\documentstyle[titlepage,leqno,11pt]{article} % equation numbering section.number \makeatletter \@addtoreset{equation}{section} \def\theequation{\thesection.\arabic{equation}} \makeatother \makeatletter \newcommand{\no}{\noindent} \newcommand{\r}{{\bf R}} \newcommand{\eps}{\varepsilon} \newcommand{\rn}[1]{{\bf R}^{#1}} \newcommand{\hh}{{\cal H}^{m}} \newcommand{\h}[2]{{\cal H}^{#1}\bigl({#2}\bigr)} \newcommand{\dis}[2]{{\rm dist}^2({#2},{#1})} \newcommand{\E}[2]{\int_{#2}|\nabla {#1}|^2\,dx} \newcommand{\F}[2]{\int_{{#2}\cap S_{#1}}|T^\perp x|^2\,d{\cal H}^m} \newcommand{\LL}{{\cal A}} \newcommand{\mm}[1]{{\cal M}_\omega(#1)} \newcommand{\dev}[2]{{\rm Dev}(#1,#2)} \newcommand{\res}{\mathop{\hbox{\vrule height 7pt width .5pt depth 0pt \vrule height .5pt width 6pt depth 0pt}}\nolimits} \newcommand{\qed}{\thinspace\null\nobreak\hfill\hbox{\vbox{\kern-.2pt\hrule height.2pt depth.2pt\kern-.2pt\kern-.2pt \hbox to2.5mm{\kern-.2pt\vrule width.4pt \kern-.2pt\raise2.5mm\vbox to.2pt{}\lower0pt\vtop to.2pt{}\hfil\kern-.2pt \vrule width.4pt\kern-.2pt}\kern-.2pt\kern-.2pt\hrule height.2pt depth.2pt \kern-.2pt}}\par\medbreak} \newtheorem{definition}{Definition}[section] \newtheorem{lemma}[definition]{Lemma} \newtheorem{theorem}[definition]{Theorem} \newtheorem{proposition}[definition]{Proposition} \newtheorem{corollary}[definition]{Corollary} \newtheorem{remark}[definition]{Remark} \title{Partial regularity of free discontinuity sets I\hskip -1pt I} %\title{} \author{} \author{{\Large Luigi Ambrosio} \\ Dipartimento di Matematica \\ Via Abbiategrasso 205, 27100 Pavia, Italy \and {\Large Nicola Fusco} \\ Dip. di Matematica e Applicazioni \\ ``R. Caccioppoli'', Via Cinthia \\ 80126 Napoli, Italy \and {\Large Diego Pallara} \\ Dip. di Matematica \\ Universit\`a di Lecce \\ C.P. 193, 73100 Lecce, Italy} \begin{document} \maketitle \begin{abstract} We prove a regularity theorem for minimizers of a class of free discontinuity problems including the following example: given $\Omega\subset\rn{n}$ open, $g\in L^\infty(\Omega)$, consider the functional $$ G(u,K)=\int_{\Omega\setminus K}\bigl[|\nabla u|^2 + (u-g)^2\bigr] \, dx+\h{n-1}{K}, $$ where $K\subset\Omega$ is relatively closed, $u\in C^1(\Omega\setminus K)$ and ${\cal H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure in $\rn{n}$. If $(K,u)$ is a minimizer then $K$ is a $C^{1,\alpha}$ hypersurface outside a closed ${\cal H}^{n-1}$-negligible set. \end{abstract} \hbox{}\vfill \tableofcontents \vfill\eject \section{Introduction} In this paper we continue the study, started in \cite{ap}, of the regularity of (quasi) minimizers of a class of free discontinuity problems including the minimization of the Mumford-Shah functional \begin{equation} \label{msfun} G(u,K)=\int_{\Omega\setminus K}\bigl[|\nabla u|^2 + \alpha(u-g)^2\bigr] \,dx+\beta\h{n-1}{K}, \end{equation} where $\Omega\subset\rn{n}$ is an open set, $g\in L^\infty(\Omega)$, $\alpha,\,\beta>0$ and $\h{n-1}{K}$ is the $(n-1)$-dimensional Hausdorff measure of $K$. In (\ref{msfun}), $K$ varies in the class of relatively closed subsets of $\Omega$ and $u\in C^1(\Omega\setminus K)$ (see \cite{mumsh}, \cite{bz2}, \cite{morsol}, \cite{dcl}, \cite{dms}). Notice that, given $K$, the optimal function $u$ is the solution of the Neumann problem \begin{equation}\label{neu} \Delta u=\alpha(u-g),\qquad\qquad \frac{\partial u}{\partial n}=0 \end{equation} in $\Omega\setminus K$. Namely, we prove in Theorem \ref{main} that any optimal free discontinuity set $K$ is a $C^{1,\alpha}$ hypersurface except for a closed singular set $S$ satisfying $\h{n-1}{S}=0$. Moreover, we give a characterization of singular points which could be exploited to get further information on the dimension and the structure of $S$. The starting point of this paper is a criterion, proved in \cite{ap}, for the $C^{1,\alpha}$ local regularity of optimal sets $K$. The criterion involves the rate of decay of the quantity $$ {\cal F}(x,\rho)=\rho^{-n-1}\min_{A\in\LL}\int_{B_\rho(x)\cap K} \dis{A}{y}\,d{\cal H}^{n-1}(y) $$ measuring the flatness of $K$ (here ${\cal A}$ denotes the set of affine hyperplanes in $\rn{n}$) and of the scaled Dirichlet energy $$ {\cal D}(x,\rho)=\rho^{1-n}\int_{B_\rho(x)}|\nabla u|^2\,dy. $$ The application of such a criterion to our partial regularity result is based on two decay estimates. The first one, concerning the flatness improvement has been already proved in \cite{ap} (see Theorem \ref{decay2} below). Heuristically, (see \cite{ap}) a first variation argument and the rectifiability of $K$ show that the Dirichlet energy controls the mean curvature of $K$. Hence, the proof of flatness improvement has a more geometric flavour and involves some typical arguments of minimal surfaces theory. The second decay estimate is concerned with the Dirichlet energy. Assum\-ing the decay estimate to be false, we find a sequence of quasi minimizers $(z_h,C_h)$ and a sequence of balls $B_{\rho_h}$. Scaling and normalizing, we obtain a sequence of quasi minimizers $(v_h,K_h)$ in a fixed ball $B_R$ whose gradients weakly converge to the gradient of a function $v$ which is harmonic in the upper and lower half balls, with zero normal derivative on $\{x_n=0\}$. An easy reflection argument shows that the scaled Dirichlet energy of $v$ on $B_{\rho}$ decays like $\rho$. To transfer this property to the sequence $v_h$ (thus getting a contradiction) we need to prove {\em strong} convergence of the gradients. To obtain this convergence at least in a smaller ball, we introduce functions obtained by taking the composition of $v_h$ with deformation maps pushing a large part of $K_h$ on suitable minimal surfaces. An energy comparison argument leads to the strong convergence of gradients and to the desired contradiction. A feature of our problem is that the flatness improvement theorem involves also the Dirichlet energy, and viceversa. For this reason an iteration argument leading to the improvement of {\em both} quantities is needed. The idea is to use the decay of ${\cal F}(x,\rho)$ if ${\cal F}(x,\rho)$ controls ${\cal D}(x,\rho)$ and to use the decay of ${\cal D}(x,\rho)$ if ${\cal F}(x,\rho)$ is much smaller than ${\cal D}(x,\rho)$. This ``twospeed" iteration argument seems to be a natural tool for treating the partial regularity of these free discontinuity problems. \vspace{10pt} {\bf Acknowledgement.