Viscosity Solutions of Nonlinear Second Order ... - Semantic Scholar
Funkcialaj Ekvacioj, 38 (1995) 297-328
Viscosity Solutions of Nonlinear Second Order Elliptic PDEs Associated with Impulse Control Problems II By
Katsuyuki ISHII (Kobe University of Mercantile Marine, Japan) Dedicated to Professor Sadakazu Aizawa on the occasion of his 60th birthday
§1.
Introduction
This paper is concerned with the uniqueness and existence of viscosity solutions of nonlinear second order elliptic partial differential equations (PDEs) with an implicit obstacle. , we define be a bounded domain. For any function Let $M$ as the following: the operator $¥Omega¥subset R^{N}$
$u:¥overline{¥Omega}¥rightarrow R$
$Mu(x)=x+¥inf_{¥xi¥geq 0}¥overline{¥xi}¥in¥overline{¥Omega}¥{k(¥xi)+¥mathrm{u}(¥mathrm{x}+¥xi)¥}$
and is a nonnegative and continuous function on . We consider the nonlinear elliptic PDEs of the form:
where
$k(¥xi)$
means
$¥xi¥in(R^{+})^{N}$
(1.1)
, $(R^{+})^{N}$
$¥left¥{¥begin{array}{l}¥max¥{F(x,u,Du,D^{2}u),u-Mu¥}=0¥¥¥max¥{u-g,u-Mu¥}=0¥end{array}¥right.$
$¥xi¥geqq 0$
, $¥mathrm{i}¥mathrm{o}¥mathrm{n}¥partial¥Omega ¥mathrm{n}¥Omega$
.
is a second order degenerate elliptic Here the is a given function and the operator. The problem (1.1) is associated with the impulse control problems for certain diffusion processes. For the formal derivation of (1.1) and some results on the impulse control problems, see A. Bensoussan-J. L. Lions [3], J. L. Menaldi [19], B. Perthame [20] and G. Barles [1] etc. In the case where is nondegenerate, we can interpret the boundary condition in (1.1) in the “classical” sense. In [3] the existence and uniqueness is discussed from the viewpoint of quasiof solutions of (1.1) in . B. Perthame [21] is linear and on variational inequality when . obtained the existence and uniqueness of solutions of (1.1) in After introducing the notion of viscosity solutions, B. Perthame [22] and the author [11] showed the uniqueness and existence of viscosity solutions of (1.1). is degenerate (especially on However, in the case ), we cannot interpret the boundary condition in the classical sense. Then H. Ishii [8] pointed out that in the degenerate case we should interpret the boundary condition $F$
$g$
$F$
$H_{0}^{1}(¥Omega)¥cap C(¥overline{¥Omega})$
$F$
$g$
$¥equiv 0$
$¥partial¥Omega$
$W_{loc}^{2,¥infty}(¥Omega)¥cap C(¥overline{¥Omega})$
$F$
$¥partial¥Omega$
298
Katsuyuki ISHII
in the “viscosity” sense and proved the comparison principle and existence of viscosity solutions of first order Hamilton-Jacobi equations by analytical methods. (Also see M. G. Crandall-H. Ishii-P. L. Lions [4] and references therein). In order to get the comparison principle, he assumed the continuity . Recently M. A. Katsoulakis of viscosity sub- and supersolutions near comparison principle of viscosity solutions the and have obtained [12] [13] of nonlinear second order degenerate elliptic PDEs. To show the comparison principle he has assumed the nontangential semicontinuity of viscosity sub- and supersolutions, which is a weaker assumption than that in [8]. Moreover in [12] and [13] he has established the existence of such solutions by probabilistic arguments. As to the systems of elliptic PDEs, see S. Koike [15] and M. A. Katsoulakis- S. Koike [14]. Our main purpose here is to get the comparison principle and existence of viscosity solutions of the problem (1.1). Since we deal with the case where is degenerate, we consider the boundary condition in the viscosity sense. This paper is organized in the following way. In Section 2 we give the definition of viscosity solutions of (1.1) and the equivalent propositions. In Section 3 we prove the comparison principle of viscosity solutions of (1.1). We remark that its proof is improved as compared with that of [11; Theorem 3.1]. Sections 4 and 5 provide the existence of continuous viscosity solutions of (1.1). Since it is difficult to discuss it for general elliptic operators, we consider only the case is the Hamilton-Jacobi-Bellman operator in these sections. In Section 4 we apply the iterative approximation scheme by B. HanouzetJ. L. Joly [7] to obtain the existence result, assuming the existence of continuous viscosity solutions of the usual obstacle problems. In Section 5 we show it by using the results in [13]. In Section 6 we prove that the unique solution of (1.1) obtained in Section 4 can be represented as the optimal cost function associated with the impulse control problem. In Section 7 we treat the boundary value problem of oblique type involving the operator $M$ . For the related problems, see P. L. Lions-B. Perthame [18], P. Dupuis-H. Ishii [5], [6] and H. Ishii [9]. In what follows we suppress the term “viscosity” since we are mainly concerned with viscosity sub-, super- and solutions. $¥partial¥Omega$
$F$
$F$
§2.
Definitions of solutions In this section we give the definitions of solutions of (1.1) and the equiva-
lent propositions. We begin by preparing some notations. $¥langle ¥cdot, .
¥rangle=$
the Euclidean inner product in
$R^{N}$
.
299
Second Order Elliptic PDEs $B(x, r)=$
the open ball of radius
$I$
Let
$¥mathcal{O}¥subset R^{N}$
.
$=$
centered at
for , $s>0$ and $r$
$K(r, s, n)=¥bigcup_{00$
with
$n¥in C(¥overline{¥Omega}:R^{N})$
$|n|=$
I
such that
$ z+K(r, s, n(z))¥subset¥Omega$
$y+K$
(
$r$
, ,
$¥frac{X}{|x|})¥subset¥Omega$
$t$
(A.3) There exists a mapping
(A.4)
and a mapping
for all
$ z¥in¥partial¥Omega$
for all
$x$
,
$¥in K(r, s, n(z))$
for all
$ P(x, ¥xi)=¥xi$
if
$P(¥cdot, ¥xi)¥in C(¥overline{¥Omega})$
for each
$y¥in B(z, r)¥cap¥overline{¥Omega}$
.
satisfying
$P:¥overline{¥Omega}¥times(R^{+})^{N}¥rightarrow(R^{+})^{N}$
$x+P(x, ¥xi)¥in¥overline{¥Omega}$
,
$(x, ¥xi)¥in¥overline{¥Omega}¥times(R^{+})^{N}$
$X+¥xi¥in¥overline{¥Omega}$
,
$¥xi¥geqq 0$
.
.
$F¥in C(¥overline{¥Omega}¥times R¥times R^{N}¥times ¥mathrm{S}^{N})$
(A.5) There exists a function $F(y, r, p, ¥mathrm{Y})$
if
for all
$x$
,
$-3a$
$y¥in¥overline{¥Omega}$
$¥omega_{1}¥in C(R^{+})$
such that
for which
$-F(x, r, p, X)¥leqq¥omega_{1}(a|x-y|^{2}+|x-y|(|p|+1))$
$¥left(¥begin{array}{ll}I & O¥¥O & I¥end{array}¥right)¥leqq¥left(¥begin{array}{ll}X & O¥¥O & -¥mathrm{Y}¥end{array}¥right)$
,
$¥omega_{1}(0)=0$
$p¥in R^{N}$
,
(A.6) There exists a function
$a$
$>1$
and
$¥omega_{2}¥in C(R^{+})$
$X$
$¥leqq 3a$
,
$¥left(¥begin{array}{ll}I & -I¥¥-I & I¥end{array}¥right)$
$¥mathrm{Y}¥in ¥mathrm{S}^{N}$
such that
. $¥omega_{2}(0)=0$
for which
$|F(x, r, p, X)$ $-F(x, r, q, X)|¥leqq¥omega_{2}(|p-q|)$
for all
$¥in¥overline{¥Omega}$ $¥mathrm{x}$
,
$r¥in R$
,
$p$
,
$q¥in R^{N}$
(A.7) There exists a constant
$¥lambda>0$
and
$X¥in ¥mathrm{S}^{N}$
.
such that
$F(x, r, p, X)$ $-F(x, s, p, X)¥leqq¥lambda(r-s)$
for all
(A.8)
$x¥in¥overline{¥Omega}$
$k¥in C((R^{+})^{N})$
all
$¥xi¥geqq 0$
.
