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.
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.
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
- is a given vector field such that ,
Let . If
- is given ;
- such that ;
then there exists a unique solution of
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 ;