} Work partially supported by MURST 40\%. \section{Quasi minimizers} In the following a natural number $n\geq 2$ will be fixed and we omit the dependence of several constants on $n$. Since we will often deal with $(n-1)$-dimensional sets, we will use the notation $m$ for $n-1$. Let $\Omega\subset\rn{n}$ be an open set. For any function $u\in L^1(\Omega)$ the set $S_u$ denotes the the complement of the Lebesgue set of $u$, {\em i.e.}, $$ x\notin S_u\qquad\Longleftrightarrow\qquad \exists\,\,z\in\rn{}\,\,\hbox{\rm s.t. }\lim_{\rho\to 0^+} \rho^{-n}\int_{B_\rho(x)}|u(y)-z|\,dy=0. $$ The space $SBV(\Omega)$, introduced by De Giorgi and Ambrosio in \cite{dga}, consists of all functions $u$ with finite total variation in $\Omega$ such that the singular part (with respect to Lebesgue measure) $D^s u$ of the distributional derivative $Du$ is supported in $S_u$, {\em i.e.}, $$|D^s u|(\Omega\setminus S_u)=0.$$ \no In the following, $\nabla u$ stands for the density (with respect to Lebesgue measure) of $Du$. Notice that \begin{equation}\label{w12} u\in SBV(\Omega)\qquad\Longrightarrow \qquad u\in W^{1,1}(\Omega\setminus\overline{S}_u) \end{equation} \no because the singular part of the distributional derivative of $u$ is supported in $S_u$. In $SBV(\Omega)$ many variational problems involving volume and surface energies can be formulated and have a solution (see \cite{am1}, \cite{am2}, \cite{am3}, \cite{am4}). In this paper very little of the $SBV$ theory will be used, because a density lower bound for minimizers (see Proposition \ref{dgcl} below and (\ref{minK})) allows a reduction to the study of pairs $(u,K(u))$, with $u\in W^{1,1}(\Omega\setminus K(u))$ and $K(u)=\overline{S}_u$. Now we give the definitions of local minimizers and quasi minimizers assuming $\beta=1$. This assumption is not really restrictive, by a scaling argument. \begin{definition}[local minimizers] \label{locmin} We say that $u\in SBV_{\rm loc}(\Omega)$ is a local minimizer in $\Omega$ if \begin{equation}\label{enefin} \E{u}{A}+\h{m}{S_u\cap A}<+\infty\qquad\qquad \forall A\subset\subset\Omega \end{equation} \no and $$ \E{u}{A}+\h{m}{S_u\cap A} \leq \E{v}{A}+\h{m}{S_v\cap A} $$ \no whenever $v\in SBV_{\rm loc}(\Omega)$ and $\{v\not=u\}\subset\subset A\subset\subset\Omega$. \end{definition} \begin{definition}[quasi minimizers] We will call deviation from minimality $\dev{u}{\Omega}$ of a function $u\in SBV_{\rm loc}(\Omega)$ satisfying (\ref{enefin}) the smallest $\lambda\in[0,+\infty]$ such that $$ \E{u}{A}+\h{m}{S_u\cap A} \leq \E{v}{A}+\h{m}{S_v\cap A} +\lambda $$ \no for any $v\in SBV_{\rm loc}(\Omega)$ such that $\{v\not= u\}\subset\subset A\subset\subset\Omega$. Clearly, $\dev{u}{\Omega}=0$ if and only if $u$ is a local minimizer in $\Omega$. Moreover, we say that $u$ is a quasi minimizer in $\Omega$ if there exists a nondecreasing function $\omega(t):(0,+\infty)\to [0,+\infty)$ such that $\omega(t)\downarrow 0$ as $t\downarrow 0$ and \begin{equation} \label{deviation} \dev{u}{B_\rho(x)}\leq\rho^m\omega(\rho) \end{equation} \no for any ball $B_\rho(x)\subset\Omega$. We denote by $\mm{\Omega}$ the class of functions satisfying (\ref{deviation}). \end{definition} \begin{remark} {\rm The canonical example of quasi minimizer is given by the minimum $u$ of the Mumford-Shah functional \begin{equation} \label{msfunc} F(u)=\int_{\Omega}\bigl[|\nabla u|^2 + \alpha(u-g)^2\bigr] \,dx+\h{m}{S_u}, \end{equation} \no with $g\in L^\infty(\Omega)$ and $\alpha\geq 0$. Indeed, it is easy to check that $$ F(v)\geq F(M\land v\lor -M)\qquad\forall v\in SBV_{\rm loc}(\Omega)$$ \no with $M=\Vert g\Vert_\infty$. This shows that $\Vert u \Vert_\infty\leq M$ and, for any competing function $v$, the inequality $F(u)\leq F(M\land v\lor -M)$ implies that the deviation from minimality of $u$ in $B_\rho(x)$ does not exceed $4\alpha\omega_n\Vert g\Vert^2_\infty\rho^n$. Hence $u\in\mm{\Omega}$ with $$\omega(t)=4\alpha\omega_n\Vert g\Vert^2_\infty t.$$ Existence of minimizers follows by the compactness and lower semicontinuity results of \cite{am1} and \cite{am4}. Using Proposition \ref{dgcl} below it can be seen (see also \cite{dcl}) that the minimum of $F$ on $SBV_{\rm loc}(\Omega)$ is equal to the minimum of the functional originally proposed by Mumford-Shah and Blake-Zissermann: \begin{equation} \label{minG} G(u,K)=\int_{\Omega\setminus K}\bigl[|\nabla u|^2 +\alpha(u-g)^2\bigr] \, dx+\h{m}{K}. \end{equation} \no In (\ref{minG}), $K$ varies in the relatively closed subsets of $\Omega$ and $u$ varies in $C^1(\Omega\setminus K)$. }\end{remark} \begin{proposition} \label{dgcl} Let $u\in\mm{\Omega}$. Then, for any ball $B_\rho(x)\subset\Omega$ centered at $x\in\overline{S}_u$ we have $$\rho\leq\rho_\omega\qquad\Longrightarrow\qquad \hh\bigl(S_u\cap B_\rho(x)\bigr)\geq\theta_\omega\rho^m $$ \no for some positive constants $\rho_\omega$ and $\theta_\omega$ depending only on $\omega$. Moreover, $$ \hh(\overline{S}_u\cap\Omega\setminus S_u)=0. $$ \end{proposition} {\sc Proof.