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\title{Partial regularity of free discontinuity sets
I\hskip -1pt I}
%\title{}
\author{}
\author{{\Large Luigi Ambrosio} \\ Dipartimento di Matematica \\ 
Via Abbiategrasso 205, 27100 Pavia, Italy \and
{\Large Nicola Fusco} \\ Dip. di Matematica e Applicazioni
\\ ``R. Caccioppoli'', Via Cinthia \\ 80126 Napoli, Italy \and
{\Large Diego Pallara} \\ Dip. di Matematica \\ Universit\`a di
Lecce \\ C.P. 193, 73100 Lecce, Italy}


\begin{document}
\maketitle
\begin{abstract}
We prove a regularity theorem for minimizers of a class of free
discontinuity problems including the following example:
given $\Omega\subset\rn{n}$ open, $g\in L^\infty(\Omega)$, consider the
functional
$$
G(u,K)=\int_{\Omega\setminus K}\bigl[|\nabla u|^2 + (u-g)^2\bigr] \,
dx+\h{n-1}{K},
$$
where $K\subset\Omega$ is relatively closed,
$u\in C^1(\Omega\setminus K)$ and
${\cal H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure in $\rn{n}$.
If $(K,u)$ is a minimizer then $K$ is a $C^{1,\alpha}$ hypersurface
outside a closed ${\cal H}^{n-1}$-negligible set.
\end{abstract}

\hbox{}\vfill
\tableofcontents

\vfill\eject

\section{Introduction}

In this paper we continue the study, started in
\cite{ap}, of the regularity of
(quasi) minimizers of a class of free discontinuity problems
including the minimization of the Mumford-Shah functional
\begin{equation}       \label{msfun}
G(u,K)=\int_{\Omega\setminus K}\bigl[|\nabla u|^2 +
\alpha(u-g)^2\bigr] \,dx+\beta\h{n-1}{K},
\end{equation}
where $\Omega\subset\rn{n}$ is an open set, $g\in L^\infty(\Omega)$,
$\alpha,\,\beta>0$ and $\h{n-1}{K}$ is the $(n-1)$-dimensional Hausdorff
measure of $K$.  In (\ref{msfun}), $K$ varies in the class of relatively
closed subsets of $\Omega$ and $u\in C^1(\Omega\setminus K)$ (see
\cite{mumsh}, \cite{bz2}, \cite{morsol}, \cite{dcl}, \cite{dms}).  Notice
that, given $K$, the optimal function $u$ is the solution of the Neumann
problem
\begin{equation}\label{neu}
\Delta u=\alpha(u-g),\qquad\qquad \frac{\partial u}{\partial n}=0
\end{equation}
in $\Omega\setminus K$.  Namely, we prove in Theorem \ref{main} that any
optimal free discontinuity set $K$ is a $C^{1,\alpha}$ hypersurface except
for a closed singular set $S$ satisfying $\h{n-1}{S}=0$.  Moreover, we give
a characterization of singular points which could be exploited to get
further information on the dimension and the structure of $S$.

The starting point of this paper is a criterion, proved in \cite{ap},
for the $C^{1,\alpha}$ local regularity of optimal sets $K$. The criterion
involves the rate of decay of the quantity
$$
{\cal F}(x,\rho)=\rho^{-n-1}\min_{A\in\LL}\int_{B_\rho(x)\cap K}
\dis{A}{y}\,d{\cal H}^{n-1}(y)
$$
measuring the flatness of $K$ (here ${\cal A}$ denotes the set of affine
hyperplanes in $\rn{n}$) and of the scaled Dirichlet energy
$$
{\cal D}(x,\rho)=\rho^{1-n}\int_{B_\rho(x)}|\nabla u|^2\,dy.
$$
The application of such a criterion to our partial regularity
result is based on two decay estimates. The first one, concerning
the flatness improvement has been already proved in
\cite{ap} (see Theorem \ref{decay2} below). Heuristically,
(see \cite{ap}) a first variation argument
and the rectifiability of $K$ show that the Dirichlet energy controls
the mean curvature of $K$. Hence, the proof of flatness improvement
has a more geometric flavour and involves some typical arguments of
minimal surfaces theory.

The second decay estimate is concerned with the Dirichlet energy.
Assum\-ing the decay estimate to be false, we find a sequence of quasi
minimizers $(z_h,C_h)$ and a sequence of balls $B_{\rho_h}$.  Scaling and
normalizing, we obtain a sequence of quasi minimizers $(v_h,K_h)$ in a
fixed ball $B_R$ whose gradients weakly converge to the gradient of a
function $v$ which is harmonic in the upper and lower half balls, with zero
normal derivative on $\{x_n=0\}$.  An easy reflection argument
shows that the scaled Dirichlet energy of $v$ on $B_{\rho}$ decays like
$\rho$.  To transfer this property to the sequence $v_h$ (thus getting a
contradiction) we need to prove {\em strong} convergence of the gradients.
To obtain this convergence at least in a smaller ball, we introduce
functions obtained by taking the composition of $v_h$ with deformation maps
pushing a large part of $K_h$ on suitable minimal surfaces.  An energy
comparison argument leads to the strong convergence of gradients and to the
desired contradiction.

A feature of our problem is that the flatness improvement theorem involves
also the Dirichlet energy, and viceversa.  For this reason an iteration
argument leading to the improvement of {\em both} quantities is needed.
The idea is to use the decay of ${\cal F}(x,\rho)$ if ${\cal F}(x,\rho)$
controls ${\cal D}(x,\rho)$ and to use the decay of ${\cal D}(x,\rho)$ if
${\cal F}(x,\rho)$ is much smaller than ${\cal D}(x,\rho)$.  This
``twospeed" iteration argument seems to be a natural tool for treating the
partial regularity of these free discontinuity problems.

\vspace{10pt}

{\bf Acknowledgement.}
Work partially supported by MURST 40\%.

\section{Quasi minimizers}

In the following a natural number $n\geq 2$ will be fixed and we omit the
dependence of several constants on $n$.  Since we will often deal with
$(n-1)$-dimensional sets, we will use the notation $m$ for $n-1$.

Let $\Omega\subset\rn{n}$ be an open set.  For any function $u\in
L^1(\Omega)$ the set $S_u$ denotes the the complement of the Lebesgue set
of $u$, {\em i.e.},
$$
x\notin S_u\qquad\Longleftrightarrow\qquad
\exists\,\,z\in\rn{}\,\,\hbox{\rm s.t.  }\lim_{\rho\to 0^+}
\rho^{-n}\int_{B_\rho(x)}|u(y)-z|\,dy=0.
$$
The space $SBV(\Omega)$, introduced by De Giorgi and Ambrosio
in \cite{dga}, consists of all functions $u$ with finite
total variation in $\Omega$ such that the singular part (with
respect to Lebesgue measure) $D^s u$ of the distributional
derivative $Du$ is supported in $S_u$, {\em i.e.},
$$|D^s u|(\Omega\setminus S_u)=0.$$

\no In the following, $\nabla u$ stands for the density (with respect
to Lebesgue measure) of $Du$. Notice that
\begin{equation}\label{w12}
u\in SBV(\Omega)\qquad\Longrightarrow
\qquad u\in W^{1,1}(\Omega\setminus\overline{S}_u)
\end{equation}

\no because the singular part of the distributional derivative of $u$ is
supported in $S_u$.  In $SBV(\Omega)$ many variational problems involving
volume and surface energies can be formulated and have a solution (see
\cite{am1}, \cite{am2}, \cite{am3}, \cite{am4}).  In this paper very little
of the $SBV$ theory will be used, because a density lower bound for
minimizers (see Proposition \ref{dgcl} below and (\ref{minK})) allows a
reduction to the study of pairs $(u,K(u))$, with $u\in
W^{1,1}(\Omega\setminus K(u))$ and $K(u)=\overline{S}_u$.

Now we give the definitions of local minimizers and quasi minimizers
assuming $\beta=1$.  This assumption is not really restrictive, by a
scaling argument.

\begin{definition}[local minimizers] \label{locmin}
We say that $u\in SBV_{\rm loc}(\Omega)$
is a local minimizer in $\Omega$ if
\begin{equation}\label{enefin}
\E{u}{A}+\h{m}{S_u\cap A}<+\infty\qquad\qquad
\forall A\subset\subset\Omega
\end{equation}

\no and
$$
\E{u}{A}+\h{m}{S_u\cap A} \leq
\E{v}{A}+\h{m}{S_v\cap A}
$$

\no whenever $v\in SBV_{\rm loc}(\Omega)$ and
$\{v\not=u\}\subset\subset A\subset\subset\Omega$.
\end{definition}

\begin{definition}[quasi minimizers]
We will call deviation from minimality $\dev{u}{\Omega}$ of a
function $u\in SBV_{\rm loc}(\Omega)$ satisfying (\ref{enefin})
the smallest $\lambda\in[0,+\infty]$ such that
$$
\E{u}{A}+\h{m}{S_u\cap A} \leq
\E{v}{A}+\h{m}{S_v\cap A} +\lambda
$$

\no for any $v\in SBV_{\rm loc}(\Omega)$
such that $\{v\not= u\}\subset\subset A\subset\subset\Omega$.
Clearly, $\dev{u}{\Omega}=0$ if and only if $u$ is a local minimizer in
$\Omega$. Moreover, we say that $u$ is a quasi minimizer in
$\Omega$ if there exists a nondecreasing
function $\omega(t):(0,+\infty)\to [0,+\infty)$ such that
$\omega(t)\downarrow 0$ as $t\downarrow 0$ and
\begin{equation}       \label{deviation}
\dev{u}{B_\rho(x)}\leq\rho^m\omega(\rho)
\end{equation}

