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\title{\LARGE Partial regularity of free discontinuity sets I}

\author{{\Large Luigi Ambrosio} \\ Dipartimento di Matematica
\\ Via Abbiategrasso 205, 27100 Pavia, Italy \and
{\Large Diego Pallara} \\ Dip. di Matematica \\ Universit\`a di
Lecce \\ C.P. 193, 73100 Lecce, Italy}

\date{}

\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
$$
G(u,K)=\int_{\Omega\setminus K}\bigl[|\nabla u|^2 + (u-g)^2\bigr] \,
dx+\h{n-1}{K},
$$
\no 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 and $|\nabla u|$ belongs to the
Morrey space $L^{2,\lambda}(\Omega)$
for some $\lambda>n-1$, then $K$ is a
$C^{1,\alpha}$ hypersurface outside a closed ${\cal H}^{n-1}$-negligible
singular set.
\end{abstract}

\vfill\eject

\hbox{}\vfill

\tableofcontents

\vfill\eject

\section{Introduction}
In the last few years new variational problems have been studied, where the
functionals involved depend on a variable closed set $K$ and a function
which is smooth outside $K$ (see {\em e.g.} \cite{am2}, \cite{am3},
\cite{bz2}, \cite{clt1}, \cite{dgparis}, \cite{dga}, \cite{mumsh}): such
problems have been called {\em free discontinuity problems} (notice that
$K$ is not necessarily a boundary).
The canonical example of free discontinuity problem is the
minimization of the so-called Mumford-Shah functional
\begin{equation}       \label{msfunc}
G(u,K)=\int_{\Omega\setminus K}\bigl[|\nabla u|^2 + (u-g)^2\bigr] \,
dx+\h{n-1}{K},
\end{equation}
\no where $\Omega\subset\rn{n}$ is an open set, $g\in L^\infty(\Omega)$ and
${\cal H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure in $\rn{n}$.
In the two dimensional case, this functional
has been proposed (see \cite{mumsh}, \cite{morsol})
as a variational model of image segmentation: $g$ is the
grey level function giving the image to be segmented, $K$ is the
competing segmentation, and $u\in C^1(\Omega\setminus K)$ is the smoothed
approximation of $g$. The proof of the existence of minimizing pairs
$(K,u)$ is due to De Giorgi, Carriero and Leaci (see \cite{dcl}):
they relax the problem in the space $SBV$ of special functions with bounded
variation introduced in \cite{dga}, where a weak solution is known to exist
(see \cite{am2}), and then prove that from any weak minimizer in $SBV$ a
minimizing pair for the original functional can be constructed. A
different proof in the planar case is also given in \cite{dms}.

The singular sets $K$ of minimizers are known only to be
rectifiable, but are expected to be piecewise regular (see \cite{dgparis}).
The main aim of this and
of the subsequent paper \cite{afp} is to prove some results in this
direction (see \cite{davsemm} and \cite{bon}
for some results in the case $n=2$). More generally,
we study a class of perturbations of the functional
\begin{equation}       \label{enerfunc}
F(u,K)=\int_{\Omega\setminus K}|\nabla u|^2 \,dx+\h{n-1}{K}
\end{equation}
\no which is suitable to encompass $G$, and we prove that if $(K,u)$ is a
quasi minimizing pair in $\Omega$ and $|\nabla u|$ belongs to the Morrey
space $L^{2,\lambda}(\Omega)$ for some $\lambda>n-1$, then
$K$ is a $C^{1,\alpha}$ hypersurface outside a relatively closed,
${\cal H}^{n-1}$-negligible singular set (see Theorem \ref{main} and the
subsequent remarks).

The main difficulty in facing the regularity problem for (quasi) minimizers
of $F$ relies in the necessity of dealing at the same time with an elliptic
term and a surface integral; this can perhaps be more easily seen by
looking at the first variation formula (see \cite{clpp}):
\begin{equation}\label{deform}
\int_\Omega \bigl[|\nabla u|^2\diver\eta-2\langle \nabla u,\nabla u\nabla
\eta \rangle\bigr]\,dx+ \int_{K}\divt\eta\,d{\cal H}^{n-1}=0
\end{equation}
\no (where $\divt$ is the tangential divergence along $K$) which holds
for any vector field $\eta\in C^1_0(\Omega,\rn{n})$ and any local
minimizer $u$ of (\ref{enerfunc}). In fact, the surface integral in
(\ref{deform}) is the first variation of area, hence (\ref{deform})
shows that the mean curvature of $K$ is
related to the Dirichlet integral of $u$.
Thus, even though we have taken as a starting point the regularity theory
of minimal surfaces, as it has been developed in the context of varifolds
by Allard in \cite{allard}, it has been necessary to
adapt these ideas to the peculiar
form of the term controlling the mean curvature of $K$. In particular,
we borrow many ideas from Brakke's book \cite{brakke}, where the
properties of varifolds with mean curvature in $L^1$ are analyzed.
However, to make the paper more self contained, we will not directly use
results from the theory of varifolds.

In this paper we do not treat the regularity problem in full generality,
because of the a priori assumption $|\nabla u|\in L^{2,\lambda}(\Omega)$ (see
Remark \ref{radice}): this hypothesis will be removed in \cite{afp} by means
of a suitable decay lemma for the Dirichlet integral which will be the natural
counterpart of the flatness improvement theorem \ref{decay} of this paper.
Looking for decay properties of the Dirichlet integral is a natural approach,
because $u$ solves a Neumann problem in the domain $\Omega\setminus K$, which
has a highly irregular ({\em a priori} only rectifiable) boundary, hence no
estimates on the Dirichlet integral of $u$ near $K$ are directly available. A
similar approach to the the regularity of free interface problems has been
developed by Lin in \cite{lin}.

Before passing to a brief description of the paper, we point out that the
techniques developed here are suitable to be further exploited and
applied in a large class of free discontinuity problems (see also
Remark \ref{estensioni}), leading likely to a general scheme for the proof
of similar regularity results.

The content of the paper is the following.

In \S 2 we collect a few notions and notations which are needed in the
sequel, and give the definition of quasi minimizer; the main density
properties of quasi minimizers are studied in \S 4 and pave the way to the
main geometric results of \S 5, namely the Lipschitz approximation Theorem
\ref{lipappr} and Theorem \ref{c1graf}, which gives a sufficient condition
in order that a portion of $K$ be a $C^{1,\alpha}$ graph.

The main point in the
proof of Theorem \ref{main}, presented in \S 7, is checking the
hypotheses of Theorem \ref{c1graf}, and this will pass as usual
through an iterative procedure applied to the flatness improvement
theorem of \S 6.

\vfill\eject

\section{Notations and statement of the problem}

\subsection{Rectifiable sets, approximate tangent space}

The dimension $n\geq 2$ of the ambient space ${\bf R}^n$ will be fixed
throughout the paper; since we will often deal with $(n-1)$-dimensional
sets, we will use $m$ for $n-1$. We indicate by $\omega_p$ the measure of
the unit ball in ${\bf R}^p$.

In this paper $\hh$ denotes the $m$-dimensional Hausdorff measure in ${\bf
R}^n$, $\hh\res E$ the measure $\hh$ restricted to the set $E$,
$\theta_m(E,x)$ the $m$-dimensional density of $E$ at $x$ and
${\bf G}_{n,m}$ the Grassmann manifold of $m$-planes in ${\bf R}^n$.  Each
$T\in{\bf G}_{n,m}$ will be identified with the matrix $(T_{ij})$
representing the orthogonal projection onto $T$ with respect to the
canonical basis $\{e_1,\ldots,e_n\}$; the distance $\Vert S-T\Vert$ between
two $m$-planes is the euclidean norm of the matrix $S-T$.

\begin{definition}[rectifiable sets]
{\rm (see \cite[3.2.14]{fed})} We say that $E\subset{\bf
R}^n$ is countably $(\hh,m)$-rectifiable if $\hh$-almost all of $E$ can be
covered with a sequence of $C^1$ hypersurfaces $\Gamma_i$, {\em i.e.}
$$
\hh\Bigl(E\setminus\bigcup_{i=1}^\infty\Gamma_i\Bigr)=0.
$$
\no We say that $E$ is $(\hh,m)$-rectifiable if $E$ is countably
$(\hh,m)$-rectifiable and $\h{m}{E}<+\infty$.
\end{definition}
The approximate tangent space ${\rm Tan}^m(E,x)$ of a
$(\hh,m)$-rectifiable
set $E$ at $x$ is the $m$-plane $S$ such that, setting
$E_\rho=\rho^{-1}(E-x)$, we have
$$
\lim_{\rho\to 0}\int_{E_\rho}\phi(y)\, d\hh (y) = \int_S\phi(y)\, d\hh
(y)\quad\quad\forall\,\phi\in C_0^1({\bf R}^n).
$$
\no The map $x\mapsto{\rm Tan}^m(E,x)$ is defined $\hh$-a.e. on $E$ and is
$\hh$-measurable (see \cite[3.2.25]{fed}). When integrations are involved
and no confusion is possible we shorten ${\rm Tan}^m(E,x)$ to $S_x$;
for instance
$$
\int_E\Vert S_x-T\Vert^2\, d\hh
$$
\no denotes the mean square deviation of the approximate tangent space from a
given $m$-plane $T$ (this quantity, which is usual to call {\em tilt},
will be crucial in the Lipschitz approximation Theorem \ref{lipappr}).

Finally, we recall that any Lipschitz continuous function $f:{\bf R}^n \to
{\bf R}^n$ maps $(\hh,m)$-rectifiable sets into $(\hh,m)$-rectifiable sets.
If
$f$ is one-to-one then $\h{m}{f(E)}$ can be computed using the area formula
(see {\em e.g.} \cite[\S 8]{simon}):
\begin{equation}      \label{area}
\h{m}{f(E)}=\int_E{\bf J}_m(f,S_x)\, d\hh,
\end{equation}
\no where by definition ${\bf J}_m^2(f,S)={\rm det}\bigl(df|_S\bigr)^* \circ
\bigl(df|_S\bigr)$ for any $m$-plane $S$. We will often use (\ref{area})
with $f(x)=\Phi_\eps(x)=x+\eps\phi(x)$, $\phi\in C_0^1\bigl({\bf R}^n, {\bf
R}^n \bigr)$ and $\eps$ small; in this case
\begin{equation}\label{derivaj}
\frac{d}{d\eps}{\bf J}_m(\Phi_\eps,S)\Big|_{\eps=0} =\diver_S\phi=
\sum_{i,j=1}^n S_{ij}\frac{\partial\phi_i}{\partial x_j}
\end{equation}
\no for any $m$-plane $S$.

\subsection{$SBV$ functions and free discontinuity problems}
Let $\Omega\subset{\bf R}^n$ be a bounded open set and let $u\in BV(\Omega)$,
{\em i.e.} a function with bounded variation in $\Omega$. We denote by
$S_u$ the jump set of $u$, defined as the complement of the Lebesgue set of
$u$:
$$
x\not\in S_u\quad\Longleftrightarrow\quad \exists z\in\r
\,\,\hbox{\rm s.t.}\,\, \lim_{\rho\to 0}
\rho^{-n} \int_{B_\rho(x)}|u(y)-z|\, dy=0.
$$
\no It is well known (see for instance \cite[4.5.9(16)]{fed}) that $S_u$ is
countably $(\hh,m)$-rectifiable. The vector measure $Du$ representing the
distributional derivative of $u$ can be decomposed into absolutely
continuous part $\nabla u$ and singular part $D^s u$ with respect to the
Lebesgue measure ${\cal L}^n$. It is clear that $u$ belongs to the Sobolev
space $W^{1,1}(\Omega)$ if and only if $Du=\nabla u{\cal L}^n$, or $D^su=0$.

\begin{definition}[$SBV$ functions]
We say that $u$ is a special function with bounded variation in
$\Omega$, and we write $u\in SBV(\Omega)$, if $D^su$ is supported in $S_u$,
{\em i.e.}, $|D^su|(\Omega\setminus S_u)=0$.
\end{definition}

The space $SBV$ has been introduced in \cite{dga} to give a rigorous
mathemat\-ical formulation to several variational problems involving
both a ``volume'' energy and a ``surface'' energy. The main feature of such
problems, also called free discontinuity problems, see \cite{dgparis}, is
that the surface energy is supported on a set which is not fixed {\em a
priori} and is not necessarily a boundary.

The model problem has been suggested by Mumford and Shah in
\cite{mumsh}. Given $g\in L^\infty(\Omega)$ and $\alpha,\,\beta>0$,
we look for minimizers of the functional
\begin{equation}       \label{msfunct}
G(u,K)=\int_{\Omega\setminus K}\bigl[|\nabla u|^2 +\alpha (u-g)^2\bigr] \,
dx+\beta\h{m}{K}
\end{equation}
\no where $K$ is a closed subset of $\rn{n}$ and $u\in C^1(\Omega\setminus K)$.
We refer to \cite{bz2}, \cite{mumsh} for the relevance of this problem
in computer vision theory and some properties of minimizers; here, we want
only to emphasize that, by minimizing $G$, one looks for a ``piecewise
$C^1$ approximation'' of $g$.

Existence of minimizers is achieved passing through a weak formulation of
the problem in $SBV(\Omega)$, namely, minimizing the functional
$$
F(u)=\int_{\Omega}\bigl[|\nabla u|^2 +\alpha (u-g)^2\bigr] \,
dx+\beta\h{m}{S_u}.
$$
\no The equivalence between the strong and weak formulation of the problem
has
been proved in \cite{dcl}, whereas the
existence of minimizers of $F$ easily follows from a compactness property
of $SBV$ first proved in \cite{am1} (see also \cite{am4}).

\begin{theorem}[compactness theorem]
Let $(u_h)\subset SBV(\Omega)\cap L^\infty(\Omega)$, $p>1$ and assume that
$$
\sup_{h\in{\bf N}}\Bigl\{\int_\Omega |\nabla u_h|^p\, dx + \h{m}{S_{u_h}} +
\Vert u_h\Vert_\infty\Bigr\}<+\infty.
$$
\no Then, there exists a subsequence $(u_{h_k})$ converging in
$L^1(\Omega)$ to $u\in SBV(\Omega)$. Moreover, $\nabla u_{h_k}$
weakly converges to $\nabla u$ in $L^p(\Omega,\rn{n})$ and $\hh\res
S_{u_{h_k}}$ weakly converges in $\Omega$ to a measure $\mu$ such that
$\mu\geq \hh\res S_u$.
\end{theorem}

\subsection{Quasi minimizers}

We are interested in the interface regularity of solutions $u\in
SBV_{\rm loc}(\Omega)$ of variational problems whose leading term is
$$
\E{u}{\Omega}+\beta\h{m}{S_u\cap\Omega}.
$$
\no Our model is the Mumford-Shah functional described above. By a scaling
argument, it is not restrictive to assume $\beta=1$.