} See Theorem 2.7 and Proposition 2.8 of \cite{ap}. The density lower bound for minimizers of the Mumford-Shah functional has been proved also in \cite{dms} and \cite{morsol}. \qed We will denote the set $\overline{S}_u$ by $K(u)$. If $u\in\mm{\Omega}$, Proposition \ref{dgcl} and the quasi minimality condition imply \begin{eqnarray}\label{minK} &&\int_{B_\rho(x)}|\nabla u|^2\, dy+\h{m}{K(u)\cap B_\rho(x)} \\ &&\leq\int_{B_\rho(x)}|\nabla v|^2\, dy+\h{m}{K(v)\cap B_\rho(x)} +\rho^m\omega(\rho)\nonumber \end{eqnarray} \no for any $v\in SBV_{\rm loc}(\Omega)$ such that $\{v\neq u\}\subset\subset B_\rho(x)\subset\Omega$. \section{Statement of the main result} In the following, ${\cal A}$ stands for the set of affine hyperplanes in $\rn{n}$. The main result of our paper is the following: \begin{theorem} \label{main} Let $\mm{\Omega}$ be defined in Definition \ref{locmin}, $u\in\mm{\Omega}$, and assume that \begin{equation} \label{main1} \omega(\rho)\leq c_0\rho^{2\gamma} \end{equation} \no for some constants $c_0\geq 0,\,\gamma>0$ and let $\alpha=\min\{1/4,\gamma\}/(m+2)$. Then, there is a positive constant $\eps_0(c_0,\gamma)$ such that for any $x\in K(u)$ and any ball $B_\rho(x)\subset\Omega$ with $\rho<1$, the condition \begin{equation} \label{main2} \rho^2\int_{B_\rho(x)}|\nabla u|^2\,dy+ \min_{A\in\LL}\int_{B_\rho(x)\cap K(u)} \dis{A}{y}\, d{\cal H}^{m}(y)<\eps_0\rho^{m+2} \end{equation} \no implies that $B_{\rho/2}(x)\cap K(u)$ is a $C^{1,\alpha}$ hypersurface. \end{theorem} \begin{remark} \label{singtrasc} {\rm Let $R$ be the set of regular points of $K(u)$, {\em i.e.}, the set of those points $x\in K(u)$ such that (\ref{main2}) holds for a sufficiently small $\rho\in (0,1)$. Then, the complement $S=K(u)\setminus R$ is relatively closed in $\Omega$ and it is made of all points satisfying either \begin{equation}\label{noflat} \limsup_{\rho\to 0^+}\rho^{-m-2}\min_{A\in{\cal A}} \int_{B_\rho(x)\cap K(u)}\dis{A}{y}\,d{\cal H}^m(y)>0 \end{equation} \no or \begin{equation}\label{pocaene} \liminf_{\rho\to 0^+}\rho^{-m} \int_{B_\rho(x)}|\nabla u|^2\,dx\geq\eps_0. \end{equation} \no It is easy to see that $\h{m}{S}=0$. Indeed, (\ref{noflat}) is not satisfied at any point $x\in K(u)$ where $K(u)$ has an approximate tangent space $P$ (see \cite[Th. 11.6]{simon}) because, to estimate the integral in (\ref{noflat}) we can choose $A=x+P$, thus obtaining that the limit is $0$. Moreover, a differentiation theorem (see \cite[3.2(1)]{simon}) shows that $$\int_C |\nabla u|^2\,dx\geq\frac{\eps_0}{2}\h{m}{C}$$ \no for any Borel subset $C$ of the set $E$ of points $x\in\Omega$ satisfying (\ref{pocaene}). The inequality above implies that $E$ has locally finite ${\cal H}^m$-measure, hence ${\cal L}^n(E)=0$. Using the inequality again we obtain that $\h{m}{E}=0$. }\end{remark} The proof of Theorem \ref{main} is based, among other things, on a $C^{1,\alpha}$ regularity criterion proved in \cite{ap}. Let us introduce a notation for the {\em Dirichlet integral} of $u$ \begin{equation}\label{defD} {\bf D}(x,\rho)=\int_{B_\rho(x)}|\nabla u|^2\,dy \end{equation} and the {\em flatness} of $K(u)$ \begin{equation}\label{defA} {\bf A}(x,\rho)=\min_{A\in\LL}\int_{B_\rho(x)\cap K(u)} \dis{A}{y}\,d{\cal H}^{m}(y). \end{equation} \no We can state the following result: \begin{theorem}\label{c1graf} Let $u\in\mm{\Omega}$ and assume that there exist $C>0, \sigma\in (0,1)$ such that $\omega(\rho)\leq C\rho^{2\sigma}$ and $$ {\bf D}(x,\rho)+\rho^{-2} {\bf A}(x,\rho)\leq C\rho^{m+\sigma} $$ \no for any ball $B_\rho(x)\subset\Omega$ centered at $x\in K(u)$. Then, $\Omega\cap K(u)$ is a $C^{1,\alpha}$ hypersurface with $\alpha=\sigma/(m+2)$. \end{theorem} {\sc Proof.} See \cite{ap}, Theorem 5.3 and Remark 5.4. \qed To verify the assumption of Theorem \ref{c1graf} in $B_{\rho/2}(x)$ (with $x$ satisfying (\ref{main2})) we need to know that ${\bf A}$ and ${\bf D}$ satisfy good decay properties. The following typical decay property of ${\bf A}$ has been proved in Theorem 6.2 of \cite{ap}: it says that if $K(u)$ is sufficiently flat near $x$, it is flatter on a smaller scale, provided $\rho^2{\bf D}(x,\rho)$ and $\rho^{m+2}\sqrt{\omega(\rho)}$ are comparable with ${\bf A}(x,\rho)$ (remember that the Dirichlet integral is related to the curvature of $K(u)$). \begin{theorem} [flatness improvement] \label{decay2} For any choice of $\beta\in (0,1/112)$ and $\eta>0$ there exists $\eps_1(\beta,\eta,\omega)>0$ such that, for any $u\in\mm{B_\rho(x)}$, the conditions $$ {\bf A}(x,\rho)\leq\eps_1\rho^{m+2},\qquad {\bf A} (x,\beta\rho)\geq\eta\rho^2\bigl[ {\bf D}(x,\rho)+ \rho^{m}\sqrt{\omega(\rho)}\bigr]$$ \no and $x\in K(u)$ imply $${\bf A}(x,\beta\rho)\leq C\beta^{m+4}{\bf A}(x,\rho)$$ \no with $C$ depending only on $\omega$. \end{theorem} The following theorem is the natural counterpart of the flatness improvement theorem: it says that in the regions where ${\bf A}$ is much smaller than ${\bf D}$ there is improvement of ${\bf D}$. \begin{theorem} [energy improvement] \label{decay1} For any choice of $\beta\in (0,6/7)$ there exist $\eps_2(\beta,\omega)>0$ and $\eta(\beta,\omega)\in (0,1)$ such that, for any $u\in\mm{B_\rho(x)}$, the conditions $${\bf D}(x,\rho)\leq\eps_2\rho^{m},\quad {\bf A}(x,\rho)\leq\eps_2\rho^{m+2},\quad \rho^{m+2}\sqrt{\omega(\rho)}+ {\bf A}(x,\beta\rho)\leq\eta\rho^2 {\bf D}(x,\rho)$$ \no and $x\in K(u)$ imply $${\bf D}\bigl(x,\frac{\beta}{25}\rho\bigr) \leq \beta^n {\bf D}(x,\rho).