, , $r$
$s¥in R$
,
$p¥in R^{N}$
,
$X¥in ¥mathrm{S}^{N}$
if
$r¥leqq s$
.
and there exists a constant
$k_{0}>0$
such that
$k(¥xi)¥geqq k_{0}$
for
Second Order Elliptic PDEs
(A.9)
$g¥in C(¥overline{¥Omega})$
303
.
Remark 3.1. (1) When is of class , we take $r=s=t>0$ sufficiently small and such that $n(x)$ is the inner normal to at . easily Then it is verified that (A.2) is satisified. The assumption is convex and regular, (2) (A.3) is not trivial. When we can take $P(x, ¥xi)$ as the projection of on . See A. Benssousan-J. L. Lions [3; Chapter 4, Remark 1.7] and J. L. Menaldi [19, Section 1]. is degenerate elliptic, (cf. [4; (3) If (A.6) holds, then the operator Remark 3.4].) is the Hamilton-Jacobi(4) A typical example of satisfying Bellman operator treated in Sections 4 and 5. $C^{2}$
$¥partial¥Omega$
$n¥in C(¥overline{¥Omega}:R^{N})$
$ x¥in¥partial¥Omega$
$¥Omega$
$¥Omega$
$(R^{+})^{N}¥cap¥overline{¥Omega}-¥{x¥}$
$¥xi$
$F$
$F$
$(¥mathrm{A}.4)-(¥mathrm{A}.7)$
We recall the properties of the operator
$M$ .
Proposition 3.2. Assume (A. ), (A.3) and (A.8) hold. Then we have the following properties. (1) If $u¥leqq v$ on , then $Mu$ $¥leqq Mv$ on . (2) $M(tu+(1-t)v)¥geqq tMu+(1-t)Mv$ for all $t¥in[0,1]$ . (3) $M(u+c)=Mu+c$ for all $c¥in R$ . , then $Mu$ . (4) If $Mu$ then , . (5) If . (6) for all ,
Let
$¥mathrm{I}$
$¥overline{¥Omega}$
.
$¥in USC(¥overline{¥Omega})$
$||Mu-Mv||_{C(¥overline{¥Omega})}¥leqq||u-v||_{C(¥overline{¥Omega})}$
Proof.
$v:¥overline{¥Omega}¥rightarrow R$
$¥in LSC(¥overline{¥Omega})$
$u¥in USC(¥overline{¥Omega})$
$M$
,
$¥overline{¥Omega}$
$u¥in LSC(¥overline{¥Omega})$
of
$u$
$u$
$v¥in C(¥overline{¥Omega})$
We only show (4) and (5) because it is obvious by the definition and (6) hold.
that
$(1)-(3)$
(4) We take $u¥in LSC(¥overline{¥Omega})$
such that implies that for each there exists a ,
$¥{x_{n}¥}_{n¥in N}¥subset¥overline{¥Omega}$
$x¥in¥overline{¥Omega}$
$x_{n}$
$¥mathrm{x}_{n}+¥xi_{n}¥in¥overline{¥Omega}$
,
. The condition such that
$x_{n}¥rightarrow ¥mathrm{x}(n ¥rightarrow+¥infty)$
$¥xi_{n}¥geqq 0$
$Mu(¥mathrm{x}_{n})=k(¥xi_{n})+u(x_{n}+¥xi_{n})$
.
Since is bounded, by taking a subsequence, if necessary, we may consider that such that . Hence we have $¥{¥xi_{n}¥}_{n¥in N}$
$x+¥xi¥in¥overline{¥Omega}$
$¥lim_{n¥rightarrow+¥infty}¥xi_{n}=¥xi¥geqq 0$
$¥lim_{n¥rightarrow}¥inf_{+¥infty}Mu(x_{n})¥geqq¥lim_{n¥rightarrow¥dagger¥infty}k(¥xi_{n})+¥lim_{n¥rightarrow}¥inf_{+¥infty}u(x_{n}+¥xi_{n})$
$¥geqq k(¥xi)+u(x+¥xi)$ $¥geqq Mu(x)$
that is,
$Mu$
$¥in LSC(¥overline{¥Omega})$
(5) We take such that
,
.
as in the proof of (4) and fix . Then we have by (A.3),
$¥{x_{n}¥}_{n¥in N}¥subset¥overline{¥Omega}$
$x+¥xi¥in¥overline{¥Omega}$
and
$x¥in¥overline{¥Omega}$
$¥xi¥geqq 0$
304
Katsuyuki ISHII $Mu(x_{n})¥leqq k(P(x_{n}, ¥xi))+u(¥mathrm{x}_{n}+P(x_{n}, ¥xi))$
.
Thus we get $¥lim_{n¥rightarrow}¥sup_{+¥infty}Mu(x_{n})¥leqq¥lim_{n¥rightarrow+¥infty}k(P(¥mathrm{x}_{n}, ¥xi))+¥lim_{n¥rightarrow}¥sup_{+¥infty}u(¥mathrm{x}_{n}+P(x_{n}, ¥xi))$
$¥leqq k(¥xi)+u(x+¥xi)$
Taking the infimum with respect to is proved.
.
$¥xi¥geqq 0$
satisfying
$¥mathrm{x}$
$+¥xi¥in¥overline{¥Omega}$
,
$Mu$
$¥in USC(¥overline{¥Omega})$
$¥blacksquare$
The comparison principle of solutions of (1.1) is stated as follows. hold. Let and , respectively, be a subTheorem 3.3. Assume (A. ) , let $K_{z}=z+K(r, s, n(z))$ . For each supersolution a solution and of (1.1). on . If any one of the following holds, then . and (1) for each $u^{*}(z) ¥ leqq g(z)$ . and (2) for each $v_{*}(z)¥geqq g(z)$ . each and (3) for $1$
$-(A.¥mathit{9})$
$u$
$v$
$ z¥in¥partial¥Omega$
$¥overline{¥Omega}$
$u^{*}¥leqq v_{*}$
$ z¥in¥partial¥Omega$
$¥lim¥inf_{K_{¥mathrm{z}}¥ni x¥rightarrow z}v_{*}(x)=v_{*}(z)$
$¥lim¥sup_{K_{¥mathrm{z}}¥ni x¥rightarrow z}u^{*}(¥mathrm{x})=u^{*}(z)$
$ z¥in¥partial¥Omega$
$¥lim¥sup_{K_{¥mathrm{z}}¥ni x¥rightarrow z}u^{*}(¥mathrm{x})=u^{*}(z)$
$ z¥in¥partial¥Omega$
$¥lim¥inf_{K_{¥mathrm{z}}xz}¥ni¥rightarrow v_{*}(¥mathrm{x})=v_{*}(z)$
Remark 3.4. We call the properties in Theorem 3.3 (1) nontangential upper- and lower semicomtinuity, respectively. See [12], [13] and [14]. We need the following lemma to deal with the term $u-Mu$ . be compact and $u¥in USC(¥mathcal{O})$ . Then, Lemma 3.5. Let $q¥in R^{N}$ , the function $u(x)+¥langle q$, x) takes its strict maximum on . $¥mathcal{O}¥subset R^{N}$
for
$a.a$
.
