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\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--1996/02\hfil A Lower Bound for the Gradient \hfil\folio}
\def\leftheadline{\folio\hfil Edi Rosset
\hfil EJDE--1996/02}
\voffset=2\baselineskip
\vbox {\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol.\ {\eightbf 1996}(1996) No.\ 02, pp. 1--7.\hfill\break
ISSN 1072-6691. URL: http://ejde.math.swt.edu (147.26.103.110)\hfil\break
telnet (login: ejde), ftp, and gopher access:
 ejde.math.swt.edu or ejde.math.unt.edu}
\footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt %
1991 {\eighti Subject Classification:} 35J70, 35B05, 35B50, 31C45.
\hfil\break
{\eighti Key words and phrases:} $\infty$-harmonic functions,
$p$-harmonic functions.
\hfil\break
\copyright 1996 Southwest Texas State University  and
University of North Texas.\hfil\break
Submitted November 20, 1995. Published February 6, 1996.\hfil\break
Partially supported by MURST.} }

\bigskip\bigskip

\centerline{A LOWER BOUND FOR THE GRADIENT}
\centerline{OF $\infty$-HARMONIC FUNCTIONS}
\smallskip
\centerline{Edi Rosset}
\bigskip\bigskip

{\eightrm\baselineskip=10pt \narrower
\centerline{\eightbf Abstract}
We establish a lower bound for the gradient of the solution to
$\infty$-Laplace equation in a strongly star-shaped annulus with 
capacity type  boundary conditions. The proof involves properties 
of the radial derivative of the solution,  so that starshapedness
of level sets easily follows.
\bigskip}

\bigbreak
\centerline{\bf \S 1. Introduction} \medskip\nobreak

In this paper we deal with solutions to the
$\infty $-Laplace equation
$$\Delta_\infty
u=\sum_{i,j=1}^nu_{x_i}u_{x_j}u_{x_ix_j}=0,
\eqno(1_\infty)$$
in a domain $\Omega$ of $\Bbb R^n$.

Equation $(1_\infty)$ was first
considered by G. Aronsson ([Ar1], [Ar2]) and naturally arises 
as the Euler equation of minimal Lipschitz extensions.
It is a highly degenerate elliptic equation which 
is formally the limit, as 
$p\rightarrow\infty$, of the $p$-Laplace equation
$$\Delta_pu=\hbox{div}(|Du|^{p-2}Du)=0.\eqno(1_p)$$
This limit process has been
recently made rigorous by R. Jensen in [J], where he
establishes the fundamental result that any Dirichlet
problem for equation $(1_\infty)$ has a unique viscosity 
solution
$u\in W^{1,\infty}(\Omega)
\cap C^0(\bar\Omega)$ which is the limit, as
$p\rightarrow\infty$, of the unique solution 
$u_p\in W^{1,p}(\Omega)$ to equation
$(1_p)$ satisfying the same Dirichlet data (which
exists and is unique by standard variational
arguments), in the sense that
$u_p\rightarrow u$ uniformly in $\bar\Omega$ and
weakly in $W^{1,q}(\Omega)$ for any $q$ such
that $q<\infty$. 
For a discussion of the related concepts of absolutely
minimizing Lipschitz extension, variational solution
and viscosity solution to equation $(1_\infty$), we also refer to
[B-D-M]. 

Concerning the critical points of $\infty$-harmonic functions,
Aronsson proved that any non-constant $C^2$ solution to
$(1_\infty)$ in the plane has non-vanishing gradient ([Ar2]), 
this result
has been recently extended to $C^4$ solutions in higher
dimensions by L. C. Evans ([E]). 
On the other hand, Aronsson gave examples of $C^1$
non-constant (viscosity) solutions to $(1_\infty)$ having an
interior critical point ([Ar3]).

We consider a strongly star-shaped annulus $\Omega$, that is
$\Omega=\Omega_2\setminus
\bar\Omega_1$, where
$\Omega_1$ and $\Omega_2$,
$\Omega_1\subset \subset\Omega_2$, are  
bounded domains with $C^2$-smooth boundaries
$\partial\Omega_1$ and $\partial\Omega_2$
satisfying the following strong starshapedness
condition with respect to the origin
$$\exists\,\alpha_0>0\hbox{ such that }{\bf n}(x) \cdot x
\geq\alpha_0|x|,\quad\forall x\in \partial\Omega_i,\ i=1,2,
\eqno(S)$$
where 
${\bf n}(x)$ is the outer unit normal to $\Omega_i$ in
$x\in\partial\Omega_i$, i=1,2.
We establish a lower bound for the gradient
of the solution $u$ to equation $(1_\infty)$ with capacity type
boundary conditions
$$ \left\{\eqalign{u=0&\qquad \hbox{on }
\partial\Omega_1\cr
u=1&\qquad \hbox{on }\partial\Omega_2\cr}\right.
\eqno(BC)$$

Let us recall that solutions to equation $(1_p)$
satisfying boundary conditions (BC) were studied by
J. A. Pfaltzgraff in [P], proving starshapedness
of the level sets, and also by J. L. Lewis in [L], who proved, 
having convexity results as objective, a lower 
bound for the gradient for any $p$, $1<p<\infty$.

