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\title{Higher regularity of solutions \\
of free discontinuity problems}
%\title{}
\author{}
\author{{\Large Luigi Ambrosio} \\ Dipartimento di Matematica \\
``F. Casorati'', Via Abbiategrasso
205 \\27100 Pavia, Italy \and
{\Large Nicola Fusco} \\ Dipartimento di Matematica
\\ ``U. Dini'', Viale Morgagni 67/A\\ 50134 Firenze, Italy \and
{\Large Diego Pallara} \\ Dip. di Matematica \\ Universit\`a di
Lecce \\ C.P. 193, 73100 Lecce, Italy}


\begin{document}
\maketitle
\begin{abstract}
In this paper we continue the analysis, started in \cite{ap}, \cite{afp} of
the regularity of solutions of free discontinuity problems.  We choose as 
a
model problem the minimization of the Mumford-Shah functional.  
Assuming
that in some region the optimal discontinuity set $\Gamma$ is the graph 
of
a $C^{1,\rho}$ function, we look for conditions ensuring the higher
regularity of $\Gamma$.  Our results are optimal in the two dimensional
case.

As an application, we prove that in the case of the Mumford-Shah 
functional
and in similar problems the Lavrentiev phenomenon does not occur.
\end{abstract}

\hbox{}\vfill
\tableofcontents

\vfill\eject

\section{Introduction}

Free discontinuity problems are variational problems where the 
functional
to be minimized depends on a pair $(u,K)$, where $K$ is a
hypersurface, or more generally a closed set, and $u$ is usually smooth
outside $K$. The model problem is the Mumford-Shah functional,
proposed in \cite{mumsh} as a variational model of image segmentation:
\begin{equation}        \label{msfunct}
F(u,K)=\int_{\Omega\setminus K} \bigl[ |\nabla u|^2 +\alpha|u-g|^2
\bigr]\, dx + \beta{\cal H}^{n-1}(\Omega\cap K);
\end{equation}
here, $\Omega\subset\rn{n}$ is an open set, $\alpha,\,\beta>0$,
$g\in L^\infty(\Omega) \cap
L^2(\Omega)$ and ${\cal H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff
measure.  We refer to \cite{mumsh}, \cite{morsol} for a comprehensive
discussion of this model.

The existence of a minimizing pair $(u,K)$, with $K$ closed and $u\in
C^1(\Omega\setminus K)$, has been proved by De Giorgi, Carriero and 
Leaci
in \cite{dcl} (see also \cite{dms}) using the theory of $SBV$ functions
introduced in \cite{dga}.  But, while the smoothness of the function $u$ 
in
$\Omega\setminus K$ can be easily investigated using classical 
arguments
based on first variation and elliptic equations techniques, as regards the
regularity of $K$ the existence theory ensures only that $K$ is 
rectifiable
-- a very mild regularity property.

Recently, we faced the problem of the partial regularity of the
discontinuity set $K$:  our main results are presented in \cite{ap},
\cite{afp} (see also \cite{ap2} for a brief survey) and are based on a 
double
blow-up technique, suitable to exploit the interaction between the
volume term and the surface term in functionals like $F$ in
(\ref{msfunct}).  Our results, which hold for a class of free discontinuity
problems including the Mumford-Shah functional, can be briefly 
summarized
by stating that if $(u,K)$ is a minimizing pair then $K$ is locally a
$C^{1,\rho}$ hypersurface up to a ${\cal H}^{n-1}$ negligible singular
set $S$.  Some other results, valid only for the Mumford-Shah functional 
in
two space dimensions, have been obtained with different methods in
\cite{davsemm}, \cite{bon}.

The aim of this paper is to improve the regularity results on the
discontinuity set $K$:  in particular, we focus
our attention on the Mumford-Shah functional and show the following
result, which is an immediate consequence of Theorem \ref{meanreg1}:

\begin{theorem}
Let $(u,K)$ be a minimizer of (\ref{msfunct}); then, up to a ${\cal
H}^{n-1}$-negligible singular set $S$, the minimizing set $K$ is (locally)
the graph of a $C^{1,1}$ function if the space dimension is 2, and
of a $H^2$ function $\phi$ with gradient in $C^{0,\rho}$ for any $\rho<1$
if $n>2$.
\end{theorem}

Moreover (see Theorem \ref{meanreg2}), if the datum $g$ is assumed
to belong to $C^{k,\rho}$ with $k\geq 1$ and $\rho\in (0,1]$, then (for
any $n\geq 2$) $\phi\in C^{k+2,s}$ for some $s\in (0,1]$.  The
basic idea can be explained as follows:  given a minimizing pair $(u,K)$,
by the first variation formula, see \cite{clpp}, \cite{mumsh}, it is known
that the function $u$ solves the equation $\Delta u=u-g$ in
$\Omega\setminus K$ with homogeneous Neumann boundary conditions 
on $K$,
and moreover, under regularity assumptions, the mean curvature of $K$
equals the jump of $|\nabla u|^2$ across it.  It follows that $\phi$ is a
weak solution of a mean curvature equation, and this can be exploited to
get further regularity.  The first step is the proof of the regularity of
$\nabla u$ up to (both sides of) $K$:  this amounts to prove a boundary
regularity result for elliptic problems in $C^{1,\rho}$ domains which
seems to be interesting by itself.  This is achieved in \S 2, proving that
$\nabla u$ has H\"older continuous traces on both sides of $K$.  In \S 3 
we
prove a regularity result for weak solutions of mean curvature 
equations
which will be used in \S 4 to prove the announced regularity theorem 
on the
discontinuity set $K$.  Finally, extending to any dimension the results of
\cite{dise}, in \S 5 we use our regularity results to
prove that in Mumford-Shah problem (and even in more general 
situations,
see Remark~\ref{Lav} below) the so called Lavrentiev phenomenon does 
not
occur, {\em i.e.} the minimum of $F$ among all the pairs $(u,K)$ with 
$K$
closed and $u\in C^1(\Omega\setminus K)$ agrees with the infimum of 
$F$
among all the pairs $(u,K)$ with $K$ piecewise smooth and $u\in
C^\infty(\Omega\setminus K)$.  This will easily follow by a general
approximation result concerning $SBV$ functions.

Finally, we remark that all the results of this paper are independent of
\cite{ap}, \cite{afp}, and of the existence of minimizers; we simply 
assume
that a minimizing pair $(u,K)$ exists and that, in some region $A$, $K$
is the graph of a $C^{1,\rho}$ function. It has been proved
in \cite{ap}, \cite{afp} that ${\cal H}^{n-1}$-all points of a minimizer
$K$ have a neighbourhood $A$ where this property holds.
This paper, except for the
last section, is independent also of the theory of $SBV$ functions, whose
main properties will be recalled at the beginning of the last section.

\section{Regularity of Neumann problems in $C^{1,\rho}$ domains}

Let us recall the definition of the Morrey and Campanato
spaces in a regular domain $\Om\subset\rn{n}$.
For more information we refer to \cite{gia}.
The space $L^{2,\lambda}(\Om)$, with $0\leq\lambda\leq n$, is the 
Morrey
space of the $L^2(\Om)$ functions $u$ satisfying
$$
\sup_{x\in\Omega,\,\varrho>0} \varrho^{-\lambda}
\int_{B_\varrho(x)\cap\Omega}|u|^2\, dy<+\infty ;
$$
for $0\leq\lambda\leq n+2$, ${\cal L}^{2,\lambda}(\Om)$ denotes the
Campanato space of the $L^2(\Om)$ functions satisfying
$$
\sup_{x\in\Omega,\,\varrho>0} \varrho^{-\lambda}
\int_{B_\varrho(x)\cap\Omega}|u-u_{x,\varrho}|^2\, dy<+\infty,
$$
where $u_{x,\varrho}$ is the mean value of $u$ in 
$B_\varrho(x)\cap\Omega$.
Finally, notice that for $\lambda>n$, ${\cal L}^{2,\lambda}(\Om)$ is
isomorphic to $C^{0,\gamma}(\overline{\Om})$ for $\gamma=(\lambda-
n)/2$.

