\documentclass[twoside,11pt]{article} \message{ Emanuele Paolini: Regularity for minimal boundaries in $R^n$ with mean curvature in $L^n$} \message{ Manuscripta Mathematica MM-No.: 64 } %makefile:TEX(MYNAME) \usepackage[english]{babel} \usepackage{a4} \usepackage{latexsym} \usepackage{graphicx} \usepackage{pstricks} %\usepackage{pst-node} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newenvironment{proof} {% % \begin{list}% {}% {% %\setlength{\rightmargin}{\leftmargin}% }% % \item% \noindent% {\it Proof:}\\% }% {% \hfill $\Box$% % \end{list}% } \def\text#1{{\mathrm{#1}}} \newcommand{\R}{{\mathbf R}} %% numeri reali \newcommand{\C}{{\mathcal C}} %% funzioni continue \renewcommand{\H}{{\mathcal H}} %% misura di Hausdorff \renewcommand{\epsilon}{{\varepsilon}} \renewcommand{\phi}{{\varphi}} \renewcommand{\subset}{{\subseteq}} \renewcommand{\supset}{{\supseteq}} \renewcommand{\:}{{\!\,:\,}} \renewcommand{\d}{{\,\text{d}}} \newcommand{\diffsim}{\triangle} \newcommand{\rem}[1]{} \newcommand{\loc}{\mathrm{loc}} \def\bea{\begin{eqnarray*}} \def\eea{\end{eqnarray*}} \pagestyle{myheadings} \markboth{Regularity for minimal boundaries with mean curvature in $L^n$ }{Emanuele Paolini} \setlength{\voffset}{1 cm} \setlength{\oddsidemargin}{1.5cm} \setlength{\textwidth}{11.5cm} \setlength{\textheight}{19.5cm} \begin{document} \thispagestyle{empty} \sloppy{\mbox{} \vspace{2cm}\\ \begin{LARGE} {Regularity for minimal boundaries in $\R^n$ with mean curvature in $L^n$} \end{LARGE} \mbox{}\vspace{0.5 cm}\\ \begin{Large}Emanuele Paolini\end{Large} \mbox{}\vspace{0.1 cm}\\ {Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 PISA, Italy} \\ \\ \\} \begin{list} {} {\addtolength{\leftmargin}{0.0cm} \addtolength{\rightmargin}{1cm} } \item \noindent \begin{small} \\ Let $E_0\subset\R^n$ be a minimal set with mean curvature in $L^n$ that is a minimum of the functional $E\mapsto P(E,\Omega)+\int_{E\cap\Omega} H$, where $\Omega\subset\R^n$ is open and $H\in L^n(\Omega)$. We prove that if $2\le n\le 7$ then $\partial E_0$ can be parametrized over the $(n-1)$-dimensional disk with a $\C^{0,\alpha}$ mapping with $C^{0,\alpha}$ inverse. \end{small} \end{list} \linespread{1} % %% non indenta i paragrafi bensi' li separa un poco % \setlength{\parindent}{0pt} % \setlength{\parskip}{3pt} % \section*{Introduction} We are concerned with the study of the functional \[ %\label{eq:fun} F_H(E) = P(E,\Omega)+\int_{E\cap\Omega} H \] where $\Omega$ is an open set of $\R^n$, $H\in L^1(\Omega)$, $E$ is a \emph{Caccioppoli set} (i.e. the distributional derivative $D\chi_E$ of the characteristic function of $E$ is a Radon vector measure) and $P(E,\Omega)=|D\chi_E|$ (the total variation of $D\chi_E$) is the \emph{perimeter} of $E$ in $\Omega$. If $E$ is a local minimizer of $F_H$ (i.e. if $F_H(E)\le F_H(X)$ for all $X$ such that $E\diffsim X$\footnote{Here $E\diffsim X=(E\setminus X) \cup (X\setminus E)$ denotes the symmetric difference between $E$ and $X$.} has compact closure in $\Omega$), then we will say that \emph{$E$ is a minimal set with mean curvature $H$ in $\Omega$}. As a matter of fact, if $H$ and $\partial E$ are sufficiently regular near a point $x\in \partial E\cap \Omega$, then $-H(x)/(n-1)$ is the usual mean curvature of $\partial E$ in $x$. In 1974 U.~Massari \cite{Mas74,Mas75} proved that if $E$ is a minimal set with mean curvature $H\in L^p$ and $p>n$ then, up to a set of dimension not greater than $n-8$, $\partial E$ is a hypersurface of class $C^{1,\alpha}$ with $\alpha=(p-n)/4p$. Then, in 1987 E.~Barozzi, E.~H.~A.~Gonzales and I.~Tamanini \cite{BarGonTam87,BarTam88} proved that given any Caccioppoli set $E$ it is possible to find a suitable $H\in L^1$ such that $E$ is a minimal set with mean curvature $H$. Moreover they noticed that if $E$ is a minimal set with curvature $H\in L^p$ with $1\le p n$. Anyway in~\cite{GonMasTam93} many properties for minima are given even in the case $H\in L^n$. In Theorem~\ref{teo:spirale} we summarize the properties we need in this paper. Given $X$ a subset of $\R^n$ and $\rho>0$ we denote by \[ (X)_\rho=\{y\in\R^n\: \exists x\in X\ |x-y|\le \rho\} \] the closed $\rho$-neighbourhood of $X$. \begin{definition}\label{def:reif} Let $S$ be a closed set of $\R^n$, $x_0\in S$, $R>0$, $\epsilon>0$, and $m\le n$ an integer. We say that $S$ satisfies \emph{$(\epsilon,R,m)$-Reifenberg property} in $x_0$ if given any ball $B(x,r)\subset B(x_0,R)$ with $x\in S$ there exists an affine $m$-plane $\Sigma$ through $x$ such that \bea S\cap B(x,r)&\subset& \left(\Sigma\cap B(x,r)\right)_{\epsilon r},\\ \Sigma\cap B(x,r) &\subset& \left(S\cap B(x,r)\right)_{\epsilon r}. \eea \end{definition} The following theorem can be found in~\cite[Chapter~4, page~64]{Rei60}. \begin{theorem}[topological disk]\label{teo:Reif} Given $0<\alpha<1$ there exists $\epsilon_0>0$ such that if $S$ is a closed set in $\R^n$ satisfying the $(\epsilon,R,m)$-Reifenberg property in $x_0\in S$ for some $R>0$, some integer $0\le m \le n$ and some $0<\epsilon<\epsilon_0$ then there exists a neighbourhood of $x_0$ in $S$ which is homeomorphic to the $m$-dimensional disk through an homeomorphism $\tau$ such that both $\tau$ and $\tau^{-1}$ are $\alpha$-H{\"o}lder maps. \end{theorem} \begin{theorem}\label{teo:spirale} Let $E\subset\R^n$ be a minimal set with mean curvature $H$ in $\Omega$, with $H\in L^n(\Omega)$. Then we can find a closed set $E_0$ which differs from $E$ by a set of Lebesgue measure zero, which is a minimal set with mean curvature $H$ in $\R^n$ and satisfies the following properties: \begin{enumerate} \item $\partial E_0$ is $(n-1)$-rectifiable and $P(E_0,A)=\H^{n-1}(A\cap\partial E_0)$ for any open set $A\subset\R^n$; \item given a sequence of points $x_k\to x_0$ with $x_0,x_k\in\partial E_0\cap \Omega$, and a sequence $\lambda_k\to \infty$ ($\lambda_k>0$) there exist a subsequence $k_j$ and a minimal set $E_\infty$ with mean curvature $0$ in $\R^n$ such that \[ \lim_{j\to\infty}\lambda_{k_j}(E_0-x_{k_j})= E_\infty \] in $L^1_\loc(\R^n)$, and \[ \lim_{j\to\infty} P(\lambda_{k_j}(E_0-x_{k_j}),B(0,1)) = P(E_\infty,B(0,1)). \] Moreover if $n\le 7$ then $E_\infty$ is an half-space. \end{enumerate} \end{theorem} \begin{proof} See Theorem~1.1 and Theorem~1.2 of~\cite{GonMasTam93}. \end{proof} The following lemma gives a link between minimal surfaces and surfaces with prescribed mean curvature, specifing in which sense the latter may be called \emph{quasi-minimal surfaces.} % \begin{lemma}\label{lemma:omegamin} Let $H\in L^n(\Omega)$ and $E_0$ be a minimal set with mean curvature $H$ in $\Omega$. Then, there exists an increasing function $\omega\:\R^+\to\R^+$ with \[ \lim_{r\to 0^+}\omega(r)=0 \] such that if $E$ is a Caccioppoli set and $\overline{E\diffsim E_0} \subset B(x,r)\subset \Omega$ then \[ P(E_0,B(x,r))\le P(E,B(x,r))+ r^{n-1} \omega(r). \] If only $E\diffsim E_0\subset B(x,r)\subset\Omega$ then \[ P(E_0,\overline{B(x,r)})\le P(E,\overline{B(x,r)}) +r^{n-1}\omega(r). \] \end{lemma} \begin{proof} If $\overline{E\diffsim E_0}\subset B(x,r)$, in view of the locality of perimeter, one can find an open set which contains $\Omega\setminus B(x,r)$ such that the measures $P(E_0,\cdot)$ and $P(E,\cdot)$ coincide on it. So $P(E_0,\Omega\setminus B(x,r))=P(E,\Omega\setminus B(x,r))$. From the minimality of $E_0$ we obtain \[ P(E_0,B(x,r))+\int_{B(x,r)\cap E_0} H \le P(E,B(x,r))+\int_{B(x,r)\cap E} H \] hence \[ P(E_0,B(x,r)) \le P(E,B(x,r))+\int_{B(x,r)}|H| \] and from H{\"o}lder inequality we get \[ \int_{B(x,r)}|H| \le ||H||_{L^n(B(x,r))} |B(x,r)|^{n-1\over n} = \gamma_n^{n-1\over n} r^{n-1} ||H||_{L^n(B(x,r))} \] (in the whole paper we denote by $\gamma_k$ the measure of the $k$-dimensional unit ball and use the notation $|X|=\H^n(X)$). So if we set \begin{equation}\label{eq:omega} \omega(r)=\sup\{\gamma_n^{n-1\over n} ||H||_{L^n(B(x,\rho))}\: B(x,\rho)\subset\Omega,\ \rho\le r\} \end{equation} using $H\in L^n$ we obtain, as needed, that $\omega$ is increasing and with limit $0$ as $r\to 0$. If $E\diffsim E_0\subset B(x,r)$ we know that $P(E_0,\cdot)$ and $P(E,\cdot)$ coincide on the open set $\Omega\setminus \overline{B(x,r)}$. With similar considerations as before, we obtain the second statement. \end{proof} \section{Technical Lemmas} Let us fix some notations. Let $\Omega\subset\R^n$, $H\in L^n(\Omega)$, $E_0$ be a minimum of $F_H$ satisfying the properties given in Theorem~\ref{teo:spirale} and $\omega$ the function described in Lemma~\ref{lemma:omegamin}. Let $2\le n\le 7$ and $m=n-1$. We adopt the following notations \bea S_0 &=& \partial E_0;\\ \phi(x,r) &=& \H^m(S_0\cap B(x,r));\\ \psi(x,r) &=& \int_0^r \H^{m-1}(S_0\cap\partial B(x,t))\d t. \eea Here by $\H^k$ we mean the $k$-dimensional spherical Hausdorff measure that is \[ \H^k(X) = \lim_{\delta\to 0} \H_\delta^k(X) \] where \[ \H^k_\delta(X) = \inf\left\{\sum_i \gamma_m r_i^m\: \{B(x_i,r_i)\}_i\ \mbox{is a covering of}\ X,\ r_i<\delta \right\}. \] Notice that Theorem~\ref{teo:spirale} was proved for Hausdorff measure, but it also holds for spherical Hausdorff measure as these two measures coincide on rectifiable sets (see \cite[3.2.26]{Fed69}). On the other hand, for a general Caccioppoli set $E$ we will only use the following property \begin{equation}\label{eq:diseq} P(E,A)\le \H^m(\partial E\cap A) \end{equation} which holds for Hausdorff measure and then also for spherical Hausdorff measure, the second being bigger than the first (see \cite[2.10.2]{Fed69}). \begin{lemma}\label{lemma:coarea} Let $S$ be a closed set of $\R^n$ and denote by $S_t$ the set of the points of $S$ which have distance $t$ from a given $k$-dimensional plane $\Sigma$ where $0\le k \le n$. Then, for every integer $m\le n$, \begin{equation}\label{eq:coarea} \int_0^\infty \H^{m-1}(S_t)\d t \le \H^m(S) \end{equation} \end{lemma} \begin{proof} See~\cite[Theorem~10.2.3]{Mor66}. \end{proof} \begin{lemma}\label{lemma:cono} Given $x\in\Omega$, for almost every $r$ with $B(x,r)\subset\Omega$ and every $z\in B(x,r)$ there exists a set $E$ such that \[ E\diffsim E_0 \subset B(x,r) \] and such that for every open set $A\subset\R^n$ the following estimate holds \bea P(E,\overline{B(x,r)}) &\le& {r_1\over m} \H^{m-1}(S_0\cap\partial B(x,r)\cap A)\\ \mbox{}&&+ {r_2\over m} \H^{m-1}(S_0\cap\partial B(x,r)\setminus A) \eea where \bea r_1&=&\sup\{|y-z|\: y\in S_0\cap\partial B(x,r)\cap A\};\\ r_2&=&\sup\{|y-z|\: y\in (S_0\cap\partial B(x,r))\setminus A\}. \eea In particular, if we choose $A=\R^n$ and $z=x$ we obtain \[ P(E,\overline{B(x,r)}) \le {r\over m} \H^{m-1}(S_0\cap\partial B(x,r)). \] \end{lemma} \begin{proof} As $S_0$ is an $m$-rectifible set, for almost every $r$ the set $S_0\cap\partial B(x,r)$ is $(m-1)$-rectifiable. Let $C=C(z,E_0\cap\partial B(x,r))$ be the cone over $E_0\cap\partial B(x,r)$ with vertex $z$ and let $E=(E_0\setminus B(x,r))\cup C$. We can easily find that \[ \partial E = (S_0 \setminus B(x,r)) \cup C(z,S_0\cap\partial B(x,r)) \] so $\partial E$ is an $m$-rectifiable set with finite $m$-dimensional measure, that is $E$ is a Caccioppoli set. Moreover, from~(\ref{eq:diseq}), \bea \lefteqn{P(E,\overline{B(x,r)})}\\ &\le& \H^m(\partial E\cap\overline{B(x,r)}) \\ &=& \H^m\left(C(z,S_0\cap\partial B(x,r))\right)\\ &=& \H^m\left(C(z,S_0\cap\partial B(x,r)\cap A)\right)\\ &&\mbox{}+ \H^m\left(C(z,(S_0\cap\partial B(x,r))\setminus A)\right)\\ &\le& {r_1\over m} \H^{m-1}(S_0\cap\partial B(x,r)\cap A) +{r_2\over m} \H^{m-1}((S_0\cap\partial B(x,r))\setminus A) \eea where in the last inequality we use the fact that if $X\subset B(z,r)$ is an $(m-1)$-rectifiable set then \[ \H^{m}\left(C(z,X)\right) \le {r\over m}\H^{m-1}(X). \] \end{proof} \begin{lemma}\label{lemma:1*} If $x\in \Omega$ and $B(x,r)\subset\Omega$ then \[ \psi(x,r)\le\phi(x,r) \] and for almost every $r$ such that $B(x,r)\subset\Omega$ \[ \phi(x,r)\le {r\H^{m-1}(S_0\cap\partial B(x,r))\over m}+ r^m \omega(r). \] \end{lemma} \begin{proof} First inequality is given by Lemma~\ref{lemma:coarea}. To prove the second inequality we consider the set $E$ given by the second part of Lemma~\ref{lemma:cono}. Then \[ P(E,\overline{B(x,r)}) \le {\frac r m}\H^{m-1}(S_0\cap\partial B(x,r)) \] and, on the other hand, from Lemma~\ref{lemma:omegamin} we get \bea \phi(x,r)&=&\H^m(S_0\cap B(x,r))=P(E_0,B(x,r))\\ &\le& P(E,\overline{B(x,r)})+r^m\omega(r). \eea \end{proof} \begin{lemma}\label{lemma:omegamul} If $B(x,r)\subset \Omega$ and $E=E_0\setminus B(x,r)$ then, for almost every $r$, \[ P(E_0,\overline{B(x,r)}) \le {1+c_n\omega(r)\over 1-c_n\omega(r)} P(E,\overline{B(x,r)}). \] where $c_n$ is a constant depending only on $n$. \end{lemma} \begin{proof} As in Lemma~\ref{lemma:omegamin} we have (being $E=E_0\setminus B(x,r)$) \[ P(E_0,\overline{B(x,r)}) \le P(E,\overline{B(x,r)}) + ||H||_{L^n(B(x,r))}|E_0\cap B(x,r)|^{n-1\over n}. \] From the isoperimetric inequality we obtain \bea |E_0\cap B(x,r)| &\le& c'_n P(E_0\cap B(x,r),\R^n)^{n\over n-1}\\ &\le& c'_n\left(P(E_0,\overline{B(x,r)})+P(E,\overline{B(x,r)})\right)^{n\over n-1}. \eea So \bea \lefteqn{ ||H||_{L^n(B(x,r))}|E_0\cap B(x,r)|^{n-1\over n}}\\ &\le& ||H||_{L^n(B(x,r))}c''_n \left( P(E_0,\overline{B(x,r)})+P(E,\overline{B(x,r)})\right) \eea and by the definition of $\omega$ we have \bea P(E_0,\overline{B(x,r)}) &\le& P(E,\overline{B(x,r)}) \\ \mbox{}&&+ c_n\omega(r) \left(P(E_0,\overline{B(x,r)})+P(E,\overline{B(x,r)})\right) \eea which is the claim of the lemma. \end{proof} \begin{lemma}\label{lemma:spirale} Let $n\le 7$ and $x_0\in S_0\cap \Omega$. Given $\alpha>0$ there exists $r_0=r_0(\alpha)>0$ such that if $x\in S_0$, $B(x,r)\subset B(x_0,r_0)$ then \[ 1-\alpha \le {\phi(x,r)\over \gamma_m r^m} \le 1+\alpha. \] In particular every point of $S_0\cap\Omega$ has $m$-density $1$ in $S_0$. \end{lemma} \begin{proof} Suppose there exist $\alpha>0$, a sequence of points $x_k\to x_0$, $x_k\in S_0$ and a sequence of radii $r_k\to 0$ such that \begin{equation}\label{eq:spirale} \left|\frac{\H^m(\partial E_0\cap B(x_k,r_k))}{\gamma_m r_k^m}-1 \right| \ge \alpha. \end{equation} By Theorem~\ref{teo:spirale}, up to a subsequence, the sequence $(1/ r_k)(E_0-x_k)$ converges to an halfspace $E_\infty$. So we obtain \bea \gamma_m &=&P(E_\infty,B(0,1)) = \lim_{k\to\infty} P((1/r_k)(E_0-x_k),B(0,1)) \\ &=&\lim_{k\to\infty} {1\over r_k^m}P(E_0,B(x_k,r_k)) =\lim_{k\to\infty} {\H^m(\partial E_0\cap B(x_k,r_k))\over