\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 43, 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  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/43\hfil Viscosity solutions and boundary conditions]
{Generalized viscosity solutions of elliptic PDEs
and  boundary conditions}
\author[G. Gripenberg\hfil EJDE-2006/43\hfilneg]
{Gustaf Gripenberg}  

\address{Gustaf Gripenberg \newline
Institute of Mathematics\\
Helsinki University of Technology, P.O. Box 1100,
FIN-02015 TKK, Finland}
\email{gustaf.gripenberg@tkk.fi}
\urladdr{www.math.tkk.fi/$\sim$ggripenb}

\date{}
\thanks{Submitted March 14, 2006. Published March 28, 2006.}
\subjclass[2000]{35K55, 35K65}
\keywords{Viscosity solutions; boundary conditions}

\begin{abstract}
 Sufficient conditions are given for a generalized viscosity
 solution  of an elliptic boundary value problem to satisfy
 the boundary values in the strong sense.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newcommand{\abs}[1]{\lvert #1 \rvert}
\newcommand{\set}[1]{\lbrace #1 \rbrace}


\section{Introduction}

When studying nonlinear elliptic equations of the form
    %%
    \begin{equation}\label{E:BasicEq}
  F(\mathbf{x},u,Du, D^2u)=0, \quad \mathbf{x} \in \Omega,
    \end{equation}
    %%
 one very fruitful approach may be to use the notion of viscosity solutions.
We say that $u$ is a viscosity solution if it is both a subsolution and
supersolution and $u$ is a subsolution in $\Omega$ if it is upper
semicontinuous in $\Omega$ and for every $\mathbf{x}_0\in \Omega$ the
following implication holds: If $\psi\in \mathcal{C}^2(\mathbb{R}^d)$ and
$u(\mathbf{x})\leq \psi(\mathbf{x})+ u(\mathbf{x}_0)- \psi(\mathbf{x}_0)$,
$\mathbf{x}\in \Omega$, $\abs{\mathbf{x}-\mathbf{x}_0} < \delta$, for
some $\delta > 0$ then $ F\big(\mathbf{x}_0,u(\mathbf{x}_0),
D\psi(\mathbf{x}_0), D^2\psi(\mathbf{x}_0)\big) \leq 0$.  Supersolutions are
defined symmetrically, for details and further information and references,
see e.g. \cite{CrandallGuide92}. Since one wants to be sure that classical
solutions are viscosity solutions, one has to assume that $F$ is
nonincreasing in its last argument (with the natural ordering for symmetric
matrices).

One way of proving existence is to use Perron's method, introduced in the
viscosity setting in \cite{Ishii87}.  That is, one proves that the supremum
of a suitable set of subsolutions is the solution. For this to work one needs
a subsolution $\underline u$ and a supersolution $\overline u$ such that
$\underline u\leq \overline u$ and a comparison result saying that a
subsolution is less than or equal to a supersolution if both lie between
$\underline u$ and $\overline u$. If $\underline u(\mathbf{x})= \overline
u(\mathbf{x})$ when $\mathbf{x}\in \partial \Omega$ then all functions
between $\underline u$ and $\overline u$ will automatically satisfy a
Dirichlet boundary condition but if this is not the case then the situation
is not so simple any more. One can, however, take another approach and
consider a boundary condition $G(\mathbf{x},u(\mathbf{x}))=0$ in the
viscosity sense which means that one considers the equation
$H(x,u,Du,D^2u)=0$ in $\overline{\Omega}$ where
   %%
    \begin{equation*}
  H(\mathbf{x},r,\mathbf{p} ,X) = \begin{cases}
    F(\mathbf{x},r,\mathbf{p},X), & \mathbf{x}\in \Omega,\\
      G(\mathbf{x},r), & \mathbf{x}\in \partial\Omega.
  \end{cases}
    \end{equation*}
    %%
  For the use of Perron's method in this case, see also
  \cite[Thm.\ 6.1]{DaLio02}.
   It turns out that when dealing with subsolutions the function $H$ should
be lower semicontinuous and when considering supersolutions $H$ should be
upper semicontinuous. Thus one has to introduce upper and lower
semicontinuous envelopes defined as follows: If $v:\mathcal{A}\to
[-\infty,\infty]$, $\mathcal{A} \subset \mathbb{R}^N$, then
$v^*(\mathbf{z})=\lim_{s\downarrow 0} \sup\set{v(\mathbf{y})\mid y\in
\mathcal{A},\quad \abs{\mathbf{y}-\mathbf{x}}\leq s}$ and
$v_*(\mathbf{z})=\lim_{s\downarrow 0}\inf\set{u(\mathbf{y})\mid y\in
\mathcal{A},\quad \abs{\mathbf{y}-\mathbf{x}}\leq s}$, for $\mathbf{z}\in
\mathbb{R}^N$. Thus a generalized viscosity solution of \eqref{E:BasicEq}
with boundary condition $G(\mathbf{x},u(\mathbf{x}))=0$ should be a
subsolution of $H_* =0$ and a supersolution of $H^* =0$. Observe that if
both $F$ and $G$ are lower semicontinuous (otherwise replace $F$ and $G$
below by $F_*$ and $G_*$, respectively) and $\Omega$ is open then
    %%
    \begin{equation*}
  H_*(\mathbf{x},r,\mathbf{p},X) = \begin{cases}
  F(\mathbf{x},r,\mathbf{p},X), & \mathbf{x}\in \Omega,\\
                 \min\{ G(\mathbf{x},r),F(\mathbf{x},r,\mathbf{p},X)\}, &
\mathbf{x}\in \partial\Omega.
  \end{cases}
    \end{equation*}
    %%
  One consequence of studying equation $H_*=0$ is that the boundary
conditions may not be satisfied at all points, and this in turn will cause
grave problems when one tries to prove comparison results. Thus the purpose
of this note is to study under what assumptions it follows that if $u$ is a
subsolution of $H_*=0$, then one actually has
$G(\mathbf{x},u(\mathbf{x}))\leq 0$ for all points $\mathbf{x}$ on the
boundary. By symmetry one can then get corresponding results for
supersolutions, because $u$ is a supersolution of $H=0$ if and only if $-u$
is a subsolution of $\overleftrightarrow H =0$ where $\overleftrightarrow H
(\mathbf{x},r,\mathbf{p},X) = - H(\mathbf{x},-r,-\mathbf{p},-X)$.

 These results improve those that can be found in \cite{DaLio02} (see also
\cite{BarlesDaLio04}), in particular concerning the assumptions on the domain
$\Omega$. Concerning the equations studied in e.g.  \cite{BarlesBusca01} on
sees that the assumptions in the theorems below are satisfied for the
$p$-Laplacian equation $-\mathop{\rm div}(\abs{Du}^{p-2}Du) =0$ when $1 < p
<\infty$ where one thus has $F(\mathbf{x},r,\mathbf{q},X) = -
\abs{\mathbf{q}}^{p-2}\mathop{\rm tr}(X) -
(p-2)\abs{\mathbf{q}}^{p-4}\langle{\mathbf{q}},{X\mathbf{q}}\rangle$ when
$\mathbf{q}\neq 0$ and for the infinity-Laplacian equation $-\Delta_\infty
u=0$ where $ F(\mathbf{x},r,\mathbf{q},X) =
-\langle{\mathbf{q}},{X\mathbf{q}}\rangle$. (Here $\langle\cdot,\cdot\rangle$
denotes the standard inner product in $\mathbb{R}^d$.)  However, the
assumptions are not satisfied for the $1$-Laplacian equation, nor for the
minimal surface equation $-\mathop{\rm div} \big(\frac 1{\sqrt{1+\abs{Du}^2}}
Du\big)=0$ where $ F(\mathbf{x},r,\mathbf{q},X) = - \frac {\mathop{\rm
tr}(X)}{\sqrt{1+\abs{\mathbf{q}}^2}} +
\frac{\langle{\mathbf{q}},{X\mathbf{q}}\rangle}{\sqrt{1
+\abs{\mathbf{q}}^2}^3}$.



