\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 130, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/130\hfil Extreme values of infinity-harmonic functions] {Some properties of the extreme values of infinity-harmonic functions} \author[T. Bhattacharya\hfil EJDE-2009/130\hfilneg] {Tilak Bhattacharya} \address{Tilak Bhattacharya \newline Department of Mathematics, Western Kentucky University\\ Bowling Green, KY 42101, USA} \email{tilak.bhattacharya@wku.edu} \thanks{Submitted August 4, 2009. Published October 9, 2009.} \subjclass[2000]{35J60, 35J70} \keywords{local behavior, extreme values} \begin{abstract} We study local behavior of infinity-harmonic functions, in particular, the extreme values of such functions on a ball. We show that the extreme values obey certain relationships, and use this to derive an interior growth estimate. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this work we study some properties of infinity-harmonic functions. The questions of local behavior and regularity form the main motivation for our present work, and our hope is that the results in this work will provide some insight into these matters. We introduce notation for our discussion. We take $\Omega\subset\mathbb{R}^n$, $n\ge 2$ to stand for a domain. Our results, being local in nature, would apply even when $\Omega$ is unbounded. We will often use $x,y,z$ to denote points on $\mathbb{R}^n$, and $o$ will stand for the origin. We will occasionally write a point as $x=(x_1,x_2,\dots ,x_n)$. A ball of radius $r$ and center $x$ will be denoted by $B_r(x)$. We define $u:\Omega\to \mathbb{R}$, to be infinity-harmonic in $\Omega$ if it satisfies, in the sense of viscosity, $$\label{e1} \Delta_{\infty}u=\sum_{i,j=1}^n \frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}\frac{\partial^2 u}{\partial x_i\partial x_j}=0\quad \text{in } \Omega.$$ For motivation and a detailed discussion of various properties of such solutions, see \cite{acj,tb2,tdm, c,ceg,pssw}. An important aspect of these functions is that they are completely characterized by the cone comparison property", a fact first observed in \cite{ceg}. Like all other results, our work too exploits this property to achieve its ends. We refer to \cite{acj,c,ceg} for a detailed discussion of this issue. Our motivation for this work is related to the question of local regularity. Despite great recent progress the matter of local regularity remains somewhat unresolved. It is well-known that $u$ is locally Lipschitz continuous \cite{tb2,c,ceg,j}, and it has been shown that $u$ is $C^{1,\alpha}$ when $n=2$, see \cite{es,s}. However, differentiability is still open when $n\ge 3$. The work \cite{es} proves deep results regarding this matter and states a conjecture, whose proof would lead to $C^{1,\alpha}$ regularity of such functions. Our goal in this work is to derive new properties of the extreme values of such functions which we hope may lead to some additional insight into this problem. In what follows, we will limit our discussion to a ball $B_r(x)$ in $\Omega$. Let $\overline{A}$ denote the closure of a set $A$. Define $M_x(r)=\sup_{\bar{B}_r(x)}u$ and $m_x(r)=\inf_{\bar{B}_r(x)}u$. We will drop the subscript if $x=o$. It is well-known that $u$ satisfies the strong maximum principle, and these values are attained on the boundary of the ball $B_r(x)$ \cite{tb5,ceg}. It is also well-known that $M_x(r)-u(x)$ and $u(x)-m_x(r)$ are both convex in $r$, and the following limit exists: $$\label{e2} \lim_{r\downarrow 0}\frac{M_x(r)-u(x)}{r} =\lim_{r\downarrow 0}\frac{u(x)-m_x(r)}{r}:=\Lambda(x).$$ Moreover, the limit $\Lambda(x)$ would equal $|Du(x)|$, if $u$ is differentiable at $x$, see \cite{acj,tb2,c,ce,ceg}. In addition, it was shown in \cite{ce} that if $\omega\in S^{n-1}$ is such that $u(x+r\omega)=M_x(r)$ and the set of $\omega$'s has a unique limit point as $r\downarrow 0$, then $u$ is differentiable at $x$. It is also known that if $B_r(x)\subset\subset\Omega$, then $u$ is differentiable at $x+r\omega$ \cite{tb2}. Moreover, the limit $\Lambda(x)$ is upper semi-continuous \cite{acj,tb2,c,ceg} and satisfies a maximum principle, \cite{tb1,ceg}. An important question in this context is whether or not infinity-harmonic functions, defined on all of $\mathbb{R}^n$ with bounded $\Lambda$, are affine. This was proven in $n=2$ in \cite{s}, as a consequence of differentiability. Some of the facts mentioned above will play a role in this work. From hereon we assume that $o\in \Omega$ and work in a ball $B_R(o)\subset \Omega$. Additionally, we always take $u(o)=0$. To make our presentation clearer, we redefine $m(r)=-\inf_{\bar{B}_r(o)}u$, and unless otherwise mentioned, from hereon we will always take this to be the definition of $m(r)$. We now state the first of our main results. \begin{theorem} \label{thm1} Let $u$ be infinity-harmonic in $B_R(o)$. Suppose that $u(o)=0$; define for $r\in[0,R)$, $m(r)=-\inf_{\bar{B}_r(o)}u$ and $M(r)=\sup_{\bar{B}_r(o)}u$. The following two inequalities then hold: \begin{itemize} \item[(a)] $m(r)\ge \Big( \sqrt{rM'(r-)}-\sqrt{rM'(r-)-M(r)}\Big)^2$ for $r\in[0,R)$; \item[(b)] $M(r)\ge \Big( \sqrt{rm'(r-)}-\sqrt{rm'(r-)-m(r)}\Big)^2$ for $r\in[0,R)$. \end{itemize} Moreover, if $u$ is infinity-harmonic in $\mathbb{R}^n$ and equality holds in both the inequalities for every $r>0$, then $u$ is affine. \end{theorem} A proof appears in Section 2. An easy consequence of Theorem \ref{thm1} are the inequalities, $$m(r)M'(r-)\ge \frac{M(r)^2}{4r} \quad\mbox{and}\quad M(r)m'(r-)\ge \frac{m(r)^2}{4r},$$ see Remark \ref{rmk1}. The inequalities in Theorem \ref{thm1} place no restrictions on how small, for instance, $m(r)$ could be. As a matter of fact we construct an example in Lemma \ref{lem1} in Section 2 which supports this observation. Using the inequalities in Theorem \ref{thm1}, we also prove the following growth rate in Theorem \ref{thm2}. \begin{theorem} \label{thm2} Suppose that $u$ is infinity-harmonic in $B_1(o)$ and $u(o)=0$. Define for $r\in[0,1)$ $m(r)=-\inf_{\bar{B}_r(o)}u$ and $M(r)=\inf_{\bar{B}_1(o)}u$. \noindent (i) Suppose that $M(r)\le r$ for every $r\in[0,1)$. Then either $m(r)\le r$ for every $r\in[0,1)$, or there is an $a\in(0,1)$ such that $m(a)>a$ and $$m(r)\ge r(1+k\log(r/a)),\;\forall\;a< r< 1,$$ where $k=(m(a)-a)^2/4a^2$, \noindent (ii) Analogously, if $m(r)\le r$ for every $r\in[0,1)$, then either $M(r)\le r$ for every $r\in[0,1)$, or $M(a)>a$ for some $a\in(0,1)$ and $$M(r)\ge r(1+k\log(r/a)),\;\forall\;a< r< 1,$$ where $k=(M(a)-a)^2/4a^2$. \end{theorem} We provide a proof in Section 2. It is to be noted that the hypothesis $M(r)\le r$, or for that matter $m(r)\le r$, is not restrictive. One can scale $u$ to obtain this inequality, see Remark \ref{rmk2} for a general version. We now bring up a related matter. If $u$ is infinity-harmonic in $B_1(o)$, by convexity, $M'(r+)$ and $M'(r-)$ exist everywhere and $M'(r)$ exists for almost every $r$. As pointed out before (see \cite{tb2}), if $p\in \partial B_r(o)$ for $r<1$, is a point of maximum then $u$ is differentiable at $p$ and $$M'(r-)\le |Du(p)|\le M'(r+).$$ Moreover, as Lemma \ref{lem2} shows there are points $p$ on $\partial B_r(o)$ where the values $M'(r+)$ and $M'(r-)$ are attained by $|Du(p)|$. Our question is: does $M'(r)$ exist at every $r\in[0,1)$, or equivalently, is $M'(r-)=M'(r+)$ ? Clearly, this equality holds at $r=0$. We have been unable to settle the matter, however, the example in Lemma \ref{lem4} shows that these may disagree on the boundary $\partial B_1(o)$ in the event $u\in C(\bar{B}_1(o))$. In particular, we show that $M'(r)$ exists for every $r\in[0,1)$ but there are points $p$ on $\partial B_1(o)$ such that $u(p)=M(1)$, and the one sided gradient $\Lambda(p)>M'(1-)$. As our calculations will show this difference can be made arbitrarily large. We have divided our work as follows. Proofs of Theorems \ref{thm1} and \ref{thm2}, and Lemma \ref{lem1} appear in Section 2. Section 3 contains proofs of Lemmas \ref{lem2}, \ref{lem3} and \ref{lem4}. In this work, all sets are subsets of $\mathbb{R}^n,\;n\ge 2$, unless otherwise mentioned. \section{Proofs of Theorems \ref{thm1} and \ref{thm2}} We first state a few properties of infinity-harmonic functions that could be thought of as "monotonicity" along radial segments, see \cite{tb4,c}. For completeness, we provide short proofs of these, also see Exercise 16 in \cite{c}. The proofs will use the comparison principle \cite{acj,bb,c,cgw,j}. Let $v$ be a positive infinity-harmonic function in a domain $\Omega$ and $B_{\rho}(o)\subset \Omega$. Take $y\in B_{\rho}(o)$ and let $d>0$ be chosen to be any value that does not exceed the distance of $y$ from $\partial \Omega$. By comparing $v(x)$ to the infinity-harmonic function $v(y)(1-|x-y|/d)$, we see $$v(x)\ge v(y)\Big(1-\frac{|x-y|}{d}\big),\quad \text{for }x\in B_d(y),$$ In this discussion, we refer to the above inequality as {\em cone comparison in $B_{d}(y)$}. Taking $d=\rho-|y|$, a rearrangement leads to the inequality $$\frac{v(y)-v(x)}{|x-y|}\le \frac{v(x)}{d-|x-y|}.