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\title{\bf Regularity results for some $1$-homogeneous functionals}
\author{Matteo Novaga and Emanuele Paolini}

\begin{document} 
\maketitle 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
We consider local minimizers for a class of $1$-homogeneous 
integral functionals
defined on $BV_\loc(\Omega)$, with $\Omega\subset \R^2$. 
Under general assumptions on the functional, we prove that the boundary of the
subgraph of such minimizers is (locally) a lipschitz graph in a
suitable direction. 
The proof of this statement relies on a regularity result  
holding for boundaries in $\R^2$ which minimize an anisotropic perimeter.
This result is applied to the boundary of sublevel sets 
of a minimizer $u\in BV_\loc(\Omega)$.

We also provide an example which shows that such regularity result is optimal.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{secintro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this paper we study the regularity properties of local minimizers
of functionals of the type
%
\begin{equation}\label{eqintro}
 F_\phi(u) = \int_\Omega \phi(Du) + \int_\Omega fu \d x,
\end{equation}
%
where $\Omega\subset \R^2$, $u\in BV_\loc(\Omega)$, $f\in
L^\infty_\loc(\Omega)$ and $\phi: \R^2\to \R^+$ is a generic positively
$1$-homogeneous convex function. 
Since $Du$ is general only a measure we shall give 
a precise meaning to the integral $\int \phi(Du)$.

A serious difficulty in considering such minimizers comes from the
fact that the functional $F_\phi$ is not strictly convex in $u$ and
has a linear growth, hence we cannot apply the usual techniques
\cite{BoDeMi:69}, \cite{LaUr:70}, which lead to $C^{1,\alpha}$
regularity in the case of the prescribed mean curvature problem, i.e.
when $\phi(Du)=\sqrt{1+|Du|^2}$~\cite{Massari:74}, \cite{Bo:82}.
Indeed, in the case $\phi(Du)=|Du|$, it is easy to find minimizers
of~(\ref{eqintro}) which are not even continuous. However, in the case
$f=0$ and $\phi(Du)=|Du|$ it has been proved \cite{Mi:65} that, if we
prescribe sufficiently regular boundary conditions, a lipschitz
minimum always exists.

However, the homogeneity property of the functional allows us to
conclude that if $u$ is a minimizer of $F_\phi$ then $\chi_{\{ u<t\}}$ is also 
a minimizer for any $t\in\R$. 
This implies that each sublevel set of $u$ is itself a minimum of an
anysotropic prescribed curvature problem 
\cite{EmGoTa:83}, \cite{Fi:86}, \cite{Mo:94}.
Such minimizers have been considered in~\cite{AmNoPa:00}, 
\cite{NoPa:00} and it is known (in dimension $n=2$) 
that their boundary is locally the graph of 
a lipschitz function.

Using this information we are able to conclude that the boundary of
the subgraph of $u$, as a subset of $\Omega\times \R$ is itself
locally a lipschitz graph, even if not necessarily in the vertical
direction.  To perform this step we need an assumption on the convex
set $\{\phi < 1\}$, which we call \emph{fatness} condition (see
Section~\ref{secnot}), and we show with an explicit example that such
condition is necessary to get this kind of regularity. Moreover, such
an example shows that in some cases the minimizers of a crystalline
perimeter (in dimension $n=3$) are not locally lipschitz graphs.

We conjecture that, without any assumption on $\phi$, the graph of a
minimizer is locally parameterizable by means of a bilipschitz map.

The plan of the paper is the following.  In Section~\ref{secnot} we
describe the notation that we shall use in the sequel.  In
Section~\ref{seclocmin} we introduce the class of local minimizers for
$F_\phi$, and we prove a compactness results for sequences in this
class (\ref{compattezza}).  Moreover, in~\ref{teoreg} we prove the
main result of the paper, which states that the graph of a local
minimizer is itself (locally) the grapf of a lipschtz function, 
when $\phi$ satisfies the \emph{fatness} condition. Finally,  
in Section~\ref{secsex} we give an example which shows that this
condition is really necessary to conclude that the graph of the
minimizer is locally a lipschitz graph. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Notations}\label{secnot}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\phi\:\R^2\to \R^+$ be a function such that
\begin{enumerate}
\item 
        $\phi(x)=0 \Leftrightarrow x=0$ (coercivity);
\item
        $\phi(t x)=t \phi(x)\qquad \forall t\ge 0$ (positive $1$-homogenity);
\item
        $\phi(x+y)\le \phi(x)+\phi(y)$ (convexity).
\end{enumerate}

We define $\phip\:\R^2\to \R^+$ as
\[
        \phip(v)=\sup_{\xi\ne 0} \frac{\langle \xi, v\rangle}{\phi(\xi)}
\]
where $\langle \cdot,\cdot\rangle$ is the usual scalar product of $\R^2$.
It is not difficult to check that $\phip$ satisfies the same properties of 
$\phi$ and that
\[
        \phi(\xi)=\sup_{v\ne 0} \frac{\langle \xi, v\rangle}{\phip(v)}.
\]

We call \emph{Wulff shape} the set $W_\phi:=\{x\in\R^2\: \phip(x)<1\}$ and
\emph{Frank diagram} the set $F_\phi:=\{\xi\in\R^2\: \phi(\xi)<1\}$.
We define the following (multivalued) duality maps 
\bea
v^*    &:=& \{\xi\in\R^2\: \phi(\xi)=\phip(v),\ \langle \xi, v\rangle
        = \phi(\xi)\phip(v)\}\\
\xi^*  &:= &\{v \in \R^2\: \phip(v)=\phi(\xi),\ \langle \xi,v\rangle
        = \phi(\xi)\phip(v)\}.
\eea

We say that $W_\phi$ is \emph{slim} if there exists an edge 
$l\subset \partial W_\phi$ such that the sum of the two angles 
of $W_\phi$ adiacent to $l$ is less than or equal to $\pi$. 
We say that $W_\phi$ \emph{fat} if it is not slim.

For example, all triangles and quadrilaterals are slim, 
whereas strictly convex Wulff shapes are fat.

Given $E\subseteq\R^n$, we let $\chi_E: \R^n\to\R$ be the 
characteristic function of $E$, i.e. $\chi_E(x)=1$ if $x\in E$, 
and $\chi_E(x)=0$ otherwise.

