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

Each result cited is universally quantified over the data in its own statement. The set $H$ is nonempty, since it contains $0_{H}$.

Step 1 (the two constants are classical, hence viscosity, sub- and supersolutions). Let $f,g:H\to\mathbb{R}$ be the functions with constant values $-C$ and $C$. By Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (i) both belong to $C^{2}(H)$, and at every $x\in H$ their gradients are $0_{H}$ and their Hessians are the zero form of $\mathrm{Sym}(H)$. For $x\in D(A)$ the restriction to $V$ of the zero form of $\mathrm{Sym}(H)$ has value $0$ at every pair of elements of $V$, hence is $0_{\mathrm{Sym}}$. Since $W=D(A)$, the hypothesis therefore gives

$$F\bigl(x,f(x),Df(x),D^{2}f(x)|_{V}\bigr)=F\bigl(x,-C,0_{H},0_{\mathrm{Sym}}\bigr)\le0\qquad\text{for every }x\in W,$$

so $f$ is a classical subsolution of $F$ on $H$, and likewise $0\le F(x,g(x),Dg(x),D^{2}g(x)|_{V})$ for every $x\in W$, so $g$ is a classical supersolution of $F$ on $H$. As $F$ is degenerate elliptic, Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on a Hilbert Triple are Viscosity Sub- and Supersolutions (iii) shows that $f$ is a viscosity subsolution of $F$ on $H$ and Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on a Hilbert Triple are Viscosity Sub- and Supersolutions (v) shows that $g$ is a viscosity supersolution of $F$ on $H$.

Step 2 (Perron's method). Since $0\le C$ we have $f(x)=-C\le C=g(x)$ for every $x\in H$. By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (iv) the bound $-C\le f$ makes $f$ bounded below near each point of $H$, and the bound $g\le C$ makes $g$ bounded above near each point of $H$. Let $\mathcal{G}$ and $w$ be the family and the function of Perron's Method on a Hilbert Triple: Existence of a Viscosity Solution Between a Subsolution and a Supersolution for the data $F$, $f$ and $g$ on $U=H$. By Perron's Method on a Hilbert Triple: Existence of a Viscosity Solution Between a Subsolution and a Supersolution (i) we have $-C\le w(x)$ and $w(x)\le C$ for every $x\in H$, and by Perron's Method on a Hilbert Triple: Existence of a Viscosity Solution Between a Subsolution and a Supersolution (ii) the function $w$ is a viscosity solution of $F$ on $H$.

Step 3 (the restriction of $w$ to $V$ is uniformly continuous). By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (iii) the function $w$ is both a viscosity subsolution and a viscosity supersolution of $F$ on $H$, and by Step 2 it satisfies $w(x)\le C$ and $-C\le w(x)$ for every $x\in H$. Let $\epsilon\in\mathbb{R}$ be positive. Applying A Comparison Principle on a Hilbert Triple under the First-Order Structure Condition (i) with $w$ in the role of both the subsolution and the supersolution, with the constant $C$ and with the positive number $\tfrac{\epsilon}{2}$, we obtain a positive $\theta$ such that all $x,y\in V$ with $|x-y|_{H}\le\theta$ satisfy $w(x)-w(y)\le\tfrac{\epsilon}{2}$. For such $x$ and $y$ the symmetry of the metric $d_{H}$ gives $|y-x|_{H}=|x-y|_{H}\le\theta$, so also $w(y)-w(x)\le\tfrac{\epsilon}{2}$, and claim 6 of Properties of the Absolute Value in an Ordered Field yields

$$|w(x)-w(y)|\le\tfrac{\epsilon}{2}<\epsilon .$$

Let $w_{V}:V\to\mathbb{R}$ be the restriction of $w$ to $V$. Since every $x,y\in V$ with $|x-y|_{H}<\theta$ satisfy $|x-y|_{H}\le\theta$, the display shows that $\theta$ witnesses the condition of uniform continuity for $\epsilon$; as $\epsilon$ was an arbitrary positive real, $w_{V}$ is uniformly continuous on $V$.

Step 4 (extension to $H$ and conclusion). The subspace $V$ is nonempty and dense in $H$ by Hilbert Triples: Standing Notation and Background (i). By Extension of a Uniformly Continuous Real Function from a Dense Subset (i), applied in the metric space $(H,d_{H})$ with the dense subset $V$ and the function $w_{V}$, there is a function $u:H\to\mathbb{R}$ that is uniformly continuous on $H$, continuous on $H$, and satisfies $u(x)=w(x)$ for every $x\in V$. Uniform continuity is property 3.

By Step 2 and claim 6 of Properties of the Absolute Value in an Ordered Field, $|w(x)|\le C$ for every $x\in H$, so $|u(x)|\le C$ for every $x\in V$; since $u$ is continuous on $H$, Extension of a Uniformly Continuous Real Function from a Dense Subset (iv) gives $|u(x)|\le C$ for every $x\in H$, which is property 2.

Finally, $|u(x)|\le C$ and $|w(x)|\le C$ for every $x\in H$ give, again by claim 6 of Properties of the Absolute Value in an Ordered Field, the bounds $-C\le u\le C$ and $-C\le w\le C$ on $H$, so by Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (iv) both $u$ and $w$ are bounded above near each point of $H$ and bounded below near each point of $H$. The two functions agree on $V=V\cap H$, and $w$ is a viscosity solution of $F$ on $H$ by Step 2, so The Viscosity Property Depends Only on the Values on the Trace of $V$ (iv) shows that $u$ is a viscosity solution of $F$ on $H$, which is property 1.