$¥mathcal{O}$
$ $
For the proof, see. H. Ishii-S. Koike [10; Lemma 3.3.].
. We and Theorem 3.3. We may assume easily observe that $u¥leqq Mu$ on . First let the condition (1) hold. We suppose $¥sup_{x¥in¥overline{¥Omega}}(u-v)=5¥theta>0$ and shall get a contradiction. Let . We take $q¥geqq 0$ be the standard basis for and let such that
Proof of
$u¥in USC(¥overline{¥Omega})$
$v¥in LSC(¥overline{¥Omega})$
$¥overline{¥Omega}$
$L=¥sup_{x¥in¥overline{¥Omega}}|x|$
(3.1)
$R^{N}$
$¥{e_{i}¥}_{1¥leqq i¥leqq N}$
$00$
,
$ 0¥leqq¥omega_{2}(|q|)¥leqq¥lambda¥theta$
for each
$i=1$ ,
and fix it. Then by Lemma 3.5 the function . We easily see strict maximum at
$¥ldots$
,
,
$N$
$ u(x)-v(x)+¥langle q, ¥mathrm{x}¥rangle$
$z(=z_{q})¥in¥overline{¥Omega}$
(3.3)
$ u(z)-v(z)+¥langle q, z¥rangle¥geqq 4¥theta$
,
We claim
(3.4)
$v(z)<Mv(z)$ .
$u(z)>v(z)$
.
attains its
305
Second Order Elliptic PDEs
To prove this, suppose $v(z)¥geqq Mv(z)$ . Since satisfying of $M$ and (A.8), we can find $k(¥xi_{z})+v(z+¥xi_{z})$ . Thus $u(z)¥leqq Mu(z)$ and $v(z)¥geqq
, using the definition and $Mv(z)=$ imply
$v¥in LSC(¥overline{¥Omega})$
$¥xi_{z}¥neq 0$
$¥xi_{z}¥geqq 0$
,
$Z+¥xi_{z}¥in¥overline{¥Omega}$
Mv(z)$
$ u(z)-v(z)+¥langle q, z¥rangle¥leqq u(z+¥xi_{z})-v(z+¥xi_{z})+¥langle q, z+¥xi_{z}¥rangle-¥langle q, ¥xi_{Z}¥rangle$
Then we obtain a contradiction because get the claim (3.4). We divide our consideration into three cases. Case 1. Let
$ z¥in¥partial¥Omega$
and
$v(z)0$ satisfies be a maximum point of Let we get
$¥omega_{1}(s_{0}^{2})0$
.
$¥Phi$
.
Since
.
,
,
$¥Phi(z_{n}, z)¥leqq¥Phi(x_{n}, y_{n})$
$u(z_{n})-v(z)+¥langle q, z¥rangle¥leqq u(z_{n})-v(z)+¥langle q, z¥rangle+¥frac{a_{n}}{2}|X_{n}-y_{n}-z_{n}+z|^{2}$
$¥leqq u(x_{n})-v(y_{n})+¥langle q, y_{n}¥rangle$
.
because , The function $u(x)-v(y)$ is bounded above on . Hence (3.5) implies $|x_{n}-y_{n}-z_{n}+z|¥rightarrow 0$ as is compact in and . Then there as . Moreover we easily observe . as , point such that sequence a and a exist that It follows from this, (3.5) and the semicontinuity of and $¥overline{¥Omega}¥times¥overline{¥Omega}$
$u$
$-v¥in USC(¥overline{¥Omega})$
$R^{2N}$
$¥overline{¥Omega}¥times¥overline{¥Omega}$
$|x_{n}-y_{n}|¥rightarrow 0$
$ n¥rightarrow+¥infty$
$ n¥rightarrow+¥infty$
$¥overline{z}¥in¥overline{¥Omega}$
$¥{n_{k}¥}¥subset N$
$X_{n_{k}}$
$u$
$y_{n_{k}}¥rightarrow¥overline{Z}$
$ k¥rightarrow+¥infty$
$v$
$ u(z)-v(z)+¥langle q, z¥rangle¥leqq u(¥overline{z})-v(¥overline{z})+¥langle q,¥overline{z}¥rangle$
is a unique maximum point of the function Since and , it follows from this inequality that
$ u(x)-v(x)+¥langle q, x¥rangle$
$z$
$¥overline{z}=z$
$¥overline{¥Omega}$
(3.6)
$x_{n}$
,
$y_{n}¥rightarrow Z$
$(n ¥rightarrow+¥infty)$
.
Thus, by (3.5) we get $¥lim_{n¥rightarrow¥infty}(u(x_{n})-v(y_{n}))=u(z)-v(z)$
Using (3.5), this equality and the semicontinuity of
$u$
.
and , we have $v$
on
306
Katsuyuki ISHII
(3.7)
$u(x_{n})¥rightarrow u(z)$
,
We may consider
$v(y_{n})¥rightarrow v(z)$
,
$a_{n}|x_{n}-y_{n}-z_{n}+z|^{2}¥rightarrow 0$
$(n ¥rightarrow+¥infty)$
.
for sufficiently large $n¥in N$ because (3.7) implies $|x_{n}-y_{n}-z_{n}+z|0$
,
.
characteristic function for
$¥tilde{g}=¥min¥{g, ¥psi¥}$
,
$A$
.
We consider the penalized problem for (2.1).
,
$+n(u_{n}-¥psi)^{+}=0$ $¥{_{u_{n}}F(¥mathrm{x}=’ u_{n}¥tilde{g}, Du_{n}, D^{2}u_{n})$
$¥mathrm{i}¥mathrm{o}¥mathrm{n}¥partial¥Omega ¥mathrm{n}¥Omega$
’
where $n¥in N$ and $r^{+}=¥max¥{r, Noting $ r^{+}=¥sup$ to the following :
0¥}$ .
$¥{¥gamma r|0¥leqq¥gamma¥leqq 1¥}$
, it is easily seen that (5.1) is equivalent
$¥mathrm{P}¥mathrm{D}¥mathrm{E}$
(5.2)
,
$¥left¥{¥begin{array}{l}¥sup_{a¥in¥Lambda}¥gamma¥in[0,1]¥{-+(c(x,a)+n¥gamma)u_{n}-f(x,¥alpha)-n¥gamma¥psi¥}=0tr(^{t}¥sigma(x,¥alpha)¥sigma(x,a)Du_{n})+¥langle b(x,a),Du_{n}¥rangle¥¥ u_{n}=¥tilde{g}¥end{array}¥right.$
$¥mathrm{i}¥mathrm{o}¥mathrm{n}¥partial¥Omega ¥mathrm{n}¥Omega$
.
