\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 200, pp. 1--21.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/200\hfil H\"older regularity for signed solutions] {H\"older regularity for signed solutions to singular porous medium type equations} \author[S. Puglisi \hfil EJDE-2012/200\hfilneg] {Simona Puglisi} \address{Simona Puglisi \newline Dipartimento di Matematica e Informatica \\ University of Catania \\ Viale A. Doria 6, 95125 Catania, Italy} \email{spuglisi@dmi.unict.it} \thanks{Submitted July 17, 2012. Published November 15, 2012.} \subjclass[2000]{35K67, 35B65, 35B45} \keywords{Singular parabolic equations; H\"older continuity} \begin{abstract} We prove H\"older regularity for bounded signed solution to singular porous medium type equations, whose prototype is $$u_t-\operatorname{div}m|u|^{m-1}Du=0\quad\text{weakly in }E_T,$$ with $m\in(0,1)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction and statement of main result} Let $E$ be an open set in $\mathbb{R}^N$, for $T>0$ denote the cylindrical domain $$E_T=E\times (0,T]$$ and let $\Gamma=\partial E_T\setminus\bar E\times\{T\}$ be its parabolic boundary. We consider quasi-linear homogeneous singular parabolic partial differential equation $$\label{pm} u_t-\operatorname{div} \mathcal{A}(x,t,u,Du)=0 \quad \text{weakly in }E_T,$$ where $\mathcal{A}:E_T\times\mathbb{R}^{N+1}\to\mathbb{R}^N$ is measurable and subject to the structure conditions $$\label{struct cond} \left\{\begin{gathered} \mathcal{A}(x,t,z,\xi)\cdot\xi\geq C_0m|z|^{m-1}|\xi|^2 \\ |\mathcal{A}(x,t,z,\xi)|\leq C_1m|z|^{m-1}|\xi| \end{gathered}\right.$$ for a.e. $(x,t)\in E_T$, for every $z\in\mathbb{R}$, $\xi\in\mathbb{R}^N$, where $C_0$, $C_1$ are given positive constants and $01$, such a modulus vanishes when $u$ vanishes, and for this reason we say that the equation \eqref{pm}-\eqref{struct cond} is \emph{degenerate}. Whenever $01$, or \emph{fast diffusion}, when $01$ and $\alpha\in(0,1)$ such that for every compact set $\mathcal{K}\subset E_T$ $|u(x_1,t_1)-u(x_2,t_2)|\leq c\,\|u\|_{\infty,E_T} \Big(\frac{\|u\|_{\infty,E_T}^{\frac{1-m}{2}}|x_1-x_2|+|t_1-t_2|^{1/2}} {\operatorname{m-dist}(\mathcal{K},\Gamma)}\Big)^\alpha,$ for every pair of points $(x_1,t_1),(x_2,t_2)\in\mathcal{K}$. \end{theorem} The constant $c$ depends only upon the data, the norm $\|u\|_{\infty,\mathcal{K}}$ and $\operatorname{m-dist}(\mathcal{K},\Gamma)$; the constant $\alpha$ depends only upon the data and the norm $\|u\|_{\infty,\,\mathcal{K}}$. In some physical applications it is natural to consider positive solutions to quasi-linear parabolic equations of the form \eqref{pm}, and it is also a very useful simplification from the mathematical point of view. Therefore, most of the papers directly deal with positive solutions. A H\"older regularity result for signed solutions was obtained first by DiBenedetto in \cite{dib86} for degenerate ($p>2$) $p$-laplacian type equations and then by Chen and DiBenedetto in \cite{chen dib} for singular ($12$) and the degenerate porous medium type equations (for $p=2$ and $m>1$). As a consequence, it only remained open the case of the singular porous medium type equations. We want to point out that the difficulty in our case is due to the presence of the term $|u|^{m-1}$ in the modulus of continuity; indeed, the fact that $u$ changes sign plays a crucial role here. In the $p$-laplacian case, the modulus of continuity is $|Du|^{p-2}$, thus the proof does not change if $u$ is positive or if it changes sign. One could think to follow the lines of \cite{chen dib} with minor modification, but at some point it will appear $|u|^{m-1}$ that one cannot control from above in a sublevel of the modulus of $u$, being 0\frac1m\,, \] and define $|v|^{n-1}v=u,$ which is equivalent to $v=|u|^{\frac{1}{n}-1}u.