Convergence of Solutions of Viscous Hamilton-Jacobi Equations with Monotone Nonlinearities on a Hilbert Triple under Convergence of the Operators

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, with $W=D(A)$ and $V\cap H=V$ there; $\mathrm{Sym}(H)$, $\mathrm{Sym}(V)$, the order $\preceq$, the zero form $0_{\mathrm{Sym}}$ and the identity form $I_{H}$ are as in Hilbert Triples: Standing Notation and Background §restriction. Assume that $H$, as a vector space over $\mathbb{R}$, is not finite-dimensional, and that every sequence in $V$ that is bounded in $V$ has a subsequence that converges in $H$.

Let $\lambda_{0},\nu,C_{g},\ell_{g},C'\in\mathbb{R}$ with $0<\lambda_{0}$ and $\nu,C_{g},\ell_{g},C'$ nonnegative. Let each of the quadruples $(B,f,\Gamma,g)$ and $(B_{N},f^{N},\Gamma_{N},g_{N})$, for $N\in\mathbb{N}$, consist of a monotone nonlinearity $V\to H$ for $(H,V,A)$, a sequence square-summable in $V$, a form in $\mathrm{Sym}(H)$ lying between $0_{\mathrm{Sym}}$ and $I_{H}$ for $\preceq$, and a function $V\to\mathbb{R}$ bounded in absolute value by $C_{g}$ and Lipschitz with constant $\ell_{g}$ for $d_{H}$, that is $|g(x)-g(y)|\le\ell_{g}|x-y|_{H}$ for $x,y\in V$, and likewise for each $g_{N}$. Let $F$ and $F_{N}$ be the operators on $D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ given by

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

and by the same expression with $(B_{N},f^{N},\Gamma_{N},g_{N})$ in place of $(B,f,\Gamma,g)$, where $\mathrm{Tr}_{f}$ and $\mathrm{Tr}_{f^{N}}$ are the traces along $f$ and $f^{N}$. These are the operators of A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses with the Lipschitz drift there the zero map of $H$ and the modulus $\omega_{g}(t)=\ell_{g}t$, admissible because $|x-y|_{H}\le|x-y|_{V}$ for $x,y\in V$ (Hilbert Triples: Standing Notation and Background §triple); so they are second-order equation operators on $H$ relative to $(H,V,A)$ by A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §operator. Assume that $(F_{N})_{N\in\mathbb{N}}$ converges to $F$ on bounded test data.

Let $u,u_{N}:H\to\mathbb{R}$, for $N\in\mathbb{N}$, be continuous on $H$, bounded in absolute value by $C'$, and such that $u$ is a viscosity solution of $F$ on $H$ and $u_{N}$ is a viscosity solution of $F_{N}$ on $H$ for every $N\in\mathbb{N}$. Then the following hold.

1. (Uniform Lipschitz bound)¶ For every $N\in\mathbb{N}$ and all $x,y\in H$,

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

and the same bound holds for $u$.

2. (Pointwise convergence)¶ For every $x\in H$ the sequence $(u_{N}(x))_{N\in\mathbb{N}}$ converges to $u(x)$.

3. (Uniform convergence on bounded sets of $V$)¶ For all real numbers $R>0$ and $\varepsilon>0$ there is $N_{0}\in\mathbb{N}$ such that $|u_{N}(x)-u(x)|\le\varepsilon$ for every $N\in\mathbb{N}$ with $N\ge N_{0}$ and every $x\in V$ with $|x|_{V}\le R$.