数理解析研究所講究録
第 898 巻 1995 年 132-148

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On the nonlinear degenerate elliptic PDEs with obstacles$\dagger\# t\overline{p}\ovalbox{\tt\small REJECT} WAX^{A}\mp$

Jfi#$E\not\equiv$(KATSUYUKI ISHII)\S 1. Introduction
This article is a part of [10]. We are concemed with the nonlinear degenerate
elliptic partial differential equations (PDEs) with obstacles.
Let
(1.1)

$\Omega\subset \mathbb{R}^{N}$

be a bounded domain. We consider the following elliptic PDE:$\{\begin{array}{ll}\max\{u+F(x, Du, D^{2}u), u-\psi\}=0 in \Omega,u=g \end{array}$on$\partial\Omega$.

Here $F$ is the Hamilton-Jacobi-Bellman operator:$F(x,p, X)= \sup_{\alpha\in\Lambda}\{-tr({}^{t}\sigma(x, \alpha)\sigma(x, \alpha)X)+\{b(x, \alpha),p\rangle-f(x, \alpha)\}$where$\Lambda$

is a compact metric space and $trA$ andand the transposed matrix of$A$${}^{t}A$,denote, respectively, the trace. The problem (1.1) is derived from the optimal

stopping problems for diffusion processes. See [2] for more backgrounds.
It is easily seen by a simple example that, in general, the problem (1.1) has

no classical solution.
In the case $F$ is nondegenerate, we obtained the existence and uniqueness ofsolutions satisfying the boundary condition in the classical sense. By [11] and [17]there exists a unique solution of (1.1) in$W^{2,\infty}(\Omega)\cap C$(St). Applying the resultsin [4], we have a unique viscosity solution of (1.1).
However, in the case $F$ is degenerate, especially on$\partial\Omega$, we cannot interpretthe boundary condition in (1.1) in the classical sense. In [6] H. Ishii pointed outthat in this case we should interpret the boundary condition in the “viscosity
sense”, which is naturally derived from the dynamic programming principle in

the optimal control theory. Moreover he obtained the comparison principle and133

existence of viscosity solutions for Hamilton-Jacobi equations. Recently In [12],[13] M. A. Katsoulakis have proved the ones for second order degenerate elliptic

PDEs without obstacles.

Our main aim here is to discuss the uniqueness and existence of viscostiy
solutions of (1.1) and to apply them to the implicit boundary value problems.
Since we consider the case $F$ is degenerate, we interpret the boundary conditionin the viscosity sense.
This article is organized in the following way. In Section 2 we state ourassumptins and recall the notion of viscosity solutions of (1.1). In Section 3 weprove the comparison principle of viscosity solutions of (1.1). Section 4 is devoted

to the existence of viscosity solutions of (1.1). In Section 5 we treat some implicit
boundary value problems.
In what follows we surpress the term “viscosity” since we are mainly concernedwith viscocity sub-, super- and solutions.

\S 2. Preliminaries
In this section we state our assumptions and give the definition of solutionsof (1.1). We make the following assumptions.

is a bounded domain with the smooth boundary(A.1)$\Omega\subset \mathbb{R}^{N}$(A.2)$\sup_{\alpha\in\Lambda}\{\Vert\sigma(\cdot, \alpha)\Vert_{W^{1,\infty}(\overline{\Omega})},$ $\Vert b(\cdot, \alpha)\Vert_{W^{1,\infty}(\overline{\Omega})},$(A.3)$\psi,$$g\in C(\overline{\Omega})$(A.4) For each
(i)

and

$z\in\partial\Omega$$\psi\geqq g$(iii)$\partial\Omega$, there exist$\alpha(\cdot)\in W^{1,\infty}(\overline{\Omega})$

(ii)

on

$\partial\Omega$.

$\Vert f(\cdot, \alpha)\Vert_{C(\overline{\Omega})}\}=K<+\infty$.

.

$\alpha=\alpha(z)\in\Lambda$

satisfying,$tr({}^{t}\sigma(z, \alpha(z))\sigma(z, \alpha(z))D^{2}\rho(z))-\langle b(z, \alpha(z)),$ $D\rho(z))\geqq\eta$$\langle{}^{t}\sigma(z, \alpha(z))\sigma(z, \alpha(z))D\rho(z),$ $D\rho(z)\rangle=0$(iv) There are unit vectors

for some $\eta>0$ ,,$\{\hat{e}_{l}\}_{1\leqq l\leqq N-1}\subset \mathbb{R}^{N}$by which the tangent

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plane at$z$

is spanned such that$\langle t\sigma(z, \alpha(z))\sigma(z, \alpha(z))\hat{e}_{l},\hat{e}\iota\rangle=0$

except at most two vectors(A.5) For each(i)$z\in\partial\Omega$$\{\hat{e}_{l_{1}},\hat{e}_{l_{2}}\}$.