Then applying the results in M. A. Katsoulakis [13], for each exists a unique solution of (5.2). Next we consider the following problem: $u_{n}¥in C(¥overline{¥Omega})$
(5.3)
$¥left¥{¥begin{array}{l}F(¥mathrm{x},v_{n},Dv_{n},D^{2}v_{n})+n(u_{n}-¥psi)^{+}=0¥¥v_{n}=¥tilde{g}¥end{array}¥right.$
, $¥mathrm{i}¥mathrm{o}¥mathrm{n}¥partial¥Omega ¥mathrm{n}¥Omega$
’
, there
$n¥in N$
312
Katsuyuki ISHII
where is the function obtained above. Using the results in [13] again, for $n ¥ in N$ each , there exists a unique solution of (5.3) and it is characterized as follows: $u_{n}$
$v_{n}¥in C(¥overline{¥Omega})$
$v_{n}(x)=¥inf_{a¥in d}E_{x}¥{¥int_{0}^{¥tau}(f(X_{t}, ¥alpha_{t})-n(u_{n}(X_{t})-¥psi(X_{t}))^{+})¥cdot¥exp(-¥int_{¥mathrm{o}}^{t}c(X_{s}, ¥alpha_{s})ds)dt$
$+¥tilde{g}(X_{¥tau})¥exp(-¥int_{0}^{¥tau}c(X_{s}, a_{s})ds)¥}$
.
Since (5.1) and (5.2) are equivalent to each other and the uniqueness of solutions of (5.1) holds in the class , we get $C(¥overline{¥Omega})$
(5.4)
$u_{n}(x)=¥inf_{a¥in d}E_{X}¥{¥int_{0}^{¥tau}(f(X_{t}, ¥alpha_{t})-n(u_{n}(X_{t})-¥psi(X_{t}))^{+})$
. $¥exp(-¥int_{0}^{t}c(X_{s}, ¥alpha_{s})ds)dt+¥tilde{g}(X_{¥tau})¥exp(-¥int_{¥mathrm{o}}^{¥tau}c(X_{s}, a_{s})ds)¥}$
.
Using (C.4) and the barrier argument, we have
(5.5)
on
$u_{n}¥leqq¥tilde{g}$
$¥partial¥Omega$
for all
$n¥in N$
Since the operator is monotone with respect to large $C>0$ , we obtain $nr^{+}$
(5.6)
. $n¥in N$
on
$-C¥leqq¥cdots¥leqq u_{n}¥leqq¥cdots¥leqq u_{2}¥leqq u_{1}$
and
$u_{n}¥geqq-C$
for
$¥overline{¥Omega}$
by the comparison principle of solutions of (5.1). (cf. M. G. Crandall-H. Ishii-P. L. Lions [4; Theorem 7.9].) Hence we can define the function by $u$
(5.7)
$u(x)=¥lim_{n¥rightarrow+¥infty}u_{n}(x)(=¥lim_{y}n¥rightarrow+¥infty¥sup_{¥rightarrow x}u_{n}(y))$
.
Then we get the following lemma.
Lemma 5.1.
The above
Proof. Since
function
$u$
is a u.s.c. subsolution
the sequence is decreasing by (5.6), we easily observe . Using (5.5) and letting , we have on . any For , we assume that attains a local maximum at . We may consider and that is a strict local maximum point . Then there exists $¥delta>0$ such that $¥{u_{n}¥}_{n¥in N}$
$u¥in USC(¥overline{¥Omega})$
$ n¥rightarrow+¥infty$
$¥varphi¥in C^{2}(¥overline{¥Omega})$
$x_{0}¥in¥overline{¥Omega}$
of
of (2.1). $¥partial¥Omega$
$u¥leqq¥tilde{g}$
$ u-¥varphi$
$ x_{0}¥in¥Omega$
$x_{0}$
$ u-¥varphi$
(5.8)
$u(¥mathrm{x}_{0})-¥varphi(x_{0})>u(x)-¥varphi(x)$
for all
be a maximum point of on sequence such that Let
$ u_{n}-¥varphi$
$x_{n}$
$¥{x_{n_{k}}¥}_{k¥in N}¥subset¥{x_{n}¥}_{n¥in N}$
$x¥in¥overline{B(x_{0},¥delta)}(¥subset¥Omega)$
$¥overline{B(¥chi_{0},¥delta)}$
.
,
$¥mathrm{x}¥neq x_{0}$
.
Then there exists a sub-
313
Second Order Elliptic PDEs $x_{n_{k}}¥rightarrow¥overline{X}¥in¥overline{B(¥chi_{¥dot{0}},¥delta)}$
,
$(k ¥rightarrow+¥infty)$
$u_{n_{k}}(x_{n_{k}})¥rightarrow¥beta¥in R$
.
Since for all
$u_{n_{k}}(¥mathrm{x})-¥varphi(¥mathrm{x})¥leqq u_{n_{k}}(X_{n_{k}})-¥varphi(x_{n_{k}})$
,
$¥in B(x_{0}, ¥delta)$
$x$
we get $u(¥mathrm{x}_{0})-¥varphi(x_{0})¥leqq¥lim_{x¥rightarrow}k¥rightarrow+¥infty¥sup_{x_{¥mathrm{O}}}(u_{n_{k}}(¥mathrm{x})-¥varphi(x))$
$¥leqq¥lim_{k¥rightarrow}¥sup_{+¥infty}(u_{n_{k}}(x_{n_{k}})-¥varphi(x_{n_{k}}))$
$=¥beta-¥varphi(¥overline{x})$
$¥leqq¥lim_{x¥rightarrow}k¥rightarrow+¥infty¥sup_{¥overline{X}}(u_{n_{k}}(¥mathrm{x})-¥varphi(x))$
$=u(¥overline{x})-¥varphi(¥overline{x})$
.
Therefore using (5.8) and the above inequality, we obtain
(5.9)
$x_{n}¥rightarrow x_{0}$
,
$u_{n}(x_{n})¥rightarrow u(x_{0})$
(cf. G. Barles-B. Perthame [2; Lemma A.3].) Since (5.1), we get (5. 10)
.
$(n ¥rightarrow+¥infty)$
is a subsolution of
$u_{n}$
$F(x_{n}, u_{n}(x_{n}), D¥varphi(x_{n}), D^{2}¥varphi(x_{n}))+n(u_{n}(x_{n})-¥psi(x_{n}))^{+}¥leqq 0$
.
It follows from (C.I) and (5.9) that there exists a constant $C>0$ such that
for all
$n(u_{n}(x_{n})-¥psi(x_{n}))^{+}¥leqq C$
Thus passing to the limit as
$ n¥rightarrow+¥infty$
.
, we have
$¥mathcal{U}(¥chi_{0})-¥psi(x_{0})¥leqq 0$
Moreover, (5.10) implies
$n¥in N$
.
$F(x_{n}, u_{n}(¥mathrm{x}_{n}), D¥varphi(x_{n}), D^{2}¥varphi(x_{n}))¥leqq 0$
.
Sending
$ n¥rightarrow+¥infty$
,
we obtain
$F(x_{0}, u(x_{0}), D¥varphi(x_{0}), D^{2}¥varphi(x_{0}))¥leqq 0$
Therefore we have completed the proof.
.
$¥blacksquare$
Remark 5.2. We notice that we cannot apply the results for the limit operations in [4; Section 5] to (2.1) and (5.1) directly since the term $n(r-¥psi(x))^{+}$ does not converge to 0 locally uniformly on as . $¥overline{¥Omega}¥times R$
We return to the formula (5.4). we get the following lemma.
$ n¥rightarrow+¥infty$
According to N. V. Krylov [16; p. 37],
314
Katsuyuki ISHII
Lemma 5.3.
(5. 11)
formula (5.4) can be
The
rewritten as
follows.
