数理解析研究所講究録
第 1545 巻 2007 年 1-12

1

Maximum principle via the iterated comparison
function method(小池 茂昭)Shigeaki Koike(埼玉大学)
Saitama University

1

Introduction

In this note, we present several maximum principles for -viscosity soIutions of fully
nonlinear but uniformly $elliptic/parabolic$ partial differential equations (PDEs for short).
Our maximum principles are extentions of Aleksandrov-Bakelman-Pucci (ABP for short)
type for elliptic case, and of ABP-Krylov-Tso for parabolic case.
We will work in a bounded open set
for the elliptic case, and in $Q:=\Omega x(0,T$]
with a fixed $T>0$ for the parabolic case. We will denote by
the open ball with center.
at the origin and the radus $r>0$ .
We denote by
the set ofsymmetrIc matrices with the standard ordering. ;$L^{p}$$\Omega\subset R^{n}$$B_{r}$$S^{n}$$n\cross n$$X\leq Y$$\leq$$\Leftrightarrow$$\langle X\xi,\xi\rangle’\leq 0$

for$\forall\xi\in R^{n}$.

Throughout this paper, we at least suppose$p> \frac{n}{2}$for the elliptic case and, $p> \frac{n+2}{2}$ for the parabolic case.

We use the standard -norm in a domain
we denote byboth
andfollowing notation:$L^{p}$$\Vert\cdot\Vert_{p}$$\Vert\cdot\Vert_{L^{p}(\Omega)}$$L_{+}^{p}(U)=$$U\subset R^{m}$$\Vert\cdot\Vert_{L^{p}(Q)}$( $m=n$ or $n+1$ ); II .

$\Vert_{L^{p}(U)}$. However,if there is no confusion. We also use the{ $u\in L^{p}(U)|u\geq 0a.e$ . in }.
$U$

In what follows, given a function : $Uarrow R$ , when we discuss it in a larger set , weutilize the zero extention of by the same .
Reezing the uniform ellipticity constants, we denote by
the set of all$A\in S^{n}$ such
that.
Then, we define the Pucci operators: for $X\in S^{n}$ ,
$f$

$V$

$f$

$f$

$0<\lambda\leq\Lambda$$S_{\lambda,\Lambda}^{n}$$\lambda I\leq A\leq\Lambda I$$\mathcal{P}^{\pm}$$\mathcal{P}^{+}(X)=\max\{-trace(AX)|A\in S_{\lambda,\Lambda}^{n}\}$An easy observation is that for $X,$

$Y\in S^{\mathfrak{n}}$,.

$\mathcal{P}^{-}(X)=\min\{-trace(AX)|A\in S_{\lambda,\Lambda}^{n}\}$,$\mathcal{P}^{-}(X)+\mathcal{P}^{-}(Y)\leq \mathcal{P}^{-}(X+Y)\leq \mathcal{P}^{-}(X)+\mathcal{P}^{+}(Y)\leq \mathcal{P}^{+}(X+Y)\leq \mathcal{P}^{+}(X)+\mathcal{P}^{+}.(Y)$which has a roll of “linearity” of fully nonlinear operators$\mathcal{P}^{\pm}$

.

,

2

2

Elliptic caseWithout loss of generality, we may suppose that.
Let us consider the most general PDEs of second-order In the elliptic case:$\Omega\subset B_{1}$$F(x, u, Du, D^{2}u)=f(x)$where $F:\Omega\cross R\cross R^{n}\cross S^{n}arrow R$ and :is continuous in the last three variables.
$f$Definition. We call $u\in C(\Omega)$ an$L^{p}$$\Omegaarrow R$

in$\Omega$,(1)

are given measurable functions, and$F$-viscosity subsolution (resp., supersolution) of (1)ess $\lim_{yarrow}\inf_{x}\{F(y, u(y), D\phi(y), D^{2}\phi(y))-f(y)\}\leq 0$$(resp.$ ,ess $\lim_{yarrow}\sup_{x}\{F(y, u(y), D\phi(y), D^{2}\phi(y))-f(y)\}.\geq 0)$

wheneverandis a local maximum (resp., minimum) point of $u-\phi$ .
We then call $u\in C(\Omega)$ an -viscosity solution of (1) if it is an -viscosity subsolution
and an $IP$ -viscosity supersolution of (1).
In order to memorize the right inequality, we will often say that is an -viscosity
subsolution of$F(x,u, Du, D^{2}u)\leq f(x)$ etc.
$x\in\Omega$$\phi\in W_{1oc}^{2,p}(\Omega)$$L^{p}$$L^{p}$$L^{p}$$u$Definition. We also callof (1) if satisfies$u\in W_{1oc}^{2,p}(\Omega)$an$L^{p}$-strong subsolution (resp., supersolution)$u$$F(x, u(x),$ $Du(x),$ $D^{2}u(x))-f(x)\leq 0$

We then call$u\in W_{loc}^{2,p}(\Omega)$(resp.,$\geq 0$)a.e. in .
$\Omega$an $IP$ -strong solution of (1) if the equality holds in the above.

