A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses

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 $\mathrm{Sym}(H)$, with their orders $\preceq$, be as in Hilbert Triples: Standing Notation and Background (vi), let $h$ be the penalty function, and let $\mathbb{N}$ be the set of natural numbers.

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 $0\le\ell$, and let $L:H\to H$ be Lipschitz with constant $\ell$ from $(H,d_{H})$ to itself. Put $c_{L}=|L(0_{H})|_{H}$.

Let $\Gamma\in\mathrm{Sym}(H)$ satisfy $0_{\mathrm{Sym}}\preceq\Gamma\preceq I_{H}$, where $0_{\mathrm{Sym}}$ and $I_{H}$ are the zero and identity forms of $\mathrm{Sym}(H)$ fixed in Hilbert Triples: Standing Notation and Background (vi). Let $\nu\in\mathbb{R}$ satisfy $0\le\nu$, let $f=(f_{k})_{k\in\mathbb{N}}$ be square-summable in $V$, with sum $\sigma(f)$, 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

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

which is defined because $x\in D(A)$ gives $x\in V$ and $Ax\in H$ by Hilbert Triples: Standing Notation and Background (iii), whence $B(x)\in H$ and $g(x)\in\mathbb{R}$, and $x\in H$ by Hilbert Triples: Standing Notation and Background (i), whence $L(x)\in H$, while $\mathrm{Tr}_{f}X\in\mathbb{R}$ by Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them (ii). Then the following hold.

1. (A degenerate elliptic operator)(i) $F$ is a second-order equation operator on $H$ relative to $(H,V,A)$ and is degenerate elliptic.

2. (Strict properness)(ii) For every positive $R\in\mathbb{R}$, $\lambda_{0}$ is a properness constant for $F$ at $R$; in particular $F$ is locally strictly proper. No further restriction on $\lambda_{0}$ is imposed anywhere below.

3. (The second-order structure condition)(iii) Let $\bar{\omega}_{g}$ be the nondecreasing envelope of $\omega_{g}$ truncated at $M=2C_{g}$, which satisfies $\bar{\omega}_{g}(t)\le2C_{g}$ for every nonnegative $t$ by The Nondecreasing Envelope of a Truncated Modulus of Continuity (ii), let $\omega_{g}^{\ast}$ be its quadratic reparametrisation at $2C_{g}$, let $\omega_{1}$ be the function on the nonnegative reals with

$$\omega_{1}(t)=\omega_{g}^{\ast}(t)+\ell\,t,$$

and let $\omega_{2}$ be the function with value

$$\omega_{2}(t,\alpha)=\bigl(K\alpha^{2}+\nu\,\sigma(f)\bigr)\,t$$

at each pair $(t,\alpha)$ of real numbers with $0\le t$ and $1<\alpha$, where $K=64+4c_{L}^{2}+4\ell^{2}$. Then $\omega_{1}$ is a modulus of continuity, for every real $\alpha>1$ the function $t\mapsto\omega_{2}(t,\alpha)$ on the nonnegative reals is a modulus of continuity, and for every positive $R\in\mathbb{R}$ the pair $(\omega_{1},\omega_{2})$ is a second-order structure pair for $F$ at $R$; in particular $F$ satisfies the second-order structure condition.

4. (The tail-insensitivity condition)(iv) Let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$ with $e_{k}\in V$ for every $k\in\mathbb{N}$. Then $F$ is tail-insensitive along $(e_{k})_{k\in\mathbb{N}}$. Consequently, if $H$, as a vector space over $\mathbb{R}$, is not finite-dimensional, then $F$ satisfies the tail-insensitivity condition.

5. (The shift-continuity condition)(v) $F$ satisfies the shift-continuity condition.

6. (Comparison for this operator)(vi) Assume in addition that $H$, as a vector space over $\mathbb{R}$, is not finite-dimensional. Let $u,v:H\to\mathbb{R}$ and $C\in\mathbb{R}$ satisfy $u(x)\le C$ and $-C\le v(x)$ for every $x\in H$, let $u$ be a viscosity subsolution of $F$ on $H$ and let $v$ be a viscosity supersolution of $F$ on $H$. Then $u(x)\le v(x)$ for every $x\in V$.

Taking $\Gamma=\theta I_{H}$ with $0\le\theta\le1$ gives the operator with quadratic term $\tfrac{\theta}{2}|p|_{H}^{2}$, and taking moreover $\theta=1$ and $\nu=0$ the operator of A Hamilton-Jacobi Operator with a Monotone Nonlinearity and a Lipschitz Perturbation Satisfies the Comparison Hypotheses. The upper bound $\Gamma\preceq I_{H}$ keeps the diagonal terms of the $\delta$-shifted quadratic term dominated by the dissipation of the $\delta$-shift of the penalty function; the cross term is what makes $\omega_{2}$ quadratic in $\alpha$.