The definition of the viscosity solution of Dirichlet problem

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 ${O}$ of ${\mathbb R^{d}}$, we consider equation

$\displaystyle F[u] = 0, \hbox{ on } O, \quad u = g \hbox{ on } O^{c} \ \ \ \ \ (1)$

where ${F: C_{b}^{\infty}(\mathbb R^{d}) \mapsto C(\mathbb R^{d})}$ and ${g:\mathbb R^{d} \mapsto \mathbb R}$ are two given functions. If ${O}$ is a proper subset of ${\mathbb R^{d}}$, then we say (1) is a Dirichlet problem and the task is to find a solution ${u: \bar O \mapsto \mathbb R}$.

Let’s first recall the viscosity property at a point ${x}$ for a function ${u: \mathbb R^{d} \mapsto \mathbb R}$, ( here for details). We say

1. ${F[u] (x) \le_{v} 0}$ if ${F [\phi](x) \le 0}$ for any super test function ${\phi \in J^{+} (u,x).}$
2. ${F[u] (x) \ge_{v} 0}$ if ${F [\phi](x) \ge 0}$ for any sub test function ${\phi \in J^{-} (u,x).}$

Regarding the solution of (1) given above, ${u}$ is only defined to be a mapping of ${\bar O \mapsto \mathbb R}$, while ${F[u](x) \le_{v} 0}$ is defined for ${\mathbb R^{d} \mapsto \mathbb R}$, hence the meaning of ${F[u](x) \le_{v} 0}$ of (1) becomes obscure. Therefore, we shall extend ${u}$ to the whole space.

Definition 1

Given ${u: \bar O \mapsto \mathbb R}$, we define ${\bar u = u I_{\bar O} + g I_{\bar O^{c}}}$. We say ${u}$ is

1. a generalized subsolution of (1), if

$\displaystyle F[\bar u] \le_{v} 0 \ \hbox{ on } O; \quad \hbox{ and } \min\{ F[ \bar u ], \ \bar u - g\} \le_{v} 0 \ \hbox{ on } \partial O. \ \ \ \ \ (2)$

If in addition ${\bar u^{*} \le g}$ holds for all ${x\in \partial O}$, then ${u}$ is called a classical subsolution.

2. a generalized supersolution of (1), if

$\displaystyle F[\bar u] \ge_{v} 0 \ \hbox{ on } O; \quad \hbox{ and } \max\{ F[\bar u], \ \bar u - g\} \ge_{v} 0 \ \hbox{ on } \partial O. \ \ \ \ \ (3)$

If in addition ${\bar u_{*} \ge g}$ holds for all ${x\in \partial O}$, then ${u}$ is called a classical supersolution. – END

In the above,

$\displaystyle \min\{ F[ \bar u ], \ \bar u - g\} \le_{v} 0 \ \hbox{ on } \partial O$

shall be equivalent to

$\displaystyle \partial O = \{x\in \partial O: F[\bar u] \le_{v} 0\} \cup \{x\in \partial O: \bar u^{*} - g \le 0\}.$

One can define a generalized or classical solution of (1) in a obvious fashion.