $x\in M$ and satisfy
%
$$
E u \LL M = (u^+ - u^-) {\odot} \nu_{\scriptscriptstyle M}\,\Hn \LL M\,,
\leqno(\frm{rappJ})
$$
%
where $\nu_M^{}$ is a unit normal to $M$ and
$\odot$ is the symmetric product between vectors
(see Section~\rf{secnot}). It is easy to extend these 
properties to any countably
$(\Hn,{n-1})$ rectifiable Borel set $M\subset\Om$.

(iii) If $\Om$ is a bounded open set with Lipschitz boundary
$\partial\Om$ and $u\in BD(\Om)$, then the function
%
$$
\bar u(x):= \cases{u(x)\,, & if $x\in\Om\,$,\cr
0\,, & if $x\notin\Om\,$,}
\leqno(\frm{estensu})
$$
%
belongs to $BD(\Rn)$ (see~[\rf{Te85}], Chapter~II, Proposition~2.1)
and, for $\Hn$ almost every $x\in\partial\Om$, the one sided Lebesgue 
limits $\bar{u}^+(x)$ and $\bar{u}^-(x)$ with respect to the outer unit 
normal are respectively equal to~$0$ and to the trace of~$u$ on 
$\partial\Om$, that will be denoted by~$u_*$. Moreover we have the 
estimate $\V u_* \V_{L^1(\partial\Om)} \leq
c(\Om) \V u\V_{BD(\Om)}$ for a suitable constant $c(\Om)$
depending on $\Om$ (see [\rf{Te85}], Chapter~II, Theorem~2.1, for the 
estimate of the trace, and [\rf{Mi67}], Theorem~1, for an argument
which proves the existence of the one sided Lebesgue limits at the
boundary).
Given a smooth symmetric matrix valued function 
$\varphi=\{\varphi_{lm}\}$
with compact support in $\Rn$, applying (\rf{rappJ}) to
$\bar{u}$ and using the coincidence of $Eu$ and $E\bar{u}$ in $\Om$
we obtain the {\it divergence theorem\/} in $BD$:
%
$$
\sum_{l,m=1}^n \int_\Om \varphi_{lm}\,dE_{lm}u+
\int_\Om (u,{\rm div}\varphi)\,dx\,
=\,\int_{\partial\Om}
(u_*{\odot}\nu^{}_\Om,\varphi)\,d\Hn\,,
\leqno(\frm{divbd})
$$
%
where $\nu^{}_\Om$ is the outer unit normal to $\partial\Om$ and
${\rm div} \varphi$ is the $\R^n$ valued function defined by
%
$$({\rm div} \varphi)^l\,=\,
\sum_{m=1}^n D_m \varphi_{lm}\qquad\qquad
l=1,\ldots,n\,.
$$
%

(iv) Let ${\cal R}$ be the class of {\it rigid motions\/} in
$\Rn$, i.e., affine maps of the form $A x+d$ with $A$
skew symmetric $n{\times} n$ matrix and $d\in\Rn$. The following
``Poincar\'e type'' inequality for $BD$ functions follows from 
Proposition~2.2 and Remark~1.1 of Chapter~II of [\rf{Te85}]. For a 
complete proof see [\rf{Ko}], Part~II, Proposition 3.11.

\th{\thm{TS}} {Let $\Om$ be a bounded connected open set with
Lipschitz boundary and let $R\colon BD(\Om)\to {\cal R}$
be a continuous linear map which leaves the elements of ${\cal R}$ 
fixed. Then there exists a constant $c(\Om,R)$ such that
%
$$
\int_\Om |u-R(u)|\,dx\,\leq\, c(\Om,R)|Eu|(\Om)
\qquad\qquad\forall u\in BD(\Om)\,.
$$}


Now we shall see that the space $BD(\Om)$ can be characterized using
suitable one dimensional sections
(see~[\rf{Mi64}], Theorem~3.3, and [\rf{Fed68}], Theorem~4.5.9, for the 
analogous property in $BV$; see also
[\rf{BeCoDM95}], Theorem 3.1 and Proposition 3.2, where
a slightly weaker property is proved for~$BD$).
We shall use the notation introduced
in (\rf{sezioni}).
Given a function $u\colon B\to \R^n$, defined on a subset $B$ of $\R^n$,
for every $y,\,\xi \in \Rn$, $\xi\neq 0$, the
function $u_y^\xi \colon B_y^\xi \rga \R$ is defined by
%
$$
\uyx(t)\, =\, u^\xi(y+t\xi)
\,=\, (u(y + t \xi), \xi)
\qquad \forall t\in B^\xi_y\,.
\leqno(\frm{uyxi})
$$
%
When $\xi=e_i$, the $i^{th}$ vector of the canonical basis, we write 
$u^i_y$ instead of $u_y^{e_i}$.
If $u \in L^1(\Om;\Rn)$, then for every $\xi\in\Rn$, $\xi\neq 0$,
and every $y\in \Omx$ the one dimensional section $\tu_y^{\xi}$
of the function $\tu$ introduced
in (\rf{puntoLeb}) is defined for every
$t\in (\Om_u)_y^{\xi}$ and
the corresponding pointwise variation $V\tu_y^{\xi}$
is evaluated using only points $t\in (\Om_u)_y^{\xi}$.
%
\prop{\thm{tsezioni}}{Let $u\in BD(\Om)$ and let $\xi\in\Rn$ with
$\xi\neq 0$. Then the following two conditions 
hold for $\Hn$ almost every $y\in\Om^\xi$:
\smallskip
\item{\rm(i)} $\tu_y^\xi$ is defined and coincides with $u_y^\xi$ 
$\Le^1$ almost everywhere in $\Om^\xi_y$;
\smallskip
\item{\rm(ii)} $u^\xi_y\in BV(\Om^\xi_y)$ and the measures $|D\uyx|$ 
and 
$V{\tilde u_y^\xi}$ coincide on $\Om^\xi_y$.
\smallskip
\noindent Moreover
%
$$
(Eu \xi,\xi)
\,=\,\int_{\Om^\xi} Du^\xi_y \,d\Hn(y) \,,
\qquad
|(Eu \xi,\xi)|
\,=\,\int_{\Om^\xi} |Du^\xi_y| \,d\Hn(y)
\leqno(\frm{wow})
$$
%
as measures in $\Om$.

Conversely, let $u\in L^1(\Om;\Rn)$ and let $\xi_1,\ldots,\xi_n$ be a 
basis of $\Rn$. Assume that for every~$\xi$ of the form ${\xi_i+\xi_j}$ 
$$
\displayliness{
u_y^\xi \in BV(\Om_y^\xi) \quad \hbox{for }\Hn \hbox{ a.e. } 
y\in\Om^\xi \,,
\cr
\int_{\Om^\xi} |Du_y^\xi|(\Om_y^\xi) \,d\Hn(y) \,<\, +\infty \,.
\cr}
\leqno(\frm{finite})
$$
Then $u\in BD(\Om)$.
}
\proof
Since all properties are local, by (\rf{estensu})
we may assume $\Om=\Rn$. Let $u\in BD(\Rn)$;
as $\Le^n(S_u)=0$ and $u=\tu$ $\Le^n$ almost everywhere in $\R^n$,
Fubini's theorem implies~(i).

Let $u_\eps:=u\star\psi_\eps$ be
a regularization of $u$; it is well known (see, e.g., [\rf{Zie89}], 
Theorem~1.6.1) that $u_\eps$ converges pointwise to $\tilde{u}$
at every Lebesgue point of~$u$; moreover,
since $(Eu_\eps \xi,\xi)=(Eu \xi,\xi)\star\psi_\eps$, Fubini's 
theorem and (\rf{asscont}) imply that
%
$$
|(Eu \xi,\xi)|(\Rn)\geq |(Eu_\eps \xi,\xi)|(\Rn)=
\int_{\pi_\xi} V[(u_\eps)_y^\xi] (\R)\,d\Hn(y)\,.
$$
%
Now we pass to the limit as $\eps\to 0^+$, noticing that, by (\rf{defVu}),
$v\mapsto Vv (\R)$ is lower semicontinuous with respect to pointwise
convergence. By Fatou's lemma we get
%
$$
|(Eu \xi,\xi)|(\Rn)\geq
\int_{\pi_\xi} V\tilde{u}_y^\xi (\R)\,d\Hn(y)\,.
\leqno(\frm{dis2})
$$
%
In particular, by (\rf{DuleVu}) we infer that $u^\xi_y\in BV(\R)$
for $\Hn$ almost every $y\in\pi_\xi$ and
$\int_{\pi_\xi} \v D\uyx \v (\R)\,d\Hn(y) < + \infty$. For every 
function~$v$ defined on $\Rn$ we set $v_{y,\xi}(t):=v(y+t\xi)$ for every 
$y,\,\xi\in \Rn$ and for every $t\in\R$. Since 
$D_\xi\varphi({y+t\xi})=D\varphi_{y,\xi}(t)$,
by Fubini's theorem we get
%
$$
\eqalign{
\int_{\Rn} \varphi\,d (Eu \xi,\xi)\,
&=-\int_{\Rn} u^\xi D_\xi\varphi\,dx
=\,-\int_{\pi_\xi} \bigl[ \int_{\R} 
u^\xi_y D\varphi_{y,\xi}\,dt \,\bigr]\,d\Hn(y)
\cr
&=\,\int_{\pi_\xi} \bigl[ \int_{\R}\varphi_{y,\xi}\,d(Du^\xi_y)
\,\bigr]\,d\Hn(y)
}\leqno(\frm{fubini})
$$
%
for any $\varphi\in \C^1_c(\Rn)$. Therefore, a density argument implies 
the first equality in
(\rf{wow}); the second equality is a consequence of~(\rf{vartot}).
{}From (\rf{wow}), (\rf{dis2}), and (\rf{DuleVu}) it follows that
the measures $|Du^\xi_y|$ and $V\tilde{u}_y^\xi$ coincide
for $\Hn$ almost every $y\in\pi_\xi$, and the proof of~(ii) is complete.

Conversely, assume that $u\in  L^1(\Om;\Rn)$ and that (\rf{finite}) is 
satisfied for every~$\xi$ of the form ${\xi_i+\xi_j}$, where 
$\xi_1,\ldots,\xi_n$ be a basis of $\Rn$. By~(\rf{fubini}) for these 
vectors~$\xi$ there exist constants $C(\xi)$ such that
$$
\int_{\Rn} u^\xi D_\xi\varphi\,dx \,\le\, C(\xi) 
\|\varphi\|_{L^\infty(\Rn)} \qquad
\forall \varphi\in \C^1_c(\Rn) \,.
$$
This implies that $D_\xi u^\xi$ is a bounded Radon measure on $\Rn$ 
and, consequently, $u$ belongs to $BD(\Rn)$ (property~(i) at the 
beginning of this section).
\endproof
%
\rem{\thm{nulli}} {Using (\rf{wow}) we obtain that $|(Eu \xi,\xi)|(B)=0$
for every $B\in\B(\Om)$ with $\Hn(B)=0$.
Taking (\rf{matrici}) into account we conclude that
%
$$
\Hn(B)=0 \qquad\Longrightarrow\qquad |Eu|(B)=0
\leqno (\frm{nullo})
$$
%
for every $B\in \B(\Om)$. More generally, $|Eu|(B)=0$ for every
$B\in\B(\Om)$ such that $\Hn\bigl(\pi_\xi(B)\bigr)=0$ for $\Hn$ 
almost every $\xi\in\Sn$ (see also [\rf{Ko}], Part~II, Proposition~4.16 
and Lemma~4.2).
}

\spazio

The next proposition partially extends Proposition 3.2 to a more general
situation.
For every linear subspace $V$ of $\Rn$ we can define the space $BD(V)$ 
by introducing a suitable orthogonal coordinate system on~$V$. This
definition does not depend on the choice of the coordinate system (see
property (i) at the beginning of this section).

\prop{\thm{projBD}}{Let $V$ be a linear subspace of~$\Rn$ of
dimension~$k$, let  $V^\perp$ be the orthogonal subspace, and let
$p^{}_V$ be the orthogonal projection onto~$V$. Given a function
$u\in BD(\Rn)$, for every $z\in V^\perp$ let $v_z\colon V \to V$ be
the function defined by $v_z(y)=p^{}_V u({y+z})$. Then $v_z\in BD(V)$
for $\H^{n-k}$ almost every $z\in V^\perp$, and for every
$B\in\B(\Rn)$ and every $\xi\in v$ we have
$$
\leqalignno{(Eu\xi,\xi)(B)\,=&
\, \int_{V^\perp}(Ev_z \xi,\xi)(B_z)\,d\H^{n-k}(z)\,,&
(\frm{sezV1})
\cr
|(Eu\xi,\xi)|(B)\,=&
\, \int_{V^\perp}|(Ev_z \xi,\xi)|(B_z)\,d\H^{n-k}(z)\,,&
(\frm{sezV2})
}
$$
where $B_z=\{y\in V: y+z\in B\}$.
}

\proof Let $\xi_1, \ldots, \xi_k$ be a basis of $V$ and let $\xi$ be a
vector of the form $\xi_i+\xi_j$. Recalling property~(i) at the
beginning of this section, it is enough to prove that
 $D_\xi (v_z)^\xi$ is a Radon measure on $V$ for $\H^{n-k}$ almost
every $z\in V^\perp$. This property follows from Theorem~3.3 of
[\rf{Mi64}], which proves also that
$$
|D_\xi u^\xi|(B)\,=
\, \int_{V^\perp}|D_\xi (v_z)^\xi|(B_z)\,d\H^{n-k}(z)\,.
$$
By~(\rf{Dxiuxi}) this formula is equivalent to~(\rf{sezV2}).
Now~(\rf{sezV1}) can be obtained as in the proof of~(\rf{wow}).
\endproof

\spazio

For every $u\in BD(\Om)$ let us consider now the set $\Theta_u$
defined in
(\rf{Thetau}) and introduced by Kohn in [\rf{Ko}].
First of all we claim that $J_u \subset \Theta_u$. Indeed,
let $x$ be a point in the jump set $J_u$; using the lower semicontinuity 
of the map
$v\mapsto |Ev|(B_1(0))$ with respect to the $L^1(B_1(0);\Rn)$
convergence, from (\rf{uoo}) and (\rf{rappJ}) we infer
%
$$
\eqalign{&\liminf_{\ro\to 0^+}{|Eu|(\Brx)\over\ro^{n-1}}\,=\,
\liminf_{\ro\to 0^+}|Eu_\ro|(B_1(0))\,\geq\,|Eu_0|(B_1(0))\cr
&=\,\omega_{n-1}|(u^+(x)-u^-(x)) {\odot} \nu(x)|\,\ge\,
{\omega_{n-1}\over\sqrt2} |u^+(x)-u^-(x)| \,>\,0\,.}
$$
%
Hence $x\in\Theta_u$ and the claim is proved.

\spazio

The following proposition concerns the rectifiability of the sets
$\Theta_u$ and $J_u$.