, there exist a constant $\eta>0$ and$\beta(\cdot)\in W^{1,\infty}(\overline{\Omega})$$\beta=\beta(z)\in\Lambda$

satisfying,(ii) $tr({}^{t}\sigma(z, \beta(z))\sigma(z, \beta(z))D^{2}\rho(z))-\langle b(z, \beta(z)),$ $D\rho(z)\}\leqq-\eta$or$\{{}^{t}\sigma(z, \beta(z))\sigma(z, \beta(z))D\rho(z),$ $D\rho(z)\rangle\geqq\eta$.

Remark 2.1. As to the assumption (A.4), see [12] and [13].
Next we give the definition of solutions of (1.1) and the equivalent proposi-tion. For any function$u$:$\overline{\Omega}arrow \mathbb{R},$$u^{*}$

and$u_{*}$

denote, respectively, the uppersemicontinuous $(u.s.c.)$ envelope and the lower semicontinuous $(l.s.c.)$ envelope of$u$:$u^{*}(x)= \lim_{rarrow 0}\sup\{u(y)||y-x|<r, y\in St\}$$u_{*}(x)= \lim_{rarrow 0}\inf\{u(y)||y-x|<r, y\in St\}$We define$J_{\frac{2}{\Omega’}}^{+}u(x),$$J_{\frac{2}{\Omega’}}^{-}u(x)$$+ \frac{1}{2}(Xh,$$u(x+h)\leqq u(x)+\langle p,$ $h)$$h\}+o(|h|^{2})$$J_{\frac{2}{\Omega’}}^{-}u(x)=\{(p,X)\in \mathbb{R}^{N}\cross\^{N}$$(p, X)\in J_{\frac{2}{\Omega’}}^{+}u(x)$$\varphi\in C^{2}(\overline{\Omega})$as$x+h\in\overline{\Omega}$

and $harrow 0$ ,$u(x+h)\geqq u(x)+(p,$$+ \frac{1}{2}\{Xh, h\}+o(|h|^{2})$a function.

by$J_{\frac{2}{\Omega’}}^{+}u(x)=\{(p,X)\in \mathbb{R}^{N}\cross S^{N}$

It is observed that if

,

$($

as

resp.,$x+h\in\overline{\Omega}$$h\rangle$

and $harrow 0$$\in J_{\frac{2}{\Omega’}}^{-}u(x))$, then there exists

such that $u-\varphi$ takes a local maximum (resp., a local

135

minimum) at$\varphi\in C^{2}(\overline{\Omega})$

, if

$x$

and $(D\varphi(x), D^{2}\varphi(x))=(p, X)$ . Conversly it is seen that, for any$u-\varphi$

takes a local maximum (resp., local minimum) at$(D\varphi(x), D^{2}\varphi(x))\in J_{\frac{2}{\Omega’}}^{+}u(x)$Definition 2.2. Let(1) We say

$u$

$($

resp.,$\in J_{\frac{2}{\Omega’}}^{-}u(x))$$u:\overline{\Omega}arrow \mathbb{R}$$u^{*}$.

(2) We say$u$$u_{*}$$ailx\in\overline{\Omega}$,$\overline{\Omega}$$(x\in\partial\Omega)$

and for all$(x\in\Omega)$

or $\max\{u_{*}(x)+F(x,p, X), u_{*}(x)-\psi(x)\}\geqq 0$$u$

are the graph closure of$J_{\frac{2}{\Omega’}}^{+}u(x),$$J_{\frac{2}{\Omega’}}^{-}u(x)$$(x\in\partial\Omega)$$u^{*}(x)\leqq g(x)$$u$:$u^{*}$$\overline{\Omega}arrow \mathbb{R}$$\overline{\Omega}$,$x\in\overline{\Omega}$andsatisfi es$(x\in\Omega)$,

or $\max\{u^{*}(x)+F(x,p,X), u^{*}(x)-\psi(x)\}\leqq 0$$d$$(p, X)\in J_{\frac{\eta}{\Omega’}}^{-}u_{*}(x),$$u_{*}$$\overline{\Omega}$$(x\in\partial\Omega)$

and for all.

$x\in\overline{\Omega}$satisfies$\max\{u_{*}(x)+F(x,p,X), u_{*}(x)-\psi(x)\}\geqq 0$$u_{*}(x)\geqq g(x)$$\overline{J}_{\frac{2}{\Omega’}}^{+}u(x)$.

and for all

is a supersolution of (2.1} if an only if $u_{*}>-\infty$ on

and.

