数理解析研究所講究録
第 730 巻 1990 年 128-137

128

On the existence of viscosity solutions to nonlinear problems

involving an integro-differential operator

山田直記 (Naoki Yamada)神戸大・理

1. Introduction

This is a part of the joint work [11] with Suzanne M. Lenhart at University ofTennessee, Knoxville.

In this note we consider the existence of viscosity solutions for an obstacle prob-

lem involving an integro-differential operator associated with piecewise-deterministic

processes.
Let$Lu(x)=-g(x) \cdot\nabla u(x)+\alpha(x)u(x)-\lambda\backslash (x)\int_{\Omega}(u(y)-u(x))Q(dy, x)$

where. is the inner product in$R^{n},$$\nabla u$is the gradient vector of

$u$

and

,

$Q(\cdot,\backslash x)$

is aprobability measure.
We consider the following obstacle problem:$\min\{Lu-f, u-\psi\}=0$

(1.1)

in

$\Omega$,with the boundary condition(1.2)
The operator$u(x)= \int_{\Omega}u(z)Q(dz, x)$

$L$

on

$\partial\Omega$.

arises as a generalized infinitesimal generator of a piecewise-determinis-tic (PD in short) process. These PD processes have deterministic dynamics
1$g$

between129
randomjumps. The jump distribution is represented by transition probability measure$Q(\cdot, x)$. See Davis [4] for the detail of PD processes.

In the case that$L$

is an infinitesimal generator of a diffusion process, it is wellknown that the unilateral obstacle problem (1.1) with the Dirichlet boundary condition

arises as a dynamic programming equation associated with an appropriate optimal
control problem (see Bensoussan and Lions [1]).

The equation (1.1) is also the dynamic programming equation associated with an
optimal control problem in which the underlying process is a PD process.
In the case that the domain$\Omega$

is a bounded domain in$R^{n}$, the PD processjumps

back into the interior upon hitting the boundary which leads to the boundary condition(1.2) (see Davis [4]).

The obstacle problem (1.1), (1.2) is first treated by Lenhart and Liao [9], [10]by using singular perturbation method. After introduction of the notion of viscosity
solution by Crandall and Lions [2], Lenhart [8] has proved the existence and uniquenessof viscosity solution for a system of obstacle problems.

In these articles, it is commonly assumed that$\alpha(x)\geq\alpha_{0}>0$

for sufficiently large$\alpha_{0}$.

The perpose of this note is to eliminate the condition of largeness for the zero-thorder term by using Perron’s method which is introduced by Ishii [6].
In section 2, we state the notion of viscosity solutions and assumptions. We also

give a brief review of Perron’s method. In section 3, we shall explain how to apply the
Perron’s method to get a viscosity solution of (1.1) satisfying the boundary condition(1.2). To show the existence of super- and subsolution, which are needed to apply

Perron’s method, we consider also a linear first order PDE with the boundary condition(1.2). Our main result is Theorem 3.3.

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2. Assumptions and Perron’s method
Let$Lu(x)=-g(x) \cdot\nabla u(x)+\alpha(x)u(x)-\lambda(x)\int_{\Omega}(u(y)-u(x))Q(dy, x)$ ,(2.1)

where . is the usual inner product in$R^{n},$$\nabla u$is the gradient vector of

$u$

and

$Q(\cdot, x)$a probability measure.

We consider the following obstacle problem.
(2.2)$\min\{Lu-f, u-\psi\}=0$(2.3)$u(x)= \int_{\Omega}u(y)Q(dy, x)$inon$\Omega$,$\partial\Omega$

We assume the following conditions.
(H.1)

is a bounded domain in$\Omega$(H.2) $g(x)$ :$\alpha_{0}>0$(H.4) $\lambda(x)>0$ for$x\in\Omega$$Q(\cdot, x)$(i)with smooth boundary

is Lipschitz continuous,$\Omegaarrow R^{n}$(H.3) There exists(H.5)$R^{n}$

such that:$\alpha(x),$ $\lambda(x)$$\alpha(x)\geq\alpha_{0}$for$x\in\overline{\Omega}$$\partial\Omega$.

$\overline{\Omega}arrow R$

are continuous.

.

.

satisfies:$Q(\cdot, x)$

is a probability measure on$\Omega$for$x\in\overline{\Omega}$$| \int_{\Omega}v(y)Q(dy, x)|\leq C||v||_{L^{1}(\Omega)}$such thatfor all$v\in L^{1}(\Omega)$.