\no for any ball $B_\rho(x)\subset\Omega$. We denote by $\mm{\Omega}$ the
class of functions satisfying (\ref{deviation}).
\end{definition}

\begin{remark} {\rm The canonical example of quasi minimizer
is given by the minimum $u$ of the Mumford-Shah functional
\begin{equation}       \label{msfunc}
F(u)=\int_{\Omega}\bigl[|\nabla u|^2 +
\alpha(u-g)^2\bigr] \,dx+\h{m}{S_u},
\end{equation}

\no with $g\in L^\infty(\Omega)$ and $\alpha\geq 0$. Indeed,
it is easy to check that
$$ F(v)\geq F(M\land v\lor -M)\qquad\forall v\in SBV_{\rm loc}(\Omega)$$

\no with $M=\Vert g\Vert_\infty$. This shows that $\Vert u
\Vert_\infty\leq M$ and, for any competing function $v$,
the inequality $F(u)\leq F(M\land v\lor -M)$ implies that
the deviation from minimality of
$u$ in $B_\rho(x)$
does not exceed $4\alpha\omega_n\Vert g\Vert^2_\infty\rho^n$.
Hence $u\in\mm{\Omega}$ with
$$\omega(t)=4\alpha\omega_n\Vert g\Vert^2_\infty t.$$

Existence of minimizers follows by the compactness and lower semicontinuity
results of \cite{am1} and \cite{am4}.
Using Proposition \ref{dgcl} below it can be seen
(see also \cite{dcl}) that the minimum of $F$ on $SBV_{\rm loc}(\Omega)$
is equal to the minimum of the functional originally proposed by
Mumford-Shah and Blake-Zissermann:
\begin{equation}       \label{minG}
G(u,K)=\int_{\Omega\setminus K}\bigl[|\nabla u|^2 +\alpha(u-g)^2\bigr] \,
dx+\h{m}{K}.
\end{equation}

\no In (\ref{minG}), $K$ varies in the relatively closed subsets of
$\Omega$ and $u$ varies in $C^1(\Omega\setminus K)$.

}\end{remark}


\begin{proposition} \label{dgcl}
Let $u\in\mm{\Omega}$. Then, for any ball
$B_\rho(x)\subset\Omega$ centered at $x\in\overline{S}_u$ we have
$$\rho\leq\rho_\omega\qquad\Longrightarrow\qquad
\hh\bigl(S_u\cap B_\rho(x)\bigr)\geq\theta_\omega\rho^m
$$

\no for some positive constants $\rho_\omega$ and $\theta_\omega$
depending only on $\omega$. Moreover,
$$
\hh(\overline{S}_u\cap\Omega\setminus S_u)=0.
$$
\end{proposition}
{\sc Proof.} See Theorem 2.7 and Proposition 2.8 of \cite{ap}.
The density
lower bound for minimizers of the Mumford-Shah functional has been
proved also in \cite{dms} and \cite{morsol}.
\qed

We will denote the set $\overline{S}_u$ by $K(u)$. If $u\in\mm{\Omega}$,
Proposition \ref{dgcl} and the quasi minimality condition imply
\begin{eqnarray}\label{minK}
&&\int_{B_\rho(x)}|\nabla u|^2\, dy+\h{m}{K(u)\cap B_\rho(x)} \\
&&\leq\int_{B_\rho(x)}|\nabla v|^2\, dy+\h{m}{K(v)\cap B_\rho(x)}
+\rho^m\omega(\rho)\nonumber
\end{eqnarray}

\no for any $v\in SBV_{\rm loc}(\Omega)$ such that
$\{v\neq u\}\subset\subset B_\rho(x)\subset\Omega$.




\section{Statement of the main result}

In the following, ${\cal A}$ stands for the set of affine hyperplanes
in $\rn{n}$. The main result of our paper is the following:

\begin{theorem}    \label{main}
Let $\mm{\Omega}$ be defined in Definition \ref{locmin},
$u\in\mm{\Omega}$, and assume that

\begin{equation}       \label{main1}
\omega(\rho)\leq c_0\rho^{2\gamma}
\end{equation}

\no for some constants $c_0\geq 0,\,\gamma>0$ and let
$\alpha=\min\{1/4,\gamma\}/(m+2)$.
Then, there is a positive constant
$\eps_0(c_0,\gamma)$ such that for any $x\in K(u)$ and
any ball $B_\rho(x)\subset\Omega$ with $\rho<1$, the condition
\begin{equation}        \label{main2}
\rho^2\int_{B_\rho(x)}|\nabla u|^2\,dy+
\min_{A\in\LL}\int_{B_\rho(x)\cap K(u)}
\dis{A}{y}\, d{\cal H}^{m}(y)<\eps_0\rho^{m+2}
\end{equation}

\no implies that $B_{\rho/2}(x)\cap K(u)$ is a $C^{1,\alpha}$ hypersurface.
\end{theorem}


\begin{remark} \label{singtrasc}
{\rm Let $R$ be the set of regular points of $K(u)$,
{\em i.e.}, the set of those points $x\in K(u)$ such that
(\ref{main2}) holds for a sufficiently small $\rho\in (0,1)$. Then,
the complement $S=K(u)\setminus R$ is relatively closed in
$\Omega$ and it
is made of all points satisfying either
\begin{equation}\label{noflat}
\limsup_{\rho\to 0^+}\rho^{-m-2}\min_{A\in{\cal  A}}
\int_{B_\rho(x)\cap K(u)}\dis{A}{y}\,d{\cal H}^m(y)>0
\end{equation}

\no or
\begin{equation}\label{pocaene}
\liminf_{\rho\to 0^+}\rho^{-m}
\int_{B_\rho(x)}|\nabla u|^2\,dx\geq\eps_0.
\end{equation}

\no It is easy to see that $\h{m}{S}=0$. Indeed, (\ref{noflat}) is not
satisfied at any point $x\in K(u)$ where $K(u)$ has an
approximate tangent
space $P$ (see \cite[Th. 11.6]{simon}) because, to estimate
the integral in (\ref{noflat}) we can choose $A=x+P$,
thus obtaining that the limit is $0$.
Moreover, a differentiation theorem
(see \cite[3.2(1)]{simon}) shows that
$$\int_C |\nabla u|^2\,dx\geq\frac{\eps_0}{2}\h{m}{C}$$

\no for any Borel subset $C$ of the set $E$ of points $x\in\Omega$
satisfying (\ref{pocaene}). The inequality above implies that
$E$ has locally finite ${\cal H}^m$-measure, hence ${\cal L}^n(E)=0$.
Using the inequality again we obtain that $\h{m}{E}=0$.

}\end{remark}

The proof of Theorem \ref{main} is based, among other things,
on a $C^{1,\alpha}$ regularity criterion proved
in \cite{ap}. Let us introduce a notation for the
{\em Dirichlet integral} of $u$
\begin{equation}\label{defD}
{\bf D}(x,\rho)=\int_{B_\rho(x)}|\nabla u|^2\,dy
\end{equation}
and the {\em flatness} of $K(u)$
\begin{equation}\label{defA}
{\bf A}(x,\rho)=\min_{A\in\LL}\int_{B_\rho(x)\cap K(u)}
\dis{A}{y}\,d{\cal H}^{m}(y).
\end{equation}

\no We can state the following result:

\begin{theorem}\label{c1graf}
Let $u\in\mm{\Omega}$ and assume that there exist $C>0, \sigma\in
(0,1)$ such that $\omega(\rho)\leq C\rho^{2\sigma}$ and
$$
{\bf D}(x,\rho)+\rho^{-2}
{\bf A}(x,\rho)\leq C\rho^{m+\sigma}
$$

\no for any ball $B_\rho(x)\subset\Omega$ centered at $x\in K(u)$.
Then, $\Omega\cap K(u)$ is a $C^{1,\alpha}$ hypersurface with
$\alpha=\sigma/(m+2)$.
\end{theorem}
{\sc Proof.} See \cite{ap}, Theorem 5.3 and Remark 5.4.
\qed

To verify the assumption of Theorem \ref{c1graf} in $B_{\rho/2}(x)$
(with $x$ satisfying (\ref{main2})) we need to know that ${\bf A}$ and
${\bf D}$ satisfy good decay properties. The following typical
decay property of ${\bf A}$ has been proved in Theorem 6.2 of \cite{ap}:
it says that if $K(u)$ is sufficiently flat
near $x$, it is flatter on a smaller scale, provided
$\rho^2{\bf D}(x,\rho)$ and $\rho^{m+2}\sqrt{\omega(\rho)}$ are
comparable with ${\bf A}(x,\rho)$ (remember that the Dirichlet integral
is related to the curvature of $K(u)$).

\begin{theorem} [flatness improvement] \label{decay2}
For any choice of $\beta\in (0,1/112)$ and
$\eta>0$ there exists $\eps_1(\beta,\eta,\omega)>0$ such that,
for any $u\in\mm{B_\rho(x)}$, the conditions
$$
{\bf A}(x,\rho)\leq\eps_1\rho^{m+2},\qquad {\bf A}
(x,\beta\rho)\geq\eta\rho^2\bigl[ {\bf D}(x,\rho)+
\rho^{m}\sqrt{\omega(\rho)}\bigr]$$

\no and $x\in K(u)$ imply
$${\bf A}(x,\beta\rho)\leq C\beta^{m+4}{\bf A}(x,\rho)$$