\section{Statement of results}

We shall prove two theorems that differ only in a tradeoff between the
assumptions on the domain $\Omega$ and the nonlinearity $F$.  Note also that
we do not have to assume that $F$ and $G$ are nondecreasing in the second
variable, but this assumption is essential in comparison results.  Below
$\mathcal{S}(d)$ denotes the set of real symmetric $d\times d$-matrices with
$X\geq Y$ for $X,Y\in \mathcal{S}(d)$ if all eigenvalues of $X-Y$ are
nonnegative, and $\mathbf{n} \otimes \mathbf{n}$ is the matrix with
$(i,j)$-element $\mathbf{n}_i\mathbf{n}_j$.

\begin{theorem}\label{T:SphereCondThm}
Assume that $d\geq 1$ and that
 \begin{enumerate}
 \item[(i)] $\Omega \subset \mathbb{R}^d$ is open;
 \item[(ii)] $\mathbf{x}_\diamond\in \partial \Omega$
  satisfies an exterior ball condition, i.e., there is a vector
$\mathbf{n}_\diamond \in \mathbb{R}^d$ with $\abs{\mathbf{n}_\diamond}=1$ and
numbers $\rho_\diamond> 0$ and $\beta_\diamond> 0$ such that
$\set{\mathbf{x}\in \mathbb{R}^d\mid \abs{\mathbf{x}- \mathbf{x}_\diamond -
\rho_\diamond\mathbf{n}_\diamond} \leq \rho_\diamond, \abs{\mathbf{x} -
\mathbf{x}_\diamond}\leq \beta_\diamond}\cap \overline{\Omega} =
\{\mathbf{x}_\diamond\}$;
  \item[(iii)] $F:\overline{\Omega}\times \mathbb{R}\times \mathbb{R}^d
\times \mathcal{S}(d)\to [-\infty,\infty]$ is lower semicontinuous and
degenerate elliptic, i.e., nonincreasing in its last variable;
 \item[(iv)]  
    %%
    \begin{equation*}
   \liminf F\Big(\mathbf{x},r, \lambda \mathbf{n} , - \eta \lambda^2 \,
\mathbf{n}\otimes \mathbf{n} + \frac 1\rho_\diamond \lambda I \Big) > 0,
    \end{equation*}
where the limit is taken as
$\mathbf{x}\to \mathbf{x}_\diamond$, $\mathbf{x}\in \overline{\Omega}$,
$\mathbf{n} \to \mathbf{n}_\diamond$, $\abs{\mathbf{n}_\diamond}=1$,
$r \to u(\mathbf{x}_\diamond)$, $\lambda \to \infty$, and
$\eta\to\infty$;


 \item[(v)] $G:\partial\Omega\times \mathbb{R} \to [-\infty,\infty]$ is lower
semicontinuous;

\item[(vi)] $u: \overline{\Omega} \to \mathbb{R}$ is a subsolution of
$H_*=0$ in $\overline{\Omega}$ where
    %%
    \begin{equation*}
  H(\mathbf{x},r,\mathbf{p} ,X) = \begin{cases} F(\mathbf{x},r,\mathbf{p},X),
& \mathbf{x}\in \Omega,\\
        G(\mathbf{x},r), & \mathbf{x}\in \partial\Omega.
  \end{cases}
    \end{equation*}
    %%

 \end{enumerate}
  Then $G(\mathbf{x}_\diamond,u(\mathbf{x}_\diamond)) \leq 0$.
\end{theorem}

\begin{theorem}\label{T:ConeCondThm}
Assume that $d\geq 1$ and that
 \begin{enumerate}
 \item[(i)] $\Omega \subset \mathbb{R}^d$ is open;
 \item[(ii)]  $\mathbf{x}_\diamond\in \partial \Omega$
 satisfies an exterior cone condition, i.e., there is a vector
$\mathbf{n}_\diamond \in \mathbb{R}^d$ with $\abs{\mathbf{n}_\diamond}=1$ and
numbers $\theta_\diamond\in (0,\frac \pi 2]$ and $\beta_\diamond> 0$ such
that $\set{\mathbf{x}\in \mathbb{R}^d\mid
\langle{\mathbf{x}-\mathbf{x}_\diamond},{\mathbf{n}_\diamond}\rangle\geq
\cos(\theta_\diamond) \abs{\mathbf{x}- \mathbf{x}_\diamond},
\abs{\mathbf{x}-\mathbf{x}_\diamond} \leq \beta_\diamond}\cap
\overline{\Omega} = \{\mathbf{x}_\diamond \}$;

\item[(iii)] $F:\overline{\Omega}\times \mathbb{R}\times \mathbb{R}^d \times \mathcal{S}(d)\to [-\infty,\infty]$
is lower semicontinuous and degenerate elliptic, i.e., nonincreasing in its
last variable;

 \item[(iv)]
    %%
    \begin{equation*}
   \liminf
  F\Big(\mathbf{x},r, \lambda \mathbf{n} , - \mu\lambda^2(\eta
\mathbf{n}\otimes \mathbf{n} - I) \Big)>0,
    \end{equation*} 
     %%
  where the limit is taken as $\mathbf{x}\to
\mathbf{x}_\diamond$, $\mathbf{x}\in \overline{\Omega}$, $r \to
u(\mathbf{x}_\diamond)$ $\abs{\mathbf{n}}=1$, $ \lambda \to \infty$,
$\mu\to\infty$, and $ \eta\to\infty$; 

\item[(v)]  $G:\partial\Omega\times \mathbb{R} \to [-\infty,\infty]$
is lower semicontinuous;

  \item[(vi)] $u: \overline{\Omega} \to \mathbb{R}$ is a subsolution of
$H_*=0$ in $\overline{\Omega}$ where
    %%
    \begin{equation*}
  H(\mathbf{x},r,\mathbf{p} ,X) = \begin{cases}
   F(\mathbf{x},r,\mathbf{p},X), & \mathbf{x}\in \Omega,\\
           G(\mathbf{x},r), & \mathbf{x}\in \partial\Omega.
  \end{cases}
    \end{equation*}
    %%

 \end{enumerate}
  Then $G(\mathbf{x}_\diamond,u(\mathbf{x}_\diamond)) \leq 0$.
\end{theorem}


It seems to be quite difficult to get a counterexample showing, for example,
that one cannot replace (iv) in Theorem \ref{T:ConeCondThm} by
(iv) in Theorem \ref{T:SphereCondThm}) but a modification of a
classical example due to Lebesgue (see e.g.\ \cite[p.
303]{CourantHilbertBook62} where the notion of viscosity solutions is not
considered) shows that one cannot hope to be able to significantly weaken the
external cone condition. In this example one considers Laplace's equation
$-\Delta u =0$.


