\input amstex \documentstyle{amsppt} \loadmsbm \magnification=\magstephalf \hcorrection{1cm} \vcorrection{6mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1995/04\hfil The Harnack inequality for $\infty$-harmonic functions \hfil\folio} \def\leftheadline{\folio\hfil P. Lindqvist \& J. J. Manfredi \hfil EJDE--1995/04} \def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations\hfil\break Vol. {\eightbf 1995}(1995), No. 04, pp. 1-5. Published April 3, 1995.\hfil\break ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfil\break ftp (login: ftp) 147.26.103.110 or 129.120.3.113 \bigskip}} \topmatter \title The Harnack Inequality for $\infty$-harmonic Functions \endtitle \thanks \noindent {\it 1991 Mathematics Subject Classifications:} 35J70, 26A16.\hfil\break {\it Key words and phrases:} Harnack inequality, $p$-harmonic equations. \hfil\break \copyright 1995 Southwest Texas State University and University of North Texas.\hfil\break Submitted: February 22, 1995. \endthanks \author Peter Lindqvist \\ and \\ Juan J. Manfredi \endauthor \address Department of Mathematics \newline\indent Norwegian Institute of Technology\newline\indent N-7034 Trondheim\newline\indent Norway \endaddress \email lqvist\@imf.unit.no \endemail \address Department of Mathematics \newline\indent University of Pittsburgh \newline\indent Pittsburgh, PA 15260 \newline\indent USA \endaddress \email juanjo\@na0b.math.pitt.edu \endemail \abstract The Harnack inequality for nonnegative viscosity solutions of the equation $ \Delta_\infty u = 0 $ is proved, extending a previous result of L.C. Evans for smooth solutions. The method of proof consists in considering $ \Delta_\infty u=0$ as the limit as $p\to\infty$ of the more familiar $p$-harmonic equation $\Delta_p u=0$. \endabstract \endtopmatter \document The purpose of this note is to present a proof of the Harnack inequality for nonnegative viscosity solutions of the $\infty-$harmonic equation $$\sum\limits^n_{i=1,j=1} \frac{\partial u}{\partial x_i} \ \frac{\partial u}{\partial x_j} \ \frac{\partial^2u}{\partial x_i\partial x_j} = 0 \tag 1$$ where $u = u(x_1, \cdots, x_n).$ For classical $C^2-$solutions this has recently been obtained by Evans, see [E]. While Evans works directly with equation (1), we approximate it by the $p-$harmonic equation $$\hbox{div}(|\nabla u|^{p-2}\nabla u) = 0 \tag 2$$ and let $p \to \infty.$ (See [A], [K], and [BDMB] for background and information about the $\infty$-Laplacian.) The Harnack inequality for nonnegative $p-$harmonic functions can be proved by the now standard iteration methods of DeGiorgi and Moser, see [S] and [DB-T]. Unfortunately, in both of these methods the Harnack constants blow up as $p \to \infty.$ Another approach to the Harnack inequality, valid only when $p> n,$ follows from energy bounds for $\nabla(\log u),$ see [M] and [KMV]. We begin with a well known estimate: \proclaim{Lemma} Suppose that $u_p$ is a nonnegative weak solution of (2) in a domain \hbox{$\Omega \subset \Bbb R^n$.} Then, we have $$\int\limits_\Omega |\zeta\, \nabla \log u_p|^p dx \leqq \left(\frac{p}{p-1} \right)^p \int\limits_\Omega | \nabla \zeta |^p dx \tag 3$$ whenever $\zeta \in C^\infty_0 (\Omega)$. \endproclaim \demo{Proof} We may assume that $u_p > 0.$ (Consider $u_p(x) + \varepsilon$ and let $\varepsilon \to 0^+.$) Use the test function $|\zeta|^p u_p^{1-p}$ in the weak formulation of (2). This simple calculation is given in [L, Corollary 3.8]. \qed \enddemo Our main result states that one can take the limit as $p \to \infty$ in (3). \proclaim{Theorem} Suppose that $u$ is a nonnegative viscosity solution of (1) in a domain $\Omega \subset \Bbb R^n.