The definition of the viscosity solution property

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

We consider equation

$\displaystyle F[u] = 0, \hbox{ on } \mathbb R^{d} \ \ \ \ \ (1)$

where ${F: C_{b}^{\infty}(\mathbb R^{d}) \mapsto C(\mathbb R^{d})}$ is a given operator. One of the desired candidates for ${F}$ is

$\displaystyle F[u] = \sup_{a\in \Lambda} - \mathcal L^{a} u + \lambda u - \ell, \ \ \ \ \ (2)$

where ${\Lambda}$ is a set, ${\mathcal L^{a}}$ is some integro-differential operator parameterized by ${a\in \Lambda}$ , and ${\ell \in C(\mathbb R^{d})}$ is a function.

Example 1

Let ${F}$ be

$\displaystyle F[u] = |u'| + u - 1 \ \hbox{ on } C_{b}^{\infty}(\mathbb R) \ \ \ \ \ (3)$

Then ${F}$ is well defined for ${u \in C_{b}^{\infty}}$ , but not for ${u \notin C_{b}^{1}}$ .

– QED –

If ${u}$ does not belong to ${C_{b}^{\infty}}$ , the given domain of ${F}$ , one can not evaluate ${F[u]}$ by its very definition. Hence ${F[u] = 0}$ can not be justified. In this below, we shall give a proper meaning of ${F[u] \ge 0}$ and ${F[u] \le 0}$ in a extended sense for an irregular ${u}$ using smooth test functions. To proceed, we use

${u^{*}}$ to denote its upper-semicontinuous (USC) envelop;
${u_{*}}$ to denote its lower-semicontinuous (LSC) envelop.

For a function ${u: \mathbb R^{d} \mapsto \mathbb R}$ ,

${\phi \in J^{+} (u, x)}$ is called supertest function of ${u}$ at ${x \in \mathbb R^{d}}$ , where
$\displaystyle J^{+} (u, x) = \{\phi \in C_{b}^{\infty}(\mathbb R^{d}), \hbox{ such that } \phi \ge u^{*} \hbox{ and } \phi(x) = u^{*}(x)\}.$
${\phi \in J^{-} (u, x)}$ is called subtest function of ${u}$ at ${x \in \mathbb R^{d}}$ , where
$\displaystyle J^{-} (u, x) = \{\phi \in C_{b}^{\infty}(\mathbb R^{d}), \hbox{ such that } \phi \le u_{*} \hbox{ and } \phi(x) = u_{*}(x)\}.$

Due to ${(-u)^{*} = - u_{*}}$ , it is easy to see a symmetry in the sense of

$\displaystyle J^{-} (u, x) = -J^{+} (-u, x).$

Definition 1

We say a function ${u: \mathbb R^{d} \mapsto \mathbb R}$ satisfies

the viscosity subsolution property of ${F[u] = 0}$ at ${x\in \mathbb R^{d}}$ , denoted by ${F[u] (x) \le_{v} 0}$ , if
$\displaystyle F [\phi](x) \le 0, \ \forall \phi \in J^{+} (u,x). \ \ \ \ \ (4)$
the viscosity supersolution property of ${F[u] = 0}$ at ${x \in \mathbb R^{d}}$ , denoted by ${F[u](x) \ge_{v} 0}$ , if
$\displaystyle F [\phi] (x) \ge 0, \ \forall \phi \in J^{-} (u,x). \ \ \ \ \ (5)$
the viscosity solution property of ${F[u] = 0}$ at ${x \in \mathbb R^{d}}$ , denoted by ${F[u](x) =_{v} 0}$ , if both ${F[u](x) \ge_{v} 0}$ and ${F[u](x) \le_{v} 0}$ hold.

– QED –

Example 2

Let ${F}$ be given by (3). Verify that ${u = 1 - e^{-1 + |x|}}$ satisfies ${F[u](x) =_{v} 0}$ at any ${x\in \mathbb R}$ .
Hint: Note that ${J^{-}(u, 0)}$ is empty set due to its kink. See the graph by running Python. (here)
– QED –

By extending the meaning of ${F[u](x) \le 0}$ for an irregular ${u}$ , we shall worry about its consistencies in various perspectives.

Example 3

Suppose ${\mathcal L^{a}}$ is a linear operator satisfying Fermat’s theorem for each ${a\in \Lambda}$ , i.e.,

$\displaystyle \mathcal L^{a} \phi (x) \le 0, \ \hbox{ for any } \phi \in C^{\infty}_{b}, x \in \arg\max \phi, \hbox{ and } a\in \Lambda$

then ${F}$ of (2) satisfies the consistency in the following sense: For any ${u \in C^{\infty}_{b}}$ and ${x\in \mathbb R^{d}}$ ,

${F[u](x) \le 0}$ if and only if ${F[u](x) \le_{v} 0}$ .

– QED –

Consistency holds in another context given below.

Proposition 2

If ${F_{1}[u](x) \ge_{v} 0}$ and ${F_{2}[u](x) \ge_{v} 0}$ , then ${\max\{F_{1}[u](x), F_{2}[u](x)\} \ge_{v} 0}$ .
– 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

$\displaystyle \max\{F[u](x), u(x) - g(x)\} \ge_{v} 0$

is equivalent to

$\displaystyle F[u](x) \ge_{v} 0, \ \hbox{ or } \ u^{*}(x) \ge g(x).$

– QED – Indeed, ${F[u](x) \le_{v} 0}$ is inconsistent in the following sense.

Example 5

${F[u](x) \le_{v} 0}$ does not imply ${-F[u](x) \ge_{v} 0}$ .

– QED –

This issue brings up the definition of the properness of ${F}$ , in which makes ${-F}$ is not proper. We shall restrict the scope of the study to proper ${F}$ under the viscosity property.

Finally, it is natural to give such a definition of the viscosity solution of (1):

${u}$ is a subsolution, if ${F[u](x) \le_{v} 0}$ for all ${x\in \mathbb R^{d}}$ ;
${u}$ is a supersolution, if ${F[u](x) \ge_{v} 0}$ for all ${x\in \mathbb R^{d}}$ ;
${u}$ is a solution, if ${F[u](x) =_{v} 0}$ for all ${x\in \mathbb R^{d}}$ .

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