Last Friday, I’ve had a pleasant discussion with Professor Lions (here) on my recent paper (here). PDF is also available (here).
The main topic was the existence of the viscosity solution (here for Definition) on a given PDE with Dirichlet boundary, and I told him please stop me anytime you think it’s a garbage.
We started with three toy problems for a motivation. Consider the following three ODEs with a fixed domain .
By using the sufficient condition derived by Example 4.6 of [CIL92] (User’s Guide to Viscosity Solutions) together with some elementary analysis, the answer to the existence of the above three equations shall be
No, Yes, and No,
respectively. The main idea is to construct a sub and supersolutions, and then apply Perron’s method. However, as it is mentioned in Example 4.6 of [CIL92], it leaves an open question for the existence of a sub and supersolution in general.
Next, we take a look at two equations, which may not be a direct consequence of Example 4.6.
Let be a constant and
be a unit open ball.
-END
If in the above example, then the answers are Yes for both, according to the paper [BCI08] (here). But if
, we will answer the question in this below.
To proceed, let’s first recall some well known facts and terminologies:
- Let
be the collection of RCLL processes
equipped with Skorohod metric;
- We fix
in the rest of the notes. For
, we denote
If
, it is well known that there exists a Feller process
We know that a Feller process is strong Markovian and takes its path in
with probability one.
- Define exit times as the mappings
by
- We say
is regular w.r.t.
, and denoted by
, if there exists a strong Markov
satisfying
If
-
,
-
is a given vector field such that
,
then there exists a unique solution of
-END
Let . If
-
is given
;
-
such that
;
then there exists a unique solution of
-END
The answer to Example 1 is given below now:
- (4) is solvable if and only if
. Theorem 1 answers if direction, while an obvious reason answers only if part;
- (5) is solvable due to Theorem 2 with the choice
.
- The above answers shall remain the same if
is merely an open bounded set satisfying exterior cone condition. (At this time, Lions said, it is certainly not a garbage.)
Proof: We sketch the proof of two theorems and more details are referred to the paper.
- Theorem 2 is a consequence of Perron’s method, given that Theorem 1 is true. We shall prove Theorem 1 in this below.
- Let
of Theorem 1, and define
Given that
, then it is routine to show that
is solution of (6). It is remained to show
.
- One can rewrite
with obvious choice of
. Now, we finish the proof of
by checking the following three facts:
-
is continuous;
-
is continuous on
;
-
.
In the above,
is the collection of
satisfying
-
is continuous at
;
-
.
-