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

Throughout, $\mathbb{R}$ carries the operations and the order $\le$ of its ordered field structure, where $s-t$ abbreviates $s+(-t)$ and $s<t$ means that $s\le t$ and $s\ne t$; the order $\le$ is a total order and is in particular reflexive. Regard $\mathbb{R}^n$ as a metric space through the Euclidean distance $d_E$, which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$; this is the metric used in the local extremum notions below. For $x\in U$ write $Du(x)$ for the gradient of $u$ at $x$ and $D^2u(x)$ for the Hessian matrix of $u$ at $x$.

Step 1 (The test function $u$ itself). Take $\varphi=u$; this function is of class $C^2$ on $U$ by hypothesis, so it is admissible as a test function in either viscosity notion, and $D\varphi(x)=Du(x)$ and $D^2\varphi(x)=D^2u(x)$ for every $x\in U$. For this choice the function $u-\varphi:U\to\mathbb{R}$ of that definition has value $u(y)-u(y)$ at $y\in U$, and $u(y)-u(y)=0$ by claim 3 of Additive Cancellation and Elementary Additive Identities in a Field. Write $w:U\to\mathbb{R}$ for this function, so that $w(y)=0$ for every $y\in U$.

Step 2 (Every point of $U$ is both a local maximum and a local minimum of $w$). Let $x\in U$. By claim 6 of Elementary Order Arithmetic in an Ordered Field we have $0<1$ in $\mathbb{R}$, so $\delta=1$ is an admissible radius. Every $y\in U$ with $d_E(x,y)<1$ satisfies $w(y)=0$ and $w(x)=0$, whence $w(y)\le w(x)$ and $w(x)\le w(y)$ by reflexivity of $\le$. Therefore $w$ has a local maximum at $x$ relative to $U$ and also a local minimum at $x$ relative to $U$.

Step 3 (Proof of claim 1). Assume that $u$ is a viscosity subsolution of $F$ on $U$ and let $x\in U$. By Steps 1 and 2 the function $u-\varphi$ associated with the test function $\varphi=u$ has a local maximum at $x$ relative to $U$, so the defining condition of a viscosity subsolution, applied to this $\varphi$ and this $x$, gives

$$F(x,u(x),D\varphi(x),D^2\varphi(x))\le 0 ,$$

that is, $F(x,u(x),Du(x),D^2u(x))\le 0$. Since $x\in U$ was arbitrary and $u$ is of class $C^2$ on $U$, this is exactly the statement that $u$ is a classical subsolution of $F$ on $U$.

Step 4 (Proof of claim 2). Assume that $u$ is a viscosity supersolution of $F$ on $U$ and let $x\in U$. By Steps 1 and 2 the same function $u-\varphi$, with $\varphi=u$, has a local minimum at $x$ relative to $U$, so the defining condition of a viscosity supersolution gives

$$0\le F(x,u(x),D\varphi(x),D^2\varphi(x)) ,$$

that is, $0\le F(x,u(x),Du(x),D^2u(x))$. Since $x\in U$ was arbitrary and $u$ is of class $C^2$ on $U$, $u$ is a classical supersolution of $F$ on $U$.