In this note we show that such a lower bound is uniform
with respect to $p$ in a neighbourhood of $+\infty$ and
furthermore we prove that it extends to the solution
to equation $(1_\infty)$ satisfying boundary conditions (BC).

We note that in the two dimensional case one can prove
a lower bound for 
$|Du_p|$ in all of 
$\Omega$ uniformly as $p\rightarrow\infty$, 
without any starshapedness
assumption, since $\log |Du_p|$ satisfies a uniformly
elliptic equation, due to a
result of G. Alessandrini ([Al]), but unfortunately the
type of convergence we have for the 
$u_p$'s, as described previously, does not ensure
the convergence of $\inf |Du_p|$ as $p\rightarrow\infty$.

On the other hand the starshapedness assumption 
becomes necessary when $n\geq 3$, in fact by an example 
of A. W. J. Stoddart ([S]) with $p=2$ there exist simply
connected domains $\Omega_1$, $\Omega_2$ for
which the solution to problem $(1_p)$-(BC) has the gradient
vanishing at at least one interior point. By standard
arguments it can be seen that the example by Stoddart
can also be generalized to the case $p\neq 2$ for any
$p\in(1,\infty)$.

\medskip
\noindent
{\bf Notation.} 
We shall denote  the Euclidean scalar product of two elements
$u$, $v\in\Bbb R^n$ by $u\cdot v$.

\bigbreak
\centerline{\bf \S 2. Statement and proof of main result} \medskip\nobreak

Let us briefly sketch the steps of the proof. Let $u$,
$u_p$ denote the solutions to the Dirichlet 
problems $(1_\infty)$-(BC) and
$(1_p)$-(BC) respectively.

We establish a lower bound for $|Du_p|$ on $\partial
\Omega$, uniformly in $p$, for large $p$ (see Lemma 2). 
By the starshapedness 
assumption, this implies a lower bound on $\partial
\Omega$ for the radial derivative $v_p=Du_p \cdot x$,
uniformly in $p$. 
In Lemma 1 we prove that $v_p$ satisfies in the
distributional sense a degenerate elliptic equation,
see also the Remark following Lemma 1 and,
constructing an appropriate test function, we deduce a lower
bound for $v_p$ in all of $\Omega$. Passing to the limit
as $p\rightarrow\infty$, a lower bound for the radial 
derivative $v=Du \cdot x$ follows. Finally, starshapedness
of level sets is a consequence of the positivity of the radial
derivative.

\proclaim
Lemma 1.  Let $u_p$ be a weak solution to $(1_p)$ in a 
domain 
$\Omega$, where $p>2$. Let 
$$v_p=Du_p \cdot x=\sum_{i=1}^n x_iu_{p,x_i},$$
$$A^p=(A^p_{ij})_{i,j=1,...,n},$$
$$(A^p_{ij})=|Du_p|^{p-2}\delta _{ij}+(p-2)|Du_p|
^{p-4}u_{p,x_i}u_{p,x_j}.$$
Then $v_p$ satisfies the following identity
$$\int_\Omega A^pDv_p\cdot D\phi =0,\qquad
\forall\phi\in C^\infty_0(\Omega).\eqno(2)$$

\noindent{\bf Proof.} For simplicity of notations we shall
drop the index $p$ from $A^p$, $u_p$, $v_p$. 

Let us recall that, by a well known result of K. Uhlenbeck ([U]),
weak solutions to $(1_p)$ belong to
$C^{1,\alpha}_{loc}(\Omega)$
and satisfy
$|Du|^{{p-2 \over 2}}u_{x_ix_j}\in
L^2_{loc}(\Omega)$, so that $ADv\in L^2_{loc}(\Omega)$.
Let $\phi\in C^{\infty}_0(\Omega)$. Then, by equation $(1_p)$,
$$\int_\Omega ADv\cdot D\phi=
\int_\Omega |Du|^{p-2}x_iDu_{x_i}\cdot D\phi
+\int_\Omega |Du|^{p-2}Du\cdot D\phi+$$
$$+\int_\Omega (p-2)|Du|^{p-4}x_iu_{x_j}u_{x_ix_j}
Du\cdot D\phi
+\int_\Omega (p-2)|Du|^{p-4}|Du|^2
Du\cdot D\phi=$$
$$=
\int_\Omega x_i\left(|Du|^{p-2}Du\right)_{x_i}
\cdot D\phi=
-\int_\Omega |Du|^{p-2}Du\cdot 
\left(\sum_i (x_iD\phi)_{x_i}\right)$$
$$=-\int_\Omega |Du|^{p-2}Du\cdot Dh=0,$$
where $h=\sum_{i=1}^n x_i\phi_{x_i}+(n-1)\phi$. {\tenmsa \char003}