\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}       \label{ellinfty}
{\rm Let $u\in SBV(\Omega)$ be a solution of the Mumford-Shah problem of
minimizing (\ref{msfunct}). Then, if $g\in L^\infty(\Omega)$, the function
$u$ is a quasi minimizer. Indeed, since
$$F(v)\geq F(-M\lor v\land M)\qquad\forall v\in SBV_{\rm loc}(\Omega)$$
\no with $M=\Vert g\Vert_\infty$, we have $\Vert u\Vert_\infty\leq M$ and,
for any competing function $v$, the inequality
$F(u)\leq F(-M\lor v\land M)$ implies
$$
\dev{u}{B_\rho(x)}\leq\alpha\int_{B_\rho(x)}|-M\lor v\land M-g|^2\, dy\leq
4\alpha\Vert g\Vert_\infty^2\omega_n\rho^n
$$
\no for any ball $B_\rho(x)\subset\Omega$, hence we can take}
$$
\omega(t)=4\alpha\Vert g\Vert_\infty^2\omega_n t.
$$
\end{remark}
The following fundamental density estimate on the jump set of quasi
minimizers was proved in \cite{dcl}, \cite{cl1}
(see also \cite{dms} for
two-dimensional minimizers of (\ref{msfunct})).

\begin{theorem}[density lower bound]     \label{dgcl}
There exist constants $\rho_\omega$ and $\theta_\omega>0$ depending only on
$n$ and $\omega$ with the following property:  if $u\in\mm{\Omega}$,
we have
$$
\hh\bigl(S_u\cap B_\rho(x)\bigr)\geq\theta_\omega\rho^m
$$
\no for any ball $B_\rho(x)\subset\Omega$ centered at $x\in\overline{S}_u$
with $\rho\leq\rho_\omega$.
\end{theorem}
{\sc Proof.} By approximation, we need only to prove that
the inequality is valid for balls centered at points $x\in S_u$.
The proof can be achieved by a decay lemma (see \cite{cl1},
Proposition 1.2 and Remark 3.13). Setting
$$
F(u,B_\rho(x))=\int_{B_\rho(x)}|\nabla u|^2\,dy+
{\cal H}^m\bigl(B_\rho(x)\cap S_u\bigr)
$$
\no it has been proved that for $\alpha\in (0,1)$ and
$\eps(n,\alpha,\omega)$ small enough, the conditions
$$
{\cal H}^m\bigl(B_\rho(x)\cap S_u\bigr)<\eps\rho^m,\qquad
\rho<\eps
$$
\no imply $F(u,B_{\alpha\rho}(x))\leq (\alpha)^{m+1/2}F(u,B_\rho(x))$. Now,
let $\alpha\in (0,1)$ and choose $\alpha'\in (0,1)$ so small that
$$
2\sqrt{\alpha'}n\omega_n<\eps(n,\alpha,\omega).
$$
\no We define $\theta_\omega=\eps(n,\alpha',\omega)$ and $\rho_\omega$
satisfying the conditions
$$
\rho_\omega<\min\bigl\{\eps(n,\alpha,\omega),
\eps(n,\alpha',\omega)\bigr\},\qquad
\omega(\rho_\omega)\leq n\omega_n.
$$
\no If ${\cal H}^m(S_u\cap B_\rho(x))<\theta_\omega\rho^m$, using the energy
upper bound (see Remark \ref{mst})
and the decay property with $\alpha'$, we have
$$
F(u,B_{\rho\alpha'}(x))\leq (\alpha')^{m+1/2}
F(u,B_\rho(x))\leq 2\sqrt{\alpha'}n\omega_n(\rho\alpha')^m
<\eps(n,\alpha,\omega)(\rho\alpha')^m.
$$
\no Applying now $k$ times the decay property with $\alpha$ we get
$$
F(u,B_{\rho\alpha^k\alpha'}(x))\leq \alpha^{k(m+1/2)}
F(u,B_{\rho\alpha'}(x)).
$$
\no It follows that $\eta^{-m}F(u,B_\eta(x))$ converges to $0$,
as $\eta\downarrow 0$, hence (see \cite{dcl}) $x\notin S_u$.
\qed

A first consequence of Theorem \ref{dgcl} is the following

\begin{proposition}     \label{closure}
Let $u\in SBV_{\rm loc}(\Omega)$ be a quasi minimizer in $\Omega$. Then
$$
\h{m}{\overline{S}_u\cap\Omega\setminus S_u}=0.
$$
\end{proposition}
{\sc Proof.} Let $\mu$ be the restriction of $\hh$ to $S_u$ and
$B=\overline{S}_u\cap\Omega\setminus S_u$. Since
$$
\liminf_{\rho\to 0}\frac{\mu(B_\rho(x))}{\omega_m\rho^m}\geq
\frac{\theta}{\omega_m}\qquad\forall\,x\in\overline{S}_u
$$
\no by \cite[Th.3.2]{simon} we infer that $\mu\geq(\theta/\omega_m) \hh\res
\overline{S}_u$, hence
$$
0=\mu(B)\geq\frac{\theta}{\omega_m}\h{m}{B\cap\overline{S}_u}=
\frac{\theta}{\omega_m}\h{m}{B}.
$$
\qed

\begin{remark} \label{pairs} {\rm
Let $u\in\mm{\Omega}$ and let us denote by $K(u)$ the set $\overline{S}_u$.
Thanks to Proposition \ref{closure},
the quasi minimality of $u$ implies:
\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 ball $B_\rho(x)\subset\Omega$ and any function $v\in
SBV(\Omega)$
such that $\{v\neq u\}\subset\subset B_\rho(x)$. In the following, we will
always work with pairs $(u,K(u))$, using the (weaker) minimality condition
(\ref{minK}) and avoiding the $SBV$
theory (except in Step 5 of Theorem \ref{planes}).
}\end{remark}

From now on, we will use the notation
$\mu(u)$ for $\hh\res K(u)$ and we denote by $\divt$
the tangential divergence along the approximate tangent space to $K(u)$.

\vfill\eject

\section{Statement of the main result}

The main result of this paper is the following (here $L^{p,\lambda}$ are
the classical Morrey spaces, see {\em e.g.} \cite{Giaq}):

\begin{theorem}    \label{main}
Let $u\in SBV(\Omega)$, assume that $|\nabla u|\in L^{2,\lambda}
(\Omega)$ for some $\lambda>m$ and that
constants $c_0>0$, $s\in (0,2)$ exist such that
\begin{equation}       \label{main1}
\dev{u}{B_\rho(x)}\leq c_0\rho^{2s}
\end{equation}
\no for any ball $B_\rho(x)\subset\Omega$. Then, setting $\alpha=
\min\{\lambda-m,s\}$, there is a constant
$\eps_0(n,c_0,\alpha,\Vert\nabla u\Vert_{L^{2,\lambda}})$
such that for any $x\in K(u)$ and
any ball $B_\rho(x)\subset\Omega$, the conditions
\begin{equation}        \label{main2}
\rho<\eps_0,\quad \min_{T\in{\bf G}_{n,m}}\int_{B_\rho(x)\cap S_u}
|T^\perp(y-x)|^2\, d\mu(u)<\eps_0\rho^{m+2}
\end{equation}
\no imply that $B_{\rho/2}(x)\cap K(u)$ is a $C^{1,\alpha/(m+2)}$
hypersurface.
\end{theorem}

\begin{remark}
{\rm Let $R(u)$ 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 $\rho>0$. It is easy to see that
$R(u)$ is relatively open in $K(u)$. Moreover, using in the definition of
approximate tangent space a function $\phi=|T^\perp y|\zeta(y)$ with a
cut-off function $\zeta$, it is easy to check that $R(u)$ contains the set
of the points where the approximate tangent space exists. In particular,
$$
\h{m}{K(u)\setminus R(u)}=0.
$$
\no Theorem \ref{main} applies in particular to minimizers $u$ of the
Mumford-Shah functional (\ref{msfunct}) under the assumption $|\nabla u|
\in L^{2,\lambda}(\Omega)$ for some $\lambda>m$. Indeed, arguing as in
Remark \ref{ellinfty}, we see that (\ref{main1}) holds with
$s=1/2$.}
\end{remark}

\begin{remark} \label{radice} {\rm
We remark that an easy comparison argument (see Remark \ref{mst}
below) and (\ref{main1}) imply that $|\nabla u|\in L^{2,m}(\Omega)$. Our
regularity theorem requires the stronger assumption
$|\nabla u|\in L^{2,\lambda}(\Omega)$ with $\lambda>m$;
by H\"older's inequality,
this assumption is satisfied if $|\nabla u|\in L^p(\Omega)$ for some
$p>2n$.

At the moment, no nontrivial example of local minimizer of (\ref{enerfunc})
is known. However, there is some evidence (see \cite{clpp}, \cite{mumsh})
that the function
(in polar coordinates) $u(\rho,\theta)=\sqrt{2\rho/\pi}\sin(\theta/2)$ is a
local minimizer in $\rn{2}$ and that it presents the typical behaviour of a
local minimizer near the tip of the singular set. Elementary computations
show that $|\nabla u| \in L^{2,1}(\rn{2})$, hence the hypothesis $|\nabla u|
\in L^{2,\lambda}$, with $\lambda=m$, rather than $\lambda>m$, is
suitable to cover the general case. This question will be tackled in a
successive paper \cite{afp}.}
\end{remark}

\begin{remark} \label{estensioni} {\rm The techniques of this paper
apply, more generally, to quasi minimizers of functionals
\begin{displaymath}
F(u)=\int_\Omega H(\nabla u)\,dx+\h{m}{S_u}
\qquad u\in SBV_{\rm loc}\bigl(\Omega,\rn{k}\bigr)
\end{displaymath}
\no assuming that the following three condition hold:

1) $H$ is continuous, $p$-homogeneous for some $p>1$ and $H(z)>0$
for $z\neq 0$;

2) $|\nabla u|\in L^{p,\lambda}(\Omega)$ for some $\lambda>m$;

3) the density lower bound of Theorem \ref{dgcl} holds.

Up to now, using the blow-up method of \cite{dcl}, \cite{cl1} the density
lower bound has been proved only in the cases $H(z)=|z|^p$ for $k\geq 1$,
even under the constraint $|u|=1$ (in the vectorial case, see
\cite{cl2}). Its general validity is still an open problem.
}\end{remark}

\section{Some properties of quasi minimizers}

We start by recalling an elementary upper bound and the natural
scaling properties of quasi minimizers.

\begin{remark}[energy upper bound] \label{mst}
{\rm For every $u\in SBV(\Omega)$ and every ball
$B_\rho(x_0)\subset\Omega$, comparing with functions constant inside
$B_{\rho'}(x_0)\subset\subset B_\rho(x_0)$ and letting
$\rho'\uparrow\rho$ we get
$$
\E{u}{B_\rho(x_0)}+\hh\bigl(K(u)\cap B_\rho(x_0)\bigr)\leq n\omega_n\rho^m
+\dev{u}{B_\rho (x_0)}.
$$
}\end{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
$$u_\rho(y)={1\over\sqrt{\rho}}u(x_0+\rho y)$$
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*}
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}

The following theorem shows the asymptotic behaviour of $K(u)$ in
regions where both the Dirichlet energy and the tilt of tangent planes
are small. Using essentially a first variation argument we show that
$\mu(u)$ is close to a locally finite sum of measures $\mu_i$ supported in
parallel planes $T_i$. Then, lower semicontinuity of the
energy and the minimality imply that $\mu_i=\hh\res T_i$.

\begin{theorem} \label{planes}
Let $(u_h)\subset\mm{\Omega}$ be a sequence and let us assume that
$$
\lim_{h\to+\infty}\E{u_h}{A}+\T{u_h}{A}+\dev{u_h}{A}=0
$$
for any open
set $A\subset\subset\Omega$.  Then, a subsequence of $\mu(u_h)$ weakly
converges in $\Omega$ to a measure $\mu$ which is locally sum of affine
$m$-planes parallel to $T$ with multiplicity $1$.
\end{theorem}
{\sc Proof.} {\sc Step 1.}
Let ${\bf G}_m(\Omega)$ be the product of $\Omega$ and
${\bf G}_{n,m}$; the so-called varifold measures $V_h$ associated
to $\mu(u_h)$ are defined by
$$\int_{{\bf G}_m(\Omega)}\phi(x,S)\,dV_h(x,S)=\int_\Omega
\phi(x,S_x)\,d\mu(u_h)$$
for any bounded Borel function $\phi(x,S)$ with compact support in $x$.
Since $V_h$ are locally equibounded in ${\bf G}_m(\Omega)$,
we can assume that $V_h$ are
weakly converging in ${\bf G}_m(\Omega)$ to a measure $V$.

