Consistency of the Classical and Viscosity Notions for Functions of Class $C^2$

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $U\subseteq\mathbb{R}^{n}$ be open, let $F$ be a second-order equation operator on $U$ that is degenerate elliptic, and let $u:U\to\mathbb{R}$ be of class $C^{2}$ on $U$.

Then the following equivalences hold.

1. (Subsolutions) $u$ is a classical subsolution of $F$ on $U$ if and only if $u$ is a viscosity subsolution of $F$ on $U$.

2. (Supersolutions) $u$ is a classical supersolution of $F$ on $U$ if and only if $u$ is a viscosity supersolution of $F$ on $U$.

3. (Solutions) $u$ is a classical solution of $F$ on $U$ if and only if $u$ is a viscosity solution of $F$ on $U$.