数理解析研究所講究録 1242 巻 2002 年 16-29

16Interior H\"older continuity for viscosity solutions of fully
nonlinear second-0rder uniformly elliptic PDEs withmeasurable ingredients
Shigeaki Koike (Saitama University)
小池

茂昭

埼玉大学)

Contents
1.
1.1
1.2
1.3
2.
2.1
2.2
2.3
2.4

Introduction
Hypotheses
Known results
Two strategies to show Harnack inequality
Main results
Preliminaries
Local maximum principle
Weak Harnack inequality
Concluding remarks
References

1Introduction
In this note, we obtain the Harnack inequality for “weak” solutions of the following fully
nonlinear, second-0rder, uniformly ellptic partial differential equations (PDEs for short):
F (x,Du,
where,$\Omega\subset \mathrm{R}^{n}$$D^{2}u$) $=f$

in 0,(1)

cm for simplicity, andis abounded domain with smooth boundaryand :are given functions. Here,$\Omega \mathrm{x}\mathrm{R}^{||}\mathrm{x}S^{n}arrow \mathrm{R}$$f$$\Omegaarrow \mathrm{R}$symmetric $n\cross n$ real matrices with the standard ordering.

$S^{n}$:
denotes the set of all$F$

It is well-known that the Harnack inequality implies the Holder continuity ofsolutions. We note that this yields an equi-continuity of solutions since the Holder exp0nent and the Holder semi-norm depend only on the space-dimension, the uniform ellipticity
constants and given data in (1).
This research is jointly done with N. S. Trudinger.

17
1.1

Hypotheses

In our mind, we consider the case when the coefficients of the second derivatives are merely. Moreover, we allow $F$ to havemeasurable, and inhomogenious term belongs to only
the quadratic growth in the first derivatives.
However, $F$ is supposed to be uniformly elliptic in the second derivatives.
Thus, our hypotheses are as follows:$L^{n}(\Omega)$

Hypotheses

$\{$

(A1)
(A2)
(A3)

$xarrow F(x,p,X)$ ;measurable$(p\in \mathrm{R}^{n},X\in S^{1}’)$$|F(x,p, O)|\leq\gamma|p|^{2}$$(x\in\Omega,p\in \mathrm{R}^{n})$$P^{-}(X-\mathrm{Y})\leq F(x,p, X)-F(x,p, \mathrm{Y})\leq P^{+}(X-\mathrm{Y})$$(x\in\Omega,p\in \mathrm{R}^{n},X, \mathrm{Y}\in S^{n})$

(A4)

,
,

$f\in L^{n}(\Omega)$,,

where, in (A2), $\gamma>0$ is aconstant, and in (A3),
operators defined by$p\pm:S^{n}arrow \mathrm{R}$are the s0-called Pucci$P^{+}(X)= \max\{-\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(AX)|\lambda I\leq A\leq\Lambda I\}$$p-(X)= \min\{-\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(AX)|)I\leq A\leq\Lambda I\}$,.

are fixed.
In what follows, the above constants for uniform ellipticity
Under these hypotheses, we note that if is asubsolution (resp., supersolution) of (1),then it is asubsolution (resp., supersolution) of$0<\lambda\leq \mathrm{A}$$u$$P^{-}(D^{2}u)-\gamma|Du|^{2}\leq f$(resp.,$P^{+}(D^{2}u)+\gamma|Du|^{2}\geq f)$.

We will give the definition of sub- and supersolutions of (1) later.
true.
It is immediate to see that the following properties on$p\pm \mathrm{h}\mathrm{o}1\mathrm{d}$Proposition(1)(2)$\mathrm{V}-(\mathrm{X})\leq P^{+}(X)$,$P^{+}(X)=-\mathrm{p}-(-\mathrm{X})$,$P^{\pm}(\alpha X)=\alpha P^{\pm}(X)(\alpha\geq 0)$$P^{-}(X)+P^{-}(\mathrm{Y})\leq P^{-}(X+\mathrm{Y})\leq P^{+}(X)+P^{-}(\mathrm{Y})\leq P^{+}(X+\mathrm{Y})\leq P^{+}(X)+P^{+}(\mathrm{Y})$Remark] In view of (1) and (2) in the above, it is easy to see that $p+$ is convex, and $p-$

is concave.