{\sc Step 2.}
Now we prove that $V$ is stationary in the varifold sense, {\em i.e.},
\begin{equation}     \label{stationarity}
\int_{{\bf G}_m(\Omega)}S_{ij}{\partial \varphi_i\over\partial x_j}
\,dV(x,S)=0\qquad\forall\,\varphi\in C^1_0(\Omega,\rn{n}).
\end{equation}
Indeed, we fix an open set $A\subset\subset\Omega$ containing the support
of $\varphi$ such that $V(\partial A\times {\bf G}_{n,m})=0$.  For $\eps$
sufficiently small, the map $\Phi_\eps(x)=x+\eps\varphi(x)$ is a $C^1$
diffeomorphism of $\Omega$; comparing $u_h$ with $u_h\circ\Phi^{-1}_\eps$,
taking into account the strong convergence of the gradients and the
identities (see (\ref{area})
\begin{eqnarray*}
\h{m}{A\cap K(u_h\circ\Phi_\eps^{-1})}&=& \h{m}{A\cap\Phi_\eps(K(u_h))} \\
&=&\int_A{\bf J}_m(\Phi_\eps,S_x)\, d\mu(u_h) (x)
\end{eqnarray*}
we find
$$
\eta_h + \dev{u_h}{A} + \int_{A\times{\bf G}_{n,m}}
{\bf J}_m(\Phi_\eps,S)\, dV_h (x,S) \geq V_h\bigl(A\times{\bf
G}_{n,m}\bigr),
$$
for a suitable infinitesimal $\eta_h$. Passing to the limit with respect to
$h$ we obtain
$$
\int_{A\times{\bf G}_{n,m}}
{\bf J}_m(\Phi_\eps,S)\, dV (x,S) \geq V\bigl(A\times{\bf G}_{n,m}\bigr).
$$
Differentiation with respect to $\eps$ and (\ref{derivaj})
yield (\ref{stationarity}). Moreover, denoting
by $\mu(C)=V(C\times{\bf G}_{n,m})$ the projection of $V$ on $\Omega$,
the measures $\mu(u_h)$ weakly converge to $\mu$
and the density lower bound implies $\mu(B_\rho(x))\geq
\theta_\omega\rho^m$ for any $x\in{\rm spt}\mu$ and any ball
$B_\rho(x)\subset
\Omega$ with $\rho<\rho_\omega$.
Hence, by \cite[Th.3.2]{simon}, the support of $\mu$ is a
set with locally ${\cal H}^m$-finite measure in $\Omega$.
The density lower bound also
implies that the supports of $\mu(u_h)$ converge to the support of
$\mu$ in the sense of Kuratowski. Finally, by lower semicontinuity
we have
$$\int_{{\bf G}_m(\Omega)}\Vert S-T\Vert^2\,dV(x,S)=0$$
that means that $V$ is supported in $\Omega\times\{T\}$.

{\sc Step 3.} The next step is to
prove that $V$ is invariant by translations along
$T$. For, let $b\in T$, and $\tau_t(x)=x+tb$, $V_t=
(\tau_t)_\#V$; for any $\phi\in C_0^1(\Omega)$ we have
\begin{eqnarray*}
\frac{d}{dt}\int_{{\bf G}_m(\Omega)} \phi(x)\, dV_t(x) &=&
\frac{d}{dt}\int_{{\bf G}_m(\Omega)} \phi(x-tb)\,
dV (x,S)     \\
&=& -\int_{{\bf G}_m(\Omega)}\langle\nabla\phi(x-tb),b\rangle\, dV(x,S)
\\
&=& -\int_{{\bf G}_m(\Omega)}
{\partial \phi\over\partial x_i}(x-tb)T_{ij}b_j\, d V(x,S)
\\
&=& -b_j\int_{{\bf G}_m(\Omega)} S_{ij}{\partial\phi\over\partial x_i}
dV(x,S)=0
\end{eqnarray*}
hence $\mu$ is invariant by
translations along $T$ and then ${\rm spt}\,\mu$ is contained in a locally
finite union of affine $m$-planes parallel to $T$.

{\sc Step 4.}
It remains to prove that the density of $\mu$ is equal to 1.
To this end, assume that $T=e_n^{\perp}$ and
consider a $m$-plane $P$ parallel to $T$ such that $P\cap{\rm
spt}\,\mu\not=\emptyset$. For every $x\in P\cap{\rm spt}\,\mu$,
we choose $\rho_0\in (0,\rho_\omega)$
such that for every $\rho\in(0,\rho_0)$ the inclusion
$Q_\rho\cap{\rm spt}\,\mu \subset P$ holds, where
$Q_\rho$ is a cube symmetric with respect to $P$ with centre $x$ and side
$\rho$. Fix $\rho\in(0,\rho_0/2)$ and let $w_h(y)=\rho^{-1/2}u_h(x+\rho y)$;
then,
for $Q=(-1,1)^n$, $H=Q\cap\{x_n=0\}$, we have a sequence
$(w_h)$ such that
\begin{equation}        \label{hplim}
\int_Q |\nabla w_h|^2 dx + \dev{w_h}{2Q}\to 0
\end{equation}
and $K(w_h)\to H$ in the sense of Kuratowski.
For $a\in (0,2^{-n}\theta_\omega/m)$, set
$Q_a^+=Q\cap\{x_n>a\}$, $Q_a^-=Q\cap\{x_n<-a\}$ and $Q_a=Q_a^+\cup Q_a^-$,
if $h$ is large enough we have $w_h\in W^{1,2}(Q_a)$; furthermore,
translating $w_h$ if necessary, we can suppose that
\begin{equation}     \label{medianulla}
\frac{1}{{\cal H}^n(Q_a^+)}\int_{Q_a^+}w_h(x)\, dx=0 \qquad\forall h\in
{\bf N},
\end{equation}
which implies $w_h\to 0$ strongly in $W^{1,2}(Q_a^+)$.  Similarly,
setting
$$
c_h=\frac{1}{{\cal H}^n(Q_a^-)}\int_{Q_a^-}w_h(x)\, dx \qquad\forall h\in
{\bf N},
$$
we have $u_h-c_h\to 0$ strongly in $W^{1,2}(Q_a^-)$.
Up to subsequences and a change of sign we can suppose $c_h\to c\in
[0,+\infty]$. We first show that $c=0$ is impossible. In fact, for $a$ as
above and $b\in (a,2^{-n}\theta_\omega/m)$
we can find $a_1,a_2\in (a,b)$ such that (here $Q=Q'\times (-1,1)$)
\begin{equation}     \label{nullenerg}
\left\{
\begin{array}{ll}
\displaystyle\int_{Q'} \bigl[|\nabla w_h|^2(x',a_1)+|\nabla
w_h|^2(x',-a_2)\bigr]\, dx'\to 0 \\ & \\
\displaystyle\int_{Q'} \bigl[|w_h|^2(x',a_1)+|w_h|^2(x',-a_2)\bigr]\, dx'\to 0
\end{array}
\right.
\end{equation}
where $x=(x',x_n)$. Let us define the functions
$$
v_h(x)=\left\{
\begin{array}{ll}
w_h(x) &{\rm if\ }x\in Q_a,
\\  & \\
\displaystyle
w_h(x',a_1)-\frac{x_n-a_1}{a_1+a_2}\bigl[w_h(x',-a_2)-w_h(x',a_1)
\bigr] \quad &{\rm otherwise.}
\end{array}
\right.
$$
Taking into account that (\ref{nullenerg}) implies
$$
\lim_{h\to +\infty}\int_{-a_2}^{a_1}dx_n\int_{Q'}|\nabla v_h|^2\, dx'
=0,
$$
and that the area of the subset of $\partial Q$ where $v_h$ can be
different from $w_h$ does not exceed $m2^m(a_1+a_2)$, by the
quasi minimality of $w_h$ and (\ref{hplim}) we get
\begin{eqnarray*}
0&\leq& \limsup_{h\to +\infty}\int_{Q}|\nabla {v_h}|^2 +\h{m}{K(v_h)\cap Q}
\\
&&- \int_{Q}|\nabla {w_h}|^2
-\h{m}{K(w_h)\cap Q}+\dev{w_h}{2Q}  \\
&\leq& m2^m(a_1+a_2)-\liminf_{h\to +\infty}\hh (K(w_h)\cap Q)
\leq m2^nb -\theta_\omega,
\end{eqnarray*}
a contradiction. Hence, $c>0$.

{\sc Step 5.} Consider the sequence
$h\mapsto \bar{w}_h=(-1)\vee (w_h\wedge 1)$, which is compact in $SBV(Q)$,
hence (up to subsequences) converges to a function, say $v$, in $L^1(Q)$
and a.e. in $Q$. From (\ref{medianulla}) it follows that
$v\equiv 0$ in $Q\cap\{x_n>0\}$, whereas
$$
\int_{Q_a^-}|w_h-c_h|\, dx \geq \int_{Q_a^-}\bigl|\bar{w}_h-(-1)\vee
(c_h\wedge 1)\bigr|\, dx
$$
implies
$$
\int_{Q_a^-}\bigl|v-(-1)\vee (c\wedge 1)\bigr|\, dx =0;
$$
since $c>0$, it holds $(-1)\vee (c\wedge 1)
\not= 0$, whence $v\not\equiv 0$ in $Q$. On the other hand,
since
$$
\int_{Q_a^-}|\nabla w_h|^2\, dx\to 0\qquad{\rm as\ }h\to +\infty,
$$
by semicontinuity $v$ is constant in $Q\cap\{x_n<0\}$. Therefore
$$
\liminf_{h\to +\infty}\hh\bigl(K(w_h)\cap Q\bigr)\geq \liminf_{h\to +\infty}
\hh\bigl(K(\bar{w}_h)\cap Q\bigr) \geq \hh\bigl(K(v)\cap Q\bigr)=2^m.
$$
Coming back to the sequence $u_h$, the previous inequality  shows that
$$
\mu(\overline{Q}_\rho)\geq
\liminf_{h\to +\infty}\hh\bigl(Q_\rho\cap K(u_h)\bigr) \geq 2^m \rho^m,
$$
for every $\rho\in(0,\rho_0/2)$, {\em i.e.} that the density of $\mu$ at $x$
is $\geq 1$ for every $x\in{\rm spt}\,\mu$.  Let us prove the opposite
inequality.  For, suppose for simplicity $x=0$, consider two parameters
$a,b\in (0,\rho_0)$, and the neighbourhood of 0 given by
$Q=(-b/2,b/2)^m\times (-a,a)$.  Choose then $a_1, a_2\in (0,a)$ such
that
$$
\int_{Q\cap\{x_n=a_1\}}|\nabla_{x'}u_h|^2
+\int_{Q\cap\{x_n=-a_2\}}|\nabla_{x'}u_h|^2 \to 0,
$$
(where as above $x=(x',x_n)$) and define the comparison functions
$$
v_h(x)=\left\{
\begin{array}{lll}
u_h(x) &{\rm if\ }x_n\geq a_1\ {\rm or\ }x_n\leq -a_2,
\\
u_h(x',a_1) &{\rm if\ }0<x_n<a_1
\\
u_h(x',-a_2) \quad&{\rm if\ }-a_2<x_n<0.
\end{array}
\right.
$$
By comparing the energies of $v_h$ and $u_h$ we get
\begin{eqnarray*}
&&\mu\bigl((-b/2,b/2)^n\bigr)=\mu(Q)\leq
\liminf_{h\to +\infty}\hh\bigl(K(u_h)\cap Q\bigr)   \\
&&\leq b^m+mb^{m-1}(a_1+a_2) \leq b^m+2amb^{m-1},
\end{eqnarray*}
which gives the desired estimate letting $a\downarrow 0$.
\qed

Using Theorem \ref{planes} we can show in Corollary
\ref{densi1} below that the $m$-dimensional
density of $K(u)$ in $B_R(x)$ is less than $\tau>1$ if $K(u)$ is
sufficiently flat in some direction $T$ and both the Dirichlet energy
and the deviation from minimality are small in a larger ball.

Similarly, we can show in Corollary \ref{densi2} below
that the $m$-dimensional
density of $K(u)$ in $B_R(x)$ is greater than $\tau<1$ if $x\in K(u)$
and the tilt of tangent planes, the Dirichlet energy
and the deviation from minimality are small in a larger ball. In this
Corollary, for later use in Theorem \ref{lipappr}, we allow a small
translation in the direction $T$.

\begin{corollary}          \label{densi1}
For any $\tau>1$ and any $\beta\in (0,1)$ there exists a constant
$\gamma(\beta,\tau,\omega)\in (0,1)$
such that, for any $m$-plane $T$, any $R\in (0,1]$ and
any $u\in\mm{B_R}$, the condition
$$
\E{u}{B_R}+R^{-2}\F{u}{B_R}+\dev{u}{B_R}<\gamma R^m
$$
implies
$$\hh\bigl(K(u)\cap B_{\beta R}\bigr)<\tau\omega_m(\beta R)^m.$$
\end{corollary}
{\sc Proof.} We argue by contradiction. Suppose that $\rho\in(0,1)$,
$\tau>1$ and sequences $R_h$, $(u_h)$ exist such that, for any
$h\in{\bf N}$, $0\in K(u_h)$,
\begin{equation}            \label{cond2h}
\E{u_h}{B_{R_h}}+R_h^{-2}
\F{u_h}{B_{R_h}}+\dev{u_h}{B_{R_h}}<\frac{R_h^m}{h}
\end{equation}
and
\begin{equation}            \label{tesi1neg}
\hh\bigl(K(u_h)\cap B_{\beta R_h}\bigr)\geq\tau\omega_m(\beta R_h)^m.
\end{equation}
Using Remark \ref{scaling}, it is not restrictive to assume that
$R_h=1$ for any $h$. By compactness, (up to subsequences) $\mu(u_h)$ weakly
converges in $B_1$
to a measure $\mu$ supported in $T\cap B_1$ by (\ref{cond2h}).  In order to
get a contradiction, we need only to show that $\mu\leq \hh\res T$.  The
inequality follows by the same argument used in the final part of the proof
of Theorem \ref{planes}.  Indeed, for any box $(x_0-b/2,x_0+b/2)^n$
contained in $B_1$ and any $a<b/2$ we have
$$
\mu \bigl((x_0-b/2,x_0+b/2)^n\bigr) = \mu\bigl( (x_0-b/2,x_0+b/2)^m\times
(-a,a)\bigr) \leq b^m+2ab^{m-1}.
$$
Since $a$ and $x_0$ are arbitrary, $\mu\leq \hh\res T$.
\qed

\begin{corollary} \label{densi2}
For any $\tau\in (0,1)$ there exists a constant
$\gamma_1(\tau,\omega)>0$ such that, for any $m$-plane $T$, $R\in (0,1]$,
the conditions
$$ u\in\mm{B_{2R}},\quad 0\in K(u),
\quad b\in T,\ |b|\leq R$$
$$\E{u}{B_{2R}}+\T{u}{B_{2R}}+\dev{u}{B_{2R}}<\gamma_1 (2R)^m$$
imply
$$\hh\bigl(K(u)\cap B_R(b)\bigr)\geq\tau\omega_mR^m.$$
\end{corollary}
{\sc Proof.} If the statement were false, $\lambda\in(0,1)$ and
sequences $b_h\in T_h$, $R_h$ and
$u_h\in\mm{B_{2R_h}}$ would exist such that for any $h\in{\bf N}$
both
$$
\E{u_h}{B_{2R_h}}+\int_{B_{2R_h}}|T_h^\perp x|^2\, d\mu(u_h)
+\dev{u_h}{B_{2R_h}}<\frac{(2R_h)^m}{h}
$$
\begin{equation}           \label{tesi2neg}
\hh\bigl(K(u_h)\cap B_{R_h}(b_h)\bigr)<\tau\omega_mR_h^m
\end{equation}
hold.  Using Remark \ref{scaling} and the invariance under rotations, we
can assume that $R_h=1$ and $T_h=T$ for any $h$.  Then, up to subsequences,
$b_h$ converges to some $b\in T$ and $\mu(u_h)$ weakly converges
in $B_2$ to a measure $\mu$ satisfying
$$
\mu\bigl(B_1(b)\bigr)\leq\tau\omega_m
$$
because of (\ref{tesi2neg}). On the other hand, since $0\in{\rm spt}\,\mu$,
Theorem \ref{planes} implies that $\mu\geq{\cal H}^m\res T$ in $B_2$,
hence $\mu\bigl(B_1(b)\bigr)\geq\omega_m$ holds.
\qed