\no with $C$ depending only on $\omega$.
\end{theorem}

The following theorem is the natural counterpart of the flatness
improvement theorem: it says that in the regions where
${\bf A}$ is much smaller than ${\bf D}$ there is
improvement of ${\bf D}$.

\begin{theorem} [energy improvement]
\label{decay1} For any choice of $\beta\in (0,6/7)$ there
exist $\eps_2(\beta,\omega)>0$
and $\eta(\beta,\omega)\in (0,1)$
such that, for any $u\in\mm{B_\rho(x)}$, the conditions
$${\bf D}(x,\rho)\leq\eps_2\rho^{m},\quad
{\bf A}(x,\rho)\leq\eps_2\rho^{m+2},\quad
\rho^{m+2}\sqrt{\omega(\rho)}+
{\bf A}(x,\beta\rho)\leq\eta\rho^2 {\bf D}(x,\rho)$$

\no and $x\in K(u)$ imply
$${\bf D}\bigl(x,\frac{\beta}{25}\rho\bigr)
\leq \beta^n {\bf D}(x,\rho).$$
\end{theorem}

A large part of this paper will be devoted to the proof of the energy
improvement theorem. Using a careful choice
of the constants in both decay theorems, and using
in an essential way the independence of the constant $C$ of Theorem
\ref{decay2} on $\beta$ and $\eta$, we will be able to perform in
Lemma \ref{twospeed} and Corollary \ref{twospeed1} a joint iteration
leading to the desired estimates on ${\bf A}$ and ${\bf D}$.




\section{Preliminary results}

The following theorem is concerned with the asymptotic behaviour of
sequences of quasi minimizers assuming that Dirichlet energy tends to zero
and the jump sets become flatter and flatter.

\begin{theorem} \label{planes} Let $(u_h)\subset\mm{B_R}$, $(S_h)\subset
\LL$ be satisfying the conditions
$$\lim_{h\to +\infty}\int_{B_\rho}|\nabla u_h|^2\,dx+
{\rm Dev}(u_h,B_\rho)+\int_{B_\rho\cap K(u_h)}\dis{S_h}{x}\,
d{\cal H}^{m}=0
$$

\no for any $\rho\in (0,R)$ and
$$ \lim_{h\to +\infty}{\rm dist}\bigl(0,K(u_h)\bigr)=0,\qquad
\lim_{h\to +\infty}S_h=S.$$

\no Then $S$ contains
the origin and the following properties hold:
\smallskip

\noindent
(i) for any $\rho\in (0,R)$ the height
$$\max_{x\in \overline{B}_\rho\cap K(u_h)}{\rm dist}(x,S)$$

\no is infinitesimal as $h\to +\infty$;

\noindent
(ii) denoting by $S_{R/4}^\pm$ the two connected components of
$$\bigl\{x\in B_{R/2}: {\rm dist}(x,S)\geq R/4\bigr\},$$

\no we have
$$
\liminf_{h\to +\infty}\biggl\vert
\int_{S_{R/4}^+}u_h\,dx-\int_{S_{R/4}^-}u_h\,dx\biggr\vert>0;
$$

\noindent
(iii) the measures ${\cal H}^{m}\res K(u_h)$
weakly converge to ${\cal H}^{m}\res S$ in $B_R$.
\end{theorem}
{\sc Proof.} Let $x_h\in K(u_h)$ be converging to $0$ and
$$\rho_h=\min\biggl\{\frac{{\rm dist}(0,S_h)}{4},\rho_\omega,
\frac{R}{4}\biggr\}.$$

\no Using Proposition \ref{dgcl},
if (a subsequence of) $\rho_h$ is not infinitesimal, for $h$
large enough we get
\begin{eqnarray*}
\int_{B_{R/2}\cap K(u_h)}\dis{S_h}{x}\,d{\cal H}^{m} &\geq &
\int_{B_{\rho_h}(x_h)\cap K(u_h)}\dis{S_h}{x}\,d{\cal H}^{m}
\\
&\geq& \theta_\omega\rho_h^{m}\min_{B_{\rho_h}(x_h)\cap K(u_h)}
\dis{S_h}{x}
\\
&\geq& \theta_\omega\rho_h^{m}\min_{B_{2\rho_h}}
\dis{S_h}{x}
\geq\frac{\theta_\omega}{4}\rho^{m+2}_h
\end{eqnarray*}

\no a contradiction. Hence, the
distances of $S_h$ from $0$ are infinitesimal, $S$ contains the
origin and
$$\lim_{h\to +\infty}\int_{B_\rho}|\nabla u_h|^2\,dx+
{\rm Dev}(u_h,B_\rho)+\int_{B_\rho\cap K(u_h)}\dis{S}{x}
\,d{\cal H}^{m}=0
$$

\no for any $\rho\in (0,R)$. Then, (i) follows again by the density
lower bound (see \cite[Proposition 5.1]{ap}). The properties
(ii), (iii) are proved in Step 4 and Step 5 of Theorem 4.3 \cite{ap}
(see also Lemma 6.1 of the same paper).
\qed

The next lemma is based on a well known reflection argument.

\begin{lemma} \label{har1} Let $B^+$ be the upper unit half
ball in $\rn{n}$ and $v:B^+\to\rn{}$. For $a\in (0,1)$,
let $S_a^+=\{(z,y)\in B_1:y>a\}$ and let us assume that
$v\in H^1(S_a^+)$ for any $a>0$. Then

\smallskip

\noindent
(i) $v$ belongs to $H^1(B^+)$ if and only if
$\nabla v\in L^2(B^+)$;

\noindent
(ii) if $v\in H^1(B^+)$ and
\begin{equation}\label{weakarmo}
\int_{B^+}\langle\nabla v,\nabla\phi\rangle\,dx=0\qquad
\forall \phi\in C^1_0(B_1)
\end{equation}

\no the function $\tilde{v}(z,y)=v(z,-y)$ is the harmonic extension of $v$
to $B_1$.
\end{lemma}
{\sc Proof.} (i) One implication is trivial.
Assuming $\nabla v\in L^2(B^+)$, we need only to show that $v\in L^2(B^+)$.
Clearly, the functions $v_k=k\land v\lor -k$ belong to $H^1(B^+)$
and $\Vert\nabla v_k\Vert_2\leq\Vert \nabla v\Vert_2$ for any
$k>0$; the Poincar\'e inequality (see \cite[Th. 4.4.2]{zi})
implies
$$\int_{B^+}|v_k-\bar{v}_k|^2\,dx\leq c\int_{B^+}|\nabla v_k|^2\,dx$$

\no for some dimensional constant $c$, with $\bar{v}_k$ equal to the
average of $v_k$ on $S_{1/2}^+$. Letting $k\to +\infty$ we get
$$\int_{B^+}|v-\bar{v}|^2\,dx\leq c\int_{B^+}|\nabla v|^2\,dx$$

\no hence $v\in L^2(B^+)$.

\noindent
(ii) It is well known that $\tilde{v}\in H^1(B_1)$. For any
test function $\phi\in C^1_0(B_1)$, using two times (\ref{weakarmo})
and a change of variables we get
\begin{eqnarray*}
\int_{B_1}\langle\nabla \tilde{v},\nabla\phi\rangle\,dx&=&
\int_{B^+}\langle\nabla v,\nabla\phi\rangle\,dx+
\int_{B_1\setminus B^+}\langle\nabla\tilde{v},\nabla\phi\rangle\,dx
\\
&=&\int_{B_1\setminus B^+}\langle\nabla\tilde{v},\nabla\phi\rangle\,dx
=\int_{B^+}\langle\nabla v,\nabla\tilde{\phi}\rangle\,dx=0
\end{eqnarray*}

\no with $\tilde{\phi}(z,y)=\phi(z,-y)$. Hence, $\tilde{v}$ is harmonic.\qed

For any $1$-Lipschitz function $v$ on $\partial B_R$,
let us denote by $\tilde{v}:B_R\rightarrow\rn{}$ the solution of the
minimum problem
\begin{equation}\label{diri}
\min\biggl\{\int_{B_R}\sqrt{1+|\nabla u|^2}\,dx: u\,\,\,\hbox{\rm
1-Lipschitz}, \,\,\, u=v\,\,\,\hbox{\rm on }\partial B_R\biggr\}
\end{equation}
and let
$${\cal E}(v)=\int_{B_R}\sqrt{1+|\nabla \tilde{v}|^2}\,dx.$$

\no We are interested to estimate the convergence to $0$ of
${\cal E}(v)-\omega_nR^n$ as $\Vert v\Vert_{H^1(\partial B_1)}$ tends
to $0$.

\begin{lemma} [area excess estimate] \label{har2}
Let $v:\partial B_R\rightarrow \rn{}$ be a $1/2$-Lipschitz function.
For any $\alpha\geq 1$ we have
\begin{equation}\label{areamin}
{\cal E}(v)-\omega_n R^n\leq\biggl[
\frac{\alpha^2}{R}\Vert v\Vert^2_{L^2(\partial B_R)}
+{R\Vert \nabla v\Vert^2_{L^2(\partial B_R)}\over n+2\alpha-2}
\biggr]
\end{equation}
provided $\Vert v\Vert_\infty<R/(2\alpha)$.
\end{lemma}
{\sc Proof.} The transformation
$$ u(x)\mapsto \hat{u}(y)=\frac{1}{R}u(Ry)$$
preserves the Lipschitz constant and maps functions on $B_R$ to
functions on $B_1$. Hence, we can assume with no loss of generality
that $R=1$.

