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

Real arithmetic and order facts are those of The Real Numbers: Standing Notation and Background, and vector identities, in $H$ and in the vector spaces $\mathrm{Sym}(H)$ and $\mathrm{Sym}(V)$ (Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (i)), are those of Elementary Identities in a Vector Space. The proposition is stated for an arbitrary degenerate elliptic operator $F$ and an arbitrary $u\in C^{2}(U)$, so each claim, once proved, may be applied to other such data; claim 4 applies claim 2 in this way.

Claim 1. Every member of $C^{2}(U)$ is continuous on $U$ by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space (ii), continuity being relative to $U$ in the ambient metric space $(H,d_{H})$ with values in $(\mathbb{R},d_{\mathbb{R}})$. Hence Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (vi) shows that $u$ is bounded above and bounded below near each point of $U$, and that $u^{-}_{\delta}(x)=u(x)-\delta h(x)$ and $u^{+}_{\delta}(x)=u(x)+\delta h(x)$ for every $\delta>0$ and $x\in V\cap U$.

Claim 2. Let $\delta$, $\varphi$ and $\hat{x}$ be as in the claim. Put $\psi=u-\varphi$. By Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (iv), $\psi\in C^{2}(U)$ with $D\psi(\hat{x})=Du(\hat{x})-D\varphi(\hat{x})$ and $D^{2}\psi(\hat{x})=D^{2}u(\hat{x})-D^{2}\varphi(\hat{x})$. By claim 1, for $x\in V\cap U$,

$$u^{-}_{\delta}(x)-\varphi(x)=u(x)-\delta h(x)-\varphi(x)=\psi(x)-\delta h(x),$$

so the function $\psi-\delta h$ on $V\cap U$ has a local maximum at $\hat{x}$ relative to $V\cap U$. By First- and Second-Order Conditions at a Local Extremum of a $C^2$ Function Penalised by $h$ on the Small Space (i) with $\lambda=\delta$: $\hat{x}\in W$,

$$\delta A\hat{x}=D\psi(\hat{x})=Du(\hat{x})-D\varphi(\hat{x}),\qquad\text{and}\qquad\bigl(D^{2}u(\hat{x})-D^{2}\varphi(\hat{x})\bigr)|_{V}\preceq\delta I_{V}.$$

Writing the difference of forms as $D^{2}u(\hat{x})+(-1)D^{2}\varphi(\hat{x})$ (Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv)), Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (x) shows that the left side of the last relation is $D^{2}u(\hat{x})|_{V}-D^{2}\varphi(\hat{x})|_{V}$; adding $D^{2}\varphi(\hat{x})|_{V}$ to both sides (Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (viii)) and simplifying $(X-Z)+Z=X$ and $\delta I_{V}+Z=Z+\delta I_{V}$ in $\mathrm{Sym}(V)$ gives

$$D^{2}u(\hat{x})|_{V}\preceq D^{2}\varphi(\hat{x})|_{V}+\delta I_{V}. \tag{1}$$

Now evaluate the shift. By Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts (ii), claim 1 (which gives $u^{-}_{\delta}(\hat{x})+\delta h(\hat{x})=u(\hat{x})$), and $D\varphi(\hat{x})+\delta A\hat{x}=Du(\hat{x})$,

$$F^{-}_{\delta}\bigl(\hat{x},u^{-}_{\delta}(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x})\bigr)=F\bigl(\hat{x},\,u^{-}_{\delta}(\hat{x})+\delta h(\hat{x}),\,D\varphi(\hat{x})+\delta A\hat{x},\,D^{2}\varphi(\hat{x})|_{V}+\delta I_{V}\bigr)=F\bigl(\hat{x},\,u(\hat{x}),\,Du(\hat{x}),\,D^{2}\varphi(\hat{x})|_{V}+\delta I_{V}\bigr).$$

By (1) and Degenerate Elliptic Second-Order Equation Operator on a Hilbert Triple (i), applied with $X=D^{2}u(\hat{x})|_{V}$ and $Y=D^{2}\varphi(\hat{x})|_{V}+\delta I_{V}$,

$$F\bigl(\hat{x},u(\hat{x}),Du(\hat{x}),D^{2}\varphi(\hat{x})|_{V}+\delta I_{V}\bigr)\le F\bigl(\hat{x},u(\hat{x}),Du(\hat{x}),D^{2}u(\hat{x})|_{V}\bigr)\le0,$$

the last inequality by Classical Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i) at the point $\hat{x}\in W$. This proves the claim.