The next proposition is essentially an adaptation to our problem of the
multilayer monotonicity lemma of \cite[5.3]{brakke}: we consider two points
$x,\,x' \in K(u)\cap B_R$ where $K(u)$ has density 1; given their
vertical separation and a parameter $\lambda\in (1/2,1)$, Proposition
\ref{densi3} shows that a good control on the Dirichlet energy and the
tilt implies that $\hh\res K(u)$ satisfies
\begin{equation}\label{twolay}
\hh\bigl(K(u)\cap \bigl[ B_R(b+x)\cup B_R(b+x')\bigr]\bigr)
\geq 2\lambda\omega_m R^m
\end{equation}
where $b\in T\cap\overline{B}_R$. This proposition will be the main tool in
the proof of the Lipschitz approximation Theorem \ref{lipappr}, showing
that in regions where the $m$-dimensional density of $K(u)$ is close
to $1$ the vertical separation of a large portion of $K(u)$ can be
controlled.

The proof of the proposition follows by Theorem \ref{planes} after a
suitable blow-up argument.

\begin{proposition}\label{densi3} Given $\lambda\in (1/2,1)$ and $L\in
(0,1]$, there exists a
constant $\gamma_2(\lambda,L,\omega)\in(0,1)$ such that, for any $m$-plane
$T$,
$b\in T\cap\overline{B}_R$, $x,\,x'\in B_R$ the conditions
\begin{equation}\label{cd2}
\theta_m(K(u),x)=\theta_m(K(u),x')=1,\,\,\,
|T^\perp(x-x')|> L |T(x-x')|
\end{equation}
\begin{equation}\label{cd1}
R\leq 1,\quad u\in\mm{B_{3R}},\quad\omega(3R)\leq \gamma_2
\end{equation}
\begin{equation}      \label{cd3}
\int_{B_\rho(x)\cup B_\rho(x')}|\nabla u|^2\,dy
<\gamma_2\rho^m\qquad
\forall\rho\in (0,2R)
\end{equation}
\begin{equation}       \label{cd4}
\T{u}{B_\rho(x)\cup B_\rho(x')}<\gamma_2\rho^m\qquad\forall
\rho\in (0,2R)
\end{equation}
imply (\ref{twolay}).
\end{proposition}

{\sc Proof.} Let us argue by contradiction. Let
$T_k,\,b_k,\,R_k,\,u_k,\,x_k,\,
x_k'$ be satisfying (\ref{cd2}), (\ref{cd1}), (\ref{cd3}), (\ref{cd4})
with $\gamma_k=1/k$ and
$$\hh\bigl(K(u_k)\cap \bigl[ B_{R_k}(b_k+x_k)\cup B_{R_k}(b_k+x_k')\bigr]\bigr)
<2\lambda\omega_mR_k^m.$$
By a rotation we can assume that $T_k=T=e_n^{\perp}$.
We will denote $x\in\rn{n}$ by $(z,t)$ with $z=Tx$ and $t=T^\perp x$. Let
\begin{eqnarray*}
r_k=\sup\Bigl\{r<R_k&:&\mu(u_k)\bigl(B_s(x_k+sb_k)\cup B_s(x_k'+sb_k)
\bigr)   \\
&&\geq 2\lambda\omega_m s^m\,\,\forall s\in (0,r)\Bigr\}.
\end{eqnarray*}
Since the density of $K(u_k)$ is $1$ at $x_k$ and at $x_k'$, we have
$0<r_k<R_k\leq 1$. Now we rescale all by a factor $r_k$ and we translate
$x_k'$ to the origin, defining
$$ \bar{x}_k=(\bar{z}_k,\bar{t}_k)=\frac{x_k-x_k'}{r_k},\qquad
\bar{u}_k(x)=\frac{1}{\sqrt{r_k}}u\bigl(r_k x+x_k'\bigr).$$
Then, by Remark \ref{scaling}, $\bar{u}_k\in\mm{\Omega_k}$, where
$\Omega_k=r_k^{-1}\bigl(B_{3R_k}-x_k'\bigr)$.
Moreover, the monotonicity of $\omega(\rho)$ yields
\begin{eqnarray}      \label{deviaz}
\dev{\bar{u}_k}{B_\rho}&=&\frac{\dev{u_k}{B_{\rho r_k}(x_k')}}{r_k^m}
\leq \frac{(\rho r_k)^m\omega(\rho r_k)}{r_k^m}\nonumber  \\
&\leq&\rho^m\omega(3R_k)\leq \frac{\rho^m}{k}
\end{eqnarray}
for any $\rho\in (0,3R_k/r_k)$. Similarly, we have
\begin{equation}      \label{deviaz2}
\dev{\bar{u}_k}{B_\rho(\bar{x}_k)}\leq \frac{\rho^m}{k}
\end{equation}
for any $\rho\in (0,3R_k/r_k)$.
Notice also that
$\bar{x}_k$ and $0$ belong to $K(\bar{u}_k)$ and
(\ref{cd2}), (\ref{cd3}), (\ref{cd4}) together with the
definition of $r_k$ yield
\begin{equation}\label{cd5}
|\bar{t}_k|\geq L|\bar{z}_k|
\end{equation}
\begin{equation}\label{cd6}
\E{\bar{u}_k}{B_\rho(\bar{x}_k)\cup B_\rho}< \frac{1}{k}\rho^m
\qquad\forall\rho\in (0,2R_k/r_k)
\end{equation}
\begin{equation}      \label{cd7}
\T{\bar{u}_k}{B_\rho(\bar{x}_k)\cup B_\rho}<\frac{1}{k}\rho^m
\qquad\forall\rho\in (0,2R_k/r_k)
\end{equation}
\begin{equation}\label{cd8}
\hh\bigl(K(\bar{u}_k)\cap \bigl[ B_1(\bar{x}_k+b_k)\cup B_1(b_k)
\bigr]\bigr)=2\lambda\omega_m
%\end{array}
\end{equation}
\begin{equation}\label{cd9}
\hh\bigl(K(\bar{u}_k)\cap \bigl[ B_\rho(\bar{x}_k+b_k)\cup B_\rho(b_k)
\bigr]\bigr)\geq 2\lambda\omega_m\quad\forall\rho\in (0,1).
\end{equation}
It is easy to see that $|\bar{x}_k|\leq 2$ for $k$ large enough. Indeed,
if $|\bar{x}_k|>2$ the balls
$B_1(\bar{x}_k+b_k)$ and $B_1(b_k)$ are disjoint; since, by Corollary
\ref{densi2} and (\ref{deviaz}), (\ref{deviaz2}), (\ref{cd6}), (\ref{cd7})
$$\limsup_{k\to +\infty}\bigl[\hh\bigl(K(\bar{u}_k)\cap
B_1(\bar{x}_k+b_k)\bigr)+\hh\bigl(K(\bar{u}_k)\cap
B_1(b_k)\bigr)\bigr]\geq 2\omega_m$$
this contradicts (\ref{cd8}) for $k$ large enough. Possibly passing
to a subsequence we can assume that $\bar{x}_k$ converges to $x=(z,t)$
and $b_k$ converges to $b\in T$ as $k\to +\infty$. We note that $|t|\geq
L|z|$ because of (\ref{cd5}).

Since $B_1(\bar{x}_k+b_k)\cup B_1 (b_k)\subset\Omega_k$
by Theorem \ref{planes} and (\ref{deviaz})--(\ref{cd7}) we
can assume that $\mu(\bar{u}_k)$ weakly converge in $B_1(x+b)\cup B_1(b)$
to a measure $\mu$ which is locally sum of $m$-dimensional
Hausdorff measures associated to $m$-planes parallel to $T$.
By (\ref{cd8}) we get
\begin{equation}\label{cd10}
\mu\bigl(B_1(x+b)\cup B_1(b)\bigr)\leq 2\lambda.
\end{equation}
On the other hand, the density lower bound implies that $x$ and $0$ belong
to the support of $\mu$; we need only to show that $x\neq 0$ (hence
$t\neq 0$) to contradict (\ref{cd10}). If $\bar{x}_k$ were converging
to $0$, for any $\tau>0$ we would have
$$B_\rho(\bar{x}_k+b_k)\cup B_\rho(b_k)\subset B_{\rho+\tau}(b_k)$$
for $k$ large enough, so that (\ref{cd9}) would imply
$\mu(\overline{B}_{\rho+\tau}(b))\geq 2\lambda\rho^m$. Letting first
$\tau\downarrow 0$ then $\rho\downarrow 0$ we find that the density of
$\mu$ in $0$ is $2\lambda>1$, a contradiction.
\qed

\section{Lipschitz approximation}

In this section we show that the set $K(u)$ of a quasi minimizer $u$ can be
approximated by the graph $\Gamma(f)$
of a Lipschitz function $f$ in small balls
(depending on $\omega$ and the Lipschitz constant of $f$).
Related results
are proved in \cite{davsemm} in the two dimensional case, for minimizers of
the Mumford-Shah functional. We also estimate the measure of the symmetric
difference $K(u)\Delta\Gamma(f)$;
this estimate will be useful in the proof of Theorem \ref{decay}.

The following proposition shows that an integral estimate of
$|T^\perp (x-x_0)|$ in $B_R(x_0)$
leads to a pointwise estimate of the same quantity in
a smaller ball, via the density lower bound.

\begin{proposition}[height bound]      \label{altezza}
Let $u\in\mm{B_{2R}(x_0)}$. Denoting by
$\rho_\omega$ and $\theta_\omega$ the constants given by Theorem
\ref{dgcl}, we have
$$
\sup_{x\in B_R(x_0)\cap K(u)}|T^\perp (x-x_0)|^{m+2}
\leq \frac{2^{m+2}}{\theta_\omega}
\int_{B_{2R}(x_0)}|T^\perp(x-x_0)|^2\,d\mu(u)
$$
provided $R\leq 2\rho_\omega$.
\end{proposition}
{\sc Proof.} Let for simplicity $x_0=0$, assume
$$
\sigma= \Bigl[\frac{2^{m+2}}{\theta_\omega}
\F{u}{B_{2R}}\Bigr]^{\frac{1}{m+2}}<R $$
(otherwise the thesis is trivial), and suppose that a point $x_1\in K(u)$
with
$|x_1| \leq R$ exists such that $|T^\perp x_1|>\sigma$; then $|T^\perp
x|>\sigma /2$ in $B_{\sigma /2}(x_1)$, whence, using the density lower
bound, we deduce
\begin{eqnarray*}
&&\F{u}{B_{2R}} \geq \F{u}{B_{\sigma /2}(x_1)}   \\
&&>\theta_\omega \Bigl(\frac{\sigma}{2}\Bigr)^{m+2} = \F{u}{B_{2R}},
\end{eqnarray*}
which gives a contradiction.

\qed

\noindent
{\bf Choice of the constants.} Now we fix all the constants of the previous
density bounds. All these constants depend on $n$ and possibly on
$\omega$, $L$.

{\bf 1.} Let $r\in (1,2)$ such that $r^m<2$ and $\lambda\in (1/2,1)$
such that $r^m<2\lambda$. Now we fix $\delta>0$ so small that
\begin{equation} \label{conddelta}
1+\delta<r,\qquad \sqrt{\frac{1}{16}+\delta^2}<\frac{5}{8}.
\end{equation}

{\bf 2.} Now, let $\gamma(r/2,2\lambda/r^m,\omega)$,
$\gamma_1(1/2,\omega)$,
$\gamma_2(\lambda,L,\omega)$ be given by Corollary \ref{densi1},
Corollary \ref{densi2} and Proposition \ref{densi3} respectively.
We will impose a smallness condition on $R$, denoting by
$R_{n,\omega,L}$ the largest constant $R\leq \inf\{2\rho_\omega,1/2\}$
such that
\begin{equation} \label{romega}
\omega(6\rho)\leq\min\biggl\{\frac{\gamma}{2},\,\frac{\gamma_1}{3},
\,\gamma_2,\,n\omega_n\biggr\}\qquad\forall\rho\in (0,R].
\end{equation}

{\bf 3.} Since
$$\omega_m\bigl(2r\bigr)^m<\omega_m2^{m+1}\lambda$$
Corollary \ref{densi1} (with $\beta=r/2$ and $\tau=2\lambda/r^m$) and
our choice of $\gamma$ and of $R_{n,\omega,L}$ guarantee that the condition
$$
R^{-m}\E{u}{B_{4R}}+R^{-m-2}\F{u}{B_{4R}}<\frac{\gamma}{2}
$$
implies
$$
\hh\bigl(K(u)\cap B_{2rR}\bigr)<2^{m+1}\lambda\omega_mR^m
$$
provided $u\in\mm{B_{4R}}$ and
$R\leq R_{n,\omega,L}$.