, respectiverly.

is a $su$ bsolu tion of (2.1) if and only if $u^{*}<+\infty$ on$\max\{u^{*}(x)+F(x,p,X), u^{*}(x)-\psi(x)\}\leqq 0$

$u$

$d$

$u$

$(p,X)\in\overline{J}_{\frac{2}{\Omega’}}^{+}u^{*}(x),$

(2)

an

is a solu tion of (1.1) if is both a sub-and a supersolution of (1.1).

Proposition 2.3. Assume (A.2) and (A.3). Let(1)$x\in\overline{\Omega}$.

,

Next we mention the equivalent proposition to Definition 2.2.
$\overline{J}_{\frac{2}{\Omega’}}^{-}u(x)$andsatisfies$\max\{u_{*}(x)+F(x,p, X), u_{*}(x)-\psi(x)\}\geqq 0$

$u$

$d$

$(x\in\Omega)$

is a supersolu tion of (1.1) if $u_{*}>-\infty$ on$u_{*}(x)\geqq g(x)$an foror $\max\{u^{*}(x)+F(x,p, X), u^{*}(x)-\psi(x)\}\leqq 0$$(p, X)\in J_{\frac{2}{\Omega’}}^{-}u_{*}(x),$(3) We say$\overline{\Omega}$satisfi es$\max\{u^{*}(x)+F(x,p, X), u^{*}(x)-\psi(x)\}\leqq 0$$u^{*}(x)\leqq g(x)$, then.

is a $su$ bsolu tion of (1.1) if $u^{*}<+\infty$ on$(p, X)\in J_{\frac{2}{\Omega’}}^{+}u^{*}(x),$$x\in\overline{\Omega}$$(x\in\Omega)$,

or $\max\{u_{*}(x)+F(x,p,X), u_{*}(x)-\psi(x)\}\geqq 0$$(x\in\partial\Omega)$

.

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We omit the proofs of the above proposition. See [4; Section 7].

\S 3. Comparison principle of solutions
In this section we prove the comparison principle of solutions of (1.1). To doso, we use the similar techniques seen in [6], [4], [12] and [14] etc.
We note that, by (A.1) there exists a small constant $r_{0}>0$ such that, for any$z\in\partial\Omega$

,

(3.1)

$K_{y}=y+ \bigcup_{0<s<r_{0}}B(sn(z), s)\subset\Omega$where $-n(z)$ is the outward unit normal toopen ball centered at$x$, for all$\Omega$at$y\in B(z, r_{0})\cap\overline{\Omega}$$x\in\partial\Omega$

and $B(x, r)$ denotes thewith radius .
$r$

Theorem 3.1. Assume $(A.1)-(A.5)$ hold. Let
$su$

,

$u$

and , be, respecti$v$$i^{\gamma}ely,$$a$bsolution an a supersolu tion of (1.1). If any on of th folloivings holds, then$d$on$u^{*}\leqq v_{*}$$\overline{\Omega}$

$e$

$e$

.

(1) lim $supK_{z}\ni xarrow zu^{*}(x)=u^{*}(z)$ and lim $infK_{z}\ni xarrow zv_{*}(x)=v_{*}(z)$ for each $z\in$$\partial\Omega$.

(2) lim $supK_{z}\ni xarrow zu^{*}(x)=u^{*}(z)$ and $u^{*}(z)\leqq g(z)$ for each(3) lim $infK_{z}\ni xarrow zv_{*}(x)=v_{*}(z)$ and $v_{*}(z)\geqq g(z)$ for each$z\in\partial\Omega$$z\in\partial\Omega$.

.

Remark 3.2. We call the properties in Theorem 3.1 (1) nontangential upperand lower semicomtinuity, respectively. See [12], [13].

Proof of Theorem 3.1. We may consider thatand l.s. . on$c$$\overline{\Omega}$

$u$

and

$v$

are, respectively, u.s. .
$c$. First let the condition (1) hold.

We suppose $\sup_{\overline{\Omega}}(u-v)=\theta>0$ and get a contradiction.

$\backslash Ve$

may assume$u-v$ takes its strict maximum at $z\in$ St, because, if otherwise, we can make it doso by using some perturbation techniques.

137

We divide our consideration into three cases.
Case 1.

Let$z\in\partial\Omega$

and $v(z)<g(z)$ .