$$ It is clear by using \eqref{e2} that $\Lambda(y)\le v(y)/d$, for every $y\in B_{\rho}(o)$. Moreover, if $\omega\in S^{n-1}$, by selecting the points $x$ and $y$ on the radial ray, in $B_{\rho}(o)$ and along $\omega$, it is clear that $v(\theta \omega)/(\rho-\theta)$ is an increasing function of $\theta$ in $[0,\rho)$. Next we show that $v(\theta\omega)(\rho-\theta)$ is decreasing. To see this, take $x=\theta\omega$ with $\theta>0$, small, so that $B_{\rho}(x)\subset \Omega$ (one may need the assumption $\bar{B}_{\rho}(o)\subset \Omega$, but this is not restrictive). Using cone comparison in $B_{\rho}(x)$, we have $v(x)(\rho-|x|)\le v(o)\rho$. Taking $y=\bar{\theta}\omega$ with $\bar{\theta}>\theta$ and $\bar{\theta}$ close to $\theta$, cone comparison in $B_{\rho-|x|}(y)$, leads to $v(x)(\rho-|x|)\ge v(y)(\rho-|y|)$. Proceeding in this way we obtain our claim. An alternative is to use the Harnack inequality \cite{tb4}. Collecting our conclusions, for any $\omega\in S^{n-1}$ and $\theta\in[0,\rho)$, $$\label{e3} \parbox{9cm}{ (i) v(\theta\omega)/(\rho-\theta) is increasing in \theta, \\ (ii) v(\theta\omega)(\rho-\theta) is decreasing in \theta, and \\ (iii) \Lambda(y)\le v(y)/(\rho-|y|), for all y\in B_{\rho}(o).}$$ For applications, we will often use these properties with $v=M-u$ or $v=u-l$, where $M$ and $l$ are any numbers with $M\ge \sup_{\bar{B}_{\rho}(o)}u$ and $l\le \inf_{\bar{B}_{\rho}(o)}u$. We also note another property. For $r\le \rho$, if $\gamma,\bar{\gamma}\in S^{n-1}$ are such that $v(r\gamma)=\sup_{\bar{B}_r(o)}v$ and $v(r\bar{\gamma})=\inf_{\bar{B}_r(o)}v$, then $u$ is differentiable at $r\gamma$ and $r\bar{\gamma}$ and $$\label{e4} Du(r\gamma)=\alpha \gamma,\quad Du(\bar{\gamma}r)=\beta\bar{\gamma}.$$ for some $M'(r-)\le \alpha\le M'(r+)$ and $m'(r-)\le\beta\le m'(r+)$. For a proof see \cite{tb2}, also see Lemma \ref{lem2} for a related statement. We thank the referee for his/her suggestions that have simplified the following proof. \begin{proof}[Proof of Theorem \ref{thm1}] We prove inequality (a) of the theorem, as inequality (b) follows by symmetry. First observe that $(r-t)(M(t)+m(r))$ is non-increasing in $t$. This may be argued by using 3(ii) with $v=u+m(r)$. Alternatively, one could use \cite[Lemma 4.6]{c} or \eqref{e3}(iii) to first derive the following differential inequality $$M'(t+)\le \frac{M(t)+m(r)}{r-t},$$ and then conclude the same. In any case, since $u(o)=0$, M(t)\le \Big(\frac{t}{r-t}\Big)m(r),\quad 0\le t0. Now applying the tight on a line" result in Section 7.2 in \cite{c}, it follows that u is affine. \end{proof} \begin{remark} \label{rmk1} \rm We discuss briefly the inequalities in Theorem \ref{thm1}. From \eqref{e7}, it follows easily that m(r)M'(r-)\ge M(r)^2/4r, and, similarly, M(r)m'(r-)\ge m(r)^2/4r. Next in \eqref{e7}, we set x=rM'(r-) and y=M(r), and use an expansion for \sqrt{x-y}. Working with \sqrt{x-y}\le\sqrt{x}-\frac{y}{2x^{1/2}}-\frac{y^2}{8x^{3/2}} -\frac{y^3}{16x^{5/2}},\quad 00 if x_n>0, and \psi(\pm\pi/2)=v(\bar{x},0)=0;\\ (ii) \sup_{\partial B_r(o)}v=\psi(0)/r^{1/3};\\ (iii) \psi(\theta)=\psi(-\theta), and \psi(\theta) is decreasing in [0,\pi/2].