Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$

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 $C^{2}(U)$ as in Hilbert Triples: Standing Notation and Background (v), and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$. For a form $X$ on $V$ or on $H$, $-X$ is the form $(-1)X$ of Hilbert Triples: Standing Notation and Background (vi), with $(-Y)|_{V}=-(Y|_{V})$ for $Y\in\mathrm{Sym}(H)$ and $\lVert -Y\rVert=\lVert Y\rVert$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (x) and Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (iii); for a function $u$ on $U$, $-u$ is the function with value $-u(x)$ at $x$, and for $u\in C^{2}(U)$ one has $-u\in C^{2}(U)$ with $D(-u)(x)=-Du(x)$ and $D^{2}(-u)(x)=-D^{2}u(x)$ for $x\in U$ by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (iii). For $\delta>0$ the $\delta$-envelopes of $u$ and of $-u$ are related by Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (ii). Define $\tilde{F}:W\times\mathbb{R}\times H\times\mathrm{Sym}(V)\to\mathbb{R}$ by

$$\tilde{F}(x,r,p,X)=-F(x,-r,-p,-X).$$

Then the following hold.

1. (Operator)(i) $\tilde{F}$ is a second-order equation operator on $U$ relative to $(H,V,A)$, and applying the same construction to $\tilde{F}$ returns $F$.

2. (Ellipticity)(ii) $\tilde{F}$ is degenerate elliptic if and only if $F$ is.

3. (Shifts)(iii) For every real $\delta>0$ and every $(x,r,p,Y)\in W\times\mathbb{R}\times H\times\mathrm{Sym}(H)$, the $\delta$-shifts satisfy

$$\tilde{F}^{+}_{\delta}(x,r,p,Y)=-F^{-}_{\delta}(x,-r,-p,-Y)\qquad\text{and}\qquad \tilde{F}^{-}_{\delta}(x,r,p,Y)=-F^{+}_{\delta}(x,-r,-p,-Y).$$

4. (Classical solutions)(iv) Let $u\in C^{2}(U)$. Then $u$ is a classical subsolution of $F$ on $U$ if and only if $-u$ is a classical supersolution of $\tilde{F}$ on $U$; $u$ is a classical supersolution of $F$ on $U$ if and only if $-u$ is a classical subsolution of $\tilde{F}$ on $U$; and $u$ is a classical solution of $F$ on $U$ if and only if $-u$ is a classical solution of $\tilde{F}$ on $U$.

5. (Viscosity solutions)(v) Let $u:U\to\mathbb{R}$. Then $u$ is a viscosity subsolution of $F$ on $U$ if and only if $-u$ is a viscosity supersolution of $\tilde{F}$ on $U$; $u$ is a viscosity supersolution of $F$ on $U$ if and only if $-u$ is a viscosity subsolution of $\tilde{F}$ on $U$; and $u$ is a viscosity solution of $F$ on $U$ if and only if $-u$ is a viscosity solution of $\tilde{F}$ on $U$. In each equivalence the local boundedness required of $u$ by the one definition holds if and only if that required of $-u$ by the other holds.