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

Throughout, $\mathbb{R}$ carries the order $\le$ of its ordered field structure, which is a total order and is in particular reflexive and antisymmetric. 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 $x$.

Claims 1 and 2. Both implications from left to right are claims 1 and 2 of Classical Sub- and Supersolutions of a Degenerate Elliptic Operator are Viscosity Sub- and Supersolutions, whose hypotheses hold because $F$ is degenerate elliptic and $u$ is of class $C^2$ on $U$. Both implications from right to left are claims 1 and 2 of A Viscosity Subsolution or Supersolution of Class $C^2$ is Classical, whose hypotheses hold because $u$ is of class $C^2$ on $U$.

Claim 3, from left to right. Suppose $u$ is a classical solution of $F$ on $U$, so that $F(x,u(x),Du(x),D^2u(x))=0$ for every $x\in U$. Since $0\le 0$ by reflexivity of $\le$, for every $x\in U$ both

$$F(x,u(x),Du(x),D^2u(x))\le 0\qquad\text{and}\qquad 0\le F(x,u(x),Du(x),D^2u(x))$$

hold. Thus $u$ is both a classical subsolution and a classical supersolution of $F$ on $U$, so by claims 1 and 2 it is both a viscosity subsolution and a viscosity supersolution of $F$ on $U$, which is precisely what it means to be a viscosity solution of $F$ on $U$.

Claim 3, from right to left. Suppose $u$ is a viscosity solution of $F$ on $U$. By that definition $u$ is both a viscosity subsolution and a viscosity supersolution of $F$ on $U$, so by claims 1 and 2 it is both a classical subsolution and a classical supersolution of $F$ on $U$. Hence for every $x\in U$ both

$$F(x,u(x),Du(x),D^2u(x))\le 0\qquad\text{and}\qquad 0\le F(x,u(x),Du(x),D^2u(x))$$

hold, and antisymmetry of $\le$ gives $F(x,u(x),Du(x),D^2u(x))=0$ for every $x\in U$. Since $u$ is of class $C^2$ on $U$, this says that $u$ is a classical solution of $F$ on $U$.