Proof of Gauge Covariance under Subtraction of a Multiple of a $C^2$ Function: Viscosity and Classical Sub- and Supersolutions and the Structural Properties of the Operator

Each result cited is universally quantified over the data in its own statement. For a function $\psi$ of class $C^{2}$ on $D$, claims 3 and 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied to first and then to second partial derivatives, show that $\psi+cP$ and $\psi-cP$ are of class $C^{2}$ on $D$ with

We also note the inversion identity: $F_{c}$ is a second-order equation operator on $D$, so $(F_{c})_{-c}$ is defined by the same formula, and for all $(x,r,p,X)$ the vector-space axioms give

Finally, $-cP$ is of class $C^{2}$ on $D$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence continuous at every point of $D$ relative to $D$ by Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §value, hence upper and lower semicontinuous on $D$ by claim 2 of Semicontinuity Under Negation and Characterization of Continuity.

Claim 1. If $u$ is a viscosity subsolution of $F$, then $u-cP$ is one of $F_{c}$. The function $u-cP$ is upper semicontinuous on $D$ by claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions. Let $\psi$ be of class $C^{2}$ on $D$ and let $x_{0}\in D$ be a point at which $(u-cP)-\psi$ has a local maximum relative to $D$. Since $(u-cP)-\psi=u-(\psi+cP)$ pointwise, $u-(\psi+cP)$ has a local maximum at $x_{0}$, and $\psi+cP$ is of class $C^{2}$ on $D$; so the subsolution property of $u$ and (1) give

Conversely, if $u-cP$ is a viscosity subsolution of $F_{c}$, the implication just proved, applied to the operator $F_{c}$, the constant $-c$ and the function $u-cP$, shows that $(u-cP)-(-c)P=u$ is a viscosity subsolution of $(F_{c})_{-c}$, which is $F$ by (2).

Claim 2. Identical, with lower semicontinuity (claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions), local minima and the reversed inequality.

Claim 3. By (1) with $\psi=\varphi$ and the definition of $F_{c}$, for every $x\in D$,

so the inequalities of Classical Subsolution and Supersolution of a Second-Order Equation for $\varphi$ and $F$ and for $\varphi-cP$ and $F_{c}$ hold at the same points.

Claim 4. Fix $x\in D$ and write $\Phi_{x}(r,p,X)=(r+cP(x),p+cDP(x),X+cD^{2}P(x))$, so that $F_{c}(x,r,p,X)=F(x,\Phi_{x}(r,p,X))$.

Degenerate ellipticity. If $X\preceq Y$, then $X+cD^{2}P(x)\preceq Y+cD^{2}P(x)$ by Second-Order Equations on Euclidean Open Sets §matrices, so degenerate ellipticity of $F$ gives $F_{c}(x,r,p,Y)\le F_{c}(x,r,p,X)$.

Strict properness. Condition 1 of Strictly Proper Second-Order Equation Operator is the previous paragraph. If $s\le r$, then $s+cP(x)\le r+cP(x)$ and $(r+cP(x))-(s+cP(x))=r-s$, so condition 2 for $F$ at $(x,s+cP(x),p+cDP(x),X+cD^{2}P(x))$ gives $\gamma(r-s)\le F_{c}(x,r,p,X)-F_{c}(x,s,p,X)$.

Convexity. For $0\le t\le1$ the vector-space axioms give $\Phi_{x}\bigl((1-t)(r,p,X)+t(s,q,Y)\bigr)=(1-t)\Phi_{x}(r,p,X)+t\Phi_{x}(s,q,Y)$ componentwise, since $(1-t)+t=1$; so the convexity inequality of Second-Order Equation Operator Convex in the Value, Gradient and Matrix Variables for $F$ at the triples $\Phi_{x}(r,p,X)$ and $\Phi_{x}(s,q,Y)$ is the convexity inequality for $F_{c}$ at $(r,p,X)$ and $(s,q,Y)$.

Continuity. Let $(x_{0},r_{0},p_{0},X_{0})$ be a quadruple and $\varepsilon$ positive. By continuity of $F$ at $(x_{0},\Phi_{x_{0}}(r_{0},p_{0},X_{0}))$ (Continuity of a Second-Order Equation Operator §at-point) there is a positive $\eta$ such that every quadruple whose four components are within $\eta$ of those of $(x_{0},\Phi_{x_{0}}(r_{0},p_{0},X_{0}))$, in $d_{E}$, $|\cdot|$, $\lVert\cdot\rVert$ and $d_{\mathcal{S}(n)}$ respectively, has $F$-value within $\varepsilon$ of $F(x_{0},\Phi_{x_{0}}(r_{0},p_{0},X_{0}))$. Put $\eta'=\eta\,(2(|c|+1))^{-1}$, positive. By claims Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §value and Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §hessian of Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function there is a positive $\delta_{1}$ such that every $y\in D$ with $d_{E}(y,x_{0})<\delta_{1}$ satisfies $|P(y)-P(x_{0})|<\eta'$, $\lVert DP(y)-DP(x_{0})\rVert<\eta'$ and $d_{\mathcal{S}(n)}(D^{2}P(y),D^{2}P(x_{0}))<\eta'$. Let $\delta$ be the least of $\delta_{1}$ and $\tfrac{\eta}{2}$. If $d_{E}(y,x_{0})$, $|s-r_{0}|$, $\lVert q-p_{0}\rVert$ and $d_{\mathcal{S}(n)}(Y,X_{0})$ are all less than $\delta$, then by the triangle inequalities and homogeneity (claims 4 and 5 of Properties of the Absolute Value in an Ordered Field, claims 5 and 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, claim 5 of Properties of the Norm of a Symmetric Real Matrix, with $d_{\mathcal{S}(n)}(A,B)=\lVert A-B\rVert$ by Second-Order Equations on Euclidean Open Sets §matrices)

and in the same way $\lVert(q+cDP(y))-(p_{0}+cDP(x_{0}))\rVert<\eta$ and $d_{\mathcal{S}(n)}(Y+cD^{2}P(y),X_{0}+cD^{2}P(x_{0}))<\eta$, while $d_{E}(y,x_{0})<\eta$. Hence $|F_{c}(y,s,q,Y)-F_{c}(x_{0},r_{0},p_{0},X_{0})|<\varepsilon$, and $F_{c}$ is continuous by Continuity of a Second-Order Equation Operator §continuous.