We will denote with $\H^k$, $k>0$, the $k$-dimensional Hausdorff measure in $\R^n$, 
and we let $|E|:=\H^n(E)$ be the Lebesgue measure of the set $E\subseteq \R^n$.

Given $v\in\R^3\setminus\{0\}$ we say that a set $S\subset\R^3$ 
is a \emph{graph along
$v$}, if it is not possible to find two different points $x,y\in S$
such that $x-y=\lambda v$ for some $\lambda\in \R$.

The anisotropic perimeter of a set $E$ in the open set $A\subset \R^2$
is defined by
\[
        P_\phi(E,A) := \sup\left\{\frac{\pi}{|W_\phi|}\int_E {\rm div}\, \psi(x) \d x \:
        \psi\in \C_c^1(A;\R^2), \phip(\psi(y))\le 1 \quad\!\! \forall y\in
        A\right\}.
\]
The usual notion of perimeter of $E$ in $A$ will still be denoted by $P(E,A)$.

We let $B_\rho(x):=\{y\: ||x-y||<\rho\}$ be the usual
euclidean ball of $\R^2$ and we set for simplicity  $B_\rho:= B_\rho(0)$. 

Given a set $E\subseteq \R^n$ of locally finite perimeter, we define 
        \bea
        \partial E&:=&\big\{x\in \R^n\: \forall \rho>0\quad |E\cap
        B_\rho(x)|\in \left]0,|B_\rho(x)|\right[\big\},\\
        \overline{E}&:=&\big\{x\in \R^n\: \forall \rho>0\quad |E\cap
        B_\rho(x)| \not =  0\big\},\\
        \open{E}&:=&\big\{x\in \R^n\: \forall \rho>0\quad |E\cap
        B_\rho(x)| =  |B_\rho(x)|\big\}.
        \eea
It holds, as usual, that $\overline{E},\partial E$ are closed sets, 
whereas $\open{E}=\overline{E}\setminus\partial E$ is an open set.
Notice that if $|E\diffsim F|=0$ then $\partial E=\partial F$
(where $E\diffsim F:=(E\setminus F)\cup(F\setminus E)$).

We let $\partial^*E$ be the reduced boundary 
in the sense of De Giorgi \cite{DG:54}, and 
\[
\nu_E(x) := \lim_{\rho\to 0^+}
-\frac{D\chi_E(B_\rho(x))}{\vert D\chi_E(B_\rho(x))\vert}
\]
be the exterior unit normal to $\partial E$ in $x\in\partial^* E$.

Given a vector-valued Radon measure $\mu$ on $\Omega\subseteq\R^2$, we define the
measure $\phi(\mu)$ as the $\phi$-total variaton of $\mu$:
\[
    \phi(\mu)(B)\defeq \sup \sum_i \phi(\mu(B_i))
\]
where the supremum is taken over the family of all partitions
$\{B_i\}_{i\in I}$ of the Borel set $B\subseteq\Omega$.
With this definition, if $u\in BV_\loc(\Omega)$ then $\phi(Du)$ is a
positive measure on $\Omega$ and the integral $\int_\Omega \phi(Du)$
makes sense.

For $u\in BV_\loc(\Omega)$ we define $S=S_u\defeq\{(x,t)\in\Omega\times\R\:
u(x) <t \}$ and let $\Gamma=\Gamma_u\defeq \partial S_u\cap(\Omega\times\R)$.
Notice that $\Gamma_u$ is always a closed set in $\Omega\times \R$ and 
$\Gamma_u=\Gamma_v$ whenever $u=v$ a.e.

For $t\in\R$ we define $E_t \defeq \{x\in\Omega\: u(x)>t\}$ and
$F_t \defeq \{x\in\Omega\: u(x)\ge t\}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Local minimizers and 
regularity result}\label{seclocmin}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As explained in the Introduction, we shall consider local minimizers of the functional
\[
        F_\phi(u)=\int_\Omega \phi(Du) + \int_\Omega f u \d x
\]
where $\Omega$ is an open subset of $\R^2$, $u\in BV_\loc(\Omega)$, 
$f\in L^\infty_\loc(\Omega)$ and $\phi$ 
is a convex positive $1$-homogeneous function.

\begin{remark}\label{remdef}
  Observe that for any $u\in BV_\loc(\Omega)$ there holds
\begin{equation}\label{eqrappr}
F_\phi(u) = \sup \Big\{ 
-\int_\Omega (u {\rm div}\psi + fu)~dx : 
~\psi\in C^1_c(\Omega),~\phi^o(\psi)\le 1
\Big\}
\end{equation}
%
In particular, from~(\ref{eqrappr}) it follows that $F_\phi$ is lower
semicontinuous in $L^1_\loc(\Omega)$, i.e.
\[
\int_B \phi(Du) + \int_B f u \d x \le \liminf_{k\in\mathbb{N}}\int_B
\phi(D(u+\psi_k)) + \int_B f (u+\psi_k) \d x ,
\]
whenever $\psi_k\to 0$ in $L^1(B)$, $\psi_k\in BV(B)$,
$B\Subset\Omega$.
\end{remark}

\begin{definition}\label{defmin}
  We say that $u\in BV_\loc(\Omega)$ is a (local) minimizer for
  $F_\phi$ if
\[
\int_{B_\rho(x)} \phi(Du) + \int_{B_\rho(x)} f u \d x \le
\int_{B_\rho(x)} \phi(Dv) + \int_{B_\rho(x)} f v \d x,
\]
whenever $B_\rho(x)\Subset\Omega$, $v\in BV_\loc(\Omega)$ and
$\overline{\{x\in\Omega\: u(x)\neq v(x)\}} \subset B_\rho(x)$.