$ Then we have $$\|\zeta\, \nabla \log u \|_{\infty, \Omega} \; \leqq \; \| \nabla \zeta \|_{\infty, \Omega} \tag 4$$ whenever $\zeta \in C^\infty_0(\Omega)$. \endproclaim \demo{Proof} Select a bounded smooth domain $D$ such that $$\text{supp}\, \zeta \subset D \subset \overline D \subset \Omega.$$ By a fundamental result of Jensen $u \in W^{1,\infty}(D)$ and it is the unique viscosity solution of (1) with boundary values $u|_{\partial D}.$ For these results and the definition of viscosity solutions we refer to [J]. For $p > n$ let $u_p$ be the solution to the problem $$\cases \hbox{div}(|\nabla u_p |^{p-2} \nabla u_p) = 0 \qquad &\text{in} \ D\\ u_p - u \in W_0^{1,p}(D).\endcases $$ By the results of [BDBM, Section I], there exists a sequence $p_j \to \infty$ such that $u_{p_j}$ tends to a viscosity solution $v$ of (1) in $C^\alpha(\overline D)$ for any $\alpha \in [0, 1)$ and weakly in $W^{1,m}(D)$ for any finite $m.$ Since $u$ and $v$ have the same boundary values, the uniqueness theorem of Jensen [J] implies that $u\equiv v$. Note, in addition, that any other subsequence of $u_p$ has a subsequence converging to a viscosity solution of (1) and that this limit is $u$. We conclude that $$u_p \to u \qquad \text{in} \qquad C^\alpha(\overline D) \qquad \text{for any} \quad \alpha \in [0,1) \tag 5$$ and $$u_p \rightharpoonup u \qquad \text{in} \qquad W^{1,m}(D) \qquad \text{for any finite} \quad m \tag 6$$ as $p\to\infty$.\par Fix $m \ge n$ and consider $p > m.$ We have $$\align \int\limits_D |\zeta\,\nabla\log u_p|^m dx &\leqq \Big(\int\limits_D | \ \zeta\, \nabla \log u_p \ |^p dx\Big)^{m/p} | D |^{(p-m)/p}\\ &\leqq \left(\frac{p}{p-1}\right)^m\Big(\int\limits_D |\nabla\zeta|^p \, {dx}\Big)^{m/p} | D |^{(p-m)/p},\endalign $$ where we have used the Lemma in the second inequality. Therefore, we get $$\Big(\int\limits_D | \zeta\, \nabla \log u_p|^m dx\Big)^{1/m} \leqq \frac{p}{p-1} \Big(\int\limits_D |\nabla\zeta|^p dx\Big)^{1/p} |D|^{(p-m)/pm}.\tag 7$$ Assume momentarily that $\zeta\, \nabla \log u_p$ converges weakly to $\zeta\,\nabla \log u$ in $L^m(D).$ By the weak lower semi-continuity of the norm we obtain $$\Big(\int\limits_D |\zeta\,\nabla \log u \ |^mdx\Big)^{1/m} \leqq \ \| \nabla \zeta \|_{\infty, D} |D|^{1/m}.\tag 8$$ Observe that (7) holds for the translated functions $u_p(x)+\varepsilon$, where $\varepsilon>0$ is fixed, in place of $u_p$. Since these functions are bounded away from zero, it is elementary to check that $\zeta\, \nabla \log (u_p+\varepsilon)$ converges weakly to $\zeta\,\nabla \log (u+\varepsilon)$ in $L^m(D)$. It now follows from (5) and (6) that estimate (8) holds for $u(x)+\varepsilon$. We now let $\varepsilon\to 0$. By the Monotone Convergence theorem, we obtain estimate (8) for $u$. Finally, letting $m \to \infty$ we finish the proof of (4).\qed \enddemo If $B_r$ and $R_R$ are two concentric balls in $\Omega$ with radius $r$ and $R$, the usual choice of a radial test function $\zeta$ ($0 \leqq \zeta \leqq 1, \quad \zeta = 1 \ \text{in} \ B_r, \quad \zeta = 0$ outside $B_R)$ in (4) yields the estimate $$\| \nabla \log u \|_{\infty,B_r} \leqq \frac{1}{R-r} \tag 11$$ provided that $B_R \subset \Omega.$ In particular, we obtain the following result. \proclaim{Corollary 1} (a) If $u$ is a nonnegative viscosity solution of (1) in a domain $\Omega \subset \Bbb R^n$, then for a.~e. $ x \in \Omega$ $$| \nabla u(x) | \leqq \frac{u(x)}{d(x, \partial \Omega)}\cdot \tag 12$$ \medskip (b) If $u$ is a bounded viscosity solution of (1) in a domain $\Omega \subset \Bbb R^n,$ then for a.