\no Let $v_\alpha(x)=|x|^\alpha v(x/|x|)$.
By the equations (in polar coordinates)
$$\nabla_r v_\alpha=\alpha r^{\alpha-1}v(\omega),\qquad\qquad
\nabla_\omega v_\alpha=r^\alpha \nabla_\omega v(\omega)$$

\no we infer that $|\nabla_r v_\alpha|\leq 1/2$ and
$|\nabla_\omega v_\alpha /r|\leq 1/2$, hence $v_\alpha$
is a $1$-Lipschitz function. In particular
\begin{eqnarray*}
{\cal E}(v)-\omega_n &\leq& {\cal E}(v_\alpha)-\omega_n\leq
{1\over 2}\int_{B_1}|\nabla v_\alpha|^2\,dx
\\
&\leq& \biggl[\alpha^2\Vert v\Vert_{L^2(\partial B_1)}^2\int_0^1
r^{n+2\alpha-3}\,dr
+\Vert \nabla_\omega v\Vert^2_{L^2(\partial B_1)}\int_0^1
r^{n+2\alpha-3}\,dr
\biggr]
\\
&\leq&\biggl[\alpha^2 \Vert v\Vert^2_{L^2(\partial B_1)}
+{\Vert \nabla v\Vert^2_{L^2(\partial B_1)}\over n+2\alpha-2}
\biggr].
\end{eqnarray*}
\qed

\begin{lemma} [deformation lemma]
\label{har3} Let $g^+,\,g^-,\,g:\overline{B}^m_1\to\rn{}$
be Lipschitz functions such that $g^+=g^-=g$ on $\partial B^m_1$ and
$$
-2<g^-(z)\leq g(z)\leq g^+(z)<2 \qquad\qquad\forall z\in B^m_1.
$$
Let $C$ be the cylinder $B^m_1\times (-3,3)$, $\Gamma(g)\subset C$ be
the graph of $g$, $W\subset C$ be the open set above the graph of
$g^+$ and below the graph of $g^-$.

Then, for any $v\in SBV(C)$ there exists $w\in SBV(C)$ such that the traces
of $v$ and $w$ on $\partial C$ are equal and
$$
\int_C|\nabla w|^2\,dx\leq M\int_W|\nabla v|^2\,dx,\qquad\qquad
\h{m}{S_w\setminus \Gamma(g)}\leq M\h{m}{S_v\cap W}
$$
with $M$ depending only on the Lipschitz constants of $g^+$, $g^-$, $g$.
\end{lemma}
{\sc Proof.} Let $x=(z,y)$, $\Phi:C\to C$ be the map defined by
$$\Phi(z,y)=\cases{
\bigl(z,3+(3-y)(-1-g^+(z)+g(z))\bigr) & if $2<y<3$;\cr
\bigl(z,y-g^+(z)+g(z)\bigr) & if $g^+(z)<y\leq 2$;\cr
\bigl(z,g(z)\bigr) & if $g^-(z)< y\leq g^+(z)$;\cr
\bigl(z,y-g^-(z)+g(z)\bigr) & if $-2<y\leq g^-(z)$;\cr
\bigl(z,-3+(y+3)(1-g^-(z)+g(z))\bigr) & if $-3<y\leq -2$.}
$$
It is easy to check that the restriction of $\Phi$ to $\partial C$ is the
identity map, $\Phi:W\to C\setminus \Gamma(g)$ is invertible and the norm
of the Jacobian matrices of $\Phi$ and $\Phi^{-1}$ can be uniformly
estimated with the Lipschitz constants of $g^+,\,g^-,\,g$.
The function $w:=v\circ\Phi^{-1}$ (arbitrarily defined on $\Gamma(g)$)
has the same trace of $v$ on $\partial C$
and $S_w\setminus \Gamma(g)\subset \Phi(W)$, hence
$$\h{m}{S_w\setminus \Gamma(g)}\leq M\h{m}{S_v\cap W}.$$
Moreover
\begin{eqnarray*}
\int_C|\nabla w(x)|^2\,dx&=&\int_C\bigl\vert
\nabla v(\Phi^{-1}(x))\nabla \Phi^{-1}(x) \bigr\vert^2\,dx
\\
&=&\int_W \bigl\vert\nabla v(x)\nabla\Phi^{-1}(\Phi(x))\bigr\vert^2
|{\rm det} J\Phi(x)|\,dx\leq M\int_W |\nabla v|^2\,dx.
\end{eqnarray*}
\qed


\begin{theorem}[Lipschitz approximation]  \label{lipappr}
There exist constants $R(\omega)>0$ and $P(\omega)>0$ with the
following property: let $u\in\mm{B_{7r}}$ for some $r\leq R$ and let
$$
A=\F{u}{B_{6r}\cap K(u)},\hfill D=\E{u}{B_{6r}},\hfill
L=r^m\sqrt{\frac{{\rm Dev}(u,B_{6r})}{(6r)^m}}.
$$
Then, if $PA< r^{m+2}$ and $K(u)\cap B_{r/16}\neq\emptyset$
there exists a Lipschitz function $f:T\to T^\perp$ with Lipschitz
constant less than $1/2$ such that
\begin{equation}\label{additional}
\ \ \sup |f|^{m+2}\leq P A,\quad
\int_{B^m_{r/4}}|f|^2\,dz\leq A + P (\frac{A}{r^2}+D+L) (PA)^{2/(m+2)}
\end{equation}

\begin{equation}\label{additional1}
\int_{B^m_{r/4}}|\nabla f|^2\,dz\leq P\bigl[\frac{A}{r^2}+D+L\bigr]
\end{equation}
and, denoting by $\Gamma(f)$ the graph of $f$, we have
\begin{equation} \label{additional2}
\hh\bigl(B_r\cap K(u)\setminus\Gamma(f)\bigr)
\leq P\bigl[\frac{A}{r^2}+D+L\bigr].
\end{equation}
\end{theorem}
{\sc Proof.} Let $R=R(\omega,1/2)$ be the costant defined before
Theorem 5.2 of \cite{ap}. We will indicate by $c$ the generic constants
depending only on $n$ and $\omega$ appearing in the estimates of
Theorem 5.2 of \cite{ap}. Then, this theorem states that
$A/r^{m+2}<c$ and $K(u)\cap B_{r/16}\neq \emptyset$ imply the existence of
a Lipschitz continuous
function $f:T\rightarrow T^\perp$ with Lipschitz constant less than
$1/2$ such that $\sup |f|^{m+2}\leq c A$ and
\begin{eqnarray}\label{additional3}
&&\hh\bigl(B_r\cap K(u)\setminus\Gamma(f)\bigr)+
\hh\bigl(X\bigr) \nonumber
\\
&&\leq c \biggl[\E{u}{B_{5r}}+\int_{B_{5r}\cap K(u)}
\Vert S_x-T\Vert^2\,d\hh\biggr]
\end{eqnarray}

\no where $X$ is the projection on $B^m_{r/4}$ of $\Gamma(f)\setminus K(u)$
and $\Vert S_x-T\Vert$ represents the distance of the approximate tangent
space $S_x$ to $K(u)$ at $x$ from $T$ (see \cite{ap} for details).

Using the tilt lemma (see Lemma 6.1 of \cite{ap}) we obtain
(\ref{additional2}) from (\ref{additional3}) and the estimate $\h{m}{X}\leq
c(A/r^2+D+L)$.  The second inequality in (\ref{additional}) can be easily
deduced from the first one by evaluating separately the integral on $X$ and
the integral on $B^m_{r/4}\setminus X$.

Finally, the inequality (\ref{additional1}) can be proved with the same
method using the tilt lemma and noticing that $|\nabla f|\leq 1/2$ and
$$
\Vert S_x-T\Vert^2=2\bigl(1-\nu^2_n\bigr)= \frac{2|\nabla
f(z)|^2}{1+|\nabla f(z)|^2}\geq |\nabla f(z)|^2
$$
where $\nu$ is the normal to the graph of $f$ at $x=(z,f(z))$,
with $z\in X$ differentiability point of $f$.\qed


\section{Proof of the energy improvement theorem}

We assume whitout loss of generality that $\omega(t)>0$ for any $t>0$.
% (see Remark \ref{anytheta}
As usual, to prove Theorem \ref{decay1} we argue
by contradiction, getting sequences $(z_h)\subset\mm{B_{\rho_h}(y_h)}$,
$(S_h), \,(\tilde{S}_h)\subset\LL$ satisfying
\begin{equation}\label{zeqm1}
\ \ \int_{B_{\rho_h}(y_h)}|\nabla z_h|^2\,dx<\gamma_h\rho^m_h,\
\int_{B_{\rho_h}(y_h)\cap K(z_h)}\dis{S_h}{x}\,d{\cal H}^{m}<
\gamma_h\rho_h^{m+2}
\end{equation}

\begin{eqnarray}\label{zeqm2}
\rho_h^{m+2}\sqrt{\omega(\rho_h)}&+&
\int_{B_{\beta\rho_h}(y_h)\cap K(z_h)}
\dis{\tilde{S}_h}{x}\,d{\cal H}^{m} \nonumber
\\
&\leq& \gamma_h\rho_h^2\int_{B_{\rho_h}(y_h)}|\nabla z_h|^2\,dx
\end{eqnarray}

\no for a suitable infinitesimal sequence $\gamma_h$ and
\begin{equation}\label{zeqm3}
\int_{B_{\beta\rho_h/25}(y_h)}|\nabla z_h|^2\,dx>
\beta^n\int_{B_{\rho_h}(y_h)}|\nabla z_h|^2.
\end{equation}