\prop{\thm{rettifTu}} {For every $u\in BD(\Om)$ the jump set $J_u$
is a countably $(\Hn,{n-1})$ rectifiable Borel set.
Moreover, $\Hn(\Theta_u\setminus J_u)=0$.}

\proof The fact that $\Theta_u$ is countably $(\Hn,{n-1})$ rectifiable is 
proved in [\rf{Ko}], Part~II, Theorem~4.18.
For the reader's convenience we repeat that proof here.
For every $\eps >0$ let us consider the set
%
$$
\Theta^\eps_u \,:=\,\{x\in \Om :\limsup_{\ro\to 0^+} {\v 
Eu\v(B_\ro(x))\over
\ro^{n-1}} >\eps \}\,.
$$
%
It is easy to see that $\Theta^\eps_u$ is a Borel set for every $\eps >0$.
In order to show that $\Theta_u$ is countably $(\Hn,{n-1})$ rectifiable
it is enough to prove the same property for each $\Theta^\eps_u$.
By Theorem 2.10.19 of [\rf{Fed68}] we have that
%
$$
\eps \,\Hn(B) \leq \omega_{n-1} \v Eu\v (B)
\qquad\forall B\in {\cal B}(\Om),\,\,
B\subset\Theta^\eps_u\,.
\leqno(\frm{stimarettif})
$$
%
In particular,
$\Hn(\Theta^\eps_u)<+\infty$ and we can apply
Theorems 3.3.13 and 2.10.15 of [\rf{Fed68}] to obtain
a countably $(\Hn,{n-1})$ rectifiable Borel
set $R^\eps\subset\Theta^\eps_u$ such that
%
$$
\Hn(p_\xi(\Theta^\eps_u \setminus R^\eps)) =0
\quad \hbox{for } \Hn\hbox{ a.e. } \xi\in \Sn\,.
$$
%
Choosing a basis $\xi_1,\ldots,\xi_n$ of $\Rn$ satisfying
$\Hn(p_{\xi_i}(\Theta^\eps_u \setminus R^\eps)) =0$,
by (\rf{wow}) we infer $|(Eu \xi_i,\xi_i)|(\Theta^\eps_u\setminus
R^\eps)=0$ for $i=1,\ldots,n$. By (\rf{matrici}) we obtain
that $\v Eu\v (\Theta^\eps_u \setminus R^\eps) =0$.
Choosing $B=\Theta^\eps_u\setminus R^\eps$ in (\rf{stimarettif})
we obtain that $\Hn(\Theta^\eps_u \setminus
R^\eps)=0$ and the claim is proved.

Since $\Theta_u$ is countably $(\Hn,{n-1})$ rectifiable, the one sided
Lebesgue limits $u^\pm(x)$ of $u$ at $x$ are well defined for
$\Hn$ almost every $x\in \Theta_u$ (see (ii) at the beginning of the 
section).
Now, for every $\eps >0$ let us consider the Borel set
$B^\eps =\{x\in \Theta^\eps_u : u^+(x) = u^-(x) \}\,$.
By (\rf{stimarettif}) and (\rf{rappJ}) we know that
$\eps \Hn(B^\eps) \leq \omega_{n-1} \v Eu\v (B^\eps) =0$,
which implies that $\Hn(B^\eps)=0$ for every $\eps >0$.
Therefore $\Hn(\{x\in \Theta_u : u^+(x) = u^-(x) \})=0$ and the proof is 
complete.
\endproof

\rem{\thm{normalJ}} {By the definition of $J_u$ for every $x\in J_u$ 
there exists a unit vector $\nu_u(x)$ which satisfies~(\rf{twosided}).
Since $J_u$ is countably $(\Hn,{n-1})$ rectifiable,
for $\Hn$ almost every $x\in J_u$ we can define a Borel measurable 
approximate unit normal $\nu_{J_u}^{}$ characterized by the following 
property: for every $\C^1$ hypersurface $M$ we have
$$
\nu_{J_u}^{}(x)=\pm\nu_M^{}(x)
\qquad\hbox{\rm for $\Hn$ a.e. }x\in J_u\cap M\,.
$$
By property (ii), stated at the beginning of this section, the function~$u$ 
has one sided Lebesgue limits $u^{\pm}(x)$ with respect to the normal 
$\nu_{J_u}^{}(x)$ for $\Hn$ almost every $x\in J_u$. By the uniqueness 
of the one sided Lebesgue limits, this implies that $\nu_u(x)=\pm 
\nu_{J_u}^{}(x)$ for $\Hn$ almost every $x\in J_u$. In other words,
$\nu_u(x)$ is an approximate unit normal to $J_u$ for $\Hn$ almost 
every ${x\in J_u}$.
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%
\parag{\chp{secstructure}}{The structure theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%


For every $u\in BD(\Om)$ we shall consider the Radon-Nikodym
decomposition $Eu=E^au+E^s u$ of $Eu$, where $E^au$ is absolutely 
continuous and $E^s u$ is singular with respect to ${\cal L}^n$.

\defin{\thm{decomp}} {Let $u\in BD(\Om)$.\par
\item{(i)} The
{\it approximate
symmetric differential\/} of $u$ is the density ${\cal E}u$ of $E^a u$ 
with respect to ${\cal L}^n$, i.e., $E^au={\cal E}u\,{\cal L}^n$.\par

\item{(ii)} The
{\it jump part\/} of $Eu$ is the restriction $\Jp u$ of $E^s u$
to the jump set $J_u$, i.e., $\Jp u:=E^s u\LL J_u$.\par
\item{(iii)} The
{\it Cantor part\/} of $Eu$ is the restriction $\Cp u$ of $E^s u$
to the complement of $J_u$, i.e., $\Cp u:=E^s u\LL
(\Om\setminus J_u)$.\par
}

\rem{\thm{remdeco}}
{Since $J_u\subset S_u$ is Lebesgue negligible, we have also
$\Jp u=Eu\LL J_u$. The definition above implies the following
decomposition of $Eu$ into three parts, analogous with the one
adopted for $BV$ functions:
%
$$
Eu= {\cal E}u\,{\cal L}^n+\Jp u+\Cp u\,.
\leqno (\frm{decoE})
$$
%
Since $J_u$ is countably $(\Hn,{n-1})$ rectifiable,
from (\rf{rappJ}) with $M:=J_u$ we infer the representation
%
$$
\Jp u \,= \, (u^+ - u^-) {\odot} \nu_u \,\Hn \LL J_u\,.
\leqno(\frm{rappJJ})
$$
%
Since $\Theta_u\setminus J_u$ is $\Hn$ negligible, (\rf{nullo})
shows that the measure $\Jp u$ can also be obtained by restriction of
$Eu$ to the larger set $\Theta_u$.}
\spazio

%
The following theorem justifies our terminology for the density $\E u$
of the absolutely continuous part of $Eu$ (see also the Appendix of
[\rf{BeCoDM95}], where a weaker property of the same type is proved).


\th{\thm{symmdi}} {Let $u\in BD(\Om)$. For $\Le^n$ almost every 
$x\in\Om$
we have
%
$$
\lim_{\ro\to 0^+}
{1\over \ro^n}\int_{B_\ro(x)} {\v (u(y) - u(x) - \E u(x)(y-x),y-x) \v
\over |y-x|^2}\,dy \,=0\,.
$$
}
\negbigskip
\proof
Let $B_\ro(x)$ be a ball contained in $\Om$, let
$\tau\in (0,1)$, and let $v\in \C^1(B_\ro(x))$. Using the equality
%
$$
 (v(y) - v(x+\tau(y-x)),y-x)\,=\,\int_{\tau}^1
 \bigl(\E v(x+t(y-x))(y-x),y-x\bigr)\,dt
 %\quad\forall y\in B_\ro(x)
$$
%
and Fubini's theorem, a straightforward computation gives
%
$$
\int_{B_\ro(x)}
{\v (v(y) - v(x+\tau(y-x)),y-x) \v\over |y-x|^2}\,dy
\leq\int_{\tau}^1 {|Ev|(B_{t\ro}(x))\over t^n}\,dt\,.
\leqno(\frm{stima})
$$
%%
Assume now that $v\in BD(\Om)$ and let $v_\eps:=v\star\psi_\eps$ be 
the 
corresponding regularization. Since $|Ev_\eps|(B_{t\ro}(x))$
converges to $|Ev|(B_{t\ro}(x))$ as $\eps\to 0^+$ for any
$t\in (0,1)$ such that $|Ev|(\partial B_{t\ro}(x))=0$,
passing to the limit we can prove that (\rf{stima}) holds also
for $v\in BD(\Om)$.
If $x\notin S_v$, letting $\tau\to 0^+$ and dividing both sides by
$\ro^n$ we obtain
%
$$
{1\over\ro^n} \int_{B_\ro(x)}
{\v (v(y) - \tilde{v}(x),y-x) \v\over |y-x|^2}\,dy
\leq\int_0^1 {|Ev|(B_{t\ro}(x))\over (t\ro)^n}\,dt\,.
\leqno(\frm{stima2})
$$
Let us fix a Lebesgue point $x$ for $\E u$ such that
$\widetilde{\E u}(x)=\E u(x)$,
$x\notin S_u$,
$\tilde{u}(x)=u(x)$,
and $|E^s u|$ has $n$ dimensional density zero at~$x$ (all these
conditions are satisfied $\Le^n$ almost everywhere in~$\Om$); applying 
(\rf{stima2}) to $v(y):=u(y)-\E u(x)(y-x)$ we obtain
%
$$
\eqalign{
&{1\over\ro^n} \int_{B_\ro(x)}
{\v (u(y) - u(x) -\E u(x)(y-x),y-x)\v\over |y-x|^2}\,dy
\cr
&\le\, \sup_{t\in (0,1)}
\bigl[\, {|E^s u|(B_{t\ro}(x))\over (t\ro)^n} \,+\,
{1\over(t\ro)^n} \int_{B_{t\ro}(x)}|\E u(y)-\E u(x)|\,dy \, \bigr]\,.
\cr}
$$
By our choice of $x$, both terms in the right hand side vanish as 
${\ro\to 0^+}$.
\endproof
\spazio
The following proposition shows that, in (\rf{decoE}), $\Jp u$ is the only
measure concentrated on ${n-1}$ dimensional sets.

\prop{\thm{cantor0}} {For every $u\in BD(\Om)$ the measure $\Cp u$
vanishes on any Borel set $B\subset\Om$
which is $\sigma$-finite with respect to $\Hn$.}

\proof Let $B\subset\Om$ be a Borel set such that
$\Hn(B)<+\infty$. Since ${\Theta_u\setminus J_u}$ is $\Hn$ negligible
(Proposition~\rf{rettifTu}), by (\rf{nullo})
the measure $|\Cp u|$ is concentrated on ${\Om\setminus\Theta_u}$.
For this reason it is not restrictive to assume that
$B\cap\Theta_u=\emptyset$; under this assumption, the ${n-1}$ 
dimensional
density of $|\Cp u|$ at any point of $B$ is zero. A covering argument
(see, e.g., [\rf{Fed68}], 2.10.19)
shows that $|\Cp u|(B)\leq \eps\Hn(B)$ for any
$\eps>0$, hence ${|\Cp u|(B)=0}$.
\endproof

\spazio

The main result of our paper is the following structure
theorem for $BD$ functions, that will be proved in 
Section~\rf{secproofs}. Properties (i) and (ii) are proved
also in [\rf{BeCoDM95}], Proposition~3.2 and Lemma~3.3. For every 
$\xi\in\Rn$ let us define
$$
J_u^\xi\,:=\, \{x\in J_u: (u^+(x)-u^-(x),\xi)\neq0\}\,.
$$
Since $\Hn(\{\xi\in\Sn : (u^+(x)-u^-(x),\xi)=0\})=0$ for every $x\in J_u$, 
by Fubini's theorem we have
$$
\Hn(J_u\setminus J_u^\xi)=0
\qquad \hbox{ for }\, \Hn\hbox{ a.e. }\,\xi\in\Sn \,.
\leqno(\frm{Juxi})
$$
We recall that  $u_y^\xi$ is the function defined in~(\rf{uyxi}).

\th{\thm{structure}} {\sl Let $u\in BD(\Om)$ and let $\xi\in\Rn$ with
$\xi\neq 0$. Then the following properties hold:
\bigskip
\item{\rm(i)} \hfil $\displaystyle
(E^a u \xi,\xi)\,=\, \int_{\Om^\xi} D^a u^\xi_y \,d\Hn(y)\,,
\qquad
|(E^a u \xi,\xi)|\,=\, \int_{\Om^\xi} |D^a u^\xi_y| \,d\Hn(y)\,;$ \hfil \par
\bigskip
\item{\rm(ii)} 
for $\Hn$ almost every $y\in\Om^\xi$ the functions $\uyx$ and
$\tilde u_y^\xi$ belong to $BV(\Om_y^\xi)$ and coincide
$\Le^1$ almost everywhere on $\Om_y^\xi$, the measures
$\v D\uyx \v$ and $V\tilde u_y^\xi$ coincide on $\Om_y^\xi$, and
$(\E u(y+t\xi)\xi,\xi)= \nabla \uyx (t) = (\tilde u^\xi_y)'(t)$
for $\Le^1$ almost every $t \in \Om_y^\xi$;
%
\bigskip
\item{\rm(iii)} \hfil
$\displaystyle (\Jp u \xi,\xi)\,=\, \int_{\Om^\xi} \DJp u^\xi_y \,d\Hn(y)
\,,\qquad
|(\Jp u \xi,\xi)|\,=\, \int_{\Om^\xi} |\DJp u^\xi_y| \,d\Hn(y)\,;$
\hfil \par
\bigskip
\item{\rm(iv)}for $\Hn$ almost every $y\in\Om^\xi$ we have
$(J_u^\xi)^\xi_y = J_{u^\xi_y}$ and
for every $t\in (J_u^\xi)^\xi_y$
%
$$
(u^+(y+t\xi),\xi)\,=\,(\uyx)^+(t) =
\,\lim_{s\to t^+}\tilde u_y^\xi(s)\,,\quad
(u^-(y+t\xi),\xi)\,=\,(\uyx)^-(t) =\,\lim_{s\to t^-}\tilde u_y^\xi(s)\,,
$$
%
where the normals to $J_u$ and $J_{u^\xi_y}$
are oriented so that $(\nu_u,\xi)\geq 0$ and $\nu_{\uyx}=1$;
\bigskip
\item{\rm(v)} \hfil
$\displaystyle
(\Cp u \xi,\xi)\,=\, \int_{\Om^\xi} \DCp u^\xi_y \,d\Hn(y)\,,
\qquad
|(\Cp u \xi,\xi)|\,=\, \int_{\Om^\xi} |\DCp u^\xi_y| \,d\Hn(y)\,.$
\hfil \par
\medskip
}


%\rem{\thm{cons1}} {We know from Proposition~\rf{tsezioni}
%that $\tilde{u}^\xi_y$
%and $u^\xi_y$ are in the same equivalence class for $\Hn$ almost
%every $y\in\pi_\xi$. Hence, the properties stated
%in the structure theorem
%can be reformulated replacing the classical derivative of
%$\tilde{u}^\xi_y$
%with the {\it approximate\/} derivative
%(see (\rf{apdi})) of $u^\xi_y$ and the
%classical limits in (\rf{salti}) with the one sided {\it approximate\/}
%limits. Therefore, for $\Hn$ almost every $y\in\pi_\xi$ we have
%
%$$
%(u^+(y+t\xi),\xi)\,=\,\lim_{s\to t^+}{1\over s-t}\int_t^s 
%u^\xi_y(\tau)\,d\tau,
%\qquad\qquad
%(u^-(y+t\xi),\xi)\,=\,
%\lim_{s\to t^-}{1\over s-t}\int_t^s u^\xi_y(\tau)\,
%d\tau\,
%$$
%
%for any $t\in (J_u)^\xi_y$ and
%
%$$(\tilde u(\xi+ty),\xi)\,=\,\lim_{s\to t}{1\over s-t}\int_t^s
%u^\xi_y(\tau)\,d\tau
%$$
%
%for any $t\in\R\setminus (J_u)^\xi_y$.}

\spazio

Motivated by the structure theorem we can now
define the class of special functions with bounded deformation.

