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

Each result cited below is universally quantified over the data in its own statement. Elementary facts about real numbers (arithmetic, the order, absolute values, and the fact that a real number $s$ with $s\le\varepsilon$ for every real $\varepsilon>0$ satisfies $s\le0$) are used without comment; they are carried by The Real Numbers: Standing Notation and Background. Throughout, $d_{H}(x,y)=|x-y|_{H}$ is the distance of $H$, as fixed in Hilbert Triples: Standing Notation and Background §triple, and we put

$$L=\frac{\ell_{g}}{\lambda_{0}}+2C'+1 .$$

Since $\ell_{g}\ge0$, $\lambda_{0}>0$ and $C'\ge0$, we have $L\ge1>0$.

Step 1 (claim 1). Fix $N\in\mathbb{N}$. We apply Lipschitz Regularity of Bounded Continuous Viscosity Solutions of a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple §lipschitz with the data $\lambda_{0},C_{g},\ell_{g}$, the function $g_{N}$ in place of $g$, the nonlinearity $B_{N}$ in place of $B$, the number $\nu$, the sequence $f^{N}$ in place of $f$, the form $\Gamma_{N}$ in place of $\Gamma$, the constant $C'$, and the function $u_{N}$ in place of $u$. Its hypotheses hold: $H$ is not finite-dimensional by assumption; $0<\lambda_{0}$, $0\le C_{g}$, $0\le\ell_{g}$ and $0\le\nu$; $g_{N}:V\to\mathbb{R}$ satisfies $|g_{N}(x)|\le C_{g}$ and $|g_{N}(x)-g_{N}(y)|\le\ell_{g}|x-y|_{H}$ for $x,y\in V$; $B_{N}$ is a monotone nonlinearity for $(H,V,A)$; $f^{N}$ is square-summable in $V$; $\Gamma_{N}\in\mathrm{Sym}(H)$ satisfies $0_{\mathrm{Sym}}\preceq\Gamma_{N}\preceq I_{H}$; the function on $D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ that the cited theorem builds from these data is, expression for expression, the operator $F_{N}$ of the statement; and $u_{N}:H\to\mathbb{R}$ is continuous on $H$, satisfies $|u_{N}(x)|\le C'$ for every $x\in H$, and is a viscosity solution of $F_{N}$ on $H$, all by hypothesis. The cited clause therefore gives

$$|u_{N}(x)-u_{N}(y)|\le L\,|x-y|_{H}\qquad\text{for all }x,y\in H .$$

The constant $L$ involves only $\ell_{g},\lambda_{0},C'$, so the bound is the same for every $N\in\mathbb{N}$. The same application with $(B,f,\Gamma,g)$ in place of $(B_{N},f^{N},\Gamma_{N},g_{N})$ and $u$ in place of $u_{N}$, whose hypotheses hold for the same reasons, the operator built there being $F$, gives $|u(x)-u(y)|\le L|x-y|_{H}$ for all $x,y\in H$. This proves claim 1.

Step 2 (a common modulus and the limit functions). Let $\omega$ be the function on the nonnegative reals with $\omega(t)=Lt$. Since $L\ge0$, $\omega$ is a modulus of continuity by Linear Moduli of Continuity §modulus (with $c=L$). By Step 1 and the hypothesis $|u_{N}(x)|\le C'$,

$$|u_{N}(x)|\le C'\quad\text{and}\quad|u_{N}(x)-u_{N}(y)|\le\omega\bigl(d_{H}(x,y)\bigr)\qquad\text{for all }N\in\mathbb{N}\text{ and }x,y\in H .$$

For $x\in H$ let $\bar{u}(x)$ and $\underline{u}(x)$ be the limit superior and the limit inferior of the bounded sequence $(u_{N}(x))_{N\in\mathbb{N}}$. These are the functions $\bar{u},\underline{u}$ of Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets for the metric space $(H,d_{H})$, the constant $C=C'$, the modulus $\omega$ and the sequence $(u_{N})_{N\in\mathbb{N}}$, and they are also the functions $\bar{u},\underline{u}$ of Stability of Viscosity Sub- and Supersolutions on a Hilbert Triple: Limits of Uniformly Bounded, Equicontinuous Sequences under Convergence of the Operators on Bounded Test Data for the same $C$, $\omega$ and $(u_{N})_{N\in\mathbb{N}}$. By Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets §bounds, $|\bar{u}(x)|\le C'$ and $|\underline{u}(x)|\le C'$ for every $x\in H$, and by Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets §modulus,

$$|\bar{u}(x)-\bar{u}(y)|\le L\,|x-y|_{H}\quad\text{and}\quad|\underline{u}(x)-\underline{u}(y)|\le L\,|x-y|_{H}\qquad\text{for all }x,y\in H .$$

Moreover, for each $x\in H$ we have $\underline{u}(x)\le\bar{u}(x)$ by claim 1 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence.