We shall give atypical example for which (A1)-(A3)are satisfied

18
Bxmple$\rangle\rangle$

-

Here,

$A(\cdot)=(a_{\dot{|}j}(\cdot))$,$b(\cdot)$

and$f(\cdot)$

satisfy the following:$\lambda|\xi|^{2}\leq(\mathrm{A}(\mathrm{x})\mathrm{t}, \leq\Lambda|\xi|^{2}(\xi\in \mathrm{R}^{n})$where$\langle\cdot, \cdot\rangle$(2)$\sum_{:\dot{s}=1}^{n}a_{ij}(x)\frac{\partial^{2}u}{\partial x_{1}\partial x_{j}}\cdot+b(x)|Du|^{2}=f(x)$,$\sup_{x\in\Omega}|b(x)|\leq\gamma$

denotes the standard inner product in$\mathrm{R}^{n}$,$f\in L^{n}(\Omega)$,.

This kind of PDEs arises in the risk-sensitive stochastic control and certain PDEs derived
from large deviation problems.

1.2

Known results

Let us mention known-results in case when the linear growth condition is supposed in placeof (A2);$|F(x,p, O)|\leq\gamma|p|$$(x, \in\Omega,p\in \mathrm{R}^{n})$When $F$ is merely measurable in :$x$

Krylov-Safonov [21] (1979) first obtained the Holder continuity of solutions by aprobabilistic approach. Trudinger [25] (1980) showed the same result as in [21] only by tools
from PDEs. We note that in these results, solutions means “strong” solutions; they belongtoand satisfy the PDEs alomost everywhere sense.
Recently, Caffarelli [3] (1989) showed the Holder continuity of the “standard” viscosity. The reason why
solutions when is continuous but the estimate depends only on
is supposed to be continuous there is that Alexandroff-Bakelman-Pucci (ABP for short)maximum principle holds for the standard viscosity solutions only when $f\in C(\Omega)$ . However, utilizing an approximation technique, CaffareUi-CrandaU-Kocan-Swiyh [4] (1996)
for slightly restricted viscosity soluproved the ABP maximum principle whentions.
In this article, we adapt the notion in [4], $IP$-viscosity solutions, but, under
the assumption $f\in C(\Omega)$ , it is easy to check that our results below are still valid
for the standard viscosity solutions.
For higher regularity of solutions, Caffarelli [3] obtained that soultions belong towhen “the oscillation of coefficients for the second derivatives are small in -sense. However, in general, we cannot expect that solutions are in. Because, if we could get
the higher regularity, then the solution would be the unique strong solution, which contradicts the fact that there exists acounter-example for uniqueness of viscosity solutions$W_{loc}^{2,n}(\Omega)$$f$$||f||_{L^{n}(\Omega)}$$f$$f\in L^{n}(\Omega)$$W_{l\mathrm{o}\mathrm{e}}^{2,n}(\Omega)$$L^{n}$$W_{loe}^{2,n}(\Omega)$

19
by Nadirashvili [22](1997). We also refer to Safonov [23](1999), which gives an alternative
proof of [22] by aPDE approach.

When $F$ is continuous in :$x$Here, we only mention $C^{1,a}(\alpha\in(0,1))$ estimates for viscosity solutions by Trudinger[26] and [27].

1.3

Two ways to derive Harnack inequality, there exists
We recall the meaning that the Harnack inequality holds; For any 0’such that for any nonnegative solutions of (1), it follows
aconstantthat$\mathbb{C}$$\Omega$$C=C(\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\Omega’,\partial\Omega)>0$,$\leq C(_{\mathrm{f}\mathrm{f}}\mathrm{n}u+\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\Omega’)||f||_{L^{n}(\Omega)})$$\max_{\overline{\Omega}}$

Bemark] By the standard scaling argument and translation, we only have to show theabove inequality when$\Omega’$is aunit cube or aball.

We shall use the following symbols:$B_{r}:=\{y\in \mathrm{R}^{n}||y|\leq r\}$,

$B_{r}(x):=B_{\mathrm{r}}+x$,$Q_{r}:=\{y\in \mathrm{R}^{n}||y_{k}|\leq r/2\}$,$Q_{f}(x):=Q_{r}+x$

Bemark 1 We notice the following inclusions hold.
$Q_{1}\subset B_{\sqrt{n}/2}\subset Q_{\sqrt{n}}$.