$u_{n}(x)=a¥in d¥inf_{¥theta¥in ¥mathit{9}}E_{x}¥{¥int_{¥mathrm{o}}^{¥tau¥wedge¥theta}f(X_{t}, a_{t})¥exp(-¥int_{0}^{t}c(X_{s}, ¥alpha_{s})ds)dt$
$+1_{¥theta
Viscosity Solutions of Nonlinear Second Order Elliptic PDEs Associated with Impulse Control Problems II By
Katsuyuki ISHII (Kobe University of Mercantile Marine, Japan) Dedicated to Professor Sadakazu Aizawa on the occasion of his 60th birthday
§1.
Introduction
This paper is concerned with the uniqueness and existence of viscosity solutions of nonlinear second order elliptic partial differential equations (PDEs) with an implicit obstacle. , we define be a bounded domain. For any function Let $M$ as the following: the operator $¥Omega¥subset R^{N}$
$u:¥overline{¥Omega}¥rightarrow R$
$Mu(x)=x+¥inf_{¥xi¥geq 0}¥overline{¥xi}¥in¥overline{¥Omega}¥{k(¥xi)+¥mathrm{u}(¥mathrm{x}+¥xi)¥}$
and is a nonnegative and continuous function on . We consider the nonlinear elliptic PDEs of the form:
where
$k(¥xi)$
means
$¥xi¥in(R^{+})^{N}$
(1.1)
, $(R^{+})^{N}$
$¥left¥{¥begin{array}{l}¥max¥{F(x,u,Du,D^{2}u),u-Mu¥}=0¥¥¥max¥{u-g,u-Mu¥}=0¥end{array}¥right.$
$¥xi¥geqq 0$
, $¥mathrm{i}¥mathrm{o}¥mathrm{n}¥partial¥Omega ¥mathrm{n}¥Omega$
.
is a second order degenerate elliptic Here the is a given function and the operator. The problem (1.1) is associated with the impulse control problems for certain diffusion processes. For the formal derivation of (1.1) and some results on the impulse control problems, see A. Bensoussan-J. L. Lions [3], J. L. Menaldi [19], B. Perthame [20] and G. Barles [1] etc. In the case where is nondegenerate, we can interpret the boundary condition in (1.1) in the “classical” sense. In [3] the existence and uniqueness is discussed from the viewpoint of quasiof solutions of (1.1) in . B. Perthame [21] is linear and on variational inequality when . obtained the existence and uniqueness of solutions of (1.1) in After introducing the notion of viscosity solutions, B. Perthame [22] and the author [11] showed the uniqueness and existence of viscosity solutions of (1.1). is degenerate (especially on However, in the case ), we cannot interpret the boundary condition in the classical sense. Then H. Ishii [8] pointed out that in the degenerate case we should interpret the boundary condition $F$
$g$
$F$
$H_{0}^{1}(¥Omega)¥cap C(¥overline{¥Omega})$
$F$
$g$
$¥equiv 0$
$¥partial¥Omega$
$W_{loc}^{2,¥infty}(¥Omega)¥cap C(¥overline{¥Omega})$
$F$
$¥partial¥Omega$
298
Katsuyuki ISHII
in the “viscosity” sense and proved the comparison principle and existence of viscosity solutions of first order Hamilton-Jacobi equations by analytical methods. (Also see M. G. Crandall-H. Ishii-P. L. Lions [4] and references therein). In order to get the comparison principle, he assumed the continuity . Recently M. A. Katsoulakis of viscosity sub- and supersolutions near comparison principle of viscosity solutions the and have obtained [12] [13] of nonlinear second order degenerate elliptic PDEs. To show the comparison principle he has assumed the nontangential semicontinuity of viscosity sub- and supersolutions, which is a weaker assumption than that in [8]. Moreover in [12] and [13] he has established the existence of such solutions by probabilistic arguments. As to the systems of elliptic PDEs, see S. Koike [15] and M. A. Katsoulakis- S. Koike [14]. Our main purpose here is to get the comparison principle and existence of viscosity solutions of the problem (1.1). Since we deal with the case where is degenerate, we consider the boundary condition in the viscosity sense. This paper is organized in the following way. In Section 2 we give the definition of viscosity solutions of (1.1) and the equivalent propositions. In Section 3 we prove the comparison principle of viscosity solutions of (1.1). We remark that its proof is improved as compared with that of [11; Theorem 3.1]. Sections 4 and 5 provide the existence of continuous viscosity solutions of (1.1). Since it is difficult to discuss it for general elliptic operators, we consider only the case is the Hamilton-Jacobi-Bellman operator in these sections. In Section 4 we apply the iterative approximation scheme by B. HanouzetJ. L. Joly [7] to obtain the existence result, assuming the existence of continuous viscosity solutions of the usual obstacle problems. In Section 5 we show it by using the results in [13]. In Section 6 we prove that the unique solution of (1.1) obtained in Section 4 can be represented as the optimal cost function associated with the impulse control problem. In Section 7 we treat the boundary value problem of oblique type involving the operator $M$ . For the related problems, see P. L. Lions-B. Perthame [18], P. Dupuis-H. Ishii [5], [6] and H. Ishii [9]. In what follows we suppress the term “viscosity” since we are mainly concerned with viscosity sub-, super- and solutions. $¥partial¥Omega$
$F$
$F$
§2.
Definitions of solutions In this section we give the definitions of solutions of (1.1) and the equiva-
lent propositions. We begin by preparing some notations. $¥langle ¥cdot, .
¥rangle=$
the Euclidean inner product in
$R^{N}$
.
299
Second Order Elliptic PDEs $B(x, r)=$
the open ball of radius
$I$
Let
$¥mathcal{O}¥subset R^{N}$
.
$=$
centered at
for , $s>0$ and $r$
$K(r, s, n)=¥bigcup_{00$
with
$n¥in C(¥overline{¥Omega}:R^{N})$
$|n|=$
I
such that
$ z+K(r, s, n(z))¥subset¥Omega$
$y+K$
(
$r$
, ,
$¥frac{X}{|x|})¥subset¥Omega$
$t$
(A.3) There exists a mapping
(A.4)
and a mapping
for all
$ z¥in¥partial¥Omega$
for all
$x$
,
$¥in K(r, s, n(z))$
for all
$ P(x, ¥xi)=¥xi$
if
$P(¥cdot, ¥xi)¥in C(¥overline{¥Omega})$
for each
$y¥in B(z, r)¥cap¥overline{¥Omega}$
.
satisfying
$P:¥overline{¥Omega}¥times(R^{+})^{N}¥rightarrow(R^{+})^{N}$
$x+P(x, ¥xi)¥in¥overline{¥Omega}$
,
$(x, ¥xi)¥in¥overline{¥Omega}¥times(R^{+})^{N}$
$X+¥xi¥in¥overline{¥Omega}$
,
$¥xi¥geqq 0$
.
.
$F¥in C(¥overline{¥Omega}¥times R¥times R^{N}¥times ¥mathrm{S}^{N})$
(A.5) There exists a function $F(y, r, p, ¥mathrm{Y})$
if
for all
$x$
,
$-3a$
$y¥in¥overline{¥Omega}$
$¥omega_{1}¥in C(R^{+})$
such that
for which
$-F(x, r, p, X)¥leqq¥omega_{1}(a|x-y|^{2}+|x-y|(|p|+1))$
$¥left(¥begin{array}{ll}I & O¥¥O & I¥end{array}¥right)¥leqq¥left(¥begin{array}{ll}X & O¥¥O & -¥mathrm{Y}¥end{array}¥right)$
,
$¥omega_{1}(0)=0$
$p¥in R^{N}$
,
(A.6) There exists a function
$a$
$>1$
and
$¥omega_{2}¥in C(R^{+})$
$X$
$¥leqq 3a$
,
$¥left(¥begin{array}{ll}I & -I¥¥-I & I¥end{array}¥right)$
$¥mathrm{Y}¥in ¥mathrm{S}^{N}$
such that
. $¥omega_{2}(0)=0$
for which
$|F(x, r, p, X)$ $-F(x, r, q, X)|¥leqq¥omega_{2}(|p-q|)$
for all
$¥in¥overline{¥Omega}$ $¥mathrm{x}$
,
$r¥in R$
,
$p$
,
$q¥in R^{N}$
(A.7) There exists a constant
$¥lambda>0$
and
$X¥in ¥mathrm{S}^{N}$
.
such that
$F(x, r, p, X)$ $-F(x, s, p, X)¥leqq¥lambda(r-s)$
for all
(A.8)
$x¥in¥overline{¥Omega}$
$k¥in C((R^{+})^{N})$
all
$¥xi¥geqq 0$
.