$ Notice that $Du=n|v|^{n-1}Dv, \quad Dv=\frac{1}{n}|u|^{\frac{1}{n}-1}Du.$ With this substitution equation \eqref{pm} becomes $\big(|v|^{n-1}v\big)_t-\operatorname{div}\widetilde{\mathcal{A}}(x,t,v,Dv)=0 \quad \text{weakly in }E_T,$ where $\widetilde{\mathcal{A}}(x,t,v,Dv)=\mathcal{A}(x,t,u,Du)\big|_{u=|v|^{n-1}v}.$ Now, let us see what the structure conditions become. We have \begin{align*} \widetilde{\mathcal{A}}(x,t,v,Dv)\cdot Dv &=\frac{1}{n}|u|^{\frac{1}{n}-1}\mathcal{A}(x,t,u,Du)\cdot Du\\ &\geq\frac{m}{n}\,C_0|u|^{\frac{1}{n}+m-2}|Du|^2\\ &=nmC_0|v|^{1+nm-2n}|v|^{2(n-1)}|Dv|^2\\ &=nmC_0|v|^{nm-1}|Dv|^2; \end{align*} since the exponent isnm-1>0, the equation is degenerate''. In the same way \begin{align*} |\widetilde{\mathcal{A}}(x,t,v,Dv)| &=|\mathcal{A}(x,t,u,Du)|\leq mC_1|u|^{m-1}|Du|\\ &=mC_1|v|^{n(m-1)}n|v|^{n-1}|Dv| =nmC_1|v|^{nm-1}|Dv|. \end{align*} If we denote our variable withu$again, we are then led to consider equations of the type $(|u|^{n-1}u)_t-\operatorname{div}\widetilde{\mathcal{A}}(x,t,u,Du)=0 \quad \text{weakly in }E_T,$ with structure conditions $$\label{struct cond2} \left\{\begin{gathered} \widetilde{\mathcal{A}}(x,t,z,\xi)\cdot\xi\geq nmC_0|z|^{nm-1}|\xi|^2\\ |\widetilde{\mathcal{A}}(x,t,z,\xi)|\leq nmC_1|z|^{nm-1}|\xi|, \end{gathered}\right.$$ for a.e.$(x,t)\in E_T$and for every$z\in\mathbb{R}$,$\xi\in\mathbb{R}^N$. Without loss of generality, we can assume$n$to be odd; in this case $|u|^{n-1}u=u^n,$ and we can rewrite the equation as $$\label{pm2} (u^n)_t-\operatorname{div}\widetilde{\mathcal{A}}(x,t,u,Du)=0 \quad \text{weakly in }E_T.$$ Hence we have reduced problem \eqref{pm}-\eqref{struct cond} to \eqref{pm2} with structure conditions \eqref{struct cond2}. Let us now see what the notion of weak solution becomes in this new setting. A function$u$such that$u^n\in C_{\rm loc}(0,T;L^2_{\rm loc}(E))$with$|u|^{nm}\in L^2_{\rm loc}(0,T;H^1_{\rm loc}(E))$is a local weak sub(super)-solution to \eqref{pm2} if for every compact set$\mathcal{K}\subset E$and every subinterval$[t_1,t_2]\subset(0,T]$$\int_{\mathcal{K}} u^n\varphi\,dx\big|_{t_1}^{t_2} +\int_{t_1}^{t_2}\int_{\mathcal{K}}[-u^n\varphi_t +\widetilde{\mathcal{A}}(x,t,u,Du)\cdot D\varphi]\,dx\,dt\leq(\geq)\,0,$ for all non-negative test functions$\varphi\in H^1_{\rm loc}(0,T;L^2(\mathcal{K}))\cap L^2_{\rm loc} (0,T;H^1_0(\mathcal{K}))$. \section{Preliminaries} Let$r,s\geq1$and let us consider the Banach spaces \begin{gather*} V^{r,s}(E_T)=L^\infty\big(0,T;L^r(E)\big)\cap L^s\big(0,T;W^{1,s}(E)\big),\\ V_0^{r,s}(E_T)=L^\infty\big(0,T;L^r(E)\big)\cap L^s\big(0,T;W_0^{1,s}(E)\big), \end{gather*} both equipped with the norm $\|v\|_{V^{r,s}(E_T)}=\operatornamewithlimits{ess\,sup}_{0s (for a proof one can see \cite{dib}). \begin{proposition}\label{prop1} If v\in V^{r,s}_0(E_T), then there exists a positive constant \gamma, depending only upon N,r, and s, such that \[ \iint_{E_T}|v|^q \,dx\, dt\leq\gamma^q\Big(\iint_{E_T}|Dv|^s \,dx\, dt\Big) \Big(\operatornamewithlimits{ess\,sup}_{00\}\big|^\frac{r}{N+r}\,\|v\|^r_{V^r(E_T)}.