Proof of Classical Sub- and Supersolutions of a Degenerate Elliptic Operator are Viscosity Sub- and Supersolutions

Throughout, $\mathbb{R}$ carries the operations and the order $\le$ of its ordered field structure; that order is a total order and is in particular transitive. 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 semicontinuity and local extremum notions below. Let $0_{\mathbb{R}^n}$ denote the origin of $\mathbb{R}^n$, that is, the point all of whose coordinates are $0$, and let $0_n$ denote the real $n\times n$ matrix all of whose entries are $0$.

For a function of class $C^2$ on $U$ and a point $x\in U$ we write $D$ for its gradient at $x$ and $D^2$ for its Hessian matrix at $x$, as in $Du(x)$ and $D^2u(x)$. By Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian every such Hessian matrix lies in the set $\mathcal{S}(n)$ of symmetric real $n\times n$ matrices, so the positive semidefinite ordering $\preceq$ applies to the pairs of Hessian matrices considered below.

Step 1 (Semicontinuity). By claim 2 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, $u$ is both upper semicontinuous on $U$ and lower semicontinuous on $U$. This supplies the semicontinuity requirement in each of the two viscosity notions.

Step 2 (The difference of $u$ and a test function). Let $\varphi:U\to\mathbb{R}$ be of class $C^2$ on $U$ and let $w:U\to\mathbb{R}$ be the function whose value at $y\in U$ is $u(y)-\varphi(y)$. By claim 1 of Differences and Constants for Functions of Class $C^2$ on a Euclidean Open Set the function $w$ is of class $C^2$ on $U$ and, for every $x\in U$,

the first being the difference of points of $\mathbb{R}^n$ and the second the difference of real matrices.

Step 3 (A vanishing gradient difference). Suppose $x\in U$ satisfies $Dw(x)=0_{\mathbb{R}^n}$. For each coordinate index $i$, the $i$th coordinate of the point $Du(x)-D\varphi(x)$ is the real number $(Du(x))_i-(D\varphi(x))_i$, and it equals $0$ because every coordinate of $0_{\mathbb{R}^n}$ is $0$. Claim 3 of Additive Cancellation and Elementary Additive Identities in a Field therefore gives $(Du(x))_i=(D\varphi(x))_i$ for every $i$. Two points of Euclidean space $\mathbb{R}^n$ with the same coordinates are equal, so

Step 4 (Proof of claim 1). Assume that $u$ is a classical subsolution of $F$ on $U$, that is, that $F(y,u(y),Du(y),D^2u(y))\le 0$ for every $y\in U$. Let $\varphi:U\to\mathbb{R}$ be of class $C^2$ on $U$, let $w$ be as in Step 2, and let $x\in U$ be a point at which $w$ has a local maximum relative to $U$. Since $w$ is of class $C^2$ on $U$, claim 1 of First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$ gives

By Step 3 the first of these gives $Du(x)=D\varphi(x)$. By Step 2 the second reads $D^2u(x)-D^2\varphi(x)\preceq 0_n$, so claim 1 of Comparison with the Zero Matrix in the Positive Semidefinite Ordering, applied with $A=D^2u(x)$ and $B=D^2\varphi(x)$, gives

Apply now the degenerate ellipticity of $F$ at the point $x$, with the real number $u(x)$, the point $Du(x)$ of $\mathbb{R}^n$, and this last pair of symmetric matrices:

The right-hand side is at most $0$ by the classical subsolution hypothesis applied at $y=x$, so transitivity of $\le$ gives

which, after substituting $Du(x)=D\varphi(x)$, is exactly $F(x,u(x),D\varphi(x),D^2\varphi(x))\le 0$. As $\varphi$ and $x$ were arbitrary, this together with the upper semicontinuity from Step 1 shows that $u$ is a viscosity subsolution of $F$ on $U$.

Step 5 (Proof of claim 2). Assume that $u$ is a classical supersolution of $F$ on $U$, that is, that $0\le F(y,u(y),Du(y),D^2u(y))$ for every $y\in U$. Let $\varphi:U\to\mathbb{R}$ be of class $C^2$ on $U$, let $w$ be as in Step 2, and let $x\in U$ be a point at which $w$ has a local minimum relative to $U$. Claim 2 of First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$ gives

so $Du(x)=D\varphi(x)$ by Step 3, while claim 2 of Comparison with the Zero Matrix in the Positive Semidefinite Ordering, applied with $A=D^2u(x)$ and $B=D^2\varphi(x)$, gives

Degenerate ellipticity of $F$, applied at $x$ with the real number $u(x)$, the point $Du(x)$, and this pair of symmetric matrices, yields

and since $0\le F(x,u(x),Du(x),D^2u(x))$ by the classical supersolution hypothesis at $y=x$, transitivity of $\le$ gives $0\le F(x,u(x),Du(x),D^2\varphi(x))$. Substituting $Du(x)=D\varphi(x)$ turns this into $0\le F(x,u(x),D\varphi(x),D^2\varphi(x))$. As $\varphi$ and $x$ were arbitrary, this together with the lower semicontinuity from Step 1 shows that $u$ is a viscosity supersolution of $F$ on $U$.