r_k^m} \eea which is a contradiction with~(\ref{eq:spirale}). \end{proof} \begin{lemma}\label{lemma:schiacciata} Let $E$ be a closed Caccioppoli set in $\Omega$, $B(x,r)\subset\Omega$, $\Sigma$ an affine hyperplane through $x$, $\pi_\Sigma$ the orthogonal projection onto $\Sigma$ and $0<\epsilon<{1\over 2}$. If $\partial E\cap \overline{B(x,r)}$ is contained in the interior of $(\Sigma)_{\epsilon r}$, then either \begin{equation}\label{eq:schiacciata1} \pi_\Sigma(\partial E\cap B(x,r))\supset\Sigma\cap B(x,(1-\epsilon)r) \end{equation} or, if $E'=E\setminus B(x,r)$, \begin{equation}\label{eq:schiacciata2} P(E',\overline{B(x,r)}) \le 4\epsilon r\H^{m-1}(\partial E\cap\partial B(x,r)). \end{equation} \end{lemma} \begin{proof} Let $x_1,\ldots,x_n$ be a system of coordinates such that $\Sigma=\{x_n=0\}$ and define $E^+=\{x_n\ge\epsilon r\}\cap\overline{B(x,r)}$ and $E^-=\{x_n\le -\epsilon r\}\cap\overline{B(x,r)}$. From the assumptions on $E$ we know that $\partial E$ itersects neither $E^+$ nor $E^-$. So, being $\partial E$ closed and $E^+$, $E^-$ compact sets, we conclude that $\partial E$ has positive distance from $E^+$ and $E^-$. So one of the following statements must hold: \begin{enumerate} \item\label{poss1} $E\supset E^-$, $E\cap E^+=\emptyset$; \item\label{poss2} $E\supset E^+$, $E\cap E^-=\emptyset$; \item\label{poss3} $ E\cap E^+=\emptyset$, $E\cap E^-=\emptyset$; \item\label{poss4} $ E\supset E^+\cup E^-$. \end{enumerate} \emph{Suppose that~\ref{poss1} is true,} than we claim that~(\ref{eq:schiacciata1}) holds. Let $\Sigma^+=\{x_n=\epsilon r\}\cap\overline{B(x,r)}$ and $\Sigma^-=\{x_n=-\epsilon r\}\cap\overline{B(x,r)}$. Given $y\in\Sigma\cap B(x,(1-\epsilon)r)$ denote by $y^+$ and $y^-$ respectively the intersections of $\pi^{-1}_\Sigma(x)$ with $\Sigma^+$ and $\Sigma^-$. Since $y^-\in E$ while $y^+\not\in E$ there must certainly be a point in the segment $[x^-,x^+]$ which belongs to $\partial E$ (and that is contained in $B(x,r)$) so $\pi_\Sigma(\partial E\cap B(x,r))\ni y$. \emph{If statement~\ref{poss2} holds} with a symmetry with respect to $\Sigma$ we obtain~(\ref{eq:schiacciata1}) again. \emph{Suppose that~\ref{poss3} is true}. This time we claim that~(\ref{eq:schiacciata2}) holds. Let $E'=E\setminus B(x,r)$. Obviously $E'$ is a closed Caccioppoli set, $E\diffsim E'\subset B(x,r)$ and we get \begin{equation}\label{eq:sch1} \partial E'\cap\overline{B(x,r)} =\partial E'\cap\partial B(x,r) = E \cap \partial B(x,r). \end{equation} Consider now the projection \[ \pi\:\partial B(x,r)\cap(\Sigma)_{\epsilon r}\to \partial B(x,r)\cap \Sigma \] which maps a point $z$ to the point of $\partial B(x,r)\cap\Sigma$ with minimal distance from $z$. Then we have \[ \pi(E\cap \partial B(x,r))\subset \pi(\partial E\cap\partial B(x,r)) \] so that \begin{eqnarray}\label{eq:sch2} \H^m(E\cap\partial B(x,r)) &\le& \H^m(\pi^{-1}(\pi(\partial E\cap\partial B(x,r)))) \\ &\le& 2\epsilon r\H^{m-1}(\pi(\partial E\cap \partial B(x,r)))\nonumber \\ &\le& 4\epsilon r\H^{m-1}(\partial E\cap\partial B(x,r)).\nonumber \end{eqnarray} We conclude, using~(\ref{eq:sch1}) and~(\ref{eq:sch2}), \bea P(E',\overline{B(x,r)}) &\le& \H^m(\partial E'\cap\overline{B(x,r)})\\ &=& \H^m(E\cap \partial B(x,r)) \le 4\epsilon r \H^{m-1}(\partial E\cap\partial B(x,r)). \eea \emph{If statement~\ref{poss4} holds} the proof is that of case~\ref{poss3} where instead of $E$ we consider its complementary set. \end{proof} \begin{lemma}\label{lemma:7*} Let $S_0$ be a closed set, $\theta>0$, $e>0$, $\alpha>0$ and $0<\epsilon< 1-\cos\theta$. Let $B(x,r)$ be a ball such that given any arbitrarily small $\delta>0$ there exist a family of disjoint balls $\{B(x_i,r_i)\}$ and a family of $m$-dimensional affine planes $\{\Sigma_i\}$ with $x_i\in\Sigma_i$ such that every plane $\Sigma_i$ makes an angle greater than $\theta$ with the segment $[x_i;x]$ and, moreover, the following properties hold \begin{eqnarray*} B(x_i,r_i)&\subset& B(x,r) \label{eq:7*15} \\ r_i &<& \delta \label{eq:7*16}\\ \phi(x_i,r_i)&\ge&(1-\alpha)\gamma_m r_i^m \label{eq:7*17}\\ \sum_i \gamma_m r_i^m &>& e \label{eq:7*18}\\ S_0\cap B(x_i,r_i)&\subset& (\Sigma_i)_{\epsilon r_i}. \label{eq:7*19} \end{eqnarray*} Then \[ \psi(x,r)\le \frac{\phi(x,r)}{1-\alpha}-e\left({1-\cos\theta-\epsilon \over 2}\right)^m. \] \end{lemma} \begin{proof} (See~{\cite[Lemma $7^*$]{Rei60}}). Suppose $\Sigma$ is an $m$-dimensional plane and that $y_1$ and $y_2$ are two points of $\Sigma$ at a distance $l$ apart: define $d(l,\rho,t)$ and $\sigma(l,\rho,t)$ to be respectively the diameter and the $\H^{m-1}$ measure of the set $\Sigma\cap B(y_1,\rho)\cap \partial B(y_2,t)$. It follows immediately from the definition of the symbols concerned that \begin{equation}\label{eq:7*21} \int_0^r \sigma(l,\rho,t)\d t = \H^m(\Sigma\cap