Notice that, by approximation, we can restrict the class of test
functions to the functions $v=u+\psi$, where $\psi\in
C_c^\infty(B_\rho(x))$.
\end{definition}

We denote by $\M(\Omega)$ the family of all local minimizers of
$F_\phi$ in $\Omega$.  With a little abuse of notation, when $E\subset
\Omega$ is a measurable set, we write $E\in \M(\Omega)$ instead of
$\chi_E \in \M(\Omega)$.

\begin{remark}
  The semicontinuity of $F_\phi$ guarantees that (when $\Omega$ is
  bounded) minimizers do exist in the closure of any nonempty subset
  of $BV(\Omega)$ which is bounded in $L^1$.  In particular, given
  $u_0 \in BV(\Omega)$, there always exists a minimizer for $F_\phi$,
  among the functions $u\in BV(\Omega)$ which coincide with $u_0$
  outside of a set $B\Subset \Omega$.  However, it may be convenient
  for the minimizers to have jumps on $\partial B$
\end{remark}

In the following theorem we state a fundamental compactness property
of the class $\M(\Omega)$.

\begin{theorem}[compactness]\label{compattezza}
If $u_k\in\M(\Omega)$, $u\in
L^1_\loc(\Omega)$ and $u_k \to u$ in $L^1_\loc(\Omega)$ then $u\in\M(\Omega)$. 
\end{theorem}

\begin{proof}
Let $B_\rho(x)\Subset \Omega$ and $v\in BV_\loc(\Omega)$ be such that
$K=\overline{\{y\in\Omega\: u(y)\neq v(y)\}} \subset B_\rho(x)$. Let
moreover $\rho'\in\left]0,\rho\right[$ be such that $K \subset
B_\rho'(x)$. Suppose for simplicity of notation $x=0$.
Since the proof does not change significantly we shall also assume $f=0$. 

We claim that it is possible to find a sequence of radii 
$\eta_k,\eta\in\left ]\rho',\rho\right [$ with $\eta_k\nearrow \eta$
such that it holds
\begin{eqnarray}\forall k\quad
\int_{\partial B_{\eta_k}} \left\vert Du_k \right\vert &=& 0,
\label{h1}\\
\lim_{k\to\infty} \int_{B_\eta\setminus B_{\eta_k}}\left\vert D u_k
\right\vert &=& 0, \label{h2}\\
\liminf_{k\to\infty} \int_{\partial B_{\eta_k}} \left\vert
D\big((u-u_k)\chi_{B_{\eta_k}}\big)\right\vert&=& 0. \label{h3} 
\end{eqnarray}

If this is true we can conclude the proof by considering the functions
$v_k=(v-u_k)\chi_{B_{\eta_k}}+u_k$ which are variations of 
$u_k$ in $B_{\eta}$ and coincide with $v$ in $B_{\eta_k}$ so that, by
the semicontinuity of $u\mapsto \int_U \phi(Du)$, the
minimality of $u_k$ with respect to $v_k$, the locality of $u\mapsto
\int_U \phi(Du)$,
we get
\bea
\lefteqn{
\int_{B_\eta} \phi(Du) 
\le \liminf_{k\to \infty} \int_{B_\eta}\phi(Du_k)  
\le \liminf_{k\to\infty} \int_{B_\eta}\phi(Dv_k)}\\
&=& \liminf_{k\to\infty} \left[
  \int_{B_{\eta_k}} \phi(Dv)
  + \int_{\partial B_{\eta_k}} \phi(Dv_k)
  + \int_{B_{\eta}\setminus \overline{B_{\eta_k}}} \phi(Du_k)\right]\\
&\le& \int_{B_\eta}\phi(Dv) + C \liminf_{k\to\infty}
    \left[ \int_{\partial B_{\eta_k}} 
      \left\vert D\big((u-u_k)\chi_{B_{\eta_k}}+u_k\big)\right\vert 
    + \int_{ B_{\eta}\setminus  B_{\eta_k}} 
      \left\vert Du_k\right\vert\right]
\\
&\le& \int_{B_\eta}\phi(Dv) + C \liminf_{k\to\infty} 
   \int_{\partial B_{\eta_k}} 
   \left[ \left\vert D \big((u-u_k)\chi_{B_{\eta_k}}\big)\right\vert
   + \left\vert D u_k\right\vert\right] \\
&=& \int_{B_\eta}\phi(Dv),
\eea
where $C>0$ is such that $\phi(\xi)\le C \left|x\right|$.

Let us prove the claim.
Let $T_1=\bigcap_k \{t\in\left]\rho',\rho\right[\: 
    \int_{\partial B_t } \left\vert
    Du_k\right\vert = 0\}$. 
Since $u_k\in BV(B_\rho)$ the set $T_1$ is
an intersection of  countably many sets with measure $\rho-\rho'$ that
is $T_1$ has itself measure $\rho-\rho'$. So, for~(\ref{h1}) to hold, we
just need $\eta_k\in T_1$ for all $k$.

Regarding~(\ref{h3}), 
we notice that~\cite[Sec. 3.7]{AmFuPa:00} for a.e. $t>0$ we have 
\[
\int_{\partial B_t} \left\vert D\big((u-u_k)\chi_{B_t}\big)\right\vert
\le \int_{\partial B_t} |u(x)-u_k(x)|\d\H^{1}(x),
\]
where the second integral must be intended in the Lebesgue sense.

Consider the functions $f_k(t)=\int_{\partial B_t}
|u(x)-u_k(x)|\d\H^1(x)$ defined for $t\in\left]\rho',\rho\right[$. 
By Fubini-Tonelli formula we know that
$f_k\in L^1(\left]\rho',\rho\right[)$ and
\[
||f_k||_{L^1(\left]\rho',\rho\right[)} = 
\left\vert u-u_k\right\vert_{L^1(B_\rho\setminus\overline{B_{\rho'}})} \to 0 
\qquad {for}~ k\to 0.
\]
So, applying Egoroff theorem, there exists $T_2\subset
\left]\rho',\rho\right[$ with $|T_2|>(\rho-\rho')/2$ and a subsequence
of $f_k$ which converges to $0$ uniformily on $T_2$. Let now $T_3$ be the
set of points $\eta \in T_2\cap T_1$ such that there exists an
increasing sequence $\eta_j\nearrow \eta$ with $\eta_j\in T_2\cap T_1$.
Since $T_2\cap T_1$ is uncountable 
also $T_3$ is uncountable and in particular not empty. 
Therefore there exist $\eta\in T_3$ and $\eta_j\nearrow\eta$,  
with $\eta_j\in T_2\cap T_1$, such that 
\[
\liminf_{k\to \infty}\ \sup_j \int_{\partial B_{\eta_j} }
  |u(x)-u_k(x)|\d\H^1(x)
    \le \liminf_{k\to\infty} \sup_{t\in T_2} |f_k(t)|
    = 0,
\]
since (up to a subsequence) $f_k\to 0$ uniformily on $T_2$.

