On the Dirichelet problem 25
Step 4. Now take δ 0, then there exists p such that for all n, m ≥ p:
|H
m g
m
− H
g
| ≤ |H
m g
m
− H
m g
n
| + |H
m g
n
− H
m g
| + |H
m g
− H
g
| ≤ 3δ . Similarly
|H
m g
m
− H
g
| ≤ 3δ. But H
m g
m
= H
m g
m
, and this complete the proof.
2.2. Continuous dependence under local uniform convergence of the operator
In this subsection we study the continuous dependence of e-solutions with respect to perturba- tions of the operator, depending on a parameter h, that are not uniform over all
× ×
N
× SN as they were in Theorem 5, but only on compact subsets of
× ×
N
× SN. A typical example we have in mind is the vanishing viscosity approximation, but similar arguments work
for discrete approximation schemes, see [3]. We are able to pass to the limit under merely local perturbations of the operator by approximating with a nested family of open sets 2
ǫ
, solving the problem in each 2
ǫ
, and then letting ǫ, h go to 0 “with h linked to ǫ” in the following sense. D
EFINITION
4. Let v
ǫ h
, u : Y → , for ǫ 0, h 0, Y ⊆
N
. We say that v
ǫ h
converges to u as ǫ, h
ց 0, 0 with h linked to ǫ at the point x, and write lim
ǫ, hց0,0
h≤hǫ
v
ǫ h
x = ux
25 if for all γ 0, there exist a function ˜
h :]0, +∞[→]0, +∞[ and ǫ 0 such that
|v
ǫ h
y − ux| ≤ γ, for all y ∈ Y : |x − y| ≤ ˜hǫ
for all ǫ ≤ ǫ, h ≤ ˜hǫ.
To justify this definition we note that: i it implies that for any x and ǫ
n
ց 0 there is a sequence h
n
ց 0 such that v
ǫ
n
h
n
x
n
→ ux for any sequence x
n
such that |x − x
n
| ≤ h
n
, e.g. x
n
= x for all n, and the same holds for any sequence h
′ n
≥ h
n
; ii if lim
hց0
v
ǫ h
x exists for all small ǫ and its limit as ǫ ց 0 exists, then it coincides with the
limit of Definition 4, that is, lim
ǫ, hց0,0
h≤hǫ
v
ǫ h
x = lim
ǫց
lim
hց0
v
ǫ h
x . R
EMARK
4. If the convergence of Definition 4 occurs on a compact set K where the limit u is continuous, then 25 can be replaced, for all x
∈ K and redefining ˜h if necessary, with |v
ǫ h
y − uy| ≤ 2γ, for all y ∈ K : |x − y| ≤ ˜hǫ ,
and by a standard compactness argument we obtain the uniform convergence in the following sense:
D
EFINITION
5. Let K be a subset of
N
and v
ǫ h
, u : K
→ for all ǫ, h 0. We say that
v
ǫ h
converge uniformly on K to u as ǫ, h ց 0, 0 with h linked to ǫ if for any γ 0 there are
ǫ 0 and ˜
h :]0, +∞[→]0, +∞[ such that
sup
K
|v
ǫ h
− u| ≤ γ
26 M. Bardi – S. Bottacin
for all ǫ ≤ ǫ, h ≤ ˜hǫ.
The main result of this subsection is the following. Recall that a family of functions v
ǫ h
:
→ is locally uniformly bounded if for each compact set K
⊆ there exists a constant C
K
such that sup
K
|v
ǫ h
| ≤ C
K
for all h, ǫ 0. In the proof we use the weak limits in the viscosity sense and the stability of viscosity solutions and of the Dirichlet boundary condition in viscosity
sense 21 with respect to such limits. T
HEOREM
6. Assume the Comparison Principle holds,
✁
6= ∅ and let u be a continuous subsolution of 16 such that u
= g on ∂. For any ǫ ∈]0, 1], let 2
ǫ
be an open set such that 2
ǫ
⊆ , and for h ∈]0, 1] let v
ǫ h
be a non-continuous viscosity solution of the problem F
h
x, u, Du, D
2
u = 0
in 2
ǫ
, ux
= ux or F
h
x, u, Du, D
2
u = 0 on ∂2
ǫ
, 26
where F
h
: 2
ǫ
× ×
N
× SN → is continuous and proper. Suppose
{v
ǫ h
} is locally uniformly bounded, v
ǫ h
≥ u in , and extend v
ǫ h
: = u in \ 2
ǫ
. Finally assume that F
h
converges uniformly to F on any compact subset of ×
×
N
× SN as h ց 0, and 2
ǫ
⊇ 2
δ
if ǫ δ, S
0ǫ≤1
2
ǫ
= . Then v
ǫ h
converges to the e-solution H
g
of 16 with h linked to ǫ, that is, 25 holds for all x
∈ ; moreover the convergence is uniform as in Def. 5 on any compact subset of where H
g
is continuous. Proof. Note that the hypotheses of Theorem 3 are satisfied, so the e-solution H
g
exists. Consider the weak limits
v
ǫ
x :
= lim inf
hց0
∗ v
ǫ h
x : = sup
δ
inf {v
ǫ h
y : |x − y| δ, 0 h δ} ,
v
ǫ
x :
= lim sup
hց0 ∗
v
ǫ h
x : = inf
δ
sup {v
ǫ h
y : |x − y| δ, 0 h δ} .