Step 3 (stability). As recorded in the statement, $F$ and every $F_{N}$ 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, and $(F_{N})_{N\in\mathbb{N}}$ converges to $F$ on bounded test data by hypothesis; the hypothesis that every sequence in $V$ bounded in $V$ has a subsequence converging in $H$ is also assumed. For every $N\in\mathbb{N}$, $u_{N}$ is a viscosity solution of $F_{N}$ on $H$, hence by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §solution both a viscosity subsolution and a viscosity supersolution of $F_{N}$ on $H$. Together with the bounds of Step 2, this verifies the hypotheses of Stability of Viscosity Sub- and Supersolutions on a Hilbert Triple: Limits of Uniformly Bounded, Equicontinuous Sequences under Convergence of the Operators on Bounded Test Data. By Stability of Viscosity Sub- and Supersolutions on a Hilbert Triple: Limits of Uniformly Bounded, Equicontinuous Sequences under Convergence of the Operators on Bounded Test Data §subsolution, $\bar{u}$ is a viscosity subsolution of $F$ on $H$, and by Stability of Viscosity Sub- and Supersolutions on a Hilbert Triple: Limits of Uniformly Bounded, Equicontinuous Sequences under Convergence of the Operators on Bounded Test Data §supersolution, $\underline{u}$ is a viscosity supersolution of $F$ on $H$. Likewise $u$, a viscosity solution of $F$ on $H$, is both a viscosity subsolution and a viscosity supersolution of $F$ on $H$ by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §solution.

Step 4 (comparison on $V$). The operator $F$ is the operator of A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses for the data $\lambda_{0}$, $C_{g}$, $\omega_{g}(t)=\ell_{g}t$, $g$, $B$, $\ell=0$, the zero map $L_{0}:H\to H$ (with $L_{0}(x)=0_{H}$) as the Lipschitz drift, $\Gamma$, $\nu$ and $f$. Indeed: $\omega_{g}$ is a modulus of continuity by Linear Moduli of Continuity §modulus (with $c=\ell_{g}\ge0$); for $x,y\in V$ we have $|x-y|_{H}\le|x-y|_{V}$ by Hilbert Triples: Standing Notation and Background §triple, so $|g(x)-g(y)|\le\ell_{g}|x-y|_{H}\le\ell_{g}|x-y|_{V}=\omega_{g}(|x-y|_{V})$, while $|g(x)|\le C_{g}$; the map $L_{0}$ is Lipschitz with constant $0$ from $(H,d_{H})$ to itself in the sense of Lipschitz Map Between Metric Spaces, since $d_{H}(L_{0}(x),L_{0}(x'))=d_{H}(0_{H},0_{H})=0=0\cdot d_{H}(x,x')$; and since $Ax+B(x)+L_{0}(x)=Ax+B(x)$, the function defined there is exactly $F$. Since $H$ is not finite-dimensional, the comparison clause A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §comparison applies to $F$ with the constant $C=C'$ in two ways.

First, with $\bar{u}$ as the subsolution and $u$ as the supersolution: $\bar{u}(x)\le C'$ for every $x\in H$ by Step 2 and $-C'\le u(x)$ for every $x\in H$ by hypothesis, and $\bar{u}$, $u$ are a viscosity subsolution and a viscosity supersolution of $F$ on $H$ by Step 3. Hence $\bar{u}(y)\le u(y)$ for every $y\in V$.

Second, with $u$ as the subsolution and $\underline{u}$ as the supersolution: $u(x)\le C'$ by hypothesis and $-C'\le\underline{u}(x)$ by Step 2, for every $x\in H$, and $u$, $\underline{u}$ are a viscosity subsolution and a viscosity supersolution of $F$ on $H$ by Step 3. Hence $u(y)\le\underline{u}(y)$ for every $y\in V$.

