Proof of The Viscous Hamilton-Jacobi Operator with a Drift that is One-Sided Lipschitz on Sublevel Sets Satisfies the Operator Hypotheses of the Weighted-Penalty Comparison Principle

Each result cited is universally quantified over the data in its own statement. The order and arithmetic of $\mathbb{R}$ are those of the ordered field of real numbers; multiplying an inequality by a nonnegative real preserves it (claim 5 of Elementary Arithmetic in an Ordered Field), and $0<2^{-1}$ by claims 8 and 7 of Elementary Order Arithmetic in an Ordered Field, so $\tfrac{\theta}{2}$ and $\tfrac{\kappa}{2}$ are nonnegative. Write

Hypothesis Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §strictly-proper with $\gamma=\lambda$. If $X\preceq Y$ in $\mathcal{S}(n)$, then $\operatorname{tr}(X)\le\operatorname{tr}(Y)$ by The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound §monotone; multiplying by $\tfrac{\kappa}{2}\ge0$ and applying claim 4 of Elementary Order Arithmetic in an Ordered Field gives $-\tfrac{\kappa}{2}\operatorname{tr}(Y)\le-\tfrac{\kappa}{2}\operatorname{tr}(X)$, and adding the remaining terms gives

So $F$ is degenerate elliptic. For $s\le r$, all terms other than $\lambda r$ and $\lambda s$ cancel, and $F(x,r,p,X)-F(x,s,p,X)=\lambda(r-s)$ by distributivity. Thus $F$ is strictly proper with constant $\lambda$.

Hypothesis Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §convex. Fix $x\in D$, triples $(r,p,X)$ and $(s,q,Y)$, and $t$ with $0\le t\le1$. The maps $r\mapsto\lambda r$, $p\mapsto b(x)\cdot p$ (claim 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$) and $X\mapsto\operatorname{tr}(X)$ (claim 1 of Basic Properties of the Trace) are linear, and $G(x)=(1-t)G(x)+tG(x)$, so these terms of $F$ at the combined triple equal the same combination of their values at the two triples. For the quadratic term, The Squared Norm of a Convex Combination of Two Points, applied with weight $1-t$ on $p$ and $t$ on $q$, gives

since $t(1-t)\lVert p-q\rVert^{2}$ is a product of nonnegative reals (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). Multiplying by $\tfrac{\theta}{2}\ge0$ and adding the linear terms gives the inequality of Second-Order Equation Operator Convex in the Value, Gradient and Matrix Variables.

Hypothesis Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §continuity. Fix $(x_{0},r_{0},p_{0},X_{0})$ and a positive $\varepsilon$, and put $\varepsilon'=\varepsilon\cdot5^{-1}$ with $5=1+1+1+1+1$, positive by claims 3, 6, 7 and 5 of Elementary Order Arithmetic in an Ordered Field. For $(y,s,q,Y)$ put $h=q-p_{0}$. By the field axioms

