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

In the setting of Hilbert Triples: Standing Notation and Background, the set $H$ is open in $H$ by Hilbert Triples: Standing Notation and Background §open-sets; accordingly $W=D(A)\cap H=D(A)$ and $V\cap H=V$ there. Let $\mathrm{Sym}(V)$ and its zero form $0_{\mathrm{Sym}}$ be as fixed there, let $0_{H}$ be the zero vector of $H$, and let $\mathbb{N}$ be the set of natural numbers. Assume that $H$, as a vector space over $\mathbb{R}$, is not finite-dimensional.

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 $g:V\to\mathbb{R}$ satisfy

Let $B:V\to H$ be a monotone nonlinearity for $(H,V,A)$, let $\ell\in\mathbb{R}$ satisfy $0\le\ell$, and let $L:H\to H$ be Lipschitz with constant $\ell$ from $(H,d_{H})$ to itself. Let $\theta,\nu\in\mathbb{R}$ satisfy $0\le\theta\le1$ and $0\le\nu$, let $f=(f_{k})_{k\in\mathbb{N}}$ be square-summable in $V$ and let $\mathrm{Tr}_{f}$ be the trace along $f$. 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, and is a second-order equation operator on $H$ relative to $(H,V,A)$ that is degenerate elliptic, by A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §operator. Let $C=\tfrac{C_{g}}{\lambda_{0}}$, the quotient of $C_{g}$ by the nonzero $\lambda_{0}$; it is nonnegative. Then the following hold.

1. (Existence)¶ There is a function $u:H\to\mathbb{R}$ that is a viscosity solution of $F$ on $H$, satisfies $|u(x)|\le C$ for every $x\in H$, and is uniformly continuous on $H$ with respect to $d_{H}$ and the metric of Real Hilbert Spaces: Standing Notation and Background §numbers.

2. (Uniqueness among bounded continuous solutions)¶ Let $C'\in\mathbb{R}$ and let $u_{1},u_{2}:H\to\mathbb{R}$ be viscosity solutions of $F$ on $H$ that are continuous on $H$ and satisfy $|u_{1}(x)|\le C'$ and $|u_{2}(x)|\le C'$ for every $x\in H$. Then $u_{1}(x)=u_{2}(x)$ for every $x\in H$.

3. (Well-posedness)¶ Let $u$ be as in claim 1. Then $u$ is continuous on $H$; and if $u'$ is a viscosity solution of $F$ on $H$ that is continuous on $H$ and for which some $C''\in\mathbb{R}$ satisfies $|u'(x)|\le C''$ for every $x\in H$, then $u'(x)=u(x)$ for every $x\in H$.