\defin{\thm{defsbd}} {We say that $u\in BD(\Om)$ is a {\it special
function with bounded deformation\/}, and we write $u\in SBD(\Om)$, 
if
the measure $\Cp u$ in Definition~\rf{decomp} is zero, i.e., if
$|E^s u|({\Om\setminus J_u})=0$.
}

\spazio

The following proposition shows that the space $SBD(\Om)$ of
Definition \rf{defsbd} coincides with the one introduced in
[\rf{BeCoDM95}] and used to solve free discontinuity problems
related to fracture mechanics.

\prop{\thm{equiva}} {Let $u\in BD(\Om)$ and let $\xi_1,\ldots,\xi_n$
be a basis of $\Rn$.
Then the following
three conditions are equivalent:
\smallskip
\item{\rm(i)} $u\in SBD(\Om)$;
\smallskip
\item{\rm(ii)} for every $\xi=\xi_i+ \xi_j$ with $1\leq i,\,j\leq n$,
we have $u^\xi_y\in SBV(\Om^\xi_y)$ for $\Hn$ almost every 
$y\in\Om^\xi$;
\smallskip
\item{\rm(iii)} the measure $|E^s u|$ is concentrated on a Borel set
$B\subset\Om$ which is $\sigma$-finite with respect to
$\Hn$.
\par}
%
\proof
The equivalence of (i) and (ii) follows easily from (v) of
Theorem~\rf{structure} and from~(\rf{matrici}).
The implication (i)$\Rightarrow$(iii) follows from (\rf{rappJJ}),
taking into account that $J_u$ is countably
$(\Hn,{n-1})$ rectifiable. Finally, the implication (iii)$\Rightarrow$(i)
follows from Proposition~\rf{cantor0}.
\endproof

\rem{\thm{SBD&BV}}{It is easy to see that $SBV(\Om;\Rn)$ is contained 
in $SBD(\Om)$. We shall see in Example~\rf{orn} that there are functions 
in $SBD(\Om)$ which do not belong to $BV(\Om;\Rn)$.
}

\spazio

Following [\rf{BeCoDM95}], we now
introduce an integralgeometric measure
associated with the jump sets of the one dimensional sections. As shown 
in [\rf{BeCoDM95}], this measure is crucial in the proof of the 
compactness properties of $SBD(\Om)$.

\defin{\thm{deflu}}{For any $u \in BD(\Om)$ we define the
non-negative Borel measure $\la_u$ on $\Om$ as
%
$$
\la_u(B)  \,:=\, \kappa_n \int_{\Sn}
\la_u^\xi(B) \, d\Hn(\xi) \qquad \forall B \in \BOm\,,
\leqno{(\frm{deflu1})}
$$
%
where $\kappa_n\,:=\,(2\om_{n-1})^{-1}$ and
for every $\xi \in \Sn$
%
$$
\la_u^\xi(B) \,:=\, \int_{\Om^\xi}
\H^0 \bigl(J_{u_y^\xi} \cap B_y^\xi \bigr)\,d\Hn(y)
\qquad \forall B \in \BOm\,.
\leqno{(\frm{deflu2})}
$$
%
}

Another consequence of the structure theorem is the equality
$\la_u =\Hn\LL J_u$.


\th{\thm{teoluBD}}{For every $u \in BD(\Om)$ and any $\xi\in\Sn$ we 
have
%
$$
\la_u^\xi(B)\,=\,\int_{J_u^\xi\cap B}|(\nu_u,\xi)|\,d\Hn\qquad\qquad
\forall B\in{\cal B}(\Om)\,,
\leqno(\frm{rappla})
$$
%
where $\nu_u$ is the approximate unit normal to $J_u$.
Moreover $\la_u = \Hn \LL J_u$.
}
\proof
Using part (iv) of the structure Theorem~\rf{structure}, we have
%
$$
\la_u^\xi(B)\,=\,\int_{\Om^\xi}
\H^0 \bigl(\{t\in \R: y+t\xi\in B\cap J_u^\xi \} \bigr)\,d\Hn(y)\,.
$$
Let $p_\xi\colon J_u\to\pi_\xi$ be the orthogonal projection
onto $\pi_\xi$. Equality~(\rf{rappla}) follows now from the area formula 
for maps defined on a countably $(\Hn,{n-1})$
rectifiable Borel set (see~[\rf{Si83}], 12.4)
using the fact that the ${n-1}$ dimensional
area element of $p_\xi$ on $J_u$ is equal to $|(\nu_u,\xi)|$.
Finally, since
%
$$
\int_{{\bf S}^{n-1}}|(\nu,\xi)|\,d\Hn(\xi)\,=\,
\kappa_n^{-1}=2\omega_{n-1}
$$
%
for any $\nu\in \Sn$,
the equality $\la_u = \Hn \LL J_u$ can be obtained by integrating
both sides in (\rf{rappla}), taking (\rf{Juxi}) into account.
\endproof



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%
\parag{\chp{secproofs}}{Proof of the structure theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%

The main ingredient in the proof of the structure 
Theorem~\rf{structure} is the following result, which shows that, under 
some assumptions, points of ${n-1}$ dimensional density zero for $|Eu|$ 
are Lebesgue points for the restriction of the function $p_\eta u$ to 
$\pi_\eta$, where $p_\eta$ is the orthogonal projection onto the 
hyperplane $\pi_\eta$ orthogonal to $\eta\in\Sn$ and passing through 
the origin. For the sake of simplicity, since all our statements are local, 
here and in the rest of the section we assume that ${\Om=\Rn}$.

\th{\thm{main}} {Let $u\in BD(\Rn)$, $x\notin\Theta_u$,
$\eta\in {\bf S}^{n-1}$.
Assume that $\Hn$ almost every point of $x+\pi_\eta$
are Lebesgue points of $u$ in $\Rn$.
Let us consider the function
$v\colon\pi_\eta\to\pi_\eta$ given by $v(y):=p_\eta \tilde u(x+y)$
and let us assume that~$v$ has one sided Lebesgue limits at~$0$ in 
$\pi_\eta$ with respect to a suitable direction 
$\nu\in {{\bf S}^{n-1}\cap \pi_\eta}$.
Then $0$ is a Lebesgue point of $v$ in $\pi_\eta$.
}

\proof Up to a rotation and a translation, it is 
not restrictive to assume that $\eta = e_n$ and $x=0$.
Then the function $v$ is defined on the hyperplane $\pi:=\pi_{e_n}$ by 
$v(y):=(\tilde u^1(y), \ldots, \tilde u^{n-1}(y), 0)$. Up to a rotation in 
$\pi$ we can assume that $\nu^1 :=(\nu,e_1) >0$ and that the one sided 
Lebesgue limits $v_0^+$ and $v_0^-$ of~$v$ at~$0$ in~$\pi$
satisfy $\v (v_0^+ -v_0^-, e_1) \v \geq {1\over 2} \v v_0^+ - v_0^- \v$.
We have to prove that $v_0^+ = v_0^-$.

Let $Y$ be the set of all real numbers $\ro \ge0$ such that both
$y+\ro e_n$ and ${y-\ro e_n}$ are Lebesgue points of $u$ in $\Rn$
for $\Hn$ almost every $y\in \pi$. We know that 
$\Le^1({\R_+\setminus Y})=0$
and, by our assumption, $0\in Y$.

For every $\ro >0$ let $C_\ro$ be the $n$ dimensional closed cube in 
$\Rn$
with centre $0$ and sides parallel to the axes with length $2\ro$.
We set $Q_\ro := C_\ro \cap \pi$
and $Q_\ro^\pm := \{y\in Q_\ro : (y,\pm\nu)\geq 0\}$.
Let $A_\ro$ be the ${n-1}$ dimensional closed ball in $\pi$ with centre 
$0$
and radius $\nu^1\ro$, so that $A_\ro + \ro\, e_1 \subset Q_{2\ro}^+$
and $A_\ro - \ro\, e_1 \subset Q_{2\ro}^-$.

By our assumptions we have that
%
$$
\ldisplaylinesno{\limro {1\over\ro^{n-1}}\,\v Eu\v (C_\ro) \,=\, 0 \,,
&(\frm{ipotmain1})
\cr
\limro {1\over\ro^{n-1}} 
\int_{Q_\ro^\pm} \v v(y) - v_0^\pm \v \,d\Hn(y) \,=\, 0\,.
&(\frm{ipotmain2})
\cr
}
$$
%
As $\v v_0^+ - v_0^- \v \leq 2 \v (v_0^+ - v_0^-, e_1) \v$, 
for every $\ro >0$ we have
%
$$
\eqalign{
\v v_0^+ - v_0^- \v \leq
&{2k \over \ro^{n-1}}
\Bigl[ \int_{A_\ro} \v v(y+\ro e_1) - v_0^+ \v \,d\Hn(y)
+ \int_{A_\ro} \v v(y-\ro e_1) - v_0^- \v \,d\Hn(y)\cr
&+ \int_{A_\ro} \v v^1(y+\ro e_1) - v^1(y-\ro e_1) \v\,d\Hn(y) 
\,\Bigr] \,,
}
$$
%
where $v^1$ is the first component of $v$ and
$k=\bigl(\omega_{n-1}(\nu^1)^{n-1}\bigr)^{-1}$.
By simple changes of variables we obtain
%
$$
\leqalignno{
\v v_0^+ - v_0^- \v &\leq {2k \over \ro^{n-1}}
\Bigl[ \int_{A_\ro +\ro e_1} \v v(y) - v_0^+ \v\,d\Hn(y) \,+
\int_{A_\ro -\ro e_1} \v v(y) - v_0^- \v\,d\Hn(y)
&\cr
&\quad  +
\int_{A_\ro} \v v^1(y+\ro e_1) - v^1(y-\ro e_1) \v\,d\Hn(y)
\,\Bigr]&\cr
&\leq {2k \over \ro^{n-1}}
\Bigl[ \int_{Q_{2\ro}^+} \v v(y) - v_0^+ \v\,d\Hn(y)+
\int_{Q_{2\ro}^-} \v v(y) - v_0^- \v\,d\Hn(y) \,\Bigr]
&(\frm{main1})
\cr
&\quad+ {2k \over \ro^{n-1}} \int_{A_\ro} 
\v \tilde u^1(y+\ro e_1) - \tilde u^1(y-\ro e_1) \v\,d\Hn(y)\,.&\cr
}
$$
%
By (\rf{ipotmain2}) the first part of the right hand side vanishes as 
$\ro\to 0^+$.
We now estimate the last term.
% 
For any $\ro \in Y$ with $\ro>0$ we write
%
$$
\eqalign{
2 \bigl[\tilde u^1(y+\varrho e_1) - \tilde u^1(y-\ro e_1)\bigr]
&= \bigl[(\tilde u^1-\tilde u^n)(y-\ro e_n)
- (\tilde u^1-\tilde u^n)(y-\ro e_1) \bigr]\cr
&\quad -\bigl[(\tilde u^1-\tilde u^n)(y+\ro e_n)
- (\tilde u^1-\tilde u^n)(y+\ro e_1) \bigr]\cr
&\quad +\bigl[(\tilde u^1+\tilde u^n)(y+\ro e_n)
- (\tilde u^1+\tilde u^n)(y-\ro e_1) \bigr]\cr
&\quad -\bigl[(\tilde u^1+\tilde u^n)(y-\ro e_n)
- (\tilde u^1+\tilde u^n)(y+\ro e_1) \bigr]\cr
&\quad + 2 [\tilde u^n(y-\ro e_n) - \tilde u^n(y+\ro e_n)]
\cr}
\leqno(\frm{sommasottr})
$$
for $\Hn$ almost every $y\in\pi$ (see Figure~1).


\def\punto{{\scriptstyle\bullet}}

\def\destra#1#2{#1\rlap{\hbox{\kern3pt$#2$}}}

\def\sinistra#1#2{#1\llap{\hbox{$#2$\kern6pt}}}

\def\sopra#1#2{\mathop{#1}\limits^{\raise4pt\hbox{
$\displaystyle #2$}}}

\def\sotto#1#2{\mathop{#1}\limits_{\displaystyle #2}}

\def\soprdest#1#2{#1\setbox100=\hbox{$#1$}
\kern-0.5\wd100
\raise10pt\hbox{\rlap{\hbox{
$\displaystyle #2$}}}}

\def\sottdest#1#2{#1\setbox100=\hbox{$#1$}
\kern-0.5\wd100
\raise-10pt\hbox{\rlap{\hbox{
$\displaystyle #2$}}}}

\def\soprsinist#1#2{#1\setbox100=\hbox{$#1$}
\kern-0.5\wd100
\raise10pt\hbox{\llap{\hbox{
$\displaystyle #2$}}}}

\def\sottsinist#1#2{#1\setbox100=\hbox{$#1$}
\kern-0.5\wd100
\raise-10pt\hbox{\llap{\hbox{
$\displaystyle #2$\hskip 3pt}}}}


\def\superbox#1#2#3{\hbox to #1pt{\hfil \vbox to #2pt{
\vfil \hbox{$#3$}\vfil}\hfil}}

\def\disegno#1#2{
\hbox{
  \vbox{
      \hbox{\superbox{#1}{#2}{}
            \superbox{#1}{#2}{}
            \superbox{#1}{#2}{\sopra\punto{y+\rho e_n}}
            \superbox{#1}{#2}{}
            \superbox{#1}{#2}{}
            }
      \hbox{\superbox{#1}{#2}{}
            \superbox{#1}{#2}{\soprsinist{}{\tilde u^1+\tilde u^n}}
            \superbox{#1}{#2}{}
            \superbox{#1}{#2}{\soprdest{}{\tilde u^1-\tilde u^n}}
            \superbox{#1}{#2}{}
            }
      \hbox{\superbox{#1}{#2}{\sinistra\punto{y-\rho e_1}}
            \superbox{#1}{#2}{}
            \superbox{#1}{#2}{\destra{}{2\tilde u^n}}
            \superbox{#1}{#2}{}
            \superbox{#1}{#2}{\destra\punto{y+\rho e_1}}
            }
      \hbox{\superbox{#1}{#2}{}
            \superbox{#1}{#2}{\sottsinist{}{\tilde u^1-\tilde u^n}}
            \superbox{#1}{#2}{}
            \superbox{#1}{#2}{\sottdest{}{\tilde u^1+\tilde u^n}}
            \superbox{#1}{#2}{}
            }
      \hbox{\superbox{#1}{#2}{}
            \superbox{#1}{#2}{}
            \superbox{#1}{#2}{\sotto\punto{y-\rho e_n}}
            \superbox{#1}{#2}{}
            \superbox{#1}{#2}{}
            }

     }
  }
}

$$
\disegno{50}{55}
$$

\centerline{{\bf Figure~1.} The arrow $\displaystyle
a\mathop{\longrightarrow}\limits^{\textstyle \varphi} b$ denotes the
difference
$\varphi(b)-\varphi(a)$.}
\bigskip

