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\begin{document}
\null
\vspace{1cm}
\begin{center}
{\Large \bf THE CALIBRATION METHOD \\

\


FOR FREE-DISCONTINUITY PROBLEMS \\

\

ON VECTOR-VALUED MAPS \\

}

\vspace{1.5cm}

{\large Maria Giovanna \sc Mora}

\vspace{.6cm}


{\large S.I.S.S.A.}\\ \vspace{.1cm}
{\large via Beirut 2-4, 34014 Trieste, Italy}\\ \vspace{.1cm}
{\large e-mail: \tt mora@sissa.it}



\vspace{3cm}

\begin{minipage}[t]{11.5cm}
{\normalsize
\noindent
{\bf Abstract.} The calibration method is a classical minimality criterion,
which has been recently adapted to functionals with free discontinuities
by Alberti, Bouchitt\'e, Dal Maso. In this paper we present a
further generalization of this theory to functionals defined on vector-valued
maps.}\vspace{.3cm}

\noindent{\normalsize {\em Mathematics Subject Classification (2000):}  
49K10 (49Q20).} \vspace{.3cm}

\noindent{\normalsize {\em Keywords:} free-discontinuity
problems, sufficient condition for minimality, calibration method.}
\end{minipage} \end{center}

\setcounter{page}{0}
\thispagestyle{empty}


\vfill

\begin{center}
{\large Ref. S.I.S.S.A. 61/2001/M  (July 2001)}
\end{center}

\pagebreak
\clearpage
\phantom{a}
\setcounter{page}{0}
\thispagestyle{empty}

\newpage
\pagestyle{myheadings}
\markboth{\rm \small\hfill Maria Giovanna \sc Mora \hfill}{\hfill
\rm\small The calibration method for free-discontinuity problems on
vector-valued maps \hfill}

\thispagestyle{plain}
\null
\vspace{.5cm}
\begin{center}
{\large \bf THE CALIBRATION METHOD FOR FREE-DISCONTINUITY 

\vspace{.5cm}

PROBLEMS ON VECTOR-VALUED MAPS}
\vspace{1cm}

{\normalsize  Maria Giovanna \sc Mora}\\
{\normalsize  S.I.S.S.A.}\\ 
{\normalsize via Beirut 2-4, 34014 Trieste, Italy}\\ 
{\normalsize e-mail: \tt mora@sissa.it}
\vspace{1cm}

\begin{minipage}[t]{11.5cm}
{\footnotesize
\noindent
{\bf Abstract.} The calibration method is a classical minimality criterion,
which has been recently adapted to functionals with free discontinuities
by Alberti, Bouchitt\'e, Dal Maso. In this paper we present a
further generalization of this theory to functionals defined on vector-valued
maps.}\vspace{.2cm}

\noindent{\footnotesize {\em Mathematics Subject Classification (2000):}  
49K10 (49Q20).} \vspace{.2cm}

\noindent{\footnotesize {\em Keywords:} free-discontinuity
problems, sufficient condition for minimality, calibration method.}
\end{minipage}
\end{center}

\vspace{1cm}




\section{Introduction}\label{sec:int}

Many variational problems arising from several branches of
applied analysis lead to consider minimum problems for functionals of the form
\begin{equation}\label{funct-int}
F(u)=\int_{\Omega}f(x,u,\nabla u)\,dx
+\int_{S_u}\psi(x,u^-,u^+,\nu_u)\,d{\cal H}^{n-1},
\end{equation}
where $\Omega$ is a bounded open subset of $\real^n$ with Lipschitz boundary, 
$f:\Omega{\times}\real^N{\times}\real^{nN}\to [0,+\infty]$, and
$\psi:\Omega{\times}\real^N{\times}\real^N{ \times }\sph^{n-1}\to [0,+\infty]$
are Borel functions, $\sph^{n-1}:=\{v\in\real^n: \ |v|=1\}$, ${\cal H}^{n-1}$ 
is the $(n-1)$-dimensional Hausdorff measure, and the unknown function $u$
belongs to the space $[SBV(\Omega)]^N$ of special functions of bounded
variation in $\Omega$; $\nabla u$ denotes the approximate gradient of $u$,
$S_u$ is the set of essential discontinuity points of $u$, $\nu_u$ is the
approximate unit normal vector to $S_u$, and $u^-,u^+$ the approximate limits
of $u$ on the two sides of $S_u$.

Following a terminology by De Giorgi, variational problems of this form are
denoted by {\em free-disconti\-nui\-ty problems}. A typical example is
provided by the so-called Mumford-Shah functional, introduced in
\cite{Mum-Sha} in the context of image segmentation, which can be written as
$$MS(u):=\int_{\Omega}|\nabla u|^2 dx +\alpha{\cal H}^{n-1}(S_u)
+\beta\int_{\Omega}|u-g|^2\, dx,$$
where $g$ is a function in $[L^{\infty}(\Omega)]^N$, and $\alpha>0$ and
$\beta\geq 0$ are constants.

One of the main features of functionals of the form
(\ref{funct-int}) is that they are in general not convex; therefore, all
conditions which can be obtained by infinitesimal variations are necessary for
minimality, but in general not sufficient.

In \cite{Alb-Bou-DM2} Alberti, Bouchitt\'e, Dal Maso have proposed a
sufficient condition for minimality, based on the {\em calibration method}, 
which leads to many applications (see \cite{Alb-Bou-DM2}, \cite{DM-Mor-Mor}, 
\cite{mg}, \cite{Mor-Mor}, \cite{Mor}). In the form presented in
\cite{Alb-Bou-DM2}, this minimality criterion is limited to scalar
functions.

The purpose of this paper is to develop a similar theory for functionals on
vector-valued maps.

In order to describe the basic idea behind the calibration method, let us focus
our attention on {\em Dirichlet minimizers} of $F$, that is minimizers with
prescribed boundary values. Given a candidate $u$, if we are able to construct
a functional $G$ which is invariant on the class of functions having the same
boundary values and satisfies 
\begin{equation}\label{cnec}
G(u)=F(u), \qquad \hbox{and}\qquad G(v)\leq F(v)\quad \text{for every admissible $v$,}
\end{equation}
then $u$ is a Dirichlet minimizer of $F$. Indeed, if such a functional
exists, for every $v$ with the same boundary values as $u$ we have
$$F(u)=G(u)=G(v)\leq F(v).$$

In \cite{Alb-Bou-DM2} the r\^ole of $G$ is carried out by the flux of a
suitable divergence-free vectorfield $\phi:\Omega{\times}\real\to\real^{n+1}$
through the complete graph of $v$, which is defined as the union of the usual
graph of $v$ and of all segments joining $(x,v^-(x))$ and
$(x,v^+(x))$ with $x$ ranging in $S_v$.
Since $\phi$ is divergence-free, the flux is clearly invariant with respect to
the boundary values, while suitable further conditions on $\phi$ 
guarantee (\ref{cnec}) (see Remark~\ref{rmk:scal-vect}).

In this paper this theory is generalized to the vectorial case by considering
a different kind of invariant functional: the calibration is no longer 
a vectorfield, but a pair of functions $(\s,\so)$, where
$\s:\Omega{\times}\real^N\to\real^n$ is suitably regular (more
precisely, globally Lipschitz and piecewise $C^1$), while $\so$ belongs to
$L^1(\Omega)$; the comparison functional for $F$ is given by 
\begin{equation}\label{sso}
-\int_{\df\Omega}\la \s(x,v),\nu_{\df\Omega}\ra\,d{\cal H}^{n-1}
+\int_{\Omega}\so(x)\, dx,
\end{equation}
where $\nu_{\df\Omega}$ is the inner unit normal to $\df\Omega$.
It is clear that the functional (\ref{sso}) is constant on the functions having
the same values at $\df\Omega$. Moreover, by the divergence theorem we can 
rewrite (\ref{sso}) as 
$$\int_{\Omega}\,d\mu_v +\int_{\Omega}\so(x)\, dx,$$
where $\mu_v$ is the divergence (in the sense of distributions) of the
composite function $\s(\cdot,v(\cdot))$. A generalized version of the chain
rule in $BV$ (which is proved in Section~\ref{sec:pre}) implies that
$$\mu_v=\left( [\dive_x \s](x,v) + \la(D_y \s(x,v))^{\tau},\nabla v\ra
\right){\cal L}^n +  \la\s(x,v^+)-\s(x,v^-),\nu_v\ra\, {\cal H}^{n-1}\lfloor
S_v,$$ where $[\dive_x \s]$ denotes the divergence of $\s$ with respect to the
variable $x\in\Omega$, and $(D_y\s)^{\tau}$ the transpose of the Jacobian
matrix of $\s$ with respect to the variable $y\in\real^N$.
Therefore the functional (\ref{sso}) turns out to be equal to
$$\int_{\Omega}\left([\dive_x \s](x,v) + \la(D_y
\s(x,v))^{\tau},\nabla v\ra + \so(x)\right) dx +
\int_{S_v} \la\s(x,v^+)-\s(x,v^-),\nu_v\ra\, d{\cal H}^{n-1}.$$
By comparing this expression with the functional
(\ref{funct-int}), we find pointwise conditions on $\so$, $\s$, and the
derivatives of $\s$, which guarantee (\ref{cnec}), and then the Dirichlet
minimality of a given $u$. For a precise statement of these conditions see
Lemma~\ref{lm-main-0} and Lemma~\ref{lm-main}.

This formulation is related to classical field theory for multiple integrals
of the form
$$F_0(u)=\int_{\Omega}f(x,u,\nabla u)\, dx.$$
In this framework a sufficient condition for the minimality of a candidate
$u\in [C^1(\Omega)]^N$ is obtained by comparing $F_0$ with the integral of a
null-lagrangian of divergence type, which is constructed starting from a
suitably defined slope field ${\cal P}$, called {\em Weyl field\/}, and a
function $\s\in[C^2(\Omega{\times}\real^N)]^n$, the {\em eikonal map}
associated with ${\cal P}$ (cf., e.g., \cite{Gia-Hil}). In
Section~\ref{sec:gh}  we prove that, under suitable assumptions on $f$ and
$\psi$, whenever a Weyl field exists for a function $u\in [C^1(\Omega)]^N$
(so that $u$ is a Dirichlet minimizer for $F_0$), then there exists a
calibration for $u$ with respect to the functional $F$ (which is given by the
eikonal map $\s$ and by $\so\equiv 0$), so $u$ is also a Dirichlet minimizer
for $F$ among $SBV$ functions.

Some examples and applications are presented in  
Section~\ref{sec:ex}. In Examples~\ref{ex:harm}, 
\ref{ex:step}, \ref{ex:tj}, and \ref{ex:part} we deal with minimizers of the  
Mumford-Shah functional, and we generalize some results proved in 
\cite{Alb-Bou-DM2} for the scalar case. A purely vectorial example is given by 
Example~\ref{ex:grif}, where we study the minimality of continuous solutions of the Euler 
equations for a functional arising in fracture mechanics, 
which can be defined only on maps from $\Omega\subset \real^n$ into $\real^n$. 

Finally, we point out that, as mentioned in \cite{Alb-Bou-DM2}, one could try 
to generalize the calibration theory from the
scalar case to the vectorial one by replacing divergence-free
vectorfields by closed $n$-forms on $\Omega{\times}\real^N$, acting on the
graphs of the functions $v$, viewed as suitably defined surfaces in
$\Omega{\times}\real^N$.  This could lead to the idea that our choice of
writing the calibration in terms of the pair $(\s,\so)$ is somehow
restrictive. This is not at all the case, since the existence of a calibration 
expressed via differential forms implies the existence of a calibration
expressed in terms of a pair $(\s,\so)$, as shown in Section~\ref{sec:df}. 

The plan of the paper is the following: in Section~2 we fix the notation and
we recall some basic results from $BV$ functions theory; in Section~3 we present the
calibration method for functionals of the form (\ref{funct-int}) on
vector-valued maps; Section~4 is devoted to the link between calibration theory
and classical field theory; Section~5 contains some examples and applications; finally, 
in Section~6 we reformulate the
theory of calibrations in terms of differential forms and show that this
formulation does not lead to new results.


\section{Notation and preliminary results}\label{sec:pre}

In this section we fix the notation and we recall some results from $BV$
functions theory.

Given $x,y\in\real^n$, we denote their scalar product by $\la x,y \ra$, and
the euclidean norm of $x$ by $|x|$. Given $a,b\in\real$, the maximum and the
minimum of $\{a,b\}$ are denoted by $a\lor b$ and $a\land b$, respectively.

In the following $\Omega$ is a fixed bounded open subset of $\real^n$ with
Lipschitz boundary, $\nu_{\df\Omega}$ is its inner unit normal, while $U$ is a 
closed subset of $\ol{\Omega}{\times}\real^N$. The
letter $x$ usually denotes the variable in $\Omega$ (or $\real^n$), while $y$
or $z$ is the variable in $\real^N$.

We say that a function $u:\Omega\to\real^N$ has {\em bounded variation in
$\Omega$}, and we write $u\in [BV(\Omega)]^N$, if $u$ belongs to
$[L^1(\Omega)]^N$ and its distributional derivative $Du$ is a finite Radon
$\real^{nN}$-valued measure in $\Omega$.

Since $\Omega$ has Lipschitz boundary, we can speak about the {\em
trace} of $u$ on $\df\Omega$, which belongs to $L^1(\df\Omega,{\cal H}^{n-1})$
and will be still denoted by $u$.

The {\em singular set} $S_u$ of $u$ is defined as the set of all points where
$u$ does not admit an approximate limit; $S_u$ is countably $({\cal
H}^{n-1},n-1)$-rectifiable, that is, it can be covered, up to an ${\cal
H}^{n-1}$-negligible set, by countably many $C^1$-hypersurfaces. Moreover, at
${\cal H}^{n-1}$-a.e. $x_0\in S_u$ there exists a triplet
$(u^+(x_0),u^-(x_0),\nu_u(x_0))\in\real^N{\times}\real^N{\times}\sph^{n-1}$
such that $u^+(x_0)\neq u^-(x_0)$, $\nu_u(x_0)$ is normal to $S_u$ in an
approximate sense, and $$\lim_{r\to 0^+} \frac{1}{{\cal
L}^n(B^{\pm}_r( x_0))}\int_{B^{\pm}_r(x_0)}
|u(x)- u^{\pm}(x_0)|\,dx =0,$$
where $B^{\pm}_r(x_0)$ is the intersection of the ball of radius $r$
centred at $x_0$ with the half-plane $\{ x\in \real^n: 
\pm\la x-x_0,\nu_u(x_0)\ra \geq 0 \}$.
The triplet $(u^+(x_0),u^-(x_0),\nu_u(x_0))$ is uniquely determined up to a
permutation of $(u^+(x_0),u^-(x_0))$ and a change of sign of $\nu_u(x_0)$. 

The measure $Du$ can be decomposed as 
$$Du=D^a u+D^j u+ D^c u,$$
where $D^a u$ is the absolutely continuous part with respect to ${\cal L}^n$,
$D^j u$ is the {\em jump part} and satisfies $D^j u=(u^+ -u^-)\otimes \nu_u
\lfloor {\cal H}^{n-1}$, and finally $D^c u$ is the so-called {\em Cantor
part}. The density of $D^a u$ with respect to ${\cal L}^n$ is denoted by
$\nabla u$ and agrees with the {\em approximate gradient} of $u$. Moreover, we
call $D^a u+D^c u$ the {\em diffuse part} of the derivative of $u$ and we
denote it by $\tilde Du$.