Remark. Notice that we do not assume that
. Thus, if is an -viscosIty
subsolution of (1), then it is aiso an -viscosity subsolution of (1) provided $q\geq p$ .
$f\in L^{p}(\Omega)$$u$$L^{p}$$L^{q}$

Now we suppose the uniform ellipticity for $F$ :$\mathcal{P}^{-}(X-Y)\leq F(x, r,p, X)-F(x, r,p, Y)\leq \mathcal{P}^{+}(X-Y)$

for$x\in\Omega,$$r\in R,$ $p\in R^{n}$, and $X,$ $Y\in S^{n}$ . Typical examples of $F$ are$F(x, r,p, X)= \max_{1\leq i\underline{<}M}\min_{1<\lrcorner\leq N}\{-trace(A(x;i,j)X)+(b(x;i,j),p\rangle+c(x;i,j)r\}$,where for $M,$ $N>1$ , functions
and$xarrow c(x;i,j)$ are measurable $(1 \leq i\leq M, 1\leq j\leq N)$ . Notice that the aboveis
non-convex and non-concave in general.
$x\in\Omegaarrow A(x;i,j)\in S_{\lambda,\Lambda}^{n},$$x\in\Omegaarrow b(X_{1}^{\cdot}i,j)\in R^{n}$$F$3Under the uniform ellipticity assumption, we notice that iftion of (1), then it is also an -viscosity subsolution ofis an $IP$ -viscosity subsolu-$u$$L^{p}$$\mathcal{P}^{-}(D^{2}u)+F(x, u, Du, O)\leq f(x)$.

Therefore, for the sake of simplicity, instead of (1), we shall study the maximum principleforin$\mathcal{P}^{-}(D^{2}u)-\mu(x)|Du|=f(x)$$\Omega$.

(2)

Proposition 1. There exist $C_{k}=C_{k}(n, \lambda, \Lambda)>0(k=1,2)$ such that if
andis an -strong subsolution of (2), then we have$u \in C(\prod)\cap W_{1oc}^{2,n}(\Omega)$$f,\mu\in L_{+}^{n}(\Omega)$,$L^{n}$$m_{\frac{a}{\Omega}}xu\leq\max u\partial\Omega+C_{1}\exp(C_{2}\Vert\mu||_{n})||f||_{n}$.

(3)Remark. In the above statement, we can replace, wherebyisthe upper contact set of in . See Gilbarg-Trudinger’s book for the definition of.
Rom Proposition 1, it is trivial to obtain the corresponding result for $IP$ -strong supersolutions of$\Vert f||_{n}$$u$$||f||_{L^{n}(\Gamma[u])}$$\Gamma[u]$$\Omega$$\Gamma[u]$$\mathcal{P}^{+}(D^{2}u)+\mu(x)|Du|\geq f(x)$by taking $v=-u$ , which is an$L^{p}$

in$\Omega$-viscosity subsolution ofin$P^{-}(D^{2}v)-\mu(x)|Dv|\leq-f(x)$$\Omega$.

Thus, we will give results only for subsolutions.
To utihize the “iterated comparison function method”, we often use the folowing existence result for extremal equations (see [3]).

Proposition 2. There exists $p_{0}=p_{0}(n, \Lambda/\lambda)\in[n/2,n)$ satIsfying the following: If $p>$
and satisfy the uniform exterior cone condition, then there are $C=C(n,p, \lambda,\Lambda)>0$
such that for, there is an $IP$ -strong solution
of

$po$

$\Omega$$f\in L^{p}(\Omega)$$v\in C(\overline{\Omega})\cap W_{1oc}^{2,p}(\Omega)$$\{\begin{array}{ll}\mathcal{P}^{+}(D^{2}v)=f(x) in \Omega,v=0 on \partial\Omega\end{array}$such that
in$-C||f^{-}||_{p}\leq v\leq C||f^{+}||_{p}$Moreover, for each open set$\Omega’\propto\Omega$$\Omega$.

