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

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $D\subseteq\mathbb{R}^{n}$ be open, let $P:D\to\mathbb{R}$ be of class $C^{2}$ on $D$, let $c\in\mathbb{R}$ and let $F$ be a second-order equation operator on $D$. For $x\in D$ and $X\in\mathcal{S}(n)$ the matrix $X+cD^{2}P(x)$ lies in $\mathcal{S}(n)$ by Second-Order Equations on Euclidean Open Sets §matrices, so the formula

defines a second-order equation operator $F_{c}$ on $D$.

Then the following hold.

1. (Viscosity subsolutions)¶ A function $u:D\to\mathbb{R}$ is a viscosity subsolution of $F$ on $D$ if and only if $u-cP$ is a viscosity subsolution of $F_{c}$ on $D$.

2. (Viscosity supersolutions)¶ A function $u:D\to\mathbb{R}$ is a viscosity supersolution of $F$ on $D$ if and only if $u-cP$ is a viscosity supersolution of $F_{c}$ on $D$.

3. (Classical sub- and supersolutions)¶ Let $\varphi:D\to\mathbb{R}$ be of class $C^{2}$ on $D$, so that $\varphi-cP$ is of class $C^{2}$ on $D$ by Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. Then $\varphi$ is a classical subsolution (respectively supersolution) of $F$ on $D$ if and only if $\varphi-cP$ is a classical subsolution (respectively supersolution) of $F_{c}$ on $D$.

4. (Structural properties)¶ Let $\gamma\in\mathbb{R}$ be positive. If $F$ is continuous, degenerate elliptic, strictly proper with constant $\gamma$, or convex in $(r,p,X)$, then $F_{c}$ has the same property.