Let $u\in H^1(\Om)$ be satisfying, in the distribution sense
\begin{equation}\label{eq66}
-\Delta u=f\quad\hbox{\rm in }\Om,\qquad\qquad
\frac{\partial u}{\partial n}=0\quad\hbox{\rm on }\Gamma
\end{equation}
where $\Gamma\subset\partial\Om$ is a $C^{1,\rho}$ hypersurface,
relatively open in $\partial\Om$ and $f\in L^\infty(\Om)$.

In this section we investigate on the regularity of $\nabla u$ near
$\Gamma$.  In the two dimensional case the regularity of $\nabla u$ 
can be
easily achieved using conformal mappings and Shwartz's reflection
principle:

\begin{theorem}\label{neureg1}
Assume $n=2$.
Then $\nabla u$ has an extension to
$\Om\cup\Gamma$ which is locally $C^{0,s}$ for any $s<\rho$.
\end{theorem}
{\sc Proof.} It is not restrictive to assume $\rho<1$.
For any $x_0\in\Gamma$ and for $\varrho>0$ sufficiently small
there is a $C^{1,\rho}$ conformal mapping $\Phi$ between
$B_\varrho(x_0)\cap\Om$ and a half ball $B^+$ such that
$B_\varrho(x_0)\cap\partial\Om=B_\varrho(x_0)\cap\Gamma$
is mapped onto the flat part $H$ of $\partial B^+$. Setting
$$v(y)=u(\Phi^{-1}(y)), \qquad
g(y)=f(\Phi^{-1}(y))|{\rm det} J\Phi^{-1}(y)|$$
the function $v$ solves $\Delta v=g$ in $B^+$ with homogeneous 
Neumann
boundary conditions on $H$. By the boundary regularity of conformal
mappings between $C^{1,\rho}$ domains
(see for instance \cite{po}, Theorem 3.6) $g$ is bounded near
$H$. By an easy reflection argument the functions $v$ and $g$
can be extended to the whole ball $B$ in such a way that
$\Delta v=g$ in $B$. In particular, $v\in C^{1,s}_{\rm loc}(B)$
for any $s<1$. Since $u=v\circ\Phi$, again the $C^{1,\rho}$
regularity of $\Phi$ implies our statement.\qed


In higher dimensions, we still use a change of domain and a reflection
argument. However, the change of variables changes the equation hence 
the
proof is much more technical and our result is not sharp as in dimension 
2.

\begin{theorem}\label{neureg}
Under the assumptions above, the function $\nabla u$ has a locally
H\"older continuous extension to
$\Om\cup\Gamma$.
\end{theorem}

Equations (\ref{eq66}) are equivalent to

\begin{equation}\label{eq1}
\int_\Om \langle\nabla u(x),\nabla\varphi(x)\rangle\,dx=
\int_\Om f(x)\varphi(x)\,dx
\end{equation}
for any $\varphi\in C^1_0(\rn{n})$ with $\varphi=0$ on
$\partial\Omega\setminus\Gamma$.
We know that for any $x_0\in\Gamma$ there is a $C^{1,\rho}$
diffeomorphism $\Phi$ of a neighbourhood $U$ of $x_0$ and the unit 
cube
$Q$ mapping $U\cap\Om$ into $Q^+=Q\cap\{x_n>0\}$ and
$U\cap\partial\Om$ into $H=Q\cap\{x_n=0\}$.
Then, assuming that $U$ is sufficiently small to have the inclusion
$U\cap\partial\Om\subset\Gamma$, condition (\ref{eq1}) implies

\begin{equation}\label{eq2}
\int_{Q^+} A_{ij}
\frac{\partial v}{\partial y_i}
\frac{\partial\phi}{\partial y_j}\,dy=\int_{Q^+}  g(y)\phi(y)\,dy
\end{equation}
for any $\phi\in C^1_0(\rn{n})$ with $\phi=0$ on $\partial Q^+\setminus 
H$,
and with $v(y)=u\bigl(\Phi^{-1}(y)\bigr)$, $g(y)=
f\bigl(\Phi^{-1}(y)\bigr)\bigl\vert{\rm det\,} J\Phi^{-1}(y)
\bigr\vert$ and

\begin{equation}\label{eq3}
A_{ij}(y)=\sum_{k=1}^n
\frac{\partial\Phi^i}{\partial x_k}\bigl(\Phi^{-1}(y)\bigr)
\frac{\partial\Phi^j}{\partial x_k}\bigl(\Phi^{-1}(y)\bigr)
\bigl\vert{\rm det\,} J\Phi^{-1}(y)\bigr\vert.
\end{equation}

The fuctions $A_{ij}$ are H\"older continuous and $g$ is bounded.
Moreover, choosing $U$
sufficiently small and $\Phi$ sufficiently close in $C^1$ norm to a
rotation we can assume that the matrix $A$ is uniformly elliptic, {\em
i.e.},
\begin{equation}\label{eq4}
A_{ij}(y)\xi_i\xi_j\geq \theta|\xi|^2\qquad\qquad\forall\xi\in\rn{n},\,\,
\forall y\in Q
\end{equation}
for some constant $\theta>0$.
Then, since $x_0\in\Gamma$ is arbitrary, Theorem~\ref{neureg} is a
straightforward consequence of the following result:

\begin{theorem}\label{neureg2}
Let $v\in H^1(Q^+)$ be satisfying (\ref{eq2}) with the matrix $A$
$\rho$-H\"older continuous, uniformly elliptic in $Q^+$ and
$g\in L^\infty(Q^+)$. Then, $\nabla v$ has a locally H\"older
continuous extension to
$Q^+\cup H$.
\end{theorem}

The proof of Theorem \ref{neureg2} will be achieved by a reflection
argument. For any $x=(x',x_n)\in Q^-$ we define $v(x)=v(x',-x_n)$,
$g(x)=g(x',-x_n)$ and
$$
A_{ij}(x',x_n)=\cases{\,\,\,\,A_{ij}(x',-x_n) &
if either $1\leq i,\,j<n$ or $i=j=n$;
\cr\cr -A_{ij}(x',-x_n) & otherwise.}
$$
In this way the coefficients $A_{in}(x),\,A_{ni}(x)$ are (for
$i<n$) odd with respect to $x_n$, while $v$, $g$ and all the other
coefficients are even with respect to the same variable. We point out 
that

\smallskip
(a) $A\in C^{0,\rho}(Q^+)\cap C^{0,\rho}(Q^-)$;

(b) $A$ is uniformly elliptic.
\smallskip

\noindent
This extension of $v$, $g$ and $A$ to $Q$ is motivated by the equation

\begin{equation}\label{eq5}
\int_Q A_{ij}
\frac{\partial v}{\partial y_i}
\frac{\partial\phi}{\partial y_j}\,dy=\int_\Om g(y)\phi(y)\,dy
\qquad\qquad\forall\phi\in C^1_0(Q)
\end{equation}
that immediately follows by (\ref{eq2}). Then, Theorem~\ref{neureg2}
is a direct consequence of the following interior regularity result:

\begin{theorem}\label{neureg3}
Let $v\in H^1(Q)$ be satisfying (\ref{eq5}) and assume that
$A$ fulfils (a), (b). Let

$$
D_\tau v=\bigl(\nabla_1 v,\ldots,\nabla_{n-1}v\bigr),
\qquad\qquad
\dc v=\sum_{i=1}^n A_{in}\nabla_i v.$$
Then, $D_\tau v$ and $\dc v$ are locally H\"older continuous
in $Q$.
\end{theorem}
{\sc Proof.} In the following,
$c$ stands for a generic (computable) constant depending only on
$n$, $\rho$,
the ellipticity constant $\theta$, the H\"older norm
of $A$ in $Q^+$ and in $Q^-$ and the $L^\infty$ norm of $g$ in $Q$.