, there is $C’=C’(n,p, \lambda,\Lambda, dist(\Omega’, \partial\Omega))>0$ such that$||v||_{W^{2.p}(\Omega’)}\leq C’||f||_{p}$.

In this section, A C $B$ means $Z\subset B$ .
To show Proposition 1 for -viscosity solutions, when is unbounded (i.e.
with $1\leq q<\infty$ in our case), it is not trivial even if we suppose. (When
we may apply a techinque as in our first paper [10].)$L^{P}$$\mu$$f\equiv 0$$\mu\in L^{q}(\Omega)$$\mu\in L^{\infty}(\Omega)$,

4
The next proposition is a restatement of Lemma 2.11 of [8] although our assumption
that
seems restrictive (cf. [8]).
$supp\mu(\subset\Omega$

Proposition 3. Let

satisfy the uniform exterior cone condition. For$\Omega$$q\geq p>n$

we suppose$f\in L^{p}(\Omega)$supersolution$u$(resp.,, and$L^{p}$$0\dot{r}$with-strong subsolution )$\mu\in L_{+}^{q}(\Omega)$$q>p=n$ ,$supp\mu \mathbb{C}\Omega$$v$,$(resp.,$$\{\mathcal{P}^{-}(D^{2}u)-\mu(x)|Du|\geq f(x)u=0on\partial\Omega in\Omega$(4). Then, there exist anof$L^{p}$-strong$\in C(\prod)\cap W_{1oc}^{2,p}(\Omega)$$\{\mathcal{P}^{+}(D^{2}v)+\mu(x)|Dv|\leq f(x)v=0on\partial\Omega in\Omega,)$such that(resp.,$||u\Vert_{\infty}$

where
and
we have$C_{1}$$C_{2}$$\Vert v\Vert_{\infty}$)$\leq C_{1}\exp(C_{2}\Vert\mu\Vert_{n})\Vert f|.|_{n}$,are the constants from Proposition 1. Moreover, for each open$||u\Vert_{W^{2.p}(\Omega’)}(re\bm{s}p.,$Now, we present an$L^{p}$$\Vert v\Vert_{W^{2,p}(\Omega’)})\leq C$( $n,p,$$\lambda,$$\Lambda,$$\Vert\mu\Vert_{q}$, dist$(\Omega’,$$\partial\Omega)$)$||f\Vert_{p}$$\Omega’\subset\Omega$,.

-viscosity version of Proposition 1.

Proposition 4. Assume (4). Then, there exist $C_{k}=C_{k}(n, \lambda, \Lambda)>0(k=1,2)$ suchthat if, and $u\in C(D)$ is an -viscosity subsolution of (2), then wehave$f\in L_{+}^{p}(\Omega),$$\mu\in L_{+}^{q}(\Omega)$$L^{p}$$m_{\frac{a}{\Omega}}xu\leq\max u\partial\Omega+C_{1}\exp(C_{2}\Vert\mu\Vert_{n})\Vert f\Vert_{n}$Proof. Fix. Recalling$v\inC(F_{2})\cap
W_{loc}^{2.p}(B_{2})$tionof$\epsilon>0$$\Omega\subset B_{1}$.

, from Proposition 2, we find an$L^{P}$-strong$subsolurightarrow$$\{\begin{array}{ll}\mathcal{P}^{+}(D^{2}v)+\mu(x)|Dv|\leq-f(x)-\epsilon in B_{2},v=0 on \partial B_{2}\end{array}$such thatin$0\leq-v\leq C_{1}\exp(C_{2}||\mu||_{n})(||f||_{n}+\epsilon)$

It is easy to check that $w:=u+v$ is an$L^{p}$

attains its maximum at
a contradictlon. Thus, we have$w$$x\in\Omega$.

-viscosity subsolution of$\mathcal{P}^{-}(D^{2}w)-\mu(x)|Dw|\leq-\epsilon$

Hence, if$B_{2}$

in, the defnition of$\max_{B}w=\max w\partial\Omega$which implies that$m_{\frac{a}{\Omega}}xu\leq\max u\partial\Omega+\max_{\Omega}(-v)$.

$\Omega$$L^{P}$.

-viscosity subsolutions yields

5This gives the result follows by letting.
Next, we consider the case of $p_{0}<p<n$ , which extends that in [8] and [9].
$\epsilonarrow 0$$\square$

Theorem 5. Assume $p_{0}<p<n<q$ , and $m=1$ . There exist an integer $N=N(n,p, q)$
and $C=C(n, \lambda, \Lambda,p, q)>0$ such that if
, and
is an
viscosity subsolution of (2), then we have$f\in L_{+}^{p}(\Omega),$$\mu\in L_{+}^{q}(\Omega)$$u\in C(\overline{\Omega})$$L^{p_{-}}$.

