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

Throughout, $\mathrm{Sym}(V)$ and $\mathrm{Sym}(H)$ are vector spaces over $\mathbb{R}$ under the operations of Hilbert Triples: Standing Notation and Background (vi) (Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (i)), so identities such as $-(-X)=X$, $-(X-Z)=(-X)+Z$ and $(-1)(\lambda X)=(-\lambda)X$ hold for forms by Elementary Identities in a Vector Space, exactly as they hold in $H$ and in $\mathbb{R}$. Restriction is linear: $(X+Z)|_{V}=X|_{V}+Z|_{V}$ and $(\lambda X)|_{V}=\lambda(X|_{V})$ for $X,Z\in\mathrm{Sym}(H)$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (x). Real order facts are those of The Real Numbers: Standing Notation and Background; in particular $s\le t$ if and only if $-t\le-s$ (claim 4 of Elementary Order Arithmetic in an Ordered Field). The lemma is stated for an arbitrary operator $F$ and, in claims 4 and 5, an arbitrary function $u$; every claim is therefore proved for all such data at once, and once proved it may be applied to other data, in particular to the pair $(\tilde{F},-u)$, with $\tilde{F}$ an operator by claim 1. Such applications are marked below.

Claim 1. For $(x,r,p,X)\in W\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ the quadruple $(x,-r,-p,-X)$ lies in the same set, so $\tilde{F}(x,r,p,X)=-F(x,-r,-p,-X)$ is a well-defined real number and $\tilde{F}$ is a second-order equation operator on $U$ relative to $(H,V,A)$. Applying the construction to $\tilde{F}$ gives the operator $(x,r,p,X)\mapsto-\tilde{F}(x,-r,-p,-X)=-\bigl(-F(x,-(-r),-(-p),-(-X))\bigr)=F(x,r,p,X)$.

Claim 2. Suppose $F$ is degenerate elliptic, and let $x\in W$, $r\in\mathbb{R}$, $p\in H$ and $X,Y\in\mathrm{Sym}(V)$ with $X\preceq Y$. Multiplying by $-1\le0$ (Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (viii)) gives $-Y\preceq-X$, so Degenerate Elliptic Second-Order Equation Operator on a Hilbert Triple (i) for $F$ yields $F(x,-r,-p,-X)\le F(x,-r,-p,-Y)$. Negating, $\tilde{F}(x,r,p,Y)=-F(x,-r,-p,-Y)\le-F(x,-r,-p,-X)=\tilde{F}(x,r,p,X)$; hence $\tilde{F}$ is degenerate elliptic. Conversely, if $\tilde{F}$ is degenerate elliptic, then applying what was just proved to the operator $\tilde{F}$ in place of $F$ shows that the operator obtained from $\tilde{F}$ by the same construction is degenerate elliptic, and that operator is $F$ by claim 1.

Claim 3. Let $\delta>0$ and $(x,r,p,Y)\in W\times\mathbb{R}\times H\times\mathrm{Sym}(H)$. By Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts (ii) for $\tilde{F}$ and the definition of $\tilde{F}$,

$$\tilde{F}^{+}_{\delta}(x,r,p,Y)=\tilde{F}\bigl(x,\,r-\delta h(x),\,p-\delta Ax,\,Y|_{V}-\delta I_{V}\bigr)=-F\bigl(x,\,-(r-\delta h(x)),\,-(p-\delta Ax),\,-(Y|_{V}-\delta I_{V})\bigr).$$

Now $-(r-\delta h(x))=(-r)+\delta h(x)$, $-(p-\delta Ax)=(-p)+\delta Ax$, and $-(Y|_{V}-\delta I_{V})=-(Y|_{V})+\delta I_{V}=(-Y)|_{V}+\delta I_{V}$ by the linearity of restriction. Hence the right side is $-F\bigl(x,(-r)+\delta h(x),(-p)+\delta Ax,(-Y)|_{V}+\delta I_{V}\bigr)=-F^{-}_{\delta}(x,-r,-p,-Y)$, again by Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts (ii), now for $F$ at $(x,-r,-p,-Y)$. Similarly

$$\tilde{F}^{-}_{\delta}(x,r,p,Y)=-F\bigl(x,\,-(r+\delta h(x)),\,-(p+\delta Ax),\,-(Y|_{V}+\delta I_{V})\bigr)=-F\bigl(x,(-r)-\delta h(x),(-p)-\delta Ax,(-Y)|_{V}-\delta I_{V}\bigr)=-F^{+}_{\delta}(x,-r,-p,-Y).$$