\noindent
The last term in (\rf{sommasottr}) is an increment of the function
$\tilde u_n$ in the direction $e_n$. Therefore, using~(ii) of
Proposition~\rf{tsezioni}, we have that
%
$$
\v \tilde u^n(y-\ro e_n) - \tilde u^n(y+\ro e_n) \v
\leq V \tilde u_y^n ([-\ro,\ro])
= \v Du^n_y \v ([-\ro,\ro])
$$
%
for $\Hn$ almost every $y\in A_\varrho$.
Hence (\rf{wow}) implies that
%
$$
\int_{A_\ro} \v \tilde u^n(y-\ro e_n) - \tilde u^n(y+\ro e_n) \v
\,d\Hn(y) \leq 
\v (Eu e_n,e_n) \v (C_{\ro}) \leq \v Eu \v (C_{\ro})\,
$$
%
for any $\ro\in Y$ with $\ro>0$.
Analogously, the other terms in (\rf{sommasottr}) are increments
of the function ${\tilde u^1-\tilde u^n}$ in the direction ${e_1 -e_n}$
or of the function ${\tilde u^1+\tilde u^n}$ in the direction ${e_1+e_n}$.
In the same way they can
be controlled by $\v Eu\v (C_{2\ro})$ up to a multiplicative constant.
For instance, setting $\xi=e_1 +e_n$, $p(y,\ro)=
p_\xi(y-\ro e_1)$, and $\Sigma_\ro =
p_\xi (A_\ro -\ro e_1)$,
we have
%
$$
\eqalign{
&\int_{A_\ro} \v \tilde u^\xi(y+ \ro e_n)
- \tilde u^\xi(y-\ro e_1) \v\,d\Hn(y)
\leq \int_{A_\ro} \v Du^\xi_{p(y,\ro)}\v ([-\ro,\ro])\,d\Hn(y)\cr
=&\sqrt{2} \int_{\Sigma_\ro} \v Du^\xi_y \v ([-\ro,\ro])
\,d\Hn(y) \leq \sqrt{2} \v (Eu \xi,\xi)\v (C_{2\ro})
\leq 2 \sqrt2 \v Eu\v (C_{2\ro})\,.\cr
}
$$
%
Therefore there exists a constant $c>0$ such that
%
$$\int_{A_\ro} \v \tilde u^1(y+\ro e_1) - \tilde u^1(y-\ro e_1) \v 
\,d\Hn(y)
\leq c \, \v Eu\v (C_{2\ro})\,
$$
%
for any $\ro\in Y$  with $\ro>0$.
Thus, by (\rf{ipotmain1}), also the last term at the right hand side of 
(\rf{main1}) vanishes
as $\ro \to 0^+$, hence $v_0^+ =v_0^-$ and the proof is complete.
\endproof

\spazio

\proofof{Theorem~\rf{structure}}Since the properties
we have to prove are local, we may 
assume $\Omega=\Rn$. Let $u\in BD(\Rn)$ and let $\xi\in\Rn$ with
${\xi\neq 0}$. Let $Y$ be the set of points $y\in\pi_\xi$
such that $u^\xi_y\in BD(\R)$, $\tilde u^\xi_y$ is defined and agrees 
with $u^\xi_y$ ${\cal L}^1$ almost everywhere in~$\R$, and
$V\tilde{u}^\xi_y =|Du^\xi_y|$ as measures on~$\R$.
By Proposition~\rf{tsezioni} we have $\Hn(\pi_\xi \setminus Y)=0$.

We know from Proposition~\rf{tsezioni} that $(Eu \xi,\xi)=
\int_{\pi_\xi} Du^\xi_y\,d\Hn(y)$, and we claim that this decomposition
is still true for the absolutely continuous part, the jump part, and
the Cantor part of $Eu$. We observe that
%
$$ (E^a u \xi,\xi)\,-\, \int_{\pi_\xi} D^a u^\xi_y\,d\Hn(y)\,=\,
\int_{\pi_\xi} D^s u^\xi_y\,d\Hn(y) \,-\, (E^s u \xi,\xi)\,.
$$
%
Since the left hand side is absolutely continuous with respect to $\Le^n$,
while the right hand side is singular, both sides are $0$ and the first 
equality in~(i) is
proved. The second equality follows from~(\rf{vartot}).

Recalling that for $y\in Y$ we have
$D^a \uyx = \nabla \uyx \,\Le^1= (\tilde u^\xi_y)' \Le^1$,
we get
%
$$
(E^a u \xi,\xi)=\int_{\pi_\xi} \nabla \uyx \,\Le^1\,d\Hn(y) =
\int_{\pi_\xi} (\tilde u^\xi_y)' \Le^1\,d\Hn(y)\,.
$$
%
Since, by definition, the density of $E^a u$ is $\E u$, Fubini's
theorem implies (ii).

Now we can further decompose $E^s u$ and $D^s u^\xi_y$, obtaining
%
$$ (\Jp u \xi,\xi) \,-\, \int_{\pi_\xi} \DJp u^\xi_y\,d\Hn(y)\,=\,
\int_{\pi_\xi} \DCp u^\xi_y\,d\Hn(y) \,-\, (\Cp u \xi,\xi)\,.
\leqno (\frm{furth})
$$
%
We claim now that both sides are $0$. Let us restrict both measures
to $J_u$. We know, by definition, that $|\Cp u|(J_u)=0$;
moreover, we know (see [\rf{Fed68}], Corollary~2.10.11)
that any set $B$ which is $\sigma$-finite
with respect to $\Hn$ has the property that, for
$\Hn$ almost every $y\in\pi_\xi$, the line
$y+\R\xi$ meets $B$ in an at most countable set.
Since the Cantor measures
$\DCp u^\xi_y$ in (\rf{furth}) are non atomic, applying this property
to $B=J_u$ we obtain that the restriction to $J_u$ of the measure
$\int_{\pi_\xi} \DCp u^\xi_y\,d\Hn(y)$ is zero. Hence, the restriction
to $J_u$ of the right hand side in (\rf{furth}) is $0$. To prove the
claim, we need only to show that the left hand side is concentrated 
on~$J_u$.

By definition, $(\Jp u \xi,\xi)$ is concentrated on $J_u$. The main
difficulty is to prove that $\int_{\pi_\xi} \DJp u^\xi_y\,d\Hn(y)$
is concentrated on $J_u$.
Our proof is by induction on $n\geq 2$.

Assume $n=2$. The measure $\mu:=\int_{\pi_\xi} \DJp 
u^\xi_y\,d\Hn(y)$
is concentrated on the set
of points of the form $y+t\xi$ with $y\in Y$ and $t$ in
the jump set of $u^\xi_y$. By Theorem~\rf{main}, these
points belong to $\Theta_u$, hence $\mu$
is concentrated on $\Theta_u$. Since $\mu$ vanishes on
$\Hn$ negligible sets and $\Hn(\Theta_u\setminus J_u)=0$ (Proposition
\rf{rettifTu}),
$\mu$ is also concentrated on $J_u$.

Assume the statement valid for $n-1\geq 2$. Choosing $\eta\in\Sn$
orthogonal to $\xi$, so that the hyperplane $\pi_\eta$ contains $0$ and 
$\xi$, for any $s\in\R$
we can consider the functions $v_s\colon\pi_\eta\to\pi_\eta$ defined 
by $v_s(y):=p_\eta u(s\eta+y)$.
By Proposition~\rf{projBD} we have
$v_s\in BD(\pi_\eta)$ for ${\cal L}^1$ almost every $s\in\R$. Moreover,
since $\xi$ belongs to $\pi_\eta$, for any
$\om \in \pi_{\eta\xi}:=\pi_\eta\cap\pi_\xi$ and any
$s,\,t\in\R$ we have
%
$$
(v_s)^\xi_\om (t)\,=\,(v_s(t\xi+\om),\xi)\,=
\,(p_\eta u(t\xi+s\eta+\om),\xi)\,=\,
u^\xi_{s\eta+\om}(t)\,.
$$
%
For any $B\in \B(\Rn)$ and $s\in \R$ let $B_s = \{z\in \pi_\eta\ :
s\eta + z \in B\}$. Since $\xi\in \pi_\eta$, for any $\om\in 
\pi_{\eta\xi}$
we have $(B_s)^\xi_\om = B^\xi_{s\eta +\om}$. Therefore
Fubini's theorem yields
%
$$
\int_{\pi_\xi} \DJp \uyx (B_y^\xi)\,d\Hn(y) =
\int_\R \bigl[ \int_{\pi_{\eta\xi}} \DJp (v_s)^\xi_\om ((B_s)^\xi_\om)
\,d{\cal H}^{n-2}(\om) \,\bigr]\, ds\,.
\leqno(\frm{decomp-2})
$$
%
Let $S$ be the set of points $s\in \R$ such that $v_s \in BD(\pi_\eta)$
and $v_s(z) = p_\eta \tilde u (s\eta +z)$ for $\Hn$ almost every
$z\in \pi_\eta$. By Fubini's theorem and by Proposition~\rf{projBD}
we have ${\cal L}^1({\R\setminus S})=0$.
By the induction assumption, for any $s\in S$
the measure
%
$$
\mu_s(B)\, := \,\int_{\pi_{\eta\xi}} \DJp (v_s)^\xi_\om ((B_s)^\xi_\om)
\,d{\cal H}^{n-2}(\om)\,, \qquad B\in \B(\pi_\eta)\,,
$$
%
is concentrated on $J_{v_s}$.

Hence (\rf{decomp-2}) implies that the measure
$\int_{\pi_\xi} \DJp \uyx \,d\Hn(y)$ is concentrated on the
set of points of the form $s\eta + y$, with $s\in S$ and
$y \in J_{v_s}$. By Theorem \rf{main} this set is contained
in $\Theta_u$, and this proves that $\int_{\pi_\xi} \DJp \uyx \,d\Hn(y)$
is concentrated on $\Theta_u$.
%Integration of both sides with respect to $(s,w)$ yields
%
%$$
%J u_y^\xi{\otimes}\mu=\bigl[
%[J{v_s}^\xi{\otimes}{\cal H}^{n-2}\LL\pi_{\eta\xi}]{\otimes}
%{\cal L}^1\LL L_\eta\bigr]\,,
%\leqno(\frm{lalala})
%$$
%
%where $L_\eta$ is the line spanned by $\eta$. Let
%$S\subset\R$ be
%the set of points such that $v_s(z)\in BD(\pi_\eta)$, and agrees
%$\Hn$ almost
%everywhere in $\pi_\eta$ with $\tilde{u}(s\eta+z)$.
%By the induction
%assumption, for any $s\in S$ the measure
%$\nu_s:=[J{v_s}^\xi{\otimes}{\cal H}^{n-2}\LL]$ is concentrated
%on $J_{v_s}$.
%
%Hence (\rf{lalala}) implies that the measure
%$[J u_y^\xi{\otimes}\mu]$ is concentrated on the points of the form
%$(s\eta+w)$, where $s\in S$ and $w\in J_{v_s}$. By 
%Theorem~\rf{main},
%this set is contained in $\Theta_u$, and this proves
%that $[J u_y^\xi{\otimes}\mu]$ is concentrated in $\Theta_u$.
Arguing as in the case $n=2$ we obtain that this measure
is also concentrated on $J_u$.
This proves that both sides in~(\rf{furth}) are $0$, and 
concludes the proof of the first 
equalities in~(iii) and~(v). The second equalities follow 
from~(\rf{vartot}).

We conclude by showing the validity of~(iv). Since $J_u$ is countably 
$(\Hn,{n-1})$ rectifiable and $(\nu_u,\xi)$ is the Jacobian of the map 
$p_\xi\colon J_u\to\pi_\xi$ (recall that ${(\nu_u,\xi)\ge0}$), 
using~(\rf{rappJJ}) and the area formula for rectifiable sets (see, e.g., 
[\rf{Si83}], Section~12) we get
%
$$
\eqalign{(\Jp u \xi,\xi)(B)\,&=\,\int_{B\cap J_u^\xi} (u^+(x)-u^-(x), \xi)
(\nu_u(x),\xi)\,d\Hn(x) \cr &=\,
\int_{\pi_\xi} \bigl[ \sum_{t\in (B\cap J_u^\xi)^\xi_y}
(u^+(y+t\xi)-u^-(y+t\xi),\xi) \,\bigr] \,d\Hn(y)}
$$
%
for any Borel set $B\subset \Rn$. Hence, using (iii),
we obtain the equality
of measures
%
$$
\DJp u^\xi_y\,=\,\sum_{t\in (J_u^\xi)^\xi_y}
(u^+(y+t\xi)-u^-(y+t\xi),\xi)\,\delta_t
\leqno(\frm{jump1})
$$
%
for $\Hn$ almost every $y\in Y$. On the other hand by~(\rf{decBV}) we 
have also
$$
\DJp u^\xi_y\,=\,\sum_{t\in J_{\scriptstyle u^\xi_y}}
[(u^\xi_y)^+(t)-(u^\xi_y)^-(t)]\,\delta_t
\leqno(\frm{jump2})
$$
for every $y\in Y$ (recall that $\nu_{u_y^\xi}=1$).  If $y\in Y$ satisfies 
(\rf{jump1}), from~(\rf{jump2}) we obtain
$$
\ldisplaylinesno{
(J_u^\xi)_y^\xi=J_{u_y^\xi} \,,
&(\frm{secJ})
\cr
(u^+(y+t\xi)-u^-(y+t\xi),\xi) \,=\,
(u^\xi_y)^+(t)-(u^\xi_y)^-(t) \qquad
\forall t\in (J_u^\xi)_y^\xi\,.
&
(\frm{salti2})
\cr}
$$
In order to prove the full property (iv) we shall use a localization 
argument. Since $J_u$ is countably $(\Hn,{n-1})$ rectifiable,
we can find a sequence of bounded open sets $\Om_k$ with $\C^1$ 
boundary such that the union of $\partial\Omega_k$ covers
$\Hn$ almost all of~$J_u$. Let us fix~$k$ and let~$v$ be the function 
defined by $v(x)=u(x)$, if $x\in\Om_k$, and by $v(x)=0$ otherwise. By 
property~(iii) stated at the beginning of Section~\rf{secbd} the function 
$v$ belongs to $BD(\Rn)$. We choose the sign of $\nu_{v}$ so that 
$(\nu_{v},\xi)\ge0$ in $J_v$. We define
$$
\partial_\xi^\pm \Om_k :=
\{x\in \partial\Om_k: \pm(\nu_{\Om_k}^{}(x),\xi)>0\} \,,
\qquad
\partial_\xi^0 \Om_k :=
\{x\in \partial\Om_k: (\nu_{\Om_k}^{}(x),\xi)=0\} \,,
$$
where $\nu_{\Om_k}^{}$ is the outer unit normal to 
$\partial\Omega_k$.
Since $(\nu_u,\xi)\ge0$ we have
$$
\nu_u=\pm\nu_{\Om_k}^{} \qquad\Hn\hbox{ a.e. in } J_u\cap 
\partial_\xi^\pm \Om_k
\leqno(\frm{nuomk})
$$
(see, e.g., [\rf{Si83}], Remark~11.7).

