数理解析研究所講究録
第 1779 巻 2012 年 140-150

140

THE PHRAGMEN-LINDELOF THEOREM FOR-VISCOSITY SOLUTIONS$L^{p}$KAZUSHIGE NAKAGAWA ( 中川 和重 )TOHOKU UNIVERSITY (東北大学)

ABSTRACT. The Phragm\’en-Lindel\"of theorem is established for -viscosity
solutions of fully nonlinear second order elliptic partial differential equations with unbounded ingredients.
$L^{p}$

1. INTRODUCTION
The notion of -viscosity solutions was introduced in [5] to study fully
nonlinear second order elliptic partial differential equations (PDEs for short)with unbounded inhomogeneous terms. We refer to [3] (see also [4]) asa pioneering work for the regularity theory of viscosity solutions of fully
nonlinear PDEs.
It turned out that the Aleksandrov-Bakelman-Pucci (ABP for short) maximum principle can be extended to L-viscosity solutions for fully nonlinear
second order elliptic PDEs with unbounded coefficients and inhomogeneousterms in [15]. See also [18] for a generalization.
As an application of the ABP maximum principle in [15], it is known
that the (boundary) weak Harnack inequality for $IP$ -viscosity solutions of
the associated extremal PDEs in [16] holds, which implies H\"older continuity
for -viscosity solutions of fully nonlinear elliptic PDEs with unbounded
ingredients. We also refer to [20] for H\"older continuity estimates on IPviscosity solutions by a different approach.
On the other hand, qualitative properties of viscosity solutions of fully
nonlinear elliptic PDEs have been investigated as generalizations for classical elliptic PDE theory. For instance, the ABP maximum principle inunbounded domains in [7] and [16], the Liouville property in [11, 6], theHadamard principle in [6], and the Phragm\’en-Lindel\"of theorem in [8, 14].
We refer to references in [8, 11, 6] for these qualitative properties of strong/classicalsolutions.
Our aim here is to give a sharp estimates of the Phragm\’en-Lindel\"of theorem in [14] when PDEs have unbounded coefficients (i.e. in this paper).
In view of the boundary weak Harnack inequality in [16], it is natural torelax the hypotheses on coefficients and inhomogeneous terms. However, for$I\nearrow$$L^{p}$$b$2000 Mathematics Subject Classification. $35B53,35D40,35B50$ .
Key words and phrases. Phragm\’en-Lindelof theorem; -viscosity solution; weak Harnack inequality.
$L^{p}$

141
the weak Harnack inequality, we need to suppose that the coefficient to thefirst derivatives is small enough in -norm. When we work in boundeddomains, this is not a restriction. Since we are concerned with unboundeddomains, we will need a bit more delicate analysis than those in [8].
Our paper is organized as follows: section 2 is devoted to showing the definitions and known results. In section 3, we present the ABP type estimateson -viscosity subsolutions of fully nonlinear PDEs with unbounded ingredients under appropriate geometric conditions. We show the Phragm\’enLindel\"of theorem in our setting in section 4.
$L^{n}$$L^{p}$

2. PRELIMINARIES
We consider next fully nonlinear second order PDEs in unbounded domains:$\Omega\subset \mathbb{R}^{n}$in$G(x, u, Du, D^{2}u)=f(x)$$\Omega$

,

(2.1)

:
where
and
:
are given measurable
functions. We also suppose that $(r,p, M)\in \mathbb{R}\cross \mathbb{R}^{n}\cross S^{n}arrow G(x, r,p, M)$ is
continuous for almost all. Here, denotes the set ofsymmetric
matrices with the standard order.
We will use the standard notation from [13]. We denote by
the setof all nonnegative functions in.
Throughout this paper, we assume that$G$$\Omega\cross \mathbb{R}\cross \mathbb{R}^{n}\cross S^{n}arrow \mathbb{R}$$x\in\Omega$$f$$\Omegaarrow \mathbb{R}$$S^{n}$$n\cross n$$L_{+}^{p}(\Omega)$$L^{p}(\Omega)$$p> \frac{n}{2}$

We note that if
a continuous function in.

for $p>n/2$ , then we may identify with, and $u(x)$ is twice differentiable for almost all$u\in W_{1oc}^{2,p}(\Omega)$$x\in\Omega$$\Omega$.