$\{z_{n}\}_{n\in \mathbb{N}}\subset K_{z}$be a sequence such that$z_{n}arrow z,$$(narrow+\infty)$$u^{*}(z_{n})arrow u^{*}(z)$We define the function $\Phi(x, y)$ on$\overline{\Omega}\cross\overline{\Omega}$.

by$\Phi(x, y)=u(x)-v(y)-\frac{\alpha_{n}}{2}|x-y-z_{n}+z|^{2}$

where$\alpha_{n}=s_{0}^{2}/|z_{n}-z|^{2}$

Let $(x_{n}, y_{n})\in$ St

and $s_{0}>0$ satisfies $(3K^{2}+K)s_{0}^{2}<\theta$ .

$\cross\overline{\Omega}$be a maximum point of

3 we obtain the behaviors of
$]$

(3.4)

,

$x_{n},$$y_{n},$$u(x_{n}),$ $v(y_{n})$$\Phi$. Calculating as in [4; Section

as $narrow+\infty$ :$\{\begin{array}{l}x_{n}, y_{n}arrow z, u(x_{n})arrow u(z), v(y_{n})arrow v(z),\alpha_{n}|x_{n}-y_{n}-z_{n}+z|^{2}arrow 0, \sqrt{\alpha_{n}}|x_{n}-y_{n}|arrow s_{0}.\end{array}$

We apply the maximum principle for semicontinuous functions to obtain $X$ ,$Y\in\^{N}$

satisfying$(p_{n},X)\in\overline{J}_{\frac{2}{\Omega’}}^{+}u(x_{n}),$$(p_{n}, Y)\in\overline{J}_{\frac{2}{\Omega’}}^{+}v(y_{n})$

,

and

(3.5)

$-3\alpha_{n}(oI$

$OI$

$-YO$

$\leqq(XO$

,

$-II$

$\leqq 3\alpha_{n}(-II$

where $p_{n}=\alpha_{n}(x_{n}-y_{n}-z_{n}+z)$ .
We may consider$x_{n}\in\Omega$$|x_{n}-y_{n}-z_{n}+z|<r_{0}|z_{n}-z|$

for sufficiently large
for large$n\in \mathbb{N}$$n\in \mathbb{N}$, where $r_{0}>0$ is the same constantas in (3.1). Moreover we have $v(y_{n})<g(y_{n})$ for large$v(z)<g(z)$ . Hence using the fact that

$u$

and

$v$

because (3.4) implies$n\in \mathbb{N}$by (A.3), (3.4) andare, respectively, a subsolution md

a supersolution of (1.1), we obtain the following inequalities:$\max\{u(x_{n})+F(x_{n},p_{n},X), u(x_{n})-\psi(x_{n})\}\leqq 0$$\max\{v(y_{n})+F(y_{n},p_{n}, Y), v(y_{n})-\psi(y_{n})\}\geqq 0$

.

,

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By (A.2), (A.3) and (3.5) we calculate$\theta\leqq u(x_{n})-v(y_{n})$$\leqq\max\{F(y_{n},p_{n}, Y)-F(x_{n},p_{n},X), \psi(y_{n})-\psi(x_{n})\}$$\leqq\max\{(3K^{2}+K)\alpha_{n}|x_{n}-y_{n}|^{2}+\omega(|x_{n}-y_{n}|),\omega(|x_{n}-y_{n}|)\}$

where$\omega$

is a modulus of continuity of the functions

$f$

and

$\psi$,.

Recalling (3.4) and letting $narrow+\infty$ , we obtain$\theta\leqq(3K^{2}+K)s_{0}^{2}<\theta$,

which is a contradiction.
Case 2.

$z\in\partial\Omega,$$u(z)>g(z)$ .

As in Case 1, we define the function$\Phi$

by$\Phi(x, y)=u(x)-v(y)+\{q, x\}-\frac{\alpha_{n}}{2}|x-y+z_{n}-z|^{2}$onKi$\cross\overline{\Omega}$.

We can prove the remainder similarly to the above.
Case 3.

$z\in\Omega$.

In this case the proof is standard. See [4; Section 3].
When the condition (2) (resp., (3)) holds, it is sufficiently to consider only
Case 2, 3 (resp., Case 1, 3) in the above arguments. Thus we obtain the result. 1\S 4. Existence of solutions
This section is devoted to the existence of solutions of (1.1). In doing so, theresults in [12], [13] play an important role. For the case[6]. In the following we assume(A.6)$\Lambda$

is a compact metric space.

$\sigma(\cdot, \alpha)\equiv O(\forall\alpha\in\Lambda)$

, see

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We prepare some notations.

standard $N$ –dimensional Brownian motion.

$W_{t}=${$\mathcal{A}=$$\mathcal{B}=$$X_{t}${$\alpha_{t}$$\theta$: $[0,$: progressively measurable}.

$+\infty)arrow\Lambda$: stopping time}.

