In this note, we will discuss the definition of the viscosity solution property. (PDF)
where is a given operator. One of the desired candidates for
is
where is a set,
is some integro-differential operator parameterized by
, and
is a function.
Example 1
Then is well defined for
, but not for
.
– QED –
If does not belong to
, the given domain of
, one can not evaluate
by its very definition. Hence
can not be justified. In this below, we shall give a proper meaning of
and
in a extended sense for an irregular
using smooth test functions. To proceed, we use
-
to denote its upper-semicontinuous (USC) envelop;
-
to denote its lower-semicontinuous (LSC) envelop.
For a function ,
-
is called supertest function of
at
, where
-
is called subtest function of
at
, where
Due to , it is easy to see a symmetry in the sense of
We say a function satisfies
- the viscosity subsolution property of
at
, denoted by
, if
- the viscosity supersolution property of
at
, denoted by
, if
- the viscosity solution property of
at
, denoted by
, if both
and
hold.
– QED –
Example 2
Let be given by (3). Verify that
satisfies
at any
.
Hint: Note that is empty set due to its kink. See the graph by running Python. (here)
– QED –
By extending the meaning of for an irregular
, we shall worry about its consistencies in various perspectives.
Suppose is a linear operator satisfying Fermat’s theorem for each
, i.e.,
then of (2) satisfies the consistency in the following sense: For any
and
,
if and only if
.
– QED –
Consistency holds in another context given below.
If and
, then
.
– QED –
The next example is often seen in the free boundary problems and the definition of generalized solution. It can also be regarded as a special case of Proposition 2.
is equivalent to
– QED – Indeed, is inconsistent in the following sense.
Example 5
does not imply
.
– QED –
This issue brings up the definition of the properness of , in which makes
is not proper. We shall restrict the scope of the study to proper
under the viscosity property.
Finally, it is natural to give such a definition of the viscosity solution of (1):
-
is a subsolution, if
for all
;
-
is a supersolution, if
for all
;
-
is a solution, if
for all
.
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