Concerning~(\ref{h2}), we simply note that if $\eta_j\nearrow \eta$, 
then for all $k\in\mathbb{N}$ we get 
\[
   \lim_{j\to \infty} \int_{B_\eta\setminus B_{\eta_j}} \left\vert
   Du_k\right\vert = 0
\]
since $\bigcap_j (B_\eta\setminus B_{\eta_j})=\emptyset$.
So, given $k\in\mathbb{N}$, we can find
$j(k)$ such that $\int_{B_\eta\setminus B_{\eta_j}} \left\vert
Du_k\right\vert < 1/k$. By letting $\eta_k=\eta_{j(k)}$ we have thus 
determined the sequence which satisfies (\ref{h1}), (\ref{h2}) and (\ref{h3}).
\end{proof}

The following lemma, which strictly depends on the homogeneity
property of $F_\phi$, allows us to prove that the characteristic
function of a sublevel set of a minimizer is also a minimizer for
$F_\phi$ (see \cite{Sta66}). 

\begin{lemma}\label{lemminima}
Let $u\in \M(\Omega)$. Then $u \vee C, u\wedge C, \lambda u\in
\M(\Omega)$ for any $C\in \R$ and $\lambda>0$. 
\end{lemma}

\begin{proof}
It is clear that $u+C\in \M(\Omega)$ and $\lambda u\in\M(\Omega)$
since $D(u+C)=Du$ and $\phi(D(\lambda u))=\lambda \phi(Du)$.
Write now $u=u^+ + u^-$, where $u^+ := u\vee 0$ (resp. $u^- := u\wedge 0$) is the positive 
(resp. negative) part of $u$. Given $U\Subset \Omega$, $\psi\in
C_c^\infty(U)$, we have
\bea
\lefteqn{        \int_U \phi(D u^+) + \int_U f u^+ \d x
        + \int_U \phi(D u^-) + \int_U f u^- \d x} \\
        &=& \int_U \phi(D u) + \int_U f u \d x
        \le \int_U \phi(Du + D\psi) + \int_U f (u+\psi) \d x \\
        &\le& \int_U \phi(D u^+ +D\psi) + \int_U f (u^+ + \psi) \d x 
        + \int_U \phi(Du^-) + \int_U f u^- \d x,
\eea
hence 
\[
\int_U \phi(D u^+) + \int_U f u^+ \d x
\le 
\int_U \phi(Du^+ + D\psi) + \int_U f(u^+ + \psi)\d x. 
\]
\end{proof}

\begin{theorem}\label{minima}
Let $u\in \M(\Omega)$. Then for any $t\in\R$ we have
$E_t, F_t \in \M(\Omega)$. 
\end{theorem}

\begin{proof}
Given $\epsilon>0$ consider $u_\epsilon(x)= (u(x)-t)/\epsilon
\wedge 1 \vee 0$. An easy check assures that
$u_\epsilon\to\chi_{E_t}$ pointwise as $\epsilon\to 0^+$. Since
$u_\epsilon$ are dominated by the constant $1$, by Lebesgue convergence theorem
$u_\epsilon\to \chi_{E_t}$ in $L^1_\loc(\Omega)$. So, by
compactness we get $E_t \in \M(\Omega)$.

If we instead consider $u_\epsilon(x)= (u(x)-t+\epsilon)/\epsilon
\wedge 1 \vee 0$ we conclude that $F_t\in\M(\Omega)$.
\end{proof}

We point out that, as a consequence of~\ref{minima}, from~\cite[Corollary 3.6]{AmNoPa:00}
it follows that $|\partial E_t\setminus\partial^*E_t|=0$.

\begin{lemma}\label{bordi}
Let $u\in BV_\loc(\Omega)$. Given $(x,t)\in\partial^* S_u$ then $x\in
\partial E_t$ or $\nu_S(x,t)=(0,1)$ (and both conditions can hold). 
Moreover, for a.e. $t\in\R$ and for $\H^1$-a.e. $x\in\partial^*E_t$
it holds
\begin{equation}\label{eqnorm}
  \nu_S(x,t) = \frac{(\lambda \nu,1)}{\sqrt{1+\lambda^2}}
  \quad \mathrm{or}\quad 
  \nu_S(x,t) = \nu 
\end{equation}
for some $\lambda\in\R$ and for $\nu=\nu_{E_t}(x)$.
\end{lemma}
%
\begin{proof}
Suppose that $x\not\in \partial E_t$. If $x\not\in \overline{E_t}$ then
for some $\rho>0$ we have 
\[
\left|\{y\in B_\rho(x)\: u(y)<t\}\right| = |B_\rho(x)|.
\]  
On the other hand $(x,t)\in\partial^* S_u$ means that 
\[
\frac{S_u-(x,t)}{\epsilon}\to H(\nu_{S_u}(x,t))=\{(y,s)\: 
\langle(y,s),\nu_{S_u}(x,t)\rangle < 0 \} \qquad 
{\rm in}~L^1_\loc, 
\]
for $\eps\to 0$. 
Since in this case $\epsilon^{-1}(S_u-(x,t))\cap
B_{\rho/\epsilon}\times\{t\}
\subset
H((0,1))$, we have necessarily $\nu_{S_u}=(0,1)$.
The proof is similar when $x\in\open{E_t}$. 

In order to prove the second statement, let us consider the orthogonal
projection $\Pi(y,s)=y$.  We recall that~\cite[Ch. 3]{AmFuPa:00} for
a.e. $t\in\R$ and for $\H^1$-a.e.  $x\in\partial^*E_t$ there holds
$\Pi(\nu_{S_u}(x,t))=s\nu_{E_t}(x)$ for some $s\in\R$, 
which implies (\ref{eqnorm}).
\end{proof}

\begin{lemma}\label{continuita}
Let $E,F\subset\R^2$ be Caccioppoli sets with lipschitz boundary 
such that $E\subseteq F$.
Assume that $\partial E\cap B_{\frac \rho 2}\ne\emptyset$, 
$\partial F\cap B_{\frac \rho 2}\neq \emptyset$, for some $\rho>0$, 
and let $K_1,K_2\subset\R^2$ be two convex cones (i.e. $\lambda K_1 = K_1$, $\lambda K_2=K_2$ for any $\lambda>0$) such that 
%$n, m\in\partial W_\phi$ be calibrating
%vectors for $E, F$ respectively, that is 
for $\H^1$-a.e. $x\in\partial E\cap B_\rho$ and $y\in\partial
F\cap B_\rho$ it holds $-\nu_E(x)\in K_1$ and $-\nu_F(y)\in K_2$. 
Then the following estimate holds
\begin{equation}\label{eqDd}
     \dist(K_1\cap\partial B_1,K_2\cap\partial B_1) \le 
     \frac 2 \rho \dist(\partial E\cap \overline{B_{\frac \rho 2}},
     \partial F\cap \overline{B_{\frac \rho 2}}).
\end{equation}
\end{lemma}

\begin{proof}
Let $u\in \partial E\cap \overline{B_{\frac \rho 2}}$,
$v\in \partial F \cap \overline{B_{\frac \rho 2}}$ be the points for which the minimum 
on the right hand side of the estimate is reached, and let $D=|u-v|$. 
Let also $d$ be the minimum value of the left hand side.