(ii) The function$x arrow\int_{\Omega}v(y)Q(dy, x)$

is continuous with respect to(H.6) $g(x)\cdot\eta(x)>0$ for(H.7) $f,$$\psi$$x\in\partial\Omega$$x\in\overline{\Omega}$, where, uniformly on,$v\in L^{\infty}(\Omega)$.

is the outward unit normal at$\eta(x)$$x\in\partial\Omega$are continuous on St.

We denote that
$F(x, u,p, r)= \min\{-g(x)\cdot p+(\alpha(x)+\lambda(x))u-\lambda(x)r-f(x), u-\psi(x)\}$.

3

.

is

131
for

$x\in\Omega,$$u\in R,p\in R^{n},$ $r\in R$. Notice that if we fix$v\in L^{\infty}(\Omega)$, then the equationin$F(x,$ $u(x),$ $\nabla u(x),$ $\int_{\Omega}v(y)Q(dy, x))=0$$\Omega$

is an obstacle problem with a first order Hamiltonian.
We give some notation necessary to state the definition of viscosity solution. For
bounded functions, we set$u^{*}(x)= \lim_{rarrow 0}\sup\{u(y)||x-y|<r\}$upper semi-continuous envelope of$u_{*}(x)= \lim_{rarrow 0}\inf\{u(y)||x-y|<r\}$lower semi-continuous envelope of .

$u$

and
$u$

Now we state the definition of viscosity solutions.

Definition. Let
(i)

$u$

$u$

be a bounded measurable function.

is a viscosity subsolution of (2.2) if$F(x,$ $u^{*}(x),$ $\nabla\phi(x),$ $\int_{\Omega}u^{*}(y)Q(dy, x))\leq 0$

wherever $u^{*}-\phi$ attains its maximum for
(ii)

$u$

$\phi\in C^{1}(\Omega)$.

is a viscosity supersolution of (2.2) if$F(x,$ $u_{*}(x),$ $\nabla\phi(x),$ $\int_{\Omega}u_{*}(y)Q(dy, x\rangle$$)\geq 0$

wherever $u_{*}-\phi$ attains its minimum for
(iii)

$u$

is a viscosity solution if

In the following,($u$$\phi\in C^{1}(\Omega)$.

is a viscosity sub- and supersolution.

$(sub/super)$ solution” means “viscosity (sub/super) solution”.

Assume that there exists a supersolution $W$ of (2.2) such that
(24)

$W(x) \geq\int_{\Omega}W(y)Q(dy, x)$4on$\partial\Omega$.

132
Define$S=\{v|v$ is a subsolution of (2.2) such that$v\leq W$

inand$\Omega$on$v(x) \leq\int_{\Omega}v(y)Q(dy, x)$$\partial\Omega$}.

We put$u_{0}(x)= \sup\{v(x)|v\in S\}$.

Perron’s method consists of the following two propositions:

Proposition 2.1. Assume that
Proposition 2.2. Assume$S$$S\neq\emptyset$

is not empty, then $u_{0}\in S$ .
.

If $v\in S$ is not a supersolution, then there

exists $w\in S$ such that $v(y)<w(y)$ at some$y\in\Omega$.

These two Propositions can be proved by the same idea of Ishii [6]. So we omit

the proofs. See [11] for the detail.
Note that$u_{0}$

is a viscosity solution of (2.2).

3. Main existence result

First we assume that there exists a supersolution $W$ of (2.2) satisfying (2.4).
By Perron’s method, there exists a solution$u_{0}$. Note that$u_{0}$

satisfies the boundary

inequality$u_{0}(x) \leq\int_{\Omega}u_{0}(y)Q(dy, x)$on$\partial\Omega$.

Theorem 3.1. Assume (H. ) $-(H.7)$ . Suppose that there exists a supersolution $W$$1$

of (2.2) satisfying (2.4), and a solution(3.1)$F(x,$$u_{1},$$\nabla u_{1},$$u_{1}$

of$J_{\Omega}^{-}u_{0}(y)Q(dy, x))=0$

5

in$\Omega$

133
satisfying the Dirichlet boundary condition
(3.2)

If

on

$u_{1}(x)= \int_{\Omega}u_{0}(y)Q(dy, x)$$u_{1}\leq W$, then$u_{0}$$\partial\Omega$.

is a solution of (2.2) satisfying the boundary condition (2.3).