Let $B_R(x_0)$ be a ball contained in $Q$. We denote by
$A^+$ and $A^-$ the averages of $A$ on $B_R(x_0)\cap Q^+$ and
$B_R(x_0)\cap Q^-$ respectively, for $x\in B_R(x_0)$ we set
$$
\bar{A}(x)=\cases{
A^+& if $x_n>0$;
\cr
A^- & if $x_n<0$}
$$
and we denote by $w\in H^1(B_R)$ the solution of the
problem

\begin{equation}\label{zeq1}
-{\rm div}\bigl(\bar{A}(x-x_0)\nabla w(x)\bigr)=0,\qquad\qquad
w(x)=v(x+x_0)\quad\hbox{\rm on }\partial B_R.
\end{equation}
The H\"older continuity of $A$ implies

\begin{equation}\label{zeq2}
|A(x)-\bar{A}(x)|\leq cR^\rho\qquad\qquad\forall x\in B_R(x_0).
\end{equation}
Setting

$$ \dd w(x)= \sum_{i=1}^n \bar{A}_{in}(x) \nabla_i w(x)$$
the plan of the proof is the following:
in the first step we estimate $D_\tau w$; in the second step
we estimate $\dd w$; in the last ones we get a $L^{2,\lambda}$ estimate
on $\nabla v$ and eventually ${\cal L}^{2,\mu}$ estimates which lead
to the H\"older continuity of $D_\tau w$ and $\dc w$.

{\sc Step 1.} Using the fact that $\bar{A}$ is constant  in the tangencial
directions, a standard application of Nirenberg's
difference quotient method gives
that $D_\tau w\in H^1_{\rm loc}(B_R)$ and the Caccioppoli estimate

\begin{equation}\label{zeq55}
\int_{B_\varrho}|\nabla (D_\tau w)|^2\,dx\leq c\varrho^{-2}
\int_{B_{2\varrho}}|D_\tau w-(D_\tau w)_{2\varrho}|^2\,dx
\end{equation}
for $0<\varrho\leq R/2$, where as usual $(h)_\varrho$ denotes the 
average of
the function $h$ in $B_\varrho$. Moreover, $D_\tau w$ satisfies

$$
\int_{B_R}\bar{A}_{ij} \frac{\partial D_\tau w}{\partial x_i}
\frac{\partial \phi}{\partial x_j}\,dx=0\qquad\qquad\forall\phi\in
C^1_0(B_R).
$$
The regularity theory for solutions $u$ of elliptic equations
with measurable coefficients
(see for instance \cite{gia}, page 80 and page 82)
yields the estimates
$$\varrho^n\max_{B_\varrho}|u|^2\leq c\int_{B_{2\varrho}}|u^2(x)|\,dx
\qquad 0<\varrho\leq R/2$$
$$\int_{B_\varrho}|\nabla u|^2\,dx\leq
c\biggl(\frac{\varrho}{R}\biggr)^{n-2+2\gamma}R^{-2}
\int_{B_R}|u-(u)_R|^2\,dx\qquad 0<\varrho\leq R.$$

\noindent
Applying these estimates to $u=D_\tau w$ we infer
\begin{equation}\label{zeq3}
\varrho^n\sup_{B_\varrho} |D_\tau w|^2\leq c\int_{B_{2\varrho}}
|D_\tau w|^2\,dx \qquad\qquad 0<\varrho\leq R/2
\end{equation}

\begin{eqnarray}    \label{zeq4}
&&\int_{B_\varrho}|D_\tau w-(D_\tau w)_\varrho|^2\,dx
\nonumber\\
&\leq& c\biggl(\frac{\varrho}{R}\biggr)^{n+2\gamma}
\int_{B_R}|D_\tau w-(D_\tau w)_R|^2\,dx\qquad
0<\varrho\leq R
\end{eqnarray}

\noindent
for some $\gamma>0$ depending only on $n$, the ellipticity constant
$\theta$ and the $L^\infty$ norm of $A$.

{\sc Step 2.} Since $D_\tau (\dd w)=\dd (D_\tau w)$, we have that
$\dd w$ has tangencial distributional derivatives in $L^2_{\rm 
loc}(B_R)$.
To estimate the normal derivative of $\dd w$ we use equation
(\ref{zeq1}). Indeed, the equality

$$
\nabla_n (\dd w)=-\sum_{i=1}^{n-1}\sum_{j=1}^n \bar{A}_{ij}
\nabla^2_{ij} w
$$
implies that also
$\nabla_n (\dd w)$ belongs to $L^2_{\rm loc}(B_R)$. Moreover,
we have the estimate

\begin{equation}\label{zeq5}
|\nabla (\dd w)|\leq c|\nabla (D_\tau w)|.
\end{equation}
Using Poincar\'e inequality, (\ref{zeq5}) and (\ref{zeq55})
we get

\begin{eqnarray*}
\int_{B_\varrho}|\dd w-(\dd w)_\varrho|^2\,dx&\leq&
c\varrho^2\int_{B_\varrho}|\nabla (\dd w)|^2\,dx
\\
&\leq& c\varrho^2\int_{B_\varrho}|\nabla (D_\tau w)|^2\,dx
\\
&\leq& c\int_{B_{2\varrho}}|D_\tau w-(D_\tau w)_{2\varrho}|^2\,dx,
\end{eqnarray*}
hence (\ref{zeq4}) yields

\begin{equation}\label{zeq44}
\int_{B_\varrho}|\dd w-(\dd w)_\varrho|^2\,dx\leq
c\biggl(\frac{\varrho}{R}\biggr)^{n+2\gamma}
\int_{B_R}|D_\tau w-(D_\tau w)_R|^2\,dx.
\end{equation}
The same estimates are valid in balls not centered at the origin, hence
the isomorphism between the Campanato space ${\cal 
L}^{2,n+2\gamma}$ and
$C^{0,\gamma}$ yields
$$ \Vert \dd w\Vert_{C^{0,\gamma}(B_{R/2})}\leq cR^{-n-
2\gamma}\int_{B_R}
|\nabla w|^2\,dx.$$
Since the integral of $\dd w$ on $B_{R/2}$ can be estimated with the
integral of $\nabla w$ on $B_{R/2}$ we infer
\begin{equation}\label{zeq6}
R^n\sup_{B_{R/2}} |\dd w|^2\leq c\int_{B_R} |\nabla w|^2\,dx.
\end{equation}

{\sc Step 3.} Let $\hat{v}(x)=w(x-x_0)$. By (\ref{eq5}) and (\ref{zeq1})
we infer

\begin{eqnarray*}
&&\int_{B_R(x_0)}\bar{A}_{ij}(x)\left(
\frac{\partial v}{\partial x_i}-\frac{\partial \hat{v}}{\partial x_i}
\right)\frac{\partial\phi}{\partial x_j}\,dx
\\
&=&
\int_{B_R(x_0)}\left[\bar{A}_{ij}(x)-A_{ij}(x)\right]
\frac{\partial v}{\partial x_i}\frac{\partial\phi}{\partial x_j}\,dx+
\int_{B_R(x_0)} g(x)\phi(x) \,dx
\end{eqnarray*}
for any $\phi\in H^1_0(B_R(x_0))$. Choosing $\phi=v-\hat{v}$,
the ellipticity of $\bar{A}$, (\ref{zeq2})
and Poincar\'e inequality yield

\begin{equation}\label{zeq7}
\int_{B_R(x_0)}|\nabla v-\nabla\hat{v}|^2\,dx\leq cR^{2\rho}
\int_{B_R(x_0)}|\nabla v|^2\,dx+cR^{n+2}.
\end{equation}