$\max_{\hslash}u\leq_{\theta}\max_{\Omega}u+C\{\exp(C\Vert\mu||_{n})||\mu||_{q}^{N}+\sum_{k=0}^{N-1}||\mu\Vert_{q}^{k}\}\Vert f||_{p}$Idea of proof.
ofDue to Proposition 2, we find an$L^{p}$-strong solution $v_{1}\in C(F_{R_{1}})\cap$$W_{l\alpha}^{2,p}(B_{Ra})$$\{\begin{array}{ll}\mathcal{P}^{+}(D^{2}v_{1})=-f(x) in B_{2},v_{1}=0 on \partial B_{2}\end{array}$such that $0\leq-v_{1}\leq C||f\Vert_{p}$ in$B_{2}$. By the Sobolev embedding, we have$\Vert Dv_{1}\Vert_{L(B_{8/2})}\leq C\Vert f\Vert_{p}$.

(5)Here and later, for $n>p>1$ ,$p^{*}= \frac{np}{n-p}>0$.

We will also use $C>0$ to denote various universal constants.
By setting $w_{1}=u+v_{1}$ in , it is easy to see thatis an $IP$-viscosity subsolution of$\Omega$$w_{1}$

in$\mathcal{P}^{-}(D^{2}w_{1})-\mu(x)|Dw_{1}|\leq\mu(x)|Dv_{1}(x)|=:f_{2}(x)$$\Omega$.

By (5) and the H\"older inequality yield,$\Vert f_{2}\Vert_{L^{q_{1}}(B_{8/2)}}\leq||\mu\Vert_{q}\Vert Dv_{1}||_{L^{p^{*}}(B_{3/},)}\leq C\Vert\mu\Vert_{q}\Vert f\Vert_{p}$where $q_{1}=npq/\{(n-p)q+pn\}$ . Note $q_{1}>p$ .
Let us suppose $q_{1}>n;p>nq/(2q-n)$ . In view of Proposition 4, we have$\max_{\Omega}w_{1}\leq\max_{\partial\Omega}w_{1}+C_{1}\exp(C_{2}\Vert\mu\Vert_{n})||f_{2}\Vert_{q_{1}}$,which implies

max $w1+ \max_{W}(-v1)$
max$\leq$$\max_{\text{蒼}}u$$u+C\Vert f||_{p}+C_{1}C\exp(C_{2}\Vert\mu\Vert_{n})\Vert\mu\Vert_{q}\Vert f||_{p}$$\leq$

If$q_{1}\leq n$, then we use the$L^{q_{1}}$-strong solution.

$v_{2}\in C(\overline{B}_{3/2})\cap W_{loe}^{2,q_{1}}(B_{3/2})$$\{\begin{array}{ll}\mathcal{P}^{+}(D^{2}v_{2})=-f_{2}(x) in B_{3/2},v_{2}=0 on \partial B_{3/2}\end{array}$of

6
to derive the eqaution satisfied by$w_{2}$$:=w_{1}+v_{2}$;$\mathcal{P}^{-}(D^{2}w_{2})-\mu(x)|Dw_{2}|\leq f_{3}(x)$,where $f_{3}\in L^{q_{2}}(B_{5/4})$ with $q_{2}>q_{1}$ . We keep on this procedure to arrive the situation$qN>n$ . Thus, we may apply Proposition 4 to conclude our result. ロ
Next, for $m>1$ , we consider the PDE$\mathcal{P}^{-}(D^{2}u)-\mu(x)|Du|^{m}=f(x)$in$\Omega$.

(6).

In order to show the maximum principle for (6), we need some restrictions as in [10] because
there is a counter-example (see [11]).

$0$

Theorem 6. Assume $n<p\leq q$ , and $m>1$ . Then, there exist
and $C=C(n, \lambda, \Lambda, m,p, q)>0$ such that if$\delta=\delta(n, \lambda, \Lambda, m,p, q)>$$f\in L_{+}^{p}(\Omega),$ $\mu\in L_{+}^{q}(\Omega)$,,$\Vert f\Vert_{p}^{m-1}\Vert\mu\Vert_{q}\leq\delta$

and $u \in C(\prod)$ is an$L^{p}$-viscosity subsolution of (6), then we have$; axu\leq\max_{\partial\Omega}u+C(\Vert f\Vert_{p}+\Vert f||_{p}^{m}\Vert\mu\Vert_{q})$.

