The Viscous Hamilton-Jacobi Operator is Continuous, Strictly Proper and Satisfies the Structure Condition

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets and of Bounded Open Domain in Euclidean Space, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation on which the former rests, is in force in the dimension $n$, a natural number with $1\le n$. In addition $\operatorname{tr}$ denotes the trace of a square real matrix; $T=\{t\in\mathbb{R}:0\le t\}$; we abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\lVert z\rVert$; and $\tfrac{s}{2}$ denotes the product of $s\in\mathbb{R}$ with the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field.

Let $\gamma\in\mathbb{R}$ be positive, let $\kappa\in\mathbb{R}$ be nonnegative and let $f:\overline{\Omega}\to\mathbb{R}$ be continuous on $\overline{\Omega}$, as a map into the metric space $(\mathbb{R},d_{\mathbb{R}})$ of The Absolute Value Metric on the Real Numbers. Let $\omega:T\to\mathbb{R}$ be a modulus of continuity that is nondecreasing, meaning that $\omega(s)\le\omega(t)$ whenever $s,t\in T$ satisfy $s\le t$, and that dominates the oscillation of $f$, meaning that $|f(x)-f(y)|\le\omega(t)$ for all $x,y\in\overline{\Omega}$ and every $t\in T$ with $d_{E}(x,y)\le t$. At least one such $\omega$ exists: $\overline{\Omega}$ is nonempty and compact by Bounded Open Domain in Euclidean Space (ii), so A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity applies with the metric space $(\mathbb{R}^{n},d_{E})$ and $K=\overline{\Omega}$, and A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity (i) and A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity (ii) are exactly the two properties just named.

Since $\Omega\subseteq\overline{\Omega}$ by Bounded Open Domain in Euclidean Space (ii), the formula

$$F(x,r,p,X)=\gamma r-\tfrac{\kappa}{2}\operatorname{tr}(X)+\tfrac{1}{2}\lVert p\rVert^{2}-f(x)$$

defines a function $F:\Omega\times\mathbb{R}\times\mathbb{R}^{n}\times\mathcal{S}(n)\to\mathbb{R}$, that is, a second-order equation operator on $\Omega$.

Then the following hold.

1. (Continuity)(i) $F$ is continuous.

2. (Strict properness)(ii) $F$ is strictly proper with constant $\gamma$.

3. (Structure condition)(iii) $F$ and $\omega$ satisfy the structure condition of the comparison principle for the Dirichlet problem.