\begin{example}\label{P:CounterExA} \rm
Let $d\geq 4$ and assume that $\omega$ is
a  nondecreasing continuous function on $[0,1]$ with $\omega(0)=0$  such that
      %%
    \begin{equation*}
   \int_0^1 \frac {\omega(t)}{t}\,\textup{d} t < \infty\quad\text{but}\quad
  \int_0^1  \frac {\omega(t)}{t^2}\,\textup{d} t = +\infty.
    \end{equation*}
    %%
 Furthermore, let
    %%
    \begin{gather*}
 \Omega = \set{\mathbf{x}\in \mathbb{R}^d\mid \abs{\mathbf{x}} < 1,
 \abs{(x_2,\ldots,x_d)} > \omega(\abs{x_1})^{\frac 1{d-3}}x_1},\\
  \varphi(\mathbf{x})=
   -\int_0^1 \frac
{t^{d-3}\rho(t)^{d-2}}{\abs{((x_1-t),x_2,\ldots,x_d)}^{d-2}}\,\textup{d}
t,\quad \mathbf{x}\neq (s,0,\ldots, 0),\quad s\in [0,1],\\
  F(\mathbf{x},r,\mathbf{p},X) = - \mathop{\rm tr}(X),\\
  G(\mathbf{x},r) = \begin{cases} r-\varphi(\mathbf{x}),& \mathbf{x}\in
\partial\Omega\setminus\{\mathbf{0}\},\\
   r+ \int_0^1 \frac {\omega(t)}{t}\,\textup{d} t + \frac 1{d-3}, &
\mathbf{x}= \mathbf{0},
   \end{cases}\\
   u(\mathbf{x})  = \begin{cases} \varphi(\mathbf{x}),& \mathbf{x}\in
\overline{\Omega}\setminus\{\mathbf{0}\},\\
       -\int_0^1 \frac {\omega(t)}{t}\,\textup{d} t ,& \mathbf{x}= \mathbf{0}.
  \end{cases}
    \end{gather*}
    %%
 Then all assumptions of Theorem \ref{T:ConeCondThm} are satisfied except
(ii) and the conclusion of Theorem \ref{T:ConeCondThm}
does not hold when $\mathbf{x}_\diamond=\mathbf{0}$.
 \end{example}

For $\Omega$ in the example above to satisfy the exterior cone condition one
would have to assume that $\inf_{t\in (0,1)}\omega(t) > 0$. Note also that if
$ \Omega = \set{\mathbf{x}\in \mathbb{R}^d\mid \abs{\mathbf{x}} < 1,
 \abs{(x_2,\ldots,x_d)} > (\log(\abs{\log(x_1)}))^{-1}x_1}$, then the
exterior cone condition is not satisfied at $\mathbf{0}$ but neither can the
example above be applied for any $d$.


In the next example we show that for the $1$-Laplacian equation the claim of
Theorem \ref{T:SphereCondThm} (or \ref{T:ConeCondThm}) does not hold.  This
example shows that one cannot replace assumption (iv) by an
assumption of the form $F(\mathbf{x},r,\mathbf{p}, -\eta I)> 0$ for some
$\eta > 0$ when $\mathbf{p} \neq 0$.


\begin{example}\label{P:CounterExB} \rm
Let
    %%
    \begin{gather*}
 \Omega = \set{\mathbf{x}\in \mathbb{R}^2\mid  \abs{\mathbf{x}} < 1},\\
    F = \tilde F_*\quad\text{where}\quad \tilde F(\mathbf{x},r,\mathbf{p},X)
= -\frac {\mathop{\rm tr}(X)}{\abs{\mathbf{p}}} + \frac
{\langle{\mathbf{p}},{X\mathbf{p}}\rangle}{\abs{\mathbf{p}}^3}, \quad
\mathbf{p}\neq \mathbf{0},\\
  G(\mathbf{x},r) = r, \quad
  u(\mathbf{x})= 1, \quad \mathbf{x} \in \overline{\Omega}.
    \end{gather*}
    %%
 Then all assumptions of Theorem \ref{T:SphereCondThm} are satisfied except
(iv)  and  the conclusion of Theorem \ref{T:SphereCondThm}
does not hold.
  \end{example}



We need a special case of the following lemma, which is closely related to
\cite[Lemma 3.1]{CrandallGuide92}. Here inequalities for vectors are to be
taken component wise and $\boldsymbol 1$ is the vector with all components
equal to $1$.

\begin{lemma}\label{L:ComparLemmaC}
Assume that $\mathcal{A}\subset \mathbb{R}^d$,
$w:\mathcal{A}\to \mathbb{R}$ is upper semicontinuous and
$\Psi:[1,\infty)^m\times\mathcal{A} \to [0,\infty)$
nondecreasing in its first $m$ arguments and lower semicontinuous with
respect to the last one. Suppose furthermore that
    %%
    \begin{gather*}
  \mathcal{N}=\cap_{\boldsymbol\alpha \geq \boldsymbol 1}\set{\mathbf{z}\in
\mathcal{A}\mid \Psi(\boldsymbol \alpha,\mathbf{z})=0}\neq \emptyset, \\
  \lim_{\boldsymbol\alpha\to
\boldsymbol\infty}\Psi(\boldsymbol\alpha,\mathbf{z})= \infty, \quad
\mathbf{z} \in \mathcal{A}\setminus \mathcal{N},
    \end{gather*}
    %%
  and that
    %%
    \begin{equation*}
  \sup_{\mathbf{z}\in \mathcal{A}} \big(w(\mathbf{z})-\Psi(\boldsymbol
1,\mathbf{z})\big) < \infty.
    \end{equation*}
    %%
 Let $M_{\boldsymbol\alpha}= \sup_{\mathbf{z}\in \mathcal{A}}(w(\mathbf{z})-
\Psi({\boldsymbol \alpha},\mathbf{z}))$ for ${\boldsymbol\alpha} \geq 1$ and
assume that $\mathbf{z}_{\boldsymbol\alpha}\in \mathcal{A}$ is such that
    %%
    \begin{equation*}
  \lim_{{\boldsymbol\alpha} \to \boldsymbol\infty}
\Big(M_{\boldsymbol\alpha}- \Big(w(\mathbf{z}_{\boldsymbol\alpha})
-  \Psi({\boldsymbol\alpha},\mathbf{z}_{\boldsymbol\alpha})\Big)\Big) = 0.
    \end{equation*}
    %%
 If $ {\mathbf{z}_\diamond}\in \mathcal{A}$ is a cluster point of
$\mathbf{z}_{\boldsymbol\alpha}$ as ${\boldsymbol\alpha} \to
\boldsymbol\infty$, then ${\mathbf{z}_\diamond}\in \mathcal{N}$ and
$w(\mathbf{z}) \leq w({\mathbf{z}_\diamond})$ for all $\mathbf{z}\in
\mathcal{N}$ and if $\lim_{j\to\infty} \mathbf{z}_{\boldsymbol\alpha_j}=
\mathbf{z}_\diamond$ then $\lim_{j\to \infty}
\Psi({\boldsymbol\alpha_j},\mathbf{z}_{\boldsymbol\alpha_j}) = 0$.
  Moreover, if $\mathcal{A}$ is compact, then
   %%
    \begin{equation*}
  \lim_{{\boldsymbol\alpha}\to \boldsymbol\infty}
\Psi({\boldsymbol\alpha},\mathbf{z}_{\boldsymbol\alpha}) = 0.
    \end{equation*}
    %%
 \end{lemma}

If $\mathcal{A}$ is not compact then it is not necessarily true that
$\lim_{{\boldsymbol\alpha}\to \boldsymbol\infty}
\Psi({\boldsymbol\alpha},\mathbf{z}_{\boldsymbol\alpha}) = 0$ as can be seen
by taking $\mathcal{A}=[0,\infty)$, $w(z) = \frac {2z}{z+1}$ and
$\Psi(\alpha,z)= z\max\{0,\alpha-z\} + \frac z{z+1}$.