$ \end{proposition} Given$(y,s)\in E_T$, and$\lambda,R>0$, we will denote by$K_R(y)$the cube centered at$y$with edge$2R$; i.e., $K_R(y)=\Big\{x\in\mathbb{R}^N:\max_{1\leq i\leq N}|x_i-y_{i}|l\}|\leq\gamma\,\frac{\rho^{N+1}}{|\{v1 and \alpha>0. If \[ Y_0\leq C^{-\frac1\alpha}b^{-\frac1{\alpha^2}},$ then$Y_n$converges to 0, as$n$tends to$+\infty$. \end{lemma} Let us prove energy estimates we will need later. We start with estimates for super-solutions, then we will state the analogous ones for sub-solutions. \begin{proposition}[Energy estimates for super-solutions] Let$u$be a local, weak super-solution to \eqref{struct cond2}-\eqref{pm2} in$E_T$. There exists a positive constant$\gamma$, depending only upon the data, such that for every cylinder$(y,s)+Q_R(\lambda)\subset E_T$, every level$k\in\mathbb{R}$and every non-negative, piecewise smooth cutoff function$\zeta$vanishing on$\partial K_R(y)$, \label{ee super} \begin{split} &\operatornamewithlimits{ess\,sup}_{s-\lambda0$ $\log^+s=\max\{\log s,0\}.$ \begin{proposition}[Logarithmic estimates] Let $u$ be a local, weak solution to \eqref{struct cond2}-\eqref{pm2} in $E_T$. There exists a positive constant $\gamma$, depending only upon the data, such that for every cylinder $(y,s)+Q_R(\lambda)\subset E_T$, every level $k\in\mathbb{R}$ and every non-negative, piecewise smooth cutoff function $\zeta=\zeta(x)$ \label{log estim} \begin{split} &\operatornamewithlimits{ess\,sup}_{s-\lambda0$such that $(y,s)+ Q_{2\theta\rho}\Big( \frac{(2\rho)^2}{\omega^{nm-1}}\Big)\subset E_T\,, \quad \operatornamewithlimits{ess\,osc}_{(y,s)+Q_{2\theta\rho}\Big( \frac{(2\rho)^2}{\omega^{nm-1}}\Big)} u \leq \omega\,,$ where $\theta=\omega^{\frac{1-n}2}.$ Then, there exist$\eta_*,\,c_0\in (0,1)$, depending only upon data, such that $\operatornamewithlimits{ess\,osc}_{\mathcal{Q}^*} u\leq \eta_* \omega\,,$ where $\mathcal{Q}^* = (y,s)+Q_{\theta\rho} \left( \theta_* \rho^2 \right)\,, \quad \theta_* = \frac{c_0}2\, \omega^{1-nm}\,.$ \end{theorem} As we show at the end, the local H\"older continuity of locally bounded solutions is a straightforward consequence of Theorem \ref{red}. The proof of this theorem splits into two alternatives. Let$\epsilon\in(0,1),R>0$, and$(y,s)\in E_T$. Consider the cylinder $Q_\epsilon:=K_{R^{1-\epsilon\frac{n-1}{2}}}(y) \times(s-R^{2-\epsilon(nm-1)},s]\subset E_T,$ and set $\mu_+\geq\operatornamewithlimits{ess\,sup}_{(y,s)+Q_\epsilon}u\,,\quad \mu_-\leq\operatornamewithlimits{ess\,inf}_{(y,s)+Q_\epsilon}u\,, \quad \omega=\mu_+-\mu_-\,.$ Let us recall that, without loss of generality, we can assume$\mu_+>0$,$\mu_-<0$and $\mu_+\geq|\mu_-|.$ Indeed, otherwise just change the sign of$u$and work with the new function. If we take$2\rhoR^\epsilon, then we guarantee that $(y,s)+Q_{2\theta\rho}\Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\subset Q_\epsilon.$ \section{The first alternative} We distinguish two alternatives; the first of them consists in assuming $$\label{lem 1alt} \Big|\Big\{u<\mu_-+\frac{\omega}{2}\Big\}\cap \Big\{(y,s)+Q_{2\theta\rho}\Big(\frac{(2\rho)^2}{\omega^{nm-1}}\big)\Big\} \Big| \leq c_0\Big|Q_{2\theta\rho}\Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\Big|,$$ being $c_0\in (0,1)$ a constant to be determined later. Let us prove now the following De Giorgi type lemma. \begin{lemma} There exists a number $c_0\in (0,1)$, depending only upon data, such that if \eqref{lem 1alt} holds, then $$\label{tesi lem1} u\geq \mu_-+\frac{\omega}{4}\quad\text{a.e. in }(y,s)+Q_{\theta\rho} \Big(\frac{\rho^2}{\omega^{nm-1}}\Big).