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