\section{Proofs}


\begin{proof}[Proof of Lemma \ref{L:ComparLemmaC}]
  If $\boldsymbol \alpha \geq \boldsymbol \beta\geq \boldsymbol 1$ then we
have $\Psi(\boldsymbol \alpha,\mathbf{z}) \geq \Psi(\boldsymbol
\beta,\mathbf{z})$ for all $\mathbf{z}\in \mathcal{A}$ and hence
$M_{\boldsymbol\alpha} \leq M_{\boldsymbol\beta}$ as well. On the other hand,
$\mathcal{N}$ is nonempty and $\sup_{\mathbf{z}\in \mathcal{N}} w(\mathbf{z})
\leq M_{\boldsymbol\alpha}$, so $M_{\boldsymbol\alpha}$ is bounded from below
and $M_{\boldsymbol\infty}:=\lim_{\boldsymbol\alpha\to \boldsymbol\infty}
M_{\boldsymbol\alpha}$ exists (and is finite).


Assume next that $\mathbf{z}_{{\boldsymbol\alpha}_j}\to
\mathbf{z}_\diamond\in \mathcal{A}$ where ${\boldsymbol\alpha}_j\to
\boldsymbol\infty$. If $\mathbf{z}_\diamond\notin \mathcal{N}$ then there
exists a vector $\boldsymbol\alpha_\diamond$ such that
$\Psi(\boldsymbol\alpha_\diamond,\mathbf{z}_\diamond) \geq
w(\mathbf{z}_\diamond) - M_{\boldsymbol\infty} + 2$. Since
$\mathbf{z}\to\Psi(\boldsymbol\alpha_\diamond,\mathbf{z})$ is lower
semicontinuous and $w$ is upper semicontinuous we see that for sufficiently
large $j$ we have
$\Psi(\boldsymbol\alpha_\diamond,\mathbf{z}_{{\boldsymbol\alpha}_j}) >
w(\mathbf{z}_{{\boldsymbol\alpha}_j}) - M_{\boldsymbol\infty} + 1$. Since
$\boldsymbol\alpha_j \geq \boldsymbol\alpha_\diamond$ for sufficiently large
values of $j$ and $\Psi$ is nondecreasing in the first variables we see that
$\Psi({\boldsymbol\alpha}_j,\mathbf{z}_{{\boldsymbol\alpha}_j}) >
w(\mathbf{z}_{{\boldsymbol\alpha}_j}) - M_{\boldsymbol\infty} + 1$ for all
sufficiently large $j$. But this contradicts the assumption that
$0=\lim_{\boldsymbol\alpha\to\boldsymbol\infty} (M_{\boldsymbol\alpha} -
w(\mathbf{z}_{\boldsymbol\alpha})+
\Psi(\boldsymbol\alpha,z_{\boldsymbol\alpha})) =
M_{\boldsymbol\infty}-\lim_{\boldsymbol\alpha\to\boldsymbol\infty}
(w(\mathbf{z}_{\boldsymbol\alpha})-
\Psi(\boldsymbol\alpha,z_{\boldsymbol\alpha}))$.

Since $\Psi$ is nonnegative and $w$ is upper semicontinuous we have
    %%
    \begin{align*}
  w(\mathbf{z}_\diamond) &\geq \limsup_{j\to\infty}
  \Big(w(\mathbf{z}_{\boldsymbol\alpha_j})
-\Psi(\boldsymbol\alpha_j, \mathbf{z}_{\boldsymbol\alpha_j})\Big) \\
&= \limsup_{j\to\infty}\Big(w(\mathbf{z}_{\boldsymbol\alpha_j})
-\Psi(\boldsymbol\alpha_j, \mathbf{z}_{\boldsymbol\alpha_j}) -
M_{\boldsymbol\alpha_j}\Big) + \lim_{j\to\infty} M_{\boldsymbol\alpha_j} \\
&= M_{\boldsymbol\infty} \geq \sup_{\mathbf{z}\in \mathcal{N}} w(\mathbf{z}).
\end{align*}
    %% 
  Since $\mathbf{z}_\diamond\in \mathcal{N}$ and $w(\mathbf{z}_\diamond)
\geq \limsup_{j\to\infty} w(\mathbf{z}_{{\boldsymbol\alpha}_j})$ this
inequality implies in addition that $\lim_{j\to\infty}
\Psi(\boldsymbol\alpha_j, \mathbf{z}_{\boldsymbol\alpha_j}) =0$.


Finally, if we assume that $\mathcal{A}$ is compact then every subsequence
$(\boldsymbol\alpha_j)_{j=1}^\infty$ has a subsequence for which
$\lim_{k\to\infty} \mathbf{z}_{\boldsymbol\alpha_{j_k}} \to
\mathbf{z}_\diamond\in \mathcal{A}$ and the claim follows from the results
already proven.  \end{proof}


\begin{proof}[Proof of Theorem \ref{T:SphereCondThm}] Suppose to the contrary
that $G(\mathbf{x}_\diamond,u(\mathbf{x}_\diamond)) > 0$.  Let
$\mathbf{n}_\diamond$ and $\rho_\diamond$ be the unit vector and number in
assumption (ii), let $\eta> 1$ be such that (possible by
(iv))
     %%
    \begin{equation}\label{E:PosLimitAB}
   \liminf F\Big(\mathbf{x},r, \lambda \mathbf{n} , - \eta \lambda^2 \,
\mathbf{n}\otimes \mathbf{n} + \frac 1\rho_\diamond \lambda I \Big) > 0,
    \end{equation}
     %%
   where the limit is taken as $\mathbf{x}\to \mathbf{x}_\diamond$,
$\mathbf{x}\in \overline{\Omega}$, $\mathbf{n} \to \mathbf{n}_\diamond$,
$\abs{\mathbf{n}_\diamond}=1$, $r \to u(\mathbf{x}_\diamond)$, and $ \lambda
\to \infty$.