using claims 2 and 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ for $b(y)\cdot q-b(x_{0})\cdot p_{0}$. We bound the six terms, assuming $\lVert h\rVert\le1$. (i) $|\lambda(s-r_{0})|=\lambda|s-r_{0}|$ by claim 4 of Properties of the Absolute Value in an Ordered Field. (ii) $|\lVert q\rVert^{2}-\lVert p_{0}\rVert^{2}|\le(2\lVert p_{0}\rVert+1)\lVert h\rVert$, exactly as in bound (iii) of the proof of claim 1 of The Viscous Hamilton-Jacobi Operator is Continuous, Strictly Proper and Satisfies the Structure Condition: expand $\lVert p_{0}+h\rVert^{2}$ by Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$, bound $|p_{0}\cdot h|$ by Cauchy-Schwarz Inequality for the Euclidean Dot Product, and use $\lVert h\rVert^{2}\le\lVert h\rVert$. (iii) $\lVert q\rVert\le\lVert p_{0}\rVert+1$ by claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so $|(b(y)-b(x_{0}))\cdot q|\le(\lVert p_{0}\rVert+1)\lVert b(y)-b(x_{0})\rVert$ by Cauchy-Schwarz Inequality for the Euclidean Dot Product. (iv) $|b(x_{0})\cdot h|\le\lVert b(x_{0})\rVert\,\lVert h\rVert$ by Cauchy-Schwarz Inequality for the Euclidean Dot Product. (v) $|\operatorname{tr}(Y)-\operatorname{tr}(X_{0})|\le\beta_{n}\,d_{\mathcal{S}(n)}(Y,X_{0})$ by claim 1 of Basic Properties of the Trace and The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound §norm-bound, with $d_{\mathcal{S}(n)}(Y,X_{0})=\lVert Y-X_{0}\rVert$ (Second-Order Equations on Euclidean Open Sets §matrices). (vi) By continuity of $G$ and of $b$ at $x_{0}$ there is a positive $\delta_{0}$ such that every $y\in D$ with $d_{E}(y,x_{0})<\delta_{0}$ satisfies $|G(y)-G(x_{0})|<\varepsilon'$ and $\lVert b(y)-b(x_{0})\rVert<\varepsilon'(\lVert p_{0}\rVert+1)^{-1}$. Group the six terms into five: $A=\lambda(s-r_{0})$, $B=\tfrac{\theta}{2}(\lVert q\rVert^{2}-\lVert p_{0}\rVert^{2})+b(x_{0})\cdot h$, $C=(b(y)-b(x_{0}))\cdot q$, $E=\tfrac{\kappa}{2}(\operatorname{tr}(Y)-\operatorname{tr}(X_{0}))$ and $G(y)-G(x_{0})$. By (ii), (iv) and the triangle inequality, $|B|\le c_{B}\lVert h\rVert$ with $c_{B}=\tfrac{\theta}{2}(2\lVert p_{0}\rVert+1)+\lVert b(x_{0})\rVert+1$. Let $\delta$ be the least (claim 9 of Elementary Order Arithmetic in an Ordered Field) of the positive numbers $1$, $\delta_{0}$, $\varepsilon'(\lambda+1)^{-1}$, $\varepsilon'c_{B}^{-1}$ and $\varepsilon'(\tfrac{\kappa}{2}\beta_{n}+1)^{-1}$. If $d_{E}(y,x_{0})$, $|s-r_{0}|$, $\lVert h\rVert$ and $d_{\mathcal{S}(n)}(Y,X_{0})$ are less than $\delta$, then $\lVert h\rVert\le1$, and by (i), (iii), (v), (vi) and the bound on $B$ each of $|A|$, $|B|$, $|C|$, $|E|$ and $|G(y)-G(x_{0})|$ is less than $\varepsilon'$ (claim 10 of Elementary Order Arithmetic in an Ordered Field, multiplying the bound on $\delta$ by the relevant positive factor). By the triangle inequality (claim 5 of Properties of the Absolute Value in an Ordered Field) and claim 3 of Elementary Order Arithmetic in an Ordered Field, $|F(y,s,q,Y)-F(x_{0},r_{0},p_{0},X_{0})|<5\varepsilon'=\varepsilon$. So $F$ is continuous at $(x_{0},r_{0},p_{0},X_{0})$, and as the quadruple was arbitrary $F$ is continuous.

Hypothesis Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §structure. Fix $R$ with $D_{R}$ nonempty and put $\Omega=D_{R}$. By Basic Properties of the Sublevel Sets of a Penalty §sublevel-sets, $\Omega$ is nonempty, open and bounded, so it satisfies Bounded Open Domain in Euclidean Space §domain, and $\overline{\Omega}$ is compact, nonempty and contained in $D$. The part without drift. By (1), $G_{0}$ is degenerate elliptic in the matrix variable in the sense of Structure Condition: a Degenerate Elliptic Operator with a Continuous Inhomogeneity. The restriction $f$ of $G$ to $\overline{\Omega}$ is continuous on $\overline{\Omega}$ by claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map, so A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity gives a modulus of continuity $\omega_{1}$ that is nondecreasing (A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity §monotone) and dominates the oscillation of $f$ (A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity §domination). By Structure Condition: a Degenerate Elliptic Operator with a Continuous Inhomogeneity §structure, the operator $F_{1}(x,r,p,X)=G_{0}(r,p,X)-f(x)=G_{0}(r,p,X)-G(x)$ on $\Omega$ and $\omega_{1}$ satisfy the structure condition of Comparison Principle for the Dirichlet Problem for Second-Order Equations §structure. The drift. By the one-sided bound on $D_{R}$, the restriction of $b$ to $\Omega$ and $c=c_{R}$ satisfy the hypothesis of Structure Condition: a Linear First-Order Operator with an Almost Monotone Drift, so by Structure Condition: a Linear First-Order Operator with an Almost Monotone Drift §structure the operator $F_{2}(x,r,p,X)=b(x)\cdot p$ on $\Omega$ and the modulus $\omega_{c_{R}}(\tau)=c_{R}\tau$ satisfy the same structure condition. The sum. $F|_{\Omega}=F_{1}+F_{2}$ pointwise. For data $x,y,r,\alpha,X,Y$ as in Comparison Principle for the Dirichlet Problem for Second-Order Equations §structure, with $\tau=\alpha\lVert x-y\rVert^{2}+\lVert x-y\rVert$, adding the two structure inequalities gives

$\omega_{R}$ is a modulus of continuity: it is nonnegative as a sum of nonnegative functions (claim 2 of Elementary Arithmetic in an Ordered Field), and given a positive $\varepsilon$, if $\delta_{1}$ and $\delta_{2}$ serve $\varepsilon\cdot2^{-1}$ for $\omega_{1}$ and for $\omega_{c_{R}}$ (a modulus by Linear Moduli of Continuity §modulus), the least of them serves $\varepsilon$ for $\omega_{R}$. This is hypothesis Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §structure.