By a standard result in the theory of viscosity solutions, see [10, 18, 8, 4], v
ǫ
and v
ǫ
are respec- tively supersolution and subsolution of
Fx, u, Du, D
2
u = 0
in 2
ǫ
, ux
= ux or Fx, u, Du, D
2
u = 0 on ∂2
ǫ
. 27
We claim that v
ǫ
is also a subsolution of 16. Indeed v
ǫ h
≡ u in \2
ǫ
, so v
ǫ
≡ u in the interior of
\ 2
ǫ
and then in this set it is a subsolution. In 2
ǫ
we have already seen that v
ǫ
= v
ǫ ∗
is a subsolution. It remains to check what happens on ∂2
ǫ
. Given ˆx ∈ ∂2
ǫ
, we must prove that for all p, X
∈ J
2,+
v
ǫ
ˆx we have F
h
ˆx, v
ǫ
ˆx, p, X ≤ 0 . 28
1st Case: v
ǫ
ˆx u ˆx. Since v
ǫ
satisfies the boundary condition on ∂2
ǫ
of problem 27, then for all p, X
∈ J
2,+ 2
ǫ
v
ǫ
ˆx 28 holds. Then the same inequality holds for all p, X ∈ J
2,+
v
ǫ
ˆx as well, because J
2,+
v
ǫ
ˆx ⊆ J
2,+ 2
ǫ
v
ǫ
ˆx. 2nd Case: v
ǫ
ˆx = u ˆx. Fix p, X ∈ J
2,+
v
ǫ
ˆx, by definition v
ǫ
x ≤ v
ǫ
ˆx + p · x − ˆx + 1
2 X x
− ˆx · x − ˆx + o|x − ˆx|
2
On the Dirichelet problem 27
for all x → ˆx. Since v
ǫ
≥ u and v
ǫ
ˆx = u ˆx, we get ux
≤ u ˆx + p · x − ˆx + 1
2 X x
− ˆx · x − ˆx + o|x − ˆx|
2
, that is p, X
∈ J
2,+
u ˆx. Now, since u is a subsolution, we conclude
F ˆx, v
ǫ
ˆx, p, X = F ˆx, u ˆx, p, X ≤ 0 . We now claim that
u
ǫ
≤ v
ǫ
≤ v
ǫ
≤ H
g
in , 29
where u
ǫ
is defined by 20. Indeed, since v
ǫ
is a supersolution in 2
ǫ
and v
ǫ
≥ u, by the Comparison Principle v
ǫ
≥ w in 2
ǫ
for any w ∈
such that w = u on ∂2
ǫ
. Moreover v
ǫ
≡ u on
\2
ǫ
, so we get v
ǫ
≥ u
ǫ
in . To prove the last inequality we note that H
g
is a supersolution of 16 by Theorem 3, which implies v
ǫ
≤ H
g
by Comparison Principle. Now fix x
∈ , ǫ 0, γ 0 and note that, by definition of lower weak limit, there exists h
= hx, ǫ, γ 0 such that v
ǫ
x − γ ≤ v
ǫ h
y for all h
≤ h and y ∈ ∩ Bx, h. Similarly there exists k = kx, ǫ, γ 0 such that v
ǫ h
y ≤ v
ǫ
x + γ
for all h ≤ k and y ∈ ∩ Bx, k. From Remark 1, we know that H
g
= sup
ǫ
u
ǫ
, so there exists ǫ
such that H
g
x − γ ≤ u
ǫ
x, for all ǫ ≤ ǫ .
Then, using 29, we get H
g
x − 2γ ≤ v
ǫ h
y ≤ H
g
x + γ
for all ǫ ≤ ǫ, h ≤ ˜h := min{h, k} and y ∈ ∩ Bx, ˜h, and this completes the proof.
R
EMARK
5. Theorem 6 applies in particular if v
ǫ h
are the solutions of the following vanish- ing viscosity approximation of 10
−h1v + Fx, v, Dv, D
2
v = 0 in 2
ǫ
, v
= u on ∂2
ǫ
. 30
Since F is degenerate elliptic, the PDE in 30 is uniformly elliptic for all h 0. Therefore we can choose a family of nested 2
ǫ
with smooth boundary and obtain that the approximating v
ǫ h
are much smoother than the e-solution of 16. Indeed 30 has a classical solution if, for instance, either F is smooth and Fx,
·, ·, · is convex, or the PDE 10 is a Hamilton-Jacobi- Bellman equation 3 where the linear operators
α
have smooth coefficients, see [21, 24, 31]. In the nonconvex case, under some structural assumptions, the continuity of the solution of
30 follows from a barrier argument see, e.g., [5], and then it is twice differentiable almost everywhere by a result in [43], see also [17].
28 M. Bardi – S. Bottacin
2.3. Continuous dependence under increasing approximation of the domain