&difference of proofs bewteen Trudinger’s and Caffarelli’s))
Let us formally explain the difference of proofs between Trudinger’s and Caffarelli’s.
Trudinger’s proof: We first derive the weak Harnack inequality for nonnegative supersolutions of (1). That is to find $>0$ (possibly smaller than 1) and $C>0$ such that$\kappa$$||u||_{L^{\kappa}(Q_{1})} \leq C(\min_{Q_{1}}u+||f||_{L^{n}(Q_{R})})$

for some $R>1$ which only depends on .
We remark that we obtain this estimate on cubes in place of balls since we essentially
use Calder\’on-Zygmund’s cube-decomposition lemma.
Next, we show the local maximum principle for nonnegative subsolutions; That is tofind $C>0$ such that with the above $>0$ in the weak Harnack inequality for some $R>1$ ,$n$$\kappa$$\max u\leq CB_{1}(||u||_{L^{\kappa}(B_{R})}+||f||_{L^{n}(B_{R})})$

20
Combining these, it is easy to show the Harnack inequality.
Caffarelli’s proof: We use the (essentially) same argument as that of Trudinger to get the
we Harnack inequality for nonnegative supersolutions.
Next, for nonnegative solutions, we get acontradiction if we suppose that the Harnack
inequality fails. To this end, we adapt ablow-up arugument. We note that we needproperties of subsolutions and supersolutions.
$\mathrm{k}$2Main results
Our aim is to show that any solutions of (1), for which assumptions (A1) $-(A3)$ arefulfilled, have the same equi-Holder continuity. However, without further hypothesis, wecannot expect to prove such aresult.
Let us present an example to show that we need further hypothesis.
Example)) Let $n=1$ and $\Omega=(0,1)$ . Set $u(x)=Ax(A\geq 0)$ . Notice that $-\triangle u=0$ in$(0, 1)$. By setting$v(x)=e^{\mathrm{u}(x)}$, it follows thatin $(0, 1)$$-\triangle v+e^{-Ax}|Dv|^{2}=0$

Since $u\geq 0$ in $(0, 1)$ , the “coefficient” in front
is bounded for any $A\geq 0$ . However,$\in(0,1/2)$, it is impposible to find $C=C(\epsilon)>0$ independent of $A>0$
for any fixed small
such that$\mathrm{o}\mathrm{f}|Dv|^{2}$$\epsilon$.

$x \in[\epsilon,1-\epsilon]x\in[\epsilon,1-\epsilon]\max v(x)\leq C\mathrm{m}\dot{\mathrm{m}}v(x)$We notice that is not “equ\"i-H\"older continuous when.
As will be seen, since our estimate depends only on , , and 7in the hytotheses, thisexample explains why we need the further hypothesis on-bound for solutions.
$Aarrow\infty$$v$$\lambda$$\Lambda$$n$$L^{\infty}$

Now, we shall recall the definitions of viscosity solutions.
Throughout this article, we suppose that(A5)

$p> \frac{n}{2}$

Under (A5), it is well-known that any function inalmost all.

$W_{lo\mathrm{c}}^{2p}(\Omega)$

has second-0rder derivatives$x\in\Omega$

Definition(1) We call $u\in C(\Omega)$ an -viscosity subsolution (subsolution for short) of (1) if for any, it follows that$L^{\mathrm{p}}$$\phi$$\in W_{lo\mathrm{c}}^{2,p}(\Omega)$$\lim_{\epsilonarrow 0}\mathrm{e}\mathrm{s}\mathrm{s}\inf_{y\in x)}\{F(y, D\phi(y),D^{2}\phi(y))-f(y)\}\leq 0$