Now, we can state the following Lipschitz approximation theorem (similar
to \cite[5.4]{brakke}, see also \cite{allard} and \cite{simon}). The proof
is based on the density estimates of the previous section, the
height bound and a covering argument.

\begin{theorem}[Lipschitz approximation]  \label{lipappr}
Let $L\in (0,1]$, $u\in\mm{B_{7R}}$ for some
$R\leq R_{n,\omega, L}$. Suppose also that $K(u)\cap B_{R/16}\neq
\emptyset$ and
\begin{equation}            \label{smallf}
R^{-m-2}\F{u}{B_{4R}}<\min\biggl\{
\theta_\omega\left(\frac{\delta L}{4}\right)^{m+2},\frac{\gamma}{4}\biggr\}
\end{equation}
for some $m$-plane $T$.
Then, there exists a Lipschitz function $f:T\to T^\perp$ with Lipschitz
constant less than $L$ such that
\begin{equation}\label{additional}
\sup|f|^{m+2}\leq {2^{m+2}\over\theta_\omega}\FF{B_{2R}}
\end{equation}
and, denoting by $\Gamma(f)$ the graph of $f$, we have
\begin{eqnarray}\label{error}
&&\hh\bigl(B_R\cap K(u)\setminus\Gamma(f)\bigr)+
\hh\bigl(B^m_{R/4}\cap T[\Gamma(f)\setminus K(u)]\bigr)
\\
&&\leq P(n,L,\omega)
\biggl[\E{u}{B_{5R}}+\T{u}{B_{5R}}\biggr].\nonumber
\end{eqnarray}
\end{theorem}
{\sc Proof.} It is not restrictive to assume that $T$ is spanned by
$e_1,\ldots,e_m$, and set $x=(z,t)$, $z=Tx$, $t=T^\perp x$.
Since the function $u$ will be fixed throughout the proof we set
$K=K(u)$, $\mu=\mu(u)$ and denote by $K^*$ the set of points
of density $1$ of $K$.

Since $L\leq 1$ and $R\leq 2\rho_\omega$, by the height bound and
(\ref{smallf}) we get
\begin{equation}                   \label{alt1}
|T^\perp x|\leq 2\left(\frac{1}{\theta_\omega}\FF{B_{2R}}\right)^{1/(m+2)}
<\delta R
\end{equation}
for any $x\in K\cap B_R$. We can also assume that
\begin{equation}\label{oneshee}
\hh\bigl(K\cap B_{2rR}\bigr)< 2^{m+1}\lambda\omega_mR^m.
\end{equation}
Indeed, because of our choice of $\gamma$ and (\ref{smallf}),
if (\ref{oneshee}) does not hold then
$$R^{-m}\E{u}{B_{4R}}\geq \frac{\gamma}{4}$$
and we can choose the constant
$P$ so large that (\ref{error}) trivially holds
with $f\equiv 0$.
Now, let $\sigma=\min\{\gamma_1,\gamma_2\}$ and define
\begin{eqnarray*}
A&=&\bigl\{x\in B_R\cap K^*:
\\
&&\int_{B_\rho(x)}|\nabla u|^2\, dy+
\int_{B_\rho(x)}\Vert S_y-T\Vert^2\, d\mu
<\frac{\sigma}{2}\rho^m\,\,\forall\rho\in (0,4R)\bigr\},
\end{eqnarray*}
Let
$$m(z)={\rm card}\bigl\{t: (z,t)\in A\bigr\}$$
be the multiplicity function of $A$ with respect to $T$.

{\sc Step 1.} We claim that $m(z)\leq 1$ on $B^m_R$.  Indeed, let $z\in
B^m_R$ and let $x=(z,t),\,x'=(z,t')\in A$.  Recalling our choice of
$R_{n,\omega,L}$ and the definition of $A$, we can apply Proposition
\ref{densi3} with $b=-z$, to get
$$
\hh\bigl(K\cap \bigl[B_{2R}((0,t))\cup B_{2R}((0,t'))\bigr]\bigr)
\geq 2^{m+1}\lambda\omega_mR^m.
$$
Since $t,\,t'\in (-\delta R,\delta R)$ and $1+\delta/2<r$ we have
$$
B_{2R}((0,t))\cup B_{2R}((0,t'))\subset B_{2rR}
$$
hence
$\hh(K\cap B_{2rR})\geq 2^{m+1}\lambda\omega_mR^m$
and this contradicts (\ref{oneshee}).

{\sc Step 2.} Let $E=\{z\in B_R^m:m(z)=1\}$ and let $\phi:E\to\rn{}$
such that $(z,\phi(z))\in A$ for any $z\in E$.

We now claim that $\phi$ satisfies the following condition:
\begin{equation}\label{lippro}
z,\,z'\in E\quad\Longrightarrow\quad
|\phi(z)-\phi(z')|\leq L|z-z'|
\end{equation}
Indeed, let $z,\,z'\in E$; let us assume that $|\phi(z)-\phi(z')|>L|z-z'|$;
if $|z-z'|>\delta R$ we get by Proposition \ref{altezza}
$$
(\delta R)^{m+2}<2^{m+2}\sup_{x\in K\cap B_R}|T^\perp x|^{m+2}\leq
{4^{m+2}\over\theta_\omega}\FF{B_{2R}},
$$
contradicting assumption (\ref{smallf}). Hence $|z-z'|\leq\delta R$.
Let $x=(z,\phi(z)),\,x'=(z',\phi(z'))$; by applying Proposition
\ref{densi3} with $b=-z$ we get
$$
\hh\bigl(K\cap\bigr[B_{2R}(0,\phi(z))\cup
B_{2R}(z'-z,\phi(z))\bigr]\bigr) \geq 2^{m+1}\lambda\omega_mR^m.
$$
By (\ref{alt1}) and the inequalities $|z-z'|\leq\delta R$,
$1+\delta/\sqrt{2}<r$ we infer
$$
B_{2R}(0,\phi(z))\subset B_{2rR},\qquad
B_{2R}(z'-z,\phi(z))\subset B_{2rR}
$$
and we find a contradiction with (\ref{oneshee}) as in Step 1.

{\sc Step 3.} Now we estimate $\hh(K\cap B_R\setminus A)$:
if $x\in K^*\cap B_R$ does not belong to $A$ we can find
$\rho(x)\in (0,4R)$ such that
$$
\int_{B_{\rho(x)}(x)}|\nabla u|^2\, dy+
\int_{B_{\rho(x)}(x)}\Vert S_y-T\Vert^2\, d\mu
\geq\frac{\sigma}{2}\rho^m(x),
$$
hence, by using the density upper bound and (\ref{romega})
\begin{eqnarray*}
\hh\bigl(K^*\cap B_{\rho(x)}(x)\bigr) &\leq& n\omega_n\rho^m(x)
+\rho^m\omega(\rho(x))                         \\
&\leq& \bigl(n\omega_n+\omega(4R)\bigr)
\rho^m\leq 2n\omega_n\rho^m
\end{eqnarray*}
we get
$$
\int_{B_{\rho(x)}(x)}|\nabla u|^2\, dy+
\int_{B_{\rho(x)}(x)}\Vert S_y-T\Vert^2\, d\mu
\geq\frac{\sigma}{4n\omega_n}
\hh\bigl(K^*\cap B_{\rho(x)}(x)\bigr).
$$
Using a standard covering argument and $\hh\bigl(K\setminus
K^*\bigr)=0$,  we get
\begin{equation}                 \label{estima1}
\ \ \hh\bigl(B_R\cap K\setminus A\bigr)\leq\frac{4n\omega_n\xi(n)}{\sigma}
\biggl[\E{u}{B_{5R}}+\TT{B_{5R}}\biggr]
\end{equation}
where $\xi(n)$ is the constant of Besicovitch's theorem. In particular we
can choose the constant $P$ in (\ref{error}) so large that
$$
\frac{4n\omega_n\xi(n)}{\sigma}
\biggl[\E{u}{B_{5R}}+\TT{B_{5R}}\biggr]\geq\theta_\omega (R/16)^m
$$
implies (\ref{error}) with $f\equiv 0$. Hence, in the following we can
assume
$$
\frac{4n\omega_n\xi(n)}{\sigma}
\biggl[\E{u}{B_{5R}}+\TT{B_{5R}}\biggr]<\theta_\omega (R/16)^m$$
and Theorem \ref{dgcl} and the assumption $K\cap B_{R/16}\neq\emptyset$
yield
\begin{equation}\label{cea}
A\cap B_{R/8}\neq\emptyset.
\end{equation}

{\sc Step 4.} Because of (\ref{alt1}), (\ref{lippro}) we can find
a Lipschitz extension $f$ of $\phi$ satisfying (\ref{additional}).
Now we claim that $f$ satisfies (\ref{error}). Indeed, since
$$
B_R\cap K\setminus \Gamma(f)\subset B_R\cap K\setminus A,$$
by (\ref{estima1}) we get
\begin{equation}\label{estima2}
\begin{array}{ll}
\hh\bigl(B_R\cap K\setminus\Gamma(f)\bigr)    \\
\displaystyle
\leq\frac{4n\omega_n\xi(n)}{\sigma}
\biggl[\E{u}{B_{5R}}+\TT{B_{5R}}\biggr].
\end{array}
\end{equation}
Now we estimate
$$
\hh\bigl(B^m_{R/4}\cap T(\Gamma(f)\setminus K)\bigr).
$$
Recalling that the multiplicity of $A$ does not exceed 1 on $B^m_R$, we
have
$$
B^m_{R/4}\cap T(\Gamma(f)\setminus K)\subset
B^m_{R/4}\cap T(\Gamma(f)\setminus A)\subset
B^m_{R/4}\setminus E
$$
so that we need only to estimate the measure of $F=B^m_{R/4}\setminus E$.
To this aim, we fix a compact set $G\subset A$ such that $G\cap B_{R/8}\neq
\emptyset$ (see (\ref{cea})).  For any $z\in F$ let $C_r(z) =B^m_r(z)
\times \rn{}$ be the largest cylinder which does not intersect $G$.  Since
$G\cap B_{R/8}\neq\emptyset$ we have $r\leq 3R/8$ and we can find
$x=(z',f(z'))\in\partial C_r(z)\cap G$.
By applying Corollary \ref{densi2} with $b=z-z'$ we get
$$
\hh\bigl(K\cap B_r((z,f(z'))\bigr)\geq\frac{\omega_m}{2} r^m.
$$
Since (see (\ref{alt1}) and (\ref{conddelta}))
$$
R^{-1}\bigl |(z,f(z'))\bigr|
\leq\sqrt{\frac{1}{16}+\delta^2}<\frac{5}{8}
$$
we have the inclusion $K\cap B_r((z,f(z'))\subset B_R\cap C_r(z)$,
hence
$$
\hh\bigl(B_R\cap K\cap C_r(z)\setminus G)\bigr)\geq\frac{\omega_m}{2} r^m.
$$
By Besicovitch's theorem, we can find
a disjoint collection $\{B_{r_i}^m(z_i)\}_{i\in I}$ such that
$$
\xi(n)\hh\biggl(\bigcup_{i\in I} B_{r_i}^m(z_i)\biggr)\geq \hh(F)
$$
and
$$
\hh\bigl(B_R\cap K\cap C_{r_i}(z_i)\setminus G\bigr) \geq\frac{\omega_m}{2}
r_i^m\qquad\forall i\in I.
$$
By summing with respect to $i$, since the cylinders $C_{r_i}(z_i)$ are
disjoint, we get
$$
\hh(F)\leq 2\xi(n)\hh\big(B_R\cap K\setminus G\bigr).
$$
Since $G\subset A$ is arbitrary, by (\ref{estima2}) we get the estimate
\begin{eqnarray}    \label{estima3}
&&\hh\bigl(B^m_{R/4}\cap T(\Gamma(f)\setminus K)\bigr) \nonumber \\
&&\leq\frac{8n\omega_n\xi^2(n)}{\sigma}
\biggl[\E{u}{B_{5R}}+\TT{B_{5R}}\biggr].
\end{eqnarray}
The statement now follows from (\ref{estima2}) and (\ref{estima3}).

\qed

A first consequence of the Lipschitz approximation theorem is the following
$C^{1,\alpha}$ regularity criterion for $K(u)$. Basically, we need to
control with a power strictly greater than $\rho^m$ the deviation from
minimality and the Dirichlet energy in $B_\rho(x)$; moreover, we
need a power stricly greater than $\rho^{m+2}$ to control the quantity
\begin{equation}\label{defT}
{\bf T}(x,\rho)=\min_{T\in{\bf G}(n,m)}\int_{B_\rho(x)}|T^\perp (y-x)|^2
\,d\mu(u)
\end{equation}
which measures the flatness of $K(u)$. The tilt lemma (see \S6)
guarantees also a control on the oscillation of tangent planes.

\begin{theorem}\label{c1graf}
Let $\Omega\subset\rn{n}$ be an open set and let $u\in\mm{\Omega}$.
Assume that, for some constants
$C>0$, $s>0$, $\omega(t)\leq C t^{2s}$ and
$$
\int_{B_\rho(x)}|\nabla u|^2\, dy+
\rho^{-2}{\bf T}(x,\rho) \leq C\rho^{m+s}
$$
for any ball $B_\rho(x)\subset\Omega$ and any $x\in K(u)$.
Then $\Omega\cap K(u)$ is a $C^{1,s/2}$-hypersurface.
\end{theorem}
{\sc Proof.} Let $K=K(u)$, $\mu=\mu(u)$ and $\alpha=s/2$.
By the tilt Lemma \ref{tilt} of \S6 with $\beta=1/2$ we have also
$$
\sup\biggl\{\rho^{-m-\alpha}\min_T\int_{B_\rho(x)}\Vert S_y-T\Vert^2
\,d\mu(y):B_{2\rho}(x)\subset \Omega\biggr\}<+\infty.
$$
We fix an open set $D\subset\subset \Omega$ and we
prove the regularity of $K$ in $D$.

