In this note, we will discuss the definition of the viscosity solution property. (PDF)

We consider equation

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** * *

Let be

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

**Definition 1** * *

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.

**Example 3** * *

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.

**Proposition 2** * *

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.

**Example 4** * *

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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