\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 38, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/38\hfil A new proof of Harnack's inequality] {A new proof of Harnack's inequality for \\ elliptic partial differential equations \\ in divergence form} \author[R. Crescimbeni, L. Forzani, A. Perini\hfil EJDE-2007/38\hfilneg] {Raquel Crescimbeni, Liliana Forzani, Alejandra Perini} % in alphabetical order \address{Raquel Crescimbeni\newline Departmento de Matem\'atica, Universidad Nacional del Comahue, Neuqu\'en(8300), Argentina} \email{rcrescim@uncoma.edu.ar} \address{Liliana Forzani \newline Departmento de Matem\'atica, Universidad Nacional del Litoral, Santa Fe (3000), Argentina} \email{liliana.forzani@gmail.com } \address{Alejandra Perini \newline Departmento de Matem\'atica, Universidad Nacional del Comahue, Neuqu\'en (8300), Argentina } \email{alejandraperini@gmail.com \quad aperini@uncoma.edu.ar} \thanks{Submitted January 9, 2006. Published March 1, 2007.} \thanks{The first and third authors were supported by Departamento de Matematica, Fa.E.A., \hfill\break\indent Universidad Nacional del Comahue. The second author was supported by CONICET and \hfill\break\indent Prog: CAID+D-UNL} \subjclass[2000]{35B65, 35B45} \keywords{Elliptic equations; divergence form; weak solutions; \hfill\break\indent Harnack inequality; Holder continuity} \begin{abstract} In this paper we give a new proof of Harnack's inequality for elliptic operator in divergence form. We imitate the proof given by Caffarelli for operators in nondivergence form. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{property}[theorem]{Property} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction}\label{intro} At the end of the 1950's De Giorgi \cite{DeG:57} showed that weak solutions of the second order elliptic partial differential equations in divergence form \begin{equation}\label{divergence} Lu= \sum_{i,j=1}^n \frac{\partial}{\partial x_i}\Big(a_{ij}(x)\frac{\partial u}{\partial x_j} \Big)=0, \end{equation} satisfy pointwise estimations, which allowed him to prove that all weak solutions of \eqref{divergence} are locally H\"older continuous. In 1961, Moser \cite{MO1} proved that non\-negative weak solutions of \eqref{divergence} satisfy the so called Harnack's inequality: Let $\Omega \subset \mathbb{R}^n$ be an open set, for all $Q'$ and $Q$ open cubes in $\mathbb{R}^n$ such that $Q' \subset Q \subset \Omega$, $Q' = \frac{1}{4}Q$, there exists a constant $C>1$, which depends on $Q, Q' $ and the uniform ellipticity of \eqref{divergence}, such that \begin{equation*} \sup_{Q'}u \leq C \inf_{Q'} u \quad \hbox{(Harnack inequality)} \end{equation*} for any nonnegative weak solution $u$ of \eqref{divergence} in $Q$. As a consequence of Harnack inequality, Moser obtained H\"older regularity for all weak solutions of \eqref{divergence}, and so Moser's method became the