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
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.