21provided u
attains its maximum at xE0.
an -viscosity supersolution (supersolution for short) of (1) if for(2) We call ue, it follows thatany p6$-\langle/\rangle$$\mathrm{f}$$\mathrm{C}?(\mathrm{O})$$W_{E}\ovalbox{\tt\small REJECT}\cdot(Cl)$$\lim_{\epsilonarrow 0}\mathrm{e}\mathrm{s}\mathrm{s}\sup_{y\in B_{e}(x)}\{F(y, D\phi(y), D^{2}\phi(y))-f(y)\}\geq 0$.
provided $u-\phi$ attains its minimum at(3) We call $u\in C(\Omega)$ an -viscosity solution (solution for short) of (1) if it is anviscosity sub- and supersolution of (1).
$x\in\Omega$$L^{p}$$L^{p_{-}}$-viscosity(sub-, super-) solution if the above properties hold
. Since //-viscosity solutions are more restrictive thanbyby replacing$C$-viscosity solutions, //-viscosity solutions are, indeed, $C$ -viscosity solutions.
We remark that the opposite inclusion is true when $F$ and are continuous. See [4] for
Remark] We call

$u$

a

$C$

$W_{loc}^{2,p}(\Omega)$$C^{2}(\Omega)$$f$this fact.
We recall the notion of strong solutions here:

Definition
We call $u\in C(\Omega)$ astrong subsolution (resp., supersolution) of (1) if Du(x) and $D^{2}u(x)$andexist for almost all$x\in\Omega$$F$( , Du(x), $D^{2}u(x)$ ) $-f()\leq 0$$x$(resp.,$\geq 0$)$a.e$

. in .
$\Omega$

We also call $u\in C(\Omega)$ astrong solution of (1) if it is astrong sub- and supersolution of(1).
In what follows, we mainly discuss about$L^{n}$-viscosity solutions.

Our main result is as follows:
Theorem, if asolution satisfies
For any $N>0$ and asubdomainthat $|u|\leq N$ in 0, then there is $C>0$ such that$\mathrm{f}l’\mathbb{C}\mathrm{f}l$$\max uB_{r}(x)\leq C$$\min_{B_{r}(x)}u+r||f||_{L^{n}(\Omega)}$(for$x\in\Omega’$

Remark] This result does not affect the counter-example$u$and small $r>0$ )22We shall give asufficient condition for (2) in the above example undercondition on
for which the L”-estimate is apriori obtained.

an

Bxample$\mathrm{z}\mathrm{e}\mathrm{r}\mathrm{o}\ovalbox{\tt\small REJECT}$

irichlet$\rangle\rangle$$0\leq b(x)\leq\gamma$

and

in Q.

$f\geq 0$

In fact, in this case, since 0is aclassical subsolution of (2), in view of the comparison
principle between astrong subsolution and an -supersolution in [18], we obtain that the-solution of (2) is nonnegative.
To obtain the upper bound, we find astrong supersolutionof$L^{n}$$L^{n}$$u$$w\in C(\overline{\Omega})\cap W_{lo\mathrm{c}}^{2,n}(\Omega)$$\{$$p-(D^{2}w)\geq f^{+}$

$a.e$

$w=0$

on

$0\leq w\leq C||f||_{L^{n}(\Omega)}$in. in 0,$\partial\Omega$$\Omega$.

,

See the existence of strong solutions below for the proof of the existence of . Since
astrong supersolution of (2), the comparison principle again yields that$w$$u\leq w\leq C||f||_{L^{n}(\Omega)}$in$\Omega$

$w$

is

.

We modify the proof of Trudinger’s in [27]. In fact, if we directly apply CafFarelli’s blowup argument, we can only succeed to prove the assertion in the case when the growth-0rder
in -variables is strictly less than 2. See [18] for this approach.
$p$

Idea of proof)(1) Use two different transformations to subsolutions and supersolutions, respectively, to simplify the original PDEs.
(2) Show the local maximum principle for transformed subsolutions and the weak
Harnack inequality for transformed supersolutions.

2.1

Preliminaries

Two key tools are the ABP maximum principle and the existence of strong solutions.
To this end, we introduce the upper contact set
for $u\in C(\Omega)$ ;$\Gamma_{\Omega}[u]$$\Gamma_{\Omega}[u]=${$x\in\Omega|\exists p\in \mathrm{R}^{n}$such that $u(y)\leq u(x)+\langle p,y-x\rangle$ for

fLemark] Roughly speaking, it holds that “ $D^{2}u\leq 0$”on

ABP maximum principle (Proposition 3.3 in [4])$\Gamma_{\Omega}[u]$.