To estimate the Dirichlet integral we notice that (we use (\ref{zeq3}),
(\ref{zeq6}) and (\ref{zeq7})) for $0<\varrho\leq R/2$ we have

\begin{eqnarray*}
&&\int_{B_\varrho(x_0)}|\nabla v|^2\,dx
\\
&\leq& 
2\int_{B_\varrho(x_0)}|\nabla\hat{v}|^2\,dx+2\int_{B_\varrho(x_0)}
|\nabla v-\nabla\hat{v}|^2\,dx
\\
&\leq& c\varrho^n\sup_{B_{R/2}(x_0)}|\nabla\hat{v}|^2+
cR^{2\rho}\int_{B_R(x_0)}|\nabla v|^2\,dx+cR^{n+2}
\\
&\leq& 
c\left(\frac{\varrho}{R}\right)^n\int_{B_R(x_0)}|\nabla\hat{v}|^2\,dx+
cR^{2\rho}\int_{B_R(x_0)}|\nabla v|^2\,dx+cR^{n+2}
\\
&\leq& c\left(\frac{\varrho}{R}\right)^n\int_{B_R(x_0)}|\nabla v|^2\,dx+
cR^{2\rho}\int_{B_R(x_0)}|\nabla v|^2\,dx+cR^{n+2}
\end{eqnarray*}
hence, by applying Lemma \ref{Lp} below, with $\alpha=n$, $\beta<n$, 
to

$$f(\varrho)=\int_{B_\varrho(x_0)}|\nabla v|^2\,dx$$
we infer

\begin{equation}\label{zeq8}
|\nabla v|\in L^{2,\lambda}_{\rm loc}(Q)\qquad\qquad\forall
\lambda<n.
\end{equation}
In the following we will use (\ref{zeq8}) with $\lambda$ sufficiently
close to $n$, such that $n-\lambda<\rho$.

{\sc Step 4.} Let $Q'$ be a cube strictly
contained in $Q$ and assume that $2R<\delta$, $\delta$ being the
distance of $Q'$ from $\partial Q$.
For $x_0\in Q'$, $0<\varrho\leq R$ and $d\in\rn{}$, using
(\ref{zeq2}), (\ref{zeq7}), (\ref{zeq8}) we have
\begin{eqnarray*}
&&\int_{B_\varrho(x_0)}|\dc v-d|^2\,dx
\\
&\leq& 2\int_{B_\varrho(x_0)}|\dd v- d|^2\,dx
+c\varrho^{2\rho}\int_{B_\varrho(x_0)}|\nabla v|^2\,dx
\\
&\leq& c\int_{B_\varrho}|\dd w- d|^2\,dx
+cR^{n+\rho}.
\end{eqnarray*}
Using $d=(\dd w)_\varrho$ in the inequality above
and also (\ref{zeq44}) we get

\begin{eqnarray}\label{zeq47}
&&\int_{B_\varrho(x_0)}|\dc v-(\dc v)_{x_0,\varrho}|^2\,dx
\nonumber\\
&\leq&c\left(\frac{\varrho}{R}\right)^{n+2\gamma}
\int_{B_R(x_0)}|D_\tau v-(D_\tau v)_{x_0,R}|^2\,dx+
cR^{n+\rho}.
\end{eqnarray}
By a similar argument,
based on (\ref{zeq4}), we get

\begin{eqnarray}\label{zeq48}
&&\int_{B_\varrho(x_0)}|D_\tau v-(D_\tau v)_{x_0,\varrho}|^2\,dx
\nonumber\\
&\leq&
c\left(\frac{\varrho}{R}\right)^{n+2\gamma}
\int_{B_R(x_0)}|D_\tau v-(D_\tau v)_{x_0,R}|^2\,dx+
cR^{n+\rho}.
\end{eqnarray}
Inequalities (\ref{zeq47}), (\ref{zeq48}) together with
Lemma \ref{Lp}, where we take $\alpha\!=\!n+2\gamma\!$, $\beta<
\min\{n+\rho,\alpha\}$ imply
that $\dc v$ and $D_\tau v$ belong to the Campanato space
${\cal L}^{2,n+\sigma}(Q')$, with any $\sigma<\min\{\rho,2\gamma\}$,
hence they are  H\"older continuous in $Q'$.
\qed



\begin{lemma} \label{Lp}
Let $f:(0,R_0]\to [0,+\infty)$ be a nondecreasing
function and assume that

$$
f(\varrho)\leq A\biggl[\left(\frac{\varrho}{R}\right)^\alpha+R^{\rho}
\biggr]f(R)+BR^\beta\qquad\qquad 0<\varrho\leq R/2<R\leq R_0
$$
for some constants $A,\,B\geq 0<\beta<\alpha,\,\rho>0$. Then,
the supremum of $f(\varrho)/\varrho^\beta$ in $(0,R_0]$
is bounded by a constant depending only on $A,\,B,\,\alpha,\,
\beta,\,\rho$, $R_0$ and $f(R_0)$.
\end{lemma}
{\sc Proof.} Let us fix $\beta<\gamma<\alpha$ and $0<\tau\leq 1/2$ 
such
that $2A\tau^\alpha \leq\tau^\gamma$. If we choose $R_1\leq R_0$
such that $R_1^\rho \leq\tau^\alpha$ from the assumption we get for
any $i=1,2,\ldots$
$$
f(\tau^{i+1}R_1)\leq\tau^\gamma f(\tau^iR_1)+B\tau^{i\beta}R_1^{\beta}.
$$
Therefore, iterating this estimate, we obtain for any positive integer
$k$
\begin{eqnarray*}
f(\tau^{k+1}R_1) &\leq& \tau^{\gamma k}f(\tau R_1)+B\tau^{k\beta}
R_1^{\beta}\sum_{i=1}^{k-1}\tau^{i(\gamma-\beta)}
\\
&\leq& c\tau^{k\beta}\left[ f(R_1)+BR_1\right] .
\end{eqnarray*}
If $\varrho\leq R_1/2$, choosing $k$ such that $\tau^{k+1}R_1<\varrho
\leq\tau^k R_1$, we then get
$$
\frac{f(\varrho)}{\varrho^\beta}\leq c\biggl[\frac{f(R_1)}{R_1^\beta}+
B\biggr]\leq c\left[ f(R_0)+B\right]
$$
and from this inequality the result immediately follows.
\qed


\section{Regularity of solutions of the mean curvature equation}

In this section we prove an interior regularity result for
solutions $\phi\in C^1(D)$ of the prescribed mean curvature equation
\begin{equation}\label{mcm1}
-{\rm div}\left(\frac{\nabla\phi}{\sqrt{1+|\nabla\phi|^2}}\right)=H
\qquad\qquad\hbox{\sl in }D.
\end{equation}
Here $D\subset\rn{m}$ is an open set; coupling the results of this
section (with $m=(n-1)$) with the results of the previous one, the
higher regularity of the discontinuity set will be achieved in the
forthcoming section.