To rescale all the functions to a fixed ball we will use the following
remark.

\begin{remark} [scaling] \label{scaling}
{\rm Let $u\in SBV(\Omega)$ and let $B_\rho(x_0)\subset\Omega$.
Then, it is easy to check that
$$u_\rho(y)={1\over\sqrt{\rho}}u(x_0+\rho y)$$

\no belongs to $SBV(\Omega_\rho)$ with
$\Omega_\rho=\rho^{-1}\bigl(\Omega-x_0\bigr)$. Moreover, for any ball
$B_\eta(y)\subset\Omega_\rho$ we have
\begin{eqnarray*}
\E{u_\rho}{B_\eta(y)}&=&\rho^{-m}\E{u}{B_{\eta\rho}(x_0+\rho y)},
\\
\h{m}{S_{u_\rho}\cap B_\eta (y)}&=&\rho^{-m}
\h{m}{S_u\cap B_{\eta\rho}(x_0+\rho y)}
\\
\dev{u_\rho}{B_\eta(y)}&=&\rho^{-m}\dev{u}{B_{\eta\rho}(x_0+\rho y)}.
\end{eqnarray*}

\no In particular, if $\rho\leq 1$, the monotonicity of $\omega(\rho)$
shows that
$$
u\in\mm{\Omega}\qquad\Longrightarrow\qquad u_\rho\in\mm{\Omega_\rho}.
$$
}\end{remark}
\bigskip
Let $R=R(\omega)$ be given by Theorem \ref{lipappr}.  By (\ref{zeqm1}),
(\ref{zeqm2}), (\ref{zeqm3}) and our assumption on $\omega$ we obtain that
$\rho_h\leq R$ for $h$ large enough.  Hence, scaling the functions by a
factor $\rho_h/R$ as in Remark \ref{scaling}, we obtain new sequences
$(u_h)\subset\mm{B_{R}(x_h)}$, $(T_h)$, $(\tilde{T}_h)$ satisfying the
conditions
\begin{equation}\label{zeq1}
\ \ \ \int_{B_R(x_h)}|\nabla u_h|^2\,dx<R^m\gamma_h,\
\int_{B_{R}(x_h)\cap K(u_h)}\dis{T_h}{x}\,d{\cal H}^{m}<\gamma_hR^{m+2}
\end{equation}

\begin{equation}\label{zeq2}
\ \ \sqrt{\omega(\rho_h)}+
\int_{B_{\beta R}(x_h)\cap K(u_h)}\dis{\tilde{T}_h}{x}\,d{\cal H}^{m}
\leq\gamma_h\int_{B_{R}(x_h)}|\nabla u_h|^2\,dx
\end{equation}
and
\begin{equation}\label{zeq3}
\int_{B_{\beta R/25}(x_h)}|\nabla u_h|^2\,dx>\beta^n\int_{B_{R}(x_h)}
|\nabla u_h|^2\,dx.
\end{equation}

\no The deviation from minimality of $u_h$ in $B_R(x_h)$ satisfies
\begin{eqnarray}\label{zeq33}
{\rm Dev}(u_h,B_R(x_h))&=&\frac{{\rm Dev}(z_h,B_{\rho_h}(y_h))}
{\rho^{m}_h/R^m}\leq R^m\omega(\rho_h) \nonumber
\\
&\leq& R^m\gamma_h^2\biggl(\int_{B_R}|\nabla u_h|^2\,dx\biggr)^2.
\end{eqnarray}

\no Denoting by $T$ the hyperplane $\{x_n=0\}$, by a rotation and a
translation we can assume with no loss of generality that $\tilde{T}_h=T$,
$T(x_h)=0$ and $T_h$ converges to some hyperplane $S$.  Using (\ref{zeq2})
and the density lower bound it is easy to see, by the same argument used in
Theorem \ref{planes}, that $x_h\to 0$ (notice that $x_h\in K(u_h)$).  By
Theorem \ref{planes}(iii) and (\ref{zeq1}), (\ref{zeq33}) we obtain that
${\cal H}^{m}\res K(u_h)$ weakly converges in $B_R$ to ${\cal H}^{m}\res S$
and $S$ contains the origin.  Since

\begin{eqnarray*}
0&=&\lim_{h\to +\infty}\int_{B_{\beta R}(x_h)\cap K(u_h)}\dis{\tilde{T}_h}{x}
\,d{\cal H}^{m}
\\
&=&\lim_{h\to +\infty}\int_{B_{\beta R}\cap K(u_h)}\dis{T}{x}
\,d{\cal H}^{m}
=\int_{B_{\beta R}\cap S}\dis{T}{x}\,d{\cal H}^{m}
\end{eqnarray*}

\no we infer that $T=S$. Setting now
$$\eps_h=\biggl(\int_{B_R}|\nabla u_h|^2\,dx\biggr)^{1/2}$$
and defining $v_h=u_h/\eps_h$ we can rewrite (\ref{zeq1}),
(\ref{zeq2}), (\ref{zeq3}) as follows
\begin{equation}\label{zeq4}
\lim_{h\to +\infty} \eps_h^2+
\int_{B_R(x_h)\cap K(v_h)}\dis{T}{x}\,d{\cal H}^{m}=0
\end{equation}

\begin{equation}\label{zeq5}
\int_{B_{\beta R}(x_h)\cap K(v_h)}
\dis{T}{x}\,d{\cal H}^{m}=o(\eps_h^2)
\end{equation}

\begin{equation}\label{zeq6}
\int_{B_R(x_h)}|\nabla v_h|^2\,dx=1,\qquad
\int_{B_{\beta R/25}(x_h)}|\nabla v_h|^2\,dx>\beta^n.
\end{equation}

\no We point out that, by (\ref{minK}),
the functions $v_h$ are quasi minimizers in $B_R(x_h)$
of the rescaled functionals

\begin{equation}\label{newfunct}
{\cal F}_h(v)=\int_{B_\rho}|\nabla v|^2\,dx+\frac{1}{\eps_h^2}
{\cal H}^{m}(B_\rho\cap K(v))
\end{equation}

\no and, denoting by ${\rm Dev}_h$ the deviation from minimality of
${\cal F}_h$, (\ref{zeq33}) yields
\begin{equation} \label{zeq333}
\lim_{h\to +\infty}{\rm Dev}_h(v_h,B_\rho)\leq
\limsup_{h\to +\infty}\frac{{\rm Dev}(u_h,B_R(x_h))}{\eps_h^2}=0
\end{equation}

\no for any $\rho<R$ (notice that $B_\rho\subset B_R(x_h)$ for $h$ large
enough).

{\sc Step 1.} Let us use $y=x_n$ to denote the ``vertical'' coordinate.
By (\ref{zeq4}) and Theorem \ref{planes}(i) we obtain
\begin{equation}\label{zeq7}
\lim_{h\to +\infty}\sup_{x\in D\cap K(v_h)}|y|=0\qquad
\forall D\subset\subset B_R
\end{equation}

\no In particular, the maximal height of $K(v_h)$ in $\overline{B}_{R/2}$ is
infinitesimal. For any $a>0$ we
denote by $S_a^+$ (respectively $S_a^-$) the set
of points $x\in B_{R/2}$ such that $y>a$ (respectively
$y<-a$), we denote by $c_h^\pm$ the average of $v_h$ on
$S_{R/4}^\pm$ and we assume, up to a change of sign, that
$c_h^+\geq c_h^-$. By Theorem \ref{planes}(ii) we obtain that
the sequence $\eps_h(c_h^+-c_h^-)$ has no infinitesimal
subsequences, hence $(c_h^+-c_h^-)$ diverges.

For any $a>0$ (\ref{zeq7}) implies the
existence of $h_a\in{\bf N}$ such that $v_h-c_h^{\pm}$ belongs
to $H^1(S_a^{\pm})$ for $h\geq h_a$. Moreover, if $a<R/4$, the average of
$v_h-c_h^{\pm}$ is zero on $S_{R/4}^\pm\subset S_a^\pm$
and the Poincar\'e
inequality (see \cite[Th. 4.4.2]{zi}) implies that the family
$\{v_h-c_h^\pm\}_{h\geq h_a}$ is bounded in $H^1(S_a^\pm)$.
Possibly extracting a subsequence,
by a diagonal argument we can find a function
$v:B_{R/2}\setminus T\to\rn{}$ such that
$v_h-c_h^{\pm}$ weakly converges to $v$ in $H^1(S_a^\pm)$,
and also pointwise in $S_a$,
for any $a\in (0,R/4)$. By lower semicontinuity of the Dirichlet integral
and the first equality in (\ref{zeq6}) we infer
\begin{equation}\label{zeq8}
\int_{B_R}|\nabla v|^2\,dx\leq 1.
\end{equation}

\no In particular, Lemma \ref{har1}(i) yields $v\in H^1(S_0^\pm)$.

{\sc Step 2.} Now we show that $v$ in harmonic in $B_{R/2}\setminus T$
and its normal derivative on $T$ is zero.

Using (\ref{zeq333}) and comparing
${\cal F}_h(v_h)$ with ${\cal F}_h(v_h+\phi)$ for $\phi\in
C^1_0(B_{R/2}\setminus T)$, it is
easy to check that $v$ is harmonic in $B_{R/2}\setminus T$.

Possibly passing to a subsequence we can assume that
the measures $|\nabla v_h|^2{\cal L}^n$ are weakly converging in
$B_{R/2}$ to some measure $\mu$. Since $\nabla v_h$ weakly
converges to $\nabla v$ in $B_{R/2}$ we have
$\mu\geq |\nabla v|^2{\cal L}^n$.