We say that a function $u:\Omega\to\real^N$ is a {\em special function of
bounded variation}, and we write $u\in[SBV(\Omega)]^N$, if $u\in[BV(\Omega)]^N$
and $D^c u=0$.

Finally, we define as {\em graph of $u$} the set $\reg{u}:=\{(x,\tilde u(x)): \
x\in\Omega\setminus S_u\}$, where $\tilde u$ is the precise representative
of $u$ (which is defined everywhere on $\Omega\setminus S_u$ by definition of
$S_u$). 

For more details on $BV$ functions theory see \cite{Amb-Fus-Pal}.

We conclude this section with the proof of a generalized chain rule in $BV$. 
If $u\in[BV(\Omega)]^N$ and $\s$ is a Lipschitz continuous
function from $\real^N$ into $\real^M$, it is known that $\s\circ u$ belongs to
$[BV(\Omega)]^M$. When in addition $\s\in [C^1(\real^N)]^M$, the following
chain rule formula can be written:
\begin{equation}\label{fml}
\begin{array}{rcl}
\tilde{D}(\s\circ u)& = & D\s(\tilde u(x))\tilde Du(x) \quad \text{on
$\Omega\setminus S_u$}, \vspace{.2cm}\\ D^j (\s\circ u)& = &
[\s(u^+)-\s(u^-)]\otimes \nu_u{\cal H}^{n-1}\lfloor S_u,
\end{array}\end{equation}
(see Theorem~3.96 in \cite{Amb-Fus-Pal}).
Following an idea by \cite{Mur-Tro}, we generalize formula (\ref{fml}) to the
case of a function $\s$, which may depend also on the variable $x$ and is
only piecewise $C^1$ in the sense of the following definition. 

\begin{dfn}\label{dfn-pwc1}
We say that a Lipschitz continuous function $\s:U\to \real^M$ is
{\em piecewise $C^1$} if $\s$ can be written as
\begin{equation}\label{pwc1}
\s(x,y)=\sum_{\alpha\in A}\sa(x,y)1_{U^{\alpha}}(x,y),
\end{equation}
where $(U^{\alpha})_{\alpha\in A}$ is a finite family of pairwise disjoint 
Borel sets such that $\cup_{\alpha\in A}U^{\alpha}=U$, and
$(\sa)_{\alpha\in A}$ is a family of Lipschitz continuous
functions belonging to $[C^1(\ol{\Omega}{\times}\real^N)]^M$.   
\end{dfn} 


\begin{lm}\label{lm-chain}
Let $\s:U\to \real^M$ be a Lipschitz continuous
function, piecewise $C^1$ in the sense of Definition~\ref{dfn-pwc1}, and satisfying
(\ref{pwc1}), and let $u\in [BV(\Omega)]^N$ be such that $\reg{u}\subset U$.
Then, $v:=\s(\cdot,u(\cdot))$ belongs to $[BV(\Omega)]^M$ and 
\begin{eqnarray}\label{pint}
\tilde{D}v & = & \sum_{\alpha\in A}1_{U^{\alpha}}(x,\tilde
u)(D_x \sa(x,\tilde u){\cal L}^n+D_y \sa(x,\tilde u)\tilde Du) \quad
\text{on $\Omega\setminus S_u$}, \\ D^j v & = & [\s(x,u^+)-\s(x,u^-)]\otimes
\nu_u{\cal H}^{n-1}\lfloor S_u. \label{jmp} \end{eqnarray}
\end{lm}

\proof
Since the function $\s$ can be extended to a Lipschitz function on the whole 
$\ol{\Omega}{\times}\real^N$, by Theorem~3.101 in \cite{Amb-Fus-Pal} we have
that the function $v=\s(\cdot,u(\cdot))$ belongs to $[BV(\Omega)]^M$ and formula
(\ref{jmp}) holds true.

Since $\sa$ is globally Lipschitz and of class $C^1$ on
$\ol{\Omega}{\times}\real^N$,  by Theorem~3.96 in \cite{Amb-Fus-Pal} the
function $v^{\alpha}:=\sa(\cdot,u(\cdot))$ belongs to $[BV(\Omega)]^M$
and the diffuse part of its derivative satisfies the following equality:
\begin{equation}\label{class}
\tilde D v^{\alpha} = D_x \sa(x,\tilde u){\cal L}^n+ D_y
\sa(x,\tilde u)\tilde D u. \end{equation}
Consider now the set
$$E^{\alpha}:=\{x\in\Omega\setminus S_u: \ \tilde v(x)=\tilde v^{\alpha}(x)\}.$$
Since $v$ and $v^{\alpha}$ are both $BV$ functions and their
jump sets are both contained in $S_u$, by the locality property of the
derivative of a $BV$ function (see Remark~3.93 in \cite{Amb-Fus-Pal}) it
follows that $Dv \lfloor E^{\alpha}=Dv^{\alpha}\lfloor E^{\alpha}$.
Since $E^{\alpha}\subset\Omega\setminus S_u$, the previous equality can 
be rewritten as
\begin{equation}\label{coin}
\tilde Dv\lfloor E^{\alpha}=\tilde Dv^{\alpha} \lfloor
E^{\alpha}.
\end{equation} 
If we define
$$P^{\alpha}:=\{ x\in \Omega\setminus S_u: \ (x,\tilde u(x))\in
U^{\alpha} \},$$  
since $P^{\alpha}\subset E^{\alpha}$, by
(\ref{coin}) and (\ref{class}) we can conclude that $$\tilde Dv\lfloor
P^{\alpha}= \tilde Dv^{\alpha}\lfloor P^{\alpha}= D_x \sa(x,\tilde
u){\cal L}^n\lfloor P^{\alpha}+ D_y \sa(x,\tilde u)\tilde D u\lfloor
P^{\alpha},$$ 
which immediately gives formula (\ref{pint}).
\qed


\section{Calibrations for functionals on vector-valued maps}\label{sec:vcal}


In this section we develop the theory of calibrations for functionals 
depending on vector-valued maps $u\in [SBV(\Omega)]^N$
and of the following form:
\begin{equation}\label{funct}
F(u)=\int_{\Omega}f(x,u,\nabla u)\,dx
+\int_{S_u}\psi(x,u^-,u^+,\nu_u)\,d{\cal H}^{n-1},
\end{equation}
where $f:\Omega{\times}\real^N{\times}\real^{nN}\to [0,+\infty]$, and
$\psi:\Omega{\times}\real^N{\times}\real^N{ \times }\sph^{n-1}\to [0,+\infty]$
are Borel functions, and $\psi$ satisfies the condition
$\psi(x,y,z,\nu)=\psi(x,z,y,-\nu)$.

In the sequel we will refer to the following definition of minimizers of $F$.


\begin{dfn}\label{dfn-min}
An {\em absolute minimizer of (\ref{funct}) in $\Omega$} is a function $u\in
[SBV(\Omega)]^N$ such that $F(u)\leq F(v)$
for all $v\in [SBV(\Omega)]^N$, while a {\em Dirichlet minimizer in
$\Omega$} is a function $u\in [SBV(\Omega)]^N$ such that $F(u)\leq F(v)$
for all $v\in [SBV(\Omega)]^N$ with the same trace on 
$\df\Omega$ as $u$. 
A function $u\in [SBV(\Omega)]^N$ is a {\em $U$-minimizer}  
if the graph of $u$ is contained in $U$ and $F(u)\leq F(v)$
for all $v\in [SBV(\Omega)]^N$ whose graph is contained in $U$, 
while $u$ is a {\em Dirichlet $U$-minimizer} if we add the 
requirement that the competing functions $v$ have the same trace on 
$\df\Omega$ as $u$. 
\end{dfn}

Before proving the key lemma about calibrations, we fix some further notation.

Given two functions $\s:U\to\real^n$, and $u:\Omega\to\real^N$, 
we will denote the divergence of the composite function 
$\s(\cdot,u(\cdot))$ by $\dive_x[\s(x,u(x))]$,
while the divergence of $\s$ with respect to the variable $x$ computed at 
the point $(x,u(x))$ by $[\dive_x \s](x,u(x))$. 
The Jacobian matrix of $\s$ with respect to $y$ will be denoted by
$D_y\s$ and its transpose by $(D_y\s)^{\tau}$.
Note that if $\s$ and $u$ are sufficiently regular,
$$\dive_x[\s(x,u(x))]=[\dive_x \s](x,u)+\la(D_y\s(x,u))^{\tau},\nabla u\ra.$$

We call $f^{\ast}$ and $\sdf f$ the convex conjugate and the subdifferential
of $f$ with respect to the last variable. We recall that, if $g$ is a function
from $\real^{nN}$ into $[0,+\infty]$, the subdifferential of $g$ at the point
$\xi\in\real^{nN}$ is defined as the set of all matrices $\eta\in\real^{nN}$
such that $g(\xi)+\la \eta, \zeta-\xi\ra\leq g(\zeta)$ for every
$\zeta\in\real^{nN}$. It is well known that $\la\xi,\eta\ra -
g^{\ast}(\eta)\leq g(\xi)$ for every $\xi,\eta\in\real^{nN}$, and the equality
holds if and only if $\eta\in\sdf g(\xi)$.
Moreover, if $g$ is convex and differentiable, then $\sdf g(\xi)=\{\df_{\xi}
g(\xi) \}$.
Using these properties, we can prove the following lemma.


\begin{lm}\label{lm-main-0}
Let $F$ be the functional defined in (\ref{funct}). Let $\s\in
[C^1(\ol{\Omega}{\times}\real^N)]^n$ be Lipschitz continuous and let $\so\in
L^{1}(\Omega)$. Assume that the following conditions are satisfied:  
\begin{description}
	\item{(a)} $[\dive_x\s](x,y)+\so(x) \leq 
-f^{\ast}(x,y,(D_y\s(x,y))^{\tau})$ for ${\cal L}^n$-a.e. $x\in\Omega$ and for
every $y$ with $(x,y)\in U;$   	 	
	\item{(b)} $\la\s(x,z)-\s(x,y),\nu\ra\leq
\psi(x,y,z,\nu)$ for ${\cal H}^{n-1}$-a.e. $x\in\Omega$, for every
$\nu\in\sph^{n-1}$, and for every $y,z$ with $(x,y)\in U,$ $(x,z)\in U.$ 
\end{description} 
Then for every $u\in [SBV(\Omega)]^N$ such that $\reg{u}\subset U$ we have
that $\dive_x [\s(\cdot,u(\cdot))]$ is a Radon measure on $\Omega$, which will
be denoted as $\mu_u$, and
\begin{equation}\label{1ineq} 
F(u)\geq \int_{\Omega}d\mu_u+ \int_{\Omega}\so(x)\, dx. 
\end{equation}
Moreover, equality holds in (\ref{1ineq}) for a given $u$ if and only if
\begin{description}
	\item{(a')} $[\dive_{x}\s](x,u)+\so(x)=
-f^{\ast}(x,u,(D_y\s(x,u))^{\tau})$  and 
$(D_y\s(x,u))^{\tau}\in\sdf f(x,u,\nabla u)$ for ${\cal
L}^n$-a.e. $x\in\Omega;$ 
	\item{(b')} $\la\s(x,u^+)-\s(x,u^-),\nu_u\ra=
\psi(x,u^-,u^+,\nu_u)$ for ${\cal H}^{n-1}$-a.e. $x\in S_u,$  
\end{description}
where $u$, $u^{\pm}$, $\nabla u$, and $\nu_u$ are always computed at $x$. 
\end{lm}

\proof
Let $u\in [SBV(\Omega)]^N$ be such that $\reg{u}\subset U$. By
Theorem~3.96 in \cite{Amb-Fus-Pal} the function
$S(\cdot,u(\cdot))$ belongs to $[SBV(\Omega)]^n$, and therefore, its
divergence is a Radon measure on $\Omega$. Moreover, we
have that $$D_{x_i}[\s_i(x,u)] = \df_{x_i}\s_i(x,u)\,{\cal L}^n 
+D_y \s_i(x, u)\df_{x_i}u 
\, {\cal L}^n 
+ [\s_i(x,u^+)-\s_i(x,u^-)](\nu_u)_i\,
{\cal H}^{n-1}\lfloor S_u,$$
so that the measure $\mu_u$ can be written as
\begin{eqnarray*}
\lefteqn{\mu_u(x) \hspace{.2cm} = \hspace{.2cm} 
\sum_{i=1}^n D_{x_i}[\s_i(x,u(x))]} \\
& = & [\dive_x \s](x,u)\,{\cal
L}^n+\sum_{i} D_y \s_i(x, u)\df_{x_i}u \,{\cal
L}^n+\sum_i [\s_i(x,u^+)-\s_i(x,u^-)](\nu_u)_i \,{\cal H}^{n-1}\lfloor S_u \\
& = & [\dive_x \s](x,u)\,{\cal L}^n+\la(D_y
\s(x, u))^{\tau}, \nabla u\ra \,{\cal
L}^n+\la\s(x,u^+)-\s(x,u^-),\nu_u\ra\,{\cal H}^{n-1}\lfloor S_u,
\end{eqnarray*}
and the functional at the right-hand side of (\ref{1ineq}) has the following
expression 
\begin{multline}
\int_{\Omega}d\mu_u +\int_{\Omega}\so(x)\, dx =
\int_{\Omega}\left([\dive_x \s](x,u) + \la(D_y
\s(x,u))^{\tau},\nabla u\ra + \so(x)\right) dx  \\ +
\int_{S_u} \la\s(x,u^+)-\s(x,u^-),\nu_u\ra\, d{\cal H}^{n-1}. \label{ending}
\end{multline} 
Using assumption (a) we obtain that for ${\cal L}^n$-a.e.
$x\in\Omega$
\begin{eqnarray*}
[\dive_x \s](x,u) + \la(D_y \s(x,u))^{\tau},\nabla u\ra+ \so(x)
& \leq & - f^{\ast}(x,u,(D_y\s(x,u))^{\tau}) + \la(D_y \s(x,u))^{\tau},
\nabla u\ra \\ & \leq & f(x,u,\nabla u), \end{eqnarray*}
and consequently
\begin{equation}\label{fstep}
\int_{\Omega}\left( [\dive_x \s](x,u) + \la(D_y
\s(x,u))^{\tau},\nabla u\ra+ \so(x)\right) dx 
\leq \int_{\Omega} f(x,u,\nabla u)\, dx.
\end{equation}
Moreover, equality holds in (\ref{fstep}) if and only if 
$(D_y \s(x,u))^{\tau}\in\sdf f(x,u,\nabla u)$ and
$[\dive_x \s](x,u)+\so(x)=-f^{\ast}(x,u,
(D_y\s(x,u))^{\tau}),$ which is condition (a').

As for the second integral in (\ref{ending}), condition (b) implies that
\begin{equation}\label{sstep}
\int_{S_u} \la\s(x,u^+)-\s(x,u^-),\nu_u\ra\, d{\cal H}^{n-1}\leq
\int_{S_u} \psi(x,u^-, u^+,\nu_u)\, d{\cal H}^{n-1}.
\end{equation}
Moreover, equality holds in (\ref{sstep}) if and only if (b') is satisfied.