Our result is the following:

\begin{theorem}\label{meanreg}
Assume that (\ref{mcm1}) holds with $H\in L^\infty(D)$ and
$\phi\in C^1(D)$. Then,
$\phi\in C^{1,1}_{\rm loc}(D)$ if $m=1$ and
$\phi\in H^2_{\rm loc}(D)$,
$\nabla\phi\in C^{0,\rho}_{\rm loc}(D;\rn{m})$ for any
 $\rho<1$ if if $m>1$.
\end{theorem}
{\sc Proof.}
In the case $m=1$,
the equation trivially implies that $\phi'/\sqrt{1+|\phi'|^2}$ is locally
Lipschitz continuous in $D$, hence $\phi$ is locally $C^{1,1}$ in $D$. In
the case $m>1$, we notice that $\phi$ satisfies an equation of the
form
\begin{equation}\label{eqF}
-{\rm div} \bigl( F(\nabla\phi)\bigr)=H\in L^\infty(D)
\end{equation}
with $F(z)=z/(1+|z|^2)$. Being $F$ the gradient of $\sqrt{1+|z|^2}$,
$F$ is locally strictly monotone.
Then, the $H^2$ regularity of $\phi$ follows by Poposition~\ref{H2} 
below;
the H\"older continuity of $\nabla\phi$ follows by writing the equation 
in non
divergence form
$$
\sum_{i,\,j=1}^m
A_{ij}(x){\partial^2\phi\over\partial x_i\partial x_j}(x)=-H(x)
$$
with
$$
A_{ij}(x)=\frac{\delta_{ij}-\nu_i(x)\nu_j(x)}{(1+|\nabla\phi(x)|^2)^{1/2}},
\qquad \nu_i(x)=
\frac{\frac{\partial\phi}{\partial x_i}(x)}
{(1+|\nabla\phi(x)|^2)^{1/2}}
$$
and applying the regularity theory for equations in non-divergence 
form with continuous coefficients (see for instance \cite{gt},  Theorem 
9.15).
\qed

\begin{proposition}\label{H2} Let $\phi\in C^1(D)$ be a solution,
in the distribution sense, of equation (\ref{eqF}),
where $F\in C^1(\rn{m})$ is a locally strictly monotone function, {\em
i.e.}, for any compact set $K\subset\rn{m}$ we have
\begin{equation}\label{mono}
(F(z)-F(z'),z-z')\geq c_K|z-z'|^2\qquad\qquad\forall z,\,z'\in K
\end{equation}
for a suitable constant $c_K>0$. Then, $\phi\in H^2_{\rm loc}(D)$.
\end{proposition}
{\sc Proof.} Possibly reducing $D$ we can assume that the gradient
of $\phi$ is bounded by a constant $M$ on $D$. We also assume, to
simplify some integral estimates, that $D$ is a ball of radius less than 1.
We denote by $c$ the constant in (\ref{mono})
corresponding to the compact set $K=\overline{B}_M(0)$ and by
$L$ the Lipschitz constant of $F$ on $K$.

Given an integer $s$ in $[1,m]$, let
$\Delta_hv(x)=[v(x+he_s)-v(x)]/h$ be the difference quotient
of a generic function $v$ in the direction $e_s$. We will
use the following inequality
\begin{equation}\label{basic}
\int_{\rn{m}}\psi^2(x)|\Delta_{-h} v(x)|\,dx\leq
\int_{\rn{m}}\chi_h(x)|\nabla_s v(x)|\,dx
\end{equation}
with
$$ \chi_h(x)=\int_0^1 \psi^2(x+the_s)\,dt,$$
which is true provided $\psi$ is continuous with
compact support, and $v$ is $C^1$ in the $h$-neigbourhood of the
support of $\psi$. The proof is standard, being based on the
integration of
$$ \Delta_{-h}v(x)=\int_0^1 \nabla_s v(x-the_s)\,dt $$
and Fubini's theorem, so we omit it.

Let $\eta\in C^1_0(D)$; applying (\ref{eqF}) (for $h$ sufficiently small,
depending on the support of $\eta$) with test functions
$\eta(x)$ and $\eta(x-he_s)$ and subtracting both sides we get
\begin{eqnarray}     \label{basic1}
&&{1\over h}\int_D [F_i(\nabla\phi(x+he_s))-F_i(\nabla\phi(x))]
\nabla_i\eta(x)\,dx
\nonumber \\
&=&-\int_D H(x)\Delta_{-h}\eta(x)\,dx.
\end{eqnarray}
Now we choose $\psi\in C^1_0(D)$ such that $0\leq\psi\leq 1$
and apply (\ref{basic1})
with $\eta=\psi^2\Delta_h\phi$; the integral containing $H$
is equal to
$$\int_D H\Delta_{-h}\psi^2\Delta_{-h}\phi\,dx+
\int_D H\psi^2\Delta_{-h}\Delta_h\phi\,dx$$
and can be estimated from above,
using (\ref{basic}) with $v=\Delta_h\phi$, by
$$
\Vert H\Vert_\infty\bigl[2M\Vert\nabla\psi\Vert_\infty+
\int_D\chi_h|\Delta_h\nabla_s\phi|\,dx\bigr].
$$
Adding and subtracting $\psi^2$, and using
$\Vert\chi_h-\psi^2\Vert_\infty\leq 2h\Vert\nabla\psi\Vert_\infty$,
this term can be estimated from below by
$$
\Vert H\Vert_\infty\bigl[6M\Vert\nabla\psi\Vert_\infty+
\int_D\psi^2|\Delta_h\nabla_s\phi|\,dx\bigr].
$$
Now we estimate from below the first term in (\ref{basic1}). Since
$\nabla\eta=\nabla\psi^2\Delta_h\phi+\psi^2\Delta_h\nabla\phi$, using
(\ref{mono}) we obtain the estimate
$$ c\int_D \psi^2|\Delta_h\nabla\phi|^2\,dx-
2LM\Vert\nabla\psi\Vert_\infty\int_D \psi|\Delta_h\nabla\phi|\,dx.$$
Taking into account these estimates, we obtain that
$$ \int_D \psi^2|\Delta_h\nabla\phi|^2\,dx$$
is bounded as $h\to 0$. Hence, $\nabla_s\phi\in H^1$ in the interior
of the support of $\psi$. Since $\psi$ is arbitrary, the statement
follows.\qed

\section{Higher regularity of the discontinuity set}

Let $A\subset\rn{n}$ be an open set, $g\in L^\infty(A)$,
and let $(u,\Gamma)$ be an admissible pair for the Mumford-Shah
functional $F$, minimizing $F$ with respect to relatively compact
perturbations in $A$ of $u$ and $\Gamma$. In this section
we assume that $\Gamma\cap A$ is a $C^{1,\rho}$ hypersurface, we 
assume
for simplicity $\alpha=\beta=1$ in (\ref{msfunct}) and we
investigate the higher regularity of $\Gamma$.

There is no loss of generality if we assume that $A=D\times (-1,1)$
for some open set $D\subset\rn{n-1}$ and that $\Gamma\cap A$ is the
graph of a $C^{1,\rho}$ function $\phi$ defined on $D$ such that
$\Vert\phi\Vert_{\infty}=\varrho <1$. We denote
by $A^+\subset A$ and $A^-\subset A$ respectively the epigraph and
the hypograph of $\phi$.

By the regularity results of the preceding sections (see
Theorem~\ref{neureg} or, in the case $n=2$, Theorem~\ref{neureg1}) 
applied
with $\Om=A^\pm$, we can also assume that $u$ and $\nabla u$ are 
H\"older
continuous up to $\Gamma$.  We will denote by $(z,y)$ the generic point 
in
$A$, with $z\in D$ and $y\in (-1,1)$.  We first see what can be said
without extra regularity assumptions on $g$:

\begin{theorem}\label{meanreg1}
Under the assumptions above, $\phi$ is a solution in the distribution
sense of the
prescribed mean curvature equation (\ref{mcm1})
for some $H\in L^\infty(D)$.
In particular, by Theorem~\ref{meanreg},
$\phi\in C^{1,1}_{\rm loc}(D)$ if $n=2$ and
$\phi\in H^2_{\rm loc}(D)$,
$\nabla\phi\in C^{0,\rho}_{\rm loc}(D;\rn{n-1})$ for any
$\rho<1$ if $n>2$.
\end{theorem}
{\sc Proof.} We will first show a minimality condition satisfied by 
$\phi$;
then, a suitable choice of the comparison function will allow to conclude.

{\sc Step 1.} Let us prove, by a deformation argument,
that $\phi$ satisfies the following minimality condition:
\begin{equation}\label{qmin}
\int_D\sqrt{1+|\nabla\phi|^2}\,dz\leq
\int_D\sqrt{1+|\nabla(\phi+\psi)|^2}\,dz+
\lambda\int_D |\psi(z)|\,dz
\end{equation}
for any function $\psi\in C^1_0(D)$ such that 
$\Lambda\Vert\psi\Vert_{C^1}
\leq 1$. The constants $\lambda$,
$\Lambda$ depend only on $\phi$ but not on $\psi$.