$\forall y\in\Omega$}23
Assume $f\in L^{n}(\Omega)$ . There exists $C=C(\lambda, \Lambda, n, \Omega)>0$ such that if $u\in C(\Omega)$ is ansubsolution (resp., -supersolution)of$L^{n}$$L^{n}$$P^{+}(D^{2}u)\leq f$$(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.,$$P^{+}(D^{2}u)\geq f)$,then it follows that$\mathrm{m}_{\frac{\mathrm{a}}{\Omega}}\mathrm{x}u^{+}\leq$$(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.$,an$u^{+}+C\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\Omega)||f^{+}||L^{n}(\mathrm{p}_{\Omega}[\mathrm{u}+])$$\mathrm{m}_{\frac{\mathrm{a}}{\Omega}}\mathrm{x}u^{-}\leq\max u^{-}\partial\Omega+C\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\Omega)||f^{-}||_{L^{n}(\mathrm{p}_{\Omega}[u^{-}]))}$

remark] (i) Here, we have used the notations:$u^{+}:= \max\{u, 0\}$and$u^{-}:= \max\{-u, 0\}$(ii) If is aball or acube, then $C>0$ does not depend on 0.
(iii) We do not know if this assertion holds true for $C$-solutions unless is continuous.
(iv) The idea of proof is first to approximate by smooth functions (see the proposition“
below for an existence result when is smooth), and then, to approximate ”by the supconvolution (resp., -convolution)and the standard mollifier to apply the ABP maximum
principle for strong solutions.
$\Omega$

$f$

$f$

$u$

$f$

$\inf$

Existence of strong solutions (Lemma 3.1 in [4])There exists $C>0$ such that for $f\in L^{n}(\Omega)$ , there is anof$L^{n}$-strong subsolution $u\in$$C(\overline{\Omega})\cap W_{lo\mathrm{c}}^{2,n}(\Omega)$$\{$(1) $P^{+}(D^{2}u)\leq f$(2) $u=0$
(3)

$a.e$

. in ,$\Omega$on an,$||u||_{L}\infty(\Omega)\leq C\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\Omega)||f||_{L^{n}(\Omega)}$

Bemark] (i) Here, the constant $C>0$ is the one for the ABP maximum principle. We

may have the corresponding result for $P^{-}(D^{2}u)\geq f$ .
, assuch that(ii) The sketch of proof is as follows: Choose
is convex and independent of , in view of [12], we know the existence. Sinceofof classical solutions$P^{+}(D^{2}u_{k})=f_{k}$in ,$f_{k}\in C^{\infty}(\overline{\Omega})$$k$$arrow\infty$$P^{+}$$||f-f_{k}||_{L^{n}(\Omega)}arrow 0$$x$$u_{k}$$\Omega$$\{$$u_{k}=0$on$\partial\Omega$.

Follwoing the argument in [4], we can get auniform estimate for. We remark that the limit function
uniform convergence to some.
weakly in(1) since we may only know

and the
only satisfies$||u_{k}||_{W_{loc}^{2,n}(\Omega)}$$u\in C(\overline{\Omega})$$u_{k}arrow u$$W_{loc}^{2,n}(\Omega)$

$u$

24
2.2

Local maximum principle

Setting$w(x)=e^{*}-1u\mathrm{s}$we observe that$w$,

is anonnegative subsolution of$P^{-}(D^{2}w) \leq\underline{f}:=\frac{e+_{\gamma f}^{u}}{\lambda}$.

Since we suppose that $0\leq u\leq N$ , we do not have to worry about the right hand side of
the above.
Local maximum principle
Fix any $p>0$ and. There exists $C=C$ ($\Omega’(\subset\Omega$$\lambda,$$\Lambda$, , dist(Q’,$n$$\partial\Omega),p$) $>0$ such that.

$Q_{r}(x) \max w\leq C(||w||_{L^{p}(Q_{2\sqrt{\mathrm{n}}r}(x))}+r||\underline{f}||_{L\cdot(Q_{2\sqrt r}(x)))}$

for$x\in\Omega’$and small

$r$

$>0$

.