Let $K_1^\perp,K_2^\perp$ be the convex cones 
``orthogonal'' to $K_1, K_2$ (respectively) defined as  
\[
K_i^\perp  = \{ n\in\R^2 \: ~\langle n,v\rangle \le 0, ~{\rm for\ all\ } v\in K_i\} 
\qquad i\in\{ 1,2\}.
\]
It is not difficult to 
show that $K_1^\perp\cap (-K_2^\perp)$ is a (convex) cone of angle
$2\alpha$ such that $\sin(\alpha) = \frac d 2$. 
Moreover, since $\overline{E}\supset (-K_1^\perp + u)\cap B_\rho$ 
and $(K_2^\perp + v)\cap B_\rho\subset B_\rho\setminus \open{F}$, 
it follows that the two cones $-K_1^\perp + u$ and $K_2^\perp + v$
do not intersect within the ball $B_\rho$. On the other hand, 
they must intersect in a ball of radius 
$R \le \frac{D}{2\sin(\alpha)} + \frac{\rho}{2} = \frac{D}{d} + \frac{\rho}{2}$, 
therefore 
\[
\rho\le R\le \frac{D}{d} + \frac{\rho}{2},
\]
which gives (\ref{eqDd}).
\end{proof}

We recall the following regularity result from~\cite[Theorem 6.19]{AmNoPa:00}.

\begin{theorem}\label{curvass}
Assume that $W_\phi$ is not a triangle, and let $\Omega\subset \R^2$ be an open set.
Then, for any $E\in\M(\Omega)$, $x\in \partial E\cap\Omega$ 
and $\rho> 0$ such that $B_\rho(x)\Subset \Omega$, 
there exists a lipschitz graph 
$\Gamma$, whose lipschitz constant depends only on $F_\phi$ and $\rho$,
such that $\partial E\cap B_\rho(x)\subset\Gamma$. 

Moreover, there exists a lipschitz function $v\: \Gamma\to\partial W_\phi$ 
such that 
\[
v(y) \in \left( \frac{-\nu_E(y)}{\phi(-\nu_E(y))}\right)^*
\qquad \forall y\in\partial^*E\cap B_\rho(x).
\]
\end{theorem}

\hide{
\begin{lemma}\label{flatness}
Let $W_\phi$ be different from a triangle or a quadrilateral. 
Then there exist $\rho_0>0$ and $\beta\in\left]0,1\right[$ such that
whenever $\Omega\subset \R^2$ is an open set, $E\in\M(\Omega)$,
$x\in \partial E$, $\rho<\rho_0$ and $B_\rho(x)\Subset \Omega$, it holds
\[
  \Ecc_E(B_{\beta\rho}(x)) =0 
\]
that is there exists $v\in\partial W_\phi$ such that $\langle
v,\nu_E(y)\rangle = 1$ for $\H^1$-a.e. $y \in \partial E\cap
B_{\beta\rho}(x)$. 
\end{lemma}

\begin{proof}
First we prove the following statement.
Given $\alpha>0$ there exists $\rho_0>0$ such that given any open set
$\Omega\subset\R^2$, $E\in\M(\Omega)$ and $x$ with
$B_\rho(x)\Subset \Omega$ it holds
\[
  \Ecc_E(B_\rho(x)) < \alpha. 
\]
Suppose by contradiction that for any $k\in \N$ there exist $\Omega_k$,
$E_k\in\M(\Omega_k)$, $x_k\in\partial E_k\cap \Omega_k$, $B_{1/k}(x_k)\Subset \Omega_k$,
$\Ecc_{E_k}(B_{1/k}(x_k)) \ge \alpha$. Let $F_k=k(E_k-x_k)$ so that
$F_k\in\M(k(\Omega-x_k))$ and $\Ecc_{F_k}(B_1)\ge \alpha$.
Up to a subsequence (see \cite[compattezza]{AmNoPa:00}) 
we may suppose that $F_k$ converges in
$L^1_{\loc}$ to a set $F\in\M(\R^2)$ and (by \cite[propriet\`a
eccesso]{AmNoPa:00}) there should exist $\rho\le 1$ such that
$\Ecc_F(B_\rho)\ge \alpha$. This is impossible since in \cite[minimi
globali]{AmNoPa:00} global minimal are classified and if $W_\phi$
is neither a triangle nor a quadrilateral their excess is always
zero. The statement is proved.

Now, by \cite[decadimento]{AmNoPa:00} (recall that in this case
$\omega=0$) we get that  $\Ecc_E(B_\rho(x))<
\alpha$ 
implies
$\Ecc_E(B_{\beta\rho}(x))=0$.
\end{proof}
}

\begin{lemma}\label{grafico}
Let $\Omega\subset \R^n$ be a connected open set and $E\subset \R^n$
be a Caccioppoli set. If there exist $v\in\R^n, |v|=1$ and $\lambda\in\left]0,1\right[$
such that $\langle v,\nu_E(x)\rangle \le -\lambda$ for $\H^{n-1}$-a.e. $x\in
\partial^* E\cap\Omega$, then $\partial E$ is an $L$-lipschitz graph in
the direction $v$, with $L=\sqrt{1/\lambda^2-1}$.
\end{lemma}