Step 5 (extension to $H$ by density). Fix $x\in H$ and a real $\varepsilon>0$. The set $V$ is dense in $H$ by Hilbert Triples: Standing Notation and Background §triple, that is, its closure in $(H,d_{H})$ is $H$, so $x$ lies in that closure; by the equivalence of claims 1 and 3 of Characterization of the Closure in a Metric Space by Open Balls, applied in the metric space $(H,d_{H})$ with the subset $V$ and the radius $\varepsilon/(2L)>0$, there is $y\in V$ with $|x-y|_{H}<\varepsilon/(2L)$. Using Step 1 for $u$, Step 2 for $\bar{u}$ and $\underline{u}$, and Step 4 at the point $y\in V$,

$$\bar{u}(x)-u(x)=\bigl(\bar{u}(x)-\bar{u}(y)\bigr)+\bigl(\bar{u}(y)-u(y)\bigr)+\bigl(u(y)-u(x)\bigr)\le L|x-y|_{H}+0+L|x-y|_{H}<\varepsilon,$$ $$u(x)-\underline{u}(x)=\bigl(u(x)-u(y)\bigr)+\bigl(u(y)-\underline{u}(y)\bigr)+\bigl(\underline{u}(y)-\underline{u}(x)\bigr)\le L|x-y|_{H}+0+L|x-y|_{H}<\varepsilon .$$

As $\varepsilon>0$ was arbitrary, $\bar{u}(x)\le u(x)$ and $u(x)\le\underline{u}(x)$. Combined with $\underline{u}(x)\le\bar{u}(x)$ from Step 2 this gives $\underline{u}(x)\le\bar{u}(x)\le u(x)\le\underline{u}(x)$, so

$$\underline{u}(x)=\bar{u}(x)=u(x)\qquad\text{for every }x\in H .$$

Step 6 (claim 2). Fix $x\in H$. The sequence $(u_{N}(x))_{N\in\mathbb{N}}$ is a bounded sequence of real numbers, since $|u_{N}(x)|\le C'$, and by Step 5 its limit inferior $\underline{u}(x)$ and its limit superior $\bar{u}(x)$ both equal $u(x)$. By claim 5 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence, with $a=u(x)$, the sequence $(u_{N}(x))_{N\in\mathbb{N}}$ converges to $u(x)$. This proves claim 2.

Step 7 (claim 3). Fix real numbers $R>0$ and $\varepsilon>0$, and let $K_{R}=\{x\in V:|x|_{V}\le R\}\subseteq H$. We show that $K_{R}$ is sequentially compact in $(H,d_{H})$ in the sense of Sequentially Compact Subset of a Metric Space. Let $(x_{m})_{m\in\mathbb{N}}$ be a sequence in $H$ with $x_{m}\in K_{R}$ for every $m$. It is a sequence in $V$ with $d_{V}(x_{m},0_{H})=|x_{m}|_{V}\le R$ for every $m$, hence bounded in $V$; by the compactness hypothesis of the statement there are a strictly increasing sequence $(n_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ and a point $x\in H$ such that the subsequence $(x_{n_{k}})_{k\in\mathbb{N}}$ converges to $x$ in $H$. This subsequence is a sequence in $V$ with $|x_{n_{k}}|_{V}\le R$ for every $k$, and $(V,d_{V})$ is separable by Hilbert Triples: Standing Notation and Background §separable; so Weak Compactness and Closedness of Bounded Subsets of the Small Space of a Hilbert Triple §closure, applied with the constant $R$ to the sequence $(x_{n_{k}})_{k\in\mathbb{N}}$, gives $x\in V$ and $|x|_{V}\le R$, that is $x\in K_{R}$. Thus $K_{R}$ is sequentially compact in $(H,d_{H})$.

By Step 5, $\bar{u}(x)=\underline{u}(x)$ for every $x\in K_{R}$. With the data of Step 2, Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets §uniform, applied to $K=K_{R}$ and to $\varepsilon$, gives $N_{0}\in\mathbb{N}$ such that $|u_{N}(x)-\bar{u}(x)|\le\varepsilon$ for every $N\in\mathbb{N}$ with $N\ge N_{0}$ and every $x\in K_{R}$. Since $\bar{u}(x)=u(x)$ by Step 5, this reads $|u_{N}(x)-u(x)|\le\varepsilon$ for every $N\ge N_{0}$ and every $x\in V$ with $|x|_{V}\le R$. This proves claim 3.