Comparing ${\cal F}_h(v_h)$ with ${\cal F}_h(\phi (v+c_h^{\pm})+(1-\phi)v_h)$
with $\phi\in C^1_0(B_{R/2}\setminus T)$ and using again
(\ref{zeq333}) we get the inequality
\begin{eqnarray*}
\int_{B_{R/2}}|\nabla v_h|^2\,dx &\leq&
(1+\eps)\int_{B_{R/2}}\phi|\nabla v|^2+(1-\phi)|\nabla v_h|^2\,dx
\\
&+&\bigl(1+\frac{1}{\eps}\bigr)\int_{B_{R/2}}|\nabla\phi|^2
(v_h-c_h^\pm-v)^2\,dx+o(1)
\end{eqnarray*}

\no for any $\eps >0$. Using the strong convergence
of $v_h-c_h^{\pm}$ to $v$ in $L^2_{\rm loc}(B_{R/2}\setminus T)$, letting
first $h\to +\infty$ then $\eps \to 0$ we find
$$ \int_{B_{R/2}}\phi \,d\mu\leq \int_{B_{R/2}}\phi |\nabla v|^2\,dx.$$

\no Since $\phi$ is arbitrary, we obtain that
$\mu\res \bigl(B_{R/2}\setminus T\bigr)=
|\nabla v|^2{\cal L}^n$, hence the absolutely continuous part of
$\mu$ is obviously $|\nabla v|^2{\cal L}^n$ and the singular part
must be supported in $T$.

We now claim that $v$ has zero normal derivative on
$T\cap B_{R/2}$, {\em i.e.},
\begin{equation}\label{derino}
\int_{S_0^\pm}\langle\nabla v,\nabla\phi\rangle\,dx=0
\end{equation}

\no for any $\phi\in C^1_0(B_{R/2})$. Indeed,
let $\phi\in C^1_0(B_{R/2})$, $\eps>0$ and let $\psi:\rn{}
\to\rn{}$ be a smooth non decreasing function such that $\psi(-\infty)=0$,
$\psi(+\infty)=1$ and $\psi'(t)\leq\eps$ for any $t\in\rn{}$.
By (\ref{zeq333}) the sequence $z_h={\rm Dev}_h(v_h,B_{R/2})$ is
infinitesimal. Hence, denoting by $c_h$ the average of
$c_h^+$ and $c_h^-$,
comparing ${\cal F}_h(v_h)$ with ${\cal F}_h\bigl(v_h+
\sqrt{z_h}\phi\psi(v_h-c_h)\bigr)$ and
dividing both sides by $\sqrt{z_h}$ we obtain
$$\liminf_{h\to +\infty}
\int_{B_{R/2}}\psi(v_h-c_h)\langle \nabla v_h,\nabla\phi\rangle\,dx+
\int_{B_{R/2}}\phi\psi'(v_h-c_h)|\nabla v_h|^2\,dx\geq 0.$$

\no Since $\nabla v_h$ weakly converges to $\nabla v$
in $L^2(B_{R/2},\rn{n})$ and $\psi(v_h-c_h)$ strongly converges to
$\chi_{S_0^+}$ (here we use the fact that $c_h^+-c_h^-\to +\infty$) we obtain
$$\int_{S_0^+}\langle\nabla\phi,\nabla v\rangle
\,dx\geq -\eps\Vert\phi\Vert_\infty.$$

\no As $\eps$ and $\phi$ are arbitrary, (\ref{derino}) is proved
for $S_0^+$. The argument for $S_0^-$ is similar.
By Lemma \ref{har1}(ii) we infer that the restrictions of $v$ to $S_0^{\pm}$
can be prolonged by reflection along $T$ to harmonic
functions on $B_{R/2}$. In particular, $v$ and $\nabla v$ are
locally bounded in $B_{R/2}$ and
\begin{equation}\label{gooddec}
\int_{B_\rho}|\nabla v|^2\,dx\leq
\frac{2^n\rho^n}{R^n}\int_{B_{R/2}}|\nabla v|^2\,dx\leq
\frac{2^n\rho^n}{R^n}\qquad\forall \rho\in (0,R/2).
\end{equation}

{\sc Step 3.} Now we prove an estimate from below on the size of
$K(v_h)$, namely
$${\cal H}^{m}\bigl(C_\rho\cap K(v_h)\bigr)\geq \omega_{m}\rho^{m}
-o(\eps_h^2)$$

\no for any cylinder $C_\rho=B^m_\rho(x_0)\times (-\rho,\rho)$ contained in
$B_{R/2}$. Let $Z_h\subset B^m_\rho(x_0)$ be the
projection of $C_\rho\cap K(v_h)$ on $T$; by well know properties of
Hausdorff measures it suffices to prove that ${\cal L}^{m}(Z_h)\geq
\omega_m\rho^m-o(\eps_h^2)$. Let $G_h=B^m_\rho(x_0)\setminus Z_h$;
given $a\in (0,\rho)$ we have that the functions
$v_h(z,\cdot)$ are absolutely continuous in $(-a,a)$
for almost every $z\in G_h$, hence
\begin{eqnarray}\label{stn1}
\int_{G_h}|v_h(z,a)-v_h(z,-a)|^2 \,dz&\leq& 2a\int_{G_h}\int_{-a}^a
|{\partial v_h\over\partial y}|^2(z,y)\,dy\,dz
\\
&\leq& 2a\int_{B_{R/2}}|\nabla v_h|^2\,dx\leq 2a.\nonumber
\end{eqnarray}

\no On the other hand, the integrals
$$\int_{G_h}\bigl\vert (v_h(z,a)-c_h^+)-(v_h(z,-a)-c_h^-)\bigr\vert^2\,dz$$
are bounded because of the convergence of $(v_h-c_h^\pm)$ to $v$. Since
$(c_h^+-c_h^-)$ tends to $+\infty$ we infer that ${\cal L}^{n-1}(G_h)$ is
infinitesimal, hence
\begin{equation}\label{stn2}
\lim_{h\to +\infty}
\int_{G_h}\bigl\vert (v_h(z,a)-c_h^+)-(v_h(z,-a)-c_h^-)\bigr\vert^2\,dz=0
\end{equation}

\no for any $a\in (0,\rho)$. Assuming
by contradiction that ${\cal L}^{n-1}(G_{h_k})\geq\delta\eps_{h_k}^2$ for
some $\delta>0$ and some subsequence $(h_k)$, by (\ref{stn1}) and
(\ref{stn2}) we obtain
$$\delta\eps_{h_k}^2(c_{h_k}^+-c_{h_k}^-)^2
\leq {\cal L}^{n-1}(G_{h_k})(c_{h_k}^+-c_{h_k}^-)^2\leq 3a$$

\no for $k$ large enough depending on $a$.  Since $a$ is arbitrary we find
that $\eps_{h_k}(c_{h_k}^+-c_{h_k}^-)$ tends to $0$, a contradiction with
Theorem \ref{planes}(ii).

{\sc Step 4.} Applying Theorem \ref{lipappr} to the functions $u_h$ with
$r=\beta R/6$ we can find $1/2$-Lipschitz functions $f_h:B^m_{r/4}\to\rn{}$
such that (we use (\ref{additional}), (\ref{additional1}), (\ref{zeq5}),
(\ref{zeq33}))

\begin{equation}\label{asym1}
\lim_{h\to +\infty} \sup |f_h|=0,\qquad
\int_{B^m_{r/4}}|f_h|^2\,dz=o(\varepsilon_h^2)
\end{equation}

\begin{equation}\label{asym2}
\int_{B^m_{r/4}}|\nabla f_h|^2\,dz\leq Q \varepsilon_h^2
\end{equation}

\begin{equation}
\h{m}{B_r\cap K(v_h)\setminus \Gamma(f_h)}\leq
Q \varepsilon_h^2  \nonumber
\end{equation}

\no for some constant $Q\geq 0$.
Possibly extracting a subsequence, we can assume that the measures
$$ \nu_h={1\over\varepsilon_h^2}{\cal H}^m\res
\bigl(K(v_h)\setminus \Gamma(f_h)\bigr)$$

\no are weakly converging in $B_r$ to some measure $\nu$. Since
the maximal height of $K(v_h)$ is infinitesimal, the measure $\nu$
is supported in $T$.

{\sc Step 5.} We will prove that $\mu=\lim |\nabla v_h|^2{\cal L}^n$ is
absolutely continuous with respect to ${\cal L}^n$. The convergence of the
Dirichlet integrals and the final contradiction will follow at once.