$$ \end{lemma} \begin{proof} Without loss of generality we may assume $(y,s)=(0,0)$ and for $k=0,1,\ldots$, set $\rho_k=\rho+\frac{\rho}{2^k},\quad \widetilde{K}_k=K_{\theta\rho_k},\quad \widetilde{Q}_k=\widetilde{K}_k\times\big(-\frac{\rho_k^2}{\omega^{nm-1}},0\big].$ Let $\zeta_k$ be a piecewise smooth cutoff function in $\widetilde{Q}_k$ vanishing on the parabolic boundary of $\widetilde{Q}_k$ such that $0\leq\zeta_k\leq1$, $\zeta_k=1$ in $\widetilde{Q}_{k+1}$ and $|D\zeta_k|\leq\frac{2^{k+2}}{\rho}\omega^\frac{n-1}{2},\quad 0\leq \zeta_{k,t}\leq\frac{2^k}{\rho^2}\,\omega^{nm-1}.$ We consider the following levels $$\label{levels} \begin{gathered} h_k=\mu_-+\frac{\omega}{4}+\frac{\omega}{2^{k+2}} \quad \text{if }\mu_-\geq-\frac{\omega}{8},\\ h_k=\mu_-+\frac{\omega}{2^5}+\frac{\omega}{2^{k+5}} \quad \text{if }\mu_-<-\frac{\omega}{8}. \end{gathered}$$ We first treat the least favorable case in which $u$ might be close to zero; i.e., we assume first that $$\label{1caso} \mu_-\geq-\frac{\omega}{8}.$$ Write down the energy estimates \eqref{ee super} for $(u-h_k)_-$ over the cylinder $\widetilde{Q}_k$, to obtain \begin{align*} &\operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}\frac{\omega}{2^4}\right\}}|u|^{nm-1} \big|D[(u-h_k)_-\zeta_k]\big|^2\,dx\,d\tau\\ &\quad +\iint_{\widetilde{Q}_k\cap\left\{u\leq \frac{\omega}{2^4}\right\}} \Big(\frac{\omega}{2^4}\Big)^{nm-1}\left(\frac{\omega}{2^4}-h_k\right)_-^2|D\zeta_k|^2\,dx\,d\tau\\ &\leq \iint_{\widetilde{Q}_k}|u|^{nm-1}\big|D[(u-h_k)_-\zeta_k]\big|^2\,dx\,d\tau +\frac{2^{2(k+1)}}{\rho^2}\,\omega^{n(m+1)}|A_k|\,, \end{split} where $A_k=\{u0\,,\quad \mu_-<0\,,\quad \mu_+\geq|\mu_-|.$ Throughout this new section, let us assume that \eqref{lem 1alt} does not hold; i.e., $\Big|\big\{u\geq\mu_-+\frac{\omega}{2}\big\}\cap \Big\{(y,s)+Q_{2\theta\rho} \Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\Big\} \Big| <(1-c_0)\Big|Q_{2\theta\rho}\Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\Big|.$ For simplicity in the following we assume $(y,s)=(0,0)$. \begin{lemma} There exists a time level $t^*$ in the interval $\big(-\frac{(2\rho)^2}{\omega^{nm-1}}, -\frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}\big)$ such that $$\label{K 2alt} \Big|\big\{u(\cdot,t^*)<\mu_-+\frac{\omega}{2}\big\}\cap K_{2\theta\rho}\Big| >\frac{c_0}{2}|K_{2\theta\rho}|.$$ This in turn implies $$\label{Kreverse 2alt} \Big|\big\{u(\cdot,t^*)\geq\mu_+-\frac{\omega}{4}\big\}\cap K_{2\theta\rho}\Big| \leq\Big(1-\frac{c_0}{2}\Big)|K_{2\theta\rho}|.$$ \end{lemma} \begin{proof} By contradiction, suppose that \eqref{K 2alt} does not hold for any $t^*$ in the indicated range; then \begin{align*} \Big|\big\{u<\mu_-+\frac{\omega}{2}\big\}\cap Q_{2\theta\rho} \Big(\frac{(2\rho)^2}{\omega^{nm-1}}\Big)\Big| &=\int_{-\frac{(2\rho)^2}{\omega^{nm-1}}} ^{-\frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}} \Big|\big\{u(\cdot,t^*)<\mu_-+\frac\omega2\big\}\cap K_{2\theta\rho}\Big|dt^*\\ &\quad +\int_{-\frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}}^{0} \Big|\big\{u(\cdot,t^*)<\mu_-+\frac\omega2\big\}\cap K_{2\theta\rho}\Big|dt^*\\ &\leq\frac{c_0}{2}|K_{2\theta\rho}|\Big(1-\frac{c_0}{2}\Big) \frac{(2\rho)^2}{\omega^{nm-1}}+|K_{2\theta\rho}| \frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}\\ &\mu_+-\frac{\omega}{2^{j^*}}\big\}\cap K_{2\theta\rho}\Big| <\big(1-\frac{c_0^2}{4}\big)|K_{2\theta\rho}|, \] for all times $t^*\frac{\omega}{4}$, so $k>0$. The number $\nu$ in the definition of the logarithmic function is taken as $\nu=\frac{\omega}{2^{j+2}}$, where $j$ is a positive number to be chosen. Thus we have $\psi\left(H^n,(u^n-k^n)_+,\nu^n\right) =\log^+\Big(\frac{H^n}{H^n-(u^n-k^n)_++\frac{\omega^n}{2^{(j+2)n}}}\Big),$ where $H^n=\operatornamewithlimits{ess\,sup}_{K_{2\theta\rho}\times(t^*,0)} \left[u^n-\big(\mu_+^n-(\frac{\omega}{4})^n\big)\right]_+.