 Furthermore, let $\mathbf{y}_\diamond = x_\diamond + \rho_\diamond
  \mathbf{n}_\diamond$, and define the function $\Psi$ by
    %%
    \begin{equation*}
  \Psi(\alpha,\mathbf{x}) = \psi(\alpha(\abs{\mathbf{x}-
\mathbf{y}_\diamond}-\rho_\diamond)), \quad \quad \alpha \geq 1, \quad
\mathbf{x} \in \overline{\Omega},
    \end{equation*}
    %%
  where  $\psi$  is some twice continuously
differentiable function  with $\psi'(t)\geq \frac 12$, $t\geq 0$,
$\psi(0)=0$, $\psi'(0)=1$ and $\psi''(0) = -2\eta$.
(Take for example $\psi(t) = \frac t2 + \frac 1{8\eta-2}\big(1 - \frac
1{(1+t)^{4\eta -1}}\big)$ when $t> -\frac 12$.)
   Let $\mathcal{A} = \overline{\Omega} \cap \set{\mathbf{x}\mid
\abs{\mathbf{x}-\mathbf{x}_\diamond} \leq \beta_\diamond}$ and observe that
the only point $\mathbf{x}\in \mathcal{A}$ where $\Psi(\alpha,\mathbf{x})=0$
is $\mathbf{x}_\diamond$. Since $u$ is upper semicontinuous in the compact
set $\mathcal{A}$ it is bounded from above there and for each $\alpha\geq 1$
there is a point $\mathbf{x}_\alpha \in \mathcal{A}$ such that
    %%
    \begin{equation*}
  u(\mathbf{x}_{\alpha}) - \Psi(\alpha,x_{\alpha}) = \sup_{\mathbf{x}\in
\mathcal{A}}\Big(u(\mathbf{x}) -\Psi(\alpha, \mathbf{x})\Big).
    \end{equation*}
    %%
   It follows from Lemma \ref{L:ComparLemmaC} that $\lim_{\alpha\to\infty}
\Psi(\alpha,\mathbf{x}_{\alpha}) =0$ and that $\lim_{\alpha\to\infty}
\mathbf{x}_{\alpha}=\mathbf{x}_\diamond$. Thus we see that if $\alpha$ is
sufficiently large, then $\mathbf{x}_{\alpha}$ is a local maximum point of
$u(\mathbf{x})- \Psi(\alpha,\mathbf{x})$ in $\overline{\Omega}$.  Clearly we
have $u(\mathbf{x}_{\alpha}) \geq u(\mathbf{x}_\diamond)$ and since $u$ is
upper semicontinuous we conclude that
$\lim_{\alpha\to\infty}u(\mathbf{x}_{\alpha}) =u(\mathbf{x}_\diamond)$. Since
$G$ is lower semicontinuous this implies that if $\alpha$ is sufficiently
large and $\mathbf{x}_{\alpha}\in \partial\Omega$ then
$G(\mathbf{x}_\alpha,u(\mathbf{x}_{\alpha})) > 0$.  The assumption that $u$
is a subsolution of $H_*=0$ then implies that
    %%
    \begin{equation}\label{E:FIsNotPos}
  F(\mathbf{x}_{\alpha},u(\mathbf{x}_{\alpha}),
D_{\mathbf{x}}\Psi(\alpha,\mathbf{x}_{\alpha}),
D^2_{\mathbf{x}}\Psi(\alpha,\mathbf{x}_{\alpha})) \leq 0.
    \end{equation}
    %%
  Now a calculation shows that
    %%
    \begin{equation*}
    D_{\mathbf{x}}\Psi(\alpha,\mathbf{x}) = \alpha \frac
{\psi'(\alpha(\abs{\mathbf{x}-
\mathbf{y}_\diamond}-\rho_\diamond))}{\abs{\mathbf{x} -
\mathbf{y}_\diamond}}(\mathbf{x}-\mathbf{y}_\diamond),
    \end{equation*}
    %%
  and
     %%
    \begin{align*}
&D^2_{\mathbf{x}}\Psi(\alpha,\mathbf{x}) 
= \alpha \frac {\psi'(\alpha(\abs{\mathbf{x}- \mathbf{y}_\diamond}
  -\rho_\diamond))}{\abs{\mathbf{x} - \mathbf{y}_\diamond}} I
   \\ &\quad + \Big(\alpha^2 \frac {\psi''(\alpha(\abs{\mathbf{x}-
\mathbf{y}_\diamond}-\rho_\diamond))}{\abs{\mathbf{x} -
\mathbf{y}_\diamond}^2} - \alpha \frac{\psi'(\alpha(\abs{\mathbf{x}-
\mathbf{y}_\diamond}-\rho_\diamond))}{\abs{\mathbf{x} -
\mathbf{y}_\diamond}^3}\Big) (\mathbf{x}-\mathbf{y}_\diamond)\otimes
(\mathbf{x}-\mathbf{y}_\diamond).
      \end{align*}
    %%

Now we know that $\mathbf{x}_{\alpha}\to \mathbf{x}_\diamond$ and
$\psi(\alpha(\abs{\mathbf{x}_{\alpha}-
\mathbf{y}_\diamond}-\rho_\diamond))\to 0$ and hence
$\alpha(\abs{\mathbf{x}_{\alpha}- \mathbf{y}_\diamond}-\rho_\diamond)\to 0$
as $\alpha\to \infty$. Thus we see that if we define $\mathbf{n}_\alpha =
\frac 1{\abs{\mathbf{x}_\alpha - \mathbf{y}_\diamond}}(\mathbf{x}_\alpha -
\mathbf{y}_\diamond)$ then
    %%
    \begin{gather*}
    \mathbf{n}_\alpha  \to \mathbf{n}_\diamond,\\
   \psi'(\alpha(\abs{\mathbf{x}_\alpha- \mathbf{y}_\diamond}-\rho_\diamond))
\to 1,\\
   \frac {\psi'(\alpha(\abs{\mathbf{x}_\alpha-
\mathbf{y}_\diamond}-\rho_\diamond))}
   {\abs{\mathbf{x}_\alpha - \mathbf{y}_\diamond}} \to
\frac 1\rho_\diamond,\\
   {\psi''(\alpha(\abs{\mathbf{x}_\alpha-
\mathbf{y}_\diamond}-\rho_\diamond))} \to -2\eta,
      \end{gather*}
    %%
  as $\alpha\to \infty$.

If we let $\lambda = \alpha \psi'(\alpha(\abs{\mathbf{x}_\alpha-
\mathbf{y}_\diamond} -\rho_\diamond))$ then we see that for sufficiently
large $\alpha$ (and hence $\lambda$) we have
    %%
    \begin{equation*}
  \alpha^2 {\psi''(\alpha(\abs{\mathbf{x}_\alpha-
\mathbf{y}_\diamond}-\rho_\diamond))} \leq -\lambda^2\eta .
    \end{equation*}
    %%
  Thus we see (recall that
$\abs{\mathbf{x}_\alpha-\mathbf{x}_\diamond} \geq \rho_\diamond$) that
    %%
    \begin{equation*}
  D^2_{\mathbf{x}}\Psi(\alpha,\mathbf{x}_\alpha) \leq \frac
\lambda\rho_\diamond I - \eta\lambda^2 \mathbf{n}_\alpha \otimes
\mathbf{n}_\alpha,
    \end{equation*}
    %%
  Combining this result with the degenerate ellipticity of $F$ and
\eqref{E:PosLimitAB} we get a contradiction from inequality
\eqref{E:FIsNotPos}.
 \end{proof}


\begin{proof}[Proof of Theorem \ref{T:ConeCondThm}]
  Suppose to the contrary that $G(\mathbf{x}_\diamond,u(\mathbf{x}_\diamond))
> 0$.  In order to derive a contradiction we start by choosing a number of
parameters and points.