Proof. We calim$u_{1}\in S$. Let$\phi\in C^{1}$such that$u_{1}^{*}-\phi$attains its maximum at$y_{0)}$then$F(y_{0},$ $u_{1}^{*}(y_{0}),$ $\nabla\phi(y_{0}),$ $\int_{\Omega}u_{0}(y)Q(dy, y_{0}))\leq 0$.

Note that the comparison principle for two viscosity solutions holds for the equa-tion of a first order Hamiltonian
we have$u_{0}\leq u_{1}$in$\Omega$. Using$F$($x,$ $u$$u_{0}\leq u_{1}$, Vu,$u_{0}$). Since$u_{0}$is also a subsolution of (3.1),

and the monotonicity of$F$with respect to theargument , we have$u$$F(y_{0},$ $u_{1}^{*}(y_{0}),$ $\nabla\phi(y_{0}),$ $\int_{\Omega}u_{1}(y)Q(dy, y_{0}))\leq 0$.

Also we have$u_{1}(x)= \int_{\Omega}u_{0}(y)Q(dy, x)\leq\int_{\Omega}u_{1}(y)Q(dy, x)$

Hence, we have the claim. By the definition of$u_{0}$

and$u_{0}\leq u_{1}$on$\partial\Omega$.

, we have$u_{0}\equiv u_{1}$in$\overline{\Omega}$.

This completes the proof.
To assure the assumptions of Theorem 3.1, we consider the equation

(3.3)

(3.4)

$Lu(x)=f(x)$$u(x)= \int_{\Omega}u(y)Q(dy, x)$

Theorem 3.2. Assume $(H.1)-(H.7)$

in$\Omega$on$\partial\Omega$.

Then there exists a unique solution of the,equation (3.3) satisfying the boundary condrtion (3.4).

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Proof. First we note thatis a subsolution,$w(x)=- \frac{||f||_{\infty}}{\alpha_{0}}$and$W(x)= \frac{||f||_{\infty}}{\alpha_{0}}$is a supersolution.

of (3.3) satisfying (3.4).

Applying Perron’s method, we have that there exists a solution$u_{0}$of (3.3) satis-fying the boundary inequality$u_{0}(x) \leq\oint_{\Omega}u_{0}(y)Q(dy, x)$on$\partial\Omega$.

Next we consider the equation(3.5)$-g \cdot\nabla u_{1}+(\alpha+\lambda)u_{1}-\lambda\int_{\Omega}u_{0}(y)Q(dy, x)=f$in$\Omega$with the Dirichlet boundary condition(3.6)$u_{1}(x)= \int_{\Omega}u_{0}(y)Q(dy, x)$on$\partial\Omega$.

The comparison principle for this equation is well known $[2,3]$ . By (H.6) and themethod of [12], we can prove the existence of sub- and supersolutions. Then there

exists a continuous solution$u_{1}$of the equation (3.5) with (3.6). We can apply the same

argument in the proof of Theorem 3.1 to yield that$u_{1}\equiv u_{0}$. The uniqueness follows

from Lenhart [8]. The proof is complete.
Now we can prove the main result.

Theorem 3.3. Assume (H. ) $-(H.7)$ . Then there exists a unique solution of the$1$

obstacle problem (2.2) satisfying the boundary condition (2.3).

Proof. It is sufficient to check thehypothesis of Theorem 3.1. To do so, we consider
the obstacle problem (3.1) with (3.2).

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135
Using the boundary inequality of$u_{0}$

andin$u_{0}\geq\psi$$\Omega$on$\psi(x)\leq\int_{\Omega}u_{0}(y)Q(dy, x)$, the compatibility condition$\partial\Omega$

is satisfied.