Claim 3. By claim 1, $u$ is bounded above near each point of $U$, as Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i) requires. Let $\delta>0$, $\varphi\in C^{2}(U)$, let $\hat{x}\in V\cap U$ be a local maximum of $u^{-}_{\delta}-\varphi$ relative to $V\cap U$, and let $\varepsilon>0$. Take $y=\hat{x}$, $s=u^{-}_{\delta}(\hat{x})$, $q=D\varphi(\hat{x})$ and $Y=D^{2}\varphi(\hat{x})$. By claim 2, $y\in W$ and $F^{-}_{\delta}(y,s,q,Y)\le0\le\varepsilon$. The five closeness conditions hold because each left side is the norm or absolute value of a zero element, and $0<\varepsilon$: $y-\hat{x}=0_{H}$ and $q-D\varphi(\hat{x})=0_{H}$ (an element minus itself is zero, Elementary Identities in a Vector Space) have $|\cdot|_{H}$ equal to $0$ by Elementary Identities in a Real Inner Product Space (ii); $u^{-}_{\delta}(y)-u^{-}_{\delta}(\hat{x})=0$ and $s-u^{-}_{\delta}(\hat{x})=0$ have absolute value $0$ by claim 1 of Properties of the Absolute Value in an Ordered Field; and $Y-D^{2}\varphi(\hat{x})=0_{\mathrm{Sym}}$ in the vector space $\mathrm{Sym}(H)$ has norm $0$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (iii). Hence $u$ is a viscosity subsolution of $F$ on $U$.

Claim 4. Suppose $u$ is a classical supersolution of $F$ on $U$, and let $\tilde{F}(x,r,p,X)=-F(x,-r,-p,-X)$ be the operator of Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$. We check that the pair $(\tilde{F},-u)$ satisfies the hypotheses of this proposition and of claim 2: $\tilde{F}$ is a second-order equation operator on $U$ relative to $(H,V,A)$ by Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$ (i); it is degenerate elliptic by Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$ (ii); $-u\in C^{2}(U)$ by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (iii); and $-u$ is a classical subsolution of $\tilde{F}$ on $U$ by Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$ (iv), because $u$ is a classical supersolution of $F$. (Claim 2 was proved without using claim 4, so this application is not circular.) For $\varphi\in C^{2}(U)$ we also have $-\varphi\in C^{2}(U)$ with $D(-\varphi)(\hat{x})=-D\varphi(\hat{x})$ and $D^{2}(-\varphi)(\hat{x})=-D^{2}\varphi(\hat{x})$ by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (iii). Let $\delta$, $\varphi$ and $\hat{x}$ be as in the claim. By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (ii), $(-u)^{-}_{\delta}=-u^{+}_{\delta}$ on $V\cap U$, so $(-u)^{-}_{\delta}-(-\varphi)=-(u^{+}_{\delta}-\varphi)$; negating the defining inequality of the local minimum of $u^{+}_{\delta}-\varphi$ at $\hat{x}$ (Local Minimum of a Function Relative to a Subset of a Metric Space, claim 4 of Elementary Order Arithmetic in an Ordered Field) shows that $(-u)^{-}_{\delta}-(-\varphi)$ has a local maximum at $\hat{x}$ relative to $V\cap U$. Claim 2, applied to $-u$, $\tilde{F}$ and the test function $-\varphi$, gives $\hat{x}\in W$ and

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

By Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$ (iii), the left side equals $-F^{+}_{\delta}(\hat{x},u^{+}_{\delta}(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x}))$, since $-(-s)=s$ for real numbers, vectors and forms. Hence $0\le F^{+}_{\delta}(\hat{x},u^{+}_{\delta}(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x}))$.

Claim 5. By claim 1, $u$ is bounded below near each point of $U$. Given $\delta>0$, $\varphi\in C^{2}(U)$, a local minimum $\hat{x}\in V\cap U$ of $u^{+}_{\delta}-\varphi$ relative to $V\cap U$, and $\varepsilon>0$, take $(y,s,q,Y)=(\hat{x},u^{+}_{\delta}(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x}))$. By claim 4, $y\in W$ and $-\varepsilon\le0\le F^{+}_{\delta}(y,s,q,Y)$, and the five closeness conditions hold exactly as in claim 3, with $u^{+}_{\delta}$ in place of $u^{-}_{\delta}$ in the two value conditions: each left side is again the norm or absolute value of a zero element. Hence $u$ is a viscosity supersolution of $F$ on $U$ by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (ii).

Claim 6. A classical solution is both a classical subsolution and a classical supersolution (Classical Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (iii)), so by claims 3 and 5 it is both a viscosity subsolution and a viscosity supersolution, and it is bounded above and below near each point of $U$ by claim 1; that is, it is a viscosity solution of $F$ on $U$ (Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (iii)).