\begin{proof}
Let us choose mollifiers function $\rho_\epsilon\in \C_c^\infty(\R^n)$
such that $\spt \rho_\epsilon \subset B_\epsilon$ and
consider the functions $u_\epsilon \defeq \chi_E * \rho_\eps$.
Since
$E$ has locally finite perimeter, for sufficiently small $\epsilon$ the
integral $\alpha_\epsilon^x=\int_{\R^n}\rho_\epsilon(y-x) \d
\left\vert D\chi_E\right\vert(y)$ is finite. Hence the measures
$\mu_\epsilon^x$ defined by $\d \mu_\epsilon^x(y) \defeq
(\rho_\epsilon(y-x)/\alpha_\epsilon^x)\d \left\vert
  D\chi_E\right\vert(y)$ are probability measures.
By the hypothesis we know that for 
$\H^{n-1}$--a.e. $y\in\partial E$ 
and hence for $\mu_\epsilon^x$--a.e. $y$ the
vector $-\nu_E(y)$ lies in the convex
set $K=\{\xi\in\R^n\: \langle \xi, v\rangle \ge \lambda\}$.
As $\d D\chi_E(y) = -\nu_E(y) \d \left\vert D\chi_E\right\vert(y)$, we obtain
$\nabla u_\epsilon(x) = -\alpha_\epsilon^x \int_{\R^n}
\nu_E(y) \d \mu_\epsilon^x(y)$ that is $\nabla u_\epsilon(x)/\alpha_\epsilon^x$ is a
weighted mean value of $\nu_E(y)$ and hence $\nabla u_\epsilon\in
\alpha_\epsilon^x K$.

Suppose now by simplicity that $v=(0,1)\in \R^{n-1}\times \R$ and let
$(z,t)$ be the two variables of $\R^{n-1}\times \R$. Since 
$\nabla u_\epsilon(x)\in\alpha_\epsilon^x K$ we notice that
$\frac{\partial u_\epsilon}{\partial t}(x) > 0$ for all $x\in
\Omega$. By the Implicit Function Theorem we obtain that 
$\{u_\epsilon=1/2\}\cap
\Omega$ is the graph of a function $f_\epsilon\:\R^{n-1}\to \R$ and we
know that (letting $x=(z,f_\epsilon(z))$)
\[
\left|\nabla_z f_\epsilon\right| = \left|-\nabla_z u_\epsilon /
  \left(\frac{\partial u_\epsilon}{\partial t}\right) \right|
\le \frac{\alpha_\epsilon^x}{(\lambda\alpha_\epsilon^x)}=
\frac{1}{\lambda}.
\]

Up to a subsequence, we may suppose that there exists a lipschitz
function $f\: \R^{n-1}\to\R$ such that locally $f_\eps \to f$
uniformily. 
On
the other hand since $u_\epsilon\to \chi_E$ in $L^1_\loc$ 
we have $\{u_\eps \le 1/2\} \to \R^n\setminus E$ 
locally in measure, that is the
subgraphs of $f_\eps$ converge in measure to $\R^n\setminus E$, 
hence $\partial E$ is the graph of $f$.

Moreover, since the relation $\langle \nu_E,v\rangle =
-\frac{1}{\sqrt{1+(\nabla f)^2}}$ holds almost everywhere, 
we get the estimate $|\nabla
f|\le\sqrt{1/\lambda^2 -1}$, which gives the value of the lipschitz constant.
\end{proof}

\smallskip 

The following lemma provides a characterization of
fat Wulff shapes in $\R^2$.

\begin{lemma}\label{fat}
The set $W_\phi$ is fat if and only if the following property holds: 
there exist $\epsilon_0>0$, $\delta_0>0$ and $0<\delta<1$ such that
given any $X\subset \partial W_\phi$ the following property holds:
\begin{equation}\label{eqfat}
\dist(K_1,K_2)\le \delta_0
\quad\forall v_1,v_2\in X  
\qquad\Longrightarrow\qquad
\exists p\in\partial W_\phi ~{\rm s.t.}~
\langle p,\xi\rangle > \epsilon_0
\quad \forall \xi\in X^*,
\end{equation}
where $K_i\defeq \{\xi\in\partial \F_\phi\: \langle \xi,v_i\rangle \ge
1-\delta\}$
and
 $X^* = \bigcup_{v\in X} v^*$.
\end{lemma}

\begin{proof}
Notice that $W_\phi$ is fat if and only if there exists
$\epsilon_0>0$ such that for all $v_1,v_2\in \partial W_\phi$ with
$v_1^*\cap v_2^*\neq \emptyset$ (that is $v_1,v_2$ belong to the same edge) 
there exists $p\in \partial W_\phi$ such that given any $\xi\in v_1^*\cup
v_2^*$ it holds $\langle \xi ,p\rangle > \eps_0$. The ``if'' part of
the statement is then simply proved (take $X=\{v_1,v_2\}$).

For the ``only if'' part, reasoning by contradiction with $\delta_0=\delta=1/k$, we can find
a sequence of sets $X_k\subset \partial W_\phi$ such that
(\ref{eqfat}) is false.
Up to a subsequence we may assume that $X_k\to X$ (in the sense of
Kuratowski). 
We claim that there exist $v_1,v_2\in X$ such that $X^*=v_1^* \cup v_2^*$.
In fact given any couple $v_1,v_2 \in X$ there exist $v_i^k\in X_k$ ($i=1,2$) such that $v_i^k\to v_i$. 
By (\ref{eqfat}) we can find $\xi_i^k\in K_i^k=\{\xi\in\partial \F_\phi\:
\langle \xi,v_i^k \rangle \ge 1-1/k\}$ 
such that $|\xi_1^k-\xi_2^k|\le 1/k$. Up to a subsequence we may suppose that $\xi_i^k\to \xi_i$, hence
\[
        \langle \xi_i,v_i \rangle = \lim_k \langle \xi_i^k, v_i^k\rangle 
        \ge \lim_k (1-1/k) = 1          
\]
that is $\xi_i \in v_i^*$. On the other hand it holds $\xi_1=\xi_2$, hence we have proved that $v_1^*\cap v_2^* \neq \emptyset$ and this is true for any couple $v_1,v_2 \in X$. 
Since $W_\phi$ is different from a triangle (because the triangle is \emph{slim}) 
and $n=2$ the claim is proved.

Let now $p\in\partial W_\phi$ and choose $v_i\in X$, $v_i^k\in X_k$, 
$\xi_i^k\in K_i^k$, $\xi_1=\xi_2 \in v_1^*\cap v_2^*$
as before. Since we have supposed that (\ref{eqfat}) is false, 
there exist $\xi_p^k\in X_k^*$ such that $\langle\xi_k^p,p\rangle
\le \epsilon_0$.