$$ \end{theorem} A large part of this paper will be devoted to the proof of the energy improvement theorem. Using a careful choice of the constants in both decay theorems, and using in an essential way the independence of the constant $C$ of Theorem \ref{decay2} on $\beta$ and $\eta$, we will be able to perform in Lemma \ref{twospeed} and Corollary \ref{twospeed1} a joint iteration leading to the desired estimates on ${\bf A}$ and ${\bf D}$. \section{Preliminary results} The following theorem is concerned with the asymptotic behaviour of sequences of quasi minimizers assuming that Dirichlet energy tends to zero and the jump sets become flatter and flatter. \begin{theorem} \label{planes} Let $(u_h)\subset\mm{B_R}$, $(S_h)\subset \LL$ be satisfying the conditions $$\lim_{h\to +\infty}\int_{B_\rho}|\nabla u_h|^2\,dx+ {\rm Dev}(u_h,B_\rho)+\int_{B_\rho\cap K(u_h)}\dis{S_h}{x}\, d{\cal H}^{m}=0 $$ \no for any $\rho\in (0,R)$ and $$ \lim_{h\to +\infty}{\rm dist}\bigl(0,K(u_h)\bigr)=0,\qquad \lim_{h\to +\infty}S_h=S.$$ \no Then $S$ contains the origin and the following properties hold: \smallskip \noindent (i) for any $\rho\in (0,R)$ the height $$\max_{x\in \overline{B}_\rho\cap K(u_h)}{\rm dist}(x,S)$$ \no is infinitesimal as $h\to +\infty$; \noindent (ii) denoting by $S_{R/4}^\pm$ the two connected components of $$\bigl\{x\in B_{R/2}: {\rm dist}(x,S)\geq R/4\bigr\},$$ \no we have $$ \liminf_{h\to +\infty}\biggl\vert \int_{S_{R/4}^+}u_h\,dx-\int_{S_{R/4}^-}u_h\,dx\biggr\vert>0; $$ \noindent (iii) the measures ${\cal H}^{m}\res K(u_h)$ weakly converge to ${\cal H}^{m}\res S$ in $B_R$. \end{theorem} {\sc Proof.} Let $x_h\in K(u_h)$ be converging to $0$ and $$\rho_h=\min\biggl\{\frac{{\rm dist}(0,S_h)}{4},\rho_\omega, \frac{R}{4}\biggr\}.$$ \no Using Proposition \ref{dgcl}, if (a subsequence of) $\rho_h$ is not infinitesimal, for $h$ large enough we get \begin{eqnarray*} \int_{B_{R/2}\cap K(u_h)}\dis{S_h}{x}\,d{\cal H}^{m} &\geq & \int_{B_{\rho_h}(x_h)\cap K(u_h)}\dis{S_h}{x}\,d{\cal H}^{m} \\ &\geq& \theta_\omega\rho_h^{m}\min_{B_{\rho_h}(x_h)\cap K(u_h)} \dis{S_h}{x} \\ &\geq& \theta_\omega\rho_h^{m}\min_{B_{2\rho_h}} \dis{S_h}{x} \geq\frac{\theta_\omega}{4}\rho^{m+2}_h \end{eqnarray*} \no a contradiction. Hence, the distances of $S_h$ from $0$ are infinitesimal, $S$ contains the origin and $$\lim_{h\to +\infty}\int_{B_\rho}|\nabla u_h|^2\,dx+ {\rm Dev}(u_h,B_\rho)+\int_{B_\rho\cap K(u_h)}\dis{S}{x} \,d{\cal H}^{m}=0 $$ \no for any $\rho\in (0,R)$. Then, (i) follows again by the density lower bound (see \cite[Proposition 5.1]{ap}). The properties (ii), (iii) are proved in Step 4 and Step 5 of Theorem 4.3 \cite{ap} (see also Lemma 6.1 of the same paper). \qed The next lemma is based on a well known reflection argument. \begin{lemma} \label{har1} Let $B^+$ be the upper unit half ball in $\rn{n}$ and $v:B^+\to\rn{}$. For $a\in (0,1)$, let $S_a^+=\{(z,y)\in B_1:y>a\}$ and let us assume that $v\in H^1(S_a^+)$ for any $a>0$. Then \smallskip \noindent (i) $v$ belongs to $H^1(B^+)$ if and only if $\nabla v\in L^2(B^+)$; \noindent (ii) if $v\in H^1(B^+)$ and \begin{equation}\label{weakarmo} \int_{B^+}\langle\nabla v,\nabla\phi\rangle\,dx=0\qquad \forall \phi\in C^1_0(B_1) \end{equation} \no the function $\tilde{v}(z,y)=v(z,-y)$ is the harmonic extension of $v$ to $B_1$. \end{lemma} {\sc Proof.} (i) One implication is trivial. Assuming $\nabla v\in L^2(B^+)$, we need only to show that $v\in L^2(B^+)$. Clearly, the functions $v_k=k\land v\lor -k$ belong to $H^1(B^+)$ and $\Vert\nabla v_k\Vert_2\leq\Vert \nabla v\Vert_2$ for any $k>0$; the Poincar\'e inequality (see \cite[Th. 4.4.2]{zi}) implies $$\int_{B^+}|v_k-\bar{v}_k|^2\,dx\leq c\int_{B^+}|\nabla v_k|^2\,dx$$ \no for some dimensional constant $c$, with $\bar{v}_k$ equal to the average of $v_k$ on $S_{1/2}^+$. Letting $k\to +\infty$ we get $$\int_{B^+}|v-\bar{v}|^2\,dx\leq c\int_{B^+}|\nabla v|^2\,dx$$ \no hence $v\in L^2(B^+)$. \noindent (ii) It is well known that $\tilde{v}\in H^1(B_1)$. For any test function $\phi\in C^1_0(B_1)$, using two times (\ref{weakarmo}) and a change of variables we get \begin{eqnarray*} \int_{B_1}\langle\nabla \tilde{v},\nabla\phi\rangle\,dx&=& \int_{B^+}\langle\nabla v,\nabla\phi\rangle\,dx+ \int_{B_1\setminus B^+}\langle\nabla\tilde{v},\nabla\phi\rangle\,dx \\ &=&\int_{B_1\setminus B^+}\langle\nabla\tilde{v},\nabla\phi\rangle\,dx =\int_{B^+}\langle\nabla v,\nabla\tilde{\phi}\rangle\,dx=0 \end{eqnarray*} \no with $\tilde{\phi}(z,y)=\phi(z,-y)$. Hence, $\tilde{v}$ is harmonic.\qed For any $1$-Lipschitz function $v$ on $\partial B_R$, let us denote by $\tilde{v}:B_R\rightarrow\rn{}$ the solution of the minimum problem \begin{equation}\label{diri} \min\biggl\{\int_{B_R}\sqrt{1+|\nabla u|^2}\,dx: u\,\,\,\hbox{\rm 1-Lipschitz}, \,\,\, u=v\,\,\,\hbox{\rm on }\partial B_R\biggr\} \end{equation} and let $${\cal E}(v)=\int_{B_R}\sqrt{1+|\nabla \tilde{v}|^2}\,dx.$$ \no We are interested to estimate the convergence to $0$ of ${\cal E}(v)-\omega_nR^n$ as $\Vert v\Vert_{H^1(\partial B_1)}$ tends to $0$. \begin{lemma} [area excess estimate] \label{har2} Let $v:\partial B_R\rightarrow \rn{}$ be a $1/2$-Lipschitz function. For any $\alpha\geq 1$ we have \begin{equation}\label{areamin} {\cal E}(v)-\omega_n R^n\leq\biggl[ \frac{\alpha^2}{R}\Vert v\Vert^2_{L^2(\partial B_R)} +{R\Vert \nabla v\Vert^2_{L^2(\partial B_R)}\over n+2\alpha-2} \biggr] \end{equation} provided $\Vert v\Vert_\infty0$ and $P(\omega)>0$ with the following property: let $u\in\mm{B_{7r}}$ for some $r\leq R$ and let $$ A=\F{u}{B_{6r}\cap K(u)},\hfill D=\E{u}{B_{6r}},\hfill L=r^m\sqrt{\frac{{\rm Dev}(u,B_{6r})}{(6r)^m}}. $$ Then, if $PA< r^{m+2}$ and $K(u)\cap B_{r/16}\neq\emptyset$ there exists a Lipschitz function $f:T\to T^\perp$ with Lipschitz constant less than $1/2$ such that \begin{equation}\label{additional} \ \ \sup |f|^{m+2}\leq P A,\quad \int_{B^m_{r/4}}|f|^2\,dz\leq A + P (\frac{A}{r^2}+D+L) (PA)^{2/(m+2)} \end{equation} \begin{equation}\label{additional1} \int_{B^m_{r/4}}|\nabla f|^2\,dz\leq P\bigl[\frac{A}{r^2}+D+L\bigr] \end{equation} and, denoting by $\Gamma(f)$ the graph of $f$, we have \begin{equation} \label{additional2} \hh\bigl(B_r\cap K(u)\setminus\Gamma(f)\bigr) \leq P\bigl[\frac{A}{r^2}+D+L\bigr]. \end{equation} \end{theorem} {\sc Proof.