The statement follows now from (\ref{ending}), (\ref{fstep}), and
(\ref{sstep}).
\qed

The assumption of $C^1$-regularity for $\s$ is often too
strong for many applications. We prove now a new version of
Lemma~\ref{lm-main-0} under weaker regularity assumptions for $\s$.

\begin{lm}\label{lm-main}
Let $F$ be the functional defined in (\ref{funct}). Let 
$\s:U\to\real^n$ be a Lipschitz continuous function, piecewise $C^1$ in
the sense of Definition~\ref{dfn-pwc1}, and satisfying (\ref{pwc1}). 
Let $\so\in L^{1}(\Omega)$.  Assume that the following conditions are
satisfied:   \begin{description}
	\item{(a)} $[\dive_x \sa](x,y)+\so(x) \leq
-f^{\ast}(x,y,(D_y\sa(x,y))^{\tau})$ for every $\alpha\in A$, for ${\cal
L}^n$-a.e. $x\in\Omega$, and for every $y\in\real^N$ with $(x,y)\in
U^{\alpha};$
   	 \item{(b)} $\la\s(x,z)-\s(x,y),\nu\ra\leq
\psi(x,y,z,\nu)$ for ${\cal H}^{n-1}$-a.e. $x\in\Omega$, for every
$\nu\in\sph^{n-1}$, and for every $y,z$ with $(x,y)\in U,$ $(x,z)\in U.$      
\end{description} 
Then for every $u\in [SBV(\Omega)]^N$ such that $\reg{u}\subset U$ we have
that $\dive_x [\s(\cdot,u(\cdot))]$ is a Radon measure on $\Omega$, which will
be denoted as $\mu_u$, and
\begin{equation}\label{1ineq*} 
F(u)\geq \int_{\Omega}d\mu_u+ \int_{\Omega}\so(x)\, dx. 
\end{equation}
Moreover, equality holds in (\ref{1ineq}) for a given $u$ if and only if
\begin{description}
	\item{(a')} $[\dive_{x}\sa](x,u)+\so(x)=
-f^{\ast}(x,u,(D_y\sa(x,u))^{\tau})$  and 
$(D_y\sa(x,u))^{\tau}\in\sdf f(x,u,\nabla u)$ for every $\alpha\in A$, for
${\cal L}^n$-a.e. $x\in\Omega$ such that $(x,u(x))\in U^{\alpha}:$ 
	\item{(b')} $\la\s(x,u^+)-\s(x,u^-),\nu_u\ra=
\psi(x,u^-,u^+,\nu_u)$ for ${\cal H}^{n-1}$-a.e. $x\in S_u,$  
\end{description}
where $u$, $u^{\pm}$, $\nabla u$, and $\nu_u$ are always computed at $x$. 
\end{lm}

\proof
Let $u\in [SBV(\Omega)]^N$ be such that $\reg{u}\subset U$. By
Lemma~\ref{lm-chain} the function $S(\cdot,u(\cdot))$ belongs to 
$[SBV(\Omega)]^n$, and therefore, its divergence is a Radon measure on
$\Omega$. By (\ref{pint}) and (\ref{jmp}) we have that the measure $\mu_u$
can be written as 
\begin{multline}\nonumber
\mu_u(x) = \sum_{\alpha\in A}1_{U^{\alpha}}(x,u)[\dive_x \sa](x,u)\,{\cal
L}^n+\sum_{\alpha\in A}1_{U^{\alpha}}(x,u) \la(D_y\sa(x, u))^{\tau}, \nabla
u\ra \,{\cal L}^n \\ 
+\la\s(x,u^+)-\s(x,u^-),\nu_u\ra\,{\cal H}^{n-1}\lfloor
S_u. \end{multline}
The proof of Lemma~\ref{lm-main-0} can be now repeated simply replacing 
$[\dive_x \s]$ with $\sum_{\alpha\in
A}1_{U^{\alpha}}[\dive_x \sa]$, and 
$D_y \s$ with $\sum_{\alpha\in A}1_{U^{\alpha}}D_y
\sa$.
\qed



\begin{dfn}\label{dfn-cal}
We say that a pair of functions $(\s,\so)$ is a {\em calibration for $u\in
[SBV(\Omega)]^N$ on $U$ with respect to the functional (\ref{funct})} if
$\s:U\to\real^n$ is a Lipschitz continuous function, piecewise $C^1$ in 
the sense of Definition~\ref{dfn-pwc1}, $\so\in
L^{1}(\Omega)$, and they satisfy assumptions (a), (b), (a'), and (b')
in Lemma~\ref{lm-main}. 
\end{dfn}

We can now prove the main result of this section.

\begin{thm}\label{thm-main}
Let $u$ be a function in $[SBV(\Omega)]^N$ whose graph is contained in
$U$. Assume that there exists a calibration $(\s,\so)$ for $u$ on 
$U$ with respect to the functional (\ref{funct}).  
Then $u$ is a Dirichlet $U$-minimizer of $F$. If,
in addition, the normal component of $\s$ at $\df
U\cap(\df\Omega{\times}\real^N)$ does not depend on $y$, namely  
for ${\cal H}^{n-1}$-a.e. $x\in\df\Omega$ there exists a constant
$a(x)\in\real$ such that 
\begin{equation}\label{norm0}
\la\s(x,y),\nu_{\df\Omega}(x)\ra=a(x) \qquad \text{for every $y$ such that
$(x,y)\in U$,} \end{equation} 
then $u$ is also an absolute $U$-minimizer of $F$.
\end{thm}

\proof
Let $v$ be a function in $[SBV(\Omega)]^N$ such that $v=u$ on $\df\Omega$ and
$\reg{v}\subset U$. Then the definition of the measure $\mu_v$ and 
the divergence theorem imply that
$$\int_{\Omega}d\mu_v =
-\int_{\df \Omega}\la\s(x,v),\nu_{\df\Omega}\ra\, d{\cal H}^{n-1}.$$
If $v$ has the same trace on $\df\Omega$ as $u$, from this identity it follows
that
\begin{equation}\label{pass}
\int_{\Omega}d\mu_v =\int_{\Omega}d\mu_u,
\end{equation}
and by applying Lemma~\ref{lm-main} we obtain 
$$F(v) \geq \int_{\Omega}d\mu_v +\int_{\Omega}\so(x) \, dx  
= \int_{\Omega}d\mu_u+ \int_{\Omega}\so(x)\, dx = F(u).$$
We have thus proved that $u$ is a Dirichlet $U$-minimizer of $F$.

If we assume, in addition, that (\ref{norm0}) holds true, then 
$\int_{\Omega}d\mu_v=-\int_{\Omega}a\, d{\cal H}^{n-1}$ for every $v\in
[SBV(\Omega)]^N$ whose graph is contained in $U$; so, the equality
(\ref{pass}) is fulfilled even if the traces of $u$ and $v$ on $\df\Omega$
differ. This proves that $u$ is an absolute $U$-minimizer of $F$. 
\qed

\begin{rmk}\label{rmk:scal-vect}
It is natural to wonder what is the link in the case $N=1$ between 
our vectorial theory and the calibration method for the
scalar case, developed in \cite{Alb-Bou-DM2}, which involves a
divergence-free vectorfield $\phi$ (see \cite[Theorem~3.10]{Alb-Bou-DM2}). 

Let $N=1$.
Let us suppose that $(\s,\so)$ is a calibration for $u$ and assume furthermore
that $\s$ is globally $C^1$. 
Take the vector field $\phi=(\phi^{x},\phi^{y}):U\to\real^n{\times}\real$ 
defined by $\phi^{x}(x,y):=\df_{y}\s(x,y)$ and $\phi^{y}(x,y):=-[\dive_{x} 
\s](x,y)-\so(x)$. Then $\phi$ satisfies all the assumptions of Theorem~3.10 in
\cite{Alb-Bou-DM2}. Indeed, by Remark~2.3 in \cite{Alb-Bou-DM2} $\phi$ is approximately
regular on $U$ in the sense of Definition~2.1 in \cite{Alb-Bou-DM2};
moreover, $\phi$ satisfies all the following conditions:
\begin{description} 
	\item{(A)} $\phi^{y}(x,y)\geq f^{\ast}(x,y,\phi^{x}(x,y))$
for ${\cal  L}^{n}$-a.e. $x\in\Omega$ and for every $y\in\real$ with $(x,y)\in
U$;
	\item{(A\!')} $\phi^{x}(x,u)\in\sdf f(x,u,\nabla u)$
and $\phi^{y}(x,u)=f^{\ast}(x,u,\phi^{x}(x,u))$ for ${\cal 
L}^{n}$-a.e. $x\in\Omega$;
 	\item{(B)} $\displaystyle 
\int_{y_{1}}^{y_{2}}\la\phi^{x}(x,y),\nu\ra\,dy\leq 
\psi(x,y_{1},y_{2},\nu)$ for ${\cal H}^{n-1}$-a.e. $x\in\Omega$, 
every $\nu\in \sph^{n-1}$, and every $y_{1}<y_{2}$ such that
$(x,y_1)\in U$, $(x,y_2)\in U$; 	
	\item{(B')} $\displaystyle 
\int_{u^{-}}^{u^{+}}\la\phi^{x}(x,y),\nu_{u}\ra\,dy=
\psi(x,u^{-},u^{+},\nu_{u})$ for ${\cal H}^{n-1}$-a.e. $x\in S_{u}$;
	\item{(C)} $\phi$ is divergence-free in the sense of
distributions  in $U$.
\end{description}
Conditions (A) and (A\!') directly follow from (a) and (a'),
respectively. By definition of $\phi$ we have that
$$\int_{y_{1}}^{y_{2}}\phi^{x}(x,y)\,dy = \s(x,y_{2})-\s(x,y_{1}),$$
so that conditions (b) and (b') imply (B) and (B'), respectively. If $\s$ is
$C^2$ and $\so$ is $C^1$, then it is trivial that $\phi$ is $C^1$ and
$\dive\phi=0$; in the general case, condition (C) can be obtained by an
approximation argument. 

Analogously it is easy to see that, if $\phi$ is a bounded Lipschitz 
$C^1$-vectorfield satisfying (A), (B), (A\!'), (B'), and (C), then we can
construct a calibration $(\s,\s_0)$. Take indeed 
$$\s(x,y):=\int_{\tau(x)}^y\phi^x(x,t)\,dt \qquad \text{and} \qquad
\s_0(x):=\la \phi^x(x,\tau(x)),\nabla\tau(x)\ra - \phi^y(x,\tau(x)),$$ 
where $\tau$ is any smooth function satisfying $(x,\tau(x))\in U$ for every
$x\in\ol{\Omega}$. 
\end{rmk}

\section{An application related to classical field theory}\label{sec:gh}

We recall now some classical results from field theory for multiple 
integrals of the form
\begin{equation}
F_{0}(u)=\int_{\Omega}f(x,u,\nabla u)\, dx,
\end{equation}
where $u\in [C^{1}(\ol{\Omega})]^{N}$ and $f\in 
C^{2}(\ol{\Omega}{\times}\real^{N}{\times}\real^{nN})$.

We will call {\em extremals} of $F_{0}$ or {\em \fex} the 
solutions $u$ of class $C^{2}$ of the Euler equations for the integral $F_{0}$,
i.e. \begin{equation}\label{eul-eq}
\sum_{i=1}^n
D_{x_{i}}[\df_{\xi_{ij}} f(x,u(x),\nabla u(x))]-\df_{u_{j}}
f(x,u(x),\nabla u(x))=0, \qquad 1\leq j\leq N. \end{equation}

In the classical field theory for multiple integrals several
sufficient conditions for the minimality of an \fex have been proposed. 
Among the others, we recall Weyl field
theory, which is strictly related to the calibration theory for vector-valued
functionals and ensures that a given \fex $u$ is in fact a minimizer of $F_0$
among all functions of class $C^1$, with
the same boundary values as $u$ and whose graph is contained in a suitable
neighbourhood of the graph of $u$. It consists in the construction of a 
suitable slope field ${\cal P}$, called Weyl field, and of a smooth function
$\s$, called the eikonal map associated with the field, satisfying the system
of equations (\ref{weq1}) -- (\ref{weq2}). 
This set of conditions arises from the comparison of $F_0$ with an invariant
functional of divergence type, which is nothing but the functional
$$\int_{\Omega}\dive_x[\s(x,v)]\, dx,$$
where $\s$ is the eikonal map (see, e.g. \cite[Chapter~7, Section~4]{Gia-Hil}). 

We will show via calibrations that,
if a Weyl field exists for an \fex $u$ (and then there exists a neighbourhood
$U$ of the graph of $u$ such that $u$ minimizes $F_{0}$ among
$C^1$-functions with the same boundary values as $u$ and with graph contained
in $U$), then $u$ is also a Dirichlet $U$-minimizer of the functional
(\ref{funct}) in the sense of Definition~\ref{dfn-min}, provided $U$ is a
sufficiently small neighbourhood of $G_u$ and the function $\psi$ satisfies
the estimate (\ref{psi-est}); moreover, if $\s$ is the eikonal map associated
with the Weyl field, then the pair $(\s,\so)$ with $\so\equiv 0$ is a
calibration for $u$ on $U$.  

\begin{dfn}
Let $U$ be a closed domain in $\ol{\Omega}{\times}\real^N$.
A mapping $p:U\to U{\times}\real^{nN}$ is called a {\em slope
field} on $U$ if it is of class $C^1$ and of the form
$$p(x,y)=(x,y,{\cal P}(x,y)) \qquad \text{for every $(x,y)\in U$;}$$ 
we denote ${\cal P}(x,y)=({\cal P}_{ij}(x,y))$ as the {\em slope function} of
the field $p$. We say that {\em a map $u\in [C^1(\ol{\Omega})]^N$ fits the
slope field $p$} if $\reg{u}\subset U$ and
$$\df_{x_i}u_j(x)={\cal P}_{ij}(x,u(x)) \qquad \text{for every
$x\in\ol{\Omega}$.}$$ 
Finally, a slope field $p$ is said to be a {\em Weyl
field} if there is a map $\s\in [C^2(U)]^n$ such that $\{\s,\cal{P}\}$
solves the Weyl equations:
\begin{eqnarray}\label{weq1}
[\dive_x \s](x,y) & = & f(x,y,{\cal P}(x,y))-\la{\cal
P}(x,y),\df_{\xi}f(x,y,{\cal P}(x,y))\ra, \\
\df_{y_j}\s_i(x,y) & = & \df_{\xi_{ij}}f(x,y,{\cal P}(x,y)).\label{weq2}
\end{eqnarray}
The function $\s$ is called the {\em eikonal map} associated with $p$.
\end{dfn}

The main results in Weyl field theory can be stated as follows. For a proof we
refer to \cite{Gia-Hil}.

\begin{thm}\label{gia-hil}
\begin{itemize}
	\item[\bf (1)] 
Assume that the function $f$ satisfies 
$$f(x,y,\xi)-f(x,y,\eta)-\la\xi-\eta,\df_{\xi}f(x,y,\eta)\ra\geq 0$$
for every $(x,y)\in U$ and $\xi,\eta\in\real^{nN}$, and let
$u\in [C^{2}(\ol{\Omega})]^{N}$ fit a Weyl field 
$p:U \to U{\times}\real^{nN}$ with the eikonal map 
$\s:U\to\real^{n}$. Then $u$ is a minimizer of $F_{0}$ among all
$v\in[C^1(\ol{\Omega})]^N$ such that $v\vert_{\df\Omega}=u\vert_{\df\Omega}$
and $\reg{v}\subset U$; in particular, $u$ is an \fex\!\!.
Moreover, if there is a constant $\mu>0$ such  that 
\begin{equation}\label{str-conv}
\sum_{i,j,h,k}\df^2_{\xi_{ij}\xi_{hk}}f(x,y,\xi)\zeta_{ij}\zeta_{hk}\geq\mu
|\zeta|^2 \qquad \forall (x,y)\in\ol{\Omega}{\times}\real^N,\, \xi,\zeta\in\real^{nN}, 
\end{equation}
then $u$ is a strict minimizer of $F_{0}$ in the same class. 
	\item[\bf (2)] 
Vice-versa, if $f$ satisfies the strict convexity
condition  (\ref{str-conv}), then every \fex fits at least locally a Weyl field
and is therefore locally minimizing $F_{0}$. In other words,
for every $x_0\in\Omega$ there exist $\eps>0$ and an open neighbourhood
$A$ of $x_0$ such that $u$ minimizes $F_{0}$ among all $v\in[C^1(\ol{A})]^N$
such that $v\vert_{\df A}=u\vert_{\df A}$ and
$\reg{v}\subset\{(x,y)\in \ol{A}{\times}\real^N: \ |y-u(x_0)|\leq\eps\}$.  
\end{itemize}
\end{thm}

