Restriction of the Equation and the Locality of the Viscosity Sub- and Supersolution Properties 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), and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$. Products of sets are Cartesian products.

For a nonempty $U'\subseteq U$ open in $H$ we write $W'=D(A)\cap U'$, and $F|_{U'}$ for the function on $W'\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ whose value at $(x,r,p,X)$ is $F(x,r,p,X)$; for a function $u$ on $U$ we write $u|_{U'}$ for the function on $U'$ whose value at $z\in U'$ is $u(z)$. Then the following hold.

1. (The restricted operator)(i) Let $U'\subseteq U$ be nonempty and open in $H$. Then $W'\subseteq W$, and $F|_{U'}$ is a second-order equation operator on $U'$ relative to $(H,V,A)$. For every real $\delta>0$ its $\delta$-shifts satisfy

$$\bigl(F|_{U'}\bigr)^{-}_{\delta}(x,r,p,Y)=F^{-}_{\delta}(x,r,p,Y)\quad\text{and}\quad\bigl(F|_{U'}\bigr)^{+}_{\delta}(x,r,p,Y)=F^{+}_{\delta}(x,r,p,Y)$$

for every $(x,r,p,Y)\in W'\times\mathbb{R}\times H\times\mathrm{Sym}(H)$. If $F$ is degenerate elliptic, then so is $F|_{U'}$.

2. (Restriction of sub- and supersolutions)(ii) Let $U'\subseteq U$ be nonempty and open in $H$ and let $u:U\to\mathbb{R}$. If $u$ is a viscosity subsolution of $F$ on $U$, then $u|_{U'}$ is a viscosity subsolution of $F|_{U'}$ on $U'$. If $u$ is a viscosity supersolution of $F$ on $U$, then $u|_{U'}$ is a viscosity supersolution of $F|_{U'}$ on $U'$.

3. (A local subsolution is a subsolution)(iii) Let $u:U\to\mathbb{R}$ and suppose that for every $x\in U$ there is a set $U_{x}\subseteq U$, open in $H$ and containing $x$, such that $u|_{U_{x}}$ is a viscosity subsolution of $F|_{U_{x}}$ on $U_{x}$. Then $u$ is bounded above near each point of $U$ and is a viscosity subsolution of $F$ on $U$. Likewise, if for every $x\in U$ there is such a set $U_{x}$ with $u|_{U_{x}}$ a viscosity supersolution of $F|_{U_{x}}$ on $U_{x}$, then $u$ is bounded below near each point of $U$ and is a viscosity supersolution of $F$ on $U$.