Proof of Viscosity Inequalities Pass to Limits of Test-Function Data

Write $c=F\bigl(x_{0},u(x_{0}),p,X\bigr)$.

Claim 1. Suppose, seeking a contradiction, that $c\le 0$ fails. Since the order on $\mathbb{R}$ is a total order, this means $0<c$.

Apply the continuity hypothesis with $\varepsilon=c$: there is $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y\in U$, $s\in\mathbb{R}$, $q\in\mathbb{R}^{n}$ and $Y\in\mathcal{S}(n)$ with

$$d_{E}(y,x_{0})<\delta,\quad |s-u(x_{0})|<\delta,\quad \lVert q-p\rVert<\delta,\quad d_{\mathcal{S}(n)}(Y,X)<\delta$$

satisfies $\bigl|F(y,s,q,Y)-c\bigr|<c$.

Apply the approximation hypothesis with $\varepsilon=\delta$: there are $y\in U$ and $\varphi:U\to\mathbb{R}$ of class $C^{2}$ on $U$ such that $u-\varphi$ has a local maximum at $y$ relative to $U$ and

$$d_{E}(y,x_{0})<\delta,\quad |u(y)-u(x_{0})|<\delta,\quad \lVert D\varphi(y)-p\rVert<\delta,\quad d_{\mathcal{S}(n)}\bigl(D^{2}\varphi(y),X\bigr)<\delta .$$

Thus the quadruple $\bigl(y,u(y),D\varphi(y),D^{2}\varphi(y)\bigr)$ satisfies the four conditions above, with $s=u(y)$, $q=D\varphi(y)$ and $Y=D^{2}\varphi(y)$; recall that $Y$ lies in $\mathcal{S}(n)$ by claim 2 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian. Writing

$$a=F\bigl(y,u(y),D\varphi(y),D^{2}\varphi(y)\bigr),$$

we therefore have $|a-c|<c$, and claim 9 of Properties of the Absolute Value in an Ordered Field gives $-c<a-c$, whence $0<a$ by claim 1 of Elementary Order Arithmetic in an Ordered Field.

On the other hand $u$ is a viscosity subsolution of $F$ on $U$, and $\varphi$ is of class $C^{2}$ on $U$ with $u-\varphi$ having a local maximum at $y\in U$ relative to $U$, so Viscosity Subsolution and Supersolution of a Second-Order Equation gives $a\le 0$. Together with $0<a$ this yields $0<0$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, which is false. Hence $c\le 0$, which is claim 1.

Claim 2. Suppose, seeking a contradiction, that $0\le c$ fails; by totality of the order, $c<0$, and by claim 4 of Elementary Order Arithmetic in an Ordered Field the element $-c$ satisfies $0<-c$.

Apply the continuity hypothesis with $\varepsilon=-c$, obtaining $\delta\in\mathbb{R}$ with $0<\delta$ as above, and then the approximation hypothesis with $\varepsilon=\delta$, obtaining $y\in U$ and $\varphi:U\to\mathbb{R}$ of class $C^{2}$ on $U$ such that $u-\varphi$ has a local minimum at $y$ relative to $U$ and the four displayed conditions hold. With

$$a=F\bigl(y,u(y),D\varphi(y),D^{2}\varphi(y)\bigr)$$

we get $|a-c|<-c$, and claim 9 of Properties of the Absolute Value in an Ordered Field gives $a-c<-c$, whence $a<0$ by claim 1 of Elementary Order Arithmetic in an Ordered Field.

On the other hand $u$ is a viscosity supersolution of $F$ on $U$ and $u-\varphi$ has a local minimum at $y$ relative to $U$, so Viscosity Subsolution and Supersolution of a Second-Order Equation gives $0\le a$. Together with $a<0$ this yields $0<0$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, which is false. Hence $0\le c$, which is claim 2.