$u$

At first, we denote the definition of $IP$ -viscosity solutions of (2.1).
Definition 2.1. We call $u\in C(\Omega)$ an $IP$ -viscosity subsolution (resp., supersolution) of (2.1) if$ess_{x}\lim_{arrow x0}\inf\{G(x, u(x), D\phi(x), D^{2}\phi(x))-f(x)\}\leq 0$$(resp.,$$ess.\lim_{xarrow x_{0}}\sup\{G(x, u(x), D\phi(x), D^{2}\phi(x))-f(x)\}\geq 0)$

wheneverandis a local maximum (resp., minimum)
point of $u-\phi$ . A function $u\in C(\Omega)$ is called an -viscosity solution of (2.1)
if it is both an -viscosity subsolution and an -viscosity supersolution of$\phi\in W_{1oc}^{2,p}(\Omega)$$x_{0}\in\Omega$$L^{p}$$L^{p}$$L^{p}$(2.1).

To make easier recalling the right inequality, we will often say that

an $IP$ -viscosity solution of$G(x, u, Du, D^{2}u)\leq f(x)$$($resp., $G(x,$$u,$ $Du,$ $D^{2}u)\geq

f(x))$

,

$u$

is

142
if it is an $IP$ -viscosity subsolution (resp., supersolution) of (2.1).
In what follows, instead of (2.1), we mainly consider PDEs which do notdepend on u-variable:in$F(x, Du, D^{2}u)=f(x)$$\Omega$.

(2.2)

We will assume that $F$ is (degenerate) elliptic:$F(x,p, M)\leq F(x,p, N)$

for all(2.3)

provided $M\geq N$.
, we also assume thatsuch that$(x,p, M, N)\in\Omega\cross \mathbb{R}^{n}\cross S^{n}\cross S^{n}$

For fixed ellipticity constants
there exists$0<\lambda\leq\Lambda$$b\in L_{+}^{q}(\Omega)$$\mathcal{P}^{-}(M)-b(x)|p|\leq F(x,p, M)$

where the Pucci operators$\mathcal{P}^{-}(M)=\min\{$for$\mathcal{P}^{\pm}:S^{n}arrow \mathbb{R}$$(x,p, M)\in\Omega\cross \mathbb{R}^{n}\cross S^{n}$

(2.4)

,

are defined by-trace(AM) : $A\in S^{n}:\lambda I\leq M\leq\Lambda I\}$ ,(2.5)

and $\mathcal{P}^{+}(M)=\max\{$ -trace(AM) : $A\in S^{n}:\lambda I\leq M\leq\Lambda I\}$ .
We will use the Escauriaza’s constant $p_{0}=p_{0}(n, \lambda, \Lambda)\in[n/2, n)$ , forwhich we refer to [12]. It is known that for $p>p_{0}$ , and $f\in L^{p}(B_{r}(z))$ ,where $B_{r}(x)=\{y\in \mathbb{R}^{n} : |x-y|<r\}$ , there exists a strong solutionof$u\in C(\overline{B}_{r}(z))\cap W_{1oc}^{2,p}(B_{r}(z))$$\mathcal{P}^{-}(D^{2}v(x))=f(x)$under $v(x)=0$ for$x\in\partial B_{r}(z)$a.e. in$B_{r}(z)$with estimates:$-C\Vert f^{-}\Vert_{L^{p}(B_{r}(z))}\leq v(x)\leq C\Vert f^{+}\Vert_{L(B_{r}(z))}p$

in $B_{r}(z)$and$\Vert v\Vert_{W_{1oc}^{2,p}(B_{r}(z))}\leq C’\Vert f\Vert_{LP(B_{r}(z))}$,where $C=C(n, \lambda, \Lambda,p)>0$ and $C’=C’(n, \lambda, \Lambda,p, r)>0$ are positive

