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

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 ;
- .