\

Let us now state and prove a similar result for free-discontinuity problems.

\begin{thm}\label{mor}
Let $f:\ol{\Omega}{\times}\real^N{\times}\real^{nN}\to[0,+\infty]$  be a
function of class $C^{2}$ satisfying (\ref{str-conv}) and let
$\psi:\Omega{\times}\real^N{\times}\real^N{\times}\sph^{n-1}\to [0,+\infty]$
be a Borel function satisfying 
\begin{equation}\label{psi-est} 
\psi(x,y,z,\nu)\geq c\, \theta(|y-z|), 
\end{equation}  
where $c$ is a positive constant, while $\theta$ is such that 
$\lim_{t\to 0^+}\frac{\theta(t)}{t}=+\infty$. Let $u$ be an \fex\!\!.
Then for every $x_0\in\Omega$ there exist $\eps>0$, an open neighbourhood
$A$ (with Lipschitz boundary) of $x_0$, and a pair $(\s,\so)$ such that
$(\s,\so)$ is a calibration for $u$ with respect to the functional
(\ref{funct}) on the set \begin{equation}\label{zc}
U:=\{(x,y)\in \ol{A}{\times}\real^N: \ |y-u(x_0)|\leq\eps\};
\end{equation}
therefore $u$ is a Dirichlet $U$-minimizer of the functional (\ref{funct}). 
\end{thm}

\proof 
Let $u$ be an \fex\!\!.
By the second part of Theorem~\ref{gia-hil} for every $x_0\in\Omega$ there
exist $\eps>0$ and an open neighbourhood $A$ (with Lipschitz boundary) of $x_0$
such that $u$ fits a Weyl fiel in the set (\ref{zc}).
Denote the Weyl field by $p(x,y)=(x,y,{\cal P}(x,y))$ and the eikonal map
associated with $p$ by $\s$. 

We claim that, if we take $\so(x):=0$ for every
$x\in\Omega$, then the pair $(\s,\so)$ is a calibration for $u$ on $U$
with respect to the functional $F$ defined in (\ref{funct}), provided $\eps$ is
sufficiently small. Let us prove it. Since $f$ is convex, for every 
$\eta\in\real^{nN}$ we have that 
$$f(x,y,\eta)-\la\eta,\df_{\xi}f(x,y,\eta)\ra= 
-f^{\ast}(x,y,\df_{\xi}f(x,y,\eta));$$
this fact, jointly with (\ref{weq1}), implies that
\begin{eqnarray}  \nonumber
[\dive_{x}\s](x,y) & = & - f^{\ast}(x,y,\df_{\xi}f(x,y,{\cal P}(x,y))) \\
& = & - f^{\ast}(x,y, (D_{y}\s(x,y))^{\tau}), \label{aeq} 
\end{eqnarray}
where the second equality follows from (\ref{weq2}).
Therefore, condition (a) is satisfied.

Condition (a') follows from (\ref{aeq}) and (\ref{weq2}), using the fact 
that $u$ fits the field ${\cal P}$, hence ${\cal P}(x,u(x))=\nabla u(x)$ for
every $x\in\ol{\Omega}$.

If we call $L$ the $L^{\infty}$-norm of the Jacobian matrix of $\s$ on
$U$, then we have that
\begin{equation}\label{w0}
\la \s(x,z)-\s(x,y),\nu \ra \leq L\,|z-y|
\end{equation}
for every $x\in\ol{\Omega}$, $y,z\in\real^N$ such that $(x,y)\in U$,
$(x,z)\in U$, and $\nu\in\sph^{n-1}$. By the assumption on the function
$\theta$ there exists $\delta>0$ such that $\theta(t)\geq Lt/c$ for every
$t\in(0,\delta)$; then from (\ref{psi-est}) it follows that
\begin{equation}\label{w00}
\psi(x,y,z,\nu)\geq L|y-z| \qquad \text{for $|y-z|<\delta$.}
\end{equation}
Taking $\eps<\delta/2$, from (\ref{w0}) and (\ref{w00}) we have that
condition (b) is satisfied.

Since $S_{u}=\emptyset$, condition (b') is trivial. 

The conclusion follows now from Theorem~\ref{thm-main}.
\qed

As made precise in the next proposition, when the function $f$ depends only
on the variables $x,\xi$, we are able to prove the minimality of an \fex $u$ on
the whole domain $\ol{\Omega}$ and to give an estimate of the width $\eps$ of
the neighbourhood of $\reg{u}$ where the minimality holds.

\begin{prp}\label{prp-easy}
In addition to the assumptions of Theorem~\ref{mor}, suppose that $f=f(x,\xi)$.
Let $u$ be an \fex\!\!. For every $(x,y)\in\ol{\Omega}{\times}\real^N$ define
\begin{equation}\label{calgen}
\s(x,y):=[\df_{\xi}f(x,\nabla u(x))]^{\tau}(y-u(x))+\sigma(x),
\end{equation}
where $\sigma:\ol{\Omega}\to\real^n$ is a solution of the equation $\dive
\sigma=f(x,\nabla u)$.
Then the pair $(\s,\so)$ with $\so\equiv 0$ is a calibration for $u$ with
respect to the functional (\ref{funct}) on the set
\begin{equation}\label{set-gn} 
U:=\{(x,y)\in \ol{\Omega}{\times}\real^N: \
|y-u(x)|\leq \eps(x)\}, \end{equation}
where
\begin{equation}\label{eps-est}
\eps(x)< \frac{1}{2}\inf\left\{ t>0:\ c\,\frac{\theta(t)}{t}< |\df_{\xi}
f(x,\nabla u(x))|\right\},
\end{equation}
and $c,\theta$ are the quantities appearing in (\ref{psi-est}).
Therefore $u$ is a Dirichlet $U$-minimizer of the functional
(\ref{funct}).
\end{prp}

\proof 
Note that by the assumption on $\theta$, the infimum in (\ref{eps-est}) is
strictly positive for every $x\in \ol{\Omega}$.

Let us prove that $(\s,\so)$ satisfies all the conditions in
Lemma~\ref{lm-main-0}.

By direct computations we have that $D_y\s(x,y)=[\df_{\xi}f(x,\nabla
u)]^{\tau}$; using the Euler equations (\ref{eul-eq}), the definition of
$\sigma$, and the convexity of $f$, we find out that
\begin{eqnarray*}
[\dive_x\s](x,y) & = & \sum_{ij}D_{x_i}(\df_{\xi_{ji}}f(x,\nabla u))(y_j-u_j)
-\la [\df_{\xi}f(x,\nabla u)]^{\tau},\nabla u \ra +\dive\sigma \\
& = & -\la [\df_{\xi}f(x,\nabla u)]^{\tau},\nabla u \ra + f(x,\nabla u) \\
& = & -f^{\ast}(x,[\df_{\xi}f(x,\nabla u)]^{\tau}).
\end{eqnarray*}
Conditions (a) and (a') are therefore satisfied.

By the definition of $\s$ we obtain 
$$|\s(x,z)-\s(x,y)|\leq |\df_{\xi}f(x,\nabla u(x))|\cdot|z-y|;$$
since $|z-y|\leq 2\eps(x)$, (\ref{eps-est}) implies that
$$|\df_{\xi}f(x,\nabla u(x))|\cdot|z-y|\leq c\, \theta(|z-y|);$$
so condition (b) follows now from (\ref{psi-est}).

Condition (b') is trivial since $S_u$ is empty. This concludes the proof.
 
We notice that the thesis can be proved also in the following way: if we define
${\cal P}(x,y):=\nabla u(x)$ for every  $(x,y)\in\ol{\Omega}{\times}\real^N$,
it is easy to see that the field $p(x,y):=(x,y,{\cal P}(x,y))$ is a Weyl
field,  $\s$ is the eikonal map associated with $p$, and $u$ fits $p$. Then we
can follow the proof of Theorem~\ref{mor}: the check of condition (a), (a'),
(b') remains the same, while the estimate on the size of $\eps(x)$ is given
by a more careful proof of condition (b). 
\qed
 
\begin{rmk}\label{rmk-ext}
When the functional (\ref{funct}) satisfies some special further conditions,
it is enough to prove the Dirichlet minimality of a given $u$ on a
neighbourhood of its graph to conclude that $u$ is in
fact a Dirichlet minimizer on the whole cylinder $\ol{\Omega}{\times}\real$,
reducing the domain $\Omega$ if needed. For istance, in addition to the
assumptions of Proposition~\ref{prp-easy}, suppose that the two following
conditions are satisfied: \begin{itemize}
 	 	\item[(1)] $f(x,\xi)\geq
f(x,(I-e_j\!\otimes\! e_j)\,\xi)$ for every $x\in\Omega$, $\xi\in\real^{nN}$,
$j=1,\ldots,N$, where $\{e_1,\ldots,e_N\}$ is the canonical basis of $\real^N$;
	\item[(2)] $\psi(x,y,z,\nu)\geq \psi(x,T_{a}^{b}(y),T_{a}^{b}(z),\nu)$
for every $(x,y)\in\Omega{\times}\real^N$, $\xi\in\real^{nN}$,
$\nu\in\sph^{n-1}$, $a,b\in\real^{N}$, where we have set
$$T_{a}^{b}:\real^{N}\to\real^{N},\qquad
(T_{a}^{b})_{j}(y):=(y_j\land a_j)\lor b_{j}.$$
\end{itemize}
If $u$ is an \fex\!\!, then by Proposition~\ref{prp-easy} we know that $u$ is
a Dirichlet $U$-minimizer of $F$, where $U$ is the set (\ref{set-gn}). 
We want to show that for every $x_0\in \Omega$ there exists an open
neighbourhood $A$ (with Lipschitz boundary) of $x_0$ such that $u$ is a
Dirichlet minimizer of $F$ in $A$.

First of all, we can find an open neighbourhood $A$ (with Lipschitz boundary)
of $x_0$ and two vectors $m,M\in\real^N$ such that $|M-m|<\eps(x)$ for every
$x\in \ol{A}$ and \begin{equation}\label{bounds} m_j\leq u_j(x)\leq M_j\qquad
\forall x\in \ol{A},\ 1\leq j\leq N. \end{equation}
Let $v$ be a function in $[SBV(A)]^N$ with the same trace on $\df A$ as
$u$ and define $\hat v:=T_m^M(v)$, which still belongs to $[SBV(A)]^N$. 
Note that $\nabla \hat v_j=1_{\{m_j< v_j<M_j\}}\nabla v_j$ for every $j$, so
that, if we call $J_0(x)$ the set of all indexes $j$ such that $v_j(x)\not\in
(m_j,M_j)$, the matrix $\nabla \hat v(x)$ can be written as
$$\nabla \hat v(x)=\nabla v(x) -\sum_{j\in J_0}(e_j\!\otimes\! e_j)\,
\nabla v(x).$$ 
By using iteratively condition (1), we obtain that $f(x,\nabla \hat v)\leq
f(x,\nabla v)$, which implies
\begin{equation}\label{cx}
\int_A f(x,\nabla \hat v)\, dx \leq \int_A f(x,\nabla v)\, dx.
\end{equation} 
Since $S_{\hat v}\subset S_v$, and $\hat v^-=T_m^M(v^-)$, $\hat
v^+=T_m^M(v^+)$ on $S_{\hat v}$, by condition (2) we obtain
\begin{equation}\label{cx1} 
\int_{S_{\hat v}\cap A} \psi(x,\hat v^-,\hat v^+,\nu_{\hat v})\, d{\cal
H}^{n-1} \leq \int_{S_v \cap A} \psi(x,v^-,v^+,\nu_v)\, d{\cal H}^{n-1}. 
\end{equation} 
On the other hand, by (\ref{bounds}) the function
$\hat v$ has the same trace on $\df A$ as $u$, and its graph is contained in
the set $$\{(x,y)\in \ol{A}{\times}\real^N:\ |y-u(x)|\leq\eps(x)\}.$$
Since $u$ is a Dirichlet minimizer on this
set, we have that
\begin{equation}\label{cy}
\int_A f(x,\nabla u)\, dx \leq \int_A f(x,\nabla \hat v)\, dx
+\int_{S_{\hat v}} \psi(x,\hat v^-,\hat v^+,\nu_{\hat v})\, d{\cal
H}^{n-1}.
\end{equation}
Therefore by (\ref{cx}), by (\ref{cx1}), and (\ref{cy}), $u$ is a Dirichlet
minimizer of $F$ in $A$. 

The same result can be achieved by calibration: indeed, we can extend the
function $\s$ in (\ref{calgen}) to the whole $\ol{\Omega}{\times}\real^N$
simply by taking $\hat\s(x,y):=\s(x,T_m^M(y))$; it is easy to see that
assumptions (1) -- (2) guarantee that the pair $(\hat\s,\so)$ provides a
calibration for $u$ on $\ol{A}{\times}\real^{N}$.

We conclude the remark with some comments on conditions (1) -- (2). 
Condition (1) ensures that the functional decreases when any row of the matrix
$\nabla u$ is annihilated, which is what occurs when a component of $u$ is
truncated. For istance, (1) is fulfilled for $f(\xi)=\sum_{ij}\varphi_{ij}
(\xi_{ij})$ where $\varphi_{ij}$ are convex and positive, and
$\varphi_{ij}(0)=0$.  As for condition (2), note that it is satisfied whenever
$\psi$ depends on $y,z$  only through the distance $|z-y|$.
\end{rmk}


\section{Some further applications}\label{sec:ex}
 
In this section we present some examples and applications. In Examples~\ref{ex:harm},
\ref{ex:step}, \ref{ex:tj}, and \ref{ex:part} we deal with minimizers of the 
Mumford-Shah functional, and we generalize some results proved in
\cite{Alb-Bou-DM2} for the scalar case.   Example~\ref{ex:grif} is a purely
vectorial example, since it involves a functional arising in fracture mechanics
which can be defined only on maps from $\Omega\subset \real^n$ into $\real^n$.

\begin{ex}\label{ex:harm}
Let $u:\Omega\to\real^N$ be a harmonic function. It is well known that
$u$ is an extremal of the functional $\int_{\Omega}|\nabla u|^2$, and a
Dirichlet minimizer of it. We can prove via calibrations that $u$ is a
Dirichlet minimizer also of the homogeneous Mumford-Shah functional 
\begin{equation}\label{ms}
MS_0(u)=\int_{\Omega}|\nabla u|^2 dx+\alpha{\cal H}^{n-1}(S_u),
\end{equation}
if the following condition is satisfied:
\begin{equation}\label{cham}
\osc_{\Omega}u \cdot \sup_{\Omega}|\nabla u|\leq\alpha,
\end{equation}
where $\osc u$ denotes the modulus of the vector in $\real^N$ whose
components are the oscillations of the components of $u$. 
When (\ref{cham}) is not fulfilled, $u$ is still a
Dirichlet $U$-minimizer of the functional $MS_0$, where 
\begin{equation}\label{phph}
U:=\left\{(x,y)\in\ol{\Omega}{\times}\real^N:\
|y-u(x)|\leq\frac{\alpha}{4|\nabla u(x)|} \right\}.
\end{equation}

This second result directly follows from Proposition~\ref{prp-easy}, where 
$f(\xi)=|\xi|^2$ and $\psi\equiv\alpha$. Moreover, a calibration is given by $(\s,\so)$ with
$\so\equiv 0$ and 
$$\s(x,y)=2[\nabla u(x)]^{\tau}(y-u(x))+\sigma(x),$$
where $\sigma:\Omega\to\real^n$ is a solution of the equation
$\dive\sigma=|\nabla u|^2$. Since $u$ is harmonic in $\Omega$, it is easy to
see that we can take $\sigma(x):=[\nabla u(x)]^{\tau}u(x)$, so that
\begin{equation}\label{expl-cal}
\s(x,y)=2[\nabla u(x)]^{\tau}\left(y-\frac{u(x)}{2}\right).
\end{equation}

As for the Dirichlet minimality of $u$, we can show that, under the assumption
(\ref{cham}), the calibration $(\s,\so)$ can be extended to the whole
$\ol{\Omega}{\times}\real^N$, applying a similar argument to the one
used in Remark~\ref{rmk-ext}.