constants.
We remark that to prove the ABP maximum principle [15, Theorem 2.9],
which implies the boundary weak Harnack inequality [16, Theorem 6.1], it
suffices to obtain the existence of strong solutions of the above extremal
equation only in balls although this fact is not clearly mentioned in [15, 16].
-estimates for more general
In fact, this existence result holds with local
domains satisfying the uniform exterior cone property but the$W^{2,p}$$p_{0} \in[\frac{n}{2}, n)$associated with general domains might be bigger than the above. We also
notice that we may replace cubes by balls in the (boundary) weak Harnack
inequality in [16] by Cabr\’e covering argument.
. Let $T,$ $T’\subset B_{R}(z)$ be domains such that
Fix $R>0$ and$s$$z\in \mathbb{R}^{n}$$\overline{T}\subset T’$,and$\theta_{0}\leq\frac{|T|}{|T|}\leq 1$

for some$\theta_{0}>0$.

When we apply our weak Harnack inequality below, our choice ofalways satisfies the above condition.

$T$and $T’$143
For a given domain$T’\cup A$ by$A\subset \mathbb{R}^{n}$and a function $v\in C(A)$ , we define$v_{\overline{m}}$on$v_{\overline{m}}(x)=\{\begin{array}{ll}\min\{v(x), m\} if x\in A,m if x\in T’\backslash A,\end{array}$where$m= \lim_{xarrow T},\inf_{\cap\partial A}v(x)$.

We note that if, thenis a real-valued function and that if
is a nonnegative $IP$-viscosity supersolution of (2.2) and $f\leq 0$
in $T’\cap A$ , thenis a nonnegative $IP$ -viscosity supersolution of (2.2) in $T’$ .
Next, we recall the boundary weak Harnack inequality when PDEs haveunbounded coefficients and inhomogeneous terms.
$T’\cap\partial A\neq\emptyset$$v_{\overline{m}}$$T’\cap\partial A\neq\emptyset,$ $v$$v_{\overline{m}}$Lemma 2.2 ([16, Theorem 6.1]). Let $T,$ $T’,$andand that$T\cap A\neq\emptyset$$A$

be as above. Assume that$T’\backslash A\neq\emptyset$$q>n$ ,

Then, there exist constants$q\geq p>p_{0}$

.

(2.6)

and
satisfying the following property: if $b\in L_{+}^{q}(T’\cap A)$ ,$f\in L_{+}^{p}(T’\cap A)_{f}$ a nonnegative-viscosity solution $w\in C(T’\cap A)$ of$\epsilon_{0}=\epsilon_{0}(n, \lambda, \Lambda)>0,$$r=r(n, \lambda, \Lambda,p, q)>0$$C_{0}=C_{0}(n, \lambda, \Lambda,p, q)>0$$L^{p}$$\mathcal{P}^{+}(D^{2}w)+b(x)|Dw|\geq-f(x)$in $T’\cap A$ ,and$\Vert b\Vert_{L^{n}(T’\cap A)}\leq\epsilon_{0}$

,

(2.7)

then it follows thatfll$( \frac{1}{|T|}\int_{T}(w_{T,A}^{-})^{r}dx)^{1/r}\leq C_{0}(\inf_{T}w_{T,A}^{-}+R\Vert$$L^{n}(T’\cap A))$

provided that $q>n$ and $q\geq p\geq n$ , and$( \frac{1}{|T|}\int_{T}(w_{T’,A}^{-})^{r}dx)^{1/r}$$\leq C_{0}(\inf_{T}w_{T’,A}^{-}+R^{2-\frac{n}{p}}\Vert f\Vert_{L^{p}(T’\cap A)}\sum_{k=0}^{M}R^{(1-\frac{n}{q})k}\Vert\mu\Vert_{L^{q}(T’\cap A)}^{k})$provided that $q>n>p>p_{0}$ , where $M=M(n,p, q)\geq 1$ is an integer.

In the next section, we will establish some local and global ABP type estimates on $IP$ -viscosity subsolutions for (2.2). Finally, we recall the notations
concerning the shape of domains from [8].