$ The cutoff function $x\to\zeta(x)$ is taken such that $\zeta =1 \quad \text{on K_{(1-\sigma)2\theta\rho} for }\sigma\in(0,1),\quad |D\zeta|\leq\frac{1}{\sigma\theta\rho}\,.$ With these choices, inequality \eqref{log estim} yields $$\label{1 up to 0} \begin{split} &\int_{K_{(1-\sigma)2\theta\rho}}\psi^2(x,t)\,dx\\ &\leq\int_{K_{2\theta\rho}}\psi^2(x,t^*)\,dx +\gamma\int_{t^*}^0\int_{K_{2\theta\rho}}|u|^{n(m-1)} \psi|D\zeta|^2\,dx\,d\tau, \end{split}$$ for all $t^*\leq t\leq 0$. Let us observe that $\psi\leq\log\bigg(\frac{\frac{\omega^n}{2^{2n}}}{\frac{\omega^n}{2^{(j+2)n}}}\bigg) =jn\log2.$ To estimate the first integral on the right-hand side of \eqref{1 up to 0}, observe that $\psi$ vanishes on the set $\{u^n\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\}; \] on such a set, since$\psi$is a decreasing function of$H^n$, we have $\psi^2\geq\log^2\Big(\frac{\frac{\omega^n}{2^{2n}}}{\frac{\omega^n}{2^{(j+1)n}}}\Big) =(j-1)^2n^2\log^22;$ hence, for all$t^*\leq t\leq0, we obtain $\Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\} \cap K_{(1-\sigma)2\theta\rho}\Big| \leq \Big\{\big(\frac{j}{j-1}\big)^2\Big(1-\frac{c_0}{2}\Big) +\frac{\gamma}{\sigma^2j}\Big\}|K_{2\theta\rho}|.$ On the other hand, \begin{align*} &\Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\} \cap K_{2\theta\rho}\Big|\\ &\leq \Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\} \cap K_{(1-\sigma)2\theta\rho}\Big|+|K_{2\theta\rho}\setminus K_{(1-\sigma)2\theta\rho}|\\ &\leq \Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\} \cap K_{(1-\sigma)2\theta\rho}\Big|+N\sigma|K_{2\theta\rho}|. \end{align*} Then $\Big|\Big\{u^n(\cdot,t)>\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big\}\cap K_{2\theta\rho}\Big| \leq\Big\{\Big(\frac{j}{j-1}\Big)^2\Big(1-\frac{c_0}{2}\Big) +\frac{\gamma}{\sigma^2j}+N\sigma\Big\}|K_{2\theta\rho}|.$ for allt^*\leq t\leq0$. Now choose$\sigma$so small and then$jso large as to obtain \begin{align*} \Big| &\big\{u(\cdot,t)>\Big(\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big) ^{1/n}\big\}\cap K_{2\theta\rho}\Big| \leq\big(1-\frac{c_0^2}{4}\big)|K_{2\theta\rho}|&\forall& t^*\leq t\leq0. \end{align*} Notice that our hypotheses imply\mu_+\geq\frac{\omega}{2},\,\mu_+<\omega; therefore, \begin{align*} \Big(\mu_+^n-\frac{\omega^n}{2^{(j+2)n}}\Big)^{1/n} &<\Big(\mu_+^n-\frac{\mu_+^n}{2^{(j+2)n}}\Big)^{1/n} =\mu_+\Big(1-\frac{1}{2^{(j+2)n}}\Big)^{1/n}\\ &\leq \mu_+\Big(1-\frac{1}{2^{(j+2)n}\,n}\Big) \leq\mu_+-\frac{\omega}{2^{(j+2)n+1}\,n}\,. \end{align*} The proof is completed once we choosej^*$as the smallest integer such that $\mu_+-\frac{\omega}{2^{(j+2)n+1}\,n}\leq\mu_+-\frac{\omega}{2^{j^*}}. \qedhere$ \end{proof} \begin{corollary}\label{coroll} For all$j\geq j^*$and for all times$-\frac{c_0}{2}\frac{(2\rho)^2}{\omega^{nm-1}}\mu_+-\frac{\omega}{2^{j}}\Big\} \cap K_{2\theta\rho}\Big|<\Big(1-\frac{c_0^2}{4}\Big)|K_{2\theta\rho}|. \end{corollary} Motivated by Corollary \ref{coroll}, introduce the cylinder $Q_*=K_{2\theta\rho}\times\big(-\theta_*(2\rho)^2,0\big], \quad \text{with } \theta_*=\frac{c_0}{2}\omega^{1-nm}.