, , $r$
$s¥in R$
,
$p¥in R^{N}$
,
$X¥in ¥mathrm{S}^{N}$
if
$r¥leqq s$
.
and there exists a constant
$k_{0}>0$
such that
$k(¥xi)¥geqq k_{0}$
for
Second Order Elliptic PDEs
(A.9)
$g¥in C(¥overline{¥Omega})$
303
.
Remark 3.1. (1) When is of class , we take $r=s=t>0$ sufficiently small and such that $n(x)$ is the inner normal to at . easily Then it is verified that (A.2) is satisified. The assumption is convex and regular, (2) (A.3) is not trivial. When we can take $P(x, ¥xi)$ as the projection of on . See A. Benssousan-J. L. Lions [3; Chapter 4, Remark 1.7] and J. L. Menaldi [19, Section 1]. is degenerate elliptic, (cf. [4; (3) If (A.6) holds, then the operator Remark 3.4].) is the Hamilton-Jacobi(4) A typical example of satisfying Bellman operator treated in Sections 4 and 5. $C^{2}$
$¥partial¥Omega$
$n¥in C(¥overline{¥Omega}:R^{N})$
$ x¥in¥partial¥Omega$
$¥Omega$
$¥Omega$
$(R^{+})^{N}¥cap¥overline{¥Omega}-¥{x¥}$
$¥xi$
$F$
$F$
$(¥mathrm{A}.4)-(¥mathrm{A}.7)$
We recall the properties of the operator
$M$ .
Proposition 3.2. Assume (A. ), (A.3) and (A.8) hold. Then we have the following properties. (1) If $u¥leqq v$ on , then $Mu$ $¥leqq Mv$ on . (2) $M(tu+(1-t)v)¥geqq tMu+(1-t)Mv$ for all $t¥in[0,1]$ . (3) $M(u+c)=Mu+c$ for all $c¥in R$ . , then $Mu$ . (4) If $Mu$ then , . (5) If . (6) for all ,
Let
$¥mathrm{I}$
$¥overline{¥Omega}$
.
$¥in USC(¥overline{¥Omega})$
$||Mu-Mv||_{C(¥overline{¥Omega})}¥leqq||u-v||_{C(¥overline{¥Omega})}$
Proof.
$v:¥overline{¥Omega}¥rightarrow R$
$¥in LSC(¥overline{¥Omega})$
$u¥in USC(¥overline{¥Omega})$
$M$
,
$¥overline{¥Omega}$
$u¥in LSC(¥overline{¥Omega})$
of
$u$
$u$
$v¥in C(¥overline{¥Omega})$
We only show (4) and (5) because it is obvious by the definition and (6) hold.
that
$(1)-(3)$
(4) We take $u¥in LSC(¥overline{¥Omega})$
such that implies that for each there exists a ,
$¥{x_{n}¥}_{n¥in N}¥subset¥overline{¥Omega}$
$x¥in¥overline{¥Omega}$
$x_{n}$
$¥mathrm{x}_{n}+¥xi_{n}¥in¥overline{¥Omega}$
,
. The condition such that
$x_{n}¥rightarrow ¥mathrm{x}(n ¥rightarrow+¥infty)$
$¥xi_{n}¥geqq 0$
$Mu(¥mathrm{x}_{n})=k(¥xi_{n})+u(x_{n}+¥xi_{n})$
.
Since is bounded, by taking a subsequence, if necessary, we may consider that such that . Hence we have $¥{¥xi_{n}¥}_{n¥in N}$
$x+¥xi¥in¥overline{¥Omega}$
$¥lim_{n¥rightarrow+¥infty}¥xi_{n}=¥xi¥geqq 0$
$¥lim_{n¥rightarrow}¥inf_{+¥infty}Mu(x_{n})¥geqq¥lim_{n¥rightarrow¥dagger¥infty}k(¥xi_{n})+¥lim_{n¥rightarrow}¥inf_{+¥infty}u(x_{n}+¥xi_{n})$
$¥geqq k(¥xi)+u(x+¥xi)$ $¥geqq Mu(x)$
that is,
$Mu$
$¥in LSC(¥overline{¥Omega})$
(5) We take such that
,
.
as in the proof of (4) and fix . Then we have by (A.3),
$¥{x_{n}¥}_{n¥in N}¥subset¥overline{¥Omega}$
$x+¥xi¥in¥overline{¥Omega}$
and
$x¥in¥overline{¥Omega}$
$¥xi¥geqq 0$
304
Katsuyuki ISHII $Mu(x_{n})¥leqq k(P(x_{n}, ¥xi))+u(¥mathrm{x}_{n}+P(x_{n}, ¥xi))$
.
Thus we get $¥lim_{n¥rightarrow}¥sup_{+¥infty}Mu(x_{n})¥leqq¥lim_{n¥rightarrow+¥infty}k(P(¥mathrm{x}_{n}, ¥xi))+¥lim_{n¥rightarrow}¥sup_{+¥infty}u(¥mathrm{x}_{n}+P(x_{n}, ¥xi))$
$¥leqq k(¥xi)+u(x+¥xi)$
Taking the infimum with respect to is proved.
.
$¥xi¥geqq 0$
satisfying
$¥mathrm{x}$
$+¥xi¥in¥overline{¥Omega}$
,
$Mu$
$¥in USC(¥overline{¥Omega})$
$¥blacksquare$
The comparison principle of solutions of (1.1) is stated as follows. hold. Let and , respectively, be a subTheorem 3.3. Assume (A. ) , let $K_{z}=z+K(r, s, n(z))$ . For each supersolution a solution and of (1.1). on . If any one of the following holds, then . and (1) for each $u^{*}(z) ¥ leqq g(z)$ . and (2) for each $v_{*}(z)¥geqq g(z)$ . each and (3) for $1$
$-(A.¥mathit{9})$
$u$
$v$
$ z¥in¥partial¥Omega$
$¥overline{¥Omega}$
$u^{*}¥leqq v_{*}$
$ z¥in¥partial¥Omega$
$¥lim¥inf_{K_{¥mathrm{z}}¥ni x¥rightarrow z}v_{*}(x)=v_{*}(z)$
$¥lim¥sup_{K_{¥mathrm{z}}¥ni x¥rightarrow z}u^{*}(¥mathrm{x})=u^{*}(z)$
$ z¥in¥partial¥Omega$
$¥lim¥sup_{K_{¥mathrm{z}}¥ni x¥rightarrow z}u^{*}(¥mathrm{x})=u^{*}(z)$
$ z¥in¥partial¥Omega$
$¥lim¥inf_{K_{¥mathrm{z}}xz}¥ni¥rightarrow v_{*}(¥mathrm{x})=v_{*}(z)$
Remark 3.4. We call the properties in Theorem 3.3 (1) nontangential upper- and lower semicomtinuity, respectively. See [12], [13] and [14]. We need the following lemma to deal with the term $u-Mu$ . be compact and $u¥in USC(¥mathcal{O})$ . Then, Lemma 3.5. Let $q¥in R^{N}$ , the function $u(x)+¥langle q$, x) takes its strict maximum on . $¥mathcal{O}¥subset R^{N}$
for
$a.a$
.