Definition 2.3 (Local geometric condition). Let
point of if there exist $R=R_{y}>0$ anda$G_{\sigma,\tau}$$\Omega$$y\in B_{R}(z)$,and$\sigma,$$\tau\in(0,1)$$z=z_{y}\in \mathbb{R}^{n}$. We call$y\in\Omega$such that$|B_{R}(z)\backslash \Omega_{y,B_{R}(z),\tau}|\geq\sigma|B_{R}(z)|$

,

(2.8)

144
containing . For
is the connected component of$\tau\in(0,1)$ , and $R_{0}>0,$, we callpoint in if is aa
$R=\mathscr{F}>0$
satisfying
point in withand $z=z_{y}$where$\sigma,$$\Omega_{y,B_{R}(z),\tau}$$y$$B_{\frac{R}{\tau}}(z)\cap\Omega$$y\in\Omega$$\eta\geq 0$$G_{\sigma,\tau}^{R_{0},\eta}$$\Omega$$y$$\Omega$$G_{\sigma,\tau}$$R\leq R_{0}+\eta|y|$

.

(2.9)

Definition 2.4 (Global geometric condition). We callpoint in .
if anyis a$y\in\Omega$a weak-G domain$\Omega$$\Omega$$G_{\sigma,\tau}^{R_{0},\eta}$,
point
Remark 2.5. For the sake of simplicity of notations, for a, whereandare from Definition
we will writefor$y\in\Omega$$G_{\sigma,\tau}$$B_{y}$$\mathscr{F}>0$$B\mathscr{N}(z_{y})$$z_{y}\in \mathbb{R}^{n}$2.3.
We refer the reader to [21] and [8] for examples of weak-G domains
We also refer to [1] for a generalization.

$\Omega$.

3. ABP TYPE ESTIMATES
In this section, we first present pointwise estimates on -viscosity subsolutions of (2.2), which is often referred as the Krylov-Safonov growth lemma.
For simplisity, throughout this paper, we assume that $p\geq n$ . In what fol.
point withlows, we fix $\tau\in(0,1)$ and $R_{4}>0$ . Let
be a, which is $hom$
It is possible to apply our weak Harnack inequality in. Here and later,is the constantDefinition 2.3, iffrom Lemma 2.2.
, we may use Cabr\’e covering argument; we
Even ifdivideinto small pieces so that we may apply the weak Harnack inequalby
ity in each piece. We then obtain the weak Harnack inequality in
combining all the inequalities for small pieces.
, this argument
However, since we need the estimates uniform inwhendoes not work immediately because of unboundedness of$L^{p}$$y\in\Omega$$\sigma,$$\eta\geq 0$$G_{\sigma,\tau}^{R_{O},\eta}$$B_{y}$$\epsilon_{0}>0$$\Vert b\Vert_{L^{n}(B_{y}\cap\Omega)}\leq\epsilon_{0}$$s$$\Vert b\Vert_{L^{n}(B_{y}\cap\Omega)}>\epsilon_{0}$$B_{y}$$B_{y}$$y\in\Omega$$\{\mathscr{F}\}_{y\in\Omega}$$\eta>0$.

To avoid this difficulty, we will suppose a decay rate of :, there exists
such thatfor any$b$$\delta>0$$\epsilon>0$$\sup_{R>1}\int_{E}R^{n}b(Rx)^{n}dx<\epsilon$

for $E\subset A,$ $|E|<\delta$,(3.1)

where $A= \Omega\cap\{x\in \mathbb{R}^{n}|\frac{1}{4}\min\{1/(1+\eta), (\sigma/4)^{1/n}\}<|x|<2+1/\tau\}$ .
Lemma 3.1 (pointwise estimate). Assume that (2.3), (2.6) and (2.4) holdpoint in with $R=\mathscr{K}>0$. Let $\eta>0$ and
be a
with
and
. Then, there exist
andsatisfying the following property: if $w\in C(\Omega)$ is an, then we have the following
viscosity subsolution of (2.2) with$|y|\leq
R_{0}$$p\geqn$,properties: (i) Ifandthen$b\in L_{+}^{q}(\Omega)$$z=z_{y}\in \mathbb{R}^{n}$$y\in\Omega$$G_{\sigma,\tau}^{R_{0},\eta}$$\Omega$$\kappa=\kappa(n, \lambda, \Lambda, \sigma, \tau, R_{0}, \eta)\in(0,1)$$Iy_{-}$$\epsilon=\epsilon(n, \sigma, \eta)>0$$f\in L_{+}^{p}(\Omega)$fll$w(y) \leq\kappa\sup_{B_{y}\cap\Omega}w^{+}+(1-\kappa)\lim_{xarrow B_{y}}\sup_{\cap\partial\Omega}w^{+}+R_{0}\Vert$$L^{n}(B_{y}\cap\Omega)$.