} Let $R=R(\omega,1/2)$ be the costant defined before Theorem 5.2 of \cite{ap}. We will indicate by $c$ the generic constants depending only on $n$ and $\omega$ appearing in the estimates of Theorem 5.2 of \cite{ap}. Then, this theorem states that $A/r^{m+2}0$ for any $t>0$. % (see Remark \ref{anytheta} As usual, to prove Theorem \ref{decay1} we argue by contradiction, getting sequences $(z_h)\subset\mm{B_{\rho_h}(y_h)}$, $(S_h), \,(\tilde{S}_h)\subset\LL$ satisfying \begin{equation}\label{zeqm1} \ \ \int_{B_{\rho_h}(y_h)}|\nabla z_h|^2\,dx<\gamma_h\rho^m_h,\ \int_{B_{\rho_h}(y_h)\cap K(z_h)}\dis{S_h}{x}\,d{\cal H}^{m}< \gamma_h\rho_h^{m+2} \end{equation} \begin{eqnarray}\label{zeqm2} \rho_h^{m+2}\sqrt{\omega(\rho_h)}&+& \int_{B_{\beta\rho_h}(y_h)\cap K(z_h)} \dis{\tilde{S}_h}{x}\,d{\cal H}^{m} \nonumber \\ &\leq& \gamma_h\rho_h^2\int_{B_{\rho_h}(y_h)}|\nabla z_h|^2\,dx \end{eqnarray} \no for a suitable infinitesimal sequence $\gamma_h$ and \begin{equation}\label{zeqm3} \int_{B_{\beta\rho_h/25}(y_h)}|\nabla z_h|^2\,dx> \beta^n\int_{B_{\rho_h}(y_h)}|\nabla z_h|^2. \end{equation} To rescale all the functions to a fixed ball we will use the following remark. \begin{remark} [scaling] \label{scaling} {\rm Let $u\in SBV(\Omega)$ and let $B_\rho(x_0)\subset\Omega$. Then, it is easy to check that $$u_\rho(y)={1\over\sqrt{\rho}}u(x_0+\rho y)$$ \no belongs to $SBV(\Omega_\rho)$ with $\Omega_\rho=\rho^{-1}\bigl(\Omega-x_0\bigr)$. Moreover, for any ball $B_\eta(y)\subset\Omega_\rho$ we have \begin{eqnarray*} \E{u_\rho}{B_\eta(y)}&=&\rho^{-m}\E{u}{B_{\eta\rho}(x_0+\rho y)}, \\ \h{m}{S_{u_\rho}\cap B_\eta (y)}&=&\rho^{-m} \h{m}{S_u\cap B_{\eta\rho}(x_0+\rho y)} \\ \dev{u_\rho}{B_\eta(y)}&=&\rho^{-m}\dev{u}{B_{\eta\rho}(x_0+\rho y)}. \end{eqnarray*} \no In particular, if $\rho\leq 1$, the monotonicity of $\omega(\rho)$ shows that $$ u\in\mm{\Omega}\qquad\Longrightarrow\qquad u_\rho\in\mm{\Omega_\rho}. $$ }\end{remark} \bigskip Let $R=R(\omega)$ be given by Theorem \ref{lipappr}. By (\ref{zeqm1}), (\ref{zeqm2}), (\ref{zeqm3}) and our assumption on $\omega$ we obtain that $\rho_h\leq R$ for $h$ large enough. Hence, scaling the functions by a factor $\rho_h/R$ as in Remark \ref{scaling}, we obtain new sequences $(u_h)\subset\mm{B_{R}(x_h)}$, $(T_h)$, $(\tilde{T}_h)$ satisfying the conditions \begin{equation}\label{zeq1} \ \ \ \int_{B_R(x_h)}|\nabla u_h|^2\,dx\beta^n\int_{B_{R}(x_h)} |\nabla u_h|^2\,dx. \end{equation} \no The deviation from minimality of $u_h$ in $B_R(x_h)$ satisfies \begin{eqnarray}\label{zeq33} {\rm Dev}(u_h,B_R(x_h))&=&\frac{{\rm Dev}(z_h,B_{\rho_h}(y_h))} {\rho^{m}_h/R^m}\leq R^m\omega(\rho_h) \nonumber \\ &\leq& R^m\gamma_h^2\biggl(\int_{B_R}|\nabla u_h|^2\,dx\biggr)^2. \end{eqnarray} \no Denoting by $T$ the hyperplane $\{x_n=0\}$, by a rotation and a translation we can assume with no loss of generality that $\tilde{T}_h=T$, $T(x_h)=0$ and $T_h$ converges to some hyperplane $S$. Using (\ref{zeq2}) and the density lower bound it is easy to see, by the same argument used in Theorem \ref{planes}, that $x_h\to 0$ (notice that $x_h\in K(u_h)$). By Theorem \ref{planes}(iii) and (\ref{zeq1}), (\ref{zeq33}) we obtain that ${\cal H}^{m}\res K(u_h)$ weakly converges in $B_R$ to ${\cal H}^{m}\res S$ and $S$ contains the origin. Since \begin{eqnarray*} 0&=&\lim_{h\to +\infty}\int_{B_{\beta R}(x_h)\cap K(u_h)}\dis{\tilde{T}_h}{x} \,d{\cal H}^{m} \\ &=&\lim_{h\to +\infty}\int_{B_{\beta R}\cap K(u_h)}\dis{T}{x} \,d{\cal H}^{m} =\int_{B_{\beta R}\cap S}\dis{T}{x}\,d{\cal H}^{m} \end{eqnarray*} \no we infer that $T=S$. Setting now $$\eps_h=\biggl(\int_{B_R}|\nabla u_h|^2\,dx\biggr)^{1/2}$$ and defining $v_h=u_h/\eps_h$ we can rewrite (\ref{zeq1}), (\ref{zeq2}), (\ref{zeq3}) as follows \begin{equation}\label{zeq4} \lim_{h\to +\infty} \eps_h^2+ \int_{B_R(x_h)\cap K(v_h)}\dis{T}{x}\,d{\cal H}^{m}=0 \end{equation} \begin{equation}\label{zeq5} \int_{B_{\beta R}(x_h)\cap K(v_h)} \dis{T}{x}\,d{\cal H}^{m}=o(\eps_h^2) \end{equation} \begin{equation}\label{zeq6} \int_{B_R(x_h)}|\nabla v_h|^2\,dx=1,\qquad \int_{B_{\beta R/25}(x_h)}|\nabla v_h|^2\,dx>\beta^n. \end{equation} \no We point out that, by (\ref{minK}), the functions $v_h$ are quasi minimizers in $B_R(x_h)$ of the rescaled functionals \begin{equation}\label{newfunct} {\cal F}_h(v)=\int_{B_\rho}|\nabla v|^2\,dx+\frac{1}{\eps_h^2} {\cal H}^{m}(B_\rho\cap K(v)) \end{equation} \no and, denoting by ${\rm Dev}_h$ the deviation from minimality of ${\cal F}_h$, (\ref{zeq33}) yields \begin{equation} \label{zeq333} \lim_{h\to +\infty}{\rm Dev}_h(v_h,B_\rho)\leq \limsup_{h\to +\infty}\frac{{\rm Dev}(u_h,B_R(x_h))}{\eps_h^2}=0 \end{equation} \no for any $\rho0$ we denote by $S_a^+$ (respectively $S_a^-$) the set of points $x\in B_{R/2}$ such that $y>a$ (respectively $y<-a$), we denote by $c_h^\pm$ the average of $v_h$ on $S_{R/4}^\pm$ and we assume, up to a change of sign, that $c_h^+\geq c_h^-$. By Theorem \ref{planes}(ii) we obtain that the sequence $\eps_h(c_h^+-c_h^-)$ has no infinitesimal subsequences, hence $(c_h^+-c_h^-)$ diverges. For any $a>0$ (\ref{zeq7}) implies the existence of $h_a\in{\bf N}$ such that $v_h-c_h^{\pm}$ belongs to $H^1(S_a^{\pm})$ for $h\geq h_a$. Moreover, if $a0$. Using the strong convergence of $v_h-c_h^{\pm}$ to $v$ in $L^2_{\rm loc}(B_{R/2}\setminus T)$, letting first $h\to +\infty$ then $\eps \to 0$ we find $$ \int_{B_{R/2}}\phi \,d\mu\leq \int_{B_{R/2}}\phi |\nabla v|^2\,dx.