Indeed, let $\Gamma'$ be the graph of $\phi+\psi$ and assume
that $\Lambda$ is so large that $\Lambda\Vert\psi\Vert_{C^1}\leq 1$
implies $\Vert\phi\Vert_\infty+2\Vert\psi\Vert_\infty\leq 1$ and
$\Vert\nabla\psi\Vert_\infty\leq 1$. Let $x=(z,y)$ and let
$\Phi:A\to A$ be the map defined by $\Phi(z,y)=(z,L_z(y))$ where
$L_z(y):[-1,1]\to [-1,1]$ is a piecewise linear bijection,
$1/2\leq L_z'\leq 3/2$,
and for any $z\in D$ satisfies the properties
$$ L_z(\phi(z))=\phi(z)+\psi(z),\qquad
L_z(y)=y\quad\hbox{\rm if }|y-\phi(z)|\geq 2|\psi(z)|.
$$
It is easy to check that the restriction of $\Phi$ to $\partial A$
is the identity map, and that $\Phi$ is invertible. Moreover,
since the coefficients of $L_z(y)$ are linear functions of
$\phi(z)$, $\psi(z)$, the Lipschitz constant of
$\Phi^{-1}$ can be estimated with the
Lipschitz constants of $\phi,\,\phi+\psi$. In addition,
\begin{equation}\label{Phiid}
{\cal L}^n\bigl(\{x\in A:\Phi(x)\neq x\}\bigr)\leq
4\int_D |\psi(z)|\,dz.
\end{equation}

The function $v=u\circ\Phi^{-1}$
has the same trace of $u$ on $\partial A$
and its discontinuity set is contained in $\Gamma'$. Taking into
account the boundedness of $u$ and $\nabla u$, comparing the
energy of $(u,\Gamma)$ with the energy of $(v,\Gamma')$ and taking
into account
$$
{\cal L}^n\bigl(\{x\in A:u(x)\neq v(x)\}\bigr)\leq
4\int_D |\psi(z)|\,dz.
$$
we obtain
$$ {\cal H}^{n-1}(\Gamma)\leq {\cal H}^{n-1}(\Gamma')+
4\lambda\int_D |\psi(z)|\,dz$$
for $\lambda$ large enough (depending on the bounds on $\nabla u$ 
and on
$\nabla\Phi^{-1}$), as claimed.

{\sc Step 2.} Let $\varphi\in C^1_0(D)$ and $\eps>0$.
Taking $\psi=\eps\varphi$ in the minimality
condition (\ref{qmin}), differentiation in $\eps$ gives
$$ -\int_D \frac{\langle\nabla\phi,\nabla\varphi\rangle}
{\sqrt{1+|\nabla\phi|^2}}\,dz
\leq 4\lambda\Vert\varphi\Vert_1.$$
Since $\varphi$ is arbitrary, equation (\ref{mcm1}) is satisfied
in the distribution sense for some function $H$ whose $L^\infty$
norm does not exceed $4\lambda$.\qed

Now we make additional regularity assumptions on $g$. We need
at least $C^1$ regularity to start the bootstrap argument.

\begin{theorem}\label{meanreg2} If $g\in C^{k,\rho}_{\rm loc}(A)$
for some integer $k\geq 1$ and some $\rho\in (0,1]$, then there exists
$s\in (0,1]$ depending on $n$,\, $\rho$ such that $\phi$ belongs to
$C^{k+2,s}_{\rm loc}(D)$.

In particular, the smoothness of $g$ implies the smoothness
of the free discontinuity set $\Gamma$.
\end{theorem}
{\sc Proof.} For simplicity's sake, let be $\alpha = \beta = 1$. We first
assume $k=1$. We know that $u$ satisfies the equation
\begin{equation}\label{neu0}
\Delta u=(u-g)\quad\hbox{\rm in }A^\pm,\qquad
{\partial u\over\partial \nu}=0\quad\hbox{\rm in }\Gamma.
\end{equation}
We recall that,
by Theorem~\ref{neureg}, $u$ and $\nabla u$ are
H\"older continuous up to $\Gamma$, on both sides. Orienting
with the inner normal to $A^+$
$$
\nu=\frac{(-\nabla\phi ,1)}{\sqrt{1+|\nabla\phi|^2}}
$$
we will prove by a deformation argument that the scalar mean 
curvature of $\Gamma$
at $x$ is given by the H\"older continuous function
\begin{equation}\label{mcm2}
H(x)=\bigl[|\nabla u|^2+(u-g)^2\bigr](x),
\end{equation}
where $[\ \cdot\ ]$ denotes the jump across $\Gamma$. Indeed, if 
$\eta\in
C_0^\infty(A)$ is given, setting for $\eps$ small $\tau_\eps (x)=
x+\eps\eta(x)$ and $u_\eps(\tau_\eps(x))=u(x)$, computing the
first variation along $u_\eps$ leads to the following equation:
\begin{eqnarray}    \label{firstvar}
&&\int_A\bigl[\bigl(|\nabla u|^2+(u-g)^2\bigr){\rm div}\eta-
2(u-g)\langle\nabla g,\eta\rangle
- 2\langle\nabla u,\nabla u\nabla\eta\rangle\bigr] dx
\nonumber \\
&&=-\int_\Gamma\delta_i\eta_i d{\cal H}^{n-1}
\end{eqnarray}
where $\delta$ stands for the tangencial differential operator.
Fix a cut-off function $\chi\in C_0^1(-1,1)$ such that $\chi\equiv 1$
in $[-\varrho,\varrho]$ (recall that $\varrho=\Vert\phi\Vert_\infty$)
and define
$\eta_i(x)=0$ for $i<n$ and
$$\eta_n(x_1,\ldots,x_n)=\varphi(x_1,\ldots,x_{n-1})\chi(x_n)$$
with $\varphi\in C^1_0(D)$. An easy computation shows that
\begin{equation}\label{firstvar1}
\int_\Gamma\delta_i\eta_i d{\cal H}^{n-1}=
\int_D \frac{\langle\nabla\phi,\nabla\varphi\rangle}
{\sqrt{1+|\nabla\phi|^2}}\,dz.
\end{equation}
Integrating by parts on $A^+$ and using the equation $\Delta u=(u-g)$
and the homogeneous Neumann boundary conditions
we obtain
\begin{eqnarray*}
&&\int_{A^+}\bigl[\bigl(|\nabla u|^2+(u-g)^2\bigr){\rm 
div}\eta\bigr]\,dx
-2\int_{A^+}\langle\nabla u,\nabla u\nabla\eta\rangle\,dx
\\
&=&-\int_\Gamma
\bigl[|\nabla u|^2+(u-g)^2\bigr]^+\langle\eta,\nu\rangle\,d{\cal H}^{n-1}
-\int_{A^+}\bigl\langle\eta,\nabla \bigl(|\nabla u|^2+(u-g)^2\bigr)
\bigr\rangle\,dx
\\
&&+2\int_{A^+}\Delta u\langle\eta,\nabla u\rangle+
\langle\eta,\nabla u\nabla^2 u\rangle\,dx
\\
&=&-\int_\Gamma
\bigl[|\nabla u|^2+(u-g)^2\bigr]^+\langle\eta,\nu\rangle\,d{\cal H}^{n-1}
+2\int_{A^+}(u-g)\langle\eta,\nabla g\rangle\,dx.
\end{eqnarray*}
By a similar argument on $A^-$ we get
\begin{eqnarray}\label{firstvar2}
&&\int_{A^-}\bigl[\bigl(|\nabla u|^2+(u-g)^2\bigr){\rm 
div}\eta\bigr]\,dx
-2\int_{A^-}\langle\nabla u,\nabla u\nabla\eta\rangle\,dx
\\
&=&\int_\Gamma
\bigl[|\nabla u|^2+(u-g)^2\bigr]^-\langle\eta,\nu\rangle\,d{\cal H}^{n-1}
+2\int_{A^-}(u-g)\langle\eta,\nabla g\rangle\,dx.\nonumber
\end{eqnarray}
The above argument works if $u$ is of class $H^2$ up to
$\Gamma$. However, in our case $u$ is only $H^2_{\rm loc}$ in
$A^\pm$. This difficulty can be overcome by lifting up or pulling
down the graph of $\phi$, so getting integration by parts formulas
similar to the ones above, with extra terms which disappear in
the limit because of the (classical) Neumann boundary condition
satisfied by $u$.