classical method for proving the regularity of weak solutions. The next big step in the study of H\"older regularity was given by Krylov and Safonov \cite{Kry;Saf:80} in 1980. They proved the Harnack inequality for the case of strong solutions of parabolic equations with elliptic part in nondivergence form. In 1986, Caffarelli \cite{CAF:86} gave a proof of the Harnack inequality for nonnegative smooth solutions of second order elliptic partial differential equations in nondivergence form \begin{equation}\label{nondivergence} Lu= \sum_{i,j=1}^n a_{ij}(x) \frac{\partial^2 u}{\partial x_i \partial x_j} =0. \end{equation} In this proof, and as a consequence of the Maximum Principle of Alexandroff-Bakelman-Pucci (see \cite{SAL:95}), Caffarelli used two properties of the nonnegative solutions of the equation \eqref{nondivergence} on cubes of $\mathbb{R}^n$; namely: \begin{property}\label{prop1.1} There exist constants $\gamma_0 >0$ and $0 1 \}|> \gamma_0 r^n, $$ then $\inf_{Q_{\frac{r}{2},x_0}}u(x) >C_1$, where $|\cdot|$ is the Lebesgue measure in $\mathbb{R}^n$ and $Q_{r,x_0}$ an open cube of size $r$ and center $x_0$, (i.e $Q_{r,x_0}= \{x \in \mathbb{R}^n : \|x -x_0\|_{\infty}< r/2\}$ where $\|x \|_{\infty}= \max_{1 \leq i \leq n}|x_i|$) . \end{property} \begin{property}\label{prop1.2} Let $M>2$, there exists $C_2>0$ such that if $u$ is a nonnegative solution of $Lu=0$ in $Q_{Mr,x_0}$ and $\inf_{Q_{r,x_0}}u\geq 1$, then $\inf_{Q_{\frac{Mr}{2},x_0}}u(x)> C_2$. \end{property} These two properties and the Calderon-Zygmund decomposition are the main tools that Caffarelli used to prove the weak Harnack inequality for nonnegative solutions of \eqref{nondivergence}. As a consequence of this inequality, Caffarelli obtained an oscillation property for all solutions, which together with Property \ref{prop1.2} allowed him to prove the Harnack inequality. Again, as a consequence of the Harnack inequality, Caffarelli proved the H\"older continuity for all solutions in nondivergence form (see \cite{CAF:86}). To prove the Harnack inequality for nonnegative weak solutions of \eqref{divergence}, Moser used an iterative argument for the functions given by $$ \Phi(p, h)= \Big( \frac{1}{|Q_{h,0}|} \int_{Q_{h,0}} u^p dx \Big)^{1/p} $$ with $ p \in \mathbb{R}$, $00$ and $01\}|> \gamma_0 r^n$ then $\inf_{Q_{\frac{r}{2},x_0}}u > C_1$. A vectorial space $U$, of functions, satisfies \textbf{Property \ref{prop1.2}} if, given $M>2$, there exists $C_2 = C_2(\lambda, \Lambda,M)>0$ such that if $u \in U$ is nonnegative in $Q_{Mr,x_0}$ and $\inf_{Q_{r,x_0}}u \geq 1$, then $\inf_{Q_{\frac{Mr}{2},x_0}}u >C_2$. A vectorial space $U$, of functions, satisfies the \textbf{weak Harnack inequality} if there exist $p>0$ and $C=C(\lambda, \Lambda,p)>0$ such that \begin{equation} \label{weakharnackp} \Big( \frac{1}{|Q_{2r,x_0}|}\int_{Q_{2r,x_0}} u^p dx \Big)^{1/p} \leq C \inf_{Q_{r,x_0}}u \end{equation} for all $u \in U$, nonnegative in $Q_{4r,x_0} \subset \Omega$. A set $U$ is \textbf{locally bounded} if $\sup_{Q}|u|< \infty$ for all $u \in U$ and for each cube $Q$ in $\Omega$, that is, if all $u \in U$ belong to $L_{\rm loc}^{\infty}(\Omega)$. We will refer to this property by saying that $U \in L_{\rm loc}^{\infty}(\Omega)$. A vectorial space $U \in L_{\rm loc}^{\infty}(\Omega)$ satisfies the \textbf{oscillation property} if there exists $0< \theta <1$ such that \begin{equation} \label{oscillation} \hbox{osc}_{Q_{r,x_0}}u \leq \theta \hspace{.1cm} \hbox{osc}_{Q_{4r,x_0}}u, \end{equation} for all $u \in U$ and $Q_{r,x_0}$ such that $Q_{4r,x_0} \subset \Omega$, where $\hbox{osc}_{Q_{r,x_0}}u = \sup_{Q_{r,x_0}}u - \inf_{Q_{r,x_0}}u$. A vectorial space $U$, of functions, satisfies the \textbf{H\"older continuity property} if there exist positive constants $C$ and $\alpha$ such that $|u(x) - u(y)| \leq C |x-y|^{\alpha}$ for all $u \in U$ and for all $Q_{4r,x_0}\subset \Omega$, with $x,y \in Q_{r,x_0}$. A vectorial space $U$, of functions, satisfies the \textbf{Harnack inequality} if there exist $\beta = \beta(\lambda, \Lambda,n)>0$ such that \begin{equation} \label{harnackp} \sup_{Q_{r,x_0}} u \leq \beta \inf_{Q_{r,x_0}} u \end{equation} for all $u \in U$, nonnegative in $Q_{4r,x_0} \subset \Omega$. \section{Statement of the main result}\label{mainresult} In \cite{AFT:01} the authors proved, that in the general setting of spaces of homogeneous type, the Properties \ref{prop1.1} and \ref{prop1.2} mentioned above, are sufficient conditions to establish the weak Harnack inequality. The technique used by Aimar, Forzani and Toledano for the weak Harnack inequality proof is like the Caffarelli's steps to prove the weak Harnack inequality for nonnegative solutions of the elliptic operator in nondivergence form given by \eqref{nondivergence}. More precisely they obtained the following theorem. \begin{theorem} \label{teoprin} For $U$ a vectorial space of functions, we have \begin{enumerate} \item if $U$ satisfies the Properties \ref{prop1.1} and \ref{prop1.2} then $U$ satisfy the weak Harnack inequality. \item if $U$ satisfies the weak Harnack inequality then $U$ satisfy the oscillation property. Moreover if $U \in L_{\rm loc}^{\infty}(\Omega)$ then $U$ satisfies the H\"older $\alpha$- continuity. \item if $U$ satisfies the Property \ref{prop1.2} and the oscillation property then $U$ satisfies the Harnack inequality. \end{enumerate} \end{theorem} Now we present the main result of this work, where the vectorial space $U$ in $\mathbb{R}^n$ is \begin{equation} \label{space} U=\{ u \in W^{1,2}(\Omega) \hbox{ such that $u$ is a weak solution of } Lu = 0\}, \end{equation} where $L$ is given by \eqref{divergence}. \begin{theorem}\label{teoprinc} The vectorial space of functions $U$ given by (\ref{space}) satisfies the Property \ref{prop1.1} and the Property \ref{prop1.2} defined in Section 2. \end{theorem} The proof of this Theorem will be given in Section 5. Using this result and the Theorem \ref{teoprin} for our particular case of the vectorial space of functions $U$ in $\mathbb{R}^n$ given by (\ref{space}), we obtain a new proof of the Harnack inequality for nonnegative weak solutions of the operator in divergence form, that follows the lines of Caffarelli's proof for nonnegative smooth solutions of the operator in nondivergence form. \section{ Previous Results} \label{propdiv} First of all, we present some classic results about Sobolev Spaces and differential equations. \begin{theorem}[Sobolev Inequality] \label{thm4.1} Let $u \in {W_0}^{1,2}(\Omega)$. Then there exists a constant $\beta=\beta(n)$ such that \begin{equation} \label{SobolevC0} \Big( \int_{\Omega} |u|^{2^*} dx \Big)^{1/2^*} \leq \beta \Big( \int_{\Omega} |{\nabla u}|^2 dx \Big) ^{1/2}, \end{equation} where $ 2^* = 2n/(n-2)$. \end{theorem} For a proof the above theorem, see for example \cite{EV:98}. \begin{theorem}[Caccioppoli Inequality] \label{thm4.2} Let $M>1$ and $u$ a positive weak subsolution of $Lu=0$ in $Q_{Mr,x_0}$ and $\Phi \in W_0^{1,2}(Q_{Mr,x_0})$. Then \begin{equation} \label{Caccioppoli} \int_{Q_{Mr,x_0}} |\nabla u|^2 \Phi^2 dx \leq C \int_{Q_{Mr,x_0}} |\nabla \Phi|^2 u^2 dx, \end{equation} where $C=C(\lambda, \Lambda, n)$. \end{theorem} For a proof of the above theorem, see for example \cite{MO1}. Caccioppoli estimates will permit us to prove other results such as the boundedness of the norm $L^{\infty}$ of the solutions. \begin{theorem} \label{thm4.3} Let $u$ be a positive weak subsolution of $Lu=0$ in $Q_{4r,x_0}$. Then \begin{equation} \label{acotsup} \|u\|_{L^{\infty}(Q_{r,x_0})} \leq \frac{C}{r^{\frac{n}{2}}}\|u\|_{ L^2(Q_{2r,x_0})}, \end{equation} where $C=C(\lambda, \Lambda)$. \end{theorem} \begin{proof} By the scale argument it is sufficient to prove \begin{equation} \label{maximoQ1} \|u\|_{L^{\infty}(Q_1,\frac{x_0}{r})} \leq C\|u\|_{L^2(Q_{2,\frac{x_0}{r}})}, \end{equation} where $C=C (\Lambda , n)$. For $ k =\frac{n}{n-2}$ and for all $j \in \mathbb{N}$ we define the number \begin{equation} \label{Nj} N_j = \Big( \int_{Q_{r_j,\frac{x_0}{r}}}u^{2k^j}dx\Big) ^{1/(2k^j)}, \end{equation} where the $r_j$ are such that the succession of cubes $Q_{r_j,\frac{x_0}{r}}$ satisfying \begin{equation} Q_{2,\frac{x_0}{r}} \supset Q_{r_1,\frac{x_0}{r}} \supset Q_{r_2,\frac{x_0}{r}} \supset\dots \supset Q_{r_{j-1}, \frac{x_0}{r}} \supset Q_{r_{j},\frac{x_0}{r}}\supset \dots\supset Q_{1,\frac{x_0}{r}}, \end{equation} with $\mathop{\rm dist} (\partial Q_{r_{j},\frac{x_0}{r}}, \partial Q_{r_{j-1},\frac{x_0}{r}}) \sim j^{-2}$, where $\sim$ denote equivalent. First we have to see that \begin{equation} \label{limsup} \|u\|_{L^{\infty} ( Q_{1,\frac{x_0}{r}} )} \leq \limsup_{j \to \infty} N_j. \end{equation} In fact, let us suppose