Using the H\"older inequality and Remark \ref{mst} we find a constant
$M>0$ such that
$$
\min_T\int_{B_\rho(x)}\Vert S_y-T\Vert\,d\mu(y)\leq M\rho^{m+\alpha}
$$
for any ball $B_{2\rho}(x)\subset\Omega$ with $\rho<1$.
We denote by $T^x_\rho$ any $m$-plane such that
$$\min_T\int_{B_\rho(x)}\Vert S_y-T\Vert\,d\mu(y)=
\int_{B_\rho(x)}\Vert S_y-T^x_\rho\Vert
\,d\mu(y).$$

{\sc Step 1.} We claim that $T^x_\rho$ converges as $\rho\to 0^+$ to
some $T^x$ and $x\mapsto T^x$ is $\alpha$-H\"older continuous in $K\cap D$.
Indeed, let $\rho_0<\min\{2\rho_\omega,1\}$ be less than
the distance of $D$ from $\partial\Omega$ and
$x\in K\cap D,\,\rho\in (0,\rho_0/2)$; we have
\begin{eqnarray}\label{oscpla}
&&\Vert T^x_\rho-T^x_{\rho/2}\Vert\leq\frac{2^m}
{\theta_\omega\rho^m}\int_{B_{\rho/2}(x)}
\Vert T^x_\rho-T^x_{\rho/2}\Vert\,d\mu(y)
\\
&&\leq\frac{2^m}{\theta_\omega\rho^m}\left[\int_{B_\rho(x)}
\Vert T^x_\rho-S_y\Vert\,d\mu(y)+\int_{B_{\rho/2}(x)}\Vert
T^x_{\rho/2}-S_y\Vert\,d\mu(y)\right]\nonumber
\\
&&\leq\frac{2^mM}{\theta_\omega\rho^m}2\rho^{m+\alpha}\leq
\frac{2^n M}{\theta_\omega}\rho^{\alpha}.\nonumber
\end{eqnarray}
Using (\ref{oscpla}) it is easy to see that $T^x_{2^{-k}}$ converges
as $k\to +\infty$ to some $T^x$. Moreover, a similar argument shows
that $T^x_\rho$ converges to $T^x$ as $\rho\downarrow 0$ and
$$
\Vert T^x_\rho -T^x\Vert\leq c\rho^\alpha\qquad\forall x\in K\cap D,
\,\,\rho\in (0,\rho_0/2)
$$
with $c$ depending only on $n$, $\theta_\omega$, $M$, $\alpha$. In order
to check the H\"older continuity of $T^x$ we choose $x,\,z\in
D\cap K$ with $\rho=|x-z|\leq\rho_0/4$ and we estimate
\begin{eqnarray}\label{oscpla2}
\Vert T^x_\rho-T^z_\rho\Vert&\leq&\frac{1}
{\theta_\omega\rho^m}\int_{B_{\rho}(x)}
\Vert T^x_\rho-T^z_{\rho}\Vert\,d\mu(y)\nonumber
\\
&\leq&\frac{1}{\theta_\omega\rho^m}\left[\int_{B_\rho(x)}
\Vert T^x_\rho-S_y\Vert\,d\mu(y)+\int_{B_{2\rho}(z)}\Vert
T^z_\rho-S_y\Vert\,d\mu(y)\right]\nonumber
\\
&\leq&\frac{M}{\theta_\omega\rho^m}\bigl[(\rho)^{m+\alpha}+(2\rho)^{m+
\alpha}
\bigr]\leq\frac{2^{m+1+\alpha}M}{\theta_\omega}|x-z|^{\alpha}.
\end{eqnarray}
Using (\ref{oscpla}) and (\ref{oscpla2}) the H\"older continuity
of $x\mapsto T^x$ in $K\cap D$ follows.

{\sc Step 2.} For $\hh$-almost every $x\in K\cap D$ the $m$-plane $T^x$
of Step 1 coincides with the approximate tangent space to $K$. Indeed,
let $x_0\in K\cap D$ be a Lebesgue point for the approximate tangent space
map $S_x$; for any $\rho\in (0,\rho_0/2)$ we have
\begin{eqnarray*}
&&\Vert T^x_\rho-S_{x_0}\Vert
\leq\frac{1}{\theta_\omega\rho^m}\int_{B_\rho(x_0)}
\Vert T^x_\rho-S_{x_0}\Vert\,d\mu(y)
\\
&&\leq\frac{1}{\theta_\omega\rho^m}\left[\int_{B_\rho(x_0)}
\Vert T^x_\rho-S_y\Vert\,d\mu(y)+\int_{B_\rho(x_0)}
\Vert S_y-S_{x_0}\Vert\,d\mu(y)
\right]
\\
&&\leq\frac{M}{\theta_\omega}\rho^\alpha+\frac{1}{\theta_\omega\rho^m}
\int_{B_\rho(x_0)}\Vert S_y-S_{x_0}\Vert\,d\mu(y)
\end{eqnarray*}
and letting $\rho\downarrow 0$ we find $S_{x_0}=T^{x_0}$.

{\sc Step 3.} Now we see that $K\cap D$ is locally the graph of a
Lipschitz continuous function.  To this aim, we remark that Corollary
\ref{densi1}
and Corollary \ref{densi2} imply that $K$ has density 1 everywhere.  We
define $\sigma$ as in the proof of Theorem \ref{lipappr}.

Let $x_0\in K\cap D$ and let $\rho_1\leq\min\{\rho_0/7,R_{n,\omega,1}\}$
so small that
\begin{eqnarray*}
A&=&\Bigl\{x\in B_{\rho_1}(x_0)\cap K:
\\
&&\E{u}{B_\rho(x_0)}+ \TT{B_\rho(x_0)}<\frac{\sigma}{2}\rho^m\,\,
\forall\rho\in (0,4\rho_1)\Bigr\}
\end{eqnarray*}
coincides with $B_{\rho_1}(x_0)\cap K$ and
$$
\rho_1^{-m-2}\int_{B_{4\rho_1}(x_0)}|T^\perp (x-x_0)|^2\,d\mu
<\min\biggl\{\theta_\omega\left(\frac{\delta}{4}\right)^{m+2},
\frac{\gamma}{4}\biggr\}.
$$
Let $f:T\to T^\perp$ be the 1-Lipschitz function given by Theorem
\ref{lipappr}.  By construction, the graph of $f$ contains $A$, hence
$B_{\rho_1}(x_0)\cap K$ is contained in the graph of $f$.  We now claim
that $\overline{B}_{\rho_1/2}(x_0)\cap K$ {\it contains} the graph of
$f$ on $B_{\rho_1/4}^m(T(x_0))$:  indeed if there were some $z\in
B_{\rho_1/4}^m(T(x_0))$ such that $(z,f(z))\notin\overline{B}_{\rho_1/2}
(x_0) \cap K$ we could take the largest cylinder
$C_r=B^m_r(z)\times\rn{}$
that does not intersect $\overline{B}_{\rho_1/2}(x_0)\cap K$ and some
point $x_1\in\partial C_r\cap K\cap\overline{B}_{\rho_1/2}(x_0)$.  Since
$|z-T(x_0)|<\rho_1/4$ we have $r<\rho_1/4$, hence $x_1\in B_{\rho_1/2}(x_0)$
because the Lipschitz constant of $f$ does not exceed 1.

Since $K\cap B_{\rho_1}(x_0)$ is contained in the graph of $f$ and since
$C_r$ does not intersect $K\cap B_{\rho_1/2}(x_0)$, the density of
$K$ at $x_1$ can be at most $\sqrt{2}/2<1$, a contradiction.

{\sc Step 4.} In order to show that the function $f$ of Step 3 is actually
$C^{1,\alpha}$ we need only to remark that at any differentiability point
$z\in B^m_{\rho_1/4}(T(x_0))$ of $f$ the approximate tangent space to
$K$ ({\em i.e.}, the graph of $f$ ) exists and is related in the
classical
way to $\nabla f(x_0)$.  In particular, Step 1 and Step 2 imply that the
map $\nabla f$ (defined at differentiability points of $f$) has a
$\alpha$-H\"older continuous extension to $B^m_{\rho_1/4}(T(x_0))$.  It is
easy to check, {\em e.g.} using mollifiers, that any Lipschitz map with
this property is actually in $C^{1,\alpha}$. \qed

\begin{remark} \label{anchecosi} {\rm
Let ${\bf A}_{n,m}$ be the set of {\it affine} $m$-planes in $\rn{n}$
and define, in analogy with (\ref{defT})
\begin{equation}\label{defA}
{\bf A}(x,\rho)=\min_{A\in{\bf A}_{n,m}}
\int_{B_\rho(x)}|A^\perp y|^2\,d\mu(u)
\end{equation}
where $|A^\perp y|$ is the distance of $y$ from $A$. Then, clearly
${\bf A}\leq {\bf T}$. However, it is not hard to see that
Theorem \ref{c1graf} still holds if ${\bf A}$ instead of ${\bf T}$
is controlled. Indeed, the following inequality holds
\begin{equation}\label{TconA}
{\bf T}(x,\rho)\leq 2A(x,\rho)+16n\omega_n\rho^m\biggl(\frac{
{\bf A}(x,\rho)}{\theta_\omega}\biggr)^{2/(m+2)}
\end{equation}
for any $\rho<2\rho_\omega$ such that $\omega(\rho)\leq n\omega_n$.
In particular, if ${\bf A}(x,\rho)\leq C\rho^{m+2+\alpha}$ for some
$\alpha\leq s$, the assumptions of Theorem \ref{c1graf} are
satisfied with $s'=2\alpha/(m+2)$.

To check (\ref{TconA}), let $A\in{\bf A}_{n,m}$ be satisfying
$$\int_{B_\rho(x)}|A^\perp y|^2\,d\mu(u)={\bf A}(x,\rho).$$
Denoting by $T\in{\bf G}_{n,m}$ the $m$-plane parallel to $A$ and by
$\delta$ the distance of $A$ and $x$, by the energy upper bound we have
$${\bf T}(x,\rho)\leq 2{\bf A}(x,\rho)+2\delta^2\mu(u)(B_\rho(x))
\leq
2{\bf A}(x,\rho)+4\delta^2n\omega_n\rho^m.$$
Hence, we need only to estimate $\delta/\rho$; using the density lower
bound we get
\begin{displaymath}
\int_{B_\rho(x)}|A^\perp y|^2\,d\mu(u)
\geq \Bigl(\frac{\delta}{2}\Bigr)^2\mu(u)\bigl(B_{\delta/2}(x)\bigr)
\geq \theta_\omega\Bigl(\frac{\delta}{2}\Bigr)^{m+2}
\end{displaymath}
and this implies (\ref{TconA}).
}\end{remark}

\vfill\eject

\section{Flatness improvement}

We begin this section proving a classical (compare \cite{allard},
\cite{simon}, \cite{brakke}) estimate on the tilt of tangent planes.
The proof is achieved by a deformation argument.

\begin{lemma}[tilt estimate]   \label{tilt} For any $\beta\in (0,1)$
there exist $c(\beta)>0$, $\eps_\beta>0$
such that for any $u\in SBV(B_R)$ and any $m$-plane $T$ we have
\begin{equation}               \label{tiltest}
\int_{B_{\beta R}\cap S_u}\Vert S_x-T\Vert^2\,d\hh(x)
\end{equation}
$$
\leq c(\beta)\biggl[\E{u}{B_R} +R^{-2} \int_{B_R\cap S_u}|T^\perp
x|^2\,d\hh(x) +R^m \sqrt{\frac{\dev{u}{B_R}}{R^m}}\biggr]
$$
provided $\dev{u}{B_R}<\eps_\beta^2 R^m$.
\end{lemma}
{\sc Proof.} Let $\mu=\hh\res S_u$,
$R=1$, and let $\zeta\in C_0^1(B_1)$ be a function
such that $0\leq \zeta\leq 1$, $\zeta =1$ in $B_\beta$ and
$|\nabla\zeta|\leq 2/(1-\beta)$.  Consider the vector field
$\phi(x)=\zeta^2 (x)T^\perp(x)$,
set $\tau_\eps(x)=x+\eps\phi(x)$, and remark that positive
constants $\eps_\beta$, $c_1$ exist such that for $0<|\eps|<\eps_\beta$ the
inequality
\begin{equation}          \label{t1}
\bigl\vert {\bf J}_m(\tau_\eps,S)- 1-\eps\diver_S\phi\bigr\vert
\leq c_1\eps^2
\end{equation}
holds uniformly with respect to $S\in{\bf G}_{n,m}$ and $\tau_\eps$
is a $C^1$ diffeomorphism of $B_1$. Moreover, for
$u_\eps(x)=u(\tau_\eps(x))$, the definition of $\dev{u}{B_1}$ implies
\begin{equation}\label{t2}
\ \ \E{u_\eps}{B_1}-\E{u}{B_1}+\hh\bigl(S_{u_\eps}\bigr)-
\hh\bigl(S_u\bigr) +\dev{u}{B_1}\geq 0.
\end{equation}
Now compute (cf. \cite[\S 22]{simon})
\begin{eqnarray*}
\diver_S T^\perp &=& \frac{1}{2}\Vert S-T\Vert^2     \\
\diver_S\phi\,\, &=& 2\zeta\langle\nabla_S\zeta,T^\perp\rangle +\zeta^2
\diver_S T^\perp
= 2\zeta\langle\nabla_S\zeta,T^\perp\rangle +\frac{\zeta^2}{2} \Vert
S-T\Vert^2,
\end{eqnarray*}
(where $\diver_S$ and $\nabla_S$ are the tangential operators along $S$)
and remark that
\begin{eqnarray}        \label{t3}
2\zeta \langle\nabla_S\zeta ,T^\perp\rangle &\geq& -2\zeta\Vert
S-T\Vert\,|T^\perp x|\, |\nabla\zeta|\nonumber   \\
&\geq& -\frac{1}{4}\zeta^2\Vert
S-T\Vert^2 - 4|\nabla\zeta|^2\, |T^\perp x|^2 .
\end{eqnarray}
To check (\ref{t3}), suppose that $T={\rm span}\,\{e_1,\ldots
,e_{n-1}\}$, and compute
$$
|\langle\nabla_S\zeta ,T^\perp\rangle (x)|= \Bigl|x_nS_{nj}
\frac{\partial\zeta}{\partial x_j}\Bigr| = \Bigl|x_n(S_{nj}-T_{nj})
\frac{\partial\zeta}{\partial x_j}\Bigr| ,
$$
whence (\ref{t3}) readily follows. Notice also that
$$
\E{u_\eps}{B_1}=\int_{B_1} |\nabla u(x)
\nabla\tau_\eps^{-1}(x)|^2 |{\rm det}\, \nabla\tau_\eps (x)|dx ,
$$
whence
\begin{equation}       \label{t4}
\biggl\vert
\frac{1}{\eps}\biggl[\E{u_\eps}{B_1}-\E{u}{B_1}\biggr] \biggr\vert\leq
c_2 \E{u}{B_1},\ \  0<|\eps|<\eps_\beta
\end{equation}
(where $c_2$ depends only on $\phi$ and $\eps_\beta$). Since
$$
\hh\bigl(S_{u_\eps}\bigr)=\int_{B_1}{\bf J}_m(\tau_\eps,S_x)\, d\mu,
$$
from (\ref{t1}) the inequality
$$
\E{u_\eps}{B_1}-\E{u}{B_1}+\eps\int_{S_u}\divt\phi\, d\hh
+\dev{u}{B_1}\geq -c_1\eps^2
$$
follows. Using (\ref{t3}), (\ref{t4}) we obtain for $-\eps_\beta<\eps<0$:
\begin{eqnarray}         \label{t5}
\frac{1}{4}\int_{B_\beta}\Vert S_x-T\Vert^2\, d\mu &\leq&
\frac{16}{(1-\beta)^2}\int_{B_1}|T^\perp x|^2\, d\mu + c_2\E{u}{B_1}
\nonumber  \\
&&-\frac{\dev{u}{B_1}}{\eps} +c_1|\eps|.
\end{eqnarray}
Finally, assuming $\dev{u}{B_1}\leq\eps_\beta^2$ we choose
$\eps=-\sqrt{\dev{u}{B_1}}$ and use (\ref{t5}) to get
\begin{eqnarray}       \label{t6}
\int_{B_\beta}\Vert S_x-T\Vert^2\, d\mu &\leq&
\frac{64}{(1-\beta)^2}\int_{B_1}|T^\perp x|^2\, d\mu
\\
&&+ 4c_2\E{u}{B_1} +4(1+c_1)\sqrt{\dev{u}{B_1}},\nonumber
\end{eqnarray}
and the thesis follows for $R=1$. To recover the general case, exploit the
scaling properties of the various quantities in (\ref{t6}) and conclude.
\qed

