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},

  1. {\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)\}.

  2. {\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

  1. 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)

  2. 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)

  3. 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):

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

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One comment

  1. Pingback: The definition of the viscosity solution of Dirichlet problem | 01law's Blog


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