 Since $G$ is lower
semicontinuous there is a number $\epsilon > 0$ such that
    %%
    \begin{equation}\label{E:ChoiceOfEpsilon}
  G(\mathbf{x},r) > 0, \quad \abs{\mathbf{x}- \mathbf{x}_\diamond} \leq
\epsilon, \quad x\in \partial \Omega,\quad
  \abs{r-u(\mathbf{x}_\diamond)} \leq \epsilon.
    \end{equation}
    %%
By assumption (iv) there is a number $k$ so that
    %%
    \begin{multline}\label{E:FPositive}
    F(\mathbf{x},r,\lambda\mathbf{n},-\mu\lambda^2(\eta \mathbf{n}\otimes
\mathbf{n} - I)) > 0\quad \text{when}\\ \text{$\mathbf{x}\in
\overline{\Omega}$, $\abs{\mathbf{x}-\mathbf{x}_\diamond} <\frac 1{k}$,
$\abs{r-u(\mathbf{x}_\diamond)} < \frac 1{k}$, $\lambda> k$,
$\abs{\mathbf{n}}=1$, $\mu > k$, and $\eta > k$}.
    \end{multline}
    %%
Choose a number $m\geq 2$ so that
    %%
    \begin{equation}\label{E:ChoiceOfK}
    m > \frac {2(k-1)}{\sin(\theta_\diamond)},\quad
   \frac {2^{m+2}}{3\epsilon(m+1)(2-\sin(\theta_\diamond))^{m}} >
k,\quad \epsilon\Big(1-\frac{\sin(\theta_\diamond)}2\Big)^{m+1} < \frac 1{k}.
    \end{equation}
    %%
 Since $u$ is upper semicontinuous
there is a number $\delta\in (0,\min\{\epsilon,\beta_\diamond\})$ such that
    %%
    \begin{equation}\label{E:uNotTooLarge}
  u(\mathbf{x}) < u(\mathbf{x}_\diamond)+
   2^{-m-2}\epsilon, \quad \abs{\mathbf{x}-\mathbf{x}_\diamond} < \delta,
  \quad \mathbf{x} \in \overline{\Omega}.
     \end{equation}
    %%
  and hence it follows from \eqref{E:ChoiceOfEpsilon} that
    %%
    \begin{equation}\label{E:GPosWhenUBig}
  \text{if $\abs{\mathbf{x}- \mathbf{x}_\diamond} \leq \delta$, $x\in
\partial \Omega$, and $G(\mathbf{x},u(\mathbf{x}))\leq 0$ then
$u(\mathbf{x})< u(\mathbf{x}_\diamond) - \epsilon.$}
    \end{equation}
    %%
  Choose $\beta \in (0,\beta_\diamond)$ so that
    %%
    \begin{equation}\label{E:ChoiceOfBeta}
  \beta < \frac \delta 3,\quad \beta < \frac {2}{5k}, \quad \beta < \frac
{\epsilon (m+1)}{2^{2m +1} \,k},
    \end{equation}
    %%
and  define
    %%
    \begin{equation*}
  \mathbf{y}_\beta = \mathbf{x}_\diamond + \beta\mathbf{n}_\diamond, \quad
\beta \in (0,\beta_\diamond).
    \end{equation*}
    %%


As a test function we take
     %%
    \begin{equation*}
  \psi(\mathbf{x}) = \begin{cases}
    - \epsilon(2\beta)^{-m-1} \big(2\beta - \abs{\mathbf{x}-
\mathbf{y}_\beta}\big)^{m+1},& \abs{\mathbf{x}- \mathbf{y}_\beta}\leq
2\beta,\\
  0,& \abs{\mathbf{x}- \mathbf{y}_\beta}> 2\beta.\\
  \end{cases}
    \end{equation*}
    %%
  Since $m \geq 2 $ this function is twice continuously differentiable when
$\mathbf{x} \neq \mathbf{y}_\beta$.


Since $u$ is upper semicontinuous there is a point $\mathbf{x}_\beta \in
\overline{\Omega}$, so that $\abs{\mathbf{x}_\beta-y_\beta }\leq 2\beta$ and
   %%
    \begin{equation*}
  u(\mathbf{x}_\beta) - \psi(\mathbf{x}_\beta) \geq u(\mathbf{x}) -
\psi(\mathbf{x}), \quad \abs{\mathbf{x}-\mathbf{y}_\beta}\leq 2\beta, \quad
\mathbf{x}\in \overline{\Omega}.
    \end{equation*}
    %%
If  $\abs{\mathbf{x}-\mathbf{x}_\diamond} < \delta < \beta_\diamond$ and
$\mathbf{x} \in \overline{\Omega}$, then
 if follows from assumption (ii) that
    %%
    \begin{equation}\label{E:LowConeBound}
  \abs{\mathbf{x}- \mathbf{y}_\beta} \geq
\beta\sin(\theta_\diamond),
    \end{equation}
    %%
and hence  we have
    %%
    \begin{equation*}
 u(\mathbf{x}_\diamond)-\psi(\mathbf{x}_\diamond) \leq
u(\mathbf{x}_\beta)-\psi(\mathbf{x}_\beta)
 \leq
  u(\mathbf{x}_\beta) +
\epsilon \Big(1-\frac {\sin(\theta_\diamond)}2\Big)^{m+1},
    \end{equation*}
    %%
  so that we conclude, since $\psi(\mathbf{x}_\diamond)\leq 0$ that
    %%
    \begin{equation}\label{E:uNotTooSmall}
  u(\mathbf{x}_\beta) \geq u(\mathbf{x}_\diamond) - \epsilon
\Big(1-\frac {\sin(\theta_\diamond)}2\Big)^{m+1}.
    \end{equation}
    %% If $\frac 32\beta \leq \abs{\mathbf{x}-\mathbf{y}_\beta}\leq 2\beta$
then $\abs{\mathbf{x} - \mathbf{x}_\diamond }< \delta $ since $\beta < \frac
\delta 3$ and hence by \eqref{E:uNotTooLarge}
    %%
    \begin{equation*}
  u(\mathbf{x}) - \psi(\mathbf{x}) < u(\mathbf{x}_\diamond) +
2^{-m-2}\epsilon + 4^{-m-1}\epsilon < u(\mathbf{x}_\diamond)
-\psi(\mathbf{x}_\diamond).
    \end{equation*}
    %%
 Thus we see that we must have
    %%
    \begin{equation}\label{E:OptUpperBound}
  \abs{\mathbf{x}_\beta - \mathbf{y}_\beta} < \frac 32 \beta,
    \end{equation}
    %% 
   that is, $\mathbf{x}_\beta$ is a local maximum point for $u-\psi$ in
$\overline{\Omega}$.


Furthermore, since $\frac 32\beta < \epsilon$ we note by
\eqref{E:GPosWhenUBig} that if $\mathbf{x} \in \partial \Omega$,
$\abs{\mathbf{x}-\mathbf{y}_\beta} < \frac 32\beta$ and
$G(\mathbf{x},u(\mathbf{x})) \leq 0$, then $u(\mathbf{x}) <
u(\mathbf{x}_\diamond)-\epsilon$ so that by \eqref{E:LowConeBound} we have
$u(\mathbf{x}) -\psi(\mathbf{x}) < u(\mathbf{x}_\diamond)-\epsilon +
\epsilon\big(1-\frac {\sin(\theta_\diamond)}2\big)^{m+1} <
u(\mathbf{x}_\diamond) < u(\mathbf{x}_\diamond)-\psi(\mathbf{x}_\diamond)$ so
we conclude that if $\mathbf{x}_\beta\in \partial\Omega$, then
$G(\mathbf{x}_\beta,u(\mathbf{x}_\beta))> 0$.  Thus it follows from the
assumption that $u$ is a subsolution of the equation $H_*=0$ that we in fact
have
    %%
    \begin{equation*}
  F\big(\mathbf{x}_\beta,u(\mathbf{x}_\beta), D\psi(\mathbf{x}_\beta),
D^2\psi(\mathbf{x}_\beta)\big) \leq 0.
    \end{equation*}
    %%
  It remains to show that this is a contradiction.