The following theorem (whose statement is similar to \cite[5.6]{brakke}) shows
a decay property of the quantity ${\bf A}(x,\rho)$ defined in (\ref{defA}),
assuming that ${\bf A}(x,\rho)/\rho^{m+2}$ is sufficiently small and that the
Dirichlet energy and the deviation from minimality are comparable with ${\bf
A}(x,\rho)$. The proof is obtained by the classical harmonic comparison
argument of De Giorgi (see also \cite{simon}), {\em i.e.}, we approximate
$K(u)$ by the graph of a Lipschitz function $f$ and show that a rescaled
function $g$ is close to a harmonic function.

\begin{theorem}             \label{decay}
For any $\beta\in (0,1/112)$ and $M>0$ there exists $\eps(\beta,M,\omega)>0$
such that, for any $u\in\mm{B_\rho(x)}$, the conditions
$$
x\in K(u),\quad {\bf A}(x,\rho)\leq\eps\rho^{m+2},
\quad M{\bf A}(x,\rho)\geq\rho^2
\Bigl[\int_{B_\rho(x)}|\nabla u|^2+\rho^m\sqrt{\omega(\rho)}\Bigr]
$$
imply
\begin{equation} \label{migliora}
{\bf A}(x,\beta\rho)\leq C\beta^{m+4} {\bf A}(x,\rho)
\end{equation}
with $C$ depending only on $n$ and $\omega$.
\end{theorem}
{\sc Proof.} Let us fix $\beta$, $M$, $\omega$, and assume for convenience
that $\omega(\rho)>0$ for any $\rho>0$ (see Remark \ref{anyomega});
let also be $R=R_{n,\omega,1}$, where $R_{n,\omega,1}$ is the
constant defined before Theorem \ref{lipappr}.  During the proof, we shall
denote by $c$ a positive constant (which can change from a line to another)
depending only on $n$, $M$, $\omega$, $R$
and the constants given by Theorem \ref{lipappr}.

Arguing by contradiction, sequences $x_h$,
$\rho_h$, $v_h\in\mm{B_{\rho_h}(x_h)}$
and $A_h$ (affine $m$-planes) exist such that
\begin{equation}     \label{dec1}
\rho_h^{-m-2}\int_{B_{\rho_h}(x_h)}|A^\perp_h x|^2\, d\mu(v_h)=
{\eta_h'}^2\to 0,
\end{equation}
\begin{equation}     \label{dec2}
M\rho_h^{m+2}{\eta_h'}^2 \geq
\rho_h^2\Bigl[\E{v_h}{B_{\rho_h}(x_h)}+ \rho_h^m\sqrt{\omega(\rho_h)}\Bigr]
\end{equation}
and for every affine $m$-plane $S$ and $h\in{\bf N}$
\begin{equation}      \label{dec3}
\int_{B_{\beta\rho_h}(x_h)} |S^\perp x|^2\, d\mu(v_h) \geq
C\beta^{m+4}{\eta_h'}^2 \rho_h^{m+2},
\end{equation}
where the constant $C$ will be specified later (see (\ref{defC}) below).

By a rotation and a translation we can assume that $A_h=T=e_n^\perp$ for
any $h$. Moreover, translating $x_h$ in a direction parallel to $T$ we can
assume that $T(x_h)=0$.

By (\ref{dec1}), (\ref{dec2}) we infer that $\rho_h\to 0$. In particular,
$\rho_h\leq 8R$ for $h$ large enough and using Remark \ref{scaling} with
$\rho=\rho_h/8R$ and $x_0=0$ we have a new sequence $u_h\in
\mm{B_{8R}(y_h)}$
such that, for $\eta_h^2={\eta_h'}^2(8R)^{m+2}$
\begin{equation}     \label{dec5}
\int_{B_{8R}(y_h)}|T^\perp x|^2\, d\mu(u_h)=\eta_h^2\to 0,
\end{equation}
\begin{equation}     \label{dec6}
M\eta_h^2 \geq
\Bigl[\E{u_h}{B_{8R}(y_h)}+(8R)^m\sqrt{\omega(\rho_h)}\Bigr]
\end{equation}
and for every affine $m$-plane $S$ and $h\in{\bf N}$
\begin{equation}      \label{dec7}
\int_{B_{\beta 8R}(y_h)} |S^\perp x|^2\, d\mu(u_h) \geq
C\beta^{m+4}\eta_h^2 .
\end{equation}
By the tilt estimate (\ref{tiltest}) and (\ref{dec2}) we obtain
$$
\rho_h^{-m}\T{v_h}{B_{\tau\rho_h}(x_h)}\leq c(M,\tau){\eta_h'}^2 \qquad\forall
\,\tau\in(0,1)
$$
hence
\begin{equation}      \label{dec4}
\T{u_h}{B_{\tau 8R}(y_h)}\leq c(M,\tau)\eta_h^2 \qquad\forall
\,\tau\in(0,1).
\end{equation}
Finally, using (\ref{dec6})
we can estimate the deviation from minimality of $u_h$ in
$B_{8R}(y_h)$ as follows:
\begin{equation} \label{decz}
\dev{u_h}{B_{8R}(y_h)}=\frac{(8R)^m}{\rho_h^m}\dev{v_h}{B_{\rho_h}(x_h)}
\leq c\eta^4_h\to 0
\end{equation}
and we observe that the density lower bound of Theorem \ref{dgcl}
and (\ref{dec5}) easily imply that $y_h=T^\perp y_h$
tends to $0$ as $h\to +\infty$ (recall that $y_h\in K(u_h)$).

Now, we shall construct for every $h$ a Lipschitz continuous function $f_h$
whose graph approximates $K(u_h)$ according to Theorem \ref{lipappr}
(Step 1) and prove (Step 2) that a subsequence of
$g_h=\eta_h^{-1}f_h$ converges to a harmonic (Step 3) function $g$.
Using estimates on $g$ we
shall find (Step 4) for sufficiently large $h$
a $m$-plane $S_h$ violating (\ref{dec7}).

{\sc Step 1.}
For $h$ large enough $y_h$ belongs to $B_{R/16}$, hence $B_{7R}
\subset B_{8R}(y_h)$. Using (\ref{dec5}) we see that for $h$ large
enough (\ref{smallf}) holds and Theorem \ref{lipappr} gives a
1-Lipschitz function $f_h:T\to T^\perp$ such that
\begin{equation}          \label{dec8}
\sup |f_h| \leq c\eta_h^{2/(m+2)}
\end{equation}
and, setting
\begin{eqnarray*}
&X_h=B_R\cap \bigl[K(u_h)\setminus \Gamma(f_h)\bigr]\quad\quad
&F_h=B_R\cap \bigl[\Gamma(f_h)\cap K(u_h)\bigr]  \\
&Q_h=B^m_{R/4}\cap T\bigl[\Gamma(f_h)\setminus K(u_h)\bigr]
&E_h=B^m_{R/4}\setminus Q_h=T(F_h)\cap B^m_{R/4},
\end{eqnarray*}
the inequality
\begin{eqnarray}     \label{dec9}
\hh\bigl(Q_h\bigr)+\hh\bigl(X_h\bigr)&\leq& P\Bigl[\E{u_h}{B_{5R}} +
\int_{B_{5R}} \Vert S_x-T\Vert^2\, d\mu(u_h)\Bigr] \nonumber    \\
&\leq& c\eta^2_h
\end{eqnarray}
follows from the tilt estimate (\ref{dec4});
moreover, (\ref{dec6}) implies
\begin{equation}       \label{dec10}
\E{u_h}{B_{7R}} =o(\eta_h).
\end{equation}

{\sc Step 2.} Let us show that (up to subsequences) $g_h=\eta_h^{-1}f_h$
weakly converges in $W^{1,2}(B_{R/4}^m)$. For, recall that
$|\nabla f_h|\leq 1$ a.e. by
Rademacher's theorem. Thus, by (\ref{dec8}), (\ref{dec5}) and (\ref{dec9}):
\begin{eqnarray*}
\int_{B^m_{R/4}}|f_h(z)|^2\, dz &\leq& c\eta_h^{4/(m+2)}
\hh\bigl(Q_h\bigr)+ \int_{E_h} |f_h|^2\, dz \nonumber \\
&\leq&  c\eta_h^{2+4/(m+2)}+\int_{B_R}|T^\perp x|^2\, d\mu(u_h) \leq
o(\eta_h^2)+\eta_h^2,
\end{eqnarray*}
whence in particular
\begin{equation}      \label{defcbar}
\limsup_{h\to +\infty}\int_{B^m_{R/4}}|g_h(z)|^2\, dz\leq 1.
\end{equation}
Regarding the gradients, first remark that
\begin{equation}   \label{dec12}
\int_{Q_h}|\nabla f_h(z)|^2\, dz\leq c\eta_h^2
\end{equation}
by (\ref{dec9}), and then notice that for ${\cal H}^m$-a.e.
$x\in F_h$ the equality
$$
S_x={\rm span}\Bigl\{e_i+\frac{\partial f_h}{\partial z_i}e_n:\
i=1,\ldots,m\Bigr\}
$$
holds. A simple computations shows that
$$\Vert S_x-T\Vert^2=2\bigl(1-\nu_n\bigr)={2\over 1+\nu_n}(1-\nu_n^2)
\geq \frac{|\nabla f_h(z)|^2}{1+|\nabla f_h(z)|^2}\geq
\frac{1}{2}|\nabla f_h(z)|^2$$
where $\nu$ is the normal to the graph of $f$ at $x$.
We can then infer that (see (\ref{dec4}))
\begin{eqnarray}
\int_{E_h}|\nabla f_h(z)|^2\, dz &\leq& 2\int_{E_h}
\Vert S_{(z,f_h(z))}-T\Vert^2\, dz \nonumber \\
&\leq& 2\int_{B_R} \Vert S_x-T\Vert^2\, d\mu(u_h) \leq
c\eta_h^2.
\end{eqnarray}
Therefore, the sequence $g_h$ is bounded in $W^{1,2}(B_{R/4})$
and we can assume that it is weakly convergent to a function $g\in
W^{1,2}(B_{R/4})$.

{\sc Step 3.} In order to see that $g=\lim\eta_h^{-1}f_h$ is harmonic in
$B_{R/4}^m$, we shall show that
\begin{equation}        \label{dec14}
\lim_{h\to +\infty}\frac{1}{\eta_h} \int_{B_{R/4}^m}\langle\nabla f_h
,\nabla\psi \rangle\, dz=0\qquad\forall \psi\in C_0^1\bigr(B_{R/4}^m\bigr).
\end{equation}
Let us consider the vector field $\phi=(0,\ldots,0,\psi)$; since $\psi\in
C^1_0(B^m_{R/4})$ and (by Proposition \ref{altezza}) the maximum of
$|T^\perp x|$ on $\overline{B}_R\cap K(u_h)$ is infinitesimal, we can find
for sufficiently large $h$ a vector field $\hat{\phi}$ with compact
support in $B_{R/4}$ and coinciding with $\phi$ on $K(u_h)\cap B_R$
(just take $\hat{\phi}(z,y)=\phi(z)\chi(y)$ for a suitable $\chi$).