For simplicity, we shall obtain the assertion when $x=0$ and $r=1$ . Let us write $B$ forfor simplicity.
We first note that it is sufficient to show the case when
. Indeed, letting
be the strong subsolution of$B_{\sqrt{n}}^{o}$$\underline{f}=0$$\psi$$\in$$C(\overline{B})\cap W_{loc}^{2,n}(B)$$a.e$$P^{+}(D^{2}\psi)\leq-\underline{f}$$\{$$\psi$

on$=0$$||\psi||_{L(B)}\infty\leq C||\underline{f}||_{L^{n}(B)}$. in ,$B$$\partial B$,,

we need to show the assertion for $w+\psi$ , which is an$L^{n}$$P^{-}(D^{2}(w+\psi))\leq 0$

subsolution of.

We notice here that even for $p\in(0,1)$ , we have the following inequality in place of thetriangle inequality for $p\geq 1$ :for$||f_{1}+f_{2}||_{L^{p}(\Omega)}\leq 2^{\frac{1}{p}}(||f_{1}||_{L^{\mathrm{p}}(\Omega)}+||f_{2}||_{L^{\mathrm{p}}(\Omega))}$Next, we introduce the following “cut-0ff’ function$\eta(|x|)=(n-|x|^{2})^{\frac{2\mathrm{n}}{\mathrm{p}}}$.

$f_{1}$,$f_{2}\in L^{p}(\Omega)$.

25
It is not hard to verify that $W(x):=\eta(x)w(x)$ satisfies$P^{-}(D^{2}W)\leq C(\eta^{-\mathrm{L}}\overline{2}n|DW|+\eta^{-R}nW)$.

Since an easy geometrical observation implies thatfor$|DW(x)| \leq\frac{W(x)}{\sqrt{n}-|x|}\leq C\eta^{-_{n}}(\epsilon x)W(x)$Thus, since$Q_{1}\not\subset$$B_{\sqrt{n}}$$x\in\Gamma_{B}[W^{+}]$.

, the ABP maximum principle yields that$\max w\leq C\max W^{+}\leq C||\eta^{-_{n}}W^{+}|\epsilon|_{L^{n}(B)}\leq\frac{1}{2}\max W^{+}+C||w||_{L^{\mathrm{p}}(B)}Q_{1}BB_{1}^{\cdot}$More precisely, we first regularize by the $\sup$-convolution.
estimate in asmaller ball , where $r=r(\epsilon)arrow\sqrt{n}$ as$w$$B_{r}$$\epsilon$of it.

$w^{\epsilon}$of -solutions. In fact, we only know that the above
$C$

2.3

get the$arrow 0$

Bemark]To deduce PDEs to homogenious ones, we need to work with$\psi$$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n},\mathrm{w}\mathrm{e}$belongs to$L^{n}$-solutions instead
but $C^{2}(B)$ .

$W_{l\mathrm{o}\epsilon}^{2,n}(B)$Weak Harnack inequality

We shall adapt Caffarell’s argument in [2] to show the weak Harnack inequality for supersoultions while in [19] we adapt the argument in [25].
We fisrt use the following transformation for :$u$$v(x)=1-e^{-\frac{\gamma u(x)}{\lambda}}$It is easy to see that$v$

is asupersolution of$P^{+}(D^{2}v) \geq\overline{f}:=\frac{e^{-\mathrm{L}^{u}})\gamma f}{\lambda}$.

Weak Harnack inequality. There exist $p>0$ and $C=C(\lambda,\Lambda, n, \Omega’)>0$ such that
Fix$\Omega’\mathrm{c}\subset\Omega$$||v||_{L^{\mathrm{p}}(Q_{r}(x))} \leq C(\min_{Q_{r}(x)}v+r||\overline{f}||_{L^{n}(Q_{2\sqrt{n}r}(x))})$

for$x\in\Omega’$and small $r>0$ .

As before, we may suppose that $x=0$ and $r=1$ .