We want to prove that for $\Hn$ almost every
$y\in \pi_\xi$ we have
$$
(u^-(y+t\xi),\xi)\,=\,(\uyx)^-(t) \qquad
\forall t\in (J_u^\xi \cap \partial_\xi^+ \Om_k)_y^\xi \,.
\leqno(\frm{umeno})
$$
For every $x\in J_u^\xi \cap \partial_\xi^+ \Om_k$ with 
$\nu_u(x)=\nu_{\Om_k}^{}(x)$ the function~$v$ 
has one sided Lebesgue limits with respect to $\nu_u(x)$ and
$$
v^+(x)=0\,,\qquad
v^-(x)=u^-(x)\,.
\leqno(\frm{vm=um})
$$
If $y\in Y$ and $t\in (J_u^\xi \cap \partial_\xi^+ \Om_k)_y^\xi$, using 
the fact that $(\nu_{\Om_k}^{}({y+t\xi}),\xi)>0$, we can prove that the 
functions $u_y^\xi$ and $v_y^\xi$ have one sided Lebesgue limits with 
respect to $\nu=1$ and
$$
(v_y^\xi)^+(t)=0\,,\qquad
(v_y^\xi)^-(t)=(u_y^\xi)^-(t)\,.
\leqno(\frm{vyxim=uyxim})
$$
Let us consider separately the sets
$$
M:=\{x\in J_u^\xi \cap \partial_\xi^+ \Om_k : (u^-(x),\xi)\neq 0\}\,,
\qquad
N:=\{x\in J_u^\xi \cap \partial_\xi^+ \Om_k : (u^-(x),\xi)= 0\}\,.
$$
By~(\rf{vm=um}) and~(\rf{nuomk}) for $\Hn$ almost every 
$y\in\pi_\xi$ we have $M_y^\xi\subset (J_v^\xi)_y^\xi$ and 
$\nu_v(y+t\xi)=\nu_u(y+t\xi)$ for every $t\in M_y^\xi$ (recall that 
${(\nu_v,\xi)\ge0}$). If we apply~(\rf{salti2}) to the function~$v$, 
taking~(\rf{vm=um}) and~(\rf{vyxim=uyxim}) into account, for
$\Hn$ almost every $y\in\pi_\xi$ we obtain
$$
(u^-(y+t\xi),\xi)\,=\,(\uyx)^-(t) \qquad
\forall t\in M_y^\xi \,.
\leqno(\frm{umenoM})
$$
If we apply~(\rf{secJ}) to the function~$v$, we obtain that 
$(J_v^\xi)_y^\xi=J_{v_y^\xi}$ for $\Hn$ almost every $y\in \pi_\xi$.
As $(u^-(x),\xi)=0$ for every ${x\in N}$, by~(\rf{vm=um}) 
and~(\rf{nuomk}) for $\Hn$ almost every $y\in\pi_\xi$ we have 
$N_y^\xi\cap J_{v_y^\xi}=N_y^\xi\cap (J_v^\xi)_y^\xi=\emptyset$, hence
$(v_y^\xi)^+=(v_y^\xi)^-$ in $N_y^\xi$. By~(\rf{vyxim=uyxim}) this 
implies that 
$(u_y^\xi)^-=0$  in $N_y^\xi$ for $\Hn$ almost every 
$y\in \pi_\xi$. As $(u^-(x),\xi)=0$ for every $x\in N$, for
$\Hn$ almost every $y\in \pi_\xi$ we get
$$
(u^-(y+t\xi),\xi)\,=\,0\,=\,(\uyx)^-(t) \qquad
\forall t\in N_y^\xi \,.
\leqno(\frm{umenoN})
$$
{}From (\rf{umenoM}) and (\rf{umenoN}) we obtain (\rf{umeno}), and 
from (\rf{umeno}) and (\rf{salti2}) for $\Hn$ almost every
$y\in \pi_\xi$ we obtain that
$$
(u^+(y+t\xi),\xi)\,=\,(\uyx)^+(t)\,,
\qquad
(u^-(y+t\xi),\xi)\,=\,(\uyx)^-(t) \qquad
\leqno(\frm{depiu})
$$
for every $t\in (J_u^\xi\cap\partial_\xi^+\Om_k)_y^\xi$.
In the same way we can prove that~(\rf{depiu}) holds for
$\Hn$ almost every $y\in \pi_\xi$ and for every
$t\in (J_u^\xi\cap\partial_\xi^-\Om_k)_y^\xi$.
Since, by the area formula, 
$\Hn(p_\xi({J_u^\xi\cap\partial_\xi^0\Om_k}))=0$, 
(\rf{depiu}) holds for $\Hn$ almost every $y\in \pi_\xi$  and for every
$t\in (J_u^\xi\cap\partial\Om_k)_y^\xi$.
As the union of the sets $\partial\Om_k$ covers $\Hn$ almost all of 
$J_u$, for $\Hn$ almost every $y\in \pi_\xi$ we have
$$
(u^+(y+t\xi),\xi)\,=\,(\uyx)^+(t)\,,
\qquad
(u^-(y+t\xi),\xi)\,=\,(\uyx)^-(t) \qquad
\forall t\in (J_u^\xi)_y^\xi \,.
\leqno(\frm{finalmente})
$$

For every $y\in Y$ the function $\tilde u_y^\xi$ has bounded pointwise 
variation and coincides with $u_y^\xi$ $\Le^1$ almost everywhere 
on~$\R$. Therefore
$$
(u^\xi_y)^+(t) \,=\, \lim_{s\to t^+}\tilde{u}_y^\xi(s)\,,
\qquad
(u^\xi_y)^-(t) \,=\, \lim_{s\to t^-}\tilde{u}_y^\xi(s)
\leqno(\frm{jump3})
$$
for every $t\in J_{u_y^\xi}$. Together with (\rf{finalmente}), 
this concludes the proof of~(iv).
\endproof


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%
\parag{\chp{sumenfu}}{Discontinuity points which are not jump points}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%

In this section we study the set ${S_u\setminus J_u}$ of
Lebesgue discontinuity points which are not jump points.
For $BV$ functions this set is $\Hn$ negligible (see, e.g., [\rf{EvGa93}], 
Section~5.9, Theorem~3); this property is not known for $BD$ functions.
The main result of this section is the following theorem, which shows
that ${S_u\setminus J_u}$ is negligible for all measures related to
$BD$ functions.

\th{\thm{suju}} {If $u,\, v\in BD(\Om)$, then 
$|Ev|({S_u\setminus J_u})=0$. In other words, $|Ev|$ almost every point 
$x\in\Om$ is either a Lebesgue point or a jump point of~$u$.}

\rem{\thm{exce}}
{Among other things, Kohn proved in [\rf{Ko}] that the set 
$S_u\setminus J_u$ has Hausdorff dimension at most ${n-1}$, i.e.,
${\cal H}^{n-1+\eps}(S_u\setminus J_u)=0$
for any ${\eps>0}$. It is not known whether $\Hn(S_u\setminus J_u)=0$ 
for every $u\in BD(\Om)$ or whether there are examples
of $BD$ functions for which this set has positive $\Hn$ measure. In the 
latter case one can not obtain Theorem~\rf{suju} from~(\rf{nullo}).
}




\rem{\thm{unrect}}
{Using property~(ii) of Section~\rf{secbd}, it is easy to 
check that for every $u\in BD(\Om)$ the set ${S_u\setminus J_u}$ is 
purely $(\Hn,{n-1})$ unrectifiable, i.e.,
$\Hn\bigl({({S_u\setminus J_u})\cap M}\bigr)\allowbreak=0$
for any $\C^1$ hypersurface ${M\subset\Rn}$ (see [\rf{Ko}], Part~II, 
Theorem~5.2). For a $BV$ function the coarea formula implies that 
$|Du|$ vanishes on all purely $(\Hn,{n-1})$ unrectifiable Borel sets.
For a $BD$ function we can prove only that $|Eu|(B)=0$ for every purely
$(\Hn,{n-1})$ unrectifiable Borel set~$B$ on which $\Hn$ is finite (or, 
more generally, $\sigma$-finite). Indeed, in this case
$|Eu|(B)=|E^ju|(B)=|Eu|(B\cap J_u)$ by Proposition~\rf{cantor0}. Since
$J_u$ is countably $(\Hn,{n-1})$ rectifiable, $\Hn(B\cap J_u)=0$ and the
conclusion follows from~(\rf{nullo}).
%This is an easy consequence of Federer's structure theorem
%(see [\rf{Fed68}], Theorem~3.3.13) and of the 
%final statement of Remark~\rf{nulli}.
The extension of this property to a more general class
of  unrectifiable sets is still an open problem. On the other hand for a 
function $u\in BD(\Om)$ is not known whether the set 
${S_u\setminus J_u}$ is $\sigma$-finite for the measure~$\Hn$. 
Therefore these general arguments can 
not be used in the proof of Theorem~\rf{suju}.
}


\spazio

Starting from the (trivial) one dimensional case,
we will be able to prove Theorem~\rf{suju} by induction on~$n$.
The induction step will be based on the following refinement
of Theorem~\rf{main}, which allows to prove that
Lebesgue points for the restriction of $u$ to a
hyperplane $\pi_\eta$ are Lebesgue points for $p_\eta u$ in~$\Rn$.

\th{\thm{main2}} {Under the assumptions
of Theorem~\rf{main}, we have that $x$ is a Lebesgue point
for $p_\eta u$ in~$\Rn$.}

\proof We shall use the same notation adopted in the proof
of Theorem~\rf{main}, assuming $x=0$ and $\pi$
equal to the hyperplane orthogonal to $e_n$ and passing
through the origin. We already know that $0$ is a Lebesgue point of the 
function $v(y):=(\tilde u^1(y), \ldots, \tilde u^{n-1}(y), 0)$ in~$\pi$. The 
first component of the Lebesgue representative in $\pi$ of $v$ at $0$ 
will be denoted by $v_0^1$. We shall prove that $v_0^1$ 
coincides with $\tilde u^1(0)$, the Lebesgue representative in $\Rn$ of 
$u^1$ at~$0$.
A similar argument
works for the other components.

Since $0\notin\Theta_u$, Corollary~\rf{Kohn2} below
yields that the quantity
%
$$
\omega(\eps):={1\over\eps^n}\int_{Q_\eps} 
\bigl[ \int_{-\eps}^{\eps}
|u^1(y + t e_n)-d_\eps|\,dt\,\bigr] \,d\Hn(y)
$$
%
vanishes as $\eps\to 0^+$,
for suitable real constants $d_\eps$. We have to prove
that $d_\eps$ converges to $v_0^1$ as ${\eps\to 0^+}$.

For any
$\eps>0$ we choose $\ro=\ro(\eps)\in (\eps/2,\eps)\cap Y$ such that
%
$$
{1\over\eps^{n-1}}\int_{Q_\eps} |\tilde u^1(y+\ro e_n)-d_\eps|+
|\tilde u^1(y-\ro e_n)-d_\eps|\,d\Hn(y)
\,\leq\,2\omega(\eps)
\leqno (\frm{last})
$$
%
and we write
%
$$
\eqalign{
\tilde u^1(y+\ro e_n) +\tilde u^1(y-\ro e_n)
&-2v_0^1\,=\, \bigl[(\tilde u^1+\tilde u^n)(y+\ro e_n)
- (\tilde u^1+\tilde u^n)(y-\ro e_1)\bigr]\cr
\,&+\bigl[(\tilde u^1-\tilde u^n)(y-\ro e_n)
- (\tilde u^1-\tilde u^n)(y-\ro e_1)\bigr]\cr
\,&+ \bigl[\tilde u^n(y-\ro e_n) - \tilde u^n(y +\ro e_n)\bigr]
+2\bigl[v^1(y-\ro e_1)-v_0^1\bigr]\,.\cr
}
$$
If we integrate both sides of this equality on
$Q_\eps$, the same argument exploited in
Theorem~\rf{main} and the Lebesgue continuity of $v^1$ at $0$ in $\pi$ 
show that
%
$$
\lim_{\eps\to 0^+} {1\over\eps^{n-1}}
\int_{Q_\eps}|\tilde u^1(y+\ro e_n) +\tilde u^1(y-\ro e_n)
-2v_0^1|\,d\Hn(y)
=0\,.
$$
%
Taking (\rf{last}) into account, we obtain that $d_\eps$ converges to
$v_0^1$ as $\eps\to 0^+$.
\endproof

\spazio

The proof of Theorem~\rf{main2} has been based on (\rf{Kohn}) of
Corollary~\rf{Kohn2}. We will obtain this and related formulas following 
the lines of Section~5 of Part~II of Kohn's thesis [\rf{Ko}], relying on an 
explicit version of Poincar\'e's inequality for $BD$ functions. Given
a matrix of measures $\{\mu_{lm}^{}\}$ on $\Rn$, a tensor
$\{T^{lm}_{ij}(x)\}$ of order~$4$, and a Borel set $B\subset\Rn$, we 
introduce the short notation
%
$$
\alpha\,=\,\int_{B} T\sep d\mu
\qquad\hbox{\rm for}\qquad
\alpha_{ij}=\sum_{l,m=1}^n
\int_{B}T_{ij}^{lm}\,d\mu_{lm}^{}\,.
$$
Let $\Gamma(x)=\{\Gamma_{ij}^{lm}(x)\}$ be the fourth order tensor 
defined by
$$
\cases{
\Gamma^{ii}_{ij}(x)=-\Gamma^{jj}_{ij}(x)=2x_ix_j\,,&
if $i\neq j$,
\cr
\Gamma^{ij}_{ij}(x)=\Gamma_{ij}^{ji}(x)=x_j^2-x_i^2\,,&
\cr
\Gamma_{ij}^{ir}(x)\,=\,\Gamma_{ij}^{ri}(x)=x_jx_r \,,&
if $i\neq j$ and $r\notin\{i,j\}$,
\cr
\Gamma_{ij}^{jr}(x)\,=\,\Gamma_{ij}^{rj}(x)=-x_ix_r \,,&
if $i\neq j$ and $r\notin\{i,j\}$,
\cr
\Gamma_{ij}^{lm}(x)\,=\,0 \,,&
if $i=j$ or $\{l,m\}\cap\{i,j\}=\emptyset$.
\cr}
$$
Note that for every $i,\,j,\,l,\,m$ we have
$$
\Gamma_{ij}^{lm}(x)=\Gamma_{ij}^{ml}(x)
=-\Gamma_{ji}^{lm}(x)
\qquad \hbox{and} \qquad
\int_{\Sn} \Gamma_{ij}^{lm} \,d\Hn \,=\,0 \,.
\leqno(\frm{sym-skewsym})
$$
%
\th{\thm{kohn1}} {Let $u\in BD(\Rn)$, $n\geq 2$,
and for any $x\in\Rn$ and $\ro>0$ define
%
$$
\beta_\ro(x)=-{1\over2\omega_n} \int_{\{|y-x|\ge \ro\}}
{\Gamma(y-x)\over |y-x|^{n+2}} \sep d Eu(y) \,.
\leqno (\frm{defbeta})
$$
%
Then, for suitable constants $d_\ro(x)$, we have the estimate
%
$$
\int_{B_\ro(x)}
|u(y)-d_\ro(x)-\beta_\ro(x)(y-x)|\,dy\leq
c_n \ro\,|E u|(B_\ro(x))\,,
\leqno(\frm{expsti})
$$
%
where $c_n$ depends only on~$n$.
}

\proof The proof can be obtained from Proposition~5.7 of Part~II of 
[\rf{Ko}] with obvious changes. For the reader's convenience we give the 
complete proof in the case ${x=0}$. For every $\ro>0$ we set 
$B_\ro=B_\ro(0)$.
Recalling that ${\cal R}$ is the class of rigid motions in $\Rn$
introduced in Section~3,
we define a linear map $R_\ro\colon BD(B_\ro)\to {\cal R}$ as follows:
$R_\ro(v)(y)=\beta_\ro y+d_\ro$ with
%
$$
\beta_\ro\,=\,{1\over 2\omega_n\ro^{n}}
\int_{\partial B_\ro}(v_*{\otimes}\nu-\nu{\otimes} v_*)\,d\Hn,\qquad 
d_\ro\,=\,{1\over n\omega_n\ro^{n-1}}\int_{\partial B_\ro} v_*\,d\Hn
\,,
\leqno(\frm{intsfe})
$$
%
where $v_*$ is the trace of $v$ on $\partial B_\ro$ and $\nu$ is the 
exterior unit normal to $\partial B_\ro$.
It is easy to check that $R_\ro$ is the identity on ${\cal R}$, so that 
(\rf{expsti}) follows from Theorem~\rf{TS}. The fact that the constant 
$c(B_\ro,R_\ro)$ has the special form $c_n\ro$ can be easily proved by 
a rescaling argument.