} \begin{lemma} \label{lem1} Let \varepsilon>0 and B_1(o)\subset\mathbb{R}^n, n\ge 2. There exists a function u_\varepsilon\in C(\bar{B}_1(o)), infinity-harmonic in B_1(o) with u_{\varepsilon}(o)=0 and \sup_{\bar{B}_1(o)}u_{\varepsilon}=1, such that \inf_{\bar{B}_1(o)}u_{\varepsilon}\ge -\varepsilon. \end{lemma} \begin{proof} To construct our example u_{\varepsilon} on B_1(o), we employ a translate of the Aronsson singular function v and use the properties stated in \eqref{e8}. Let \delta>0 and let p_{\delta}=(0,0,0,\dots ,-(1+\delta)). Define v_{\delta}(x)=\frac{\psi(\theta)}{|x-p_{\delta}|^{1/3}} =\frac{\psi(\theta)}{|x+(1+\delta)e_n|^{1/3}},\quad \theta=\theta(x)=\frac{\langle x-p_{\delta},\;e_n\rangle}{|x-p_{\delta}|}. We scale \psi(0)=1. By \eqref{e8}(i) v>0 on B_1(o), and by \eqref{e8}(ii) and (iii), \sup_{\partial B_{\delta}(p_{\delta})}v_{\delta}=\sup_{B_1(o)}v=1/\delta^{1/3}. Next for x\in B_1(o), define \begin{align*} w_{\delta}(x) &=\frac{\delta^{1/3}(1+\delta)^{1/3}(v_{\delta}-v(o))}{(1+\delta)^{1/3} -\delta^{1/3}}\\ &=\frac{\delta^{1/3}(1+\delta)^{1/3}}{(1+\delta)^{1/3}-\delta^{1/3}} \Big(\frac{\psi(\theta)}{|x+(1+\delta)e_n|^{1/3}}-\frac{1}{(1+\delta)^{1/3}}\Big). \end{align*} Then w_{\delta} is infinity-harmonic in x_n>-(1+\delta) and clearly so in B_1(o). Also \sup_{B_1(o)}w_{\delta}=1,\;w_{\delta}(o)=0 and w_{\delta}\in C(\bar{B}_1(o)). Noting that \psi(\theta)>0 when x\in B_1(o) (see \eqref{e8}(i)), we see that 0\ge \inf_{\bar{B}_1(o)}w_{\delta}\ge \frac{-\delta^{1/3}}{(1+\delta)^{1/3}-\delta^{1/3}}. Choosing \delta small enough, we obtain our desired infinity-harmonic function u_{\varepsilon}. \end{proof} Next we present a proof of the growth estimate in Theorem \ref{thm2}. We utilize the inequalities proven in Theorem \ref{thm1}. \begin{proof}[Proof of Theorem \ref{thm2}] We will only prove part (i), part (ii) will follow analogously. To achieve our goal we use inequality (b) of Theorem \ref{thm1}. Using the hypothesis, we note $$\label{e9} \Big(\sqrt{rm'(r-)}-\sqrt{rm'(r-)-m(r)}\Big)^2\le r,\quad 0\le r<1.$$ Let us assume that m(a)> a, for some 0 1,\quad m(r)>r,\quad \forall\;aa, $$\label{e10} m'(r-)\ge \frac{(m(r)+r)^2}{4r^2}.$$ Setting w=m(r)/r in the second inequality in \eqref{e10}, we obtain the differential inequality 4rw'\ge (w-1)^2. An integration from c to r, for any a\le c< 1 yields \frac{c}{m(c)-c}-\frac{r}{m(r)-r}\ge \frac{\log(r/c)}{4},\quad c\le r<1. Noting that m(r)-r\ge m(c)-c, a further rearrangement yields \[ \frac{m(r)}{r}\ge\frac{m(c)}{c}+\Big(\frac{m(c)-c}{c}\Big)^2 \frac{\log(r/c)}{4},\quad c\le r<1. Selecting k=(m(a)-a)^2/4a^2 we have m(r)\ge \frac{r}{a} m(a)+k r\log(r/a)\ge r+k r\log(r/a),\quad a\le r<1. $$The theorem follows. \end{proof} \begin{remark} \label{rmk2}\rm Firstly, we note that if M(1)=1, by convexity M(r)\le r. We now state Theorem \ref{thm2} for the general case. If v is infinity-harmonic in B_R(o) and \sup_{B_R(o)}v<\infty, then we scale v and define u(y)=(v(x)-v(o))/(\sup_{B_R(o)}v-v(o)), where y=x/R. Then \sup_{B_1(o)}u=1 and u satisfies the conditions of Theorem \ref{thm2}. Thus either$$ \inf_{B_r(o)}v\ge v(o)-\frac{r}{R}(\sup_{B_R(o)}u-u(o)),\quad \forall\;r\in[0,R), $$or for some 0r and \rho\downarrow r. Clearly for each p\in\partial B_{\rho}(o), |Du(p)|\ge M'(\rho-). Using the convexity of M(r) and the upper semicontinuity of \Lambda \cite{tb2, ceg}, we see that |Du(p^+)|\ge M'(r+). Hence equality follows. Now let p^- be a limit point of a sequence of such points p\in \partial B_{\rho}(o) with \rho0 with y\pm t\kappa\in B_1(o), u(y-t\kappa)\ge u(y+t\kappa).} The assertion in \eqref{e12} follows quite easily if the plane P_{\kappa} does not intersect \hat{C}_{\alpha}. In case it does, we recall that \phi is axially symmetric and decreasing in x_n. These properties allows us to compare boundary data after reflection and \eqref{e12} follows. Let S be a great semicircle, centered at o, with end points re_n and -re_n. Applying \eqref{e12} with suitable planes P_{\kappa} one sees that for x\in S, u(x) decreasing in x_n. Thus for every r\in(0,1), u(re_n)=-m(r) and M(r)=u(-re_n). We have thus proven the first parts of (i) and (ii) but for the estimates. To prove the rest of part (i), we note that the function$$ v(x)=\frac{u(x)+m(1)}{1+m(1)} $$satisfies the hypothesis of Lemma \ref{lem2}. For \pi-\alpha small, v(o)\ge 1-c(\pi-\alpha)^{1/3}. Since u(o)=0, it is clear that for \alpha close to \pi, m(1)\ge \hat{c}(\pi-\alpha)^{-1/3}, for some \hat{c}>0. Next by taking x=re_n and applying the inequality in Lemma \ref{lem3} to v, we see $m(1)-m(r)\ge(1+m(1))\Big(1-\frac{c(\pi-\alpha)^{1/3}}{(1+k(\alpha)-r)^{1/3}} \Big).$ Using the convexity of m the above yields, for any 00, u(x+tx/|x|) is increasing in t. Since u(o)=0, we have u(x)\ge 0 whenever x\in T. We now prove the rest of the lemma. From the foregoing, B_{|\cos \alpha|}(-e_n)\cap B_1(o)\subset T. If we set w(x)=(1-|x+e_n| |\sec \alpha|) then by comparison u(x)\ge w(x),\;x\in B_{|\cos \alpha|}(-e_n)\cap B_1(o), and M(r)=u(-re_n)\ge 1-(1-r)|\sec\alpha |. Working with 1-u, applying \eqref{e3} (i) and convexity, the one-sided gradient M'(1-) exists and$$ 1\le M'(1-)=\lim_{r\uparrow 1}\frac{1-u(-re_n)}{1-r} \le |\sec\alpha|\to 1\quad \text{as }\alpha\to \pi.  This proves part (ii). To show the last part, recall from (i) that $m(r)=-u(re_n)$. To estimate the one-sided gradient $L(p)$, we apply \eqref{e3}(i) to $1-u$ near $p$, \eqref{e4} and the Harnack inequality \cite{acj,tb4,lm} to conclude $L(p)\ge \frac{1-u(p-\varepsilon p)}{\varepsilon} \ge \frac{1+m(1-\varepsilon)}{\varepsilon}\exp \Big(\frac{-(1-\varepsilon)(\pi-\alpha)}{\varepsilon}\Big).$ We may take $\pi-\alpha=\varepsilon$ and the conclusion now follows by taking $\varepsilon\to 0$. \end{proof} \subsection*{Acknowledgments} We thank the referee for several comments and suggestions that have led to a clearer presentation. This work was partially supported by a Junior Faculty Scholarship Award funded by Western Kentucky University. \begin{thebibliography}{00} \bibitem{a} Aronsson, G.; \emph{On certain singular solutions of the partial differential equation $u_x^2u_{xx}+2u_xu_yu_{xy}+u_y^2u_{yy}=0$}, Manuscripta