Lipschitz Regularity of Bounded Continuous Viscosity Solutions of 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 $W=D(A)$ and $V\cap H=V$ there. Let $\mathrm{Sym}(H)$, with its order $\preceq$, zero form $0_{\mathrm{Sym}}$ and identity form $I_{H}$, and $\mathrm{Sym}(V)$ be as in Hilbert Triples: Standing Notation and Background §restriction, 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 $\lambda_{0},C_{g},\ell_{g}\in\mathbb{R}$ satisfy $0<\lambda_{0}$, $0\le C_{g}$ and $0\le\ell_{g}$, and let $g:V\to\mathbb{R}$ satisfy

$$|g(x)|\le C_{g}\quad\text{and}\quad|g(x)-g(y)|\le\ell_{g}\,|x-y|_{H}\qquad\text{for all }x,y\in V .$$

Let $B:V\to H$ be a monotone nonlinearity for $(H,V,A)$, let $\nu\in\mathbb{R}$ satisfy $0\le\nu$, let $f=(f_{k})_{k\in\mathbb{N}}$ be square-summable in $V$ with trace $\mathrm{Tr}_{f}$, and let $\Gamma\in\mathrm{Sym}(H)$ satisfy $0_{\mathrm{Sym}}\preceq\Gamma\preceq I_{H}$. Let $F$ be the function on $D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ with

$$F(x,r,p,X)=\lambda_{0}\,r-\tfrac{\nu}{2}\,\mathrm{Tr}_{f}X+\tfrac12\,\Gamma(p,p)+\langle Ax+B(x),p\rangle_{H}-g(x),$$

which is the operator of A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses with the Lipschitz drift $L$ the zero map of $H$ (Lipschitz with constant $\ell=0$) and the modulus $\omega_{g}(t)=\ell_{g}t$, admissible there because $|x-y|_{H}\le|x-y|_{V}$ for $x,y\in V$ (Hilbert Triples: Standing Notation and Background §triple); in particular $F$ is a second-order equation operator on $H$ relative to $(H,V,A)$ by A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §operator. Let $C'\in\mathbb{R}$ be nonnegative and let $u:H\to\mathbb{R}$ be continuous on $H$, satisfy $|u(x)|\le C'$ for every $x\in H$, and be a viscosity solution of $F$ on $H$. Then the following hold.

1. (The penalised estimate)¶ For all $x,y\in V$ and every real $\alpha>1$,

$$u(x)-u(y)\le\frac{\alpha}{2}\,|x-y|_{H}^{2}+\frac{\ell_{g}^{2}}{2\lambda_{0}^{2}\,\alpha}.$$

2. (Lipschitz continuity)¶ For all $x,y\in H$,

$$|u(x)-u(y)|\le\Bigl(\frac{\ell_{g}}{\lambda_{0}}+2C'+1\Bigr)|x-y|_{H}.$$

3. (Sharp constant at short range)¶ For all $x,y\in H$ with $|x-y|_{H}\le\frac{\ell_{g}}{\lambda_{0}}$, one has $|u(x)-u(y)|\le\frac{\ell_{g}}{\lambda_{0}}|x-y|_{H}$.