$$ \no Since $\phi$ is arbitrary, we obtain that $\mu\res \bigl(B_{R/2}\setminus T\bigr)= |\nabla v|^2{\cal L}^n$, hence the absolutely continuous part of $\mu$ is obviously $|\nabla v|^2{\cal L}^n$ and the singular part must be supported in $T$. We now claim that $v$ has zero normal derivative on $T\cap B_{R/2}$, {\em i.e.}, \begin{equation}\label{derino} \int_{S_0^\pm}\langle\nabla v,\nabla\phi\rangle\,dx=0 \end{equation} \no for any $\phi\in C^1_0(B_{R/2})$. Indeed, let $\phi\in C^1_0(B_{R/2})$, $\eps>0$ and let $\psi:\rn{} \to\rn{}$ be a smooth non decreasing function such that $\psi(-\infty)=0$, $\psi(+\infty)=1$ and $\psi'(t)\leq\eps$ for any $t\in\rn{}$. By (\ref{zeq333}) the sequence $z_h={\rm Dev}_h(v_h,B_{R/2})$ is infinitesimal. Hence, denoting by $c_h$ the average of $c_h^+$ and $c_h^-$, comparing ${\cal F}_h(v_h)$ with ${\cal F}_h\bigl(v_h+ \sqrt{z_h}\phi\psi(v_h-c_h)\bigr)$ and dividing both sides by $\sqrt{z_h}$ we obtain $$\liminf_{h\to +\infty} \int_{B_{R/2}}\psi(v_h-c_h)\langle \nabla v_h,\nabla\phi\rangle\,dx+ \int_{B_{R/2}}\phi\psi'(v_h-c_h)|\nabla v_h|^2\,dx\geq 0.$$ \no Since $\nabla v_h$ weakly converges to $\nabla v$ in $L^2(B_{R/2},\rn{n})$ and $\psi(v_h-c_h)$ strongly converges to $\chi_{S_0^+}$ (here we use the fact that $c_h^+-c_h^-\to +\infty$) we obtain $$\int_{S_0^+}\langle\nabla\phi,\nabla v\rangle \,dx\geq -\eps\Vert\phi\Vert_\infty.$$ \no As $\eps$ and $\phi$ are arbitrary, (\ref{derino}) is proved for $S_0^+$. The argument for $S_0^-$ is similar. By Lemma \ref{har1}(ii) we infer that the restrictions of $v$ to $S_0^{\pm}$ can be prolonged by reflection along $T$ to harmonic functions on $B_{R/2}$. In particular, $v$ and $\nabla v$ are locally bounded in $B_{R/2}$ and \begin{equation}\label{gooddec} \int_{B_\rho}|\nabla v|^2\,dx\leq \frac{2^n\rho^n}{R^n}\int_{B_{R/2}}|\nabla v|^2\,dx\leq \frac{2^n\rho^n}{R^n}\qquad\forall \rho\in (0,R/2). \end{equation} {\sc Step 3.} Now we prove an estimate from below on the size of $K(v_h)$, namely $${\cal H}^{m}\bigl(C_\rho\cap K(v_h)\bigr)\geq \omega_{m}\rho^{m} -o(\eps_h^2)$$ \no for any cylinder $C_\rho=B^m_\rho(x_0)\times (-\rho,\rho)$ contained in $B_{R/2}$. Let $Z_h\subset B^m_\rho(x_0)$ be the projection of $C_\rho\cap K(v_h)$ on $T$; by well know properties of Hausdorff measures it suffices to prove that ${\cal L}^{m}(Z_h)\geq \omega_m\rho^m-o(\eps_h^2)$. Let $G_h=B^m_\rho(x_0)\setminus Z_h$; given $a\in (0,\rho)$ we have that the functions $v_h(z,\cdot)$ are absolutely continuous in $(-a,a)$ for almost every $z\in G_h$, hence \begin{eqnarray}\label{stn1} \int_{G_h}|v_h(z,a)-v_h(z,-a)|^2 \,dz&\leq& 2a\int_{G_h}\int_{-a}^a |{\partial v_h\over\partial y}|^2(z,y)\,dy\,dz \\ &\leq& 2a\int_{B_{R/2}}|\nabla v_h|^2\,dx\leq 2a.\nonumber \end{eqnarray} \no On the other hand, the integrals $$\int_{G_h}\bigl\vert (v_h(z,a)-c_h^+)-(v_h(z,-a)-c_h^-)\bigr\vert^2\,dz$$ are bounded because of the convergence of $(v_h-c_h^\pm)$ to $v$. Since $(c_h^+-c_h^-)$ tends to $+\infty$ we infer that ${\cal L}^{n-1}(G_h)$ is infinitesimal, hence \begin{equation}\label{stn2} \lim_{h\to +\infty} \int_{G_h}\bigl\vert (v_h(z,a)-c_h^+)-(v_h(z,-a)-c_h^-)\bigr\vert^2\,dz=0 \end{equation} \no for any $a\in (0,\rho)$. Assuming by contradiction that ${\cal L}^{n-1}(G_{h_k})\geq\delta\eps_{h_k}^2$ for some $\delta>0$ and some subsequence $(h_k)$, by (\ref{stn1}) and (\ref{stn2}) we obtain $$\delta\eps_{h_k}^2(c_{h_k}^+-c_{h_k}^-)^2 \leq {\cal L}^{n-1}(G_{h_k})(c_{h_k}^+-c_{h_k}^-)^2\leq 3a$$ \no for $k$ large enough depending on $a$. Since $a$ is arbitrary we find that $\eps_{h_k}(c_{h_k}^+-c_{h_k}^-)$ tends to $0$, a contradiction with Theorem \ref{planes}(ii). {\sc Step 4.} Applying Theorem \ref{lipappr} to the functions $u_h$ with $r=\beta R/6$ we can find $1/2$-Lipschitz functions $f_h:B^m_{r/4}\to\rn{}$ such that (we use (\ref{additional}), (\ref{additional1}), (\ref{zeq5}), (\ref{zeq33})) \begin{equation}\label{asym1} \lim_{h\to +\infty} \sup |f_h|=0,\qquad \int_{B^m_{r/4}}|f_h|^2\,dz=o(\varepsilon_h^2) \end{equation} \begin{equation}\label{asym2} \int_{B^m_{r/4}}|\nabla f_h|^2\,dz\leq Q \varepsilon_h^2 \end{equation} \begin{equation} \h{m}{B_r\cap K(v_h)\setminus \Gamma(f_h)}\leq Q \varepsilon_h^2 \nonumber \end{equation} \no for some constant $Q\geq 0$. Possibly extracting a subsequence, we can assume that the measures $$ \nu_h={1\over\varepsilon_h^2}{\cal H}^m\res \bigl(K(v_h)\setminus \Gamma(f_h)\bigr)$$ \no are weakly converging in $B_r$ to some measure $\nu$. Since the maximal height of $K(v_h)$ is infinitesimal, the measure $\nu$ is supported in $T$. {\sc Step 5.} We will prove that $\mu=\lim |\nabla v_h|^2{\cal L}^n$ is absolutely continuous with respect to ${\cal L}^n$. The convergence of the Dirichlet integrals and the final contradiction will follow at once. Let us fix a cylinder $C=B^m_\rho(x_0)\times (-3\rho,3\rho)$ contained in $B_{r/4}$, and assume that $\rho$ and $x_0$ fulfil the conditions \begin{equation}\label{sceltaro} \mu(\partial C)=\nu(\partial C)=0 \end{equation} \begin{equation}\label{sceltaro1} \liminf_{h\to +\infty}\frac{1}{\varepsilon_h^2} \int_{\partial B^m_\rho(x_0)}|\nabla f_h|^2\,dz\leq Q'<+\infty. \end{equation} \no Let $h(k)$ be a subsequence such that \begin{displaymath} \lim_{k\to +\infty}\frac{1}{\varepsilon_{h(k)}^2} \int_{\partial B^m_\rho(x_0)}|\nabla f_{h(k)}|^2\,dz\leq Q' \end{displaymath} \no and let $g_k:\overline{B}^m_\rho(x_0)\to\rn{}$ be the $1$-Lipschitz solutions of the least area problems (\ref{diri}) with $B^m_\rho(x_0)$ in place of $B_R$ and $f_{h(k)}$ in place of $v$. Let $$ g_k^+(x)=\inf_{y\in\partial B^m_\rho(x_0)}f_{h(k)}(y)+|x-y|, \qquad g_k^-(x)=\sup_{y\in\partial B^m_\rho(x_0)}f_{h(k)}(y)-|x-y|.