The idea of proof of Theorem 5 is a combination of those in [10] and Theorem 4.
Following the argument used in the proof of Theorem 5, we can now extend Theorem 6
to the case when $p\in(p_{0}, n$].

Assume $p_{0}<p\leq n<q$ , and $m>1$ . Denote $a_{0}=0$ and $a_{k}=$. Then, there exist an integer $N=N(n, m,p, q)\geq 1,$$\delta(n, \lambda, \Lambda,m,p, q)>0$ and $C=C(n, \lambda, \Lambda, m,p,q)>0$,
such that if
Theorem 7.

$1+m+\cdots+m^{k-1}$ for $k\geq 1$$\delta=$$f\in L_{+}^{p}(\Omega),$$p> \frac{nq(m-1)}{mq-n}$and$u\in C(\overline{\Omega})$is an$L^{p}$(7),$||f\Vert_{p}^{m^{N}(m-1)}\Vert\mu||_{q^{N}}^{a(m-1)+1}\leq\delta$$\mu\in L_{+}^{q}(\Omega)$,-viscosity subsolution of (6), then we have$\max_{\hslash}u\leq\max u\partial\Omega+C\sum_{k=0}^{N+1}||\mu||_{q^{k}}^{a}||f||_{p}^{m^{k}}$.

Remark. When $1<m\leq 2-n/q,$ (7) is automaticaUy satisfied.

7

DIAGRAM 1$\mathcal{P}^{-}(D^{2}u)-\mu(x)|Du|^{m}\leq f(x)\Rightarrow m_{\frac{a}{\Omega}}xu-$

max $u\leq C\cross RHS$

We notice that when $m\geq 1,$ $p_{0}<p$ and $q=\infty$ , we obtained the maximum principle$with/without$ restriction in [10].

3Parabolic equations

In this section, we consider parabolic PDEs in $:=\Omega x(0, T$], whereagain, and$0<T\leq 1$ for simplicity. For $1\leq p\leq\infty$ , the parabolic
Sobolev space $W^{2,1,p}(Q)$ is definedby$Q$$W^{2,1,p}(Q)=\{u\in L^{p}(Q)$:$u_{t},$$\Omega\subset B_{1}$$Du,$ $D^{2}u\in L^{p}(Q)\}$.

In this section, we denote the parabolic boundary by.
$W_{1oc}^{2,1,p}(Q)=$We will ako use the space{ $u:Qarrow R$ : $u\in W^{2,1,p}(Q’)$ for all $Q’\propto Q$ },where in this section, $Q’\propto Q$ means dist $(Q’, \partial_{p}Q)>0$ .
The parabolic distance between $(x, t)$ and $(y, s)$ is defined by$\partial_{p}Q:=\Omega\cross\{0\}\cup\partial\Omega\cross[0, T]$

dist $((x,t),$ $(y, s))=(|x-y|^{2}+|t-s|)^{\int}$ .

We recall the definition of$L^{p}$-viscosity solution of general fully nonlinear parabolic PDEs.

Definition. We call $u\in C(Q)$ an$L^{p}$-viscosity subsolution (resp., supersolution) of$u_{t}+F(x, t,u, Du, D^{2}u)=f(x, t)$

in

$Q$

,

if

ess $(y_{l}) \in Qarrow\lim\inf_{(x,t)}\{\phi_{t}(y, s)+F(y, s, u(y, s), D\phi(y, s), D^{2}\phi(y, s))-f(y, s)\}\leq 0$

(8)

8

$(resp.$ ,ess $\lim_{(y,s)\in Q}\sup_{arrow(xt)},\{\phi_{t}(y, s)+F(y, s, u(y, s), D\phi(y, s), D^{2}\phi(y, s))-f(y, s)\}\geq 0)$

wheneverand $(x, t)\in\Omega\cross(O, T)$ is a local maximum (resp., minimum) pointof $u-\phi$ .
We call $u\in C(Q)$ an -viscosity solution of (8) if it is an If-viscosity sub- and supersolution of (8).
As in the elliptic case, we call $u\in W_{1oc}^{2,1,p}(Q)$ an -strong solution of (8) if satisfies$\phi\in W_{1oc}^{2,1,p}(Q)$$L^{p}$$L^{p}$$u$$u_{t}(x, t)+F(x, t, u(x, t), Du(x, t), D^{2}u(x, t))=f(x, t)$$a.e$. in .
$Q$