~e. $x \in \Omega$ we have $$| \nabla u(x) | \leqq \frac{2 \| u \|_\infty}{d(x, \partial \Omega)}\cdot \tag 13$$ \endproclaim \demo{Proof} It remains to consider only the second case, which follows from the first by considering $v = u + \|u\|_\infty.$ \qed \enddemo Next, we state the Harnack inequality, which follows from (11). \proclaim{Corollary 2} Suppose that $u$ is a nonnegative viscosity solution of (1) in $B_R(x_0).$ Then if $x, y \in B_r(x_0), 0 \leqq r < R,$ we have $$u(x) \leqq e^{|x-y|/(R-r)}u(y). \tag 14$$ \endproclaim \demo{Proof} By integrating (11) on a line segment from $x$ to $y$ we obtain $$|\log u(x) -\log u(y)|\le \frac{|x-y|}{R-r},$$ from which (14) follows by exponentiating.\qed \enddemo \remark{Remarks} \par $\S$1. The Lemma holds for nonnegative super-solutions of the $p-$Laplacian by exactly the same proof. Thus for $p > n$ we get an estimate like (10) with $m$ replaced by $p$, from which a Harnack inequality follows easily. This suggests the possibility that corollary 2 holds, indeed, for nonnegative viscosity super-solutions of (1). $\S$2. If one uses the estimate in [L, (4.10)] $$\int\limits_{\Omega} | \nabla u_p |^pu_p^{-1-\varepsilon} \zeta^pdx \leqq \Big(\frac{p}{\varepsilon}\Big)^p \int\limits_{\Omega} u_p^{p-1-\varepsilon}| \nabla\zeta|^pdx$$ where $0 < \varepsilon < p - 1$ instead of (3), we obtain the estimate $$\| \zeta u^{-\alpha} \nabla u \|_{\infty, \Omega} \leqq \frac 1 \alpha \| u^{1-\alpha} \nabla \zeta \|_{\infty, \Omega}$$ for any $\alpha > 0$ and for any nonnegative viscosity solution $u$ of (1) in $\Omega.$ Roughly speaking, estimates for the $p-$Laplacian that are independent of $p$, always yield estimates for $\infty-$harmonic functions. \endremark \bigskip \Refs \widestnumber\key{BDBM} \ref \key A \by Aronsson, G. \paper On the partial differential equation $u^2_xu_{xx}+2u_xu_yu_{xy}+u^2_yu_{yy}=0$ \jour Arkiv f\"ur Matematik \vol 7 \yr 1968 \pages 395--425 \endref \ref \key BDBM \by Batthacharya, T., Di Benedetto, E. and Manfredi, J. \paper Limits as $p \to \infty \ \text{of} \ \triangle_p u_p = f$ and related extremal problems \inbook Classe Sc. Math. Fis. Nat., Rendiconti del Sem. Mat. Fascicolo Speciale Non Linear PDE's \yr 1989 \pages 15--68 \publaddr Univ. de Torino \endref \ref \key DB-T \by Di Benedetto, E and Trudinger, N. \paper Harnack inequalities for quasiminima of variational integrals \jour Analyse nonlin\'eaire, Ann.\ Inst.\ Henri Poincar\'e \vol 1 \yr 1984 \pages 295--308 \endref \ref \key E \by Evans, L. \paper Estimates for smooth absolutely minimizing Lipschitz extensions \jour Electronic Journal of Differential Equations \vol 1993 No. 3 \yr 1993 \pages 1--10 \endref \ref \key J \by Jensen, R. \paper Uniqueness of Lipschitz extensions: Minimizing the sup-norm of the gradient \jour Arch.\ for Rational Mechanics and Analysis \vol 123 \yr 1993 \pages 51--74 \endref \ref \key K \by Kawohl, B. \paper On a family of torsional creep problems \jour J.\ Reine angew.\ Math.\ \vol 410 \yr 1990 \pages 1--22 \endref \ref \key KMV \by Koskela, P., Manfredi, J. and Villamor, E. \paper Regularity theory and traces of $\Cal A$-harmonic functions \jour Transactions of the American Mathematical Society \paperinfo to appear \endref \ref \key L \by Lindqvist, P. \paper On the definition and properties of $p$-superharmonic functions \jour J.\ Reine angew.\ Math.\ \vol 365 \yr 1986 \pages 67--79 \endref \ref \key M \by Manfredi, J. J. \paper Monotone Sobolev functions \jour J. Geom. Anal. \vol 4 \yr 1994 \pages 393--402 \endref \ref \key S \by Serrin, J. \paper Local behavior of solutions of quasilinear elliptic equations \jour Acta Math. \vol 111 \yr 1964 \pages 247--302 \endref \endRefs \newpage \enddocument