Proof of Well-Posedness for a Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple

Each result cited is universally quantified over the data in its own statement.

The operator. By A Hamilton-Jacobi Operator with a Monotone Nonlinearity and a Lipschitz Perturbation Satisfies the Comparison Hypotheses (i) the function $F$ is a second-order equation operator on $H$ relative to $(H,V,A)$ and is degenerate elliptic; by A Hamilton-Jacobi Operator with a Monotone Nonlinearity and a Lipschitz Perturbation Satisfies the Comparison Hypotheses (ii) it is locally strictly proper; by A Hamilton-Jacobi Operator with a Monotone Nonlinearity and a Lipschitz Perturbation Satisfies the Comparison Hypotheses (iii) it satisfies the first-order structure condition; and by A Hamilton-Jacobi Operator with a Monotone Nonlinearity and a Lipschitz Perturbation Satisfies the Comparison Hypotheses (iv) it satisfies the shift-continuity condition. These are exactly the hypotheses placed on the operator in Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple and in Uniqueness of a Bounded Continuous Viscosity Solution on a Hilbert Triple, both of which concern an operator on $H$ relative to $(H,V,A)$.

The constant. As $\lambda_{0}$ is positive it is nonzero, so $C=\tfrac{C_{g}}{\lambda_{0}}$ is defined, and $\tfrac{1}{\lambda_{0}}$ is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field; since $0\le C_{g}$, claim 5 of Elementary Arithmetic in an Ordered Field gives $0\le C$. Moreover $\lambda_{0}C=C_{g}$, and therefore $\lambda_{0}(-C)=-C_{g}$.

The values of $F$ at the two constants. Let $x\in D(A)$ and $r\in\mathbb{R}$. Then $x\in V$ by Hilbert Triples: Standing Notation and Background (iii), so $g(x)$ is defined. By Elementary Identities in a Real Inner Product Space (ii) we have $|0_{H}|_{H}=0$, whence $\tfrac{1}{2}|0_{H}|_{H}^{2}=0$, and $\langle Ax+B(x)+L(x),0_{H}\rangle_{H}=0$. Since the defining expression of $F$ does not contain its fourth argument,

$$F(x,r,0_{H},0_{\mathrm{Sym}})=\lambda_{0}r-g(x).$$

By claim 6 of Properties of the Absolute Value in an Ordered Field the bound $|g(x)|\le C_{g}$ gives $-C_{g}\le g(x)$ and $g(x)\le C_{g}$, so claim 3 of Elementary Arithmetic in an Ordered Field yields $-C_{g}-g(x)\le0$ and $0\le C_{g}-g(x)$. With the two values of $\lambda_{0}C$ computed above,

$$F(x,-C,0_{H},0_{\mathrm{Sym}})=-C_{g}-g(x)\le0,\qquad 0\le C_{g}-g(x)=F(x,C,0_{H},0_{\mathrm{Sym}}).$$

As $x\in D(A)$ was arbitrary, these are the hypotheses placed on the constant in Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple, whose remaining hypothesis $0\le C$ was checked above.

Claim 1. By Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple, applied to $F$ and $C$, there is a function $u:H\to\mathbb{R}$ which is a viscosity solution of $F$ on $H$ by Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple (i), satisfies $|u(x)|\le C$ for every $x\in H$ by Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple (ii), and is uniformly continuous on $H$ with respect to $d_{H}$ and the metric of the real numbers by Existence of a Bounded, Uniformly Continuous Viscosity Solution on a Hilbert Triple (iii).

Claim 2. This is Uniqueness of a Bounded Continuous Viscosity Solution on a Hilbert Triple (i), applied to $F$, to $C'$ and to $u_{1}$ and $u_{2}$; the hypotheses that corollary places on the operator are those collected in the first paragraph, and its hypotheses on the two functions are those assumed in the claim.

Claim 3. Let $u$ be as in claim 1. Applying A Uniformly Continuous Map Between Metric Spaces Is Continuous (i) with the metric space $(H,d_{H})$, with $H$ itself as the subset, and with the real numbers carrying the metric of Real Hilbert Spaces: Standing Notation and Background (i) as the target, the uniform continuity of $u$ gives that $u$ is continuous on $H$.

Let $u'$ and $C''$ be as in the claim, and let $C'$ be the greater of $C$ and $C''$, as provided by Real Hilbert Spaces: Standing Notation and Background (i); then $C\le C'$ and $C''\le C'$. By transitivity $|u(x)|\le C'$ and $|u'(x)|\le C'$ for every $x\in H$. Both $u$ and $u'$ are viscosity solutions of $F$ on $H$ that are continuous on $H$, so claim 2, applied with this $C'$ and with $u_{1}=u$ and $u_{2}=u'$, gives $u(x)=u'(x)$ for every $x\in H$.