We recall that, in the case of the functional (\ref{ms}), conditions (a), (a'),
(b), and (b') in Lemma~\ref{lm-main} become
\begin{itemize}
	\item[(a)\phantom{'}] $[\dive_x\sg](x,y)+\so(x)\leq
-\frac{1}{4}|D_y\sg(x,y)|^2$ for every $\gamma\in A$, for
${\cal L}^n$-a.e. $x\in\Omega$, and for every $y\in\real^N$ with $(x,y)\in
U^{\gamma}$; 
	\item[(a')] $[\dive_x\sg](x,u)+\so(x)= -|\nabla u(x)|^2$
and $(D_y\sg(x,u))^{\tau}=2\nabla u(x)$ for every $\gamma\in A$, and for ${\cal
L}^n$-a.e. $x\in\Omega$ such that $(x,u(x))\in U^{\gamma}$; 	
	\item[(b)\phantom{'}] $|\s(x,z)-\s(x,y)|\leq \alpha$ for
${\cal H}^{n-1}$-a.e. $x\in\Omega$ and for every $y,z\in\real^N$ such that
$(x,y)\in U$, $(x,z)\in U$;
	\item[(b')] $\s(x,u^+)-\s(x,u^-)=\alpha\nu_u$ for ${\cal
H}^{n-1}$-a.e. $x\in S_u$,
\end{itemize}
where $\s(x,y)=\sum_{\gamma\in A}\sg(x,y)1_{U^{\gamma}}(x,y)$. 

Let $m_j$ and $M_j$ be the infimum and the supremum of $u_j$ in
$\Omega$, respectively (then $\osc u_j=M_j-m_j$). Let $T$ be the function from
$\real^N$ into $\real^N$ defined as $T_j(y)=(y_j\lor m_j/2)\land M_j/2$. Define
$$\hat\s(x,y):=2[\nabla u(x)]^{\tau}\,T\!\left(y-\frac{u(x)}{2}\right).$$
It is easy to see that $(\hat\s,\so)$ satisfies conditions (a) and (a').
Condition (b') is trivial. Finally, for every $y,z\in\real^N$ we have
\begin{equation}\label{med}
|\hat\s(x,z)-\hat\s(x,y)|\leq 2 |\nabla u(x)|\cdot|T(z-u/2)-T(y-u/2)|.
\end{equation}
Since $T_j(z-u/2)$ and $T_j(y-u/2)$ belong to the interval $[m_j/2,M_j/2]$ 
for every $1\leq j\leq N$, we deduce that $|T(z-u/2)-T(y-u/2)|\leq |M-m|/2$;
so, condition (b) follows from (\ref{med}) and (\ref{cham}).
 
These two minimality
results generalize those obtained in \cite{Alb-Bou-DM} for scalar harmonic functions.
Note that the minimality of $u$ can be proved by applying the scalar
argument to each component $u_j$, but this provides a more restrictive
condition on the size of the domains where the minimality holds.
Indeed, by the scalar result in \cite{Alb-Bou-DM}, since $u_j$ is harmonic
for every $j$, if \begin{equation}\label{cham-scal}
\osc_{\Omega}u_j \cdot \sup_{\Omega}|\nabla u_j|\leq\frac{\alpha}{N} \qquad
1\leq j\leq N, 
\end{equation}
then
$$\int_{\Omega}|\nabla u_j|^2dx \leq \int_{\Omega}|\nabla v_j|^2dx
+\frac{\alpha}{N}{\cal H}^{n-1}(S_{v_j})$$
for every $v_j\in SBV(\Omega)$ with the same boundary values as $u_j$;
summing over $j$, we obtain the Dirichlet minimality of $u$ in $\Omega$.
On the other hand, it is easy to see that condition (\ref{cham-scal}) is
stronger than (\ref{cham}). Analogous remarks hold for the Dirichlet
minimality of $u$ in a neighbourhood of its graph. 
\end{ex} 


\begin{ex}\label{ex:grif}
In this example we consider a functional related to Griffith and
Barenblatt theories of fracture mechanics of the form 
$$H(u):=\mu\int_{\Omega}|e(u)|^2dx+\frac{\lambda}{2}\int_{\Omega}(\dive
u)^2\,dx + \int_{S_u}\theta(|u^+-u^-|)\,d{\cal H}^{n-1}$$  
where $u$ is a function from $\Omega\subset \real^n$ into $\real^n$, $e(u)$
denotes the symmetrized gradient of $u$, $\theta$ is a positive function
satisfying $\lim_{t\to 0^+}\theta(t)/t=+\infty$, and $\mu,\lambda$ are real
parameters. In the context of fracture mechanics, $\Omega$ is a reference
configuration of an elastic body, possibly subject to fracture, and $u$
parameterizes its displacement; the bulk term represents the energy
relative to the elastic deformation outside the fracture, while the surface
integral is the energy needed to produce the crack.

The functional $H$ is clearly of the form (\ref{funct}) with $f(\xi)=\mu \left|
(\xi^{\tau}+\xi)/2\right|^2+\frac{\lambda}{2}({\rm tr}\,\xi)^2$ and
$\psi(y,z)=\theta(|z-y|)$. However, since the bulk term in $H$ involves only
the symmetric part of the matrix $\nabla u$, the appropriate setting for the 
minimum problem for $H$ is not exactly the space $[SBV(\Omega)]^n$, but
the space $SBD(\Omega)$ of special functions with bounded deformation (for a
complete overview on the properties of this space see \cite{Amb-Cos-DM}).
Even if the calibration method has been developed only for $SBV$ functions,
we can actually prove by calibration that, if $u$ is an $f\!$-extremal, i.e.
$u\in [C^1(\ol{\Omega})]^n\cap [C^2(\Omega)]^n$ and $u$ solves the equation
\begin{equation}\label{Eul-eq-elast}
\mu\Delta u+(\mu+\lambda)\nabla(\dive u)=0 \qquad \text{on $\Omega$,}
\end{equation}
then $u$ minimizes $H$ among all functions $v\in SBD(\Omega)$ with the same
trace on $\df\Omega$ as $u$, and whose graph is contained in the set
$$U:=\{ (x,y)\in \ol{\Omega}{\times}\real^n: \ |y-u(x)|\leq \eps(x)\},$$
where
$$\eps(x)<\frac{1}{2}\inf\left\{t>0:\ \frac{\theta(t)}{t}<
|2\mu e(u)(x)+\lambda \dive u(x)I| \right\}.$$
Indeed, since $\df_{\xi_{ij}}f(\xi)=\mu(\xi_{ji}+\xi_{ij}) +\lambda ({\rm
tr}\,\xi)\delta_{ij}$, Proposition~\ref{prp-easy} implies that 
$u$ is a Dirichlet $U$-minimizer of $H$ in the class $[SBV(\Omega)]^n$
and a calibration is given by $(\s,\so)$ with $\so\equiv 0$ and 
\begin{equation}\label{3-0}
\s(x,y)=[2\mu e(u)(x)+\lambda\dive u(x)I]\left(y-\frac{u(x)}{2} \right);
\end{equation}
this last fact follows from formula (\ref{calgen}) where we have taken
$\sigma(x):=[\mu e(u)(x)+\frac{\lambda}{2}\dive u(x)I]\, u(x)$, which is a
solution of $\dive\sigma=f(\nabla u)$ thanks to (\ref{Eul-eq-elast}).

On the other hand, we can show that the pair $(\s,\so)$ provides a calibration also in the space
$SBD(\Omega)$ in the following sense: consider the functional
$$H_1(v):=-\int_{\df\Omega}\la\s(x,v),\nu_{\df\Omega}\ra\,d{\cal H}^{n-1},$$
which is the same used as comparison functional in the proof of
Theorem~\ref{thm-main}; then, $H_1$ is well defined on
$SBD(\Omega)$, is invariant on $SBD$ functions having the same trace on
$\df\Omega$, and satisfies the equality $H_1(u)=H(u)$ and the inequality
$H_1(v)\leq H(v)$ for every $v\in SBD(\Omega)$. This implies that $u$
is a Dirichlet minimizer of the functional $H$ in the class of $SBD$
functions.

Let us prove the properties of $H_1$ stated above.
If we set for simplicity of notation $A(x):=2\mu e(u)(x)+\lambda \dive u(x)I$,
by (\ref{3-0}) the functional $H_1$ can be rewritten as
$$H_1(v)=-\frac{1}{2}\int_{\df\Omega}\la A(2v-u),\nu_{\df\Omega}\ra\,d{\cal
H}^{n-1},$$
whence it is clear that it is well defined on $SBD(\Omega)$ and invariant on
the class of functions in $SBD(\Omega)$ having the same trace on $\df\Omega$.
By the generalized Green's formula in $SBD(\Omega)$ %(here
%we use the fact that $A$ is a symmetric matrix, $A\in [C^1(\Omega)]^{n^2}\cap
%[C^0(\ol{\Omega})]^{n^2}$ and that $\df\Omega$ is locally Lipschitz),
we have that
\begin{eqnarray}\nonumber
-\frac{1}{2}\int_{\df\Omega}\la A(2v-u),\nu_{\df\Omega}\ra\,d{\cal
H}^{n-1} & = & \frac{1}{2}\int_{\Omega}\la 2v-u,\dive A\ra\, dx
+\frac{1}{2}\int_{\Omega}A\,d(2Ev-Eu) \\ & = &
\frac{1}{2}\int_{\Omega}\la A,2e(v)-e(u)\ra\, dx+ \int_{S_v}\la A(v^+
-v^-),\nu_v\ra\, d{\cal H}^{n-1}, \label{tion} \end{eqnarray}
where the last equality follows by the fact that $\dive A=0$, by the
decomposition theorem for the measures $Ev,Eu$ and by the remark that
$S_u=\emptyset$. Using the definition of the matrix $A$ and (\ref{tion}) it is
easy to see that \begin{equation}\label{2-3}
H_1(u)=\frac{1}{2}\int_{\Omega}\la A,e(u)\ra\, dx= H(u),
\end{equation}
while, using also the elementary inequality $2(\xi,\eta)\leq 
|\xi|^2+|\eta|^2$ for every $\xi,\eta\in\real^{n^2}$, we obtain
\begin{eqnarray}\nonumber
\int_{\Omega}\la A,e(v)\ra \, dx & = & 2\mu\int_{\Omega}\la e(u),e(v)\ra\, dx
+\lambda\int_{\Omega}\dive u\,\dive v\, dx \\
& \leq & \mu\int_{\Omega}|e(v)|^2dx +\frac{\lambda}{2}\int_{\Omega}(\dive v)^2dx
+H(u).\label{2-0}
\end{eqnarray}
Since the graph of $v$ is contained in $U$, 
we have that $\la A(v^+ -v^-),\nu_v\ra\leq \theta(|v^+-v^-|)$ ${\cal
H}^{n-1}$-a.e. on $S_v$, so that
\begin{equation}\label{2-1}
\int_{S_v}\la A(v^+-v^-),\nu_v\ra \, d{\cal H}^{n-1}\leq
\int_{S_v}\theta(|v^+-v^-|)\,d{\cal H}^{n-1}.
\end{equation}
By (\ref{tion}), (\ref{2-3}), (\ref{2-0}), and (\ref{2-1}), 
we deduce that $H_1(v)\leq H(v)$
for every $v\in SBD(\Omega)$ whose graph is contained in $U$.

We conclude this example by noticing that the existence of a weak solution in
$[W^{1,2}(\Omega)]^n$ for the Dirichlet boundary value problem associated with
the equation (\ref{Eul-eq-elast}) is guaranteed if $\mu>0$ and
$2\mu+3\lambda>0$; moreover, the additional requirements of regularity for $u$
are always satisfied in any open subset $\Omega'\subset \subset \Omega$ (see
\cite{Cia}).
\end{ex} 

\begin{ex}\label{ex:step}
Let $\Omega$ be a product of the form $(0,a){\times}V$, where $V$ is a regular
domain in $\real^{n-1}$, and let $u$ be the step function defined as 
$u(x):=0$ for $0<x_1<c$, and $u(x)=h$ for $c<x_1<a$, where $c\in(0,a)$ and
$h\in\real^N$, $h\neq 0$. Then, $u$ is a Dirichlet minimizer of the
Mumford-Shah functional (\ref{ms}) in $\Omega$ if $|h|^2\geq a\alpha$.

This result generalizes Example~4.12 in \cite{Alb-Bou-DM}, where $u$ is a
scalar step function. 

We prove the statement by calibration.
Let $\{e_1,\ldots, e_n\}$ be the canonical basis of $\real^n$. A
calibration for $u$ is given by the pair $(\s,\so)$ with $\so\equiv 0$ and
\begin{equation}\label{100cal}
\s(x,y):= \begin{cases}
0 & \text{if $\la y,\frac{h}{|h|}\ra \leq\frac{\lambda}{2}\la x,e_1\ra$,} \\
2\lambda\left(\la y,\frac{h}{|h|}\ra-\frac{\lambda}{2}\la x,e_1\ra \right)e_1 
& \text{if $\frac{\lambda}{2}\la x,e_1\ra\leq
\la y,\frac{h}{|h|}\ra \leq\frac{\lambda}{2}\la x,e_1\ra +
\frac{\lambda}{2}a$,} \\  a\lambda^2 e_1 & \text{if
$\la y,\frac{h}{|h|}\ra\geq\frac{\lambda}{2}\la x,e_1\ra +
\frac{\lambda}{2}a$,}  \end{cases} \end{equation}
where $\lambda:=\sqrt{\alpha/a}$.
Some direct computations show that
\begin{eqnarray*}
|D_y\s(x,y)|^2 & = & \begin{cases}
4\lambda^2 & \text{if $\frac{\lambda}{2}\la x,e_1\ra\leq
\la y,\frac{h}{|h|}\ra \leq\frac{\lambda}{2}\la x,e_1\ra +
\frac{\lambda}{2}a$,} \\
0 & \text{otherwise,}
\end{cases} \\
\dive \s(x,y) & = & \begin{cases}
-\lambda^2 & \text{if $\frac{\lambda}{2}\la x,e_1\ra\leq
\la y,\frac{h}{|h|}\ra \leq\frac{\lambda}{2}\la x,e_1\ra +
\frac{\lambda}{2}a$,} \\
0 & \text{otherwise,}
\end{cases}
\end{eqnarray*}
so that condition (a) is trivially satisfied, while
condition (a') is true if $|h|\geq \frac{\lambda}{2}x_1+\frac{\lambda}{2}a$
for every $x_1\in [c,a)$, which is guaranteed by the assumption $|h|^2\geq a\alpha$.

