A Viscosity Subsolution or Supersolution of Class $C^2$ is Classical

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 $U$, and let $u:U\to\mathbb{R}$ be of class $C^{2}$ on $U$. No ellipticity hypothesis is imposed on $F$.

Then the following hold.

1. (Subsolutions) If $u$ is a viscosity subsolution of $F$ on $U$, then $u$ is a classical subsolution of $F$ on $U$.

2. (Supersolutions) If $u$ is a viscosity supersolution of $F$ on $U$, then $u$ is a classical supersolution of $F$ on $U$.