$¥mathcal{O}$
$ $
For the proof, see. H. Ishii-S. Koike [10; Lemma 3.3.].
. We and Theorem 3.3. We may assume easily observe that $u¥leqq Mu$ on . First let the condition (1) hold. We suppose $¥sup_{x¥in¥overline{¥Omega}}(u-v)=5¥theta>0$ and shall get a contradiction. Let . We take $q¥geqq 0$ be the standard basis for and let such that
Proof of
$u¥in USC(¥overline{¥Omega})$
$v¥in LSC(¥overline{¥Omega})$
$¥overline{¥Omega}$
$L=¥sup_{x¥in¥overline{¥Omega}}|x|$
(3.1)
$R^{N}$
$¥{e_{i}¥}_{1¥leqq i¥leqq N}$
$00$
,
$ 0¥leqq¥omega_{2}(|q|)¥leqq¥lambda¥theta$
for each
$i=1$ ,
and fix it. Then by Lemma 3.5 the function . We easily see strict maximum at
$¥ldots$
,
,
$N$
$ u(x)-v(x)+¥langle q, ¥mathrm{x}¥rangle$
$z(=z_{q})¥in¥overline{¥Omega}$
(3.3)
$ u(z)-v(z)+¥langle q, z¥rangle¥geqq 4¥theta$
,
We claim
(3.4)
$v(z)<Mv(z)$ .
$u(z)>v(z)$
.
attains its
305
Second Order Elliptic PDEs
To prove this, suppose $v(z)¥geqq Mv(z)$ . Since satisfying of $M$ and (A.8), we can find $k(¥xi_{z})+v(z+¥xi_{z})$ . Thus $u(z)¥leqq Mu(z)$ and $v(z)¥geqq
, using the definition and $Mv(z)=$ imply
$v¥in LSC(¥overline{¥Omega})$
$¥xi_{z}¥neq 0$
$¥xi_{z}¥geqq 0$
,
$Z+¥xi_{z}¥in¥overline{¥Omega}$
Mv(z)$
$ u(z)-v(z)+¥langle q, z¥rangle¥leqq u(z+¥xi_{z})-v(z+¥xi_{z})+¥langle q, z+¥xi_{z}¥rangle-¥langle q, ¥xi_{Z}¥rangle$
Then we obtain a contradiction because get the claim (3.4). We divide our consideration into three cases. Case 1. Let
$ z¥in¥partial¥Omega$
and
$v(z)0$ satisfies be a maximum point of Let we get
$¥omega_{1}(s_{0}^{2})0$
.
$¥Phi$
.
Since
.
,
,
$¥Phi(z_{n}, z)¥leqq¥Phi(x_{n}, y_{n})$
$u(z_{n})-v(z)+¥langle q, z¥rangle¥leqq u(z_{n})-v(z)+¥langle q, z¥rangle+¥frac{a_{n}}{2}|X_{n}-y_{n}-z_{n}+z|^{2}$
$¥leqq u(x_{n})-v(y_{n})+¥langle q, y_{n}¥rangle$
.
because , The function $u(x)-v(y)$ is bounded above on . Hence (3.5) implies $|x_{n}-y_{n}-z_{n}+z|¥rightarrow 0$ as is compact in and . Then there as . Moreover we easily observe . as , point such that sequence a and a exist that It follows from this, (3.5) and the semicontinuity of and $¥overline{¥Omega}¥times¥overline{¥Omega}$
$u$
$-v¥in USC(¥overline{¥Omega})$
$R^{2N}$
$¥overline{¥Omega}¥times¥overline{¥Omega}$
$|x_{n}-y_{n}|¥rightarrow 0$
$ n¥rightarrow+¥infty$
$ n¥rightarrow+¥infty$
$¥overline{z}¥in¥overline{¥Omega}$
$¥{n_{k}¥}¥subset N$
$X_{n_{k}}$
$u$
$y_{n_{k}}¥rightarrow¥overline{Z}$
$ k¥rightarrow+¥infty$
$v$
$ u(z)-v(z)+¥langle q, z¥rangle¥leqq u(¥overline{z})-v(¥overline{z})+¥langle q,¥overline{z}¥rangle$
is a unique maximum point of the function Since and , it follows from this inequality that
$ u(x)-v(x)+¥langle q, x¥rangle$
$z$
$¥overline{z}=z$
$¥overline{¥Omega}$
(3.6)
$x_{n}$
,
$y_{n}¥rightarrow Z$
$(n ¥rightarrow+¥infty)$
.
Thus, by (3.5) we get $¥lim_{n¥rightarrow¥infty}(u(x_{n})-v(y_{n}))=u(z)-v(z)$
Using (3.5), this equality and the semicontinuity of
$u$
.
and , we have $v$
on
306
Katsuyuki ISHII
(3.7)
$u(x_{n})¥rightarrow u(z)$
,
We may consider
$v(y_{n})¥rightarrow v(z)$
,
$a_{n}|x_{n}-y_{n}-z_{n}+z|^{2}¥rightarrow 0$
$(n ¥rightarrow+¥infty)$
.
for sufficiently large $n¥in N$ because (3.7) implies $|x_{n}-y_{n}-z_{n}+z|0$
,
.
characteristic function for
$¥tilde{g}=¥min¥{g, ¥psi¥}$
,
$A$
.
We consider the penalized problem for (2.1).
,
$+n(u_{n}-¥psi)^{+}=0$ $¥{_{u_{n}}F(¥mathrm{x}=’ u_{n}¥tilde{g}, Du_{n}, D^{2}u_{n})$
$¥mathrm{i}¥mathrm{o}¥mathrm{n}¥partial¥Omega ¥mathrm{n}¥Omega$
’
where $n¥in N$ and $r^{+}=¥max¥{r, Noting $ r^{+}=¥sup$ to the following :
0¥}$ .
$¥{¥gamma r|0¥leqq¥gamma¥leqq 1¥}$
, it is easily seen that (5.1) is equivalent
$¥mathrm{P}¥mathrm{D}¥mathrm{E}$
(5.2)
,
$¥left¥{¥begin{array}{l}¥sup_{a¥in¥Lambda}¥gamma¥in[0,1]¥{-+(c(x,a)+n¥gamma)u_{n}-f(x,¥alpha)-n¥gamma¥psi¥}=0tr(^{t}¥sigma(x,¥alpha)¥sigma(x,a)Du_{n})+¥langle b(x,a),Du_{n}¥rangle¥¥ u_{n}=¥tilde{g}¥end{array}¥right.$
$¥mathrm{i}¥mathrm{o}¥mathrm{n}¥partial¥Omega ¥mathrm{n}¥Omega$
.