Let us fix a cylinder
$C=B^m_\rho(x_0)\times (-3\rho,3\rho)$
contained in $B_{r/4}$, and assume that $\rho$ and $x_0$ fulfil
the conditions
\begin{equation}\label{sceltaro}
\mu(\partial C)=\nu(\partial C)=0
\end{equation}
\begin{equation}\label{sceltaro1}
\liminf_{h\to +\infty}\frac{1}{\varepsilon_h^2}
\int_{\partial B^m_\rho(x_0)}|\nabla f_h|^2\,dz\leq Q'<+\infty.
\end{equation}

\no Let $h(k)$ be a subsequence such that
\begin{displaymath}
\lim_{k\to +\infty}\frac{1}{\varepsilon_{h(k)}^2}
\int_{\partial B^m_\rho(x_0)}|\nabla f_{h(k)}|^2\,dz\leq Q'
\end{displaymath}

\no and let $g_k:\overline{B}^m_\rho(x_0)\to\rn{}$
be the $1$-Lipschitz solutions of the least area problems (\ref{diri})
with $B^m_\rho(x_0)$ in place of $B_R$ and $f_{h(k)}$ in place of $v$.
Let
$$ g_k^+(x)=\inf_{y\in\partial B^m_\rho(x_0)}f_{h(k)}(y)+|x-y|,
\qquad
g_k^-(x)=\sup_{y\in\partial B^m_\rho(x_0)}f_{h(k)}(y)-|x-y|.$$

\no It is well known (and easy to check) that $g_k^+$ and $g_k^-$ are
respectively the largest and the smallest $1$-Lipschitz
extensions of the restriction of $f_{h(k)}$ to $\partial B^m_\rho(x_0)$.
In particular
$$
g_k^-(x)\leq g_k(x)\leq g_k^+(x)\quad\hbox{\rm and}\quad
g_k^-(x)\leq f_{h(k)}(x)\leq
g_k^+(x)\qquad\forall x\in\overline{B}^m_\rho(x_0).
$$

\no Since $f_h$ uniformly converges to $0$ in $\partial B^m_\rho(x_0)$,
the functions $g_k^\pm$ uniformly converge in $\overline{B}^m_\rho(x_0)$
to $\pm {\rm dist}(x,\partial B^m_\rho(x_0))$; we set
$$
W_\infty=\bigl\{x=(z,y)\in\overline{C}: |y|\geq {\rm dist}\bigl(z,\partial
B^m_\rho(x_0)\bigr)\bigr\}.$$

For $k$ large enough we have $\Vert g_k^\pm\Vert_\infty< 2\rho$,
hence we can apply
Lemma \ref{har3}, getting functions $w_k\in SBV(C)$, with the same
trace of $v_{h(k)}$ of $\partial C$, such that
\begin{equation}\label{seq21}
\int_C |\nabla w_k|^2\,dx\leq M\int_{W_k}|\nabla v_{h(k)}|^2\,dx
\end{equation}
and

\begin{equation}
\h{m}{S_{w_k}}\leq M\h{m}{K(v_{h(k)})\cap W_k}+\h{m}{\Gamma(g_k)}
\end{equation}\label{seq22}

\no where $W_k$ denotes the open portion of $C$ above the graph of $g_k^+$
and below the graph of $g_k^-$.
Using Lemma \ref{har2} and taking into account (\ref{asym1}) and
(\ref{asym2}) we have
$$\limsup_{k\to +\infty} {\h{m}{\Gamma(g_k)}-\omega_m\rho^m\over
\varepsilon_{h(k)}^2}\leq {\rho Q'\over m+2\alpha-2}$$

\no for any $\alpha\geq 1$, hence $\h{m}{\Gamma(g_k)}\leq\omega_m\rho^m+
o(\varepsilon_{h(k)}^2)$.
Using the fact that $W_k$ does not intersect the graph of $f_{h(k)}$ we
obtain the following estimate on $\h{m}{S_{w_k}}$:

\begin{equation}\label{dura}
\h{m}{S_{w_k}}\leq \omega_m\rho^m+
M\varepsilon_{h(k)}^2\nu_{h(k)}(W_k)+o(\varepsilon_{h(k)}^2).
\end{equation}

\no Now, we define
$$\tilde{v}_k(x)=\cases{v_{h(k)}(x) & if $x\notin C$;\cr\cr
w_k(x) & if $x\in C$}$$

\no and we compare ${\cal F}_{h(k)}(v_{h(k)})$ with ${\cal F}_{h(k)}
(\tilde{v}_k)$.
Taking into account (\ref{zeq333}) and (\ref{seq21}) we get
\begin{eqnarray} \label{enercontrol}
\int_C |\nabla v_{h(k)}|^2\,dx &\leq& M\int_{W_k}|\nabla v_{h(k)}|^2\,dx
\nonumber
\\
&+& {\h{m}{S_{w_k}\cap C}-\h{m}{S_{v_{h(k)}}\cap C}\over
\varepsilon_{h(k)}^2}+o(1).
\end{eqnarray}

\no Now, by Step 3, $\h{m}{S_{v_{h(k)}}\cap C}$ exceeds $\omega_m\rho^m$ up
to an infinitesimal faster than $\varepsilon_{h(k)}^2$.  Hence, letting
$k\to +\infty$ in (\ref{enercontrol}) and using (\ref{dura}) we obtain
$$
\mu(C)\leq M\mu(W_\infty)+M\nu(W_\infty).
$$

\no Finally, our choice of $\rho$ and the fact that $\mu^s$ and $\nu$
are both supported in $T$ guarantees that
$\mu^s(W_\infty)=\nu(W_\infty)=0$ and
$$\mu(W_\infty)=\int_{W_\infty}|\nabla v|^2\,dx \leq
6\omega_m\Vert\nabla v\Vert_{L^\infty(B_{R/4})}^2\rho^n.$$

\no Hence, $\mu(C)\leq 6M\omega_m
\Vert\nabla v\Vert^2_{L^\infty(B_{R/4})}\rho^n.$

By a standard approximation argument, the inequality remains true even if
(\ref{sceltaro}) or (\ref{sceltaro1}) are not fulfilled. Indeed, the set
of radii $\rho$ for which (\ref{sceltaro}) is false is at most countable,
and the set of radii $\rho$ for which (\ref{sceltaro1}) is false is
${\cal L}^1$-negligible, by (\ref{asym2}) and the Fatou Lemma.  Since the
cylinder $C\subset B_{r/4}$ is arbitrary, it follows that $\mu^s\res
B_{r/4}=0$.

{\sc Step 6.} End of the proof.  From Step 5, it follows that $|\nabla
v_h|^2{\cal L}^n$ weakly converges to $|\nabla v|^2{\cal L}^n$ in $B_{\beta
R/24}$.  In particular,
$$
\lim_{h\to +\infty}\int_{B_\rho}|\nabla v_h|^2\,dx=
\int_{B_\rho}|\nabla v|^2\,dx\qquad\forall \rho\in (0,\beta R/24).
$$

\no Choosing $\rho=\beta R/25$ and passing to the limit as $h\to +\infty$ in
(\ref{zeq6}) we obtain
$$\int_{B_{\beta R/25}}|\nabla v|^2\,dx\geq\beta^n.$$

\no
This contradicts the energy estimate (\ref{gooddec}) with $\rho=\beta R/25$.
\qed

%\begin{remark} \label{anytheta} {\rm
%If we assume that $\omega(t)$ vanishes on some interval
%$(0,b)\subset (0,+\infty)$, then the statement of Theorem
%\ref{decay1} remains true with the additional assumption $\rho<b$.}
%\end{remark}

\section{Proof of partial regularity}

In this section we will prove the partial regularity of quasi
minimizers, stated in Theorem \ref{main}. Since $\omega$ is a fixed
function, we omit in this section the dependence of the constants on
$\omega$.
Let $c_0\geq 0$, $\gamma>0$, be given by (\ref{main1}).

If $u\in\mm{B_\rho(x)}$ and $x\in K(u)$ we define
\begin{equation}\label{newDA}
A(\rho)={\bf A}(x,\rho),\qquad D(\rho)={\bf D}(x,\rho).
\end{equation}
\no By Theorem \ref{decay2} we obtain that,
for any choice of $\beta\in (0,1/112)$,
$A$ satisfies the following decay property, that for
convenience we restate in a slightly different way:

{\bf (i)} For any $\eta>0$ there exists
$\eps_1(\beta,\eta)>0$ such that the conditions
$$A(\rho)\leq\eps_1\rho^{m+2},\qquad
A(\beta\rho)\geq\eta\max\bigl\{\rho^2 D(\rho),
\sqrt{c_0}\rho^{m+2+\gamma}\bigr\}$$

\no imply $A(\beta\rho)\leq C\beta^{m+4} A(\rho)$, with $C$ depending only
on $c_0$, $\gamma$.

Similarly, using Theorem \ref{decay1},
we obtain that $D$ satisfies for any $\beta\in (0,6/7)$
the following decay property:

{\bf (ii)} there exist
$\eps_2(\beta)>0$ and $\eta(\beta)\in (0,1)$ such that
the conditions
$$D(\rho)\leq\eps_2\rho^m,\qquad A(\rho)\leq\eps_2\rho^{m+2},
\qquad\max\bigl\{ \sqrt{c_0}\rho^{m+2+\gamma},A(\beta\rho)\bigr\}
\leq\eta\rho^2 D(\rho)$$

\no imply $D(\beta\rho/25)\leq \beta^n D(\rho)$.

\begin{lemma}\label{twospeed} Let $\bar{\rho}\in (0,1)$,
$A,\,D:(0,\bar{\rho})\to {\bf R}^+$ be nondecreasing functions
satisfying {\bf (i)} and {\bf (ii)} above. Then, for $\beta>0$
sufficiently small (see (\ref{mag3}) below)
there are constants $\eps,\,\eta,\,L>0$
such that, setting
\begin{equation}\label{defen}
E(\rho)=\biggl[\eta\frac{D(\rho)}{\rho^m}+\frac{A(\rho)}{\rho^{m+2}}
\biggr]
\end{equation}

\no the following implication holds
$$E(\beta\rho)>L\rho^\gamma,\,\,
E(\rho)<\eps\quad\Rightarrow\quad E(\tau_1\rho)\leq\tau_1^{1/4}
E(\rho)\quad\hbox{\it or}\quad E(\tau_2\rho)\leq\tau_2^{1/4} E(\rho)
$$

\no with $\tau_1=\beta/25$ and $\tau_2=\beta^{1/3}$.
\end{lemma}
{\sc Proof.} We fix a constant $\beta\in (0,1/(112)^3)$ so small that
\begin{equation}\label{mag3}
25^{m+3}\beta^{1/3}<\biggl(\frac{\beta}{25}\biggr)^{1/4},\qquad
2C\beta^{2/3}<\bigl(\beta^{1/3}\bigr)^{1/4}
\end{equation}

\no and we fix $\eta(\beta)>0$ according to {\bf (ii)}.
Finally, we set
$$
\eps=\eta(\beta)\min\biggl\{\eps_2(\beta),\eps_1\bigl(\beta,
\beta^{(3m+7)/3}\eta(\beta)\bigr),\eps_1\bigl(\beta^{1/3},
\beta^{(3m+7)/3}\eta(\beta)\bigr)\biggr\}
$$

\no and we choose $L>0$ satisfying the conditions
$$
\frac{\eta(\beta)}{L}
\bigl[\beta^{-m}+\beta^{1/3}\bigr]<c^{-1/2}_0,\qquad
\frac{\beta^{-m-2}+\beta^{-(6m+7)/3}}{L}<c_0^{-1/2}.
$$

\no Let us assume that $E(\rho)<\eps$ and $E(\beta\rho)>L\rho^\gamma$
for some $\rho\in (0,\bar{\rho})$.