$$ \no It is well known (and easy to check) that $g_k^+$ and $g_k^-$ are respectively the largest and the smallest $1$-Lipschitz extensions of the restriction of $f_{h(k)}$ to $\partial B^m_\rho(x_0)$. In particular $$ g_k^-(x)\leq g_k(x)\leq g_k^+(x)\quad\hbox{\rm and}\quad g_k^-(x)\leq f_{h(k)}(x)\leq g_k^+(x)\qquad\forall x\in\overline{B}^m_\rho(x_0). $$ \no Since $f_h$ uniformly converges to $0$ in $\partial B^m_\rho(x_0)$, the functions $g_k^\pm$ uniformly converge in $\overline{B}^m_\rho(x_0)$ to $\pm {\rm dist}(x,\partial B^m_\rho(x_0))$; we set $$ W_\infty=\bigl\{x=(z,y)\in\overline{C}: |y|\geq {\rm dist}\bigl(z,\partial B^m_\rho(x_0)\bigr)\bigr\}.$$ For $k$ large enough we have $\Vert g_k^\pm\Vert_\infty< 2\rho$, hence we can apply Lemma \ref{har3}, getting functions $w_k\in SBV(C)$, with the same trace of $v_{h(k)}$ of $\partial C$, such that \begin{equation}\label{seq21} \int_C |\nabla w_k|^2\,dx\leq M\int_{W_k}|\nabla v_{h(k)}|^2\,dx \end{equation} and \begin{equation} \h{m}{S_{w_k}}\leq M\h{m}{K(v_{h(k)})\cap W_k}+\h{m}{\Gamma(g_k)} \end{equation}\label{seq22} \no where $W_k$ denotes the open portion of $C$ above the graph of $g_k^+$ and below the graph of $g_k^-$. Using Lemma \ref{har2} and taking into account (\ref{asym1}) and (\ref{asym2}) we have $$\limsup_{k\to +\infty} {\h{m}{\Gamma(g_k)}-\omega_m\rho^m\over \varepsilon_{h(k)}^2}\leq {\rho Q'\over m+2\alpha-2}$$ \no for any $\alpha\geq 1$, hence $\h{m}{\Gamma(g_k)}\leq\omega_m\rho^m+ o(\varepsilon_{h(k)}^2)$. Using the fact that $W_k$ does not intersect the graph of $f_{h(k)}$ we obtain the following estimate on $\h{m}{S_{w_k}}$: \begin{equation}\label{dura} \h{m}{S_{w_k}}\leq \omega_m\rho^m+ M\varepsilon_{h(k)}^2\nu_{h(k)}(W_k)+o(\varepsilon_{h(k)}^2). \end{equation} \no Now, we define $$\tilde{v}_k(x)=\cases{v_{h(k)}(x) & if $x\notin C$;\cr\cr w_k(x) & if $x\in C$}$$ \no and we compare ${\cal F}_{h(k)}(v_{h(k)})$ with ${\cal F}_{h(k)} (\tilde{v}_k)$. Taking into account (\ref{zeq333}) and (\ref{seq21}) we get \begin{eqnarray} \label{enercontrol} \int_C |\nabla v_{h(k)}|^2\,dx &\leq& M\int_{W_k}|\nabla v_{h(k)}|^2\,dx \nonumber \\ &+& {\h{m}{S_{w_k}\cap C}-\h{m}{S_{v_{h(k)}}\cap C}\over \varepsilon_{h(k)}^2}+o(1). \end{eqnarray} \no Now, by Step 3, $\h{m}{S_{v_{h(k)}}\cap C}$ exceeds $\omega_m\rho^m$ up to an infinitesimal faster than $\varepsilon_{h(k)}^2$. Hence, letting $k\to +\infty$ in (\ref{enercontrol}) and using (\ref{dura}) we obtain $$ \mu(C)\leq M\mu(W_\infty)+M\nu(W_\infty). $$ \no Finally, our choice of $\rho$ and the fact that $\mu^s$ and $\nu$ are both supported in $T$ guarantees that $\mu^s(W_\infty)=\nu(W_\infty)=0$ and $$\mu(W_\infty)=\int_{W_\infty}|\nabla v|^2\,dx \leq 6\omega_m\Vert\nabla v\Vert_{L^\infty(B_{R/4})}^2\rho^n.$$ \no Hence, $\mu(C)\leq 6M\omega_m \Vert\nabla v\Vert^2_{L^\infty(B_{R/4})}\rho^n.$ By a standard approximation argument, the inequality remains true even if (\ref{sceltaro}) or (\ref{sceltaro1}) are not fulfilled. Indeed, the set of radii $\rho$ for which (\ref{sceltaro}) is false is at most countable, and the set of radii $\rho$ for which (\ref{sceltaro1}) is false is ${\cal L}^1$-negligible, by (\ref{asym2}) and the Fatou Lemma. Since the cylinder $C\subset B_{r/4}$ is arbitrary, it follows that $\mu^s\res B_{r/4}=0$. {\sc Step 6.} End of the proof. From Step 5, it follows that $|\nabla v_h|^2{\cal L}^n$ weakly converges to $|\nabla v|^2{\cal L}^n$ in $B_{\beta R/24}$. In particular, $$ \lim_{h\to +\infty}\int_{B_\rho}|\nabla v_h|^2\,dx= \int_{B_\rho}|\nabla v|^2\,dx\qquad\forall \rho\in (0,\beta R/24). $$ \no Choosing $\rho=\beta R/25$ and passing to the limit as $h\to +\infty$ in (\ref{zeq6}) we obtain $$\int_{B_{\beta R/25}}|\nabla v|^2\,dx\geq\beta^n.$$ \no This contradicts the energy estimate (\ref{gooddec}) with $\rho=\beta R/25$. \qed %\begin{remark} \label{anytheta} {\rm %If we assume that $\omega(t)$ vanishes on some interval %$(0,b)\subset (0,+\infty)$, then the statement of Theorem %\ref{decay1} remains true with the additional assumption $\rho0$, be given by (\ref{main1}). If $u\in\mm{B_\rho(x)}$ and $x\in K(u)$ we define \begin{equation}\label{newDA} A(\rho)={\bf A}(x,\rho),\qquad D(\rho)={\bf D}(x,\rho). \end{equation} \no By Theorem \ref{decay2} we obtain that, for any choice of $\beta\in (0,1/112)$, $A$ satisfies the following decay property, that for convenience we restate in a slightly different way: {\bf (i)} For any $\eta>0$ there exists $\eps_1(\beta,\eta)>0$ such that the conditions $$A(\rho)\leq\eps_1\rho^{m+2},\qquad A(\beta\rho)\geq\eta\max\bigl\{\rho^2 D(\rho), \sqrt{c_0}\rho^{m+2+\gamma}\bigr\}$$ \no imply $A(\beta\rho)\leq C\beta^{m+4} A(\rho)$, with $C$ depending only on $c_0$, $\gamma$. Similarly, using Theorem \ref{decay1}, we obtain that $D$ satisfies for any $\beta\in (0,6/7)$ the following decay property: {\bf (ii)} there exist $\eps_2(\beta)>0$ and $\eta(\beta)\in (0,1)$ such that the conditions $$D(\rho)\leq\eps_2\rho^m,\qquad A(\rho)\leq\eps_2\rho^{m+2}, \qquad\max\bigl\{ \sqrt{c_0}\rho^{m+2+\gamma},A(\beta\rho)\bigr\} \leq\eta\rho^2 D(\rho)$$ \no imply $D(\beta\rho/25)\leq \beta^n D(\rho)$. \begin{lemma}\label{twospeed} Let $\bar{\rho}\in (0,1)$, $A,\,D:(0,\bar{\rho})\to {\bf R}^+$ be nondecreasing functions satisfying {\bf (i)} and {\bf (ii)} above. Then, for $\beta>0$ sufficiently small (see (\ref{mag3}) below) there are constants $\eps,\,\eta,\,L>0$ such that, setting \begin{equation}\label{defen} E(\rho)=\biggl[\eta\frac{D(\rho)}{\rho^m}+\frac{A(\rho)}{\rho^{m+2}} \biggr] \end{equation} \no the following implication holds $$E(\beta\rho)>L\rho^\gamma,\,\, E(\rho)<\eps\quad\Rightarrow\quad E(\tau_1\rho)\leq\tau_1^{1/4} E(\rho)\quad\hbox{\it or}\quad E(\tau_2\rho)\leq\tau_2^{1/4} E(\rho) $$ \no with $\tau_1=\beta/25$ and $\tau_2=\beta^{1/3}$. \end{lemma} {\sc Proof.