One easily checks that the vector $\s(x,z)-\s(x,y)$ can always be written as
$\mu e_1$ with $|\mu|\leq\alpha$ ($\mu$ depending on $x,y,z$), so that 
condition (b) is fulfilled. As for condition (b'), since $|h|\geq
\frac{\lambda}{2}(c+a)$ by the assumption $|h|^2\geq a\alpha$, we have that
$\s(x,h)-\s(x,0)=a\lambda^2 e_1-0=\alpha e_1$ for
every $x\in S_u$.

We notice that the
minimality of $u$ can be proved by applying the scalar result to one component
of $u$. Take, indeed, $j\in\{1,\ldots,N\}$ such that $h_j\neq 0$;
we know that if $h_j^2\geq a\alpha$, then $$\alpha{\cal H}^{n-1}(S_{u_j})\leq
\int_{\Omega}|\nabla v_j|^2dx +\alpha {\cal H}^{n-1}(S_{v_j}),$$
for every $v\in SBV(\Omega)$ with the
same boundary values as $u$.
Now, the left-hand side coincides with $MS_0(u)$, while the right-hand side is
less than or equal to $MS_0(v)$, since $S_{v_j}\subset S_v$. So, the Dirichlet
minimality of $u$ is shown, but under the stronger condition $h_j^2\geq a\alpha$.

Actually, since the Mumford-Shah functional is invariant by rotation (and
then $u$ is a Dirichlet minimizer if and only if $R\circ u$ is a Dirichlet
minimizer, where $R$ is any rotation in $\real^N$), the scalar result can be
exploited in a more efficient way. Let $R$ be a rotation
in $\real^N$ transforming the vector $h/|h|$ in $e_1$ and let $\hat
u:=R\circ u$. Applying the argument above to the first component of $\hat
u$, we have that $\hat u$ is a Dirichlet minimizer of $MS_0$ if $|h|^2\geq
a\alpha$, which is the same condition we have found via vectorial calibrations
theory.
We also note that the calibration (\ref{100cal}) can be obtained 
starting from the vectorfield which calibrates $\hat u_1$ simply replacing
the one-dimensional vertical variable by the component of the vector $y$
along $h/|h|$ and following the instructions of
Remark~\ref{rmk:scal-vect}. 
\end{ex}


\begin{ex}\label{ex:tj}
Let $\Omega:=B(0,r)$ be the open ball in $\real^2$ centred at the origin with
radius $r$, and let $(A_1,A_2,A_3)$ be the partition of $\Omega$ defined as
follows:
$$A_i:=\left\{ x=(\rho\cos\theta,\rho\sin\theta): \
0\leq \rho<r,\ \frac{2}{3}\pi(i-1)\leq \theta <\frac{2}{3}\pi i \right\}.$$
Let $u\in [SBV(\Omega)]^N$ be the function defined as $u:=a_i$ in each $A_i$,
where $a_1,a_2,a_3$ are three distinct vectors in $\real^N$.
In \cite[Example~4.14]{Alb-Bou-DM2} it is proved that, when $N=1$, $u$
is a Dirichlet minimizer of the Mumford-Shah functional (\ref{ms}) if the
values $a_i$ are sufficiently far apart, more precisely if 
\begin{equation}\label{cond-tj}
\min\{|a_1-a_2|, |a_2-a_3|, |a_3-a_1|\}\geq \sqrt{2\alpha r}.
\end{equation}
This result can be generalized to the vectorial case $N>1$, where beside 
condition (\ref{cond-tj}) we require that 
\begin{equation}\label{cond-tj-v}
\max\{|a_1-a_2|, |a_2-a_3|, |a_3-a_1|\}\geq \sqrt{(2+\sqrt{3})\alpha r}.
\end{equation}
Note that when $N=1$ condition (\ref{cond-tj-v}) is implied by (\ref{cond-tj}):
indeed, without loss of generality we can assume that $a_1\leq a_2\leq a_3$,
so that the maximum in (\ref{cond-tj-v}) is $a_3-a_1$; then by (\ref{cond-tj})
we obtain
$$a_3-a_1=(a_3-a_2)+(a_2-a_1)\geq 2\sqrt{2\alpha r}>\sqrt{(2+\sqrt{3})\alpha
r}.$$

We prove the statement by calibration. 
For every $i,j$ we call $S_{ij}$ the interface between
$A_i$ and $A_j$, which is oriented by the normal $\nu_{ij}$ pointing from
$A_i$ to $A_j$ and we suppose that the maximum in (\ref{cond-tj-v}) is given
by $|a_1-a_2|$. Let $\so\equiv 0$ and
$$\s(x,y):=[\sigma_1(x,y)\lor 0]\,\nu_{31} + [\sigma_2(x,y)\lor 0]\,\nu_{32},$$
where 
$$\sigma_1(x,y):=\alpha -
\frac{|y-a_1|^2}{r-\la\nu_{31},x\ra}, \qquad \sigma_2(x,y):=\alpha -
\frac{|y-a_2|^2}{r-\la\nu_{32},x\ra}.$$
For any $r'<r$ the function $\s$ is Lipschitz in $\ol{B(0,r')}{\times}\real^N$.
By direct computations we have that
\begin{multline}\label{j2}
|D_y\s(x,y)|^2=4\frac{|y-a_1|^2}{(r-\la\nu_{31},x\ra)^2}1_{\{ \sigma_1>0\}}
+4\frac{|y-a_2|^2}{(r-\la\nu_{32},x\ra)^2} 1_{\{ \sigma_2>0\}} \\ +
4\frac{\la y-a_1,y-a_2
\ra}{(r-\la\nu_{31},x\ra)(r-\la\nu_{32},x\ra)}1_{\{\sigma_1>0,\, \sigma_2>0\}},
\end{multline}
while
\begin{equation}\label{dvd}
[\dive_x\s](x,y)=-\frac{|y-a_1|^2}{(r-\la\nu_{31},x\ra)^2}1_{\{ \sigma_1>0\}}
-\frac{|y-a_2|^2}{(r-\la\nu_{32},x\ra)^2} 1_{\{ \sigma_2>0\}}.
\end{equation}
Condition (a) is therefore fulfilled if and only if $\la y-a_1,y-a_2
\ra\leq 0$ for every $y$ such that there exists $x\in B(0,r')$ so that
$\sigma_1(x,y)>0$ and $\sigma_2(x,y)>0$. Taking into account the definition of
$\sigma_1,\sigma_2$, this is equivalent to require the following: if $y$
belongs to the intersection of the ball centred at $a_1$ with radius
$(r-\la\nu_{31},x\ra)$ and the ball centred at $a_2$ with radius
$(r-\la\nu_{32},x\ra)$, then the angle spanned by the two vectors $y-a_1$ and
$y-a_2$ is greater or equal to $\pi/2$. Some elementary geometric
considerations show that this is guaranteed if 
$$|a_1-a_2|^2\geq\alpha(2r-\la\nu_{31},x\ra-\la\nu_{32},x\ra)\qquad \forall
x\in B(0,r'),$$
which is implied by condition (\ref{cond-tj-v}).

From (\ref{cond-tj}) it follows that $\sigma_2(x,a_1)\leq 0$, so that
by (\ref{j2}) and (\ref{dvd}) we have $|D_y\s(x,a_1)|^2=0$ and
$[\dive_x\s](x,a_1)=0$. Since (\ref{cond-tj}) implies analogously
that $\sigma_1(x,a_2)\leq 0$, and $\sigma_1(x,a_3)\leq 0$, $\sigma_2(x,a_3)\leq
0$, we deduce that condition (a') is satisfied.

Let $(x,y),(x,z)\in\ol{B(0,r')}{\times}\real^N$. If neither $(x,y)$ nor
$(x,z)$ belongs to $\{\sigma_1>0,\sigma_2>0\}$, then it is easy to check that
the vector $\s(x,z)-\s(x,y)$ can be written as a linear combination
$\lambda_1\nu_{31}-\lambda_2\nu_{32}$ with either
$\lambda_1,\lambda_2\in[0,\alpha]$ or $\lambda_1,\lambda_2\in[-\alpha,0]$
(depending on $x,y,z$); since $\nu_{31}$ and $-\nu_{32}$ span an angle equal
to $2\pi/3$, the modulus of $\s(x,z)-\s(x,y)$ is in this case less than or
equal to $\alpha$. If $(x,y)\in\{\sigma_1>0,\sigma_2>0\}$, only two cases can
occur: either $\s(x,z)-\s(x,y)$ is a linear combination of $\nu_{31}$ and
$-\nu_{32}$ of the same kind as before (so, the same conclusion holds), or
$\s(x,z)-\s(x,y)$ can be written as $\mu_1\nu_{31}+\mu_2\nu_{32}$ with
$\mu_i\in[0,\sigma_i(x,y)]$ (depending on $x,y,z$). In this second case, we
obtain
$$|\s(x,z)-\s(x,y)|^2\leq
\sigma_1^2(x,y)+\sigma_2^2(x,y)+\sigma_1(x,y)\sigma_2(x,y) \leq 
(\sigma_1(x,y)+\sigma_2(x,y))^2.$$
It is easy to see that, under condition
(\ref{cond-tj-v}), $\sigma_1(x,y)+\sigma_2(x,y)\leq \alpha$ for every 
$(x,y)\in\{\sigma_1>0,\sigma_2>0\}$, so that (b) is always satisfied.

Finally, using (\ref{cond-tj}) we have that
$\s(x,a_2)-\s(x,a_1)= \alpha\nu_{32}-\alpha\nu_{31}=\alpha\nu_{12}$
for every $x\in S_{12}$, $\s(x,a_3)-\s(x,a_2)=-\nu_{32}=\nu_{23}$
for every $x\in S_{23}$, while $\s(x,a_1)-\s(x,a_3)=\nu_{31}$
for every $x\in S_{31}$; so, we can conclude that (b') holds true for every
$x\in S_u$.

We have thus proved that under conditions (\ref{cond-tj}) --
(\ref{cond-tj-v}), $u$ is a Dirichlet minimizer of $MS_0$ in $B(0,r')$ for
every $r'<r$. By an approximation argument this implies the Dirichlet
minimality of $u$ in the whole $B(0,r)$.

As in the previous example, the minimality of $u$ can be proved by using
the scalar result in \cite{Alb-Bou-DM2}: indeed, even if $S_{u_j}$ is strictly 
contained in $S_u$ for every $j$, one can always find a rotation $R$ in
$\real^N$  tranforming the range of $u$ in a set of three vectors which differ
each other for the same component and apply the scalar result to this
component. This procedure leads to the following condition:
$u$ is a Dirichlet minimizer if 
$$\max_{v\in\real^N, |v|=1}\ \min\,\{
|\la a_1-a_2,v\ra|,|\la
a_2-a_3,v\ra|,|\la a_3-a_1,v\ra|\}\geq\sqrt{2\alpha r},$$ which is
always more restrictive than (\ref{cond-tj}) -- (\ref{cond-tj-v}), except when
the vectors $a_i-a_j$ are collinear. \end{ex}

\begin{ex}\label{ex:part}
In this example we deal with the complete Mumford-Shah functional
\begin{equation}\label{cms}
MS(u):=\int_{\Omega}|\nabla u|^2 dx +\alpha{\cal H}^{n-1}(S_u)
+\beta\int_{\Omega}|u-g|^2\, dx
\end{equation}
where $\Omega\subset\real^2$, $g$ is a given function in
$[L^{\infty}(\Omega)]^N$ , and $\alpha,\beta$ are positive parameters.

Let $\{\Gamma_i\}_{i\in I}$ be a finite family of simple and
connected curves of class $C^2$ such that for every $i$ 
$\Gamma_i$ is either a closed curve contained in $\Omega$ or it orthogonally
meets $\df\Omega$. Suppose also that $\Gamma_i\cap\Gamma_h=\emptyset$ if
$i\neq h$. If $g$ is a piecewise constant function, whose discontinuity set
coincides with $\cup_{i\in I}\Gamma_i$, then for large values of $\beta$ the
function $g$ itself is an absolute minimizer of (\ref{cms}). 

We prove the statement by calibration. 
We recall that conditions (a), (a'), (b), and (b') in Lemma~\ref{lm-main} read
for the functional (\ref{cms}) as
\begin{itemize}
	\item[(a)\phantom{'}] $[\dive_x\sg](x,y)+\so(x)\leq
-\frac{1}{4}|D_y\sg(x,y)|^2+\beta|y-g(x)|^2$
for every $\gamma\in A$, for ${\cal L}^2$-a.e. $x\in\Omega$, and for every
$y\in\real^N$ with $(x,y)\in U^{\gamma}$;
	\item[(a')] $[\dive_x\sg](x,u)+\so(x)=
-|\nabla u(x)|^2+\beta|u-g|^2$ and $(D_y\sg(x,u))^{\tau}=2\nabla u(x)$ for
every $\gamma\in A$, and for ${\cal L}^2$-a.e. $x\in\Omega$ such that
$(x,u(x))\in U^{\gamma}$;
 	\item[(b)\phantom{'}] $|\s(x,z)-\s(x,y)|\leq \alpha$ for ${\cal
H}^1$-a.e. $x\in\Omega$ and for every $y,z\in\real^N$ such that
$(x,y),(x,z)\in U$;
	\item[(b')] $\s(x,u^+)-\s(x,u^-)=\alpha\nu_u$ for ${\cal
H}^1$-a.e. $x\in S_u$,
\end{itemize}
where $\s(x,y)=\sum_{\gamma\in A}\sg(x,y)1_{U^{\gamma}}(x,y)$. 
 

Let $\{E_j\}_{j\in J}$ be the partition of $\Omega$ generated by the
family of curves $\{\Gamma_i\}_{i\in I}$. Then the function $g$ can be
written as $$g(x)=\sum_{j\in J}a_j 1_{E_j}(x),$$
where $a_j\in\real^N$ and $a_j\neq a_k$ if $j\neq k$.
For $j<k$ we call $S_{jk}$ the interface between $E_j$ and $E_k$, oriented by
the normal $\nu_{jk}$ pointing from $E_j$ to $E_k$ (in other words, $S_{jk}$
is the set of all $x\in S_g$ such that $g^-(x)=a_j$ and $g^+(x)=a_k$).
In this way we have simply relabelled the curves $\Gamma_i$.

For every $j<k$ we can construct a $C^1$-vectorfield
$\psi_{jk}:\ol{\Omega}\to\real^n$ such that it agrees with $\nu_{jk}$ on
$S_{jk}$, is supported on a neighbourhood of $S_{jk}$, is tangent to the
boundary of $\Omega$, and $| \psi_{jk}|\leq 1$ everywhere.
Since the curves $S_{jk}$ are disjoint, the functions
$\psi_{jk}$ can be constructed in such a way that their supports are still
disjoint; moreover, if $S_{jk}$ is a closed curve, we can also assume that
the support of $\psi_{jk}$ is relatively compact in $\Omega$. Finally, for
every $j<k$ we define the functions $\lambda_{jk}:\real^N\to\real$ as 
$$\lambda_{jk}(y):=\sigma\left( \frac{\la y-a_j,a_k-a_j\ra}{|a_k-a_j|^2}
\right),$$ where $\sigma:\real\to [0,\alpha]$ is a non decreasing function of
class $C^2$ such that $\sigma(t):=\frac{1}{3}\alpha t^3$ for $t\in[0,1/8]$, 
$\sigma(t):=\alpha+ \frac{1}{3}\alpha(t-1)^3$ for $t\in[7/8,1]$,
$\sigma'(t)\in[0,2\alpha]$ for every $t$, and $|\sigma''(t)|\leq 16\alpha$
for every $t$. 