{\sc Case 1.} Assume that
\begin{equation}\label{mag1}
A(\beta\rho)\leq\beta^{(3m+7)/3}\eta(\beta)\rho^2D(\rho).
\end{equation}

\no Our choice of $\eps$ implies $D(\rho)<\rho^m\eps_2(\beta)$; moreover,
using the inequality $E(\beta\rho)>L\rho^\gamma$ we get
$$
\frac{\eta(\beta)D(\rho)}{\rho^m}\bigl[\beta^{-m}+
\beta^{1/3}\bigr]>L\rho^\gamma
$$

\no and our choice of $L$ implies $\sqrt{c_0}\rho^{m+\gamma}
\leq\eta(\beta)D(\rho)$.
Hence, we can apply {\bf (ii)}. This, together with (\ref{mag1}) yields
\begin{eqnarray*}
E\Bigl(\frac{\beta}{25}\rho\Bigr)&=&\frac{A(\frac{\beta}{25}\rho)}{(
\frac{\beta}{25}\rho)^{m+2}}
+\eta(\beta)\frac{D(\frac{\beta}{25}\rho)}
{\bigl(\frac{\beta}{25}\rho\bigr)^m}
\leq\frac{25^{m+2}A(\beta\rho)}{\beta^{m+2}\rho^{m+2}}+
25^m\eta(\beta)\frac{D(\frac{\beta}{25}\rho)}{\beta^m\rho^m}\nonumber
\\
&\leq&\frac{25^{m+2}\beta^{(3m+7)/3}\eta(\beta)\rho^2 D(\rho)}
{\beta^{m+2}\rho^{m+2}}+
25^m\eta(\beta)\frac{\beta^n D(\rho)}{\beta^m\rho^m} \nonumber
\\
&<& 25^{m+3} \beta^{1/3}\eta(\beta)\frac{D(\rho)}{\rho^m}
\leq 25^{m+3}\beta^{1/3}E(\rho).
\end{eqnarray*}

\no In this case (\ref{mag3}) yields $E(\tau_1\rho)\leq\tau_1^{1/4} E(\rho)$.

{\sc Case 2.} Now we assume that (\ref{mag1}) does not hold, hence
\begin{equation}\label{mag2}
A(\beta^{1/3}\rho)\geq A(\beta\rho)>
\beta^{(3m+7)/3}\eta(\beta)\rho^2 D(\rho)
\end{equation}

\no Our choice of $\eps$ implies
$$A(\rho)<\eps_1\bigl(\beta^{1/3},\beta^{(3m+7)/3}\eta(\beta)\bigr)
\rho^{m+2},
\qquad
A(\rho)<\eps_1\bigl(\beta,\beta^{(3m+7)/3}\eta(\beta)\bigr)\rho^{m+2}.$$

\no Moreover, (\ref{mag2}) and $E(\beta\rho)>L\rho^\gamma$ yield
$$\frac{A(\beta\rho)}{\rho^{m+2}}\bigl[
\beta^{-m-2}+\beta^{-(6m+7)/3}\bigr]>L\rho^\gamma$$

\no and our choice of $L$ yields $\sqrt{c_0}\rho^{m+2+\gamma}\leq
A(\beta\rho)$. Since $\beta<1$ we have also
$\sqrt{c_0}(\beta^{1/3}\rho)^{m+2+\gamma}\leq A(\beta^{1/3}\rho)$.

\no Hence we can apply {\bf (i)} with $\beta^{1/3}$ and $\beta$.
This, together with (\ref{mag2}), yields
\begin{eqnarray*}
&&E(\beta^{1/3}\rho)=\nonumber
\\
&=&\frac{A(\beta^{1/3}\rho)}{
\beta^{(m+2)/3}\rho^{m+2}}+
\eta(\beta)\frac{D(\beta^{1/3}\rho)}{\beta^{m/3}\rho^m}\leq
C\frac{\beta^{(m+4)/3} A(\rho)}{\beta^{(m+2)/3}\rho^{m+2}}+
\eta(\beta)\frac {D(\rho)}{\beta^{m/3}\rho^m}\nonumber
\\
&\leq&C\beta^{2/3}\frac{A(\rho)}{\rho^{m+2}}+\frac{A(\beta\rho)}
{\beta^{(3m+8)/3}\rho^{m+2}}\leq C\beta^{2/3}\frac{A(\rho)}{\rho^{m+2}}+
C\frac{\beta^{m+4} A(\rho)}{\beta^{(3m+8)/3}\rho^{m+2}}\nonumber
\\
&\leq&2C\beta^{2/3} E(\rho).
\end{eqnarray*}

\no In this case (\ref{mag3}) gives $E(\tau_2\rho)\leq\tau_2^{1/4}E(\rho)$.

\qed

\begin{corollary} \label{twospeed1}
With the assumptions and the notations of
Lemma \ref{twospeed}, $E(\rho)<\eps$ implies
$$E(r)\leq \biggl(\frac{25}{\beta}\biggr)^{m+2+\sigma}
\max\biggl\{\left(\frac{r}{\rho}\right)^\sigma E(\rho),
\frac{L}{\beta^{\sigma}}
r^\sigma\biggr\}\qquad \forall r\in (0,\rho]$$

\no with $\sigma=\min\{1/4,\gamma\}$.
\end{corollary}
{\sc Proof.} Let us assume that $E(\rho)<\eps$ for some $\rho\in
(0,\bar{\rho})$.
We inductively define an infinitesimal sequence of radii $(\rho_k)$
such that $\rho_0=\rho$, $\rho_{k+1}/\rho_k\in\{\beta,\tau_1,\tau_2\}$
and
\begin{equation}\label{induz}
E(\rho_k)\leq\max\biggl\{\left(\frac{\rho_k}{\rho}\right)^\sigma E(\rho),
\frac{L}{\beta^{\sigma}}\rho_k^\sigma\biggr\}.
\end{equation}

\no Clearly, (\ref{induz}) holds with $k=0$. Assume that the inequality
is valid for $\rho_k$; if
$$E(\beta\rho_k)\leq L\beta^{-\sigma}
(\beta\rho_k)^\sigma$$

\no we choose $\rho_{k+1}=\beta\rho_k$. Otherwise, $E(\beta\rho_k)\geq
L\rho_k^\sigma$ and Lemma \ref{twospeed} yields that (\ref{induz}) is
valid either with $\rho_{k+1}=\tau_1\rho_k$ or with $\rho_{k+1}=
\tau_2\rho_k$.

Given $r\in (0,\rho)$ there exists a unique $k$ such that
$\rho_{k+1}\leq r<\rho_k$; since
$\tau_1=\beta/25<\beta<\tau_2=\beta^{1/3}$ we have
\begin{eqnarray*}
E(r)&\leq&\biggl(\frac{\rho_k}{r}\biggr)^{m+2}E(\rho_k)
\leq \biggl(\frac{25}{\beta}\biggr)^{m+2} E(\rho_k)
\\
&\leq&\biggl(\frac{25}{\beta}\biggr)^{m+2}\max\biggl\{
\left(\frac{\rho_k}{\rho}\right)^\sigma E(\rho),L\beta^{-\sigma}
\rho_k^\sigma\biggr\}
\\
&\leq&\biggl(\frac{25}{\beta}\biggr)^{m+2+\sigma}\max\biggl\{
\left(\frac{r}{\rho}\right)^\sigma E(\rho),L\beta^{-\sigma}
r^\sigma\biggr\}
\end{eqnarray*}

\no and the statement follows.
\qed

\no {\sc Conclusion of the proof.} Let $\eps$ be given by Corollary
\ref{twospeed1}.
We need only to choose $\eps_0$ so small that (\ref{main2}) implies
$$\biggl[\eta\frac{{\bf D}(y,\rho/2)}{(\rho/2)^m}+\frac{{\bf A}(y,\rho/2)}
{(\rho/2)^{m+2}}\biggr]<\eps\qquad\forall y\in B_{\rho/2}(x).$$

\no By Corollary \ref{twospeed1} we
obtain that the assumptions of Theorem \ref{c1graf}
are satisfied in $\Omega=B_{\rho/2}(x)$.



\vfill\eject
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\end{document}