that $\|u\|_{L^{\infty}(Q_{1,\frac{x_0}{r}})}= M$ and let $M'< M$. We define $$ A= \{ x \in Q_{1,\frac{x_0}{r}}: |u(x)|> M' \}. $$ Then $|A|>0$. By definitions of $N_j$ and $A$ we obtain \begin{equation*} N_j = \Big( \int_{Q_{r_j,\frac{x_0}{r}}}u^{2k^j}dx\Big) ^{1/(2k^j)}\geq \Big( \int_{A} u^{2k^j}dx\Big) ^{1/(2k^j)} \geq M'|A|^{1/(2k^j)}. \end{equation*} Since $\lim_{j \to \infty} |A|^{1/(2k^j)}=1$, then $\liminf_{j \to \infty} N_j \geq M$ and (\ref{limsup}) follows. Let $\Phi \in C_0^1(Q_{r_{j-1},\frac{x_0}{r}})$ such that $\Phi \equiv 1$ in $Q_{r_j,\frac{x_0}{r}}$ and $ |\nabla \Phi| \leq \frac{c}{r_{j-1}-r_j}$ in $Q_{r_{j-1},\frac{x_0}{r}}$. Since $u \in W^{1,2}(Q_{4,\frac{x_0}{r}}) $ then $v= \Phi u \in W_{0}^{1,2}(Q_{r_{j-1},\frac{x_0}{r}})$. By the Sobolev inequality (\ref{SobolevC0}) and the Caccioppoli inequality (\ref{Caccioppoli}) we have \begin{equation} \label{desigualdad} \begin{aligned} \int_{Q_{r_j,\frac{x_0}{r}}} u^{2k} &\leq \int_{Q_{r_{j-1},\frac{x_0}{r}}} \left( \Phi u \right)^{2k} dx \\ &\leq \beta \Big( \int_{Q_{r_{j-1},\frac{x_0}{r}}} |\nabla ( \Phi u)|^2 dx \Big)^k \\ &\leq \beta \Big( \int_{Q_{r_{j-1},\frac{x_0}{r}}} |\nabla \Phi|^2 u^2 dx + \int_{Q_{r_{j-1},\frac{x_0}{r}}} |\Phi|^2 |\nabla u|^2 dx \Big)^k \\ &\leq \beta \Big( (1+ C(n, \lambda, \Lambda))\int_{Q_{r_{j-1}, \frac{x_0}{r}}} |\nabla \Phi|^2 u^2 dx \Big)^k \\ &\leq \beta \Big( \frac{c^2(1+ C(n, \lambda, \Lambda))}{{(r_{j-1}-r_j)}^2} \int_{Q_{r_{j-1},\frac{x_0}{r}}}u^2 dx \Big)^k. \end{aligned} \end{equation} By item \ref{subsol}) in Section \ref{defi} we have that $u^{k^{j-1}}$ is a positive subsolution. Applying (\ref{desigualdad}) to $u^{k^{j-1}}$ we have \begin{align*} N_j^{2k^{j-1}} &= \Big( \int_{Q_{r_j,\frac{x_0}{r}}} \big(u^{k^{j-1}} \big)^{2k}dx \Big) ^{1/k} \\ &\leq \frac{\beta^{1/k} c^2(1+ C(n, \lambda, \Lambda))}{(r_{j-1}-r_j)^2} \int_{Q_{r_{j-1},\frac{x_0}{r}}} \big( u^{k^{j-1}} \big)^{2}dx \\ &\leq Cj^4 N_{j-1}^{2k^{j-1}}. \end{align*} Then, $$ N_j \leq \left( Cj^4 \right)^{1/(2k^{j-1})}N_{j-1}. $$ Iterating this last inequality we obtain \begin{equation*} N_j \leq N_0 \prod_{i=1}^{\infty} \big(Ci^4\big)^{1/(2k^{i-1})}; \end{equation*} so that \begin{equation*} \ln N_j \leq \ln N_0 + \sum_{i=1}^{\infty} \frac{1}{2k^{i-1}} \ln \big( Ci^4 \big); \end{equation*} that is, \begin{equation*} N_j \leq e^{\sum_{i=1}^{\infty} \frac{1}{2k^{i-1}} \ln \big( Ci^4 \big)} N_0 \leq e^C N_0. \end{equation*} By (\ref{limsup}) we have \begin{align*} \|u\|_{L^{\infty}(Q_{1,\frac{x_0}{r}})} & \leq \limsup_{j \to \infty} N_j \leq e^C N_0 \\ & = e^C \Big(\int_{Q_{r_0,\frac{x_0}{r}}}u ^{2k^0} \Big)^{1/(2k^0)} \\ &= c \|u\|_{L^2(Q_{2,\frac{x_0}{r}})}; \end{align*} so we obtain (\ref{maximoQ1}). \end{proof} Our second step is to give another result which will provide us that the logarithm of a weak solution of \eqref{divergence} is a weak subsolution. Furthermore, we will