145(ii) Assume that (3.1) is satisfied and that $|y|>R_{0}$ . If $p\geq n$ , then.

$w(y) \leq\kappa\sup_{B_{y}\cap\Omega}w^{+}+(1-\kappa)\lim_{xarrow B_{y}}\sup_{\cap\Omega}w^{+}+R\Vert f\Vert_{L^{n}(B_{y}\cap\Omega\backslash B_{\text{\’{e}} R}(0))}$Remark 3.2. To get the weak maximum principle (Lemma 4.1 below), it
is important to have the term lfll
instead of Ifll
inthe estimates of the assertion (ii) above.
$L^{p}(B_{y}\cap\Omega\backslash B_{\epsilon R}(0))$$L^{p}(B_{y}\cap\Omega)$

Proof. First of all, we shall omit giving the proof in the case of$\Vert b\Vert_{L^{q}(\Omega)}=0$

because it is easy to do it, and we suppose that.
It is enough to show the assertion when $:= \lim\sup_{xarrow B_{y}\cap\partial\Omega}w^{+}(x)=0$ .
In fact, after
established the assertion when $\hat{C}=0$ , we may apply
the result to $w-C$ to prove the assertion in the general case.
Due to (2.4), is an -viscosity solution of$\Vert b\Vert_{L^{q}(\Omega)}>0$$\hat{C}$$having\wedge$$w$$L^{p}$

in$\mathcal{P}^{-}(D^{2}w)-b(x)|Dw|\leq f(x)$$\Omega$.

Setting, we immediately see that $v(x);=C_{w}-w(x)$ is
an -viscosity solution of$C_{w}= \sup_{B_{y}\cap\Omega}w^{+}$$I\nearrow$in$\mathcal{P}^{+}(D^{2}v)+b(x)|Dv|\geq-f(x)$$\Omega$.

We shall first prove (ii).
Case (ii) $|y|>R_{0}$ :Taking$\epsilon=\frac{1}{4}\min\{\frac{1}{1+\eta}, (\frac{\sigma}{4})^{\frac{1}{n}}\}\in(0, \frac{1}{2}\min\{\frac{1}{1+\eta}, (\frac{\sigma}{4})^{\frac{1}{n}}\})$

andObserve that
$1/(1+\eta)$$(2\epsilon)^{n}<\sigma/4$. We set$T=B_{R}(z)\backslash \overline{B}_{2\epsilon R}(0)$. Note thatand$2\epsilon<$$T’=B_{y}\backslash \overline{B}_{\epsilon R}(0)$.

$2 \epsilon R<\frac{R}{1+\eta}\leq\frac{R_{0}+\eta|y|}{1+\eta}<|y|$and consequently
ponent ofwhich contains$y\in T=B_{R}(z)\backslash \overline{B}_{2\epsilon R}(0)$$T’\cap\Omega$

$y$

. Let

$A$

. We havebe the connected com-$|T\backslash A|\geq|T\backslash \Omega_{y,B_{R}(z),\tau}|$$\geq|B_{R}(z)\backslash \Omega_{y,B_{R}(z),\tau}|-|B_{2\epsilon R}(0)|$$\geq\sigma|B_{R}(0)|-(2\epsilon)^{n}|B_{R}(0)|$$\geq\frac{\sigma}{2}|B_{R}(0)|$$\geq\frac{\sigma}{2}|T|$

.

Since
$T’\cap\partial A\subset T’\cap\partial(T^{l}\cap\Omega)\subset T’\cap(\partial T^{l}\cup\partial\Omega)=T^{l}\cap\partial\Omega$in view of$\hat{C}\leq 0$

,

(3.2)

, we have$\lim_{xarrow T},\inf_{\cap\partial A}v(x)=C_{w}-\lim_{xarrow T}\sup_{\cap\partial A}w(x)\geq C_{w}$Now, we verify (2.7). By (3.1), if I

$Rb(R\cdot)\Vert_{L^{n}(A)}\leq\epsilon_{0}$.

