Viscosity solutions

Viscosity solutions are a type of weak solutions for elliptic or parabolic fully nonlinear integro-differential equations based on the notion of the comparison principle. It is especially suitable for stability under uniform limits.

Definition

The following is a definition of viscosity solutions for fully nonlinear integro-differential equations of the form $Iu=0$, assuming that the functions involved are defined in the whole space. Other situations can be considered with straight forward modifications in the definition. The nonlocal operator $I$ is not necessarily assumed to be translation invariant. All we assume about $I$ is that it is black box operator such that

1. $Iv(x_0)$ is well defined every time $v \in C^2(x_0)$.
2. $Iv(x_0) \geq Iw(x_0)$ if $v(x_0)=w(x_0)$ and $v\geq w$ in $\R^n$.

An function $u : \R^n \to \R$, which is upper semicontinuous in $\Omega$, is said to be a viscosity subsolution in $\Omega$ if the following statement holds

For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \geq u$ in $U$, we construct the auxiliary function $v$ as \[ v(x) = \begin{cases} \varphi(x) & \text{if } x \in U \\ u(x) & \text{if } x \notin U \end{cases}\] Then, we have the inequality \[ Iv(x_0) \geq 0. \]

An function $u : \R^n \to \R$, which is lower semicontinuous in $\Omega$, is said to be a viscosity supersolution in $\Omega$ if the following statement holds

For any open $U \subset \Omega$, $x_0 \in U$ and $\varphi \in C^2(U)$ such that $\varphi(x_0)=u(x_0)$ and $\varphi \leq u$ in $U$, , we construct the auxiliary function $v$ as \[ v(x) = \begin{cases} \varphi(x) & \text{if } x \in U \\ u(x) & \text{if } x \notin U \end{cases}\] Then, we have the inequality \[ Iv(x_0) \leq 0. \]

A continuous function in $\Omega$ which is at the same time a subsolution and a supersolution is said to be a viscosity solution.

Note that if the operator $I$ happens to be local, the construction of the function $v$ is unnecessary since $Iv(x_0) = I\varphi(x_0)$. Thus for local equations the definition is given evaluating $I\varphi(x_0)$ instead.