As in section 2, we will establish maximum principles for the following simpler parabolicPDE$u_{t}+\mathcal{P}^{-}(D^{2}u)-\mu(x,t)|Du|^{m}=f(x,t)$

in

$Q$

,

(9)

where $m\geq 1$ .
The following version of maximum principle can be derived $hom[13]$ .
Let $m=1,$ $f\in L_{+}^{n+1}(Q)$ and $\mu\in L_{+}^{n+1}(Q)$ . Then, there exist$C_{k}=C_{k}(n, \lambda, \Lambda)>0(k=1,2)$ such that if $u\in C(\Phi)\cap W_{1oc}^{2,1,n+1}(Q)$ is an-strong
subsolution of (9), then we have
Proposition 8.

$L^{n+1}$$m_{\frac{a}{Q}}x\leq\max u+C_{1}\exp(C_{2}||\mu||_{n+1})\Vert f||_{n+1}\partial_{p}Q$We may also refine the above estimate using the upper contact set (see [13] for thedetails).
In this section, we fix $p_{1}=p_{1}(- n, \Lambda/\lambda)\in((n+2)/2, n+1)$ to be the “parabolic” constanttfat gives the range of exponents for which the following generalized maximum principle
holds (see [7]): for $p>p_{1}$ , there is a constant $C=C(n, \lambda, \Lambda,p)$ such that if $f\in L^{p}(Q)$ and$u\in o(\eta)\cap W_{1oc}^{2,1,p}(Q)$satisfies $u_{t}+\mathcal{P}^{-}(D^{2}u)\leq f(x,t)a.e$. in , then we have$Q$

$m_{\frac{a}{Q}}xu\leq\max u+C||f^{+}||_{p}\partial_{p}Q$

We recall results on solvability of extremal equations and on estimates of $Du$ .
Proposition 9.
Let $p>p_{1}$ . There exists $C=C(n, \lambda, \Lambda,p)>0$ such that for$f\in L^{p}(Q)$ , there exists an-strong solution $u\in c(\eta)\cap W_{1oc}^{2,1,p}(Q)$ of$L^{p}$$\{\begin{array}{l}w+\mathcal{P}^{+}(D^{2}u)=f(x,t)Qu=0\partial_{p}Q\end{array}$such that$-C\Vert f^{-}\Vert_{p}\leq u\leq C\Vert f^{+}||_{p}$

in

$Q$

.

9

Moreover, for each set $Q’(\subset Q$ , there exists $C’=C^{l}(n, \lambda, \Lambda,p, dist(Q’, \partial_{p}Q))>0$ such that$\Vert u\Vert_{W^{2,1,p}(Q’)}\leq C’\Vert f\Vert_{p}$.

To study (9), as in the elliptic case, it is important to know thethe embeddings:$L^{\infty}$-estimate of $Du$ from

Proposition 10. (cf. Theorem 7.3 in [5]) Let $p>p_{1}$ . For each set $Q’\mathbb{C}Q$ , there
exists $C=C(n, \lambda, \Lambda,p, dist(Q’, \partial_{p}Q))>0$ such that if $u\in C(O)\cap W_{1oc}^{2,1,p}(Q)$ is an $IP$-strong
solution of (9), then we have
if $p>n+2$ ,$\Vert Du\Vert_{L\infty(Q’)}\leq C(\Vert u||_{L\infty(\partial_{p}Q)}+\Vert f||_{p})$$\Vert Du\Vert_{L^{p^{*}}(Q’)}\leq C(||u||_{L\infty(\partial,Q)}+\Vert f||_{p})$Here and later,$p^{*}$

if $p\in(p_{1}, n+2)$ .

above is defined by$p^{\star}= \frac{p(n+2)}{n+2-p}$

for $p<n+2$ .

We present a parabolic version of Proposition 3:
Proposition 11. Let$\Omega$satisfy the uniform exterior cone condition.