Math. 47 (1984) 133-151. \bibitem{acj} Aronsson, G., Crandall, M. G., Juutinen, P.; \emph{A tour of the Theory of Absolutely Minimizing Functions}, Bull. Amer. Math. Soc. (N.S.) 41(2004), no. 4, 439--505 \bibitem{bb} Barles, G. and Busca, J.; \emph{Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order terms}. Comm. Partial Differential equations 26 (2001), no. 11-12, 2323-2337. \bibitem{tb1} Bhattacharya, Tilak; \emph{Remarks on the gradient of an infinity-harmonic function}, Electronic Journal of Differential Equations, Vol. 2007(2007), No. 148, pp. 1--11. \bibitem{tb2} Bhattacharya, Tilak; \emph{Some observations on the local behaviour of infinity-harmonic functions}, Differential and Integral Equations, vol 19, no 8(2006), 945-960. \bibitem{tb3} Bhattacharya, Tilak; \emph{A Note on Non-Negative Singular Infinity-Harmonic Functions in the Half-Space}, Revista Matematica Complutense (2005), 18, Num. 2, 377-385. \bibitem{tb4} Bhattacharya, Tilak; \emph{On the behavior of $\infty$-harmonic functions near isolated points}, Nonlinear Analysis(TMA), 58(2004), 333-349. \bibitem{tb5} Bhattacharya, Tilak; \emph{An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions}. Electronic. Journal of Differential Equations 2001, No. 44, 8 pp. \bibitem{tdm} Bhattacharya, T., DiBenedetto, E. and Manfredi, J. J.; \emph{Limits as $p\to\infty$ of $\Delta_pu_p=f$ and related extremal problems}, Rend. Sem. Mat. Univers. Politecn. Torino, Fascicolo Speciale (1989) Nonlinear PDE's. \bibitem{c} Crandall, M. G.; \emph{A visit with the $\infty$-Laplace equation}. Calculus of Variations and Nonlinear Partial Differential Equations, (C.I.M.E Summer School, Cetraro, 2005), Lecture Notes in Math., vol 1927, Springer, Berlin, 2008. \bibitem{ce} Crandall, M. G. and Evans, L. C.; \emph{A remark on infinity-harmonic functions}, Electron. J. Diff. Equations, Conf 06, 2001, 123-129. \bibitem{ceg} Crandall, M. G., Evans, L. C., Gariepy, R. F.; \emph{Optimal Lipschitz extensions and the infinity-Laplacian}, Calculus of Variations and Partial Differential Equations, vol 13, issue 2, pp 123-139. \bibitem{cgw} Crandall, M. G., Gunarson, G., and Wang, P. Y.; \emph{Uniqueness of $\infty$-harmonic functions and the eikonal equations}. Preprint. \bibitem{es} Evans, L. C., Savin, O.; \emph{$C^{1, \alpha}$ regularity for infinity harmonic functions in two dimensions}. Preprint. \bibitem{j} Jensen, Robert; \emph{Uniqueness of Lipschitz Extensions: Minimizing the Sup Norm of the Gradient}. Arch. Rat. Mech. Anal. 1993, 123, 51-74. \bibitem{lm} Lindqvist, P. and Manfredi, J. J.; \emph{The Harnack inequality for $\infty$-harmonic functions}. Electron. J. Differential Equations (1995), no 04, 5pp. \bibitem{pssw} Peres, Yuval; Schramm, Oded; Sheffield, Scott; Wilson, David B; \emph{Tug-of-war and the infinity Laplacian}. J. Amer. Math. Soc. Vol. 22 No. 1, 167-210(2009). \bibitem{s} Savin, O.; \emph{$C^1$ regularity for infinity harmonic functions in two dimensions}. Arch. Ration. Mech. Anal., 176 (2005), no. 3, 351--361. \end{thebibliography} \end{document}