: solution of$\{\begin{array}{l}dX_{t}=-b(X_{t}, \alpha_{t})dt+\sqrt{2}\sigma(X_{t}, \alpha_{t})dW_{t}, t>0,X_{0}=x\in\overline{\Omega}.\end{array}$$\tau=\inf\{t\geqq 0|X_{t}\not\in\overline{\Omega}\}$$1_{A}=$.

characteristic function for$A$.

Our existence result is stated as follows.
Theorem 4.1. Assume $(A.1)-(A.6)$ . Then there exists a unique solution$u\in C(\overline{\Omega})$of (1.1) and it is represented as the value function associated with the

optimal siopping problem:$u(x)= \alpha\in A\inf_{\theta\in B}E_{x}\{\int_{0}^{\tau\wedge\theta}f(X_{t}, \alpha_{t})e^{-t}dt+1_{\theta<\tau}\psi(X_{\theta})e^{-\theta}+1_{\theta\geqq\tau}g(X_{\tau})e^{-\tau}\}$.

To show this theorem, we consider the penalized problem for (1.1).
(4.1)$\{\begin{array}{ll}F(x, u_{n}, Du_{n}, D^{2}u_{n})+n(u_{n}-\psi)^{+}=0 in \Omega,u_{n}=g on \partial\Omega,\end{array}$where$n\in \mathbb{N}$and $r^{+}= \max\{r, 0\}$ .

Then applying the results in [13], for eachsolution$u_{n}\in C(\overline{\Omega})$$n\in \mathbb{N}$, there exists a uniqueof (4.1) and it is characterized as follows:(4.2)$u_{n}(x)= \inf_{\alpha\in A}E_{x}\{\int_{0}^{\tau}(f(X_{t}, \alpha_{t})-n(u_{n}(X_{t})-\psi(X_{t}))^{+})e^{-t}dt+g(X_{\tau})e^{-\tau}\}$Using (A.5) and the barrier argument, we have(4.3)

$u_{n}\leqq g$on$\partial\Omega$for all$n\in \mathbb{N}$

.

.

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Since the operator

is monotone with respect to$n\in \mathbb{N}$$-C\leqq\cdots\leqq u_{n}\leqq\cdots\leqq u_{2}\leqq u_{1}$$on$$nr^{+}$

and $u_{n}\geqq-C$ for large$C>0$ , we obtain(4.4)$\overline{\Omega}$by the comparison principle of solutions of (4.1). (cf. [4; Theorem 7.9].) Hence

we can define the function$u$by$u(x)=$ $\lim_{narrow+\infty}u_{n}(x)$ .

(4.5)

Then we get the following lemma.
Lemma 4.2. The above function

Proof. It is easily seen that

$u$

$u$

is u.s. . on$\varphi\in C^{2}(\overline{\Omega})$

We may consider, we assume that$x_{0}\in\Omega$

and that$\overline{\Omega}$$c$and letting $narrow+\infty$ , we have $u\leqq g$ on
For any

is a $u.s.c$ . subsolution of (1.1).

$x_{0}$$\partial\Omega$by means of (4.4). Using (4.3).

$u-\varphi$

attains a local maximum at$x_{0}\in\overline{\Omega}$.

is a strict local maximum point of $u-\varphi$ .

Then there exists a $\delta>0$ such thatLet$x_{n}$$u(x_{0})-\varphi(x_{0})>u(x)-\varphi(x)$for allbe a maximum point ofon$u_{n}-\varphi$$x\in B(x_{0}, \delta)(\subset\Omega),$$B(x_{0}, \delta)$$x\neq x_{0}$.

. Then by the same $\arg\iota iment$

as in G. Barles-B. Perthame [1; Lemma A.3]. , we get
$)$

(4.6)
Since

(4.7)

$x_{n}arrow x_{0}$$u_{n}$,$u_{n}(x_{n})arrow u(x_{0})$$(narrow+\infty)$.

is a subsolution of (4.1), we obtain,$F(x_{n}, u_{n}(x_{n}), D\varphi(x_{n}), D^{2}\varphi(x_{n}))+n(u_{n}(x_{n})-\psi(x_{n}))^{+}\leqq 0$.

It follows from (A.2) and (4.7) that there exists a constant $C>0$ such that$n(u_{n}(x_{n})-\psi(x_{n}))^{+}\leqq C$

for all$n\in \mathbb{N}$

.

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Thus passing to the limit as $narrow+\infty$ , we have, by (4.6)$u(x_{0})-\psi(x_{0})\leqq 0$.

On the other hand, (4.7) implies $F(x_{n}, u_{n}(x_{n}), D\varphi(x_{n}), D^{2}\varphi(x_{n}))\leqq 0$ . Sending$narrow+\infty$, we observe.