We need only to prove that for
$u\in BD(\Rn)$ the matrix $\beta_\ro$ in (\rf{intsfe}) can be computed 
by (\rf{defbeta}).
To this aim, we fix two indices $i,\,j$, with ${i\neq j}$, and a constant 
$\tau>\ro$. We apply the divergence formula (\rf{divbd}) in
$B_\tau\setminus \overline B_\ro$ to
the symmetric matrix valued function
$\psi_{lm}(y)=\Gamma^{lm}_{ij}(y)/(2\omega_n|y|^{n+2})$. Since
%
$$
\sum_{m=1}^n D_m \psi_{lm} (y)=0
\qquad\hbox{and}\qquad
\sum_{m=1}^n\psi_{lm}(y){y_m\over |y|}=\cases{
\displaystyle{y_j\over2\omega_n|y|^{n+1}}\,, & if $l=i$,
\cr\cr
-\displaystyle{y_i\over2\omega_n|y|^{n+1}}\,, & if $l=j$,
\cr\cr 
0\,, & if $l\notin\{i,j\},$}
$$
%
using (\rf{divbd}) (and also (\rf{rappJ}) if the inner and the outer trace 
of~$u$ on $\partial B_\ro$ are different) we obtain
%
$$
\int_{\{\tau>|y|\ge\ro\}}{\Gamma(y)\over 2\omega_n|y|^{n+2}}
\sep d Eu(y) \,=\, {1\over 2\omega_n\tau^n}
\int_{\partial B_\tau}(u_*{\otimes}\nu-\nu{\otimes} u_*)\,d\Hn \,-\,
\beta_\ro\,,
\leqno(\frm{incalpha})
$$
%
where~$\nu$ is the exterior unit normal to $\partial B_\tau$.
Using the integrability of $u$ in $\Rn$, it is not hard to see that
%
$$
\liminf_{\tau\to +\infty}
{1\over\tau^{n}}\int_{\partial B_\tau} |u_*|\,d\Hn\,=\,0\,.
$$
%
Hence, letting $\tau\to +\infty$ we obtain (\rf{defbeta}).
\endproof

\rem{\thm{compdro}} {By a similar argument it can be seen
that a constant $d_\ro(x)$ in (\rf{expsti}) can be explicitly
given by the formula
%
$$
d_\ro^i(x)\,=\,\sum_{l,m=1}^n\int_{\{|y-x|\ge\ro\}}
{\Lambda_i^{lm}(y-x)\over n\omega_n|y-x|^n}\,d E_{lm}u(y) \,,
$$
%
with $\Lambda_i^{ri}(y)=\Lambda_i^{ir}(y)=y_r$ if $1\leq r\leq n$,
$\Lambda_i^{rr}(y)=-y_i$ if $r\neq i$ and $\Lambda_i^{lm}(y)=0$
in all other cases.}


\spazio

\cor{\thm{Kohn2}} {Let $u\in BD(\Om)$. Then
%
$$
\limro \min_{d\in\Rn}
{1\over \ro^n} \int_{B_\ro(x)} |u(y)-d|\,dy=0
\leqno(\frm{Kohn})
$$
for every $x\in\Om\setminus \Theta_u$.}
%
\proof The first proof of this property is given in Theorem~5.13 of 
Part~II  
of [\rf{Ko}]. For the reader's convenience we reproduce it with some 
variants. 

Since the statement is local, we can assume $\Om=\Rn$ and $x=0$.
By Theorem~\rf{kohn1}, we need only to prove that $\ro\beta_\ro$
converges to $0$ as $\ro\to 0^+$. Let us fix ${\tau>0}$ and let
$\ro\in (0,\tau)$; using the identity (\rf{incalpha})
we find that the upper limit of $\ro|\beta_\ro|$ can be estimated,
up to a multiplicative constant, by
%
$$
\limsup_{\ro\to 0^+}\ro\int_{\{\tau>|y|\ge\ro\}}{1\over |y|^n}\,
d|E u|(y)\,.
$$
%
Hence, we need only to show that the upper limit above tends to zero
as $\tau\to 0^+$. Setting $\mu:=|Eu|\LL {\{\tau>|y|\ge\ro\}}$ and using 
the change of variables $t=1/s^n$ we obtain
%
$$
\eqalign{\int_{\{\tau>|y|\ge\ro\}}{1\over |y|^n}\,d|E u|(y)
&=\,\int_0^{+\infty}\mu\bigl(\{ {1\over |y|^n}>t\}\bigr)\,dt
\cr
&=\,n\int_0^{+\infty}{\mu( \{|y|<s\} ) \over s^{n+1}}\,ds
\cr
&=\,n \int_\ro^\tau {|Eu|(\{s>|y|\ge\ro\}) \over s^{n+1}}\,ds+
{|E u|(\{\tau>|y|\ge\ro\})\over \tau^n}\,.
\cr}
$$
%
Therefore, writing $s^{n+1}=s^{n-1}s^2$, we get the estimate
%
$$
\limsup_{\ro\to 0^+}\ro\int_{\{\tau>|y|\ge\ro\}}{1\over |y|^n}\,
d|E u|(y)\leq n\sup_{t\in (0,\tau)} {|E u|(\{|y|<t\})\over t^{n-1}}\,.
$$
%
By our assumption, the right hand side vanishes as $\tau\to 0^+$.
\endproof

\spazio

We are now in a position to prove Theorem~\rf{suju} in the special case 
${u=v}$.

\prop{\thm{cantor}} {For every $u\in BD(\Om)$ we have
$|Eu|({S_u\setminus J_u})=0$.}

\proof Since the property to be proved is local, 
we may assume that $u\in BD(\Rn)$.
The statement is trivial for $n=1$; assume it valid for $n-1\geq 1$ and 
let us prove it for~$n$.

For any $\xi\in\Rn$ we claim that $|(Eu \xi,\xi)|$ almost every point of 
${\Rn\setminus J_u}$ is a Lebesgue point of $u^\xi$. By 
Lemma~\rf{final} below with $\mu=Eu\LL (\Rn\setminus J_u)$, this 
property implies that $|E u|$ almost every point in $\Rn\setminus J_u$ 
is a Lebesgue point for~$u$.

Let $\eta\in \Sn$ be orthogonal to~$\xi$, so that the hyperplane
$\pi_\eta$ contains $0$ and $\xi$, and let $S$ be the set of points 
$s\in\R$ such that the 
function $v_s(y)\,:=\,p_\eta \tilde u(s\eta+y)$ is defined for
$\Hn$ almost every $y\in \pi_\eta$ and belongs to $BD(\pi_\eta)$.
By Proposition~\rf{projBD} we have ${\cal L}^1({\R\setminus S})=0$.
By the induction assumption
%and Theorem~\rf{rettifTu}
$v_s$ has one sided (possibly equal) Lebesgue limits $v_s^+$ and $v_s^-$ 
in $\pi_\eta$ at $|(E v_s \xi,\xi)|$ almost every point of $\pi_\eta$.
By Proposition~\rf{projBD}, for every Borel set $B$ of $\Rn$ we have
%
$$
(Eu \xi,\xi)(B) \,= \,\int_{-\infty}^{+\infty} (Ev_s \xi,\xi)(B_s)\,ds \,,
$$ 
%
where $B_s = \{y\in \pi_\eta\ : s\eta + y \in B\}$.
Therefore $|(Eu \xi,\xi)|$ is concentrated on the set of all points $x$ of 
the form
$s\eta+y$, with $s\in S$ and $y$ such that $v_s^+$ and $v_s^-$
exist at~$y$. By Theorem~\rf{main2}, for any
point $x$ of this type, either $x\in\Theta_u$ or
$x$ is a Lebesgue point of $p_\eta u$ in $\Rn$. 
As $\xi\in\pi_\eta$, this implies that $|(Eu \xi,\xi)|$ almost every
point either belongs to $\Theta_u$ or is a Lebesgue point of $u^\xi$.
Since $\Hn(\Theta_u\setminus J_u)=0$, the claim follows 
from~(\rf{nullo}).
\endproof

\lemma{\thm{final}} {Let $u\in L^1({\bf R}^n;{\bf R}^n)$, let
$\mu=\{\mu_{ij}\}$ be a symmetric 
matrix of Radon measures in $\Rn$ and assume that, for any
$\xi\in\Rn$, $|(\mu\xi,\xi)|$ almost every point of $\Rn$ is a Lebesgue
point of $u^\xi$. Then $|\mu|$ almost every point of $\Rn$ is a
Lebesgue point for~$u$.
}

\proof Let $\eta\in\Rn$ be given. We have to prove
that $|\mu|$ almost every point of $\Rn$ is a Lebesgue point
of $u^\eta$.
By (\rf{matrici}), we need only to prove that for any choice of
$\xi$ in $\Rn$, $|(\mu\xi,\xi)|$ almost every point is a
Lebesgue point for~$u^\eta$.

Applying our hypothesis to the vectors
$\xi,\,\eta,\,\xi\pm\eta$ and setting for brevity
$\mu_{\xi\xi}:=(\mu\xi ,\xi)$, $\mu_{\eta\eta}:=(\mu\eta ,\eta)$,
$\mu_{\xi\eta}:=(\mu\xi ,\eta)$, we obtain the following properties:

\item{(a)}$|\mu_{\xi\xi}|$ almost every $x\in\Rn$ is a Lebesgue point 
for $u^\xi$;

\item{(b)}$|\mu_{\eta\eta}|$ almost every $x\in\Rn$ is a Lebesgue 
point for $u^\eta$;

\item{(c)}$|\mu_{\xi\xi}\pm 2\mu_{\xi\eta}+\mu_{\eta\eta}|$ almost 
every $x\in\Rn$ is a Lebesgue point for $u^\xi\pm u^\eta$.

\noindent
Let $\lambda:=|\mu_{\xi\xi}|+|\mu_{\eta\eta}|+|\mu_{\xi\eta}|$. Since 
$\mu_{\xi\xi}$, $\mu_{\eta\eta}$, and $\mu_{\xi\eta}$ are absolutely 
continuous with respect to $\lambda$, there exist three locally 
$\lambda$ integrable functions $h_1$, $h_2$, and $h_3$ such that 
$\mu_{\xi\xi}=h_1\lambda$, $\mu_{\eta\eta}=h_2\lambda$, and
$\mu_{\xi\eta}=h_3\lambda$.
Let us consider the mutually disjoint sets
%
$$
A_1\,:=\,\bigl\{x\in\Rn:h_1(x)+h_2(x)=0\bigr\},\qquad
A_2\,:=\,\bigl\{x\in\Rn: h_1(x)+h_2(x)\neq 0\bigr\}\,,
$$
%
whose union is~$\Rn$.
On $A_1$ we have $\mu_{\xi\xi}=-\mu_{\eta\eta}$, hence the total 
variations are the same and (b) implies that $|\mu_{\xi\xi}|$ almost 
every $x\in A_1$ is a Lebesgue point for $u^\eta$.
On $A_2$, by condition (c), $\lambda$ almost every $x$
is a Lebesgue point either for $u^\xi+u^\eta$ or for $u^\xi-u^\eta$, since 
$h_1+h_2+2h_3$ and $h_1+h_2-2h_3$ do not vanish at the same points. 
Together with~(a), this implies that 
$|\mu_{\xi\xi}|$ almost every $x\in A_2$ is a Lebesgue point 
for~$u^\eta$.
\endproof

\spazio
Using Proposition~\rf{cantor} we can now prove Theorem~\rf{suju} in 
the 
general case.

\proofof{Theorem~\rf{suju}}Let us fix $u,\, v\in BD(\Om)$. By the 
Lebesgue decomposition theorem there exists a Borel set 
$N\subset\Om$ such that $|Eu|(N)=0$ and $|Ev|\LL({\Om\setminus N})$ 
is absolutely continuous with respect to $|Eu|$. Since 
$|Eu|\bigl({(S_u\setminus J_u)\setminus N}\bigr)=0$ by 
Proposition~\rf{cantor}, we deduce that 
$|Ev|\bigl({(S_u\setminus J_u)\setminus N}\bigr)=0$. It remains to 
prove that $|Ev|\bigl({(S_u\setminus J_u)\cap N}\bigr)=0$.
Since
$$
S_u\setminus J_u\,\subset\,
(S_v\setminus J_v) \,\cup\,
[(S_u\setminus J_u)\setminus S_v ] \,\cup\,
[ (S_u\setminus J_u)\cap J_v ]
$$
and $|Ev|(S_v\setminus J_v)=0$ (Proposition~\rf{cantor}), it is enough to 
show that
$$
\ldisplaylinesno{
|Ev|\bigl(((S_u\setminus J_u)\setminus S_v)\cap N\bigr)=0 \,,
&(\frm{sujun1})
\cr
|Ev|\bigl((S_u\setminus J_u)\cap J_v\cap N\bigr)=0 \,.
&(\frm{sujun2})
\cr}
$$
It is easy to see that a Lebesgue point of~$v$ is a Lebesgue (resp.\ 
jump) 
point of~$u$ if and only if it is  a Lebesgue (resp.\ jump) point of 
${u+v}$; this shows that ${(S_u\setminus J_u)\setminus S_v} 
={(S_{u+v}\setminus J_{u+v})\setminus S_v}$, so that, by 
Proposition~\rf{cantor}, $|E({u+v})|\bigl((S_u\setminus J_u)\setminus 
S_v\bigr)=0$.
As $Ev$ and $E({u+v})$ coincide as measures on $N$, we 
obtain~(\rf{sujun1}).

Let $x$ be a jump point of~$v$. If $x$ is a Lebesgue point of ${u+v}$, 
then $x$ is a jump point of~$u$; if $x$ is a jump point of ${u+v}$ and 
$\nu_{u+v}(x)=\pm\nu_v(x)$, then $x$ is either a Lebesgue point or a 
jump point of~$u$. As $J_v$ and $J_{u+v}$ are countably $(\Hn,{n-1})$ 
rectifiable, the condition on 
$\nu_{u+v}$ and $\nu_v$ is satisfied $\Hn$ almost everywhere in 
$J_{u+v}\cap J_v$ (see Remark~\rf{normalJ}). This implies that 
$(S_u\setminus J_u)\cap J_v 
\subset S_{u+v}\setminus J_{u+v}$ up to an $\Hn$ negligible set. 
Consequently from 
Proposition~\rf{cantor} and Remark~\rf{nulli} we get 
$|E({u+v})|\bigl((S_u\setminus J_u)\cap J_v\bigr)=0$. As $Ev$ and 
$E({u+v})$ coincide as measures on $N$, we 
obtain~(\rf{sujun2}).
\endproof



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%
\parag{\chp{apdiff}}{Approximate differentiability}
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we prove that $BD$ functions are approximately
differentiable in $\Le^n$ almost every point of their domain. We will 
also see that the same proof can be used to establish Korn's inequality.
We start by recalling the definition of approximate limit 
(see~[\rf{Fed68}], 2.9.19).