When we use the notation
    %%
    \begin{equation*}
  \mathbf{n}_\beta =\frac 1{\abs{\mathbf{x}_\beta-
\mathbf{y}_\beta}}(\mathbf{x}_\beta-\mathbf{y}_\beta),
    \end{equation*}
    %%
 we get
      %%
    \begin{align*}
    D\psi(\mathbf{x}_\beta)
&= \epsilon(2\beta)^{-m-1}(m+1)\big(2\beta - \abs{\mathbf{x}_\beta
  - \mathbf{y}_\beta}\big)^{m}  \mathbf{n}_\beta,\\
 D^2\psi(\mathbf{x}_\beta) & = \epsilon(2\beta)^{-m-1}(m+1)
 \big(2\beta - \abs{\mathbf{x}_\beta- \mathbf{y}_\beta}\big)^{m}
 \frac 1{\abs{\mathbf{x}_\beta- \mathbf{y}_\beta}} (I-\mathbf{n}_\beta
\otimes \mathbf{n}_\beta) \\ &\quad - \epsilon(2\beta)^{-m-1}m(m+1)
  \big(2\beta - \abs{\mathbf{x}_\beta- \mathbf{y}_\beta}\big)^{m-1}
\mathbf{n}_\beta \otimes \mathbf{n}_\beta.
    \end{align*}
    %%
Let
     %%
    \begin{gather*}
    \lambda = \epsilon (2\beta)^{-m-1}(m+1)(2\beta-
\abs{\mathbf{x}_\beta-\mathbf{y}_\beta})^{m}, \\
   \mu =\frac 1{\lambda\abs{\mathbf{x}_\beta-\mathbf{y}_\beta}},\\
  \eta = m\frac{\abs{\mathbf{x}_\beta-\mathbf{y}_\beta}}{2\beta-
\abs{\mathbf{x}_\beta-\mathbf{y}_\beta}} + 1.
    \end{gather*}
    %%
  Thus we see that
    %%
    \begin{equation*}
 D\psi(\mathbf{x}_\beta) = \lambda \mathbf{n}_\beta\quad\text{and}\quad
 D^2\psi(\mathbf{x}_\beta) = - \mu\lambda^2(\eta\, \mathbf{n}_\beta \otimes
\mathbf{n}_\beta - I).
     \end{equation*}
    %%

 From
\eqref{E:LowConeBound} and \eqref{E:OptUpperBound} a we get
    %%
    \begin{gather*}
  \frac{\epsilon (m+1)(2-\sin(\theta_\diamond))^{m}}{2^{m+1}\beta} \geq
\lambda \geq  \frac {\epsilon (m+1)}{2^{2m +1} \,\beta},\\
   \mu  \geq  \frac {2^{m+2}}{3\epsilon (m+1)(2-\sin(\theta_\diamond))^{m}},\\
   \eta  \geq m \frac{\sin(\theta_\diamond)}{2-\sin(\theta_\diamond)} + 1.
    \end{gather*}
    %%
  Now it follows from \eqref{E:ChoiceOfK} that $\eta > k$ and $\mu > k$ and
from \eqref{E:ChoiceOfBeta} that $\lambda > k$. Furthermore, the last
inequality in \eqref{E:ChoiceOfK} guarantees by \eqref{E:uNotTooLarge} and
\eqref{E:uNotTooSmall} that $\abs{u(\mathbf{x}_\beta)-u(\mathbf{x}_\diamond)}
<\frac 1{k}$.  Finally, since $\abs{\mathbf{x}_\beta-\mathbf{y}_\beta} <
\frac 32 \beta$ and $\beta < \frac 2{5k}$ by \eqref{E:ChoiceOfBeta} we have
$\abs{\mathbf{x}_\beta- \mathbf{x}_\diamond} < \frac 1{k}$. Thus it follows
from \eqref{E:FPositive} that $ F\big(\mathbf{x}_\beta,u(\mathbf{x}_\beta),
D\psi(\mathbf{x}_\beta), D^2\psi(\mathbf{x}_\beta)\big) > 0$ and we have a
contradiction.  \end{proof}


\begin{proof}[Proof of the claims in Example \ref{P:CounterExA}] A
straightforward calculation shows that $\varphi$ is harmonic in the set
$\set{\mathbf{x}\in \mathbb{R}^d \mid \mathbf{x}\neq (s,0,\ldots, 0), s\in
[0,1]}$ and thus we see that the only point which may cause problems for the
assumptions or the claim is the origin.

Using Fatou's lemma we
immediately conclude that $u$ is upper semicontinuous.
 If we define $k(s)=\varphi((s,s,0,\ldots,0))$, then we deduce that
$\lim_{s\to 0} k(s) = -\int_0^1 \frac{\omega(t)}t\,\textup{d} t$ by the
dominated convergence theorem and the fact that $(t-s)^2+ s^2 \geq \frac 12
t^2$, but $\lim_{t\uparrow 0} k'(t) = - (d-2)\int_0^1
\frac{\omega(t)}{t^2}\,\textup{d} t=-\infty$.  Since $(s,s,0,\ldots,0)\in
\overline{\Omega}$ for $s$ sufficiently small we conclude that there cannot
be a function $\psi\in \mathcal{C}^2(\mathbb{R}^d)$ such that $u-\psi$ has a
local maximum in $\overline{\Omega}$ at $\mathbf{0}$ and hence there is
nothing to check in the definition of a viscosity subsolution at $\mathbf{0}$.