\medskip
\noindent
{\bf Remark.} Let us point out that (2) means that
$v_p$ is a distributional solution to the following degenerate
elliptic equation
$$\hbox{div}(|Du_p|^{p-2}Dv_p
+(p-2)|Du_p|^{p-4}Du_p \cdot Dv_pDu_p)=0.$$
The degeneracy of this equation prevents to replace 
$C^\infty_0$ with $W^{1,2}_0$ test functions and
therefore a maximum principle is not straightforward.
The suitable maximum principle will be derived in the 
proof of Theorem 1, under hypotheses which ensure
that a Hopf-type Lemma holds.

\proclaim
Lemma 2 (A uniform Hopf-type Lemma).
Let $\Omega=\Omega_2\setminus
\bar\Omega_1$, where
$\Omega_1$ and $\Omega_2$,
$\Omega_1\subset \subset\Omega_2$ are two simply
connected bounded domains in $\Bbb R^n$
with $C^2$ boundaries $\partial\Omega_1$,
$\partial\Omega_2$. Let $u_p$ be the solution
to $(1_p)$-(BC).
Then there exist a constant $\tilde C>0$, $\tilde C$
independent of $p$, and $\bar p>n$ such that
$$|Du_p(x)|\geq \tilde C\qquad\forall x\in\partial\Omega,
\quad\forall p\geq\bar p.\eqno(3)$$

\noindent
{\bf Proof.} There exists $R>0$ such that for any $x_0\in
\partial\Omega$ there exists a ball of center $y\in\Omega$
and radius $R$ such that $B_R(y)\subset\Omega$ and
$x_0\in\partial B_R(y)$. Let, for instance, 
$x_0\in\partial\Omega_1$ and $y$ as above. 
Let us define, for $p>n$, 
$w(x)=K\left(R^{{p-n \over p-1}}-
r^{{p-n \over p-1}}\right)$, where $r=|x-y|$, 
${R \over 2}<r<R$ and $K$ is a positive constant 
to be chosen later. Let
$A=\{x\in\Omega\ |\ {R \over 2}<|x-y|<R\}$,
$\Omega_d=\{x\in\Omega\ |\ d(x,\partial\Omega)>d\}$,
for $d>0$. 
Let us consider the unique viscosity solution $u$ 
to the Dirichlet problem
$(1_\infty)$-(BC). Then the comparison principle ([J])
and Harnack inequality ([L-M]) 
or viscosity solutions to equation $(1_\infty)$
imply that $0<u<1$ in $\Omega$. Let 
$\gamma=\max_{\bar\Omega_{R \over 2}
\setminus\Omega_{{3 \over 2}R}}u>0$.
For sufficiently large $p$, $u_p\geq{\gamma \over 2}$
in $\partial B_{R \over 2}$ so that, choosing
$K={\gamma \over 2}\left(R^{{p-n \over p-1}}-
\left({R \over 2}\right)^{{p-n \over p-1}}\right)^{-1}$,
we have $w\leq u_p$ on $\partial B_{R \over 2}$.
Moreover, $w\equiv 0\leq u_p$ on  $\partial B_R$
and $\Delta_p w=0$ in $A$. The comparison principle
for solutions to equation $(1_p)$ implies that $w\leq u_p$
in $A$. Since $w(x_0)=
u_p(x_0)=0$, we have
$${\partial u_p \over \partial {\bf n}}(x_0)\geq
-w_r(x_0)={\gamma \over 2}\left({p-n \over p-1}\right)
R^{1-n \over p-1}
\left(R^{p-n \over p-1}-{R \over 2}^{p-n \over p-1}\right)
^{-1}\rightarrow{\gamma \over R},$$
as $p\rightarrow\infty$, where ${\bf n}$
is the outer unit normal to $\Omega_1$, and (3) follows.