, we see that$\Vert b\Vert_{L^{n}(T’\cap A)}\leq\Vert Rb(R\cdot)\Vert_{L^{n}(A)}\leq\epsilon_{0}$

.

(3.3)

146
Setting$m= \lim\inf_{xarrow T^{l}\cap\partial A}v(x)$, we use (3.3) to show for any $r>0$ ,$( \frac{\sigma}{2})^{1/r}C_{w}\leq(\frac{|T\backslash A|}{|T|})^{1/r}C_{w}\leq(\frac{1}{|T|}\int_{T\backslash A}m^{r}dx)^{1/r}\leq(\frac{1}{|T|}\int_{T}(v_{m}^{-})^{r}dx)^{1/r}$

Since $y\in A$ , we have$\inf_{T}v_{\overline{m}}\leq v(y)=C_{w}-w(y)$

.

(3.4)

Thus, letting $r>0$ be the constant from Lemma 2.2, we have$( \frac{\sigma}{2})^{1/r}C_{w}\leq C_{0}(\inf_{T}v_{\overline{m}}+R\Vert f\Vert_{L^{n}(T’\cap A))}\leq C_{0}(C_{w}-w(y)+R\Vert f\Vert_{L^{n}(T’\cap\Omega)})$.

Therefore, we conclude that the assertion (ii) holds with $\kappa=1-(\frac{\sigma}{2})^{1/r}\min\{C_{0}^{-1},1\}>$.
in the case where 1. In this case, we can show that
Next assume thatusing a Cabr\’e’s covering argument.
Case (i) $|y|\leq R_{4}$ :$0$$Rb(R\cdot)\Vert_{L^{n}(A)}\leq\epsilon_{0}$$\Vert Rb(R\cdot)\Vert_{L^{n}(A)}>\epsilon_{0}$Since we have $R\leq(1+\eta)R_{0}$ in this case, we may regard as a boundeddomain. Thus, we can use the standard covering argument by Cabr\’e without, we haveandusing (3.1). Setting $T=B_{R}(z),$$\Omega$$A=\Omega_{y,B_{R}(z),\tau}$$T^{l}=B_{\frac{R}{\tau}}(z)$$|T \backslash A|=|B_{R}(z)\backslash \Omega_{y,B_{R}(z),\tau}|\geq\sigma|B_{R}(z)|\geq\frac{\sigma}{2}|T|$.

.
We shall only give a proof when
Following the same argument as in case (ii) with the above inequality,and new $A,$ $T,$ $T’$ , we have$\Vert b\Vert_{L^{n}(T’\cap A)}\leq\epsilon_{0}$$( \frac{\sigma}{2})^{1/r}C_{w}\leq C_{0}(\inf_{T}v_{\overline{m}}+R_{0}$IIfll$L^{n}(B_{y}\cap\Omega))$$\leq C_{0}(C_{w}-w(y)+R_{0}\Vert f\Vert_{L^{n}(B_{y}\cap\Omega)})$.

Therefore, we conclude that the assertion holds with the same $\kappa\in(0,1)$ as
in case (ii).
$\square$is a weeak-G domain, we derive the ABP maximum prinWhen
ciple for If-viscosity subsolutions bounded from above of (2.2).
$\Omega\subset \mathbb{R}^{n}$

Theorem 3.3 (ABP maximum principle in unbounded domains). Assume
satisfying (3.1). Let $\eta>0$ and(2.6), (2.3) and (2.4) withbe a weak-G domain. Assume also$\Omega\subset \mathbb{R}^{n}$$b\in L_{+}^{q}(\Omega)$$\sup$$\mathscr{K}\Vert f\Vert_{L^{n}(A_{y}\cap\Omega)}<\infty$$y\in\Omega,|y|>R_{0}$Let$\frac{1}{4}\min\{\frac{1}{1+\eta}, (\frac{\sigma}{4})^{1/n}\}\leq\epsilon<\min\{\frac{1}{1+\eta}, (\frac{\sigma}{4})^{1/n}\}$. Then, there exists$C=C(n, \lambda, \Lambda,p, q,\epsilon, \sigma, \tau, R_{0}, \eta)>0$

