The Dissipative Hamilton-Jacobi Operator on a Hilbert Triple Satisfies the Comparison Hypotheses

In the setting of Hilbert Triples: Standing Notation and Background, the set $H$ is open in $H$, since every open ball of $(H,d_{H})$ is a subset of $H$; accordingly $W=D(A)\cap H=D(A)$ and $V\cap H=V$ in the notation of Hilbert Triples: Standing Notation and Background §open-sets. Let $\mathrm{Sym}(V)$ be as in Hilbert Triples: Standing Notation and Background §restriction.

Let $\lambda_{0},C_{g}\in\mathbb{R}$ satisfy $0<\lambda_{0}$ and $0\le C_{g}$, let $\omega_{g}$ be a modulus of continuity, and let $g:H\to\mathbb{R}$ satisfy

Let $F$ be the function on $D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ whose value at $(x,r,p,X)$ is

which is defined because $x\in D(A)$, so that $Ax\in H$ by Hilbert Triples: Standing Notation and Background §operator. Then the following hold.

1. (An operator of first order)¶ $F$ is a second-order equation operator on $H$ relative to $(H,V,A)$ and is first order; consequently $F$ is degenerate elliptic by A First-Order Equation Operator is Degenerate Elliptic and Its $\delta$-Shifts Ignore the Form Argument §elliptic.

2. (Strict properness)¶ For every positive $R\in\mathbb{R}$, $\lambda_{0}$ is a properness constant for $F$ at $R$; in particular $F$ is locally strictly proper.

3. (The structure condition)¶ Let $\omega_{2}$ be the function on the set of nonnegative reals with value $0$ everywhere and let $\omega_{3}$ be the function with value $\tfrac{3}{4}t$ at $t$. Then $\omega_{2}$ and $\omega_{3}$ are moduli of continuity and, for every positive $R\in\mathbb{R}$, $(\omega_{g},\omega_{2},\omega_{3})$ is a structure triple for $F$ at $R$; in particular $F$ satisfies the first-order structure condition.

4. (The shift-continuity condition)¶ $F$ satisfies the shift-continuity condition.

5. (Comparison for this operator)¶ Let $u,v:H\to\mathbb{R}$ and $C\in\mathbb{R}$ satisfy $u(x)\le C$ and $-C\le v(x)$ for every $x\in H$, let $u$ be a viscosity subsolution of $F$ on $H$ and let $v$ be a viscosity supersolution of $F$ on $H$. Then $u(x)\le v(x)$ for every $x\in V$.