Hence, (\ref{mcm2}) follows by (\ref{firstvar}), (\ref{firstvar1}),
(\ref{firstvar2}).
By Theorem~\ref{meanreg}, $\phi\in H^2_{\rm loc}(D)$; expanding
the divergence in (\ref{mcm1}) we can use the Schauder estimates
for equations in non-divergence form (see for instance
\cite{gia}, Theorem~3.6)
to conclude that $\phi\in C^{2,s}_{\rm loc}(D)$ for some
$s\in (0,1)$.

At this point, since $\phi\in C^{2,s}_{\rm loc}(D)$, the classical
Schauder estimates for the Neumann problem imply that $u$ has the
same regularity up to $\Gamma$. Coming back to (\ref{mcm1}), 
(\ref{mcm2})
we get the $C^{3,s}_{\rm loc}$ regularity of $\phi$.

If $k>1$, the higher regularity of $u$ up to $\Gamma$ and
of $\phi$ follows by the usual bootstrap argument,
based on (\ref{neu0}), (\ref{mcm1}) and (\ref{mcm2}).
\qed

\section{Density of functions with regular jump set}

In this section we show that in Mumford-Shah problem the Lavrentiev
phenomenon does not occur, {\em i.e.}, that the infimum of the 
functional
$F$ among all pairs $(\Gamma,u)$ with $\Gamma$ piecewise smooth 
(according
to Definition~\ref{piesmo}) and $u\in 
C^\infty(\Omega\setminus\Gamma)$
coincides with the minimum of the weak form of $F$. This property
is particularly useful in connection with the numerical minimization
of the Mumford-Shah functional (see for instance \cite{cham}, 
\cite{belco}).

Our strategy is to prove a more general approximation theorem of 
$SBV$
functions by smooth functions with discontinuities contained in 
piecewise
smooth sets.  A byproduct of this theorem is the absence of the 
Lavrentiev
phenomenon for general classes of functionals, including the Mumford-
Shah
functional (see Remark~\ref{Lav}).  The proof of the approximation 
property
is achieved in two steps:  first, using the partial regularity results of
\cite{ap}, \cite{afp} and section 4, we prove in Proposition~\ref{appmin}
that the approximation property holds for minimizers of the Mumford-
Shah
functional.  Successively we see that a more general class of $SBV$
functions can be approximated by minimizers.

Let us start by recalling the definition and the main properties of the
$SBV$ functions; for more details we refer to \cite{dga}, \cite{am3},
\cite{am5}, \cite{am2}. A function $u\in L^1(\Om)$ is said to be in
$SBV(\Om)$ if its distributional gradient $Du$ is a Radon vector 
measure
with finite total variation ({\em i.e.} $u\in BV(\Om)$) and moreover
$Du=\nabla u\, dx+Ju$, where $Ju$ is the restriction of $Du$ to the
complement $S_u$ of the Lebesgue set of $u$ and $Ju$ is absolutely
continuous with respect to ${\cal H}^{n-1}\res S_u$.  By $\nabla u$ we
denote the density of $Du$ with respect to the Lebesgue measure; it can 
be
interpreted as an approximate differential of $u$.  The weak form of the
Mumford-Shah functional in $SBV(\Om)$ can then be formulated as 
follows:
$$
G(u)=\int_\Om\bigl[|\nabla u|^2+\alpha|u-g|^2\bigr]\, dx +
\beta{\cal H}^{n-1}(S_u)
$$
and it can be proved that a minimizer $u\in SBV(\Om)$ exists (see
\cite{am2}).

Let us now define the class of piecewise smooth manifolds that will be
considered.

\begin{definition}\label{piesmo} Let $K\subset\rn{n}$ be compact. We 
say
that $K$ is piecewise smooth if there exist finitely many $C^\infty$
hypersurfaces $\Gamma_1,\ldots,\Gamma_N$ with $C^\infty$ 
boundaries
$\partial\Gamma_1$, ..., $\partial\Gamma_N$ such that
$K=\bigcup\limits_{i=1}^N\Gamma_i$ and
$$
\bigl(\Gamma_i\setminus\partial\Gamma_i\bigr)\cap
\bigl(\Gamma_j\setminus\partial\Gamma_j\bigr)=\emptyset
$$
whenever $i\neq j$. We denote by ${\cal K}$ the class of these sets.
\end{definition}

The main result of this section is the following theorem, which is an
immediate consequence of Theorem \ref{lavre}.

\begin{theorem} \label{lavrent} Let $\Omega\subset\rn{n}$ be a 
bounded open
set with Lipschitz boundary, $\alpha,\,\beta$ positive and $g\in
L^\infty(\Omega)$; then
$$
\inf\bigl\{ F(u,K):\ K\in {\cal K},\ u\in C^\infty(\Omega\setminus K)
\Bigr\} = \min\bigl\{ G(u):\ u\in SBV(\Omega) \Bigr\} ,
$$
where $F$ is defined in (\ref{msfunct}).
\end{theorem}

\begin{proposition}\label{appmin} Let $\Om,\,A\subset\rn{n}$ be 
bounded
open sets with $\Om\subset\subset A$, let $w\in C^\infty(A)$ and
let $(v,\Gamma)$ be a minimizer of the Mumford-Shah functional
$$
(v,\Gamma)\mapsto \int_A\bigl[|\nabla v|^2+\alpha |v-w|^2\bigr]dx +
\beta {\cal H}^{n-1} (\Gamma\cap A).
$$
Then, for every $\eps>0$ we can find a compact set $K\in{\cal K}$ and 
$u\in
C^\infty(\overline{\Omega}\setminus K)$ such that
\begin{equation}\label{require}
\left\{\begin{array}{l}
\displaystyle{\int_\Omega \bigl[
\alpha|v-u|^2+|\nabla v-\nabla u|^2\bigr]\,dx<\eps}
\\
\\
\displaystyle{\beta\bigl\vert{\cal H}^{n-1}(K\cap\overline{\Omega})-
{\cal H}^{n-1}(\Gamma\cap\overline{\Omega})
\bigr\vert<\eps.}
\end{array}\right.
\end{equation}
\end{proposition}
{\sc Proof.} Let $S\subset\Gamma$ be the singular set of $\Gamma$,
{\em i.e.},
the set of points $x\in\Gamma$ such that $\Gamma$ is not a 
$C^{1,\rho}$
hypersurface in any neighbourhood of $x$. We know from Theorem~3.1
and Remark~3.2 of \cite{afp} that ${\cal H}^{n-1}(S)=0$; moreover,
Theorem~\ref{meanreg2} implies that $\Gamma\setminus S$ is locally a
$C^\infty$ hypersurface.

