Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be nonempty and open in $H$, with $W=D(A)\cap U$ and the class $C^{2}(U)$ as in Hilbert Triples: Standing Notation and Background (v), let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$ with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$, and let $u:U\to\mathbb{R}$. For each real $\delta>0$ the $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$ of $u$ are functions on $V\cap U$, defined when $u$ is bounded above, respectively below, near each point of $U$; local bounds, local extrema and the classes $C^{2}(U)$ are those fixed there, $\lVert\cdot\rVert$ is the norm on $\mathrm{Sym}(H)$, and $|s|$ is the absolute value of a real number $s$. Since $W\subseteq V\cap U$, the $\delta$-envelopes are defined at every point of $W$.

1. (Viscosity subsolution)(i) Suppose that $u$ is bounded above near each point of $U$. The function $u$ is a viscosity subsolution of $F$ on $U$ if for every real $\delta>0$, every $\varphi\in C^{2}(U)$, every point $\hat{x}\in V\cap U$ 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 $V\cap U$, and every real $\varepsilon>0$, there exist $y\in W$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ such that

$$|y-\hat{x}|_{H}<\varepsilon,\qquad |u^{-}_{\delta}(y)-u^{-}_{\delta}(\hat{x})|<\varepsilon,\qquad |s-u^{-}_{\delta}(\hat{x})|<\varepsilon,$$ $$|q-D\varphi(\hat{x})|_{H}<\varepsilon,\qquad \lVert Y-D^{2}\varphi(\hat{x})\rVert<\varepsilon,\qquad F^{-}_{\delta}(y,s,q,Y)\le\varepsilon .$$

2. (Viscosity supersolution)(ii) Suppose that $u$ is bounded below near each point of $U$. The function $u$ is a viscosity supersolution of $F$ on $U$ if for every real $\delta>0$, every $\varphi\in C^{2}(U)$, every point $\hat{x}\in V\cap U$ 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 $V\cap U$, and every real $\varepsilon>0$, there exist $y\in W$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ such that

$$|y-\hat{x}|_{H}<\varepsilon,\qquad |u^{+}_{\delta}(y)-u^{+}_{\delta}(\hat{x})|<\varepsilon,\qquad |s-u^{+}_{\delta}(\hat{x})|<\varepsilon,$$ $$|q-D\varphi(\hat{x})|_{H}<\varepsilon,\qquad \lVert Y-D^{2}\varphi(\hat{x})\rVert<\varepsilon,\qquad -\varepsilon\le F^{+}_{\delta}(y,s,q,Y) .$$

3. (Viscosity solution)(iii) Suppose that $u$ is bounded above near each point of $U$ and bounded below near each point of $U$. The function $u$ is a viscosity solution of $F$ on $U$ if it is both a viscosity subsolution and a viscosity supersolution of $F$ on $U$.