$F(x_{0}, u(x_{0}), D\varphi(x_{0}),D^{2}\varphi(x_{0}))\leqq 0$

Therefore we have completed the proof. 1

According to [16; p.37], the formula (4.2) can be rewritten as the following:$u_{n}(x)= \alpha\in A\inf_{\theta\in B}E_{x}\{\int_{0}^{\tau\wedge\theta}f(X_{t}, \alpha_{t})e^{-t}dt+1_{\theta<\tau}\psi_{n}(X_{\theta})e^{-\theta}+1_{\theta\geqq r}g(X_{r})e^{-\tau}\}$,where $a \wedge b=\min(a, b)$ and $\psi_{n}=\psi+(u_{n}-\psi)^{+}$ .
Since $u\leqq\psi$ on$\overline{\Omega}$

by Lemma 4.2, we have the following lemma by (4.4) andDini’s Theorem.
Lemma 4.3.

$u_{n}\Rightarrow u$on$\overline{\Omega}$

as $narrow+\infty$ and the fun ction$u$is represen $ted$ as$u(x)= \alpha\in A\inf_{\theta\in B}E_{x}\{\int_{0}^{r\wedge\theta}f(X_{t}, \alpha_{t})e^{-t}dt+1_{\theta<\tau}\psi(X_{\theta})e^{-\theta}+1_{\theta\geqq\tau}g(X_{r})e^{-\tau}\}$.

We are now in a position to prove Theorem 4.1.

Proof of Theorem 4.1. We have only to show that
For any$\varphi\in C^{2}(\overline{\Omega})$

We consider the case, we assume$x_{0}\in\partial\Omega$$u-\varphi$$u$is a supersolution of (4.1).

takes a strict local minimum at$x_{0}\in\overline{\Omega}$.

. Then we may assume $u(x_{0})<g(x_{0})$ , because, if

otherwise, we have nothing to prove. Since$u\in C(\overline{\Omega})$by Lemma 4.3, there exists

a $\delta>0$ satisfying$u(x)<g(x)$$x\in B(x_{0}, \delta)\cap\partial\Omega$$u(x)<\psi(x)$$x\in\overline{B(x_{0},\delta)}\cap\overline{\Omega}$

.

,

142

Moreover,$Lemm_{\backslash }a4.3$

implies there exists a$n_{0}\in \mathbb{N}$satisfying, for all $n>n_{0}$ ,(4.8)$u_{n}(x)<g(x)$$x\in B(x_{0}, \delta)\cap\partial\Omega$(4.9)$u_{n}(x)<\psi(x)$$x\in\overline{B(x_{0},\delta)}\cap\overline{\Omega}$Let$x_{n}\in\overline{B(x_{0},\delta)}\cap\overline{\Omega}$be a minimum point of,.

$u_{n}-\varphi$on. By the$\overline{B(x_{0},\delta)}\cap\overline{\Omega}$same argument as in the proof of Lemma 4.1, we have$x_{n}arrow x_{0}$,$(narrow+\infty)$$u_{n}(x_{n})arrow u(x_{0})$Therefore, using (4.8), (4.9) and the fact that$u_{n}$.

is a supersolution of (4.1), we

obtain$F(x_{n}, u_{n}(x_{n}), D\varphi(x_{n}), D^{2}\varphi(x_{n}))\geqq 0$.

Sending $narrow+\infty$ , we get$F(x_{0}, u(x_{0}), D\varphi(x_{0}), D^{2}\varphi(x_{0}))\geqq 0$.

Thus the proof is completed. 1\S 5. Implicit boundary value problems
In this section we apply Theorems 3.1 and 4.1 to the implicit boundary value

problems.
I. The impulse control problem. We consider the following problem:(5.1)$\{\begin{array}{ll}\max\{u+F(x, Du, D^{2}u), u- Mu\}=0 in \Omega,\max\{u-g, u-- Mu \}=0 on \partial\Omega,\end{array}$where the operator $M$ is defind by$Mu(x)= \inf\{k(\xi)+u(x+\xi)|\xi\in(\mathbb{R}^{+})^{N}, x+\xi\in\overline{\Omega}\}$.

This problem arises in the impulse control problems for diffusion processes. For
the impulse control and the related results, see [3], [19], [20] and [9] etc.