B(y_1,\rho)\cap B(y_2,r)), \end{equation} and that for $\delta > \rho$ \begin{equation}\label{eq:7*22} H_\delta^{m-1} B(y_1,\rho)\cap\partial B(y_2,t) \le \gamma_{m-1}\big(\frac 1 2 d(l,\rho,t)\big)^{m-1} \le \sigma(l,\rho,t). \end{equation} From the definition of spherical Hausdorff measure we can choose a set of balls $B(x_j',r_j')$ such that for all $j$ \bea r_j'&<&\delta \\ S_0\cap B(x,r)\setminus \bigcup_i (S_0\cap B(x_i,r_i)) &\subset& \bigcup_i B(x_j',r_j') \\ \H^m\Big((S_0\cap B(x,r))\setminus \bigcup_i(S_0\cap B(x_i,r_i))\Big) + \delta &\ge& \sum_j \gamma_m {r_j'}^m. \eea By~(\ref{eq:7*22}) and~(\ref{eq:7*21}) for any $m$-plane $\Sigma\supset\{x,x_j'\}$ \begin{eqnarray}\label{eq:7*25} \lefteqn{\int_0^r \H^{m-1}_\delta(S_0\cap\partial B(x,t)\cap B(x_j',r_j')) \d t}\\ &\le& \int_0^r \H^{m-1}_\delta(\partial B(x,t)\cap B(x_j,r_j')) \d t \nonumber\\ &\le& \int_0^r \sigma(|x-x_j'|,r_j',t)\d t \nonumber\\ &=& \H^m (\Sigma\cap B(x_j',r_j')\cap B(x,r)) \nonumber\\ &\le& \H^m(\Sigma\cap B(x_j',r_j')) = \gamma_m {r_j'}^m.\nonumber \end{eqnarray} Consider now any $i$ such that $|x_i-x|\ge r_i$. If we let $R=|x-x_i|-r_i\cos\theta -r_i \epsilon$ we obtain that $S_0\cap B(x_i,r_i)\cap B(x,R)=\emptyset$ so, for any $m$-plane $\Sigma\supset\{x,x_i\}$ and by~(\ref{eq:7*22}) and~(\ref{eq:7*21}), we obtain \bea \lefteqn{\int_0^r \H^{m-1}_\delta(S_0\cap\partial B(x,t)\cap B(x_i,r_i)) \d t} \\ &\le& \int_R^r \H^{m-1}_\delta(\partial B(x,t)\cap B(x_i,r_i))\d t\\ &\le& \int_R^r \H^{m-1}(\partial B(x,t)\cap B(x_i,r_i)) \d t\\ &=& \int_R^r \sigma(|x-x_i|,r_i,t)\d t. \eea As the set $B(x,R)\cap B(x_i,r_i)$ contains a ball of radius $\frac 1 2 (1-\cos\theta-\epsilon) r_i$ we get \bea \int_0^R \sigma(|x-x_i|,r_i,t)\d t &=& \H^m(\Sigma\cap B(x_i,r_i)\cap B(x,R))\\ &\ge& \gamma_m\left(\frac{1-\cos\theta-\epsilon}{2}\right)^m r_i^m \eea and, on the other hand, \[ \int_0^r \sigma(|x-x_i,r_i,t)\d t = \H^m(\Sigma\cap B(x_i,r_i)\cap B(x,t)) \le \gamma_m r_i^m \] so that \bea \lefteqn{\int_0^r \H^{m-1}_\delta (S_0\cap\partial B(x,t)\cap B(x_i,r_i))\d t}\\ &\le& \gamma_m r_i^m-\gamma_m\left(\frac{1-\cos\theta -\epsilon}{2} \right)^m r_i^m. \eea Since the balls $B(x_i,r_i)$ are disjoint, there is at most one $i=i_0$ such that $|x-x_i|0$, there exist positive numbers $R_0=R_0(\omega,\epsilon,n)$, $\lambda=\lambda(n,\epsilon)$, $\alpha=\alpha(n,\epsilon)$ such that for all $0}(0,0)(5;120) \uput[dl](2.5;120){$r$} \pscircle(0,8){8} \psdots(0,8) \uput[r](0,8){$y$} \psline[linewidth=0.5pt]{->}(0,8)(-8,8) \uput[d](-4,8){$r_2$} \psline[linewidth=0.5pt](0,8)(0,-7) \psdots(5;18) \uput[dr](5;18){$p$} \psline[linewidth=2pt](-10,0)(10,0) \uput[d](-7,0){$\Sigma$} \rput(7,1.2){$X$} \rput(7,-1.2){$Y$} \psdots(0,-1.5) \uput[dr](0,-1.5){$z$} \psline(0,0)(10;200) \psline(0,0)(10;-20) \psarc{<-}(0,0){1}{200}{270} \uput[dl](0.2;235){$_\theta$} \rput(6;235){$C_\theta$} \psdots(5;-20) \uput[ur](5;-20){$q$} \pscurve[linewidth=2pt](-10,1)(-7,1)(5;175)(-0.2,0)(0,0)(0.3,-0.1)(2,-3)(5;-25)(7,-2)(10,-1) \uput[u](-7,1){$S_0$} \psline(0,-1.5)(5;-25) \uput[d](5.5;-25){$l_\theta(r)$} \psline(0,-1.5)(5;175) \uput[ul](5;175){$l(r)$} \endpspicture % } \caption{The surface $S_0$, the plane $\Sigma$ and the Cone $C_\theta$.} \label{fig:Ctheta} \end{figure*} Let $\Sigma$ be the $m$-dimensional plane tangent to $B(y,r_2)$ in $x^*$ (see Figure~\ref{fig:Ctheta}), consider the cone $C_\theta$ with vertex $x^*$ and amplitude $\theta$ defined as follows \[ C_\theta=\{v\in\R^n\: (v-x^*)\cdot(x^*-y)\ge |v-x^*||x^*-y| \cos\theta\} \] and take a point $z\in C_\theta$ lying on the line $x^*y$ and at a distance $d$ from $x^*$. For a radius $r\in[r_4,r_3]$ define \bea l(r)&=&S_0\cap \partial B(x^*,r);\\ l_\theta(r)&=& S_0\cap\partial B(x^*,r)\cap C_\theta; \\ d_1(r)&=& \sup\{|z-x|\: x\in l(r)\setminus l_\theta(r)\};\\ d_2(r)&=& \sup\{|z-x|\: x\in l_\theta(r)\}. \eea For almost every $r\in[r_4,r_3]$, in view of Lemma~\ref{lemma:cono} (with $A=\R^n\setminus C_\theta$), we can find a Caccioppoli set $E$ such that \[ P(E,\overline{B(x^*,r)}) \le \frac{d_1(r)}{m} \H^{m-1}(l(r)\setminus l_\theta(r)) + \frac{d_2(r)}{m} \H^{m-1}(l_\theta(r)) \] and from Lemma~\ref{lemma:omegamin} we obtain \bea \lefteqn {m\H^m(S_0\cap B(x^*,r))=mP(E_0,B(x^*,r))}\\ &\le& m P(E,B(x^*,r)) + m r^m \omega(r) \\ &\le& d_1(r)\H^{m-1}(l(r)\setminus l_\theta(r)) +d_2(r)\H^{m-1}(l_\theta(r)) + m r^m \omega(r). \eea Given any point $p\in\partial B(y,r_2)\cap \partial B(x^*,r)$ and $q\in \partial C_\theta \cap \partial B(x^*,r)$, with simple geometric considerations we deduce \bea d_1(r) &\le& |p-z| = \sqrt{r^2+d^2+d\frac{r^2}{r_2}} \le r\sqrt{1+\frac{d^2}{r_4^2}+\frac{d}{r_2}} \\ &=& r\sqrt{1+2\alpha} \le r(1+\alpha), \\ d_2(r) &\le& |q-z| = \sqrt{r^2-2dr\cos\theta+d^2} \le r \sqrt{1-2\frac{d}{r_3}\cos\theta+\frac{d^2}{r_4^2}} \\ &=& r\sqrt{1-2\alpha^{\frac 5 6}+\alpha} \le r\sqrt{1-\alpha^{\frac 5 6}} \le r(1-\frac 1 2 \alpha^{\frac 5 6}), \eea so that we obtain \bea \lefteqn{m\H^m(S_0\cap B(x^*,r))- m r^m \omega(r)} \\ &\le& r (1+\alpha)\H^{m-1}(l(r)\setminus l_\theta(r)) + r (1-\frac 1 2\alpha^{\frac 5 6})\H^{m-1}(l_\theta(r)) \\ &=& r(1+\alpha)\H^{m-1}(l(r)) -r(1+\alpha-1+\frac{1}{2}\alpha^{\frac 5 6})\H^{m-1}(l_\theta(r))\\ &\le& r(1+\alpha)\H^{m-1}(l(r)) -\frac{1}{2}r\alpha^{\frac 5 6}\H^{m-1}(l_\theta(r)). \eea Dividing by $r$, integrating and using hypothesis~(\ref{eq:beta}) we obtain \bea \lefteqn{\frac 1 2 \alpha^{\frac 5 6}\int_{r_4}^{r_3}\H^{m-1}(l_\theta(r))\d r} \\ &\le& (1+\alpha)\int_{r_4}^{r_3}\H^{m-1}(l(r))\d r\\ &&\mbox{}- \int_{r_4}^{r_3} \frac m r \Big(\H^m(S_0\cap B(x^*,r))-r^m\omega(r)\Big) \d r \\ &\le& (1+\alpha)\Big((1+\alpha)\gamma_m r_3^m - (1-\alpha)\gamma_m r_4^m\Big)\\ &&\mbox{} - (1-\alpha - \frac {\omega(R_0)}{\gamma_m}) \gamma_m(r_3^m - r_4^m) \\ &=& (1+\alpha)(1+\alpha-\alpha^{\frac m 6} + \alpha ^{1+\frac m 6})\gamma_m r_3^m \\ &&\mbox{}-(1-\alpha-\frac{\omega(R_0)}{\gamma_m}-\alpha^{\frac m 6} + \alpha^{1+\frac m 6} + \alpha ^{\frac m 6}\frac{\omega(R_0)}{\gamma_m}) \gamma_m r_3^m \\ &=& (3\alpha + \alpha^2+\alpha^{2+\frac m 6} +\frac{\omega(R_0)}{\gamma_m}(1-\alpha^{\frac m 6}) -\alpha^{1+\frac m 6})\gamma_m r_3^m \le 6 \alpha \gamma_m r_3^m \eea that is \[ \int_{r_4}^{r_3}\H^{m-1}(l_\theta(r))\d r \le 12 \alpha^{\frac{1}{6}}\gamma_m r_3^m. \] So, by Lemma~\ref{lemma:coarea} and~(\ref{eq:beta}) \begin{eqnarray} \label{eq:ltheta} \lefteqn{\H^m(S_0\cap B(x^*,r_3)\cap C_\theta)}\\ &=& \H^m(S_0\cap B(x^*,r_3)) - \H^m((S_0\cap B(x^*,r_3))\setminus C_\theta) \nonumber \\ &\le& (1+\alpha)\gamma_m r_3^m -\int_0^{r_3}\H^{m-1}(l(r)\setminus l_\theta(r))\d r\nonumber \\ &=& (1+\alpha)\gamma_m r_3^m -\int_0^{r_3} \H^{m-1}(l(r)) \d r \nonumber \\ &&\mbox{}+\int_{r_4}^{r_3} \H^{m-1}(l_\theta(r))\d r + \int_0^{r_4} \H^{m-1}(l_\theta(r))\d r \nonumber\\ &\le& (1+\alpha)\gamma_m r_3^m- (1-\alpha)\gamma_m r_3^m + 12\alpha^{\frac 1 6}\gamma_m r_3^m + \alpha^{\frac m 6}(1+\alpha)\gamma_m r_3^m \nonumber\\ &\le& 16 \alpha^{\frac 1 6}\gamma_m r_3^m. \nonumber \end{eqnarray} \emph{Step three.} Now we claim that every point $w\in S_0\cap B(x^*,r_3/2)$ is not further than ${1\over 4}\epsilon r_3$ from $\Sigma$. First of all note that $\Sigma$ divides the space $\R^n$ into two halfspaces $X$ and $Y$, one of which (suppose $X$) contains the ball $B(y,r_2)$ while the other ($Y$) contains the cone $C_\theta$. Suppose $w\in X$. Since the ball $B(y,r_2)$ doesn't intersect $S_0$, if we consider a point $p\in\partial B(x^*,r_3/2)\cap\partial B(y,r_2)$, with simple geometrical considerations we obtain \[ d(w,\Sigma)\le d(p,\Sigma)= {\frac 1 2}\frac{{(\frac {r_3} {2}})^2}{r_2} = {1\over 8}\alpha^{\frac 1 3} r_3 \le {1\over 4}\epsilon r_3. \] Consider now $w\in Y$ and suppose $d(w,\Sigma)>{1\over 4}\epsilon r_3$. Choose $q\in \partial B(x^*,r_3/2)\cap\partial C_\theta$ and denote by $\Sigma'$ the hyperplane parallel to $\Sigma$ through $q$. Again with a geometrical argument we obtain \bea d(w,\partial C_\theta)&\ge& d(w,\Sigma') \ge {1\over 4}\epsilon r_3-{1\over 2} r_3 \cos\theta \\ &=& ({1\over 4}\epsilon - {1\over 2}\cos\theta) r_3 \eea so that \[ B\big(w,\big({1\over 4}\epsilon-{1\over 2}\cos\theta\big)r_3\big)\subset B(x^*,r_3)\cap C_\theta. \] From inequality~(\ref{eq:ltheta}) we obtain (using also~(\ref{eq:beta})) \bea \lefteqn{\gamma_m(1-\alpha) r_3 ^m\big({1\over 4}\epsilon-{1\over 2}\cos\theta\big)^m}\\ &\le& \H^m\left(S_0\cap B\big(w,({1\over 4}\epsilon-{1\over 2} \cos\theta)r_3\big)\right) \\ &\le& \H^m(S_0\cap B(x^*,r_3)\cap C_\theta) \\ &\le& 16\alpha^{\frac 1 6}\gamma_m r_3^m \eea so that (recall~(\ref{eq:defalpha})) \[ 32\alpha^{\frac 1 6} = \frac{\epsilon^m}{2\cdot 8^m}\le (1-\alpha)\big({\frac 1 4}\epsilon-{\frac 1 2}\alpha^{\frac 1 6}\big)^m \le 16 \alpha^{\frac 1 6} \] which is a contradiction. Thus, we've proved that $S_0\cap B(x^*,r_3/2)$ is within a distance of ${1\over 4}\epsilon r_3$ from $\Sigma$ that is the lemma for $\lambda={1\over 2}\lambda_1\alpha^{\frac 1 3}$. \end{proof} \section{Main theorem} In view of Theorem~\ref{teo:Reif} to prove Theorem~\ref{teo} we only need the following \begin{theorem}\label{teo:valeReif} If $x_0\in S_0=\partial E_0$ and $\epsilon_0>0$ then there exist $R>0$, $\epsilon'<\epsilon_0$ such that $(\epsilon',R,n-1)$-Reifenberg property holds in $x_0$ for the set $S_0$. \end{theorem} \begin{proof} Recall that $m=n-1$ and choose positive numbers $\epsilon'$, $\epsilon$, $\lambda$, $\alpha$ and $r_0$ such that the following hypotheses hold: \[ \begin{array}{ccc} \begin{array}{rcl} \epsilon' &<& \frac{1}{32\cdot 2^m} \\ \epsilon' &<& \epsilon_0 \\ \epsilon &=& \left(\frac{\epsilon'}{320(m+1)(m+2)}\right)^2\\ \lambda &=& \lambda(n,\epsilon) \\ \alpha &<& \alpha(n,\epsilon) \\ \alpha &<& \frac{1}{12(m+3)} \left(\frac{\lambda \epsilon}{4}\right)^m \end{array}&\hspace{1cm}& \begin{array}{rcl} \alpha &<& \frac{1}{5} \\ r_0 &<& r_0(\alpha/2) \\ \omega(r_0)&<&\gamma_m\frac{\alpha}{2} \\ \omega(r_0)&<&\frac{1}{3 c_n} \\ r_0 &<& R_0(\omega,\epsilon,n) \end{array} \end{array} \] where $\lambda(n,\epsilon)$, $\alpha(n,\epsilon)$ and $R_0(\omega,\epsilon,n)$ are the constants given by Lemma~\ref{lemma:6*}, $c_n$ is the constant given by Lemma~\ref{lemma:omegamul} and $r_0(\alpha/2)$ is the constant given in Lemma~\ref{lemma:spirale}. By Lemma~\ref{lemma:spirale} for any ball $B(x,r)$ $\subset B(x_0,r_0)$ with center $x\in S_0$ we have \[ 1-{\alpha\over 2} \le {\phi(x,r)\over \gamma_m r^m} \le 1+{\alpha\over 2}. \] On the other hand, by Lemma~\ref{lemma:1*} \bea \psi(x,r) &=& \int_0^r\H^{m-1}(S_0\cap\partial B(x,t))\d t\\ &\ge& \int_0^r {m\over t}\H^m(S_0\cap B(x,t)) - m \omega(t) t^{m-1}\d t \\ &\ge& (1-{\alpha\over 2})\gamma_m r^m - \omega(r_0)r^m \\ &=& \left(1-\frac{\alpha}{2} -{\omega(r_0)\over \gamma_m}\right) \gamma_m r^m \eea and hence the following holds: \begin{equation}\label{eq:*53} 1-\alpha \le {\psi(x,r)\over \gamma_m r^m} \le {\phi(x,r)\over \gamma_m r^m} \le 1+\alpha. \end{equation} By Lemma~\ref{lemma:6*}, given any ball $B(x,r)$ $\subset B(x_0,r_0)$ with center $x\in S_0$ there exist a point $x^*\in S_0$ and an $m$-dimensional plane $\Sigma$ through $x^*$ such that \bea S_0\cap B(x^*,\lambda r) &\subset& (\Sigma)_{\epsilon\lambda r}\\ B(x^*,\lambda r)&\subset& B(x,r). \eea Now take any ball $B(x',r')\subset B(x_0,r_0)$ with $x'\in S_0$ and consider $m+2$ points $y_1,\ldots,$ $y_{m+2}$ contained in $S_0\cap B(x',r'/4)$. By well known convering theorems, given $\delta>0$ we can find a finite family $\{B(x_{i\delta},\rho_{i\delta})\}_{i\in I_\delta}$ of disjoint balls with center on $S_0$, contained in $B(x',r'/4)$ and with radius $\rho_{i\delta}< \delta$, such that \[ \sum_{i\in I_\delta} \H^m(S_0\cap B(x_{i\delta},\rho_{i\delta}))\ge {1\over 2}\H^m(S_0\cap B(x',r'/8)). \] Thus, by~(\ref{eq:*53}) \begin{equation}\label{eq:*57} (1+\alpha)\sum_i \gamma_m\rho_{i\delta}^m \ge {1\over 2}\H^m(S_0\cap B(x',r'/8)) \ge {(1-\alpha)\gamma_m\over 2}\left({r'\over 8}\right)^m. \end{equation} Obviously for each $j\le m+2$ and for each $i\in I_\delta$ \[ B(x_{i\delta},\rho_{i\delta})\subset B(x',r'/4) \subset B(y_j,r'/2)\subset B(x',r') \] so by Lemma~\ref{lemma:6*} we can find points $x^*_{i\delta}\in S_0$ and $m$-dimensional planes $\Sigma_{i\delta}$ such that \bea S_0\cap B(x^*_{i\delta},\lambda\rho_{i\delta}) &\subset&\left(\Sigma_{i\delta}\right)_{\epsilon\lambda\rho_{i\delta}}\\ B(x^*_{i\delta},\lambda\rho_{i\delta})&\subset& B(x_{i\delta},\rho_{i\delta}). \eea Choose $\theta$ so that $1-\cos\theta=3\epsilon$, and let $I_{j\delta}$ be the set of indices $i$ such that the segment $[x^*_{i\delta},y_j]$ makes an angle greater than $\theta$ with $\Sigma_{i\delta}$. Let \[ e={\gamma_m\lambda^m\over 4(m+3)(1-\alpha)}\left({r'\over 8}\right)^m. \] If for fixed $j$ there were arbitrarily small $\delta$ such that \[ \sum_{i\in I_{j\delta}}\gamma_m \rho_{i\delta}^m \lambda^m > e \] by Lemma~\ref{lemma:7*} we had \[ \psi(y_j,r'/2)\le \frac {\phi(y_j,r'/2)}{1-\alpha}- e\left({1-\cos\theta -\epsilon\over 2}\right)^m \] hence by~(\ref{eq:*53}) \bea (1-\alpha)\gamma_m \left({r'\over2}\right)^m &\le& \psi(y_j,r'/2) \le \frac{\phi(y_j,r'/2)}{1-\alpha}-e\epsilon^m \\ &\le& \frac{1+\alpha}{1-\alpha}\left({r'\over 2}\right)^m\gamma_m - e\epsilon^m \eea that is \[ 12\alpha (m+3) \ge \left({\lambda\epsilon\over 4}\right)^m. \] But by hypotheses on $\alpha$ the previous inequality is false so we deduce that for a sufficiently small $\delta$ (using also~(\ref{eq:*57})) \bea \sum_{i\in I_{j\delta}}\gamma_m\rho_{i\delta}^m\lambda^m &\le& {\gamma_m\lambda^m\over 4(m+3)(1-\alpha)}\left({r'\over 8}\right)^m\\ &\le& \frac{\lambda^m (1+\alpha)}{2(1-\alpha)^2(m+3)}\sum_{i\in I_\delta}\gamma_m \rho_{i\delta}^m \eea so that (since $\alpha<\frac{1}{5}$) \[ \sum_{i\in I_{j\delta}}\rho_{i\delta}^m \le \frac{1}{m+3}\sum_{i\in I_\delta}\rho_{i\delta}^m \] and \[ \sum_{j=1}^{m+2}\sum_{i\in I_{j\delta}}\rho_{i\delta}^m < \sum_{i\in I_\delta}\rho_{i\delta}^m. \] Therefore there exists an index $i_0\in I_\delta$ contained in none of the sets $I_{j\delta}$. So we have proved that given any $m+2$ points $y_j$ of $S_0\cap B(x',r'/4)$ it is possible to find an $m$-dimensional plane $\Sigma=\Sigma_{i_0\delta}$ such that every $y_j$ lies within a distance of ${1\over 2}r'\sin\theta$ from $\Sigma$. Being $1-\cos\theta=3\epsilon$ we have \bea {1\over 2}r'\sin\theta &=& {1\over 2}r'\sqrt{1-\cos^2\theta} = {1\over 2}r'\sqrt{3\epsilon (1+\cos\theta)}\\ &<& {5\over 4}r'\sqrt{\epsilon}. \eea Let's consider all $m$-simplexes\footnote {Here an $m$-simplex is meant to be the convex envelope of $m+1$ points, which are the vertices of the simplex. The $m$-area of the simplex is its $m$-dimensional measure.