Well-Posedness for a 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 (v); 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, and let $0_{H}$ be the zero vector of $H$.

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

$$|g(x)|\le C_{g}\quad\text{for every }x\in V,\qquad |g(x)-g(y)|\le\omega_{g}\bigl(|x-y|_{V}\bigr)\quad\text{for all }x,y\in V .$$

Let $B:V\to H$ be a monotone nonlinearity for $(H,V,A)$, let $\ell\in\mathbb{R}$ satisfy $0\le\ell$, let $L:H\to H$ be Lipschitz with constant $\ell$ from $(H,d_{H})$ to itself, and 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

$$F(x,r,p,X)=\lambda_{0}\,r+\tfrac{1}{2}|p|_{H}^{2}+\langle Ax+B(x)+L(x),p\rangle_{H}-g(x),$$

which is defined, and is a second-order equation operator on $H$ relative to $(H,V,A)$ that is degenerate elliptic, by A Hamilton-Jacobi Operator with a Monotone Nonlinearity and a Lipschitz Perturbation Satisfies the Comparison Hypotheses (i). 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)(i) 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 (i).

2. (Uniqueness among bounded continuous solutions)(ii) 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)(iii) 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 $x\in H$.