$ \begin{lemma}\label{lem Q_*} For every $\nu_*\in(0,1)$, there exists a positive integer $q_*=q_*(\text{data},\nu_*)$ such that $\Big|\Big\{u\geq\mu_+-\frac{\omega}{2^{j_*+q_*}}\Big\}\cap Q_* \Big|\leq\nu_*|Q_*|.$ \end{lemma} \begin{proof} Write down the energy estimates \eqref{ee sub} for the truncated functions $(u-k_j)_+$, with $k_j=\mu_+-\frac{\omega}{2^j}$, for $j=j_*,\ldots,j_*+q_*$ over the cylinder $\widetilde{Q}=K_{4\theta\rho}\times\Big(-c_0\frac{(2\rho)^2}{\omega^{nm-1}},0\Big] \supset Q_*\,;$ the cutoff function $\zeta$ is taken to be one on $Q_*$, vanishing on the parabolic boundary of $\widetilde{Q}$ and such that $|D\zeta|\leq\frac{1}{\theta\rho}\,,\quad 0\leq \zeta_t\leq\frac{\omega^{nm-1}}{c_0\rho^2}\,.$ Thanks to these choices, the energy estimates \eqref{ee sub} take the form \begin{align*} &\iint_{\tilde Q}|u|^{nm-1}|D(u-k_j)_+|^2\zeta^2\,dx\,d\tau\\ &\leq \gamma\Big\{\frac{\omega^{nm-1}}{c_0\rho^2}\iint_{\tilde Q} \Big(\int_{k_j}^u(s-k_j)_+s^{n-1}ds\Big)\,dx\,d\tau\\ &\quad +\frac{\omega^{n-1}}{\rho^2}\iint_{\tilde Q}|u|^{nm-1}(u-k_j)_+^2\,dx\,d\tau\Big\}. \end{align*} Estimating $\begin{split} \int_{k_j}^u(s-k_j)_+s^{n-1}ds \leq u^{n-1}\frac{(u-k_j)_+^2}{2} \leq\omega^{n-1}\frac{(u-k_j)_+^2}{2}\,, \end{split}$ and taking into account that $(u-k_j)_+\leq\frac{\omega}{2^j}$, yields $\iint_{\tilde{Q}}|u|^{nm-1}|D(u-k_j)_+|^2\zeta^2\,dx\,d\tau \leq\gamma\Big(\frac{\omega}{2^j}\Big)^2\omega^{n-1} \frac{\omega^{nm-1}}{ c_0\,\rho^2}|Q_*|.$ Note that $u>k_j\geq\frac{\omega}{4}$: indeed the second inequality is equivalent to $\mu_+\geq|\mu_-|\Big(\frac14+\frac1{2^j}\Big)\Big(\frac34-\frac1{2^j}\Big)^{-1},$ and this is implied by our assumptions. Thus we can estimate \begin{align*} \iint_{\widetilde{Q}}|u|^{nm-1}|D(u-k_j)_+|^2\zeta^2\,dx\,d\tau &\geq\iint_{Q_*}|u|^{nm-1}|D(u-k_j)_+|^2\,dx\,d\tau\\ &\geq\Big(\frac{\omega}{4}\Big)^{nm-1}\iint_{Q_*}|D(u-k_j)_+|^2\,dx\,d\tau; \end{align*} it follows that $$\label{en ineq} \iint_{Q_*}|D(u-k_j)_+|^2\,dx\,d\tau \leq\gamma\Big(\frac{\omega}{2^j}\Big)^2\omega^{n-1}\frac{1}{c_0\rho^2}|Q_*|.$$ Next, apply the isoperimetric inequality \eqref{lemma degiorgi} to the function $u(\cdot, t)$, for $t$ in the range $(-\theta_*(2\rho)^2,0]$, over the cube $K_{2\theta\rho}$, and for the levels $k=k_jk_{j+1}\}\cap K_{2\theta\rho}|\\ &\leq \frac{(2\theta\rho)^{N+1}}{|\{u(\cdot,t)k_j\}\cap Q_*.$ Square both sides of this inequality and estimate above the term containing $|D(u-k_j)_+|$ by inequality \eqref{en ineq}, to obtain $|A_{j+1}|^2\leq\,\frac{\gamma}{\,c_0^5\,}|Q_*|\,\left(|A_j|-|A_{j+1}|\right).$ Add these recursive inequalities for $j=j_*+1,\ldots,j_*+q_*-1$, where $q_*$ is to be chosen. Majorizing the right-hand side with the corresponding telescopic series, gives $(q_*-2)|A_{j_*+q_*}|^2\leq\sum_{j=j_*+1}^{j_*+q_*-1}|A_{j+1}|^2 \leq \frac{\gamma}{c_0^5}|Q_*|^2.$ From this $|A_{j_*+q_*}|\leq\frac{1}{\sqrt{q_*-2}}\,\sqrt{\frac{\gamma}{\,c_0^5\,}\,}|Q_*|.$ The number $\nu_*$ being fixed, choose $q_*$ from $\frac{1}{\sqrt{q_*-2}}\,\sqrt{\frac{\gamma}{\,c_0^5\,}}=\nu_*. \qedhere$ \end{proof} Now let $\xi\in(0,\frac{1}{2})$, $a\in(0,1)$ be fixed numbers. \begin{lemma}\label{lem1sub} There exists a number $c_*\in (0,1)$, depending upon the data, $\xi$, and $a$, such that if $$\label{lem 2alt} |\left\{u\geq\mu_+-\xi\omega\right\}\cap Q_* |\leq c_*|Q_*|,$$ then $u\leq \mu_+-a\xi\omega\quad \text{a.e. in }Q_{\theta\rho}(\theta_*\rho^2).