$q\geq p>n+2$$f\in L_{+}^{p}(Q)$$IP$, and let$\psi\in C(\partial_{p}Q)$-strong subsolutions$u$(resp.,. Let$L^{p}$

or $q>p=n+2$ ,(11)satisfy $supp\mu CQ$ . Then, there exist-strong supersolution ) $\in c(\eta)\cap W_{1oc}^{2,p}(Q)$ of$\mu\in L_{+}^{q}(Q)$$v$$\{\begin{array}{ll}u_{t}+\mathcal{P}^{-}(D^{2}u)-\mu(x,t)|Du|\geq f(x, t) in Q,u=0 on \partial_{p}Q,\end{array}$$(resp.,$$\{v_{t}+\mathcal{P}^{+}(D^{2}v)+\mu(x,t)|Dv|\leq f(x,t)v=0on\partial_{p}Q!nQ,)$

such that$\Vert u||_{L}\infty(Q)(resp.,$where$C_{1}$

and$||u\Vert_{W^{2.1,p}(Q’)}$$C_{2}$$||v||_{L}\infty(Q))\leq C_{1}\exp(C_{2}||\mu\Vert_{\mathfrak{n}+1})\Vert f||_{n+1}$,

are constants from Proposition 8. For each $Q’\propto Q$ , we have

(resp.,$||v||_{W^{2.1,p}(Q’)})\leq C$( $n,p,$$\lambda,\Lambda,$$\Vert\mu||_{L^{q}(Q)}$, dist $(Q’,$$\partial_{p}Q)$)$||f||_{L^{p}(Q)}$

.

(12)

By following the proof of Proposition 4, Proposition 10 allows us to obtain the following
maximum principle.

Proposition 12. Assume (11) and $m=1$ . Then, there exist $C_{k}=C_{k}(n, \lambda, \Lambda)>0$$(k=1,2)$ such that if $f\in L_{+}^{p}(Q),$, and $u\in c(\Phi)$ is an -viscosity subsolutionof (9), then we have$\mu\in L_{+}^{q}(Q)$$m_{\frac{a}{Q}}x\leq$.

$maxu+C_{1}\exp(C_{2}||\mu||_{n+1})||f||_{\mathfrak{n}+1}*Q$$L^{\rho}$

10
We first show that if, then even for $m>1$ , we do not need to assume thatoris small. Recall that such a restriction is necessary in the elliptic case as
discussed in [10] and [11].
$\mu\in L_{+}^{\infty}(Q)$$||\mu\Vert_{\infty}$$||f||_{p}$Assume $n+2<p\leq q$ and $m\geq 1$ .
$C(n, \lambda, \Lambda,p, m)>0$ such that if $f\in L_{+}^{p}(Q),$, andsubsolution of (9), then we have
Theorem 13.

)$\mu\in L_{+}^{\infty}(Q)$

Then, there exixts $C=$
is an -viscosity$L^{p}$$u\in C(\overline{Q})$$\max_{\partial}u\leq\max u+C(\Vert f\Vert_{p}+\Vert\mu\Vert_{\infty}||f\Vert_{p}^{m})\partial_{p}Q$

We next extend Theorem 13 to the case $p\in(p_{1}, n+2$].

Theorem 14. Assume $p_{1}<p\leq n+2<q$ , and $m\geq 1$ . Then, there exist an integer$N=N(n,p, m)\geq 1$ and $C=C(n, \lambda, \Lambda,p, m)>0$ such that if $f\in L_{+}^{p}(Q),$,$\mu\in L_{+}^{\infty}(Q)$(13)$p> \frac{(m-1)(n+2)}{m}$and $u\in C(e)$ is an $IP$ -viscosity subsolution of (9), then we have$m_{\frac{a}{Q}}xu\leq\max u*Q+C(||f||_{p}^{m}\sum_{k=0}^{N}||\mu||_{p}^{k}+||\mu||_{\infty}^{mN+1}||f||_{p}^{m^{2}})$.

Remark. We remark that when $m\in[1,2]$ , since $p_{1}\geq(n+2)/2\geq(m-1)(n+2)/m$ ,the restriction (13) is not necessary.
Next, we discuss the case when $m=1$ in (9) but $\mu\in L^{q}(Q)$ with $q>n+2$ .

Assume $p_{1}<p\leq n+2<q$ , and $m=1$ . Then, there exist an integer$N=N(n,p, q)\geq 1$ and $C=C(n, \lambda, \Lambda,p, q)>0$ such that if $f\in L_{+}^{p}(Q),$, andis an If-viscosity subsolution of (9), then we have
Theorem 15.

$\mu\in L_{+}^{q}(Q)$$u\in C(\overline{Q})$.

$\max_{\partial}u\leq\max_{\partial_{p}Q}u+C’\{\exp(C||\mu||_{n+1})||\mu||_{q}^{N}+\sum_{k\approx 0}^{N-1}||\mu||_{q}^{k}\}\Vert f||_{p}$Finally, we give sufficient conditions under which the maximum principle for (9) with$m>1$ holds true. The first result corresponds to Theorem 6 for elliptic PDEs.