Claim 4. Let $u\in C^{2}(U)$ and $x\in W$. By the preamble, $-u\in C^{2}(U)$ with $D(-u)(x)=-Du(x)$ and $D^{2}(-u)(x)=-D^{2}u(x)$, and $(-D^{2}u(x))|_{V}=-(D^{2}u(x)|_{V})$. Therefore

$$\tilde{F}\bigl(x,(-u)(x),D(-u)(x),D^{2}(-u)(x)|_{V}\bigr)=-F\bigl(x,u(x),Du(x),-(-(D^{2}u(x)|_{V}))\bigr)=-F\bigl(x,u(x),Du(x),D^{2}u(x)|_{V}\bigr).$$

Consequently $F(x,u(x),Du(x),D^{2}u(x)|_{V})\le0$ for every $x\in W$ if and only if $0\le\tilde{F}(x,(-u)(x),D(-u)(x),D^{2}(-u)(x)|_{V})$ for every $x\in W$; by Classical Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i) and Classical Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (ii), $u$ is a classical subsolution of $F$ on $U$ if and only if $-u$ is a classical supersolution of $\tilde{F}$ on $U$. The second equivalence follows by applying the first, proved for arbitrary data, to the pair $(\tilde{F},-u)$, using claim 1 and $-(-u)=u$: $-u$ is a classical subsolution of $\tilde{F}$ if and only if $u$ is a classical supersolution of $F$. The third is the conjunction of the first two (Classical Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (iii)).

Claim 5. We first prove the two forward implications.

(a) If $u$ is a viscosity subsolution of $F$ on $U$, then $-u$ is a viscosity supersolution of $\tilde{F}$ on $U$. By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i), $u$ is bounded above near each point of $U$, so by Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (ii) $-u$ is bounded below near each point of $U$ and $(-u)^{+}_{\delta}=-u^{-}_{\delta}$ on $V\cap U$ for every $\delta>0$. Let $\delta>0$, $\varphi\in C^{2}(U)$, $\varepsilon>0$, and let $\hat{x}\in V\cap U$ be a point at which $(-u)^{+}_{\delta}-\varphi$ has a local minimum relative to $V\cap U$: there is a real $\rho>0$ with $(-u)^{+}_{\delta}(\hat{x})-\varphi(\hat{x})\le(-u)^{+}_{\delta}(y)-\varphi(y)$ for every $y\in V\cap U$ with $d_{H}(\hat{x},y)<\rho$ (Local Minimum of a Function Relative to a Subset of a Metric Space). Since $(-u)^{+}_{\delta}-\varphi=-\bigl(u^{-}_{\delta}-(-\varphi)\bigr)$, negating this inequality shows that $u^{-}_{\delta}-(-\varphi)$ has a local maximum at $\hat{x}$ relative to $V\cap U$ (Local Maximum of a Function Relative to a Subset of a Metric Space, same $\rho$). As $-\varphi\in C^{2}(U)$ with $D(-\varphi)(\hat{x})=-D\varphi(\hat{x})$ and $D^{2}(-\varphi)(\hat{x})=-D^{2}\varphi(\hat{x})$ (Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (iii)), the subsolution property of $u$ with the test function $-\varphi$ provides $y\in W$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ with