26Again, considering $vf$

instead of , where$\psi$$\psi$$\{$$\psi\in C(\overline{B}_{1})\cap W_{loe}^{2,n}(B_{1})$$\mathcal{P}^{-}(D^{2}\psi)\geq\overline{f}^{-}$$a.e$$\psi=0$on$0\leq\psi$$\leq C||\overline{f}||_{L^{\mathfrak{n}}(B_{1})}$

in

. in
$\partial B_{1}$$B_{1}$$B_{1}^{o}$is asupersolution of

,

,

,

we only need to consider the case when.
$v(x)/(\min_{Q_{1}}v+\epsilon)(\epsilon>0)$Moreover, by considering
instead of , it is sufficient to find
$p>0$ such that.
To this end, we need the following decay estimate of the distribution of ;$\overline{f}=0$$v$$||v||_{L^{q}(Q_{1})}\leq C$$v$$|\{x\in Q_{1}|v(x)>t\}|\leq Ct^{-\tau}$

Here, $C>0$ and $\tau>0$ are independent ofassertion for any integer
:
$k$

$v$

$(t\geq 0)$. Thus, it suffices to show the following$\geq 1$$|${$x\in Q_{1}$I $v(x)>M^{k}$ }$|\leq\mu^{k}$,where $M>1$ and $\mu\in(0,1)$ are independent of .
For the case of $=1$ , the above estimate is adirect consequence of the ABP maximum
$k$

$k$

principle.

To show any $k\geq 1$ , we argue by contradiction: Suppose that the assertion for holds
but fails for $k+1$ . To get acontradiction, we use the cube-decomposition lemma by
Calder\’on-Zygmund. See [2] for it.
$k$

2.4

Concluding remarks

In [19], following Escauriaza in [10], we give an extension to the case when belongs to aslightly larger space,
for some $p\in(n/2, n)$ .
In [19], we also mention the Holder estimate near the boundary, which ensures the global
Holder estimate. In afuture work, we will discuss on higher regularity for solutions of (1)utlizing this global H\"older estimate.
$f$$L^{\mathrm{p}}(\Omega)$Open questions] There must be so many open questions (at least to me) in this direction.

We only list some of them:(1) Haraack inequality near the boundary when $f\in L^{p}(\Omega)$ for $p\in(n/2, n)$ . (i.e. FabesStroock type formula near)(2) Relation between Caffarell’s class and the VMO space for higher regularity.
(3) Sufficient conditions to show the existence of solutions of (1) in compariosn to Nagum$\partial\Omega.$

27

estimate than that mentioned here,(4) More delicate sufficient conditions to derive(proposed by Prof. H. Nagai)(5) More than quadratic nonlinearity. (proposed by Prof. M. Otani)$L”(\mathrm{O})$etc.
Though some of papers listed below are not refered here, for the interested readers, wegive alist of related papers on If-solutions for fully nonlinear PDEs.

References[1] X. CABR\’E, On the Alexandroff-Bakelman-Pucci estimate and the reverse H\"older inequality for solutions of elliptic and parabolic equations, Comm. Pure AppL Math.,48 (1995), 539-570.

[2] L. A. CAFFARELLI&X. CABRi, Fully Nonlinear Elliptic Equations, Amer. Math.
Soc. Colloq. Publ. 43, Providence, 1995.
[3] L. A. CAFFARELLI, Interior apriori estimates for solutions of fully non-linear equations, Ann. Math., 130 (1989), 189-213.
[4] L. A. CAFFARELLI, M. G. CRANDALL, M. Kocan &A. WIECH, On viscosity
solutions of fully nonlinear equations with measurable ingredients, Comm. Pure AppL
Math., 49 (1996), 365-397.
$\acute{\mathrm{S}}$[5] L. A. CAFFARELLI &L. WANG, AHarnack inequality approach to the interior
regularity of elliptic equations, Indiana Univ. Math. J., 42 (1993), 145-157.

[6] M. G. CRANDALL, K. Fok, M. Kocan &A. W1ECH, Remarks on nonlinear
uniformly parabolic equations, Indiana Univ. Math. J., 47 (1998), 1293-1326.
$\acute{\mathrm{S}}$[7] M. G. CRANDALL, M. Kocan, P.-L. LIons &A. WIECH, Existence results
for boundary problems for uniformly elliptic and parabolic fully nonlinear equations,Electron. J. Differential Equations, 1999 (24), (1999), 1-20.
$\acute{\mathrm{S}}$[8] M. G. CRANDALL, M. Kocan, P. SORAVIA &A. W1ECH, On the equivalenceof various weak notions of solutions of elliptic PDE’s with measurable ingredients,
Progress in Elliptic and Parabolic Partial Differential Equations (A. Alvino et al.
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