\defin{\thm{applim}}
{Let $u$ be a real valued Borel function defined $\Le^n$ almost 
everywhere in a neighbourhood of a point $x\in \Rn$.
We say that $z\in{\bf R}$ is the {\it approximate limit\/} of~$u$ at~$x$, 
and we write $z=\aplim\limits_{y\to x}u(y)$, if, for every $\eps>0$, the 
set
$A_\eps:=\{y: |u(y)-z|>\eps\}$ has $\Le^n$ density zero at $x$, i.e.,
$$
\limro{\Le^n(A_\eps\cap B_\ro(x))\over \Le^n(B_\ro(x))}\,=\,0 \,.
$$
The {\it approximate upper  limit\/}
$\aplimsup\limits_{y\to x}u(y)$ is defined as the infimum of the set 
of all numbers $t$ such that the set $\bigl\{y: u(y)>t\bigr\}$ has
$\Le^n$ density zero at~$x$.
}

\spazio

If $u$ is locally $\Le^n$ integrable, it is easy to see that
%
$$
\lim_{\ro\to 0^+}{1\over \ro^n}\int_{B_\ro(x)}|u(y)-z|\,dy\,=\,0
\qquad\Longrightarrow\qquad
z\,=\, \aplim_{y\to x}u(y)\,,
$$
%
so that any Lebesgue point $x$ where $\tilde u(x)=u(x)$
is an approximate continuity point for~$u$. In 
general, the opposite implication does not hold, unless~$u$ is locally 
bounded. 

We can now recall the definition of approximate differentiability.

\defin{\thm{appdiff}} {Let $u\colon\Om\to {\bf R}^m$ be a Borel 
function and let $x\in\Om$.
We say that
$u$ is {\it approximately differentiable\/} at $x$ if
there exists an ${m{\times}n}$ matrix~$L$ such that
%
$$
\aplim_{y\to x}{|u(y)-u(x)-L(y-x)|\over |y-x|}\,=\,0 \,.
\leqno (\frm{wwdi})
$$
%
The matrix $L$, called
{\it approximate differential\/} of~$u$ at~$x$, is uniquely determined 
and will be 
denoted by $\nabla u(x)$.
}
%
\spazio
We recall that functions which are approximately differentiable in
$\Le^n$ almost every point of their domain can be approximated, in the 
sense of Lusin, by Lipschitz functions.
Moreover a necessary and
sufficient condition for $\Le^n$ almost everywhere approximate 
differentiability is the inequality
%
$$
\aplimsup_{y\to x}{|u(y)-u(x)|\over |y-x|}<+\infty\qquad
\hbox{\rm for }\, \Le^n\hbox{ a.e. }x\in\Rn
$$
%
(see [\rf{Fed68}], Theorem~3.1.8).
A sufficient condition for the approximate differentiability,
intermediate between the stronger condition (\rf{apdi}) and
(\rf{wwdi}), is given by the following proposition.

\prop{\thm{equiva2}} {Let $u\colon\Om\to\R^m$ be a locally $\Le^n$ 
integrable function and assume that
%
$$
\lim_{\ro\to 0^+} {1\over \ro^n}\int_{B_\ro(x)}
{|u(y)-u(x)-L(y-x)|\over\ro} \,dy=0
$$
%
for some $x\in\Om$ and for some ${m{\times}n}$ matrix~$L$.
Then~$u$ is approximately differentiable at~$x$ and
${\nabla u(x)=L}$.
}
%
\proof
We will prove that for every ${\eps>0}$ the set
%
$$
A_\eps\,:=\,\bigl\{y\in\Om:
{|u(y)-u(x)-L(y-x)|\over |y-x|}>\eps\bigr\}
$$
%
has $\Le^n$ density $0$ at $x$. For any $\ro>0$ and any $t\in (0,1)$
we can split $B_\ro(x)$ into the union of the sets
$B_\ro(x)\setminus B_{t\ro}(x)$ and $B_{t\ro}(x)$. This leads to the 
estimate
%
$$
{1\over \ro^n}{\cal L}^n\bigl(B_\ro(x)\cap A_\eps\bigr)\,\leq\,
{1\over t\eps\ro^n}\int_{B_\ro(x)}
{|u(y)-u(x)-L(y-x)|\over\ro}\,dy
+\omega_nt^n\,.
$$
%
Taking the upper limit as $\ro\to 0^+$ we obtain
%
$$
\limsup_{\ro\to 0^+}
{1\over \ro^n}{\cal L}^n\bigl(B_\ro(x)\cap A_\eps\bigr)\,\leq\,
\omega_nt^n\,.
$$
%
Since $t\in (0,1)$ is arbitrary, the statement is proved.
\endproof
\spazio
We can now state our approximate differentiability result.

\th{\thm{bddiff}} {Let $u\in BD(\Rn)$. Then for $\Le^n$ almost every
$x\in\Rn$ there exists a $n{\times} n$ matrix $\nabla u(x)$ such 
that
%
$$
\lim_{\ro\to 0^+} {1\over \ro^n}\int_{B_\ro(x)}
{|u(y)-u(x)-\nabla u(x)(y-x)|\over\ro}\,dy=0\,.
\leqno (\frm{boll})
$$
%
In particular, $u$ is approximately differentiable $\Le^n$ almost 
everywhere in $\Rn$. Moreover the function $\nabla u$ 
satisfies the weak $L^1$ estimate
%
$$
{\cal L}^n\bigl(\{x\in\Rn:|\nabla u(x)|>t\}\bigr)\leq
{c(n)\over t}|Eu|(\Rn)\qquad\qquad\forall t>0\,,
$$
where $c(n)$ is a constant which depends only on~$n$.
}
%
\rem{\thm{sympart}} {Property~(\rf{boll}), together with 
Theorem~\rf{symmdi}, implies that $\E u(x)$ is equal to the symmetric 
part of the matrix $\nabla u(x)$ for $\Le^n$ almost every 
$x\in\Rn$.
}
%
\rem{\thm{noL1}} {Property (\rf{boll}) implies that
for $\Le^n$ almost every $x\in\Rn$
the rescaled functions $v_\ro(y):=[u(x+\ro y)-u(x)]/\ro$ converge in 
$L^1(B_1(0);\Rn)$, as
${\ro\to 0^+}$, to the linear function $v_0(y):=\nabla u(x)\,y$. This 
information is very useful in 
connection with the so-called blow up method introduced in 
[\rf{FoMu92}] in order to estimate the absolutely continuous part of 
relaxed  functionals with linear growth. Under the
weaker assumption (\rf{wwdi}) the convergence occurs only in measure.
}

\ex{\thm{orn}}{Assume that $n\ge2$. Using an 
example of Ornstein~[\rf{Orn62}] (see also~[\rf{Te85}], Chapter~I, 
Section~1) we can construct a sequence of functions $\varphi_k$ in 
$\C^{\infty}_c(\Rn;\R^n)$, with supports contained in $B:=B_1(0)$, which 
do not satisfy  Korn's inequality for the 
$L^1$ norm, i.e., after a suitable normalization, we have
%
$$
2^k \, \int_{B}  \v \E \varphi_k \v\,dx \,\le\, 
\int_{B}  \v \nabla \varphi_k \v\, dx \,=\, 1\,.
$$
%
In particular, by Poincar\'e's inequality, the integrals 
$\int_{B} \v \varphi_k \v\,dx$ are bounded by a 
constant $c$ independent of~$k$.
Let us consider now a sequence of pairwise
disjoint balls $B_k:=B_{r_k}(x_k)$ 
contained in $B$ such that 
$\sum\limits_{k=1}^\infty r_k < +\infty$ and  whose centres $x_k$ 
converge to~$0$. Let 
$v_k(x):= r_k^{1-n} \varphi_k \bigl(r_k^{-1}(x-x_k)\bigr)$.
By an elementary change of variables we obtain
%
$$
2^k \, \int_{B_k} \v \E v_k \v\,dx \,\le\,
\int_{B_k} \v \nabla v_k \v\, dx \,=\, 1\,,
\qquad \int_{B_k} \v  v_k \v\,dx \,\le c\,r_k\,.
$$
%
Therefore the series $\sum\limits_{k=1}^\infty v_k$ is absolutely 
convergent in the $BD$ norm and, consequently, its sum~$u$ belongs to 
$BD(\Rn)$ and satisfies $|Eu|(\Rn)\leq 1$; moreover, $u$ has compact 
support in $B$ and belongs to $\C^\infty(\Rn\setminus\{0\})$. By 
Proposition~\rf{equiva}(iii) we have also $u\in SBD(\Rn)$. But
%
$$
\int_{B} |\nabla u|\,dx\geq\sum_{k=1}^\infty
\int_{B_k}|\nabla v_k|\,dx=+\infty\,.
$$
%
This proves that $u\notin BV(\Om;\Rn)$ and that, in general,
no strong $L^1$ estimate can be obtained for~$\nabla u$.
} 

\spazio
The proof of Theorem~\rf{bddiff} is mainly based on three tools:
the first one is the explicit version of the
``Poincar\'e type'' inequality for $BD$ functions
stated in Theorem~\rf{kohn1}. Using that
inequality we are able, in $\Le^n$ almost every point,
to estimate the difference quotient of $u$ up
to a constant vector $\xi_\ro(x)$ and an affine map
$\alpha_\ro(x)(y-x)$. The vector $\xi_\ro(x)$ can be easily estimated 
making use
of Theorem~\rf{symmdi}, while the matrix $\alpha_\ro(x)$
is represented by a singular integral. For this reason we will
also need the following well known result.

\prop{\thm{singu}} {Let $K\colon\Rn\to\R$ be a $0$-homogeneous
function, smooth and with mean value zero on the unit sphere
${\bf S}^{n-1}$. For any measure $\mu$ with finite total variation
in $\Rn$ let us define the functions
%
$$
h_\ro(x)\,:=\,\int_{\{|y-x|\ge\ro\}}
{K(y-x)\over |y-x|^n}\,d\mu(y) \,,\qquad\qquad\ro>0\,.
$$
%
Then the function $h(x):=\sup\limits_{\ro>0}| h_\ro(x)|$ satisfies
the weak $L^1$ estimate
%
$$
{\cal L}^n\bigl(\{x\in\Rn:h(x)>t\}\bigr)\leq
{c(n,K)\over t}|\mu|(\Rn)\qquad\qquad\forall t>0\,,
$$
where $c(n,K)$ is a constant depending only on $n$ and $K$.
}
%
\proof If $\mu$ is absolutely continuous with respect to~$\Le^n$, the 
proof can be found, for instance, in Theorem~4 of Chapter~II of~[\rf{St}].
In the general case the statement can be easily proved by 
approximating $\mu$ by
$\mu_\eps:={\mu\star\psi_\eps}$,
taking into account
the left continuity of $\varrho\mapsto h_\varrho(x)$ and the
continuity property
$$\lim_{\epsilon\to 0^+}
\int_{\{|y-x|\geq\varrho\}}{K(y-x)\over |y-x|^n}\,d\mu_\epsilon(y)
\,=\,
\int_{\{|y-x|\geq\varrho\}}{K(y-x)\over |y-x|^n}\,d\mu(y)
$$
for any $\varrho$ such that $\mu(\partial B_\varrho(x))=0$.
\endproof

\rem{\thm{lpbou}} {If $\mu=f{\cal L}^n$ for some $f\in L^p(\Rn)$ with
$p>1$, then we have a {\it strong} $L^p$ estimate on $h$, namely
$\Vert h\Vert_{L^p(\Rn)}\leq c(n,K)\Vert f\Vert_{L^p(\Rn)}$ 
(see, e.g., [\rf{St}], Chapter~II, Theorem~4).
}

\proofof{Theorem~\rf{bddiff}}By Theorem~\rf{symmdi} and 
Proposition~\rf{singu} 
(that can be applied to each component of $\Gamma(x)/|x|^2$
thanks to~(\rf{sym-skewsym})),
$\Le^n$ almost every $x\in\Rn$ satisfies the following properties:
\bigskip
\item{(i)} \hfil
$\displaystyle
\lim_{\ro\to 0^+}{1\over \ro^n}\int_{B_\ro(x)}
{|(u(y)-u(x)-\E u(x)(y-x),y-x)|\over\ro |y-x|}\,dy\,=\,0\,;
$ \hfil
\bigskip
\item{(ii)}$x$ is a Lebesgue point for $\E u$ and
$\widetilde{\E u}(x)=\E u(x)$;
\smallskip
\item{(iii)}$|E^s u|$ has $n$ dimensional density zero at~$x$;
\smallskip
\item{(iv)}the matrices $\beta_\ro(x)$ in (\rf{defbeta}) are 
equi-bounded with respect to~$\ro$.
\smallskip
\noindent
{\bf Step 1.} Let us fix a point $x\in\Rn$ which satisfies (i)--(iv).
We will prove that these properties imply
%
$$
\lim_{\ro\to 0^+} {1\over \ro^n}\int_{B_\ro(x)}
{|u(y)-u(x)-\alpha_\ro(x)(y-x)|\over\ro}\,dy=0
\leqno (\frm{bolll})
$$
%
for suitable $n{\times} n$ matrices $\alpha_\ro(x)$
satisfying
%
$$
\sup_{\ro>0}|\alpha_\ro(x)|\leq k_n|\E u(x)|+
\sup_{\ro>0}|\beta_\ro(x)|\,,
\leqno(\frm{n2})
$$
where $k_n$ is a constant depending only on~$n$.
To this aim, given a function $\psi\in \C^1_c(\R)$ such that
$\psi(t)= 1$ for ${|t|\le1}$, $\psi(t)= 0$ for ${|t|\ge2}$,
and $|\psi'(t)|\leq 2$ for every $t\in\R$, we define 
$\varphi(y):=\psi(|{y-x}|)$, $w(y):=\varphi(y)\E u(x)({y-x})$,
$v(y):={u(y)-w(y)}$ for every $y\in\Rn$. Applying Theorem~\rf{kohn1}
to the function~$v$, for any $\ro\in(0,1)$ we obtain
%
$$
\displaylines{
\int_{B_\ro(x)}|v(y)-\gamma_\ro(x)(y-x)-\xi_\ro(x)|\,dy
\cr
\leq \, c_n \ro\int_{B_\ro(x)}|\E u(y)-\E u(x)|\,dy \,+\, 
c_n\ro\, |E^s u|(B_\ro(x))\,
}
$$
%
for suitable constant vectors $\xi_\ro(x)$, with
%
$$
\gamma_\ro(x)=\beta_\ro(x) \,+\,
{1\over2\omega_n} \int_{\{|y-x|\ge\ro\}}
{\Gamma(y-x)\over |y-x|^{n+2}}\sep \E w(y)\,dy\,.
\leqno (\frm{forga})
$$
%
Setting $\alpha_\ro(x)=\E u(x)+\gamma_\ro(x)$, and dividing 
both sides
by $\ro^{n+1}$, properties (ii) and (iii) imply
%
$$
\lim_{\ro\to 0^+}{1\over \ro^n}\int_{B_\ro(x)}
{|u(y)-\xi_\ro(x)-\alpha_\ro(x)(y-x)|\over\ro}\,dy\,=\,0.
\leqno (\frm{qq})
$$
%
Since $\gamma_\ro(x)$ is skew symmetric by~(\rf{sym-skewsym}), 
property~(i) is valid with $\E u(x)$ replaced by $\alpha_\ro(x)$; using 
this fact together
with (\rf{qq}) we infer
%
$$
\lim_{\ro\to 0^+}{1\over \ro^n}\int_{B_\ro(x)}
{|(\xi_\ro(x)-u(x),y-x)|\over \ro|y-x|}\,dy\,=\,0
$$
%
which implies that $\xi_\ro(x)=u(x)+o(\ro)$. Inserting
this into (\rf{qq}) we eventually obtain~(\rf{bolll}).