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

where we wrote $q-(-D\varphi(\hat{x}))=q+D\varphi(\hat{x})$ and $Y-(-D^{2}\varphi(\hat{x}))=Y+D^{2}\varphi(\hat{x})$. We claim that $(y,-s,-q,-Y)$ witnesses the supersolution condition for $-u$, $\tilde{F}$, $\varphi$, $\hat{x}$ and $\varepsilon$. Indeed $y\in W$ and $|y-\hat{x}|_{H}<\varepsilon$ are unchanged; $|(-u)^{+}_{\delta}(y)-(-u)^{+}_{\delta}(\hat{x})|=|-(u^{-}_{\delta}(y)-u^{-}_{\delta}(\hat{x}))|<\varepsilon$ and $|-s-(-u)^{+}_{\delta}(\hat{x})|=|-(s-u^{-}_{\delta}(\hat{x}))|<\varepsilon$ by claim 2 of Properties of the Absolute Value in an Ordered Field; $|-q-D\varphi(\hat{x})|_{H}=|-(q+D\varphi(\hat{x}))|_{H}<\varepsilon$ by Elementary Identities in a Real Inner Product Space (iv); $\lVert-Y-D^{2}\varphi(\hat{x})\rVert=\lVert-(Y+D^{2}\varphi(\hat{x}))\rVert=\lVert Y+D^{2}\varphi(\hat{x})\rVert<\varepsilon$ by the identity $\lVert-Z\rVert=\lVert Z\rVert$ recorded in the statement of the lemma; and by claim 3, $\tilde{F}^{+}_{\delta}(y,-s,-q,-Y)=-F^{-}_{\delta}(y,s,q,Y)\ge-\varepsilon$. By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (ii), $-u$ is a viscosity supersolution of $\tilde{F}$ on $U$.

(b) If $u$ is a viscosity supersolution of $F$ on $U$, then $-u$ is a viscosity subsolution of $\tilde{F}$ on $U$. The argument is the mirror image of (a). Now $u$ is bounded below near each point of $U$, so $-u$ is bounded above near each point of $U$ and $(-u)^{-}_{\delta}=-u^{+}_{\delta}$ by Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (ii). Given $\delta$, $\varphi$, $\varepsilon$ and a local maximum $\hat{x}$ of $(-u)^{-}_{\delta}-\varphi=-\bigl(u^{+}_{\delta}-(-\varphi)\bigr)$ relative to $V\cap U$, the point $\hat{x}$ is a local minimum of $u^{+}_{\delta}-(-\varphi)$ relative to $V\cap U$, and the supersolution property of $u$ with the test function $-\varphi$ provides $(y,s,q,Y)$ with $|y-\hat{x}|_{H}<\varepsilon$, $|u^{+}_{\delta}(y)-u^{+}_{\delta}(\hat{x})|<\varepsilon$, $|s-u^{+}_{\delta}(\hat{x})|<\varepsilon$, $|q+D\varphi(\hat{x})|_{H}<\varepsilon$, $\lVert Y+D^{2}\varphi(\hat{x})\rVert<\varepsilon$ and $-\varepsilon\le F^{+}_{\delta}(y,s,q,Y)$. Then $(y,-s,-q,-Y)$ satisfies the five closeness conditions for $-u$ at $\hat{x}$: $|y-\hat{x}|_{H}<\varepsilon$ is unchanged; using $(-u)^{-}_{\delta}=-u^{+}_{\delta}$, $|(-u)^{-}_{\delta}(y)-(-u)^{-}_{\delta}(\hat{x})|=|-(u^{+}_{\delta}(y)-u^{+}_{\delta}(\hat{x}))|<\varepsilon$ and $|-s-(-u)^{-}_{\delta}(\hat{x})|=|-(s-u^{+}_{\delta}(\hat{x}))|<\varepsilon$ (claim 2 of Properties of the Absolute Value in an Ordered Field); and $|-q-D\varphi(\hat{x})|_{H}<\varepsilon$ and $\lVert-Y-D^{2}\varphi(\hat{x})\rVert<\varepsilon$ exactly as in (a). Finally $\tilde{F}^{-}_{\delta}(y,-s,-q,-Y)=-F^{+}_{\delta}(y,s,q,Y)\le\varepsilon$ by claim 3. Hence $-u$ is a viscosity subsolution of $\tilde{F}$ on $U$.

Converses. Assertions (a) and (b) were proved for an arbitrary operator and an arbitrary function, so they may be applied to the pair $(\tilde{F},-u)$, for which the construction returns $F$ (claim 1) and $-(-u)=u$. If $-u$ is a viscosity supersolution of $\tilde{F}$ on $U$, then (b) for this pair shows that $u$ is a viscosity subsolution of $F$ on $U$. Likewise, if $-u$ is a viscosity subsolution of $\tilde{F}$ on $U$, then (a) for this pair shows that $u$ is a viscosity supersolution of $F$ on $U$. This proves the first two equivalences; the third is their conjunction (Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (iii)). The final sentence of the claim is the content of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (ii): $u$ is bounded above near each point of $U$ if and only if $-u$ is bounded below, and $u$ is bounded below if and only if $-u$ is bounded above; for the third equivalence both statements are used together.