Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple

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 (v). Let $0_{H}$ be the zero vector of $H$ and let $0_{\mathrm{Sym}}$ be the zero form of $\mathrm{Sym}(V)$.

Let $F$ be a second-order equation operator on $H$ relative to $(H,V,A)$ that is degenerate elliptic and locally strictly proper, and that satisfies the second-order structure condition, satisfies the shift-continuity condition and satisfies the tail-insensitivity condition. Let $C\in\mathbb{R}$ be nonnegative and assume that

$$F\bigl(x,-C,0_{H},0_{\mathrm{Sym}}\bigr)\le0\qquad\text{and}\qquad 0\le F\bigl(x,C,0_{H},0_{\mathrm{Sym}}\bigr)\qquad\text{for every }x\in D(A).$$

Then there is a function $u:H\to\mathbb{R}$ with the following three properties.

1. (Solution)(i) $u$ is a viscosity solution of $F$ on $H$.

2. (Bounded)(ii) $|u(x)|\le C$ for every $x\in H$.

3. (Uniformly continuous)(iii) $u$ is uniformly continuous on $H$, with respect to $d_{H}$ and the metric $d_{\mathbb{R}}$ of Real Hilbert Spaces: Standing Notation and Background (i).