\proclaim
Theorem 1. Let $\Omega$ be as in the hypotheses of
Lemma 2 and moreover let us assume that
the starshapedness condition (S) holds.
Let $u$ be the solution to the Dirichlet problem
$(1_\infty)$-(BC) extended by $0$ to all of $\bar\Omega_2$.
Then there exist a constant $\delta>0$, 
$\delta$ independent of $p$,
and $\bar p>n$ such that 
$${\partial u_p \over \partial r}(x)\geq \delta
\qquad \forall x\in\Omega,\quad 
\forall p\geq\bar p,\eqno(4)$$
$${\partial u \over \partial r}(x)\geq \delta
\qquad \hbox{ a.e. } x\in\Omega,\eqno(5)$$
where $u_p$ is the solution to $(1_p)$-(BC).
Moreover the set 
$\Omega_k=\{x\in \Omega_2 \ |\ u(x)<k\}$
is star-shaped w. r. t. the origin, for any $k\in (0,1)$.

\noindent
{\bf Proof.} Let us consider the unique solution 
$u_p\in W^{1,p}(\Omega)$ to the Dirichlet problem
$(1_p)$-(BC) and notice that $u_p$ solves 
a uniformly elliptic quasilinear 
equation 
in a neighbourhood of $\partial\Omega$ in 
$\bar\Omega$, due to the 
lower bound for $|Du_p|$.
Therefore, by standard
estimates (see [L-U]),
$|Du_p|^{(p-2)/2}u_{p,x_ix_j}\in
L^2(\Omega)$ so that $ADv_p\in L^2(\Omega)$ and
(2) extends to test functions in $W^{1,2}_0(\Omega)$. From 
(3) and (S) it follows that 
$$v_p\geq C:=\tilde C\alpha_0 r_1 \qquad \hbox{on } 
\partial\Omega,\quad\forall p\geq \bar p,\eqno(6)$$ 
where $r_1=d(0,\partial\Omega_1)$.
Using the test function $\phi=(|v_p|^{p-2 \over 2}v_p -
C^{p \over 2})^-
\in W^{1,2}_0(\Omega)$ in (2),
we get
$$(p-2)\int_{v_p<C}
|Du_p|^{p-4}|v_p|^{p-2 \over 2}(Dv_p \cdot Du_p)^2
+\int_{v_p<C}
|Du_p|^{p-2}|v_p|^{p-2 \over 2}|Dv_p|^2=0.\eqno(7)$$
Let $W^p=\{x\in\Omega\ |\ v_p<C\}
=U^p\cup V^p$, where
$U^p=\{x\in\Omega\ |\ v_p<C,\ v_p\neq 0\}$, 
$V^p=\{x\in\Omega\ |\ v_p<C,\ v_p=0\}$. 
Let $U^p_j$, $j\in J\subset\Bbb N$, 
be the open connected components
of $U^p$. Since $Du_p\neq 0$ in $U^p$, we have
$u_p\in C^2(U^p)$, 
and therefore from (7) it follows that $v_p\equiv c^p_j$ 
in $U^p_j$. From $\partial U^p_j\subset
\{x\in\Omega\ |\  v_p =C\}
\cup\{x\in\Omega\ |\ v_p=0\}$ and the definition of $U^p$
it follows that $U^p=\emptyset$. Therefore 
$v_p\equiv 0$ in $W^p$.
Since $\Omega$ is connected and (6) holds it follows that
$W^p=\emptyset$, that is $v_p\geq C$ 
in $\Omega$ for $p\geq \bar p$.

Now let $C'<C$. If there exists a set $S$ of positive measure 
such that
$v:= Du \cdot x\leq C'$ in $S$, 
since $v_p\rightarrow v$
weakly in $L^q$ for any $q<\infty$, we get the contradiction
$$C\mu(S)\leq\int_\Omega \chi_Sv_p\rightarrow
\int_\Omega \chi_Sv\leq C'\mu(S).$$
Therefore $v>C'$ almost everywhere, for any $C'<C$.
We have $v\geq C$ almost everywhere and 
(4) and (5) follow with $\delta={C \over r_2}$,
where $r_2=\sup_{x\in\Omega_2}|x|$.

In order to prove that $\Omega_k$ is star-shaped 
w. r. t. the origin,
let us assume by contradiction that there exist points
$x\in\Omega_k$, $\lambda x\not\in\Omega_k$,
with $\lambda\in(0,1)$ and let $\beta=u(\lambda x)-u(x)>0$.
There exists $p\geq \bar p$ such that $u_p(\lambda x)-u_p(x)
\geq{\beta \over 2}>0$, but this contradicts (4). {\tenmsa \char003}


\bigbreak
\centerline{\bf References} \medskip\nobreak

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\item{[E]}  L. C. Evans, Estimates for smooth absolutely
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Linear and quasilinear elliptic equations, 
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{\it Acta Math.} {\bf 138}, (1977), 219--240.

\bigskip
\noindent
Edi Rosset\hfil\break
Dipartimento di Scienze Matematiche\hfil\break
Universit\` a di Trieste\hfil\break
Piazzale Europa 1\hfil\break
34100 Trieste, Italy

\noindent E-mail: rossedi@univ.trieste.it




\bye