It remains to show that $G$ is lower semicontinuous in
$\partial \Omega\times \mathbb{R}$ and it suffices to show that
    %%
    \begin{equation}\label{E:LimInfClaim}
  \liminf_{x\downarrow 0} \int_0^1 \frac {t^{d-3}\omega(t)}{\sqrt{(x-t)^2 +
\omega(x)^{\frac 2{d-3}}x^{2}}^{d-2}}\,\textup{d} t 
  \geq \int_0^1 \frac{\omega(t)}t\,\textup{d} t +  \frac{1}{d-3}.
    \end{equation}
    %%
  By the triangle inequality and a series expansion we have
    %%
    \begin{equation}\label{E:LowerBoundForIntA}
    \begin{aligned}\relax
 &\int_0^1 \frac {t^{d-3}\omega(t)}{\sqrt{(x-t)^2 +
     \omega(x)^{\frac 2{d-3}}x^{2}}^{d-2}}\,\textup{d} t \\
  &\geq
  \int_x^1 \frac {t^{d-3}\omega(t)}{(t -x(1-\omega(x)^{\frac 1{d-3}}))^{d-2}}
  \,\textup{d} t \\
   &= \int_x^1 \frac {\omega(t)}{t}\textup{d} t +
 \sum_{n=1}^\infty \binom {n+d-3}{d-3}
      \Big(1-\omega(x)^{\frac 1{d-3}}\Big)^n x^n \int_x^1
   \frac  {\omega(t)}{t^{n+1}}\textup{d} t.
 \end{aligned}
    \end{equation}
     %%
   Since $\omega$ is nondecreasing we have
    %%
    \begin{equation*}
  \int_x^1 \frac {\omega(t)}{t^{n+1}}\textup{d} t \geq \frac  {\omega(x)}n
   (x^{-n}-1).
    \end{equation*}
    %%
  This inequality implies that
    %%
    \begin{equation}\label{E:LowerBoundForIntB}
  \begin{aligned}\relax
 & \sum_{n=1}^\infty \binom {n+d-3}{d-3} \Big(1-\omega(x)^{\frac 1{d-3}}\Big)^n x^n \int_x^1
   \frac {\omega(t)}{t^{n+1}}\textup{d} t \\
 & \geq \omega(x) \sum_{n=1}^\infty \frac 1n\binom
 {n+d-3}{d-3} \Big(1-\omega(x)^{\frac 1{d-3}}\Big)^n \\
 &\quad - \omega (x) \sum_{n=1}^\infty \frac 1n\binom
 {n+d-3}{d-3} \Big(1-\omega(x)^{\frac 1{d-3}}\Big)^n x^n.
  \end{aligned}
    \end{equation}
    %%
   Because $\frac 1n\binom {n+d-3}{d-3} \geq \frac 1{d-3}\binom {n+d-4}{d-4}$
we get
     %%
   \begin{align*}
&\omega(x)\sum_{n=1}^\infty \frac 1n\binom
  {n+d-3}{d-3} \Big(1-\omega(x)^{\frac 1{d-3}}Big)^n \\
&\geq \frac {\omega(x)}{d-3}\sum_{n=0}^\infty \binom
{n+d-4}{d-4} \Big(1-\omega(x)^{\frac 1{d-3}}Big)^n - \frac {\omega(x)}{d-3}\\
&=  \frac{\omega(x)}{(d-3)(1-(1-\omega(x)^{\frac 1{d-3}}))^{d-3}}
 - \frac {\omega(x)}{d-3}= \frac {1-\omega(x)}{d-3}.
\end{align*}
    %%
 This inequality, \eqref{E:LowerBoundForIntA} and \eqref{E:LowerBoundForIntB}
imply that \eqref{E:LimInfClaim} holds since $\lim_{x\downarrow
0}\omega(x)=0$ and
    %%
    \begin{gather*}
    \lim_{x\downarrow 0} \int_x^1 \frac {\omega(t)}{t}\textup{d} t=
\int_0^1 \frac {\omega(t)}{t}\textup{d} t,\\
 \lim_{x\downarrow 0}\sum_{n=1}^\infty \frac {\omega(x)}n
\binom{n+d-3}{d-3} \Big(1-\omega(x)^{\frac 1{d-3}}\Big)^n x^n =0.
    \end{gather*}
    %%
   \end{proof}


\begin{proof}[Proof of the claims in Example \ref{P:CounterExB}]
  A straightforward calculation shows that $\tilde F$ is nonincreasing in its
last variable when $\mathbf{p} \neq \mathbf{0}$ and then $F$ has the same
property (for all $\mathbf{p}$).  Since $F(\mathbf{x},r,\mathbf{0}, X) =
-\infty$ when $X\geq 0$ it is clear that $u$ is a subsolution of $F=0$ in
$\Omega$ so it remains to check the boundary points and we may without loss
of generality assume that $\mathbf{x}_0= (1,0)$. Assume thus that
$u(\mathbf{x}) \leq \psi(\mathbf{x})$ for all $\mathbf{x}\in
\overline{\Omega}$ with $\abs{\mathbf{x}-\mathbf{x}_0} < \delta$ for some
$\delta > 0$ and $u(\mathbf{x}_0) = \psi(\mathbf{x}_0)$.  Clearly
$\psi_x(\mathbf{x}_0) \leq 0$ and since the function $t\mapsto
\psi(\cos(t),\sin(t))$ has a local minimum at $t=0$ we see that
$\psi_y(\mathbf{x}_0)=0$ and $\psi_{yy}(\mathbf{x}_0) \geq 0$. If
$\psi_x(\mathbf{x}_0)=0$ it follows from the fact that
$\psi_{yy}(\mathbf{x}_0) \geq 0$ that $F(\mathbf{x}_0,1,D\psi(\mathbf{x}_0),
D^2\psi(\mathbf{x}_0)) = -\infty$ and if $\psi_x(\mathbf{x}_0) < 0$ it
follows from that fact that $\psi_y(\mathbf{x}_0)=0$ that
   \[ 
F(\mathbf{x}_0,1,D\psi(\mathbf{x}_0), D^2\psi(\mathbf{x}_0))=
\tilde F(\mathbf{x}_0,1,D\psi(\mathbf{x}_0), D^2\psi(\mathbf{x}_0)) = -
\frac{\psi_{yy}(\mathbf{x}_0)}{\abs{\psi_x(\mathbf{x}_0)}} \leq 0. 
     \]
\end{proof}


\bibliographystyle{plain}

\begin{thebibliography}{1}

\bibitem{BarlesBusca01}
G.~Barles and J.~Busca.
\newblock Existence and comparison results for fully nonlinear degenerate
  elliptic equations without zeroth-order term.
\newblock {\em Comm. Partial Differential Equations}, 26(11-12):2323--2337,
  2001.

\bibitem{BarlesDaLio04}
G.~Barles and F.~Da~Lio.
\newblock On the generalized {D}irichlet problem for viscous
  {H}amilton-{J}acobi equations.
\newblock {\em J. Math. Pures Appl. (9)}, 83(1):53--75, 2004.

\bibitem{CourantHilbertBook62}
R.~Courant and D.~Hilbert.
\newblock {\em Methods of mathematical physics. {V}ol. {II}: {P}artial
  differential equations}.
\newblock (Vol. II by R. Courant.). Interscience Publishers, New York-London,
  1962.

\bibitem{CrandallGuide92}
M.G. Crandall, H.~Ishii, and P.L. Lions.
\newblock User's guide to viscosity solutions of second order partial
  differential equations.
\newblock {\em Bull. Amer. Math. Soc.}, 27:1--67, 1992.

\bibitem{DaLio02}
F.~Da~Lio.
\newblock Comparison results for quasilinear equations in annular domains and
  applications.
\newblock {\em Comm. Partial Differential Equations}, 27(1-2):283--323, 2002.

\bibitem{Ishii87}
H.~Ishii.
\newblock Perron's method for {H}amilton-{J}acobi equations.
\newblock {\em Duke Math. J.}, 55(2):369--384, 1987.

\end{thebibliography}


\section*{Corrigendum: Posted September 5, 2006}

In Example \ref{P:CounterExB} (page 4), replace 
``$\Omega = \{\mathbf{x}\in \mathbb{R}^2\mid | \mathbf{x}| < 1\}$''
  by 
$$
\Omega =\{(x,y)\in \mathbb{R}^2 \mid x < 1\}
$$
  and in the proof of the claims in Example \ref{P:CounterExB} (page 10),
  replace 
``the function $t\mapsto \psi(\cos(t),\sin(t))$'' 
  by
``the function $t\mapsto \psi(1,t)$''.



\end{document}