Now we set
$$\s(x,y):=\sum_{(j,k):j<k}\lambda_{jk}(y)\,\psi_{jk}(x), \qquad
\so(x):=-\,\alpha\hspace{-.1cm}\sum_{(j,k):j<k}\dive \psi_{jk}(x) 1_{E_k}(x),$$
and we claim that the pair $(\s,\so)$ is a calibration for $g$ when $\beta$ is
large enough.

First of all, independently of the choice of $\sigma$, the function $\s$ has
vanishing normal component on $\df\Omega$ because of the choice of
$\psi_{jk}$, so that condition (\ref{norm0}) of Theorem~\ref{thm-main} is
satisfied.

Using the fact that the supports of the functions $\psi_{jk}$ are disjoint, and
that $|\psi_{jk}|\leq 1$, while $\lambda_{jk}$ takes values only on
$[0,\alpha]$, it is easy to see that condition (b) is fulfilled.

Since $S_g$ is the union of the disjoint curves $\{ S_{jk}\}_{j<k}$, 
for every $x\in S_g$ there exists one and only one pair $(j,k)$ with $j<k$
such that $x\in S_{jk}$, so that
$$\s(x,g^+(x))-\s(x,g^-(x))=(\lambda_{jk}(a_k)-\lambda_{jk}(a_j))\,\psi_{jk}(x)=
(\sigma(1)-\sigma(0))\,\nu_{jk}(x)=\alpha\nu_g(x).$$
Therefore, also condition (b') is satisfied.

By direct computations we obtain that
$$[\dive_x\s](x,y)=\sum_{(j,k):j<k}\lambda_{jk}(y)\,\dive\psi_{jk}(x),$$
while
$$D_y\s(x,y)=\sum_{(j,k):j<k}\sigma'\left(\frac{\la y-a_j,a_k-a_j\ra}{|a_k-a_j|^2}
\right)\psi_{jk}(x)\otimes\frac{a_k-a_j}{|a_k-a_j|^2}.$$
If $x\in E_h$ for any $h\in J$, then
\begin{eqnarray*}
[\dive_x\s](x,g(x)) & = & [\dive_x\s](x,a_h) 
\hspace{.2cm} = \hspace{.2cm}
\sum_{j<h}\lambda_{jh}(a_h)\,\dive\psi_{jh}(x)+\sum_{k>h}\lambda_{hk}(a_h)\,\dive\psi_{hk}(x) \\
& = & \alpha\sum_{j<h}\dive\psi_{jh}(x),
\end{eqnarray*}
where the last equality follows from the fact that
$\lambda_{jh}(a_h)=\sigma(1)=\alpha$, while $\lambda_{hk}(a_h)=\sigma(0)=0$. 
Arguing analogously, since $\sigma'(0)=\sigma'(1)=0$, we have that
$D_y\s(x,g(x))=0$, so that, taking into account the definition of $\so$,
condition (a') is satisfied.

It remains to prove condition (a). Let $(x,y)\in\ol{\Omega}{\times}\real^N$. 
If $x$ does not belong to any of the supports of the functions $\psi_{jk}$,
then $[\dive_x\s](x,y)=0$, $\so(x)=0$, and $D_y\s(x,y)=0$, so (a) is trivially
satisfied.
If $x$ belongs to the support of $\psi_{jk}$ for any $j<k$, then
$$[\dive_x\s](x,y)=\lambda_{jk}(y)\,\dive\psi_{jk}(x), \qquad
\so(x)=-\alpha\,\dive \psi_{jk}(x) 1_{E_k}(x),$$
$$D_y\s(x,y)=\sigma'\left(\frac{\la y-a_j,a_k-a_j\ra}{|a_k-a_j|^2}
\right)\psi_{jk}(x)\otimes\frac{a_k-a_j}{|a_k-a_j|^2};$$
if we write the vector $y-a_j$ as the sum
$v+t(a_k-a_j)$ where $v\in\real^N$ is orthogonal to $a_k-a_j$,
and $t\in\real$, condition (a) turns to be equivalent to
\begin{equation}\label{aequiv}
\dive\psi_{jk}(x)(\sigma(t)-\alpha 1_{E_k}(x))\leq
-\frac{1}{4}|\psi_{jk}(x)|^2 |\sigma'(t)|^2+\beta |v+t(a_k -a_j) +
a_j-g(x)|^2.
\end{equation}
Since we are assuming that $x$ is in the support
of $\psi_{jk}$, $x$ belongs either to $E_j$ or to $E_k$. 
When $x\in E_j$, inequality (\ref{aequiv}) reduces to
$$\dive\psi_{jk}(x)\,\sigma(t)\leq -
\frac{1}{4}|\psi_{jk}(x)|^2
|\sigma'(t)|^2+\beta |v|^2+\beta |a_k-a_j|^2t^2,$$
which is implied by
\begin{equation}\label{aequiv2}
\dive\psi_{jk}(x)\,\sigma(t)\leq -
\frac{1}{4}|\psi_{jk}(x)|^2
|\sigma'(t)|^2+\beta |a_k-a_j|^2t^2.
\end{equation}
So, let us prove (\ref{aequiv2}) for every $t\in\real$ and $x\in E_j$.
Since in (\ref{aequiv2}) the equality holds for $t=0$, it is enough 
to show the following inequality
\begin{equation}\label{aequiv3}
\dive\psi_{jk}(x)\,\sigma'(t)<-\frac{1}{4}|\psi_{jk}(x)|^2
2\sigma'(t)\,\sigma''(t)+2\beta|a_k-a_j|^2 t \qquad \text{for $t>0$},
\end{equation}
and the opposite inequality for $t<0$. Since $\sigma'\equiv 0$ for $t>1$,
inequality (\ref{aequiv3}) is trivially satisfied for $t>1$. 
For $0<t\leq 1$, (\ref{aequiv3}) follows
immediately from
$$-\|\dive\psi_{jk}\|_{\infty}\sigma'(t)>\frac{1}{2}\sigma'(t)|\sigma''(t)|-2\beta |a_k-a^j|^2t,$$ 
which is satisfied (taking into account the structure of the function
$\sigma$) for
$$\beta|a_k-a_j|^2>8\alpha\|\dive\psi_{jk}\|_{\infty}+64\alpha^2.$$
%\max\left\{\frac{1}{16}\alpha\|\dive\psi_{jk}\|_{\infty}+\frac{1}{128}\alpha^2, \
%8\alpha\|\dive\psi_{jk}\|_{\infty}+64\alpha^2
%\right\}.
The same condition implies also the opposite inequality for $t<0$. Moreover,
the same argument can be applied in the case $x\in E_k$.

In conclusion, condition (a) is fulfilled for $\beta>\beta_0$, where $\beta_0$
is defined by
\begin{equation}\label{beta}
\beta_0:=\max_{(j,k):j<k}
\frac{1}{|a_k-a_j|^2} \left(8\alpha\|\dive\psi_{jk}\|_{\infty}+64\alpha^2
\right). \end{equation}
\end{ex}

We conclude this example by noticing that this result generalizes Example~5.5
in \cite{Alb-Bou-DM2}, where $g$ is the characteristic function of a regular
set. As in the previous examples, the vectorial statement can be proved by
applying the scalar result to one suitable component of $g$, but this leads to
a worse estimate on $\beta_0$.


\section{Calibrations in terms of closed differential forms}\label{sec:df}


In this section we develop the theory of calibrations in terms of differential
forms. The scalar method presented in \cite{Alb-Bou-DM2} involves a
divergence-free vectorfield on $\Omega{\times}\real$ (and its flux through the complete graph
of the maps $u$), which is now replaced by a closed $n$-form on
$\Omega{\times}\real^N$, acting on the graphs of the maps $u$, viewed as suitably
defined $n$-surfaces in $\Omega{\times}\real^N$.

As we will see, this formulation is indeed not preferable to the one
described in Section~\ref{sec:vcal}, since it leads to the same kind of
conditions, requiring a greater technical effort.

For simplicity we restrict our discussion to piecewise smooth functions $u\in
[SBV(\Omega)]^N$ in the sense of the following definition.

\begin{dfn} 
We say that a function $u\in [SBV(\Omega)]^N$ is {\em piecewise smooth}, and we
write $u\in\rc{\Omega}$, if the following conditions are satisfied: up to an
${\cal H}^{n-1}$-negligible set, $S_u$ is a finite union of pairwise disjoint
$(n-1)$-dimensional boundaryless $C^1$-manifolds of $\real^n$; $u$ is $C^1$ on
$\Omega\setminus S_u$ up to $S_u$, that is $u\in [C^1(\Omega\setminus S_u)]^N$ and
there exist the limits of $u$ and $\nabla u$ on both sides of (the regular part
of) $S_u$. 
\end{dfn}

For $u\in\rc{\Omega}$ we define the $n$-surfaces
$$\jump{u}:=\{(x,y)\in\Omega{\times}\real^N:\ x\in S_{u}\ \ \text{and} \ \
\exists \,t\in[0,1]\ \ \text{such that}\ \ y=tu^+(x)+(1-t)u^-(x) \},$$ 
$$\grph{u}:=\reg{u}\cup \jump{u},$$
where $\reg{u}$ is the graph of $u$ on $\Omega\setminus S_u$.
Using notation from \cite{Gia-Mod-Sou}, let us consider a $n$-form 
$$\omega:\Omega{\times}\real^N\to\wedge^n\real^{n+N},$$
$$\omega(x,y)=\sum_{|\alpha|+|\beta|=n}\omega_{\alpha\beta}(x,y)\,dx^{\alpha}\land dy^{\beta},$$
whose coefficients $\omega_{\alpha\beta}$ are of class $C^1$, and for
$u\in\rc{\Omega}$ the following functional 
\begin{equation}\label{functdf}
\int_{\grph{u}}\omega,
\end{equation}
where the orientation of $\grph{u}$ will be defined later in a precise way.

If $\omega$ is a closed form, then the functional (\ref{functdf}) 
is constant on the functions $u$ which take the same value on
$\df\Omega$. Moreover, if $F$ is the functional (\ref{funct}), and if 
\begin{equation}\begin{array}{rrcl}
& \displaystyle 
\int_{\grph{v}}\omega & \leq  & F(v)\qquad \text{for every $v\in
\rc{\Omega}$,}  \\
\text{and} & & &  \\  &\displaystyle \int_{\grph{u}}\omega & =  &
F(u)\qquad \text{for a given $u\in \rc{\Omega}$,}
\end{array}\label{2c}
\end{equation}
then $u$ is a Dirichlet minimizer of $F$ in the class $\rc{\Omega}$.

Let us now look for pointwise conditions on the coefficients of the form
$\omega$ which guarantee (\ref{2c}).

By definition we have that
\begin{equation}\label{summ}
\int_{\grph{u}}\omega = \int_{\reg{u}}\omega + \int_{\jump{u}}\omega.
\end{equation} 
On $\reg{u}$ we consider the natural orientation given by the parameterization
$x\in\Omega\setminus S_u\mapsto (x,u(x))$, so that 
\begin{equation}\label{reg-dec}
\int_{\reg{u}}\omega = \sum_{|\alpha|+|\beta|=n} \int_{\Omega} 
\omega_{\alpha\beta}(x,u(x))\mu_{\alpha\beta}(x)\, dx, 
\end{equation}
where $$\mu_{\alpha\beta}(x):=\epsilon(\alpha)\det\left( \frac{\df 
u_{\beta}}{\df x_{\hat\alpha}}(x)\right).$$
In the previous formula $\hat\alpha$ denotes the increasing complement of
$\alpha$ in $\{1,\ldots,n\}$, $\epsilon(\alpha)$ is the sign of permutation of 
$(1,\ldots,n)$ into $(\alpha,\hat\alpha)$, and $\frac{\df u_{\beta}}{\df
x_{\hat\alpha}}$ is the $|\beta|{\times}|\beta|$ matrix $\frac{\df
u_{\beta_i}}{\df x_{\hat\alpha_j}}$.


On $\jump{u}$ we consider the orientation given by the following
parameterization: since $u\in\rc{\Omega}$, without loss of generality, we may
assume that $S_u$ is an $(n-1)$-dimensional $C^1$-manifold of $\real^n$
without boundary and that $S_u$ can be covered by just one parameter patch
$\gamma:S\to S_u$, where $S$ is an $(n-1)$-dimensional domain (the general
case can be easily obtained by summing over the $C^1$-pieces). Assume that 
$\gamma$ yields $\nu_u$ as orientation, that is the vector
$$\eta(\gamma(\sigma)):=\sum_{i=1}^n(-1)^{n-i}\det\left(\frac{d\gamma_{\ih}}  {
d\sigma}(\sigma)\right)e_i$$ 
(where $\{e_1,\ldots,e_n\}$ is the canonical basis of $\real^n$) satisfies
$$\frac{\eta(\gamma(\sigma))}{|\eta(\gamma(\sigma))|}=\nu_u(\gamma(\sigma)) 
\qquad \forall\sigma\in S.$$
We consider as parameterization of $\jump{u}$ the function
$\varphi=(\varphi^x,\varphi^y):S{\times}[0,1]\to \Omega{\times}\real^{N}$
defined as $\varphi^x(\sigma,t):= \gamma(\sigma)$, $\varphi^y(\sigma,t):= 
tu^{+}(\gamma(\sigma))  +(1-t)u^{-}(\gamma(\sigma))$ for every
$(\sigma,t)\in S{\times}[0,1]$, so that the second integral in (\ref{summ})
is given by \begin{equation}\label{jump-dec} \int_{\jump{u}}\omega =
\sum_{|\alpha|+|\beta|=n}\int_0^1\int_{S}
\omega_{\alpha\beta}(\varphi(\sigma,t))\det\left(\frac{\df
\varphi_{\alpha\beta}}{\df (\sigma, t)}(\sigma,t)\right) d\sigma dt,
\end{equation}
where $\varphi_{\alpha\beta}=(\varphi^x_{\alpha_1},\ldots,\varphi^x_{\alpha_p},
\varphi^y_{\beta_1},\ldots,\varphi^y_{\beta_q})$ for $|\alpha|=p$ and
$|\beta|=q=n-p$. By direct computations one can find that 
$$\det\left(\frac{\df \varphi_{\hat{0}0}}{\df(\sigma, t)}\right) =
0,$$
while for every $1\leq i\leq n$, $1\leq j\leq N$
$$\det\left(\frac{\df \varphi_{\ih j}}{\df
(\sigma, t)}\right) =
(u^{+}_{j}-u^{-}_{j})\det\left(\frac{d\gamma_{\ih}}{d\sigma}\right) 
= (-1)^{n-i}(u^{+}_{j}-u^{-}_{j})(\nu_{u})_{i}|\eta|, $$
where all the functions at the right-hand side are computed at 
$\gamma(\sigma)$.
Finally, by straightforward computations, if we set 
$a:=\hat{\alpha}$, for $|a|=|\beta|=q\geq 2$ it results that
$$\det\left(\frac{\df \varphi_{\alpha\beta}}{\df
(\sigma, t)}\right)= 
\sum_{m,k=1}^{q}\epsilon(\alpha,a_{\widehat{k}})(-1)^{n-q+m-a_k}(u_{\beta_{m}}^{+}-u_{\beta_{m}}^{-})\det 
\left(\frac{\df(tu^{+}+(1-t)u^{-})_{\beta_{\widehat m}}}{\df x_{a_{\widehat 
k}}}\right)(\nu_{u})_{a_{k}} |\eta|,$$
where $\beta_{\widehat m}$, $a_{\widehat k}$ are the increasing complement of
$\beta_m$ in $\{\beta_1,\ldots,\beta_q \}$ and of $a_k$ in $\{a_1,\ldots,a_q
\}$, respectively, while $\epsilon(\alpha,a_{\widehat{k}})$ is the sign of
permutation of $(\alpha,a_{\widehat{k}})$ in $\widehat{a_k}$; again all
the functions at the right-hand side are computed at $\gamma(\sigma)$. Set
$w^{t}:=tu^{+}+(1-t)u^{-}$ and substitute all the above expressions in 
formula (\ref{jump-dec}); since $|\eta|\,d\sigma$ is the area element of the
manifold $S_u$ parameterized by $\gamma$, we obtain \begin{eqnarray}\nonumber
\lefteqn{\int_{\jump{u}}\omega \hspace{.2cm}  = \hspace{.2cm}  \sum_{i,j}
\int_0^1\int_{S_u} (-1)^{n-i}\omega_{\ih j}(x,
w^t)(u_j^+-u_j^-)(\nu_u)_i\,d{\cal H}^{n-1}dt} \\ 
\nonumber & +  &\sum_{\substack{|a|=|\beta|=q
\\ q\geq 2}} \int_0^1\int_{S_u} \omega_{\hat a\beta}(x, w^t)
\sum_{m,k=1}^q\epsilon(\alpha,a_{\widehat{k}})(-1)^{n-q+m-a_k}
(u_{\beta_m}^+-u_{\beta_m}^-) \det\left(\frac{\df
w^t_{\beta_{\widehat m}}}{\df x_{a_{\widehat k}}}\right)\!\!(\nu_u)_{a_k}\,d{\cal
H}^{n-1}dt \\
& =: & \int_{S_u} g_{\omega}(x,
u^-,u^+,\nabla u^-,\nabla u^+,\nu_u)\, d{\cal  H}^{n-1}, \label{www}
\end{eqnarray}
where the last equality follows from changing the order of integration and
calling $g_{\omega}$ the integrand with respect to ${\cal H}^{n-1}$.
Now we wonder what kind of conditions on $\omega_{\alpha\beta}$ guarantee that
\begin{equation}\label{h-cond}
g_{\omega}(x, u^-,u^+,\nabla u^-,\nabla u^+,\nu_u)\leq
\psi(x,u^-,u^+,\nu_u) \qquad \text{on $S_u$}
\end{equation}
for every admissible $u$. The answer is given by the following proposition.