} We fix a constant $\beta\in (0,1/(112)^3)$ so small that \begin{equation}\label{mag3} 25^{m+3}\beta^{1/3}<\biggl(\frac{\beta}{25}\biggr)^{1/4},\qquad 2C\beta^{2/3}<\bigl(\beta^{1/3}\bigr)^{1/4} \end{equation} \no and we fix $\eta(\beta)>0$ according to {\bf (ii)}. Finally, we set $$ \eps=\eta(\beta)\min\biggl\{\eps_2(\beta),\eps_1\bigl(\beta, \beta^{(3m+7)/3}\eta(\beta)\bigr),\eps_1\bigl(\beta^{1/3}, \beta^{(3m+7)/3}\eta(\beta)\bigr)\biggr\} $$ \no and we choose $L>0$ satisfying the conditions $$ \frac{\eta(\beta)}{L} \bigl[\beta^{-m}+\beta^{1/3}\bigr]L\rho^\gamma$ for some $\rho\in (0,\bar{\rho})$. {\sc Case 1.} Assume that \begin{equation}\label{mag1} A(\beta\rho)\leq\beta^{(3m+7)/3}\eta(\beta)\rho^2D(\rho). \end{equation} \no Our choice of $\eps$ implies $D(\rho)<\rho^m\eps_2(\beta)$; moreover, using the inequality $E(\beta\rho)>L\rho^\gamma$ we get $$ \frac{\eta(\beta)D(\rho)}{\rho^m}\bigl[\beta^{-m}+ \beta^{1/3}\bigr]>L\rho^\gamma $$ \no and our choice of $L$ implies $\sqrt{c_0}\rho^{m+\gamma} \leq\eta(\beta)D(\rho)$. Hence, we can apply {\bf (ii)}. This, together with (\ref{mag1}) yields \begin{eqnarray*} E\Bigl(\frac{\beta}{25}\rho\Bigr)&=&\frac{A(\frac{\beta}{25}\rho)}{( \frac{\beta}{25}\rho)^{m+2}} +\eta(\beta)\frac{D(\frac{\beta}{25}\rho)} {\bigl(\frac{\beta}{25}\rho\bigr)^m} \leq\frac{25^{m+2}A(\beta\rho)}{\beta^{m+2}\rho^{m+2}}+ 25^m\eta(\beta)\frac{D(\frac{\beta}{25}\rho)}{\beta^m\rho^m}\nonumber \\ &\leq&\frac{25^{m+2}\beta^{(3m+7)/3}\eta(\beta)\rho^2 D(\rho)} {\beta^{m+2}\rho^{m+2}}+ 25^m\eta(\beta)\frac{\beta^n D(\rho)}{\beta^m\rho^m} \nonumber \\ &<& 25^{m+3} \beta^{1/3}\eta(\beta)\frac{D(\rho)}{\rho^m} \leq 25^{m+3}\beta^{1/3}E(\rho). \end{eqnarray*} \no In this case (\ref{mag3}) yields $E(\tau_1\rho)\leq\tau_1^{1/4} E(\rho)$. {\sc Case 2.} Now we assume that (\ref{mag1}) does not hold, hence \begin{equation}\label{mag2} A(\beta^{1/3}\rho)\geq A(\beta\rho)> \beta^{(3m+7)/3}\eta(\beta)\rho^2 D(\rho) \end{equation} \no Our choice of $\eps$ implies $$A(\rho)<\eps_1\bigl(\beta^{1/3},\beta^{(3m+7)/3}\eta(\beta)\bigr) \rho^{m+2}, \qquad A(\rho)<\eps_1\bigl(\beta,\beta^{(3m+7)/3}\eta(\beta)\bigr)\rho^{m+2}.$$ \no Moreover, (\ref{mag2}) and $E(\beta\rho)>L\rho^\gamma$ yield $$\frac{A(\beta\rho)}{\rho^{m+2}}\bigl[ \beta^{-m-2}+\beta^{-(6m+7)/3}\bigr]>L\rho^\gamma$$ \no and our choice of $L$ yields $\sqrt{c_0}\rho^{m+2+\gamma}\leq A(\beta\rho)$. Since $\beta<1$ we have also $\sqrt{c_0}(\beta^{1/3}\rho)^{m+2+\gamma}\leq A(\beta^{1/3}\rho)$. \no Hence we can apply {\bf (i)} with $\beta^{1/3}$ and $\beta$. This, together with (\ref{mag2}), yields \begin{eqnarray*} &&E(\beta^{1/3}\rho)=\nonumber \\ &=&\frac{A(\beta^{1/3}\rho)}{ \beta^{(m+2)/3}\rho^{m+2}}+ \eta(\beta)\frac{D(\beta^{1/3}\rho)}{\beta^{m/3}\rho^m}\leq C\frac{\beta^{(m+4)/3} A(\rho)}{\beta^{(m+2)/3}\rho^{m+2}}+ \eta(\beta)\frac {D(\rho)}{\beta^{m/3}\rho^m}\nonumber \\ &\leq&C\beta^{2/3}\frac{A(\rho)}{\rho^{m+2}}+\frac{A(\beta\rho)} {\beta^{(3m+8)/3}\rho^{m+2}}\leq C\beta^{2/3}\frac{A(\rho)}{\rho^{m+2}}+ C\frac{\beta^{m+4} A(\rho)}{\beta^{(3m+8)/3}\rho^{m+2}}\nonumber \\ &\leq&2C\beta^{2/3} E(\rho). \end{eqnarray*} \no In this case (\ref{mag3}) gives $E(\tau_2\rho)\leq\tau_2^{1/4}E(\rho)$. \qed \begin{corollary} \label{twospeed1} With the assumptions and the notations of Lemma \ref{twospeed}, $E(\rho)<\eps$ implies $$E(r)\leq \biggl(\frac{25}{\beta}\biggr)^{m+2+\sigma} \max\biggl\{\left(\frac{r}{\rho}\right)^\sigma E(\rho), \frac{L}{\beta^{\sigma}} r^\sigma\biggr\}\qquad \forall r\in (0,\rho]$$ \no with $\sigma=\min\{1/4,\gamma\}$. \end{corollary} {\sc Proof.} Let us assume that $E(\rho)<\eps$ for some $\rho\in (0,\bar{\rho})$. We inductively define an infinitesimal sequence of radii $(\rho_k)$ such that $\rho_0=\rho$, $\rho_{k+1}/\rho_k\in\{\beta,\tau_1,\tau_2\}$ and \begin{equation}\label{induz} E(\rho_k)\leq\max\biggl\{\left(\frac{\rho_k}{\rho}\right)^\sigma E(\rho), \frac{L}{\beta^{\sigma}}\rho_k^\sigma\biggr\}. \end{equation} \no Clearly, (\ref{induz}) holds with $k=0$. Assume that the inequality is valid for $\rho_k$; if $$E(\beta\rho_k)\leq L\beta^{-\sigma} (\beta\rho_k)^\sigma$$ \no we choose $\rho_{k+1}=\beta\rho_k$. Otherwise, $E(\beta\rho_k)\geq L\rho_k^\sigma$ and Lemma \ref{twospeed} yields that (\ref{induz}) is valid either with $\rho_{k+1}=\tau_1\rho_k$ or with $\rho_{k+1}= \tau_2\rho_k$. Given $r\in (0,\rho)$ there exists a unique $k$ such that $\rho_{k+1}\leq r<\rho_k$; since $\tau_1=\beta/25<\beta<\tau_2=\beta^{1/3}$ we have \begin{eqnarray*} E(r)&\leq&\biggl(\frac{\rho_k}{r}\biggr)^{m+2}E(\rho_k) \leq \biggl(\frac{25}{\beta}\biggr)^{m+2} E(\rho_k) \\ &\leq&\biggl(\frac{25}{\beta}\biggr)^{m+2}\max\biggl\{ \left(\frac{\rho_k}{\rho}\right)^\sigma E(\rho),L\beta^{-\sigma} \rho_k^\sigma\biggr\} \\ &\leq&\biggl(\frac{25}{\beta}\biggr)^{m+2+\sigma}\max\biggl\{ \left(\frac{r}{\rho}\right)^\sigma E(\rho),L\beta^{-\sigma} r^\sigma\biggr\} \end{eqnarray*} \no and the statement follows. \qed \no {\sc Conclusion of the proof.} Let $\eps$ be given by Corollary \ref{twospeed1}. 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