First assume(3.7)$h(x)= \int_{\Omega}u_{0}(y)Q(dy, x)\in C^{1}(\Omega)\cap C(\overline{\Omega})$

and
(3.8)

on

$h(x)= \int_{\Omega}u_{0}(y)Q(dy, x)>\psi(x)$$\partial\Omega$

In this case, problem (3.1) with (3.2) is equivalent to
(3.9)

in

$\min\{-g\cdot.\nabla w_{1}+(\alpha+\lambda)w_{1}-f, w_{1}-\psi\}=0$(3.10)

where $f,$on$w_{1}(x)=0$$\psi$

satisfy the same properties as $f,$$\partial\Omega$in (3.1) and $\psi<0$ on$\psi$$\Omega$$\partial\Omega$. We show theexistence of a solution to (3.9) with (3.10) by Perron’s method. lndeed, the solutionof the linear equation

in$-g\cdot\nabla w+(\alpha+\lambda)w=f$

on

$w=0$

$\Omega$,$\partial\Omega$

is a subsolution of (3.9) with (3.10).
To construct a supersolution, we follow a barrier construction argument from

Oleinik and Radkevic [12] as in Ishii and Koike [7]. Since $\psi<0$ on
a local barrier,$\psi_{z}$neighborhood of

$z$

in

where

$C(\Omega\cap V_{z})\cap C^{2}(\Omega\cap V_{z})$$z\in\partial\Omega,$$V$satisfying$\psi_{z}(z)=0$,$\psi_{z}\geq 0$$\psi_{z}\geq||f||_{\infty}/\alpha_{0}$on$-g\cdot\nabla\psi_{z}+(\alpha+\lambda)\psi_{z}\geq f$$\psi_{z}\geq\psi$

in

8

on
$\overline{\Omega}$$\overline{\Omega\cap V_{z}}$口in$\Omega\cap V_{z}$$\partial V_{z}$,$\Omega\cap V_{z}$

.

,

, and

$\partial\Omega$, there exists

is a sufficiently small

136
Define,バ$z(z)=\{\begin{array}{l}\max\{\psi_{z}(x),\max\{||f||_{\infty}/\alpha_{0},||\psi||_{\infty}\}\}\max\{||f||_{\infty}/\alpha_{0},||\psi||_{\infty}\}\end{array}$$otherwisein\Omega\cap V_{z}$and$\hat{\psi}(x)=\inf\{\hat{\psi}_{z}(x)|z\in\partial\Omega\}$Then$\hat{\psi}$.

is a supersolution. This implies that there exists a continuous solution of (3.1)

with (3.2).
For general continuous boundary value$h$, which is not necessarily satisfy (3.7)

and (3.8), we choose an approximating sequence$h_{n}>\psi$on$\partial\Omega$

and $h_{n}arrow h$ uniformly in St. Let

associated with boundary value$h_{n}$$\{h_{n}\}$$u_{n}$

such that $h_{n}\in C(\Omega)\cap C^{1}(\Omega)$ ,be a solution of (3.1) with (3.2). By standard comparison argument, we have$\sup_{\Omega}|u_{n}(x)-u_{m}(x)|\leq\sup_{\partial\Omega}|h_{n}(x)-h_{m}(x)|$converges to someHence$\{u_{n}\}$that

is a solution of (3.1) with (3.2).

$u$$u\in C(\overline{\Omega})$.

and by stability of viscosity solutions, we have

By the comparison result for obstacle problems, we have $u_{1}\leq W$ .

Theorem 3.1,

$u_{0}$

Hence bysatisfies the boundary condition (3.2).

Since the uniqueness follows from the argument in Lenhart [10], the proof is completed.

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[2] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations,
Trans. A. M. S., 277 (1983), 1-42.
[3] M. G. Crandall, L. C. Evans and P. L. Lions, Some properties of viscosity solutionsof Hamilton-Jacobi equations, Trans. A. M. S., 282 (1984), 487-502.

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137[4] M. H. A. Davis, Piecewise-deterministic Markov procecees:A general class of nondiffusion models, J. Royal Stat., B46 (1984), 487-502.
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Stochastics, 19 (1986), 221-236.
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369-384.

[7] H. Ishii and S. Koike, Viscosity solutions of a system of nonlinear second-order
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[8] S. M. Lenhart, Viscosity solutions associated with switching control problems forpiecewisedeterministic processes, Huston M. J., 13 (1987), 405-426.
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183-207.
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[11] S. M. Lenhart and N. Yamada, Perron’s method for viscosity solutions associatedwith piecewise-deteministic processes, in preparation.
[12] A. O. Oleinik and E. V. Radkevic, Second order equations with nonnegative characterictic form, Amer. Math. Soc., Providence, hode Island and Plenum Press,
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$\perp^{\backslash }\tau_{\{}$

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