Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on a Hilbert Triple are Viscosity Sub- and Supersolutions

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be nonempty and open in $H$, with $C^{2}(U)$ and local bounds as fixed there, let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$ that is degenerate elliptic, and let $u\in C^{2}(U)$. For $\delta>0$ the $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$, the $\delta$-shifts $F^{-}_{\delta}$, $F^{+}_{\delta}$, and local extrema relative to $V\cap U$ are as defined there. Then the following hold.

1. (Local bounds and envelopes)(i) The function $u$ is continuous on $U$ by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space (ii), hence bounded above and bounded below near each point of $U$, and for every real $\delta>0$ and every $x\in V\cap U$, $u^{-}_{\delta}(x)=u(x)-\delta h(x)$ and $u^{+}_{\delta}(x)=u(x)+\delta h(x)$.

2. (Exact witnesses for subsolutions)(ii) Suppose that $u$ is a classical subsolution of $F$ on $U$. Let $\delta>0$, let $\varphi\in C^{2}(U)$ and let $\hat{x}\in V\cap U$ be a point at which the function $V\cap U\to\mathbb{R}$ with value $u^{-}_{\delta}(x)-\varphi(x)$ at $x$ has a local maximum relative to $V\cap U$. Then $\hat{x}\in W$ and

$$F^{-}_{\delta}\bigl(\hat{x},\,u^{-}_{\delta}(\hat{x}),\,D\varphi(\hat{x}),\,D^{2}\varphi(\hat{x})\bigr)\le0 .$$

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

4. (Exact witnesses for supersolutions)(iv) Suppose that $u$ is a classical supersolution of $F$ on $U$. Let $\delta>0$, let $\varphi\in C^{2}(U)$ and let $\hat{x}\in V\cap U$ be a point at which the function $V\cap U\to\mathbb{R}$ with value $u^{+}_{\delta}(x)-\varphi(x)$ at $x$ has a local minimum relative to $V\cap U$. Then $\hat{x}\in W$ and

$$0\le F^{+}_{\delta}\bigl(\hat{x},\,u^{+}_{\delta}(\hat{x}),\,D\varphi(\hat{x}),\,D^{2}\varphi(\hat{x})\bigr) .$$

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

6. (Solutions)(vi) If $u$ is a classical solution of $F$ on $U$, then $u$ is a viscosity solution of $F$ on $U$.