(3.5)

147
satisfying the following properties: $ifw\in C(\Omega)$ is an IP-viscosity subsolution
bounded from above of (2.2) with, then it follows that$f\in L_{+}^{p}(\Omega)$$\sup$$\sup_{\Omega}w\leq\lim_{xarrow}\sup_{\partial\Omega}w^{+}(x)+C$$R_{y}\Vert f\Vert_{L^{n}(A_{y}\cap\Omega)}$$y\in\Omega,|y|>R_{0}$$+CR_{0}$$\sup$$\Vert f\Vert_{L^{n}(B_{y}\cap\Omega)}$

(3.6)

.

$y\in\Omega,|y|\leq R_{0}$Here,$A_{y}=B_{\underline{R}_{A}}\tau(z_{y})\backslash B_{\epsilon R_{4}}(0)$andProof. We take the supremum over$B_{y}=B_{\underline{R}_{A}}\tau(z_{y})$$y\in\Omega$

to conclude the inequalities (3.6).

.

with the estimates in Lemma 3.1$\square$

4. $PHRAGM\text{\’{E}} N-LINDEL\ddot{O}F$ THEOREM

In this section, we show that the weak maximum principle holds for
PDEs with zero-order terms. As before, assuming thatis a weak-G
domain, for each $y\in\Omega$ , we use the notations $R_{y}>0$ and
.
Also,
,
and
respectively, denote
andfor$\Omega$$z_{y}\in \mathbb{R}^{n}$$B_{y}$$A_{y}$$B_{\frac{R_{y}}{\tau}}(z_{y})\backslash B_{\epsilon R_{y}}(0)$$B_{\frac{R_{y}}{\tau}}(z_{y})$$\epsilon\in[\frac{1}{4}\min\{\frac{1}{1+\eta}, (\frac{\sigma}{4})^{1/n}\}, \frac{1}{2}\min\{\frac{1}{1+\eta}, (\frac{\sigma}{4})^{1/n}\})$.

Lemma 4.1. Assume (2.3), (2.6) and (2.4) withsatisfying(3.1). Let $\eta>0$ andbe a weak-G domain. Then, there exists $c_{0}=$$b\in L_{+}^{q}(\Omega)$$\Omega$satisfying the following property:
is an -viscosity solution bounded from above of$c_{0}(n, \lambda, \Lambda,p, q, \sigma, \tau, R_{0}, \eta)>0$

if$c\in L_{+}^{n}(\Omega),$$w\in C(\Omega)$$L^{p}$$F(x, Dw, D^{2}w)-c(x)w^{+}\leq 0$

in

(4.1)

$\Omega$such that

lim $supw(x)\leq 0$ ,(4.2)

$xarrow\partial\Omega$and$K_{0}:= \max\{\sup_{y\in\Omega,|y|>R_{0}}\Vert\{\cdot\}c(\cdot)\Vert_{L^{n}(A_{y}\cap\Omega)},\sup_{y\in\Omega,|y|\leq R_{0}}\Vert c\Vert_{L^{n}(B_{y}\cap\Omega)}\}\leq c_{0}$

,

(4.3)

then $w\leq 0$ in$\Omega$.

Proof. Note that by (2.4),

$w$

is an

$L^{n}$$\mathcal{P}^{-}(D^{2}w)-b(x)$-viscosity solution ofI $Dw|-c(x)w^{+}\leq 0$ .

We apply Theorem 3.3 with $f=cw^{+}$ to obtain that when $|y|\leq R_{4}$ ,$R_{0} \Vert cw^{+}\Vert_{L^{n}(B_{y}\cap\Omega)}\leq R_{0}\sup_{\Omega}w^{+}\Vert c\Vert_{L^{n}(B_{y}\cap\Omega)}\leq R_{0}K_{0}\sup_{\Omega}w^{+}$.

