We have discussed the definition of viscosity property (here). In this note, we will discuss the definition of the viscosity solution of Dirichlet problem. (PDF)

Given a subset of , we consider equation

where and are two given functions. If is a proper subset of , then we say (1) is a Dirichlet problem and the task is to find a solution .

Let’s first recall the viscosity property at a point for a function , ( here for details). We say

- if for any super test function
- if for any sub test function

Regarding the solution of (1) given above, is only defined to be a mapping of , while is defined for , hence the meaning of of (1) becomes obscure. Therefore, we shall extend to the whole space.

**Definition 1** * *

Given , we define . We say is

- a generalized subsolution of (1), if
If in addition holds for all , then is called a classical subsolution.

- a generalized supersolution of (1), if
If in addition holds for all , then is called a classical supersolution. – END

In the above,

shall be equivalent to

One can define a generalized or classical solution of (1) in a obvious fashion.

### Like this:

Like Loading...

Pingback: A brief discussion on Dirichlet problem « Omega