$ \end{lemma} \begin{proof} For $k=0,1,\ldots$, set $\rho_k=\rho+\frac{\rho}{2^k},\quad K_k=K_{\theta\rho_k}, \quad Q_k=K_k\times(-\theta_*\rho_k^2,0].$ Let $\zeta(x,t)=\zeta_1(x) \zeta_2(t)$ be a piecewise smooth cutoff function in $Q_k$ such that \begin{gather*} \zeta_1=\begin{cases} 1&\text{in }K_{k+1}\\ 0&\text{in }\mathbb{R}^N\setminus K_k, \end{cases} \quad |D\zeta_1|\leq \frac{2^{k+2}}{\theta\rho}, \\ \zeta_2= \begin{cases} 1&\text{if }t\geq-\frac{\rho_{k+1}^2}{\omega^{nm-1}}\\ 0&\text{if }t<-\frac{\rho_k^2}{\omega^{nm-1}}, \end{cases} \quad 0\leq\zeta_{2,t}\leq \frac{2^k}{\theta_*\rho^2}. \end{gather*} Choose the sequence of truncating levels $h_k=\mu_+-\xi_k\omega,\quad \text{where } \xi_k=a\xi+\frac{1-a}{2^k}\,\xi,$ and write down the energy estimates \eqref{ee sub} for $(u-h_k)_+$ over the cylinder $Q_k$, \begin{align*} &\operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}h_k\geq\Big(\frac{1}{2}-\xi\Big)\omega: indeed the last inequality is equivalent to $\mu_+\geq|\mu_-|\Big(\frac{1}{2}-\xi+\xi_k\Big) \Big(\frac{1}{2}+\xi-\xi_k\Big)^{-1},$ and this follows by our hypotheses. Therefore, we obtain \label{sup grad} \begin{gathered} \operatornamewithlimits{ess\,sup}_{-\frac{\rho_k^2}{\omega^{nm-1}}h_0\}\cap Q_0|}{|Q_0|} =\frac{|\{u>\mu_+-\xi\omega\}\cap Q_0|}{|Q_0|}\\ & \leq \frac{\gamma^{-\frac{N+2}{2}}}{(1-a)^{-(N+2)}} \Big(\frac{1}{2}-\xi\Big)^{\frac{N(nm-1)+2(n-1)}{2}} 2^{-(N+2)^2}, \end{align*} which is \eqref{lem 2alt} withc_*:=\frac{\gamma^{-\frac{N+2}{2}}}{(1-a)^{-(N+2)}} \big(\frac{1}{2}-\xi\big)^{\frac{N(nm-1)+2(n-1)}{2}}\,2^{-(N+2)^2}$. This completes the proof. \end{proof} Thanks to Lemma \ref{lem Q_*}, we can apply Lemma \ref{lem1sub} with$\xi=\frac{1}{2^{j_*+q_*}}$and$a=\frac{1}{2}$, getting $u \leq \mu_+-\frac{\omega}{2^{j_*+q_*+1}}\quad \text{a.e. in }Q_{\theta\rho}\big(\theta_*\rho^2\big),$ which implies $\operatornamewithlimits{ess\,sup}_{Q_{\theta\rho}(\theta_*\rho^2)}u \leq\mu_+-\frac{\omega}{2^{j_*+q_*+1}}\,.$ Hence $\operatornamewithlimits{ess\,osc}_{Q_{\theta\rho}(\theta_*\rho^2)}u \leq\mu_+-\operatornamewithlimits {ess\,inf}_{Q_{\theta\rho}(\theta_*\rho^2)}u -\frac{\omega}{2^{j_*+q_*+1}}\leq\omega\Big(1-\frac{1}{2^{j_*+q_*+1}}\Big).$ \section{Conclusion} The two alternatives just discussed can be combined to prove Theorem \ref{red}. \begin{proof}[Proof of Theorem \ref{red}] The concluding statement of the first alternative says that $\operatornamewithlimits{ess\,osc}_{Q_{\theta\rho}(\frac{\rho^2}{\omega^{nm-1}})}u\leq \frac{3}{4}\,\omega;$ analogously, the conclusion of the second alternative is that $\operatornamewithlimits{ess\,osc}_{\mathcal{Q}^*} u =\operatornamewithlimits{ess\,osc}_{Q_{\theta\rho}(\theta_*\rho^2)}u \leq \omega\Big(1-\frac{1}{2^{j_*+q_*+1}}\Big).$ Recalling the definition of$\theta_*$, we observe that $\mathcal{Q}^* =Q_{\theta\rho}(\theta_*\rho^2) \subset Q_{\theta\rho}\Big(\frac{\rho^2}{\omega^{nm-1}}\Big).$ The thesis follows by defining $\eta_* := 1-\frac{1}{2^{j_*+q_*+1}}\,.$ \end{proof} We are now ready to prove the local H\"older regularity. \begin{proof}[Proof of Theorem \ref{teor}] Let us remind that we fixed$\epsilon\in(0,1),\,R>0$,$(y,s)\in E_T$, and we considered the cylinder $Q_\epsilon=K_{R^{1-\epsilon\frac{n-1}{2}}}(y)\times(s-R^{2-\epsilon(nm-1)},s]\subset E_T.