143

In addition to $(A.1)-(A.6)$ , we make the following assumptions.
(A.7) There exists a mapping(A.8)$x+P(x, \xi)\in\overline{\Omega}$

for all$P(x, \xi)=\xi$

if$P(\cdot, \xi)\in C(\overline{\Omega})$for each$k\in C((\mathbb{R}^{+})^{N})$all$\xi\in(\mathbb{R}^{+})^{N}$

satisfying$P:\overline{\Omega}\cross(\mathbb{R}^{+})^{N}arrow(\mathbb{R}^{+})^{N}$$(x, \xi)\in\overline{\Omega}\cross(\mathbb{R}^{+})^{N}$$x+\xi\in\overline{\Omega}$,,$\xi\geqq 0$.

and there exists a constant $k_{0}>0$ such that$k(\xi)\geqq k_{0}$.

Remark 5.1. The assumption (A.7) needs to make sure that, wheneveru.s. . on$c$$\overline{\Omega}$

for

$u$

is

, so is $Mu$ . See [9; Section 2].

We give the definition of solutions of (5.1).

Deflnition 5.2. Let(1) We say$u$$u:\overline{\Omega}arrow \mathbb{R}$.

is a $su$ bsolution of (5.1) if $u^{*}<+\infty$ on$(p, X)\in J_{\frac{2}{\Omega’}}^{+}u^{*}(x),$$u^{*}$$\overline{\Omega}$

and for all$x\in\overline{\Omega}$andsatisfi es$\max\{u^{*}(x)+F(x,p,X), u^{*}(x)-Mu^{*}(x)\}\leqq 0$$(x\in\Omega)$,$\max\{u^{*}(x)-g(x), u^{*}(x)-Mu^{*}(x)\}\leqq 0$

or $\max\{u^{*}(x)+F(x,p,X), u^{*}(x)-Mu^{*}(x)\}\leqq 0$(2) We say$u$is a supersolution of (5.1) if $u_{*}>-\infty$ on$(p, X)\in J_{\frac{2}{\Omega’}}^{-}u_{*}(x),$$u_{*}$$\overline{\Omega}$$(x\in\partial\Omega)$

and for all.

$x\in\overline{\Omega}$satisfies$\max\{u_{*}(x)+F(x,p,X), u_{*}(x)-Mu_{*}(x)\}\geqq 0$$(x\in\Omega)$,$\max\{u_{*}(x)-g(x), u_{*}(x)-Mu_{*}(x)\}\geqq 0$

or $\max\{u_{*}(x)+F(x,p,X), u_{*}(x)-Mu_{*}(x)\}\geqq 0$$(x\in\partial\Omega)$

.

and

144

(3) We say$u$is a solu tion of (5.1) if is both a sub-an a supersolution of (5.1).
$d$

$u$

We can prove the proposition equivalent to the above definition similar tothat to Definition 2.2. We have the following theorem.

Theorem 5.3. Assume (A.1), (A.2), $(A.4)-(A.8)$ an

exists a unique solution$u\in C(\overline{\Omega})$$dg\in C(\overline{\Omega})$. Then thereof (5.1).

Outline of proof. The comparson principle of solutions of (5.1) can be proved
similarly to that of Theorem 3.1. Hence we show only the existence.
We may assume$u_{0}\in C(\overline{\Omega})$$f(\cdot, \alpha),$$g\geqq 0$

on$\overline{\Omega}$. By [13] there exists a unique solutionof$\{\begin{array}{ll}F(x, u, Du, D^{2}u)=0 in \Omega,u-g=0 on \partial\Omega.\end{array}$Using Theorems 3.1 and 4.1 we can define the sequence$\{u_{n}\}_{n\in IN}\subset C(\overline{\Omega})$

induc-

tively as follows:$u_{n}$: a unique solution of$\{\begin{array}{ll}\max\{F(x, u, Du, D^{2}u), u-Mu_{n-1}\}=0 in \Omega,\max\{u-g, u-Mu_{n-1}\}=0 on \partial\Omega,\end{array}$

We see by Theorem 3.1 and the properties of the operator $M$ the following estimates.
$0\leqq\cdots\leqq u_{n}\leqq\cdots\leqq u_{2}\leqq u_{1}\leqq u_{0}$on$u_{n+1}-u_{n+2}\leqq(1-\mu)^{n}\Vert u_{0}\Vert_{C(\overline{\Omega})}$onfor some $\mu\in(0,1)$ . Thus there exists a function$\overline{\Omega}$$u\in C(\overline{\Omega})$$\overline{\Omega}$$\overline{\Omega}$,,such that$u_{n}:\supseteq u$. It follows from the stability of solutions and the comparison principle thata unique solution of (5.1). 1

For the detail, see [10].