} with each vertex in $S_0\cap B(x',r'/4)$. Let then $y_1,\ldots,y_{m+1}$ be the vertices of the $m$-simplex of greatest $m$-area among all the simplexes considered, and call $a$ its area. Let $y_{m+2}$ be any other point of $S_0\cap B(x',r'/4)$ and let $\Sigma$ be the $m$-dimensional plane, corresponding to the points $y_1,\ldots,y_{m+2}$ as we have seen before. If we denote by $\Delta$ the $(m+1)$-simplex with vertices $y_1,\ldots,y_{m+2}$, we find that the $m$-area of the faces of $\Delta$ could be at most $a$, so the area of the projection of $\Delta$ onto $\Sigma$ cannot be greater than $(m+2)a$ and the $(m+1)$-area of $\Delta$ would be less than ${5\over 2}(m+2) a r' \sqrt{\epsilon}$. Thus $y_{m+2}$ is within a distance of ${5\over 2}(m+1)(m+2)r'\sqrt{\epsilon}= \epsilon' r'/128$ from the $m$-dimensional plane through $y_1,\ldots,y_{m+1}$. If $\Sigma'$ is the $m$-dimensional plane through $x'$ parallel to the plane through $y_1,\ldots,y_{m+1}$, we know that every point of $S_0\cap B(x',r'/4)$ is whithin a distance of $\epsilon' r'/64$ from $\Sigma'$. So we have proved that given any $B(x',r')\subset B(x_0,r_0)$ with $x'\in S_0$ it is possible to find an $m$-dimensional plane $\Sigma'$ through $x'$ such that \begin{equation}\label{eq:*60} S_0\cap B\left(x',{r'\over 4}\right) \subset \left(\Sigma'\right)_{\epsilon' {r'\over 64}}. \end{equation} Since \[ \int_{{1\over 8}r'}^{{1\over 4}r'}\H^{m-1}(S_0\cap\partial B(x',t))\d t \le \psi(x',r'/4) \le (1+\alpha)\left({r'\over 4}\right)^m\gamma_m \] we are able to find a set of positive measure of radii $r$ with ${1\over 8}r' 4\epsilon' r {1+c_n\omega(r)\over 1-c_n\omega(r)}\H^{m-1}(S_0\cap\partial B(x',r)). \] If now we let $E=E_0\setminus B(x',r)$ by Lemma~\ref{lemma:omegamul} there exist $r$ such that \bea P(E,\overline{B(x',r)}) &\ge& {1-c_n \omega(r)\over 1+c_n \omega(r)} P(E_0,B(x',r)) \\ &=& {1-c_n \omega(r)\over 1+c_n \omega(r)}\H^m(S_0\cap B(x',r)) \\ &>& 4\epsilon' r \H^{m-1}(S_0\cap\partial B(x',r)) \eea and then, by Lemma~\ref{lemma:schiacciata}, we get that the projection of $S_0\cap B(x',r)$ on $\Sigma'$ must contain $\Sigma'\cap B(x',(1-{1\over 4}\epsilon')r)$. We claim that \begin{equation}\label{eq:*64} \Sigma'\cap B\left(x',{1\over 16}r'\right) \subset \left(S_0\cap B\left(x',{1\over 16}r'\right)\right)_{\epsilon' r'\over 32}. \end{equation} Given any point $p\in \Sigma'\cap B(x',{1\over 16}r')$ let $p'\in\Sigma'\cap B(x',{1\over 16}r' - {\epsilon' r'\over 64})$ be a point with minimal distance from $p$. Obviously $|p-p'|\le {\epsilon' r'\over 64}$. As we have seen previously, being $(1-{1\over 4}\epsilon') r> {r'\over 16}$, we know that there exists a point $p''\in S_0\cap B(x',r)$ whose projection is $p'$. Equation~(\ref{eq:*60}) gives $|p'-p''|\le {\epsilon' r'\over 64}$ so that $p''\in B(x',{r'\over 16})$. Thus we've found a point $p''\in S_0\cap B(x',{r'\over 16})$ whose distance from $p$ is less than ${\epsilon' r'\over 32}$ as desired. By~(\ref{eq:*60}) and~(\ref{eq:*64}) we have proved that the $(\epsilon',R,m)$-Reifen\-berg property holds in $x_0$ if we let $R={r_0\over 16}$. \end{proof} \bibliographystyle{abbrv} \begin{thebibliography}{1} \bibitem{BarGonTam87} E.~Barozzi, E.~Gonzalez, and I.~Tamanini. \newblock The mean curvature of a set of finite perimeter. \newblock {\em Proc. Amer. Math. Soc.}, 99(2):313--316, February 1987. \bibitem{BarTam88} E.~Barozzi and I.~Tamanini. \newblock Penalty methods for minimal surfaces with obstacles. \newblock {\em Ann. Mat. Pura Appl.}, 152:139--157, 1988. \bibitem{Fed69} H.~Federer. \newblock {\em Geometric Measure Theory}. \newblock Springer-Verlag, 1969. \bibitem{GonMasTam93} E.~H.~A. Gonzales, U.~Massari, and I.~Tamanini. \newblock Boundaries of prescribed mean curvature. \newblock {\em Rend. Mat. Acc. Lincei}, 9(4):197--206, 1993. \bibitem{Mas74} U.~Massari. \newblock Esistenza e regolarit\`a delle ipersuperfici di curvatura media assegnata in {$\R^n$}. \newblock {\em Arch. Rat. Mech. Anal.}, 55:357--382, 1974. \bibitem{Mas75} U.~Massari. \newblock Frontiere orientate di curvatura media assegnata in {$L^p$}. \newblock {\em Rend. Sem. Mat. Univ. Padova}, 53:37--52, 1975. \bibitem{Mor66} C.~Morrey, Jr. \newblock {\em Multiple Integrals in the Calculus of Variations}. \newblock Springer-Verlag, 1966. \bibitem{Rei60} E.~R. Reifenberg. \newblock Solution of the Plateau problem for {$m$}-dimensional surfaces of varying topological type. \newblock {\em Acta Math.}, 104:1--92, 1960. \end{thebibliography} \end{document}