Again up to a subsequence we may assume that $\xi_p^k \to \xi_p\in X^*=v_1^* \cup v_2^*$ (indeed if $\xi_k\to \xi$ and $\xi_k\in X_k^*$ 
then $\xi\in X^*$).
It holds $\langle \xi_p,p\rangle =
\langle \xi_p-\xi_p^k,p \rangle + \langle \xi_p^k,p \rangle \le
\phi(\xi_p-\xi_p^k) + \epsilon_0 \to \epsilon_0$, which  
contradicts the characterization of \emph{fatness} given at the 
beginning of the proof.
\end{proof}

We can now prove the main result of the paper.

\begin{theorem}\label{teoreg}
Assume that $W_\phi$ is fat and let  $u\in \M(\Omega)$. 
Then $\Gamma_u\subset \Omega\times \R$ is locally a lipschitz graph. 
Moreover, the lipschitz constant depends only on $\phi$, on $f$ and on 
the distance from $\partial\Omega$.
\end{theorem}

\begin{proof}
Let $x_0\in \Omega$ and let $\rho_2=\dist(x_0,\partial \Omega)/2$. 
Let $\epsilon_0$, $\delta_0$ and $\delta$ be the constants given by
\ref{fat} and set $\epsilon=\delta_0/3$. 
Let $L$ be the lipschitz constant given by \ref{curvass}
with respect to $\rho_2$. Choose 
$C\ge 1$ such that for all $\xi$ it holds $|\xi|/C \le \phi(\xi)\le C
|\xi|$ and set $\rho_1:=\delta/(2CL)$, 
$\rho_0:=\min\{\eps\rho_1/(4C),\rho_1/2\}$.

We consider the family of subsets of $\Omega$ defined by 
\[
\mathcal{F} \defeq 
\Big\{ E_t \: t\in\R,\ \partial E_t\cap B_{\rho_0}(x_0)\ne\emptyset 
\Big\} \cup 
\Big\{ F_t \: t\in\R,\ \partial F_t\cap B_{\rho_0}(x_0)\ne\emptyset 
\Big\}.
\] 

By \ref{minima} we have $\mathcal F\subset
\M(\Omega)$, and by~\ref{curvass} given $E\in \mathcal F$ there exists
an $L$-lipschitz function $v_E\: \partial E\cap B_{\rho_2}(x_0)\to\partial
W_\phi$ such that 
$\langle \nu_E(x),v_E(x)\rangle = -\phi(-\nu_E(x))$ for
$\H^1$-a.e. $x\in\partial E$. 

Consider also the set 
\[
X := \Big\{v_E(x)\: x\in\partial E\cap B_{\rho_0}(x_0),\ 
E\in{\mathcal F}\Big\}\subseteq\partial W_\phi,
\] 
we want to apply \ref{fat}
to prove that there exists $p\in\partial W_\phi$ such that
$\langle \xi,p\rangle > \epsilon_0$ for all $\xi\in X^*$.

Given $v_1,v_2 \in X$ we 
let $x_1,x_2\in B_{\rho_0}(x_0)$ and $E_1,E_2\in\mathcal
F$ be such that (from now on $i=1,2$) $x_i\in \partial E_i$ and
$\nu_i=\nu^\phi_{E_i}(x_i)$ 
(we let $\nu^\phi_E(x)\defeq \nu_E(x)/\phi(-\nu_E(x))$), 
so that $\nu_i\in v_i^*$. Define also
$K_i\defeq \{\xi\in\R^2 \: \langle \xi, v_i\rangle \ge
(1-\delta)\phi(\xi)\}$.
By the $L$-lipschitz continuity of $v_E$ we notice that for $x\in
B_{\rho_1}(x_0)\cap \partial E_i$ it holds 
$\langle \nu^\phi_{E_i}(x), v_i\rangle = \langle \nu^\phi_{E_i}(x), v_{E_i}(x)
\rangle + \langle \nu^\phi_{E_i}(x), v_i - v_{E_i}(x) \rangle
\ge 1 - C L |x-x_i| \ge 1 - 2L\rho_1 = 1-\delta$. 
We have proved that $-\nu_{E_i}(x)\in K_i^\delta$ for $\H^1$-a.e. 
$x\in B_{\rho_1}(x_0)\cap \partial E_i$, and we can
apply \ref{continuita} in the ball $B_{\rho_1}(x_0)$ 
in order to obtain $\dist(K_1^\delta\cap\partial B_1,
K_2^\delta\cap\partial B_2)\le 4 \rho_0/\rho_1 \le \epsilon/C$.
So there exist $\xi_i\in K_i^\delta\cap\partial \F_\phi$ 
such that $|\xi_1-\xi_2|\le
\epsilon$. 
By \ref{fat} we can find a vector $p\in \partial W_\phi$ such that
$\langle p, \xi \rangle > \epsilon_0$ for all $\xi\in X^*$, and in
particular for $\xi=\nu^\phi_E(x)$ with
$E\in\mathcal F$
and $x\in B_{\rho_0}\cap \partial E$.

Notice that, given $(x,t)\in\Gamma_u\cap (B_{\rho_0}(x_0)\times\R)$, 
by \ref{bordi} we get that $x\in\partial E$ for some $E\in\mathcal F$ or
$\nu_{S_u}(x,t)=(0,1)$. In both cases we can write
$\nu_{S_u}(x,t) = (\lambda \nu_E(x),1)/\sqrt{1+\lambda^2}$ 
for some $\lambda\ge 0$ and we obtain 
\[
 \langle \left(\frac{p}{|p|},1\right),\nu_{S_u}(x,t)\rangle 
 = \frac{\lambda \langle \nu_E(x),p\rangle +1}{|p| \sqrt{1+\lambda^2}}
 \ge \frac{1}{|p|}\langle \nu_E(x),p\rangle > 
 \frac{\phi(-\nu_E(x))}{|p|}\epsilon_0\ge \frac{\epsilon_0}{C^2},
\]
since $\frac{a\lambda+1}{\sqrt{1+\lambda^2}} \ge a$ 
for all $\lambda\ge 0$, $a\le 1$. 

So, applying~\ref{grafico} we conclude the proof.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Example}\label{secsex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we provide an example of a function $u\in\M(\R^2)$
such that the set of points where $\Gamma_u$ is not locally the graph
of a lipschitz function has positive $\H^2$-measure.  A point
$x\in\R^2$ will be denoted by its coordinates $x=(x_1,x_2)$.  We set
$\phi(x)=|x_1|+|x_2|$ so that $W_\phi$ is a square. It is not
difficult to show that in a similar way we can treat every other slim $W_\phi$.