Since
%
$$
\nabla w(y)\,=\,\psi(|y-x|)\E u(x) \,+\,
[\E u(x)(y-x)] {\otimes}\nabla\varphi(y)
\leqno(\frm{nablaw})
$$
%
and, by (\rf{sym-skewsym}), $\Gamma$ has mean value zero on the 
unit sphere $\Sn$, the first term in the right hand side of~(\rf{nablaw}) 
gives no contribution to the integral in~(\rf{forga}). Consequently that 
integral can be estimated by
$$
C_n |\E u(x)| \int_{B_2(x)} {dy\over |y-x|^{n-1}} \,=\,
2n\omega_n C_n |\E u(x)|\,,
$$
where $C_n$ is a constant depending only on~$n$.
As $\vert\alpha_\ro(x)\vert \le \vert\E u(x)\vert + \vert 
\gamma_\ro(x)\vert$, inequality~(\rf{n2}) follows now 
from~(\rf{forga}). 


\noindent
{\bf Step 2.} Using the same argument of Proposition~\rf{equiva2},
we obtain
%
$$
\aplimsup_{y\to x}{|u(y)-u(x)|\over |y-x|}\leq
\sup_{\ro>0}|\alpha_\ro(x)| < +\infty
$$
%
for any $x\in\Rn$ satisfying (i)--(iv), and hence~(\rf{bolll})
and~(\rf{n2}). In particular,
Theorem~3.1.8 in [\rf{Fed68}] implies that $u$ is $\Le^n$ almost 
everywhere
approximately differentiable in $\Rn$ and that $\nabla u$ satisfies
%
$$
|\nabla u(x)|\leq \sup_{\ro>0}|\alpha_\ro(x)|\leq
k_n |\E u(x)|+\sup_{\ro >0}|\beta_\ro(x)|\,.
$$
%
This pointwise estimate and the weak
$L^1$ estimate of Proposition~\rf{singu} imply the stated weak $L^1$
estimate for $\nabla u$.

\noindent
{\bf Step 3.} Now we prove the stronger condition
(\rf{boll}). Let $x$ be an approximate
differentiability point where (\rf{bolll}) holds and the matrices
$\alpha_\ro(x)$ are equi-bounded. If $L$ is the limit
of $\alpha_{\ro_k}(x)$ for some sequence $\ro_k$ converging to $0$,
we have
%
$$
\lim_{h \to +\infty}{1\over (\ro_k)^n}\int_{B_{\ro_k}(x)}
{|u(y)-u(x)-L(y-x)|\over\ro_k}\,dy=0\,.
$$
%
This means that the rescaled functions
$v_k(y):=[u(x+\ro_k y)-u(x)]/\ro_k$
converge  to $Ly$ in $L^1(B_1(0);\Rn)$; on the other hand, (\rf{wwdi})
implies that the same functions converge in measure in $B_1(0)$ to
$\nabla u(x)y$, hence $L=\nabla u(x)$.
Since the sequence $\ro_k$ is arbitrary,
this proves that $\alpha_\ro(x)$ converges to $\nabla u(x)$
as $\ro\to 0^+$, so that (\rf{bolll}) implies (\rf{boll}).
\endproof




\rem{\thm{inomega}} {Since the approximate differentiability and the 
stronger condition~(\rf{boll})
are local properties, they continue to hold, more generally, for any open 
set $\Om\subset\Rn$ and any function $u\in BD(\Om)$.
If $\Om$ is bounded with Lipschitz boundary, then, extending $u$ by 
$0$ outside $\Om$ (see Section~\rf{secbd})
and applying Theorem~\rf{bddiff} to the 
extension we obtain the estimate
%
$$
{\cal L}^n\bigl(\{x\in\Om:|\nabla u(x)|>t\}\bigr)\leq
{c(n,\Omega)\over t}\bigl[\, |Eu|(\Om) +\int_\Om |u(x)|\,dx\,\bigr]\,.
$$
}
\negbigskip
\medskip
%
\rem{\thm{korn}} {We also notice that our argument provides a proof of 
Korn's inequality
%
$$
\int_{\Rn}|\nabla u(x)|^p\,dx
\leq k(n)\int_{\Rn}|\E u(x)|^p\,dx
$$
%
in the case $E^s u=0$ and $\E u\in L^p(\Rn)$ with $p>1$. Indeed, using
Remark~\rf{lpbou}, in this
case we have a strong $L^p$ estimate on 
$\sup\limits_{\ro>0}|\beta_\ro|$.
Moreover, if we approximate~$u$ by convolutions and
apply the estimate above to the regularized functions,
we obtain also that $u$ belongs to the Sobolev space $H^{1,p}(\Rn;\Rn)$.
}
%


\intro{Acknowledgements}
%
Work partially supported by MURST 40\%. This work is part of the 
following projects:  ``Phase Transition and 
Surface Tension", contract CHRX-CT94-0608 of the program HCM of the
Commission of the European Communities; ``EURHomogenization'', 
Contract SC1-CT91-0732 of the Program SCIENCE of the Commission of 
the European  Communities; ``Relaxation and Homogenization Methods
in the Study of Composite Materials'', Progetto Strategico CNR, 1995,
``Matematica per la Tecnologia e la Societ\`a''.

We thank R.V. Kohn for having given us a copy of his Ph.D. Thesis 
[\rf{Ko}].

\tolerance=1000\hfuzz=15pt

\intro{References}

\def\interrefspace{\smallskip}

{\ninepoint\frenchspacing

%\item{[\bib{Alb}]}ALBERTI  G.:
%Rank One Property for Derivatives of
%Functions with Bounded Variation.
%{\it Proc. Roy. Soc. Edinburgh Sect. A\/} {\bf 123} (1993),
%239--274.
%\interrefspace

\item{[\bib{Am89a}]}AMBROSIO  L.:
A Compactness Theorem for a Special Class of Functions
of Bounded Variation.
{\it Boll. Un. Mat. Ital.\/} {\bf 3-B} (1989), 857--881.
\interrefspace

\item{[\bib{Am89}]}AMBROSIO  L.:
Variational Problems in $SBV$.
{\it Acta Appl. Math.\/} {\bf 17} (1989), 1--40.
\interrefspace

\item{[\bib{Am90}]}AMBROSIO  L.:
Existence Theory for a New Class of Variational Problems.
{\it Arch. Rational Mech. Anal.\/} {\bf 111} (1990), 291--322.
\interrefspace

%\item{[\bib{ADM92}]}AMBROSIO  L., DAL MASO G.:
%On the Representation in $BV(\Om;{\bf R}^m)$ of Quasi-Convex
%Integrals, {\it Journal of Functional Analysis\/} {\bf 109} (1992),
%76--97.
%\interrefspace

%\item{[\bib{AmFuPa}]}AMBROSIO  L., FUSCO  N., PALLARA  D.:
%Partial Regularity of Free Discontinuity Sets I \hskip -2pt I.
%Preprint Univ. Pisa, 1994.
%\interrefspace

%\item{[\bib{AmPa}]}AMBROSIO  L., PALLARA  D.:
%Partial Regularity of Free Discontinuity Sets I.
%Preprint Univ. Pisa, 1994.
%\interrefspace

\item{[\bib{ATa}]}AMBROSIO L., TORTORELLI V.M.:
Approximation of Functionals Depending on Jumps by Elliptic
Functionals via $\Gamma$-Convergence.
{\it Comm. Pure Appl. Math.\/} {\bf 43} (1990), 999--1036.
\interrefspace

\item{[\bib{ATb}]}AMBROSIO L., TORTORELLI V.M.:
On the Approximation of Free Discontinuity Problems.
{\it Boll. Un. Mat. Ital.\/} {\bf 6-B} (1992), 105--123.
\interrefspace

\item{[\bib{AnGi}]}ANZELLOTTI  G., GIAQUINTA  M.:
Existence of the Displacement Field for an Elasto-plastic Body
Subject to Henky's Law and Von Mises Yield Condition.
{\it Manuscripta Math.\/} {\bf 32} (1980), 101--136.
\interrefspace

\item{[\bib{BeCo93}]}BELLETTINI  G., COSCIA  A.:
Una caratterizzazione dello spazio $BD(\Om)$ per sezioni 
unidimensionali.
{\it Seminario di Analisi Matematica, Dipartimento di Matematica
dell'Universit\`a di Bologna\/}, 1993.
\interrefspace

\item{[\bib{BeCoDM95}]}BELLETTINI  G., COSCIA  A.,  DAL MASO  G.:
Special Functions of Bounded Deformation.
Preprint SISSA., Trieste, 1995.
\interrefspace

\item{[\bib{BB}]}BOUCHITTE G., BUTTAZZO G.:
New Lower Semicontinuity Results for Non Convex Functionals
Defined on Measures.
{\it Nonlinear Anal.\/} {\bf 15} (1990), 679--692.
\interrefspace

\item{[\bib{BBB}]}BRAIDES A., BOUCHITTE G., BUTTAZZO G.:
Relaxation of Free Discontinuity Problems.
{\it J. Reine Angew. Math.\/} {\bf 458} (1995), 1--18.
\interrefspace

%\item{[\bib{B}]}BUTTAZZO G.:
%{\it Semicontinuity, Relaxation and Integral Representation
%in the Calculus of Variations\/}.
%Pitman Res. Notes Math. Ser. 207, Longman, Harlow, 1989.
%\interrefspace

\item{[\bib{Cha95}]}CHAMBOLLE A.: Image Segmentation by Variational 
Methods: Mumford and Shah Functional and the Discrete 
Approximations. {\it SIAM J. Appl. Math.\/} {\bf 55} (1995), 827--863.
\interrefspace

\item{[\bib{DM}]}DAL MASO G.:
{\it An Introduction to $\Gamma$-Convergence.}
Birkh\"auser, Boston, 1993.
\interrefspace

\item{[\bib{DG88}]}DE GIORGI  E.:
Free Discontinuity Problems in Calculus of Variations.
{\it Analyse Math\'ematique et Applications (Paris, 1988)\/},
 55--62, {\it Gauthier-Villars, Paris\/}, 1988.
\interrefspace

\item{[\bib{DGAm88}]}DE GIORGI  E., AMBROSIO  L.:
Un nuovo tipo di funzionale del calcolo delle variazioni.
{\it Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.\/}
{\bf 82} (1988), 199--210.
\interrefspace

\item{[\bib{DGCaLe}]}DE GIORGI  E., CARRIERO  M., LEACI  A.:
Existence Theorem for a Minimum Problem with Free Discontinuity Set.
{\it Arch. Rational Mech. Anal.\/} {\bf 108} (1989), 195--218.
\interrefspace


\item{[\bib{EvGa93}]}EVANS  L.C., GARIEPY R.:
{\it Measure Theory and Fine Properties of Functions\/}.
CRC Press, Boca Raton, 1992.
\interrefspace

\item{[\bib{Fed68}]}FEDERER  H.:
{\it Geometric Measure Theory\/}.
Springer-Verlag, Berlin, 1969.
\interrefspace

\item{[\bib{FoMu92}]}FONSECA  I., M\"ULLER  S.:
Quasiconvex Integrands and Lower Semicontinuity in $L^1$.
{\it SIAM J. Math. Anal.\/}
{\bf 23} (1992), 1081-1098.
\interrefspace

\item{[\bib{Giu84}]}GIUSTI  E.:
{\it Minimal Surfaces and Functions of Bounded Variation\/}.
Birkh\"auser, Basel, 1983.
\interrefspace

\item{[\bib{Ha}]}HAJ\L ASZ P.: On Approximate Differentiability of 
Functions with Bounded Deformation. To appear.
\interrefspace

\item{[\bib{Ko}]}KOHN  R.V.: New Estimates for Deformations in Terms of
Their Strains.
Ph.D. Thesis, Princeton Univ., 1979.
\interrefspace

\item{[\bib{MaStCh}]}MATTHIES  H., STRANG  G., CHRISTIANSEN  E.:
The Saddle Point of a Differential Program.
{\it Energy Methods in Finite Element Analysis\/},
Wiley, New York, 1979.
\interrefspace

\item{[\bib{Mi64}]}MIRANDA  M.:
Superfici cartesiane generalizzate ed insiemi di perimetro
localmente finito sui prodotti cartesiani.
{\it Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)\/} {\bf 18} (1964), 515--542.
\interrefspace

\item{[\bib{Mi67}]}MIRANDA  M.:
Sul comportamento delle successioni convergenti di frontiere minimali.
{\it Rend. Sem. Mat. Univ. Padova\/} {\bf 38} (1967), 238--257.
\interrefspace


\item{[\bib{MoSo}]}MOREL  J.M., SOLIMINI  S.:
{\it Variational Models in Image Segmentation\/}.
Birkh\"auser, Boston, 1995.
\interrefspace

\item{[\bib{MuNiSh93}]}MUMFORD  D., NITZBERG  M., SHIOTA  T.:
{\it Filtering, Segmentation and Depth\/}.
Springer-Verlag, Berlin, 1993.
\interrefspace

\item{[\bib{MuSh85}]}MUMFORD  D., SHAH  J.:
Boundary Detection by Minimizing Functionals, I.
{\it Proc. IEEE Conf. on Computer Vision and Pattern Recognition
(San Francisco, 1985)\/}.
\interrefspace

\item{[\bib{MuSh89}]}MUMFORD  D., SHAH  J.:
Optimal Approximation by Piecewise Smooth Functions
and Associated Variational Problems.
{\it Comm. Pure Appl. Math.\/} {\bf 42} (1989), 577--685.
\interrefspace

\item{[\bib{NiMu}]}NITZBERG  M., MUMFORD  D.:
The 2.1-D Sketch.
{\it International Conference on Computer Vision.
Computer Society Press, IEEE,\/} 1990.
\interrefspace

\item{[\bib{Orn62}]}ORNSTEIN  D.:
A Non-Inequality for Differential Operators in the $L^1$ Norm.
{\it Arch. Rational Mech. Anal.\/} {\bf 11} (1962), 40--49.
\interrefspace

\item{[\bib{Si83}]}SIMON  L.:
{\it Lectures on Geometric Measure Theory\/}.
Proceedings of the Centre of Mathematical Analysis, Canberra, 1984.
\interrefspace

\item{[\bib{St}]}STEIN E.M.:
{\it Singular Integrals and Differentiability Properties of
Functions\/.} Princeton University Press, New Jersey, 1970.
\interrefspace

\item{[\bib{Su}]}SUQUET  P.M.:
Existence et r\'egularit\'e des solutions des
\'equations de la plasticit\'e parfaite.
{\it C. R. Acad. Sci. Paris S\'er. A\/} {\bf 286} (1978),
1201--1204.
\interrefspace

\item{[\bib{Su79}]}SUQUET  P.M.:
Un espace fonctionnel pour les \'equations de la plasticit\'e.
{\it Ann. Fac. Sci. Toulouse\/} {\bf 1} (1979), 77--87.
\interrefspace

\item{[\bib{Te85}]}TEMAM  R.:
{\it Probl\`emes math\'ematiques en plasticit\'e\/}.
Gauthier-Villars, Paris, 1983.
\interrefspace

\item{[\bib{TeSt80}]}TEMAM  R., STRANG  G.:
Functions of Bounded Deformation.
{\it Arch. Rational Mech. Anal.\/}
{\bf 75} (1980), 7--21.
\interrefspace

\item{[\bib{Zie89}]}ZIEMER W. P.:
{\it Weakly Differentiable Functions\/}.
Springer-Verlag, Berlin, 1989.

\par
}

\bye