$ Let now$\beta,\delta\in(0,1)$to be chosen, and let us introduce the sequences $R_k:= \beta^kR, \quad \omega_k:=\delta^k\omega, \quad \theta_k:=\omega_k^{\frac{1-n}2}, \quad \mathcal Q_k:= (y,s)+Q_{\theta_k R_k}\Big(\frac{R_k^2}{\omega_k^{nm-1}}\Big),$ for$k\in\mathbb{N}$. The thesis follows by standard arguments once we prove that $$\begin{gathered} \mathcal Q_{k+1} \subset\mathcal Q_k \subset Q_\epsilon \subset E_T \quad \forall k\in\mathbb{N}, \\ \operatornamewithlimits{ess\,osc}_{\mathcal Q_k} u \leq \omega_k. \end{gathered} \label{star}$$ The inclusion$\mathcal Q_0\subset Q_\epsilon$immediately follows by assumption \eqref{assump omega}, while$\mathcal Q_{k+1}\subset\mathcal Q_k$is equivalent to $\beta\leq\min\{\delta^{\frac{n-1}2},\,\delta^{\frac{nm-1}2}\} =\delta^{\frac{n-1}2}.$ To prove \eqref{star}, we will argue by induction. The validity for$k=0$is true by construction since $\operatornamewithlimits{ess\,osc}_{\mathcal Q_0}u\leq\operatornamewithlimits{ess\,osc}_{Q_\epsilon}u\leq\omega.$ Assume that \eqref{star} holds for$k$and apply Theorem~\ref{red} taking$\rho = \frac{R_k}2$and$\omega=\omega_k$; thanks to these choices $\theta=\theta_k\,, \quad (y,s)+ Q_{2\theta\rho}\Big( \frac{(2\rho)^2}{\omega^{nm-1}}\Big) = \mathcal Q_k\,.$ The assumptions of Theorem~\ref{red} are satisfied because \eqref{star} holds for$k$; hence, we have$\operatorname{ess\,osc}_{\mathcal{Q}^*} u\leq \eta_* \omega_k$, where in this setting $\mathcal{Q}^* = (y,s)+Q_{\theta_k\frac{R_k}2} \left( \frac{c_0}{8}\, \omega_k^{1-nm} R_k^2 \right).$ This leads us to choose$\delta=\eta_*\in (0,1)$, so that$\eta_* \omega_k=\omega_{k+1}$. It remains only to check$\mathcal Q_{k+1}\subset \mathcal{Q}^*$, which by a simple calculation is equivalent to $\beta\leq\min\Big\{ \frac 12 \,\delta^{\frac{n-1}2}\,, \sqrt{\frac{c_0}8}\, \delta^{\frac{nm-1}2}\Big\}.$ We conclude by choosing$\beta\$ small enough. \end{proof} \subsection*{Acknowledgments} The author wishes to warmly thank Ugo Gianazza for his help and support while considering the problem. The author also would like to thank the anonymous referee for the helpful comments. \begin{thebibliography}{0} \bibitem{chen dib} Y. Z. Chen, E. DiBenedetto; \emph{On the local behaviour of solutions of singular parabolic equations}, Arch. Rational Mech. Anal. 103 (1988), no. 4, 319-345. \bibitem{degiorgi} E. De Giorgi; \emph{Sulla differenziabilit\a e l'analiticit\a delle estremali degli integrali multipli regolari}, (Italian) Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957) 25-43. \bibitem{dib86} E. DiBenedetto; \emph{On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 3, 487-535. \bibitem{dib} E. DiBenedetto; \emph{Degenerate Parabolic Equations}. Universitext. Springer, New York, 1993. \bibitem{dib g v} E. DiBenedetto, U. Gianazza, V. Vespri; \emph{Harnack's inequality for degenerate and singular equations}. Springer Monographs, Springer-Verlag, New York, 2012. \bibitem{lad sol ural} O. A. Lady\v{z}enskaja, V. A. Solonnikov, N. N. Ural'tzeva; \emph{Linear and quasilinear equations of parabolic type}. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1967. \bibitem{porzio vespri} M. M. Porzio, V. Vespri; \emph{H\"older estimates for local solutions of some doubly nonlinear degenerate parabolic equations}, J. Differential Equations 103 (1993), no. 1, 146-178. \bibitem{jl} J. L. V\'azquez; \emph{The Porous Medium Equation}. Mathematical Theory, Oxford Math. Monogr., Oxford University Press, Oxford, 2007. \bibitem{vespri} V. Vespri; \emph{On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations}, Manuscripta Math. 75 (1992), no. 1, 65-80. \end{thebibliography} \end{document}