Given $a,b,t\in\R$, $a<b$, we define the sets $Q(a,b,t):=\{(x_1,x_2)\:  (3
 a +  b)/4 < x_1 < ( a + 3b)/4,\ x_2<t\}$, $L(a,b,t):=\{
(x_1,x_2): a<x_1\le (a+b)/2\}\setminus Q(a,b,t) $, $R(a,b,t):=\{(x_1,x_2)\:
(a+b)/2<x_1<b\}\setminus Q(a,b,t)$ and define 
$u_{a,b,t}=a\chi_{L(a,b,t)}
+ b\chi_{R(a,b,t)} + \frac{a+b}{2} \chi_{Q(a,b,t)}\: 
]a,b[\times \R\to\R$ (see Figure~\ref{fikex}). 

%
\begin{figure}
\center{
%\input{figex.tex}
\psset{xunit=0.7,yunit=0.7}       
\pspicture*(-9,-5)(9,5)
%
\newgray{lightgray}{0.9}
\newgray{gray}{0.8}
\newgray{darkgray}{0.7}
\psset{linestyle=none,fillstyle=solid,fillcolor=lightgray}
\pspolygon(-7,-4)(3,-4)(4,-3)(-1,-3)(-1,-2)(-5,-2)
\pspolygon(4,-1)(-1,-1)(1,1)(6,1)
\pspolygon(-5,2)(0,2)(1,3)(6,3)(7,4)(-3,4)
\psset{fillcolor=darkgray}
\pspolygon(0,0)(1,1)(1,3)(0,2)
\psset{fillcolor=gray}
\pspolygon(4,-3)(-1,-3)(-1,-2)(-5,-2)(-5,2)(0,2)(0,0)(-1,-1)(4,-1)
\pspolygon(6,1)(1,1)(1,3)(6,3)
%\pspolygon(-7,-4)(3,-4)(4,-3)(4,-1)(6,1)(6,3)(7,4)(-3,4)(-5,2)(-5,-2)
%
\psset{fillstyle=none,linestyle=solid}
\psset{linewidth=1pt}
\psline(-7,-4)(3,-4)
\psline(-5,-2)(-1,-2)
\psline(4,-3)(-1,-3)(-1,-1)(4,-1)
\psline(-1,-1)(1,1)(6,1)
\psline(0,0)(0,2)(-5,2)
\psline(1,1)(1,3)
\psline(-5,2)(0,2)(1,3)(6,3)
\psline(-3,4)(7,4)
\uput[u](-7,-4){$x_1=b$}
\rput(-5,0){$x_1=\frac{a+b}{2}$}
\uput[d](-3,4){$x_1=a$}
\rput(6,2){$x_1=\frac{3a+b}{4}$}
\rput(4,-2){$x_1=\frac{a+3b}{4}$}
%
\rput(-11.2,2){
\psline{->}(5,-1)(6,-1)
\psline{->}(5,-1)(4.5,-1.5)
\psline{->}(5,-1)(5,0)
\uput[u](6,-1){$x_2$}
\uput[r](4.5,-1.5){$x_1$}
\uput[r](5,0){$x_3$}}
\rput(-1.5,-3.5){$x_2=t$}
\rput(-3,-3){$R(a,b,t)$}
\rput(-2,3){$L(a,b,t)$}
\rput(2,0){$Q(a,b,r)$}
\endpspicture 
%\includegraphics[scale=0.6]{fikex.eps}
}
\caption{the construction of the function $u_{a,b,t}$}
\label{fikex}
\end{figure}
%

Notice that $\Gamma_{u_{a,b,t}}$ is not the graph of a lipschitz
function in any direction and in any neighbourhood of
the point $((a+b)/2,t,(a+b)/2)\in\R^3$. However it can be proved (for example using a calibration) that $u_{a,b,t}\in \M(]a,b[\times\R)$. 

\smallskip 
We now merge such functions so that the singular points accumulate in
a Cantor like set with positive measure.

Let $a_k,b_k\in[0,1]$ be a sequence of points such that the intervals
$I_k=[a_k,b_k]$ are disjoint and $K=\bigcap_k([0,1]\setminus
\left ]a_k,b_k\right[)$ is a set with positive measure such that
$K \subseteq \overline{\{b_k\: k\in\N \}}$.

We then define $a_{kj}=b_k-\frac{b_k-a_k}{2^j}$, 
$b_{kj}=b_k-\frac{b_k-a_k}{2^{j+1}}$. The intervals
$\left]a_{kj},b_{kj}\right[$ are all disjoint and accumulate 
in the points $b_k$. Consider also an
enumeration $q_j$ of the rational numbers. The sets
$L(a_{kj},b_{kj},q_j)$,
$R(a_{kj},b_{kj},q_j)$ and
$Q(a_{kj},b_{kj},q_j)$ are all disjoint so that we can define
\[
u(x_1,x_2)=\left\{\begin{array}{cl}
    a_{kj} & \mathrm{if}\ (x_1,x_2)\in L(a_{kj},b_{kj},q_j) \\
    b_{kj} & \mathrm{if}\ (x_1,x_2)\in R(a_{kj},b_{kj},q_j) \\
    (a_{kj}+b_{kj})/2 & \mathrm {if}\ (x_1,x_2)\in
       Q(a_{kj},b_{kj},q_j) \\
    x_1 & \mathrm{elsewhere}
    \end{array}
    \right.
\]
%
Every point $((a_{kj}+b_{kj})/2,q_j,(a_{kj}+b_{kj})/2)$ is a
singular point for the function $u$ and the closure of these points
contains the whole set $\{(x_1,x_2,x_3)\in\R^3\: (x_1,x_2)\in K,\ x_3=x_1\}$ whose
$\H^2$-measure is greater than $\H^2(K)>0$.

\bibliographystyle{plain}
%\bibliography{grafici}
\begin{thebibliography}{10}

\bibitem{AmFuPa:00}
L.~Ambrosio, N.~Fusco, and D.~Pallara.
\newblock {\em Functions of Bounded Variation and Free Discontinuity Problems}.
\newblock Oxford Mathematical Monographs. Clarendon Press, Oxford, 2000.

\bibitem{AmNoPa:00}
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\end{thebibliography}


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