\begin{prp}\label{prp-ptw}
Inequality (\ref{h-cond}) holds true for every $u\in\rc{\Omega}$ if and only if
the following conditions are satisfied:
\begin{itemize}
	\item[(b$_{0}$\!)] $\omega_{\alpha\beta}\equiv 0$ for every
$\alpha,\beta$ such that $|\beta|\geq 2$, $|\alpha|+|\beta|=n$;
	\item[(b$_{1}$)] $\displaystyle\sum_{i,j}\int_0^1 (-1)^{n-i}\omega_{\ih
j}(x,tz+(1-t)y)(z_j-y_j)\nu_i\, dt \leq \psi(x,y,z,\nu)$ for every
$x\in\Omega$, for every $y,z\in\real^N$, and for every $\nu\in\sph^{n-1}$.
\end{itemize} Moreover, the equality holds for a given $u$ if and only if
\begin{itemize}
	\item[(b$_{2}$)] $\displaystyle\sum_{i,j}\int_0^1
(-1)^{n-i}\omega_{\ih j}(x,tu^+ +(1-t)u^-)(u^+_j-u^-_j)(\nu_u)_i\, dt =
\psi(x,u^-,u^+,\nu_u)$ for every $x\in S_u$. \end{itemize}
\end{prp}

\proof
Let $(x,y)\in\Omega{\times}\real^N$, and let us prove that
$\omega_{\alpha\beta}(x,y)=0$ for $|\hat\alpha|=|\beta|=2$.
By renumbering the coordinates of $x$ and $y$, we may 
suppose that $\beta=(1,2)$ and $a=\hat\alpha=(1,2)$. Given $C\in\real$,
we can construct $u\in \rc{\Omega}$ such that $x\in S_u$, $\nabla
u^-(x)=\nabla u^+(x)$ (hence $\nabla w^t(x)=\nabla u^-(x)$ for every
$t\in[0,1]$), and $\df_{x_i}w^t_{j}(x)=0$ for every $(i,j)\neq (1,1)$ and
$\df_{x_1}w^t_{1}(x)=C$. With this choice we have that
\begin{multline}\nonumber g_{\omega}(x,u^-,u^+,\nabla u^-,\nabla u^+,\nu_u)=
\sum_{i,j}\int_0^1 (-1)^{n-i}\omega_{\ih j}(x,w^t)(u^+_j-u^-_j)(\nu_u)_i\, dt 
\\+
C\sum_{i\neq
1,j\neq
1}\int_0^1(-1)^i\omega_{(\widehat{1,i})(1,j)}(x,w^t)(u^+_j-u^-_j)(\nu_u)_i\,
dt. \end{multline}  Since the value of $C$ is arbitrary and independent of 
$u^-(x),u^+(x),\nu_u(x)$, inequality (\ref{h-cond}) implies that
\begin{equation}\label{c1}
\sum_{i\neq
1,j\neq 1}\int_0^1(-1)^i\omega_{(\widehat{1,i})(1,j)}(x,w^t)(u^+_j-u^-_j)(\nu_u)_i\,
dt=0 
\end{equation}
whatever are the values of $u^-(x),u^+(x),\nu_u(x)$.
Choosing $\nu_u(x)$ such that $(\nu_u(x))_i=0$ for every $i\neq 2$,
$(\nu_u(x))_2=1$, we have that (\ref{c1}) is equivalent to
\begin{equation}\label{c2}
\sum_{j\neq 1}\int_0^1\omega_{(\widehat{1,2})(1,j)}(x,w^t)(u^+_j-u^-_j)\,
dt=0 
\end{equation}
whatever are the values of $u^-(x),u^+(x)$.
Choosing $u^-(x)=y$, while $u^+_j(x)=y_j$ for every $j\neq 2$,
$u^+_2(x)=y_2+c$ with $c\neq 0$, we obtain that (\ref{c2}) is equivalent to
\begin{equation}\label{c3}
c\int_0^1\omega_{(\widehat{1,2})(1,2)}(x,y_1,y_2+ct,y_3,\ldots,y_N)\,dt=0 
\end{equation}
for every $c\neq 0$. By a change of variables, (\ref{c3}) can be rewritten as
\begin{equation}\label{c4}
\int_{y_2}^{y_2+c}\omega_{(\widehat{1,2})(1,2)}(x,y_1,s,y_3,\ldots,y_N)\,ds=0. 
\end{equation}
Since (\ref{c4}) has to be true for every $c\neq 0$, this implies that
$\omega_{(\widehat{1,2})(1,2)}(x,y)=0$.

Using the fact that the coefficients $\omega_{\alpha\beta}\equiv 0$ for every
$|\beta|=2$, we can repeat the same proof to show
that $\omega_{\alpha\beta}\equiv 0$ for every $|\hat\alpha|=|\beta|=3$, and
so on.

We have thus proved that (\ref{h-cond}) implies condition (b$_0$). At this
point, it is trivial that (\ref{h-cond}) implies also condition (b$_1$), and
that the equality holds in (\ref{h-cond}) for a given $u$ if and only if  also
(b$_2$) is satisfied. 
\phantom{a}\qed

Summarizing, if conditions (b$_0$) and (b$_1$) hold true, by
Proposition~\ref{prp-ptw} inequality (\ref{h-cond}) is satisfied, hence by
(\ref{www}) we have that
\begin{equation}\label{d1}
\int_{\jump{u}}\omega\leq \int_{S_u}\psi(x,u^-,u^+,\nu_u)\, d{\cal H}^{n-1}
\end{equation}
for every $u\in \rc{\Omega}$, while the equality holds in (\ref{d1}) for a
given $u$ if and only if also (b$_2$) is verified.

Assuming that $\omega$ satisfies condition (b$_{0}$), formula 
(\ref{reg-dec}) reduces to
\begin{eqnarray*}
\int_{\reg{u}}\omega & = & \int_{\Omega} \left(\omega_{\hat{0}0}(x,u(x))
+ \sum_{i,j}(-1)^{n-i}\omega_{\ih j}(x,u(x))\df_{x_{i}}u_{j}(x)\right) dx \\
& = & \int_{\Omega} \left(\omega_{\hat{0}0}(x,u(x)) +\la
A_{\omega}(x,u(x)),\nabla u(x) \ra \right) dx,
\end{eqnarray*}
where in the last equality $(A_{\omega}(x,y))_{ji}:= (-1)^{n-i}\omega_{\ih
j}(x,y)$.  It is easy to see that, if we require the following condition: 
\begin{itemize}
	\item[(a$_1$)] $\omega_{\hat{0}0}(x,y)\leq
-f^{\ast}(x,y,A_{\omega}(x,y))$ for ${\cal L}^n$-a.e. $x\in\Omega$ and every
$y\in\real^N,$ 
\end{itemize}  then 
$$\int_{\reg{u}}\omega \leq \int_{\Omega}f(x,u,\nabla u)\,dx$$
for every $u\in \rc{\Omega}$; moreover, the equality holds for a given $u$ if
and only if \begin{itemize}
	\item[(a$_2$)]
$(A_{\omega})_{ij}(x,u(x))\in\df_{\xi_{ij}}f(x,u(x),\nabla u(x))$  and
$\omega_{\hat{0}0}(x,u(x))=- f^{\ast}(x,u(x),A_{\omega}(x,u(x)))$ for 
${\cal L}^n$-a.e. $x\in\Omega$.
\end{itemize} 

Therefore by (\ref{summ}) we can conclude that (\ref{2c}) is guaranteed if
conditions (a$_1$), (a$_2$), (b$_0$), (b$_1$), and (b$_2$) are satisfied. 
In other words, we have proved the following theorem.

\begin{thm}\label{thm-sca}
Let $u$ be a function in $\rc{\Omega}$. Assume that there exists a closed
$n$-differential form $\omega:\Omega{\times}\real^N\to \land^n\real^{n+N}$
with coefficient of class $C^1$ and satisfying condition (a$_1$), (a$_2$),
(b$_{0}$), (b$_1$), and (b$_2$). Then $u$ is a Dirichlet minimizer of the
functional (\ref{funct}) in the class $\rc{\Omega}$.
\end{thm}

We conclude this section by proving that, if $u\in\rc{\Omega}$ and there
exists a differential form $\omega$ which calibrates $u$ in the sense of
Theorem~\ref{thm-sca}, then there exists a calibration $(\s,\so)$ for
$u$ in the sense of Definition~\ref{dfn-cal}.

\begin{prp}
Let $u$ be a function in $\rc{\Omega}$ and let
$\omega:\Omega{\times}\real^N\to \land^n\real^{n+N}$ be a closed
$n$-differential form satisfying all the assumptions of
Theorem~\ref{thm-sca}. Then there exists a calibration $(\s,\so)$ for
$u$, with $\s\in[C^2(\Omega{\times}\real^N)]^n$ and $\so\in C^1(\Omega)$. 
\end{prp}

\proof
First of all, we notice that from condition (b$_{0}$) it
follows that 
$$\omega(x,y)=\omega_{\hat 00}(x,y)\,dx +
\sum_{i,j}\omega_{\ih j}(x,y)\,dx^{\ih}\land
dy^{j}.$$
Since $\omega$ is a closed form, by computing explicitly the exterior
derivative of $\omega$, we obtain that the coefficients $\omega_{\hat 00},\omega_{\ih j}$
satisfy the two following equations: 
\begin{eqnarray} \label{qdive}
&\displaystyle \sum_{i=1}^{n}(-1)^{n-i}\frac{\df\omega_{\ih j}}{\df
x_{i}}(x,y)  -\frac{\df\omega_{\hat{0}0}}{\df y_j}(x,y)=0 \qquad 
1\leq j \leq N, & \vspace{.1cm}\\
&\displaystyle (-1)^{n-i}\frac{\df \omega_{\ih j}}{\df y_{k}}(x,y) = \label{mix-der}
(-1)^{n-i}\frac{\df \omega_{\ih k}}{\df y_{j}}(x,y) \qquad 
1\leq i \leq n,\ 1\leq j,k\leq N. &
\end{eqnarray}
The last condition is equivalent to require that for every $i$ the vector
$((-1)^{n-i}\omega_{\ih j}(x,y))_{j=1,\ldots,N}$ is the gradient with respect
to $y$ of a function of class $C^2$; more precisely, there 
exists a function $\s\in [C^2(\Omega{\times}\real^N)]^n$ such that 
\begin{equation}\label{d2}
\df_{y_{j}}\s_{i}(x,y)=(-1)^{n-i}\omega_{\ih j}(x,y)\qquad 
1\leq i \leq n,\ 1\leq j\leq N.
\end{equation}
Equation (\ref{qdive}) can be therefore rewritten as
$$0=\sum_{i=1}^{n}\frac{\df^{2}\s_i}{\df x_{i}\df y_{j}}(x,y) -
\frac{\df\omega_{\hat{0}0}}{\df y_j}(x,y)= \df_{y_{j}}\left[\sum_{i=1}^{n}
\df_{x_{i}} \s_{i}(x,y)-  \omega_{\hat{0}0}(x,y)\right] \qquad 1\leq j \leq
N,$$ and then there exists a function $\so:\Omega\to \real$ of class $C^1$
such that $\omega_{\hat{0}0}(x,y)= [\dive_{x}\s](x,y)+\so(x)$.
By substituting this equality and (\ref{d2}) in conditions (a$_1$) and
(a$_2$), we directly obtain that the pair $(\s,\so)$ satisfies conditions (a)
and (a') of Lemma~\ref{lm-main-0}. 
Since the left-hand side in (b$_1$) can be rewritten as
\begin{eqnarray*}
\lefteqn{\sum_{i,j}\int_{0}^{1}(-1)^{n+i}\omega_{\ih
j}(x,tz+(1-t)y)(z_j-y_j)\nu_i\, dt}\\
& = &  \sum_{i,j}\int_{0}^{1}\df_{y_{j}}
\s_{i}(x,tu^{+}+(1-t)u^{-})
(u^{+}_{j}-u^{-}_{j})(\nu_{u})_{i}\, dt \\
& = & \sum_{i=1}^{n}\int_{0}^{1}\frac{d}{dt}[\s_i(x,tu^{+}+(1-t)u^{-})]
(\nu_{u})_{i}\, dt \\ & = &
\sum_{i=1}^{n}[\s_i(x,u^{+})-\s_i(x,u^{-})](\nu_{u})_{i}\,  \\
& = &
\la\s(x,u^{+})-\s(x,u^{-}),\nu_{u}\ra,
\end{eqnarray*}
condition (b$_1$) implies that the function $\s$ satisfies condition (b) of
Lemma~\ref{lm-main-0}, and in the same way (b$_2$) implies (b').
\qed

\vspace{1cm}
\noindent
{\bf Acknowledgements.} The author is grateful to Gianni Dal Maso for having
suggested the study of this problem and for many helpful discussions.


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\end{document} 