Theorem 16. Assume $n+2<p\leq q$ , and $m>1$ . Then, there exist
and $C=C(n, \lambda, \Lambda, m,p, q)>0$ such that if $f\in L_{+}^{p}(Q),$,$\delta=\delta(n, \lambda, \Lambda, m,p, q)>$$0$$\mu\in L_{+}^{q}(Q)$,$||f\Vert_{p}^{m-1}\Vert\mu\Vert_{q}<\delta$

and $u\in c(p)$ is an$L^{p}$-viscosity subsolution of (9), then we have$m_{\frac{a}{Q}}xu\leq\max u+C(\Vert f||_{p}+\Vert\mu\Vert_{q}\Vert f||_{p}^{m})\partial_{p}Q$11
Our last result extends Theorem 16 to the case of $p_{1}<p\leq n+2$ .
Theorem 17. Assume $p_{1}<p\leq n+2<q$ . Denote $a_{0}=0$ and $a_{k}=1+m+\cdots+m^{k-1}$
for $k\geq 1$ . Then, there exist an integer $N=N(n, m,p, q)\geq 1,$ $\delta=\delta(n, \lambda, \Lambda, m,p, q)>0$
and $C=C(n, \lambda, \Lambda, m,p, q)>0$ such that if $f\in L_{+}^{p}(Q),$,$\mu\in L_{+}^{q}(Q)$$p> \frac{(m-1)q(n+2)}{mq-n-2}$

,

(14)

and $u\in C(O)$ is an $IP$-viscosity subsolution of (9),$||f\Vert_{p}^{m^{N}(m-1)}\Vert\mu\Vert_{q^{N}}^{a(m-1)+1}\leq\delta$,then we have$m_{\frac{a}{Q}}xu\leq\max u\partial_{p}Q+C\{\sum_{k=0}^{N+1}||\mu||_{q^{k}}^{a}||f||_{p}^{m^{k}}\}$.

Remark. If $1<m<2-(n+2)/q$ , the restriction (14) is not necessary.

DIAGRAM 2 $u_{t}+ \mathcal{P}^{-}(D^{2}u)-\mu(x, t)|Du|^{m}\leq f(x, t)\Rightarrow\max_{\partial}u-\max u\partial_{p}Q\leq C\cross$ RHS

参考文献[1] Caffarelli, L. A. and X. Cabr\’e, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, 1995.

12[2] Caffarelli, L. A., Interior a priori estimates for solutions of fully non-linear equations,Ann. Math. 130 (1989), 189-213.

[3] Caffarelli, L. A., M. G. Crandall, M. Kocan, and A. $6wi9ch$ , On viscosity solutions offully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49(1996), 365-397.
[4] Crandall, M. G., K. Fok, M. Kocan, and A. Swigch, Remarks on nonlinear uniformly
parabolic equations. Indiana Univ. Math. J. 47 (1998), no. 4, 1293-1326.

[5] Crandall, M. G., M. Kocan, and A.. $6wiech,$ -Theory for fully nonlinear uniformly
parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053.
$L^{p}$[6] Crandall, M. G. and A. Swi\cach, A note on generalized maximum principles for elliptic
and parabolic PDE, Evolution equations, 121-127, Lecture Notes in Pure and Appl.
Math., 234, Dekker, New York, 2003.

$\cdot$[7] Escauriaza, L.,a priori estimates for solutions to fully non-linear equations,Indiana Univ. Math. J. 42 (1993), 413-423.
$W^{2,n}$[8] Fok, P., Some maximum principles and continuity estimates for fully nonlinear elliptic
equations of second order, Ph.D. Thesis, UCSB, 1996.
[9] Fok, K., A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations 23(1998), no. $5arrow 6,967-983$ .
[10] Koike, S. and A. $6_{wiech}$ , Maximum principle and existence of -viscosity solutions
for fuUy nonlinear uniformly elliptic equations with measurable and quadratic terms,
Nonlinear Differential Equations Appl., 11 (2004), 491-509.
$L^{p}$[11] Koike, S. and A. Swigch, Maximum principle for fully nonlinear equations via theiterated comparison function method, submitted.
[12] Krylov, N. V., Sequences of convex functions and estimates of the maximum of thesolution of a parabolic equation, Siberian Math. J., 17 (1976), 226-236.

[13] Tso, K., On an Aleksandrov-Bakelman type maximum principle for second-orderparabolic equations, Comm.
Differential Equations, 10 (1985), no. 5, 543$Pa\hslash ial$

553.