Then applying the results in M. A. Katsoulakis [13], for each exists a unique solution of (5.2). Next we consider the following problem: $u_{n}¥in C(¥overline{¥Omega})$
(5.3)
$¥left¥{¥begin{array}{l}F(¥mathrm{x},v_{n},Dv_{n},D^{2}v_{n})+n(u_{n}-¥psi)^{+}=0¥¥v_{n}=¥tilde{g}¥end{array}¥right.$
, $¥mathrm{i}¥mathrm{o}¥mathrm{n}¥partial¥Omega ¥mathrm{n}¥Omega$
’
, there
$n¥in N$
312
Katsuyuki ISHII
where is the function obtained above. Using the results in [13] again, for $n ¥ in N$ each , there exists a unique solution of (5.3) and it is characterized as follows: $u_{n}$
$v_{n}¥in C(¥overline{¥Omega})$
$v_{n}(x)=¥inf_{a¥in d}E_{x}¥{¥int_{0}^{¥tau}(f(X_{t}, ¥alpha_{t})-n(u_{n}(X_{t})-¥psi(X_{t}))^{+})¥cdot¥exp(-¥int_{¥mathrm{o}}^{t}c(X_{s}, ¥alpha_{s})ds)dt$
$+¥tilde{g}(X_{¥tau})¥exp(-¥int_{0}^{¥tau}c(X_{s}, a_{s})ds)¥}$
.
Since (5.1) and (5.2) are equivalent to each other and the uniqueness of solutions of (5.1) holds in the class , we get $C(¥overline{¥Omega})$
(5.4)
$u_{n}(x)=¥inf_{a¥in d}E_{X}¥{¥int_{0}^{¥tau}(f(X_{t}, ¥alpha_{t})-n(u_{n}(X_{t})-¥psi(X_{t}))^{+})$
. $¥exp(-¥int_{0}^{t}c(X_{s}, ¥alpha_{s})ds)dt+¥tilde{g}(X_{¥tau})¥exp(-¥int_{¥mathrm{o}}^{¥tau}c(X_{s}, a_{s})ds)¥}$
.
Using (C.4) and the barrier argument, we have
(5.5)
on
$u_{n}¥leqq¥tilde{g}$
$¥partial¥Omega$
for all
$n¥in N$
Since the operator is monotone with respect to large $C>0$ , we obtain $nr^{+}$
(5.6)
. $n¥in N$
on
$-C¥leqq¥cdots¥leqq u_{n}¥leqq¥cdots¥leqq u_{2}¥leqq u_{1}$
and
$u_{n}¥geqq-C$
for
$¥overline{¥Omega}$
by the comparison principle of solutions of (5.1). (cf. M. G. Crandall-H. Ishii-P. L. Lions [4; Theorem 7.9].) Hence we can define the function by $u$
(5.7)
$u(x)=¥lim_{n¥rightarrow+¥infty}u_{n}(x)(=¥lim_{y}n¥rightarrow+¥infty¥sup_{¥rightarrow x}u_{n}(y))$
.
Then we get the following lemma.
Lemma 5.1.
The above
Proof. Since
function
$u$
is a u.s.c. subsolution
the sequence is decreasing by (5.6), we easily observe . Using (5.5) and letting , we have on . any For , we assume that attains a local maximum at . We may consider and that is a strict local maximum point . Then there exists $¥delta>0$ such that $¥{u_{n}¥}_{n¥in N}$
$u¥in USC(¥overline{¥Omega})$
$ n¥rightarrow+¥infty$
$¥varphi¥in C^{2}(¥overline{¥Omega})$
$x_{0}¥in¥overline{¥Omega}$
of
of (2.1). $¥partial¥Omega$
$u¥leqq¥tilde{g}$
$ u-¥varphi$
$ x_{0}¥in¥Omega$
$x_{0}$
$ u-¥varphi$
(5.8)
$u(¥mathrm{x}_{0})-¥varphi(x_{0})>u(x)-¥varphi(x)$
for all
be a maximum point of on sequence such that Let
$ u_{n}-¥varphi$
$x_{n}$
$¥{x_{n_{k}}¥}_{k¥in N}¥subset¥{x_{n}¥}_{n¥in N}$
$x¥in¥overline{B(x_{0},¥delta)}(¥subset¥Omega)$
$¥overline{B(¥chi_{0},¥delta)}$
.
,
$¥mathrm{x}¥neq x_{0}$
.
Then there exists a sub-
313
Second Order Elliptic PDEs $x_{n_{k}}¥rightarrow¥overline{X}¥in¥overline{B(¥chi_{¥dot{0}},¥delta)}$
,
$(k ¥rightarrow+¥infty)$
$u_{n_{k}}(x_{n_{k}})¥rightarrow¥beta¥in R$
.
Since for all
$u_{n_{k}}(¥mathrm{x})-¥varphi(¥mathrm{x})¥leqq u_{n_{k}}(X_{n_{k}})-¥varphi(x_{n_{k}})$
,
$¥in B(x_{0}, ¥delta)$
$x$
we get $u(¥mathrm{x}_{0})-¥varphi(x_{0})¥leqq¥lim_{x¥rightarrow}k¥rightarrow+¥infty¥sup_{x_{¥mathrm{O}}}(u_{n_{k}}(¥mathrm{x})-¥varphi(x))$
$¥leqq¥lim_{k¥rightarrow}¥sup_{+¥infty}(u_{n_{k}}(x_{n_{k}})-¥varphi(x_{n_{k}}))$
$=¥beta-¥varphi(¥overline{x})$
$¥leqq¥lim_{x¥rightarrow}k¥rightarrow+¥infty¥sup_{¥overline{X}}(u_{n_{k}}(¥mathrm{x})-¥varphi(x))$
$=u(¥overline{x})-¥varphi(¥overline{x})$
.
Therefore using (5.8) and the above inequality, we obtain
(5.9)
$x_{n}¥rightarrow x_{0}$
,
$u_{n}(x_{n})¥rightarrow u(x_{0})$
(cf. G. Barles-B. Perthame [2; Lemma A.3].) Since (5.1), we get (5. 10)
.
$(n ¥rightarrow+¥infty)$
is a subsolution of
$u_{n}$
$F(x_{n}, u_{n}(x_{n}), D¥varphi(x_{n}), D^{2}¥varphi(x_{n}))+n(u_{n}(x_{n})-¥psi(x_{n}))^{+}¥leqq 0$
.
It follows from (C.I) and (5.9) that there exists a constant $C>0$ such that
for all
$n(u_{n}(x_{n})-¥psi(x_{n}))^{+}¥leqq C$
Thus passing to the limit as
$ n¥rightarrow+¥infty$
.
, we have
$¥mathcal{U}(¥chi_{0})-¥psi(x_{0})¥leqq 0$
Moreover, (5.10) implies
$n¥in N$
.
$F(x_{n}, u_{n}(¥mathrm{x}_{n}), D¥varphi(x_{n}), D^{2}¥varphi(x_{n}))¥leqq 0$
.
Sending
$ n¥rightarrow+¥infty$
,
we obtain
$F(x_{0}, u(x_{0}), D¥varphi(x_{0}), D^{2}¥varphi(x_{0}))¥leqq 0$
Therefore we have completed the proof.
.
$¥blacksquare$
Remark 5.2. We notice that we cannot apply the results for the limit operations in [4; Section 5] to (2.1) and (5.1) directly since the term $n(r-¥psi(x))^{+}$ does not converge to 0 locally uniformly on as . $¥overline{¥Omega}¥times R$
We return to the formula (5.4). we get the following lemma.
$ n¥rightarrow+¥infty$
According to N. V. Krylov [16; p. 37],
314
Katsuyuki ISHII
Lemma 5.3.
(5. 11)
formula (5.4) can be
The
rewritten as
follows.
$u_{n}(x)=a¥in d¥inf_{¥theta¥in ¥mathit{9}}E_{x}¥{¥int_{¥mathrm{o}}^{¥tau¥wedge¥theta}f(X_{t}, a_{t})¥exp(-¥int_{0}^{t}c(X_{s}, ¥alpha_{s})ds)dt$
$+1_{¥theta