obtain an estimate in the $L^2$ norm of the $\nabla(-\log(u+ \epsilon))$, with $\epsilon \in (0,1)$ and $u$ is a nonnegative weak solutions of \eqref{divergence}. The statement of this result is as follows. \begin{lemma} \label{lemitas} Let $u$ be a nonnegative weak solution of $Lu=0$ in $Q_{2Mr,x_0}$ and $f$ is defined for $x \in \mathbb{R}_0^+$ by $f(x) = \max \{-\log (x+\epsilon),0\}$ with $\epsilon \in (0,1)$, then \begin{enumerate} \item $v=f(u)$ is a nonnegative weak subsolution of \eqref{divergence} in $Q_{2Mr,x_0}$, $M \geq 1$. \item \begin{equation} \label{cotagradiente} \frac{1}{|Q_{Mr,x_0}|} \int _{Q_{Mr,x_0}} |\nabla v|^2 dx \leq C(Mr) ^{-2}, \end{equation} where $C= C(\lambda, \Lambda, n)>0$. \end{enumerate} \end{lemma} \begin{proof} The function $f$ is differentiable in $\mathbb{R}^+ \cup \{0\}$, except in $x=1-\epsilon$. The first derivative is $$ f'(x) = \begin{cases} - \frac{1}{x+\epsilon} &\text{if } 0 \leq x < 1 - \epsilon \\ 0 &\text{if } x > 1 - \epsilon \end{cases} $$ then for a fix $\epsilon$ we have $f' \in L^{\infty}(\mathbb{R}_0^+)$. Moreover the second derivative is $$ f''(x) = \begin{cases} \frac{1}{(x+\epsilon)^2} &\text{if } 0 \leq x < 1 - \epsilon \\ 0 &\text{if } x > 1 - \epsilon \end{cases} $$ then $f''(x) = [f'(x)]^2$ for $x+ \epsilon \neq 1$. For $\Psi \in C_0^1(Q_{2Mr,x_0})$, $\Psi \equiv 1$ onto $Q_{Mr,x_0}$ and $ |\nabla \Psi| \leq \frac{c(n)}{Mr}$ in $Q_{2Mr,x_0}$, we consider $w(x) = \Psi^2(x) f'(u(x))$ if $u(x) \neq 1 - \epsilon$. Since $f$ is a piecewise smooth function with $f' \in L^{\infty}(\mathbb{R}_0^+)$, we can deduce that $f'(u) \in W^{1,2}(Q_{2Mr,x_0})$ and $\nabla(f'(u))= f''(u) \nabla u$ at almost every point of $Q_{2Mr,x_0}$, then $w = \Psi^2 f'(u) \in W_0^{1,2}(Q_{2Mr,x_0})$ and \begin{equation} \label{Lu=0} \begin{aligned} 0&=- \left< Lu,w \right> = - \left< Lu,\Psi^2 f'(u) \right> \\ &= \int \sum_{i=1}^{n}a_{ij}(x) \frac{\partial u}{\partial x_j} \frac{\partial(\Psi^2 f'(u))}{\partial x_i} dx \\ &= \int \sum_{i=1}^n a_{ij}(x) \frac{\partial u}{\partial x_j} \left( 2\Psi \frac{\partial \Psi}{\partial x_i} f'(u) + \Psi^2 f''(u) \frac{\partial u}{\partial x_i} \right) dx \\ &= \int \sum_{i=1}^n a_{ij}(x) \frac{\partial u}{\partial x_j} 2 \Psi \frac{\partial \Psi}{\partial x_i} f'(u) dx + \int \sum_{i=1}^n a_{ij}(x) \frac{\partial u}{\partial x_j} {\Psi}^2 \frac{\partial u}{\partial x_i} f''(u) dx . \end{aligned} \end{equation} By the ellipticity property given by (\ref{elipticity}), the previous identity and the inequality $ 2ab \leq \delta a^2 + \frac{b^2}{\delta}$ for all $\delta>0$ we have \begin{align*} \lambda \int \Psi^2 f''(u) |\nabla u|^2 dx &\leq \int \sum_{i=1}^{n} a_{ij}(x) \frac{\partial u}{\partial x_j} {\Psi}^2 \frac{\partial u}{\partial x_i} f''(u) dx\\ &= -2 \int \sum_{i=1}^{n} a_{ij}(x) \Psi f'(u) \frac{\partial u}{\partial x_j} \frac{\partial \Psi}{\partial x_i} dx\\ &\leq 2\int \sum_{i=1}^{n} \Big| a_{ij}(x) \Psi f'(u) \frac{\partial u}{\partial x_j} \frac{\partial \Psi}{\partial x_i}\Big| dx \\ &\leq \Lambda \Big( \int \delta^2 \Psi^2 |f'(u)|^2 |\nabla u |^2 dx + \int \frac{|\nabla \Psi|^2 }{\delta^2} dx \Big). \end{align*} Then we obtain \begin{equation*} \int \left[\lambda \Psi^2 f''(u) - \Lambda \delta^2 \Psi^2|f'(u)|^2 \right]|\nabla u|^2 dx \leq \int \frac{|\nabla \Psi|^2}{\delta^2}dx. \end{equation*} If $\nabla u \neq 0$, then $f''(u) =|f'(u)|^2$. Taking $ \delta^2 = \frac{\lambda}{2\Lambda}$ we have \begin{equation} \label{desig} \int \Psi^2 |f'(u)|^2 |\nabla u|^2 dx \leq C(\lambda,\Lambda) \int |\nabla \Psi|^2 dx. \end{equation} By (\ref{desig}) and the fact that $\Psi \equiv 1$ in $Q_{Mr,x_0}$, it results that \begin{align*} \int_{Q_{Mr,x_0}}|\nabla v|^2 dx &= \int_{Q_{Mr,x_0}} |\nabla (f(u))|^2 dx \\ &= \int_{Q_{Mr,x_0}} |f'(u)|^2 |\nabla u|^2 dx \\ &\leq \int_{Q_{2Mr,x_0}} \Psi^2 |f'(u)|^2 |\nabla u|^2 dx \\ &\leq C(\lambda, \Lambda) \int_{Q_{2Mr,x_0}} |\nabla \Psi|^2 dx \\ & \leq C(\lambda, \Lambda) (Mr)^{n-2}. \end{align*} \end{proof} We remark that Lemma \ref{lemitas} is a necessary tool for the proof of Theorem \ref{teoprinc}. \section{Proof of Theorem \ref{teoprinc}} \textbf{Property \ref{prop1.1}:} It is sufficient to prove that $v= f(u) = \max \{ -\log(u+ \epsilon),0\}$ is bounded for all $x \in Q_{\frac{r}{2},x_0}$. In fact, if this is true we have that $-\log(u+\epsilon) \leq v(x) < C$ for all $x \in Q_{\frac{r}{2},x_0}$, then $\log(u+\epsilon)^{-1} < C$ and so $ u> \frac{1}{10^{C}}= C_1$ in $Q_{\frac{r}{2},x_0}$. Let $A = \{ x \in Q_{r,x_0}: u(x) >1 \}$. If $x \in A$, we have $v(x)=0$, then \begin{equation} \label{complemento} |Q_{r,x_0} - A| = |\{x \in Q_{r,x_0} : u(x) \leq 1 \}| < (1- \gamma_0) r^n. \end{equation} By Lemma \ref{lemitas} we have that $v$ is a positive weak subsolution of $Lu=0$ in $Q_{2r,x_0}$, then by (\ref{acotsup}) we obtain that, \begin{equation} \label{acotacion} \sup_{Q_{\frac{r}{2},x_0}} v^2 \leq \frac{c}{r^n} \int_{Q_{r,x_0}} v^2 dx. \end{equation} Furthermore, if we prove that there exists a constant $\gamma_0$ such that \begin{equation} \label{v2} \frac{c}{r^n} \int_{Q_{r, x_0}} v^2 dx \leq r^{2-n} \int_{Q_{r, x_0}} |{\nabla v}|^2 dx, \end{equation} and we use the estimation (\ref{cotagradiente}) with $M = 1$ we have \begin{equation}\label{v1} \int_{Q_{r,x_0}} |\nabla v|^2 dx \leq C r^{n-2}; \end{equation} then by (\ref{acotacion}), (\ref{v2}) and (\ref{v1}) we have that $\sup_{Q_{\frac{r}{2},x_0}} v$ is bounded. Finally we have only to show (\ref{v2}). The H\"older's inequality, estimate (\ref{complemento}) and the Sobolev´s inequality allow us to obtain that \begin{align*} \frac{1}{r^n} \int_{Q_{r, x_0}} v^2 dx &= \frac{1}{r^n} \int_{Q_{r, x_0} - A} v^2 dx \\ &\leq \frac{1}{r^n} \Big( \int_{Q_{r, x_0} - A} v^{\frac{2n}{n-2}} dx \Big)^{(n-2)/n} |Q_{r, x_0} - A|^{2/n}\\ &\leq \frac{c \left((1- \gamma_0)r^n \right)^{2/n}}{r^n} \Big( \int_{Q_{r, x_0}} v^{\frac{2n}{n-2}} dx \Big) ^{(n-2)/n}\\ &\leq \frac{c \beta^2 \left((1- \gamma_0)r^n \right)^{2/n}}{r^n} \Big( \frac{1}{r^2} \int_{Q_{r, x_0}} v^2 dx + \int_{Q_{r, x_0}} |{\nabla v}|^2 dx \Big). \end{align*} Then we have \begin{equation*} \left( \frac{1}{r^n} - \frac{c \beta^2 (1 - \gamma_0)^{\frac{n}{2}}}{r^n} \right) \int_{Q_{r, x_0}} v^2 dx \leq \frac{c \beta^2 r^2(1 - \gamma_0)^{2/n}}{r^n} \int_{Q_{r, x_0}} |\nabla v|^2 dx. \end{equation*} If we choose $\gamma_0$ such that $1 - c \beta^2 (1 - \gamma_0)^{\frac{n}{2}} \geq \frac{1}{2}$ we obtain (\ref{v2}). \smallskip \textbf{ Property \ref{prop1.2}:} The main estimate that we need is the following \begin{equation} \label{fundamental} \frac{1}{(Mr)^n} \int_{Q_{Mr,x_0}} v^2 dx \leq \tilde{C} (1-M^{-n+1})(Mr)^{2-n} \int_{Q_{Mr,x_0}} \left| \nabla v \right|^2dx, \end{equation} where $v = \max \{- \log(u+ \epsilon), 0 \}$. The above estimate, (\ref{acotsup}) and (\ref{cotagradiente}) allow us to obtain the result in the following inequality \begin{align*} \|v\|^2_{L^{\infty}(Q_{\frac{Mr}{2},x_0})} &\leq \frac{c}{(Mr)^n} \|v\|^2_{L^2(Q_{Mr,x_0})}\\ &= \frac{c}{(Mr)^n} \int_{Q_{Mr,x_0}} v^2 dx\\ &\leq \tilde{C} (1-M^{-n+1})(Mr)^{2-n} \int_{Q_{Mr,x_0}} \left| \nabla v \right|^2dx\\ &\leq \tilde{C} (1-M^{-n+1})(Mr)^{2-n}(Mr)^{n-2}\\ &= \tilde{C}(1-M^{-n+1}). \end{align*} Then $\sup_{Q_{\frac{Mr}{2},x_0}} v \leq C$. As in the proof of Property \ref{prop1.1} we have $\inf_{Q_{\frac{Mr}{2},x_0}}u > C_2$. Now we need only to prove (\ref{fundamental}). Since $u \geq 1$ in $Q_{r,x_0}$ then $v = \max \{- \log(u+ \epsilon), 0 \}= 0$ in $Q_{r,x_0}$. In particular $v(x_0)=0$ then for all $x\in Q_{Mr,x_0}- Q_{r,x_0}$ and $m$ such that $ m >1 $ we can write, \begin{equation*} v(x)= \int_{\frac{1}{m}}^1 \frac{\partial v}{\partial t}(tx +(1-t)x_0) dt = \int_{\frac{1}{m}}^1 \nabla v(tx-(1-t)x_0).(x -x_0) dt . \end{equation*} By the last identity, the chain rule and the Fubini's Theorem, we obtain (\ref{fundamental}) as follows \begin{align*} \int_{Q_{Mr,x_0}} |v(x)|^2 dx &= \int_{Q_{Mr,x_0}-Q_{r,x_0}}|v(x)|^2 dx \\ &\leq \int_{Q_{Mr,x_0}-Q_{r,x_0}} \int_{\frac{1}{m}}^1 \left| \nabla v(tx +(1-t)x_0) \right|^2 \left|x-x_0 \right|^2 dtdx\\ &\leq Cn \big( \frac{Mr}{2} \big)^2 \int_{\frac{1}{m}}^1 \int_{Q_{Mr,x_0}-Q_{r,x_0}} \left| \nabla v(tx +(1-t)x_0) \right|^2 dxdt\\ &= C(Mr)^2 \int_{\frac{1}{m}}^1 \int_{Q_{Mrt,x_0}-Q_{rt,x_0}} \left| \nabla v(y) \right|^2 \frac{dy}{t^n}dt\\ &= C(Mr)^2 \int_{Q_{Mr,x_0}- Q_{\frac{r}{m},x_0}} \left| \nabla v(y) \right|^2 \Big[ \int_{\frac{2\|y-x_0\|_{\infty}}{Mr}}^{\frac{2\|y-x_0\|_{\infty}}{r}} \frac{dt}{t^n} \Big] dy\\ &\leq C (Mr)^2 (1-M^{-n+1}) \int_{Q_{Mr,x_0}} \left| \nabla v(y) \right|^2 dy. \end{align*} \begin{thebibliography}{00} \bibitem{AFT:01} H. 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