On the other hand, when $|y|>R_{0}$ , we have$R_{y} \Vert cw^{+}\Vert_{L^{n}(A_{y}\cap\Omega)}\leq\frac{R_{y}}{\sqrt{1+(\epsilon \mathscr{F})^{2}}}\sup_{\Omega}w^{+}\Vert\{\cdot\}c(\cdot)\Vert_{L^{n}(A_{y}\cap\Omega)}\leq\frac{K_{0}}{\epsilon}\sup_{\Omega}w^{+}$

.

(4.4)

148
Choosing$\epsilon_{1}=\frac{1}{4}\min\{\frac{1}{1+\eta}, (\frac{\sigma}{4})^{1/n}\}$for instance, we have$\sup_{\Omega}w\leq C_{3}\max\{R_{0},$ $\frac{1}{\epsilon_{1}}\}c_{0}\sup_{\Omega}w^{+}$

for some constant $C_{3}>0$ . Taking $c_{0}<1/(C_{3} \max\{R_{0},1/\epsilon_{1}\})$ , this end theproof.
$\square$

Theorem 4.2 (Phragm\’en-Lindel\"of theorem). Assume (2.3), (2.6) and (2.4)
satisfying (3.1). Let $\eta>0$ and be a weak-G domain. Therewithexists a positive constant $\alpha>0$ such that if $w\in C(\Omega)$ is an If-viscosity
solution of$b\in L_{+}^{q}(\Omega)$$\Omega$in$F(x, Dw, D^{2}w)\leq 0$(4.5)$\Omega$with (4.2) holds and$w^{+}(x)=O(|x|^{\alpha})$then $w\leq 0$ in$\Omega$a$|x|arrow\infty$

$s$

,

(4.6)

.

Proof of Theorem 4.2. Define a positive smooth function$\xi(x)=\{x\}^{\alpha}$,

where $\alpha>0$ will be fixed later. Setting $u=w/\xi$ , which is bounded ffom
above. A straightforward calculation shows that$\frac{|D\xi|}{\xi}(x)\leq\frac{\alpha}{\langle x\}}$,for some $C_{4}>0$ . Thus, we see that$\frac{|D^{2}\xi|}{\xi}(x)\leq\frac{C_{4}\alpha}{\{x\}^{2}}$

$u$

is an

$L^{n}$-viscosity solution of$\mathcal{P}^{-}(D^{2}u)-\gamma_{1}(x)|Du|-\alpha\gamma_{2}(x)u^{+}\leq 0$

in$\Omega$,

where$\gamma_{1}(x)=\frac{C_{5}\alpha}{\{x\rangle}+b(x)$,$\gamma_{2}(x)=\frac{C_{6}}{\{x\}}(\frac{1}{\langle x\rangle}+b(x))$$C_{6}>0$ . We easily see thatfor some positive constants
satisfies (3.1).
We next show that (4.3) holds for . Direct calculation implies$C_{5},$$\gamma_{1}$$\gamma_{2}$$\tilde{K}_{0}:=\max\{ \sup \Vert\langle\cdot\}\gamma_{2}(\cdot)\Vert_{L^{n}(A_{y}\cap\Omega)}$,$y\in\Omega,|y|>R_{4}$$\sup$$\Vert\gamma_{2}\Vert_{L^{n}(B_{y}\cap\Omega)}\}<+\infty$$y\in\Omega,|y|\leq R_{0}$(4.7)

is boounded. Thus,is small when $\alpha>0$ is small enough.
Therefore, using Lemma 4.1 with, we get $u\leq 0$ . Thisandimlies $w\leq 0$ .
$K_{0}=\alpha\tilde{K}_{0}$$b=\gamma_{1}$$c=\gamma_{2}$$\square$Acknowledgements. The author wishes to thank Professor‘s H. Ishii, S.
Koike and A. Vitolo for several comments of this article.

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KAZUSHIGE NAKAGAWA
MATHEMATICAL INSTITUTE, TOHOKU UNIVERSITY
6-3, AOBA, ARAMAKI, AOBA-KU, SENDAI 980-8578,JAPANE-mail address: $knakagawa\Phi math$ . tohoku. ac. jp