On the Dirichelet problem 29
R
EMARK
6. If ∂ is not smooth and F is uniformly elliptic Theorem 7 can be used as an approximation result by choosing
n
with smooth boundary. In fact, under some structural assumptions, the solution u
n
of 32 turns out to be continuous by a barrier argument see, e.g., [5], and then it is twice differentiable almost everywhere by a result in [43], see also [17]. If, in
addition, F is smooth and Fx, ·, ·, · is convex, or the PDE 10 is a HJB equation 3 where the
linear operators
α
have smooth coefficients, then u
n
is of class C
2
, see [21, 24, 31, 17] and the references therein. The Lipschitz continuity of u
n
holds also if F is not uniformly elliptic but it is coercive in the p variables.
2.4. Continuity at the boundary
In this section we study the behavior of the e-solution at boundary points and characterize the points where the boundary data are attained continuously by means of barriers.
P
ROPOSITION
4. Assume that hypothesis i respectively ii of Theorem 2 holds. Then the e-solution H
g
of 16 takes up the boundary data g continuously at x ∈ ∂, i.e. lim
x →x
H
g
x = gx
, if and only if there is an upper respectively lower barrier at x see Definition 3.
Proof. The necessity is obvious because Theorem 2 i implies that H
g
∈
✁
, so H
g
is an upper barrier at x if it attains continuously the data at x.
Now we assume W is an upper barrier at x. Then W ≥ H
g
, because W ∈
✁
and H
g
is the minimal element of
✁
. Therefore gx
≤ H
g
x ≤ lim inf
y→x
H
g
y ≤ lim sup
y→x
H
g
y ≤ lim
y→x
W y = gx ,
so lim
y→x
H
g
y = gx = H
g
x. In the classical theory of linear elliptic equations, local barriers suffice to characterize
boundary continuity of weak solutions. Similar results can be proved in our fully nonlinear context. Here we limit ourselves to a simple result on the Dirichlet problem with homogeneous
boundary data for the Isaacs equation
sup
α
inf
β
{−a
α,β i j
u
x
i
x
j
+ b
α,β i
u
x
i
+ c
α,β
u − f
α,β
} = 0 in , u
= 0 on ∂ .
33 D
EFINITION
6. We say that W ∈ B L SCBx
, r
∩ with r 0 is an upper local barrier for problem 33 at x
∈ ∂ if i W
≥ 0 is a supersolution of the PDE in 33 in Bx ,
r ∩ ,
ii W x = 0, Wx ≥ µ 0 for all |x − x
| = r, iii W is continuous at x
. P
ROPOSITION
5. Assume the Comparison Principle holds for 33, f
α,β
≥ 0 for all α, β, and let H
g
be the e-solution of problem 33. Then H
g
takes up the boundary data continuously at x
∈ ∂ if and only if there exists an upper local barrier W at x .
30 M. Bardi – S. Bottacin
Proof. We recall that H
g
exists because the function u ≡ 0 is a lower barrier for all points
x ∈ ∂ by the fact that f
α,β
≥ 0, and so we can apply Theorem 2. Consider a supersolution w of 33. We claim that the function V defined by
V x =
ρ W x
∧ wx if x ∈ Bx ,
r ∩ ,
w x
if x ∈ \ Bx
, r ,
is an upper barrier at x for ρ 0 large enough. It is easy to check that ρW is a supersolution
of 33 in Bx ,
r ∩ , so V is a supersolution in Bx
, r
∩ by Proposition 1 and in
\ Bx ,
r . Since w is bounded, by property ii in Definition 6, we can fix ρ and ǫ 0 such that V x
= wx for all x ∈ satisfying r − ǫ |x − x | ≤ r. Then V is supersolution even
on ∂ Bx ,
r ∩ . Moreover it is obvious that V ≥ 0 on ∂ and V x
= 0. We have proved that V is supersolution of 33 in .
It remains to prove that lim
x →x
V x = 0. Since the constant 0 is a subsolution of 33 and
w is a supersolution, we have w
≥ 0. Then we reach the conclusion by ii and iii of Definition 6.
E
XAMPLE
3. We construct an upper local barrier for 33 under the assumptions of Propo- sition 5 and supposing in addition
∂ is C
2
in a neighbourhood of x ∈ ∂ ,
there exists an α
∗
such that for all β either a
α
∗
,β i j
x n
i
x n
j
x ≥ c 0
34 or
−a
α
∗
,β i j
x d
x
i
x
j
x + b
α
∗
,β i
x n
i
x ≥ c 0
35 where n denotes the exterior normal to and d is the signed distance from ∂
dx =
dist x, ∂ if x
∈ , −dist x, ∂ if x ∈
N
\ . Assumptions 34 and 35 are the natural counterpart for Isaacs equation in 33 of the con-
ditions for boundary regularity of solutions to linear equations in Chapt. 1 of [37]. We claim that
W x = 1 − e
−δdx+λ|x−x |
2
is an upper local barrier at x for a suitable choice of δ, λ 0. Indeed it is easy to compute
−a
α,β i j
x W
x
i
x
j
x + b
α,β i
x W
x
i
x + c
α,β
x W x
− f
α,β
x =
−δa
α,β i j
x d
x
i
x
j
x + δ
2
a
α,β i j
x d
x
i
x d
x
j
x + δb
α,β i
x d
x
i
x −2δλTr [a
α,β
x ]
− f
α,β
x .
Next we choose α
∗
as above and assume first 34. In this case, since the coefficients are bounded and continuous and d is C
2
, we can make W a supersolution of the PDE in 33 in a neighborhood of x
by taking δ large enough. If, instead, 35 holds, we choose first λ small and then δ large to get the same conclusion.
On the Dirichelet problem 31
3. Applications to Optimal Control 3.1. A deterministic minimum-time problem