For any $\delta\in (0,1)$ we can find a finite
family of balls $B_{r_i}(x_i)$ centered at points
$x_i\in\overline{\Om}$ whose union contains
$S\cap\overline{\Om}$, such that $r_i<\delta$ for any $i=1,\ldots,N$
and
$$ \sum_{i=1}^N r_i^{n-1}<\delta.$$
Let $\chi\in C^\infty_0(A)$ such that $\chi\equiv 1$ on the $\delta$
neighbourhood of $\overline{\Om}$.
Denoting by $U$ the union of the balls $B_{\tau r_i}(x_i)$,
we define the function
$$ u(x)=\cases{ v(x) & if $x\notin U$
\cr 0 & if $x\in U$},$$
the compact set
$$ K=\bigl(\Gamma\setminus (U\cup \{ 1+\chi\geq\tau \})\bigr)
\cup\bigcup_{i=1}^N \partial B_{\tau r_i}(x_i)
$$
and we check that for a suitable choice of $\tau\in (1,2)$ and for
$\delta$ sufficiently small, $(u,K)$ satisfy all the stated properties.
Indeed, since $r_i<\delta<1$, we have
$$
{\cal L}^n(U)\leq \sum_{i=1}^N \omega_n(2r_i)^n<2^n\omega_n\delta,
$$
so that
$$
\int_\Om \bigl[\alpha |v-u|^2+|\nabla v-\nabla u|^2\bigr]\,dx=
\int_U |v|^2+|\nabla v|^2\,dx<\eps
$$
for $\delta$ small enough. Since the measure of the symmetric 
difference
of $K$ and $\Gamma$ in $\overline{\Om}$
does not exceed the sum of the perimeter of the balls, also the second
condition in (\ref{require}) is fulfilled for $\delta$ small enough.

Finally, by Sard's theorem, the balls $B_{\tau r_i}(x_i)$
and the smooth level set $\{\chi=\tau-1\}$ meet
transversally $\Gamma$ for a.e. $\tau$, hence a choice of $\tau\in 
(1,2)$
is possible ensuring that $K\in {\cal K}$. \qed

\begin{theorem}\label{lavre} Let $\Omega\subset\rn{n}$ be a bounded 
open
set with Lipschitz boundary. Then, for
every $u\in SBV(\Omega)\cap L^\infty(\Omega)$ satisfying
$$\int_\Omega |\nabla u|^2\,dx+{\cal H}^{n-1}(S_u)<+\infty$$
we can find sets $K_h\in {\cal K}$
and functions $u_h\in C^\infty(\overline{\Omega}\setminus K_h)$ such 
that
$$\left\{
\begin{array}{l}
\displaystyle{\lim\limits_{h\to +\infty}\int_\Omega\bigl[ |u_h-
u|^2+|\nabla u_h-\nabla u|^2
\bigr]\,dx=0},
\\
\\
\displaystyle{\lim\limits_{h\to +\infty}{\cal H}^{n-
1}(K_h\cap\overline{\Omega})=
{\cal H}^{n-1}(S_u\cap\Omega).}
\end{array}\right.$$
\end{theorem}
{\sc Proof.} As a first step, we consider a bounded open set
$A\subset\rn{n}$ such that $\overline{\Omega}\subset A$ and a 
function
$w\in SBV(A)$ such that $w=u$ on $\Omega$,
${\cal H}^{n-1}(S_w\cap\partial\Omega)=0$ and
$$\int_A |\nabla w|^2\,dx+{\cal H}^{n-1}(S_w\cap A)<+\infty.$$
An extension with these properties can be easily obtained if
$\Omega$ is a rectangle, by a reflection argument. In the general
case the extension can be obtained using
partitions of unity and a deformation argument.

Let $(w_h)\subset C^\infty(A)$ be such that $h\Vert w_h-
w\Vert_{L^2(A)}\to
0$ as $h\to +\infty$, and
let $v_h\in SBV(A)$ be minimizers of the functionals
$$ v\mapsto {\cal F}_h(v)=
\int_A \bigl[|\nabla v|^2+h(v-w_h)^2\bigr]\,dx+
{\cal H}^{n-1}(S_v\cap A).$$
Since ${\cal F}_h(v_h)\leq {\cal F}_h(w)$, we obtain
\begin{eqnarray}\label{stima1}
&&\limsup_{h\to +\infty} \biggl[h\Vert v_h-w_h\Vert^2_{L^2(A)}+
\int_A |\nabla v_h|^2\,dx+ {\cal H}^{n-1}(S_{v_h}\cap A)\biggr]
\\
&\leq& \int_A |\nabla w|^2\,dx+{\cal H}^{n-1}(S_w\cap A).\nonumber
\end{eqnarray}
In particular, $\Vert v_h-w_h\Vert_{L^2(A)}\to 0$, hence
$v_h$ converges to $w$ in $L^2(A)$.
Moreover, the $SBV$ compactness theorem (see \cite{am1}, \cite{am4})
implies
$$\left\{\begin{array}{l}
\liminf\limits_{h\to +\infty}\int_A|\nabla v_h|^2\,dx\geq
\int_A |\nabla w|^2\,dx \\
\liminf\limits_{h\to +\infty}{\cal H}^{n-1}(S_{v_h}\cap A)\geq
{\cal H}^{n-1}(S_w\cap A).
\end{array}
\right.
$$
Then, these inequalities and (\ref{stima1}) imply
$$\left\{\begin{array}{l}
\lim\limits_{h\to +\infty}\int_A |\nabla v_h|^2\,dx=\int_A |\nabla 
w|^2\,dx
\\
\lim\limits_{h\to +\infty}{\cal H}^{n-1}(S_{v_h}\cap A)=
{\cal H}^{n-1}(S_w\cap A).
\end{array}
\right.
$$
In particular, $\nabla v_h$ converges strongly to $\nabla w$ as $h\to
+\infty$ and ${\cal H}^{n-1}(S_{v_h}\cap A)$ converges to
${\cal H}^{n-1}(S_w\cap A)$. Again, the $SBV$ compactness
theorem gives
$$\left\{\begin{array}{l}
\liminf\limits_{h\to +\infty}{\cal H}^{n-1}(S_{v_h}\cap\Omega)\geq
{\cal H}^{n-1}(S_w\cap\Omega)
\\
\liminf\limits_{h\to +\infty}{\cal H}^{n-1}\bigl(S_{v_h}\cap(A\setminus
\overline{\Omega})\bigr)\geq
{\cal H}^{n-1}\bigl(S_w\cap(A\setminus\overline{\Omega})\bigr).
\end{array}
\right.
$$
Since ${\cal H}^{n-1}(S_w\cap\partial\Omega)=0$, these inequalities
imply that both liminf above are limits and that equalities hold.
In particular, ${\cal H}^{n-1}(S_{v_h}\cap\Omega)$ converges
to ${\cal H}^{n-1}(S_u\cap\Omega)$ and
\begin{equation}\label{bordo}
\lim_{h\to +\infty}{\cal H}^{n-1}(S_{v_h}\cap\partial\Omega )=0.
\end{equation}
Let $\Gamma_h=\overline{S}_{v_h}$. By the minimality
of $v_h$ and the smoothness of $w_h$, the functions $v_h$ are
$C^\infty$ in $A\setminus\Gamma_h$; moreover,
(\ref{bordo}) and (see \cite{ap}, Proposition 2.8)
$$ {\cal H}^{n-1}(A\cap\Gamma_h\setminus S_{v_h})=0$$
imply that ${\cal H}^{n-1}(\Gamma_h\cap\overline{\Omega})$ 
converges
to ${\cal H}^{n-1}(S_u\cap\Omega)$.

However, the sets $\Gamma_h$ do not necessarily
belong to ${\cal K}$; using Proposition~\ref{appmin}
we can modify $(v_h,\Gamma_h)$ to obtain pairs $(u_h,K_h)$ satisfying
the statement.\qed


\begin{remark} \label{Lav} {\rm As a consequence of the approximation
Theorem~\ref{lavre}, we obtain that the Lavrentiev phenomenon does 
not
occur for minimum problems in $SBV$ of the form
$$\min\biggl\{\int_\Omega f(x,u,\nabla u)\,dx+{\cal H}^{n-
1}(S_u)\biggr\}$$
where $f$ is any Carath\'eodory function satisfying quadratic
growth conditions. Using Lemma~4.3 in \cite{am5}, more general surface
energy densities can also be considered.
}
\end{remark}

\vfill\eject
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\end{document}