11. The optimal switching problem.

$u$

on

is

145

Next we treat the following system of elliptic PDEs:(5.2)$\{\begin{array}{ll}u=(u^{1}, \cdots, u^{m}), k\in\Gamma=\{1, \cdots, m\}, \max\{u^{k}+F^{k}(x, Du^{k}, D^{2}u^{k}), u^{k}-M^{k}[u]\}=0 in \Omega,\max\{u^{k}-g^{k}, u^{k}-M^{k}[u]\}=0 on \partial\Omega,\end{array}$Here $m(>1)\in \mathbb{N}$ and$F^{k},$$M^{k}$are defined by,$F^{k}(x,p,X)= \sup_{\alpha\in\Lambda}\{-tr({}^{t}\sigma^{k}(x, \alpha)\sigma^{k}(x, \alpha)X)+(b^{k}(x, \alpha),p\}-f^{k}(x, \alpha)\}$$M^{k}[u](x)= \min\{u^{l}(x)+h^{kl}(x)|l\in\Gamma, l\neq k\}$.

This problem is associated with the optimal switching for diffusion processes. See[5], [18], [7], [14] and [15] for the related results.

As to the coefficients of$F^{k}$

and $h^{kl}(k, l\in\Gamma)$ , we make the following assump-

tions.
(A.2)’(A.9)$\sup_{k\in\Gamma,\alpha\in\Lambda}\{\Vert\sigma^{k}(\cdot, \alpha)\Vert_{W^{1,\infty}(\overline{\Omega})},$$h^{kl}\in C(\overline{\Omega})$and $h^{kl}>0$ on$\overline{\Omega}$$\Vert b^{k}(\cdot, \alpha)\Vert_{W^{1,\infty}(\overline{\Omega})},$

for

$k,$

$\Vert f^{k}(\cdot, \alpha)\Vert_{C(\overline{\Omega})}\}<+\infty$.

$l=1,$ $\cdots,$ $m$ .

Remark 5.4. The assumption (A.9) is needed to show the comparison principleof solutions of (5.2). It is called “no loop of zero cost condition”. (cf. [18].)
We give the definition of solutions of (5.2). Let $u^{*}=(u^{1*}, \cdots, u^{m*})$ and$u_{*}=(u_{*}^{1}, \cdots, u_{*}^{m})$.

Deflnition 5.5. Let(1) We say
and

$u$

$u:\overline{\Omega}arrow \mathbb{R}^{m}$.

is a subsolution of (5.2) if $u^{*}<+\infty$ on$(p, X)\in J_{\frac{2}{\Omega’}}^{+}u^{k*}(x)$$\overline{\Omega}$

and for all$k\in\Gamma,$$x\in\overline{\Omega}$,$\max\{u^{k*}(x)+F^{k}(x,p,X),u^{k*}(x)-M^{k}[u^{*}](x)\}\leqq 0$$(x\in\Omega)$,$\max\{u^{k*}(x)-g^{k}(x), u^{k*}(x)-M^{k}[u^{*}](x)\}\leqq 0$

or $\max\{u^{k*}(x)+F^{k}(x,p,X), u^{k*}(x)-M^{k}[u^{*}](x)\}\leqq 0$$(x\in\partial\Omega)$

.

146

(2) We say

and

$u$

is a supersoluiion of (5.2) if $u_{*}>-\infty$ on$(p, X)\in J_{\frac{2}{\Omega’}}^{-}u_{*}^{k}(x)$$\overline{\Omega}$

and for all$k\in\Gamma,$$x\in\overline{\Omega}$,$\max\{u_{*}^{k}(x)+F^{k}(x,p, X), u_{*}^{k}(x)-M^{k}[u_{*}](x)\}\geqq 0$$(x\in\Omega)$,$\max\{u_{*}^{k}(x)-g^{k}(x), u_{*}^{k}(x)-M^{k}[u_{*}](x)\}\geqq 0$

or $\max\{u_{*}^{k}(x)+F^{k}(x,p, X), u_{*}^{k}(x)-M^{k}[u_{*}](x)\}\geqq 0$(3) We say$u$is a solution of (5.2) if$u$$(x\in\partial\Omega)$.

is both a sub-an a supersolutioii of (5.2).
$d$

The equivalent proposition to the above defintion can be shown similarly tothat of Definition 2.2. Then we obtain the following theorem.
Theorem 5.6. Assume (A.1), (A.2)’, $(A.4)-(A.6)$ and (A.9) and$(\forall k\in\Gamma)$. Then there exists a unique solution$u\in C(\overline{\Omega})^{m}$$g^{k}\in C(\overline{\Omega})$of (5.2).

The strategy of the proof is similar to that of Theorem 5.3. Thus we leave itto the reader.

Remark 5.7. In [8], [14] and [15] we discussed the problem (5.2) from theviewpoint of monotone systems. As to the existence of solutions of (5.2), Theorem
5.4 provides another proof.

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