We use comparison functions $w_h(\tau_h(x))=u_h(x)$, where
$\tau_h(x)=x-\eta_h^2\hat{\phi}$.
By the definition of $\dev{u_h}{B_{R/4}}$ we get (see (\ref{t2}))
\begin{eqnarray}
&&\E{w_h}{B_{R/4}}-\E{u_h}{B_{R/4}}+
\hh\bigl(K(w_h)\bigr)- \hh\bigl(K(u_h)\bigr)
\nonumber  \\
&&+\dev{u_h}{B_{R/4}}\geq 0,
\end{eqnarray}
whence arguing as in the proof of Lemma \ref{tilt}
\begin{eqnarray*}
\int_{B_{R/4}}\divt\hat{\phi}\, d\mu(u_h) &\leq& \frac{\dev{u_h}{B_{R/4}}}
{\eta_h^2} \\
&&+\frac{1}{\eta_h^2}\Bigl[\E{w_h}{B_{R/4}}-\E{u_h}{B_{R/4}}\Bigr]
+c_1\eta_h^2
\end{eqnarray*}
(where $c_1$ has the same meaning as in (\ref{t1})).
By (\ref{decz}), (\ref{t4}) and (\ref{dec10}) we deduce
$$
\limsup_{h\to +\infty}\eta_h^{-1}\int_{B_{R/4}}\divt\phi\, d\mu(u_h) \leq 0.
$$
From this inequality and the analogous one obtained taking $\tau_h(x)=
x+\eta_h^2\hat{\phi}(x)$ we get
\begin{equation}          \label{dec16}
\lim_{h\to +\infty}\eta_h^{-1}\int_{B_{R/4}}\divt\phi\, d\mu(u_h) = 0.
\end{equation}
Now, since by (\ref{dec9})
$$
\int_{Q_h}\langle\nabla f_h ,\nabla\psi \rangle\, dz= o(\eta_h),
\qquad \int_{X_h}\divt\phi\,d\mu(u_h)=o(\eta_h)
$$
for every $\psi\in C_0^1\bigr(B_{R/4}^m\bigr)$, (\ref{dec14}) can be
deduced from (\ref{dec16}) and
\begin{equation}      \label{dec17}
\lim_{h\to +\infty}\frac{1}{\eta_h}\Bigl[
\int_{F_h}\divt\phi\,d\mu(u_h) -
\int_{E_h} \langle\nabla f_h ,\nabla\psi \rangle\, dz\Bigr]= 0.
\end{equation}
To prove (\ref{dec17}), notice that ${\cal H}^m$-a.e. in $F_h$,
denoting by $\nu$ the normal unit vector to $\Gamma(f_h)$
\begin{equation}           \label{dec18}
\divt\phi=\delta_n\psi= \frac{\partial\psi}{\partial x_n}- \langle\nabla
\psi,\nu \rangle\nu_n=\frac{\langle\nabla\psi,\nabla f_h\rangle}
{1+|\nabla f_h|^2}.
\end{equation}
Inserting (\ref{dec18}) in (\ref{dec17}) and using the equality
$T(F_h)\cap B^m_{R/4}=E_h$ we obtain
\begin{eqnarray*}
&&\frac{1}{\eta_h}\Bigl|\int_{E_h}
\langle\nabla f_h ,\nabla\psi \rangle\bigl(1-(1+|\nabla
f_h|^2)^{-1/2}\bigr)\, dz\Bigr|   \\
&&\leq \frac{1}{2\eta_h}\int_{E_h}|\nabla f_h|^2|\nabla\psi|\, dz
\leq\frac{1}{2\eta_h}\Vert\nabla\psi
\Vert_\infty\int_{B^m_{R/4}}|\nabla f_h|^2\, dz
\end{eqnarray*}
which is infinitesimal by (\ref{dec10}).

{\sc Step 4.} Let $\Gamma(n,R)$ be the constant
given by \cite[5.2.5]{fed} such that
\begin{equation}        \label{dec19}
\max\bigl\{|g(0)|, R|\nabla g(0)|\bigr\}\leq\Gamma \Bigl(\int_{B^m_{R/4}}
|g(z)|^2\, dz\Bigr)^{1/2}
\end{equation}
and
\begin{equation}        \label{dec20}
|g(z)-g(0)-\langle\nabla g(0),z\rangle| \leq \Gamma \Bigl(\int_{B^m_{R/4}}
|g(w)|^2\, dw\Bigr)^{1/2}|z|^2
\end{equation}
for any $z\in B^m_{R/8}$. We can now define the constant $C$ by
\begin{equation}        \label{defC}
C=2^{m+6}\Gamma^2\int_{B^m_{7R}} |z|^4\,dz.
\end{equation}
Define also
\begin{eqnarray*}
&Z_h=\{x\in B_{7\beta R}:|T^\perp x|>\eta_h^{1/(m+2)}\}
&L_h(z)=z+\eta_h\langle\nabla g(0),z\rangle e_n\\
&K_h(x)=L_h(Tx)+\eta_h g(0)e_n
&S_h=\ {\rm image\ of\ }K_h,
\end{eqnarray*}
and remark that
\begin{equation}       \label{dec21}
Z_h\cap {\rm spt}\mu(u_h)=\emptyset\qquad{\rm for\ }h{\rm \ large\ enough}.
\end{equation}
Indeed, let $x_0\in Z_h\cap {\rm spt}\mu(u_h)$ and $\tau=|T^\perp x_0|/2$;
then $\mu(u_h)(B_\tau(x_0))\geq\theta_\omega\tau^m$, whence arguing as in
the proof of Lemma \ref{altezza}
$$
\eta_h^2\geq\tau^2\theta_\omega\tau^m=\frac{\theta_\omega}{2^{m+2}}
|T^\perp x_0|^{m+2}> \frac{\theta_\omega}{2^{m+2}} \eta_h,
$$
a contradiction for $h$ large enough. Let $\gamma=2\beta$; we now want to
estimate
$$
\int_{B_{7\gamma R}}|S_h^\perp x|^2\, d\mu(u_h)
$$
to get a contradiction with (\ref{dec7}). For every $x\in B_R$ we have
$$
x-K_h(x)=x-Tx-\eta_h\bigl[ g(0)+\langle \nabla g(0),Tx\rangle\bigr]e_n
$$
whence by (\ref{defcbar}) and (\ref{dec19})
\begin{eqnarray}       \label{dec22}
|S_h^\perp x|&\leq& |x-K_h(x)|\leq |T^\perp x|+\eta_h\bigl[|g(0)|+ |\langle
Tx,\nabla g(0)\rangle |\bigr]
\nonumber \\
&\leq& |T^\perp x|+2\Gamma\eta_h \Bigl(\int_{B^m_{R/4}}|g(z)|^2\, dz
\Bigr)^{1/2}
\nonumber  \\
&\leq&|T^\perp x|+2\Gamma\eta_h.
\end{eqnarray}
Moreover, if in particular $x=(z,f_h(z))$, then
\begin{eqnarray*}
|x-K_h(x)|&=&|f_h(z)-\eta_h\bigl[ g(0)+\langle\nabla g(0),z\rangle\bigr]|
\\
&=& |f_h(z)-\eta_hg(z)+\eta_h\bigl[g(z)-g(0)-\langle \nabla
g(0),z\rangle\bigr]|
\end{eqnarray*}
whence, using (\ref{dec20}) and (\ref{defcbar}), for $|z|\leq R/8$ and
$x=(z,f_h(z))$ we obtain
\begin{equation}     \label{dec23}
|S_h^\perp x|\leq\eta_h\bigl[|g_h(z)-g(z)|+\Gamma |z|^2\bigr].
\end{equation}
Using (\ref{dec22}) and (\ref{dec23}) (recall that $7\gamma R<R/8$ because
$\gamma<1/56$) we get
\begin{eqnarray*}
&&\int_{B_{7\gamma R}}|S_h^\perp x|^2\, d\mu(u_h)\leq
2\F{u_h}{B_{7\gamma R}\cap X_h}+ 8\Gamma^2\eta_h^2\h{m}{X_h}   \\
&&+2\eta_h^2\int_{B_{7\gamma R}\cap E_h}\sqrt{1+|\nabla f_h|^2}
\bigl[|g_h(z)-g(z)|^2+\Gamma^2 |z|^4\bigr]\, dz.
\end{eqnarray*}
By virtue of $|\nabla f_h|\leq 1$ and (\ref{dec21})
\begin{eqnarray*}
&&\int_{B_{7\gamma R}}|S_h^\perp x|^2\, d\mu(u_h)\leq
\bigl[2\eta_h^{2/(m+2)}+8\Gamma^2\eta_h^2\bigr]\h{m}{X_h}  \\
&&+2\sqrt{2}\eta_h^2\int_{B^m_{R/4}}|g_h(z)-g(z)|^2\, dz +
2\sqrt{2}\eta_h^2\Gamma^2\int_{B^m_{7\gamma R}}|z|^4\, dz.
\end{eqnarray*}
Finally, since the first two terms in the above inequality are
$o(\eta_h^2)$, we get
$$
\limsup_{h\to +\infty}\beta^{-m-4}\eta_h^{-2} \int_{B_{7\gamma R}}|S_hx|^2\,
d\mu(u_h) \leq 2^{m+5}\sqrt{2}\Gamma^2\int_{B^m_{7R}}|z|^4\, dz < C,
$$
and this contradicts (\ref{dec7}) because $B_{7\gamma R}=B_{14 \beta R}$
contains $B_{8\beta R}(y_h)$ for $h$ large enough.
\qed

\begin{remark} \label{anyomega} {\rm
If we assume that $\omega(t)$ vanishes on some interval $(0,b)\subset
(0,+\infty)$, then Theorem \ref{decay} continues to hold with the
additional assumption $\rho<b$. In this case $u$ is a local minimizer,
according to Definition \ref{locmin}, in small balls.

Notice also that we won't use the full generality of Theorem \ref{decay}
to prove Theorem \ref{main} (see the next section). In \cite{afp},
coupling Theorem \ref{decay} with a decay lemma for the Dirichlet integral
we will prove that
the $L^{2,\lambda}$ assumption on $|\nabla u|$ can be
removed. The proof is based on a suitable choice of $M$ and on the
independence of the constant $C$ in (\ref{migliora}) on $M$.
}\end{remark}

\section{Proof of partial regularity}

In this section we prove Theorem \ref{main}. By (\ref{main1}) and the
$L^{2,\lambda}$ assumption on $|\nabla u|$ we can find $N>0$
such that
\begin{equation}     \label{smalldev}
\int_{B_\rho(x)}|\nabla u|^2 +
\rho^m\sqrt{\omega(\rho)}\leq N\rho^{m+\alpha}
\end{equation}
for any ball $B_\rho(x)\subset\Omega$ with $\rho\leq 1$ (recall that
$\alpha=\min\{\lambda-m,s\}$).

Since $\alpha\leq s<2$, we can choose
$\beta\in (0,1/112)$ such that $C\beta^2\leq\beta^\alpha$
(where $C$ is defined in (\ref{defC})).
By (\ref{smalldev}) and Theorem \ref{decay} with $M=1$ we infer the
existence of $\eps>0$ such that the function
$A(\rho)={\bf A}(x,\rho)$ satisfies the property
\begin{equation}\label{decaio}
A(\rho)\leq\eps\rho^{m+2},\,\,\, A(\rho)\geq N\rho^{m+2+\alpha}
\quad\Longrightarrow\quad
A(\beta\rho)\leq \beta^{m+2+\alpha} A(\rho)
\end{equation}
for any $x\in K(u)$ and any ball $B_\rho(x)\subset\Omega$.

\begin{lemma}[iteration] \label{iterazione}
Let $A(\rho)$ be a nondecreasing function satisfying
(\ref{decaio}). Then, the conditions
$$
\label{iter1} \frac{N\rho^{\alpha}}{\beta^{m+2+\alpha}}\leq\eps\quad\quad
A(\rho)\leq\eps\rho^{m+2}
$$
imply
\begin{equation}  \label{iter2}
A(\tau)\leq \beta^{-(m+2+\alpha)}\max\Bigl\{
\frac{N\tau^{m+2+\alpha}}{\beta^{m+2+\alpha}}, \eps\rho^{m+2}
\Bigl(\frac{\tau}{\rho}\Bigr)^{m+2+\alpha} \Bigr\}
\end{equation}
for every $\tau\in (0,\rho]$.
\end{lemma}
{\sc Proof.} Let $\rho_k=\rho\beta^k$; by the monotonicity of $A$, we need
only to show that
\begin{equation}    \label{iter3}
A(\rho_k)\leq \max\Bigl\{
\frac{N\rho_k^{m+2+\alpha}}{\beta^{m+2+\alpha}},\eps\rho^{m+2}
\Bigl(\frac{\rho_k}{\rho}\Bigr)^{m+2+\alpha} \Bigr\}
\end{equation}
for every $k\geq 0$. We shall prove (\ref{iter3}) by induction on $k$.
Remark that the inequality holds trivially for $k=0$, and assume that
(\ref{iter3}) holds; then (\ref{iter1}) implies
$$
\max\Bigl\{\frac{N\rho_k^{m+2+\alpha}}{\beta^{m+2+\alpha}},
\eps\rho^{m+2} \Bigl(\frac{\rho_k}{\rho}\Bigr)^{m+2+\alpha} \Bigr\}
\leq \eps\rho_k^{m+2},
$$
hence $A(\rho_k)\leq\eps\rho_k^{m+2}$. If
$A(\rho_{k+1}) \leq N \rho_{k+1}^{m+2+\alpha}\beta^{-m-2-\alpha}$ we are
done; otherwise
$$
A(\rho_k)\geq A(\rho_{k+1})\geq
\frac{N\rho_{k+1}^{m+2+\alpha}}{\beta^{m+2+\alpha}}=
N\rho_k^{m+2+\alpha}
$$
and we are in a position to apply (\ref{decaio}), getting
$$
A(\rho_{k+1})\leq\beta^{m+2+\alpha}A(\rho_k)
$$
which shows that (\ref{iter3}) holds with $k+1$ in place of $k$.
\qed

By Lemma \ref{iterazione} we obtain (for a suitable $\eps_0<1$
depending on $\eps$, $N$, $n$, $\alpha$) that the ratio
$$\frac{{\bf A}(y,\eta)}{\eta^{m+2+\alpha}}
\qquad y\in B_{\rho/2}(x)\cap K(u),\,\,0<\eta<\rho/2$$
is bounded for any $x$ satisfying (\ref{main2}). The $C^{1,\alpha/(m+2)}$
regularity
of $K(u)\cap B_{\rho/2}(x)$ follows by Theorem \ref{c1graf} and
Remark \ref{anchecosi}.

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\end{document}


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